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diff --git a/.gitattributes b/.gitattributes index 6833f05..d7b82bc 100644 --- a/.gitattributes +++ b/.gitattributes @@ -1,3 +1,4 @@ -* text=auto -*.txt text -*.md text +*.txt text eol=lf +*.htm text eol=lf +*.html text eol=lf +*.md text eol=lf diff --git a/78255-src/78255-src.htm b/78255-src/78255-src.htm new file mode 100644 index 0000000..a8bae82 --- /dev/null +++ b/78255-src/78255-src.htm @@ -0,0 +1,50771 @@ +<!DOCTYPE html> +<html lang="en"> +<head> + <meta charset="UTF-8"> + <title> + Principia Mathematica | Project Gutenberg + </title> + <link rel="icon" href="images/cover.jpg" type="image/x-cover"> + <style> + +body { + margin-left: 10%; + margin-right: 10%; +} + +/* General headers */ + +h1 { + text-align: center; + clear: both; +} + +/* General headers */ +h2, h3 { + text-align: center; + font-weight: bold; + margin-top: 1em; + margin-bottom: 1em; + } + +p { + margin-top: .51em; + text-align: justify; + margin-bottom: .49em; + text-indent: 1.5em; +} + +.nind {text-indent:0;} + +.nindc { + text-align: center; + text-indent: 0 + } + +.large { + font-size: 125% + } + +.space-above2 { margin-top: 2em; } +.space-below2 { margin-bottom: 2em; } + +.spa1 { + margin-top: 1em + } + +hr { + width: 33%; + margin-top: 2em; + margin-bottom: 2em; + margin-left: 33.5%; + margin-right: 33.5%; + clear: both; +} + +.antiqua {font-family: sans-serif;} + +hr.tb {width: 45%; margin-left: 27.5%; margin-right: 27.5%;} +hr.chap {width: 65%; margin-left: 17.5%; margin-right: 17.5%;} +@media print { hr.chap {display: none; visibility: hidden;} } + +div.chapter {page-break-before: always;} +h2.nobreak {page-break-before: avoid;} + + +table { + margin-left: auto; + margin-right: auto; +} +table.autotable { border-collapse: collapse; } +table.autotable td { padding: 0.25em; } + +.tdl {text-align: left;} +.tdr {text-align: right;} + +.tdlh { + text-align: left; + margin-left: 2em; + text-indent: 2em + } +.tdlh2 { + text-align: left; + margin-left: 4em; + text-indent: 4em + } + +.tdlh3 { + text-align: left; + margin-left: 6.4em; + text-indent: 6.4em + } + +.hanging2 {padding-left: 6em; + text-indent: -3em; + } + +.pagenum { /* uncomment the next line for invisible page numbers */ + /* visibility: hidden; */ + position: absolute; + left: 92%; + font-size: small; + text-align: right; + font-style: normal; + font-weight: normal; + font-variant: normal; + text-indent: 0; +} /* page numbers */ + +.blockquot { + margin-left: 5%; + margin-right: 10%; +} + +.allsmcap {font-variant: small-caps; text-transform: lowercase;} + +/* Images */ + +img {max-width: 100%; width: 100%; height: auto;} +.width500 {max-width: 500px;} +.x-ebookmaker .width500 {width: 100%;} + + +.figcenter { + margin: auto; + text-align: center; + page-break-inside: avoid; + max-width: 100%; +} + +/* Footnotes */ +.footnotes {border: 1px dashed;} + +.footnote {margin-left: 10%; margin-right: 10%; font-size: 0.9em;} + +.footnote .label {position: absolute; right: 84%; text-align: right;} + +.fnanchor { + vertical-align: super; + font-size: .8em; + text-decoration: + none; +} + +/* Transcriber's notes */ +.transnote {background-color: #E6E6FA; + color: black; + font-size:small; + padding:0.5em; + margin-bottom:5em; + font-family:sans-serif, serif; +} + +/* css needed in m2svg output: displayed equations and prevention of bad breaks*/ + .align-center { + display: block; + text-align: center; + text-indent: 0; + margin-top: 1em; + margin-bottom: 1em; + } + .nowrap { + white-space: nowrap; + } + + </style> +</head> +<body> + + +<figure class="figcenter width500" id="cover" style="width: 1725px;"> +<img src="images/cover.jpg" width="1725" height="2560" alt="This volume +develops cardinal and ordinal number theory, series, and transfinite +arithmetic from the logical foundations of Volume 1, advancing +mathematical logic."> +</figure> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p class="nindc"><span class="large"> +PRINCIPIA MATHEMATICA</span> +</p> +</div> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p class="nindc space-above2 space-below2"> +CAMBRIDGE UNIVERSITY PRESS<br> +<span class="antiqua"><b>London</b></span>: FETTER LANE, E.C.<br> +C. F. CLAY, <span class="allsmcap">MANAGER</span><br> +</p> + +<figure class="figcenter width500" id="i_02" style="width: 200px;"> +<img src="images/i_02.jpg" width="200" height="183" alt="decorative"> +</figure> + +<p class="nindc space-above2 space-below2"> +<span class="antiqua"><b>Edinburgh</b></span>: 100, PRINCES STREET<br> +<span class="antiqua"><b>Berlin</b></span>: A. ASHER AND CO.<br> +<span class="antiqua"><b>Leipzig</b></span>: F. A. BROCKHAUS<br> +<span class="antiqua"><b>New York</b></span>: G. P. PUTNAM'S SONS<br> +<span class="antiqua"><b>Bombay and Calcutta</b></span>: +MACMILLAN AND CO., <span class="allsmcap">LTD.</span></p> + +<p class="nindc space-above2 space-below2"> +<i>All rights reserved</i> +</p> +</div> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<h1>PRINCIPIA MATHEMATICA</h1> + +<p class="nindc space-above2 space-below2"> +<span class="allsmcap">BY</span></p> + +<p class="nindc"><span class="large"> +ALFRED NORTH WHITEHEAD, Sc.D., F.R.S.</span></p> + +<p class="nindc">Fellow and late Lecturer of Trinity College, Cambridge</p> + +<p class="nindc space-above2 space-below2"> +<span class="allsmcap">AND</span></p> + +<p class="nindc"><span class="large"> +BERTRAND RUSSELL, M.A., F.R.S.</span></p> + +<p class="nindc">Lecturer and late Fellow of Trinity College, Cambridge</p> + +<p class="nindc space-above2 space-below2"> +VOLUME II</p> + +<p class="nindc space-above2 space-below2"> +Cambridge<br> +at the University Press<br> +1912</p> +</div> + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p class="nindc space-above2 space-below2"> +<span class="antiqua">Cambridge</span>:<br> +PRINTED BY JOHN CLAY, <span class="allsmcap">M.A.</span><br> +AT THE UNIVERSITY PRESS<br> +</p> +</div> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_v">[Pg v]</span></p> + +<h2 class="nobreak" id="CONTENTS_OF_VOLUME_II">CONTENTS OF VOLUME II</h2> +</div> + +<table class="autotable"> +<tbody><tr> +<td class="tdl"> </td> +<td class="tdr"> <span class="allsmcap">PAGE</span></td> +</tr><tr> +<td class="tdl">PREFATORY STATEMENT OF SYMBOLIC CONVENTIONS</td> +<td class="tdr"><a href="#Page_ix">ix</a></td> +</tr><tr> +<td class="tdl">PART III. CARDINAL ARITHMETIC.</td> +<td class="tdr"></td> +</tr><tr> +<td class="tdlh2">Summary of Part III</td> +<td class="tdr"><a href="#Page_3">3</a></td> +</tr><tr> +<td class="tdlh"><span class="allsmcap">SECTION A. DEFINITION AND LOGICAL PROPERTIES OF CARDINAL NUMBERS</span></td> +<td class="tdr"><a href="#Page_4">4</a></td> +</tr><tr> +<td class="tdlh2">*100. Definition and elementary properties of cardinal numbers</td> +<td class="tdr"><a href="#Page_13">13</a></td> +</tr><tr> +<td class="tdlh2">*101. On 0 and 1 and 2</td> +<td class="tdr"><a href="#Page_19">19</a></td> +</tr><tr> +<td class="tdlh2">*102. On cardinal numbers of assigned types</td> +<td class="tdr"><a href="#Page_24">24</a></td> +</tr><tr> +<td class="tdlh2">*103. Homogeneous cardinals</td> +<td class="tdr"><a href="#Page_36">36</a></td> +</tr><tr> +<td class="tdlh2">*104. Ascending cardinals</td> +<td class="tdr"><a href="#Page_42">42</a></td> +</tr><tr> +<td class="tdlh2">*105. Descending cardinals</td> +<td class="tdr"><a href="#Page_52">52</a></td> +</tr><tr> +<td class="tdlh2">*106. Cardinals of relational types</td> +<td class="tdr"><a href="#Page_60">60</a></td> +</tr><tr> +<td class="tdlh"><span class="allsmcap">SECTION B. ADDITION, MULTIPLICATION AND EXPONENTIATION</span></td> +<td class="tdr"><a href="#Page_66">66</a></td> +</tr><tr> +<td class="tdlh2">*110. The arithmetical sum of two classes and of two cardinals</td> +<td class="tdr"><a href="#Page_75">75</a></td> +</tr><tr> +<td class="tdlh2">*111. Double similarity</td> +<td class="tdr"><a href="#Page_88">88</a></td> +</tr><tr> +<td class="tdlh2">*112. The arithmetical sum of a class of classes</td> +<td class="tdr"><a href="#Page_97">97</a></td> +</tr><tr> +<td class="tdlh2">*113. On the arithmetical product of two classes or of two cardinals</td> +<td class="tdr"><a href="#Page_105">105</a></td> +</tr><tr> +<td class="tdlh2">*114. The arithmetical product of a class of classes</td> +<td class="tdr"><a href="#Page_124">124</a></td> +</tr><tr> +<td class="tdlh2">*115. Multiplicative classes and arithmetical classes</td> +<td class="tdr"><a href="#Page_135">135</a></td> +</tr><tr> +<td class="tdlh2">*116. Exponentiation</td> +<td class="tdr"><a href="#Page_143">143</a></td> +</tr><tr> +<td class="tdlh2">*117. Greater and less</td> +<td class="tdr"><a href="#Page_171">171</a></td> +</tr><tr> +<td class="tdlh3">General note on cardinal correlators</td> +<td class="tdr"><a href="#Page_185">185</a></td> +</tr><tr> +<td class="tdlh"><span class="allsmcap">SECTION C. FINITE AND INFINITE</span></td> +<td class="tdr"><a href="#Page_187">187</a></td> +</tr><tr> +<td class="tdlh2">*118. Arithmetical substitution and uniform formal numbers</td> +<td class="tdr"><a href="#Page_193">193</a></td> +</tr><tr> +<td class="tdlh2">*119. Subtraction</td> +<td class="tdr"><a href="#Page_201">201</a></td> +</tr><tr> +<td class="tdlh2">*120. Inductive cardinals</td> +<td class="tdr"><a href="#Page_207">207</a></td> +</tr><tr> +<td class="tdlh2">*121. Intervals</td> +<td class="tdr"><a href="#Page_233">233</a></td> +</tr><tr> +<td class="tdlh2">*122. Progressions</td> +<td class="tdr"><a href="#Page_253">253</a></td> +</tr><tr> +<td class="tdlh2">*123. \(\aleph_{0}\)</td> +<td class="tdr"><a href="#Page_268">268</a></td> +</tr><tr> +<td class="tdlh2">*124. Reflexive classes and cardinals</td> +<td class="tdr"><a href="#Page_278">278</a></td> +</tr><tr> +<td class="tdlh2">*125. The axiom of infinity</td> +<td class="tdr"><a href="#Page_289">289</a></td> +</tr><tr> +<td class="tdlh2">*126. On typically indefinite inductive cardinals</td> +<td class="tdr"><a href="#Page_293">293</a><span class="pagenum" id="Page_vi">[Pg vi]</span></td> +</tr><tr> +<td class="tdl">PART IV. RELATION-ARITHMETIC.</td> +<td class="tdr"></td> +</tr><tr> +<td class="tdlh2">Summary of Part IV</td> +<td class="tdr"><a href="#Page_301">301</a></td> +</tr><tr> +<td class="tdlh"><span class="allsmcap">SECTION A. ORDINAL SIMILARITY AND RELATION-NUMBERS</span></td> +<td class="tdr"><a href="#Page_303">303</a></td> +</tr><tr> +<td class="tdlh2">*150. Internal transformation of a relation</td> +<td class="tdr"><a href="#Page_306">306</a></td> +</tr><tr> +<td class="tdlh2">*151. Ordinal similarity</td> +<td class="tdr"><a href="#Page_319">319</a></td> +</tr><tr> +<td class="tdlh2">*152. Definition and elementary properties of relation-numbers</td> +<td class="tdr"><a href="#Page_330">330</a></td> +</tr><tr> +<td class="tdlh2">*153. The relation-numbers \(0_{r}\), \(2_{r}\) and \(1_{s}\)</td> +<td class="tdr"><a href="#Page_334">334</a></td> +</tr><tr> +<td class="tdlh2">*154. Relation-numbers of assigned types</td> +<td class="tdr"><a href="#Page_339">339</a></td> +</tr><tr> +<td class="tdlh2">*155. Homogeneous relation-numbers</td> +<td class="tdr"><a href="#Page_344">344</a></td> +</tr><tr> +<td class="tdlh"><span class="allsmcap">SECTION B. ADDITION OF RELATIONS, AND THE PRODUCT OF TWO RELATIONS</span></td> +<td class="tdr"><a href="#Page_347">347</a></td> +</tr><tr> +<td class="tdlh2">*160. The sum of two relations</td> +<td class="tdr"><a href="#Page_351">351</a></td> +</tr><tr> +<td class="tdlh2">*161. Addition of a term to a relation</td> +<td class="tdr"><a href="#Page_357">357</a></td> +</tr><tr> +<td class="tdlh2">*162. The sum of the relations of a field</td> +<td class="tdr"><a href="#Page_362">362</a></td> +</tr><tr> +<td class="tdlh2">*163. Relations of mutually exclusive relations</td> +<td class="tdr"><a href="#Page_369">369</a></td> +</tr><tr> +<td class="tdlh2">*164. Double likeness</td> +<td class="tdr"><a href="#Page_376">376</a></td> +</tr><tr> +<td class="tdlh2">*165. Relations of relations of couples</td> +<td class="tdr"><a href="#Page_386">386</a></td> +</tr><tr> +<td class="tdlh2">*166. The product of two relations</td> +<td class="tdr"><a href="#Page_396">396</a></td> +</tr><tr> +<td class="tdlh"><span class="allsmcap">SECTION C. THE PRINCIPLE OF FIRST DIFFERENCES, AND THE MULTIPLICATION +AND EXPONENTIATION OF RELATIONS</span></td> +<td class="tdr"><a href="#Page_403">403</a></td> +</tr><tr> +<td class="tdlh2">*170. On the relation of first differences among the sub-classes of a +given class</td> +<td class="tdr"><a href="#Page_411">411</a></td> +</tr><tr> +<td class="tdlh2">*171. The principle of first differences (continued)</td> +<td class="tdr"><a href="#Page_423">423</a></td> +</tr><tr> +<td class="tdlh2">*172. The product of the relations of a field</td> +<td class="tdr"><a href="#Page_428">428</a></td> +</tr><tr> +<td class="tdlh2">*173. The product of the relations of a field (continued)</td> +<td class="tdr"><a href="#Page_443">443</a></td> +</tr><tr> +<td class="tdlh2">*174. The associative law of relational multiplication</td> +<td class="tdr"><a href="#Page_447">447</a></td> +</tr><tr> +<td class="tdlh2">*176. Exponentiation</td> +<td class="tdr"><a href="#Page_458">458</a></td> +</tr><tr> +<td class="tdlh2">*177. Propositions connecting \(P_{\text{df}}\) with products and powers</td> +<td class="tdr"><a href="#Page_471">471</a></td> +</tr><tr> +<td class="tdlh2"><span class="allsmcap">SECTION D. ARITHMETIC OF RELATION-NUMBERS</span></td> +<td class="tdr"><a href="#Page_473">473</a></td> +</tr><tr> +<td class="tdlh2">*180. The sum of two relation-numbers</td> +<td class="tdr"><a href="#Page_477">477</a></td> +</tr><tr> +<td class="tdlh2">*181. On the addition of unity to a relation-number</td> +<td class="tdr"><a href="#Page_482">482</a></td> +</tr><tr> +<td class="tdlh2">*182. On separated relations</td> +<td class="tdr"><a href="#Page_487">487</a></td> +</tr><tr> +<td class="tdlh2">*183. The sum of the relation-numbers of a field</td> +<td class="tdr"><a href="#Page_496">496</a></td> +</tr><tr> +<td class="tdlh2">*184. The product of two relation-numbers</td> +<td class="tdr"><a href="#Page_501">501</a></td> +</tr><tr> +<td class="tdlh2">*185. The product of the relation-numbers of a field</td> +<td class="tdr"><a href="#Page_505">505</a></td> +</tr><tr> +<td class="tdlh2">*186. Powers of relation-numbers</td> +<td class="tdr"><a href="#Page_507">507</a></td> +</tr><tr> +<td class="tdl">PART V. SERIES.</td> +<td class="tdr"></td> +</tr><tr> +<td class="tdlh2">Summary of Part V.</td> +<td class="tdr"><a href="#Page_513">513</a></td> +</tr><tr> +<td class="tdlh"><span class="allsmcap">SECTION A. GENERAL THEORY OF SERIES</span></td> +<td class="tdr"><a href="#Page_516">516</a></td> +</tr><tr> +<td class="tdlh2">*200. Relations contained in diversity</td> +<td class="tdr"><a href="#Page_518">518</a></td> +</tr><tr> +<td class="tdlh2">*201. Transitive relations</td> +<td class="tdr"><a href="#Page_525">525</a></td> +</tr><tr> +<td class="tdlh2">*202. Connected relations</td> +<td class="tdr"><a href="#Page_533">533</a></td> +</tr><tr> +<td class="tdlh2">*204. Elementary properties of series</td> +<td class="tdr"><a href="#Page_547">547</a></td> +</tr><tr> +<td class="tdlh2">*205. Maximum and minimum points</td> +<td class="tdr"><a href="#Page_559">559</a></td> +</tr><tr> +<td class="tdlh2">*206. Sequent points</td> +<td class="tdr"><a href="#Page_577">577</a></td> +</tr><tr> +<td class="tdlh2">*207. Limits</td> +<td class="tdr"><a href="#Page_594">594</a></td> +</tr><tr> +<td class="tdlh2">*208. The correlation of series</td> +<td class="tdr"><a href="#Page_605">605</a><span class="pagenum" id="Page_vii">[Pg vii]</span></td> +</tr><tr> +<td class="tdlh2"><span class="allsmcap">SECTION B. ON SECTIONS, SEGMENTS, STRETCHES, AND DERIVATIVES</span></td> +<td class="tdr"><a href="#Page_612">612</a></td> +</tr><tr> +<td class="tdlh2">*210. On series of classes generated by the relation of inclusion</td> +<td class="tdr"><a href="#Page_615">615</a></td> +</tr><tr> +<td class="tdlh2">*211. On sections and segments</td> +<td class="tdr"><a href="#Page_624">624</a></td> +</tr><tr> +<td class="tdlh2">*212. The series of segments</td> +<td class="tdr"><a href="#Page_651">651</a></td> +</tr><tr> +<td class="tdlh2">*213. Sectional relations</td> +<td class="tdr"><a href="#Page_668">668</a></td> +</tr><tr> +<td class="tdlh2">*214. Dedekindian relations</td> +<td class="tdr"><a href="#Page_684">684</a></td> +</tr><tr> +<td class="tdlh2">*215. Stretches</td> +<td class="tdr"><a href="#Page_691">691</a></td> +</tr><tr> +<td class="tdlh2">*216. Derivatives</td> +<td class="tdr"><a href="#Page_700">700</a></td> +</tr><tr> +<td class="tdlh2">*217. On segments of sums and converses</td> +<td class="tdr"><a href="#Page_710">710</a></td> +</tr><tr> +<td class="tdlh"><span class="allsmcap">SECTION C. ON CONVERGENCE, AND THE LIMITS OF FUNCTIONS</span></td> +<td class="tdr"><a href="#Page_715">715</a></td> +</tr><tr> +<td class="tdlh2">*230. On convergents</td> +<td class="tdr"><a href="#Page_720">720</a></td> +</tr><tr> +<td class="tdlh2">*231. Limiting sections and ultimate oscillation of a function</td> +<td class="tdr"><a href="#Page_727">727</a></td> +</tr><tr> +<td class="tdlh2">*232. On the oscillation of a function as the argument approaches a +given limit</td> +<td class="tdr"><a href="#Page_737">737</a></td> +</tr><tr> +<td class="tdlh2">*233. On the limits of functions</td> +<td class="tdr"><a href="#Page_745">745</a></td> +</tr><tr> +<td class="tdlh2">*234. Continuity of functions</td> +<td class="tdr"><a href="#Page_753">753</a></td> +</tr> +</tbody> +</table> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_viii">[Pg viii]</span></p> +<h2 class="nobreak" id="ADDITIONAL_ERRATA">ADDITIONAL ERRATA TO VOLUME I.</h2> +</div> + + +<p class="nind"> +p. 5, line 20, <i>delete</i> "\(\pi\)."<br> +p. 34, line 20, <i>for</i> "\(yRx\)" <i>read</i> "\(xRy\)."<br> +p. 36, line 7 and line 10, <i>for</i> "\(Q\mid R\)" <i>read</i> "\(R\mid P\)."<br> +p. 44, line 17, <i>for</i> "\((p).p. \text{is false}\)" <i>read</i> "\((p). p\text{ is false}\)."<br> +p. 112, in *2·52, <i>in place of</i> "\(p \supset \sim q\)" <i>read</i> "\(\sim p \supset \sim q\)."<br> +p. 129, in *5·11, <i>in place of reference to</i> "\(*2·51\)" <i>read reference to</i> "\(*2·5\)."<br> +p. 129, in *5·12, <i>in place of reference to</i> "\(*2·52\)" <i>read reference to</i> "*2·51."<br> +p. 144, *10·23 <i>should be</i> "\(\vdash\colon\ldotp(x).\phi x \supset p.\equiv :(\exists x).\phi x.\supset.p\)."<br> +p. 157, line 11, <i>for</i> "*10" <i>read</i> "*9."<br> +p. 184, last line of <i>Dem.</i> of *14·111, <i>for second</i> "\(x=c\)" <i>read</i> "\(x=b\)."<br> +p. 228, in *23·81, <i>for</i> "\(\dot{-} R \unicode{x2abd} \dot{-} S\)" <i>read</i> "\(\dot{-} S \unicode{x2abd} \dot{-} R\)."<br> +p. 242, in *25·37, <i>for</i> "\(zRw\)" <i>read</i> "\(zSw\)."<br> +p. 242, in *25·412, <i>for</i> "\(R\)" <i>read</i> "\(S\)."<br> +p. 253, 2nd and 4th lines of <i>Dem.</i> of *31·16, <i>for</i> "*21·35" <i>read</i> "*23·35."<br> +p. 259, in note to *32·35, <i>for</i> "*32·2" <i>read</i> "*32·3."<br> +p. 263, in *33·16, 4th line of <i>Dem.</i>, <i>for</i> "*20·34" <i>read</i> "*22·34."<br> +p. 265, in *33·26, 2nd line of <i>Dem.</i>, <i>for</i> "*21·34" <i>read</i> "*23·34."<br> +p. 275, in *34·6, 4th line of <i>Dem.</i>, <i>for first</i> "\(S\)" <i>read</i> "\(R\)."<br> +p. 289, 1st line, <i>for</i> "\(=\beta \uparrow \gamma\)" <i>read</i> "\(=\alpha \uparrow \gamma\)."<br> +p. 322, in *40·18, enunciation, <i>for</i> "\(\equiv\)" <i>read</i> "\(=\)".<br> +p. 329, in *40·69, <i>Dem.</i>, <i>for</i> "\(\overrightarrow{P}\)" <i>read</i> "\(\overleftarrow{P}\)" (3 <i>times</i>).<br> +p. 387, in *55·224, 1st line of <i>Dem.</i>, <i>for</i> "\(\uparrow\)" <i>read</i> "\(\downarrow\)" (<i>twice</i>).<br> +p. 388, in *55·281, <i>for third</i> "\(=\)" <i>read</i> "\(\equiv\)."<br> +p. 410, in *60·53, last line of <i>Dem.</i>, <i>for</i> "\(\gamma\)" <i>read</i> "\(\beta\)."<br> +p. 453, in *71·25, <i>Dem.</i>, 1st line, <i>for</i> "\(xRy.xRz\)" <i>read</i> "\(ySx.zSx\)."<br> +<span style="margin-left: 2em;">" " 2nd line, <i>for</i> + "\(xRy.ySu.xRz.zSv.\supset.y=z.ySu.zSv\)"</span><br> +<span style="margin-left: 2em;"><i>read</i> "\(uRy.ySx.vRz.zSx.\supset.y=z.uRy.vRz\)."</span><br> +<span style="margin-left: 2em;">" " 3rd line, <i>for</i> + "\(ySu.ySv\)" <i>read</i> "\(uRy.vRy\)."</span><br> +<span style="margin-left: 2em;">" " 6th line, <i>for</i> + "\(xRy.ySu\)" <i>read</i> "\(uRy.ySx\)" <i>and for</i></span><br> +<span style="margin-left: 10em;">"\(xRz.zSv\)" <i>read</i> "\(vRz.zSx\)."</span><br> +<span style="margin-left: 2em;">" " 7th line, <i>for</i> + "\(x(R\mid S)u.x(R\mid S)v\)" <i>read</i> "\(u(R\mid S)x.v(R\mid S)x\)."</span><br> +p. 465, in *72·16, <i>Dem.</i>, 1st line, <i>for last</i> "\(\kappa\)" <i>read</i> "\(x\)."<br> +p. 483, in *73·44, <i>Dem.</i>, 1st line, <i>for second</i> "\(y\)" <i>read</i> "\(x\)."<br> +p. 485, in *73·511, <i>for</i> "\(\beta\)" <i>read</i> "\(\alpha\)."<br> +p. 522, in *81·23, enunciation and 2nd line of <i>Dem.</i>, <i>for</i> "\(\overrightarrow{R}\)" <i>read</i> "\(R\)."<br> +p. 592, in *91·33, <i>Dem.</i>, 1st line, <i>for</i> "\(P\)" <i>read</i> "\(R\)."<br> +p. 614, in *93·36, <i>Dem.</i>, <i>for</i> "\(R\)" <i>read</i> "\(P\)" <i>throughout</i>.<br> +p. 628, in *95·21, <i>Dem.</i>, line 6, <i>for</i> "\(Q'\)" <i>read</i> "\(T'\)."<br> +</p> + +<p class="nindc space-above2">ERRATA TO VOLUME II</p> + +<p class="nind"> +p. 82, last line but one, <i>for</i> "\(\Lambda_{\alpha}\)" <i>and</i> "\(\Lambda_{\beta}\)" <i>read</i> "\(\Lambda \cap \alpha\)" <i>and</i> "\(\Lambda \cap \beta\)."<br> +p. 101, *112·23, enunciation, the second time two dots occur, <i>read</i> one dot.<br> +p. 573, *205·7, enunciation, <i>for</i> "\(\text{max}_{P}\)" <i>read</i> "\(\overrightarrow{\text{max}}_{P}\)."<br> +</p> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_ix">[Pg ix]</span></p> + +<h2 class="nobreak" id="PREFATORY_STATEMENT_OF_SYMBOLIC"> +PREFATORY STATEMENT OF SYMBOLIC +CONVENTIONS</h2> +</div> + + +<p>THE purpose of the following observations is to bring together in one +discussion various explanations which are required in applying the +theory of types to cardinal arithmetic. It is convenient to collect +these observations, since otherwise their dispersion throughout the +several numbers of Part III makes it difficult to see what is their +total effect. But although we have placed these observations at the +beginning, they are to be read concurrently with the text of Part +III, at least with so much of the text as consists of explanations +of definitions. The earlier portion of what follows is merely a +<i>résumé</i> of previous explanations; it is only in the later +portions that the application to cardinal arithmetic is made.</p> + +<p class="nindc space-above2">I. <i>General Observations on Types.</i></p> + +<p>Three different kinds of typical ambiguity are involved in our +propositions, concerning:</p> + +<p class="hanging2">(1) the functional hierarchy,</p> + +<p class="hanging2">(2) the propositional hierarchy,</p> + +<p class="hanging2">(3) the extensional hierarchy.</p> + +<p class="nind"> +The relevance of these must be separately considered.</p> + +<p>We often speak as though the type represented by small Latin letters +were not composed of functions. It is, however, compatible with all +we have to say that it should be composed of functions. It is to +be observed, further, that, given the number of individuals, there +is nothing in our axioms to show how many predicative functions of +individuals there are, <i>i.e.</i> their number is not a function +of the number of individuals: we only know that their number +\(\geq 2^{\text{Nc}ʻ\text{Indiv}}\), where "\(\text{Indiv}\)" stands for the +class of individuals.</p> + +<p>In practice, we proceed along the extensional hierarchy after the early +numbers of the book. If we have started from individuals, the result +of this is to exclude functions wholly from our hierarchy; if we have +started with functions of a given type, all functions of other types +are excluded. Thus a fresh extensional hierarchy, wholly excluding +every other, starts from each<span class="pagenum" id="Page_x">[Pg x]</span> type of function. When we speak simply +of "<i>the</i> extensional hierarchy," we mean the one which starts +from individuals.</p> + +<p>It is to be observed that when we have the assertion of a propositional +function, say "\(\vdash.\phi x\)," the \(x\) must be of some definite +type, <i>i.e.</i> we only assert that \(\phi x\) is true whatever \(x\) +may be within some one type. Thus <i>e.g.</i> "\(\vdash.x=x\)" does not +assert more than that this assertion holds for any \(x\) of a given +type. It is true that symbolically the same assertion holds in other +types, but other types cannot be included under one assertion-sign, +because no variable can travel beyond its type.</p> + +<p>The process of rendering the types of variables ambiguous is begun in +*9, where we take the first step in regard to the <i>propositional</i> +hierarchy. Before *9, our variables are <i>elementary propositions</i>. +These are such as contain no apparent variables. Hence the only +functions that occur are matrices, and these only occur through +their values. The assumption involved in the transition from Section +A to Section B (Part I) is that, given "\(\vdash.fp\)," where +\(p\) is an elementary proposition, we may substitute for \(p\) +"\(\phi!(x,y,z,\ldots)\)," where \(\phi\) is any matrix. Thus instead +of "\(\vdash.fp\)," which contained one variable \(p\) of a given +type, we have "\(\vdash.f\{\phi!(x,y,z,...)\}\)," which contains +several variables of several types (any finite number of variables and +types is possible). This assumption involves some rather difficult +points. It is to be remembered that no <i>value</i> of \(\phi\) +contains \(\phi\) as a constituent, and therefore \(\phi\) is not a +constituent of \(fp\) even if \(p\) is a value of \(\phi\). Thus we +pass, above, from an assertion containing no function as a constituent +to one containing one or more functions as constituents. The assertion +"\(\vdash.fp\)" concerns <i>any</i> elementary proposition, whereas +"\(\vdash.f\{\phi(x,y,z,\ldots)\}\)" concerns <i>any of a certain +set</i> of elementary propositions, namely any of those that are values +of \(\phi\). Different types of functions give different sorts of ways +of picking out elementary propositions.</p> + +<p>Having assumed or proved "\(\vdash.fp\)," where \(p\) is elementary and +therefore involves no ambiguity of type, we thus assert +\[ +\vdash.f\{\phi(x,y,z,\ldots)\}, +\] +where the types of the arguments and the number of them are wholly +arbitrary, except that they must belong to the functional hierarchy +including individuals. (The assumption that propositions are incomplete +symbols excludes the possibility that the arguments to \(\phi\) are +propositions.) The noteworthy point is that we thus obtain an assertion +in which there may be any finite number of variables and the variables +have unlimited typical ambiguity, from an assertion containing one +variable of a perfectly definite type. All this is presupposed before +we embark on the propositional hierarchy.</p> + +<p>It should be observed that all elementary propositions are values +of predicative functions of one individual, <i>i.e.</i> of +\(\phi!\hat{x}\), where \(\hat{x}\) is individual.<span class="pagenum" id="Page_xi">[Pg xi]</span> Thus we need not +<i>assume</i> that elementary propositions form a type; we may replace +\(p\) by "\(\phi!x\)" in "\(\vdash.fp\)." In this way, propositions as +variables wholly disappear.</p> + +<p>In extending statements concerning elementary propositions so as +formally to apply to first-order propositions, we have to assume afresh +the primitive proposition *1·11 (*1·1 is never used), <i>i.e.</i> given +"\(\vdash.\phi x\)" and "\(\vdash.\phi x\supset \psi x\)," we have +"\(\vdash.\psi x\)" which is practically *9·12. This was asserted in +*1·11 for any case in which \(\phi x\) and \(\psi x\) are elementary +propositions. There was here already an ambiguity of type, owing to the +fact that x need not be an individual, but might be a function of any +order. <i>E.g.</i> we might use *1·11 to pass from +\[ +\unicode{x201c}\vdash.\phi!a\unicode{x201d} \quad \text{and} \quad \unicode{x201c}\vdash.\phi!a\supset \phi!b\unicode{x201d} \quad \text{to} \quad \unicode{x201c}\vdash.\phi!b,\unicode{x201d} +\] +where \(\phi\) replaces the \(x\) of *1·11, and \(\hat{\phi}!a\), +\(\hat{\phi}!b\) replace \(\phi\hat{x}\) and \(\psi\hat{x}\). Thus +*1·11, even before its extension in *9, already states a fresh +primitive proposition for each fresh type of functions considered. +The novelty in *9 is that we allow \(\phi\) and \(\psi\) to contain one +apparent variable. This may be of any functional type (including +Indiv); thus we get another set of symbolically identical primitive +propositions. In passing, as indicated at the end of *9, to more +than one apparent variable, we introduce a new batch of primitive +propositions with each additional apparent variable.</p> + +<p>Similar remarks apply to the other primitive propositions of *9.</p> + +<p>What makes the above process legitimate is that nothing in the +treatment of functions of order \(n\) <i>presupposes</i> functions +of higher order. We can deal with each new type of functions as it +arises, without having to take account of the fact that there are later +types. From symbolic analogy we "see" that the process can be repeated +indefinitely. This possibility rests upon two things:</p> + +<p>(1) A fresh interpretation of our constants—\(\lor\), \(\sim\), !, +(\(x\))., (\(\exists x\)).—at each fresh stage;</p> + +<p>(2) A fresh assumption, symbolically unchanged, of the primitive +propositions which we found sufficient at an earlier stage—the +possibility of avoiding symbolic change being due to the fresh +interpretation of our constants.</p> + +<p>The above remarks apply to the axiom of reducibility as well as to our +other primitive propositions. If, at any stage, we wish to deal with +a class defined by a function of the 30,000th type, we shall have to +repeat our arguments and assumptions 30,000 times. But there is still +no necessity to speak of the hierarchy as a whole, or to suppose that +statements can be made about "all types."</p> + +<p>We come now to the extensional hierarchy. This starts from some +one point in the functional hierarchy. We usually suppose it to +start from<span class="pagenum" id="Page_xii">[Pg xii]</span> individuals, but any other starting-point is equally +legitimate. Whatever type of functions (including \(\text{Indiv}\) +we start from, all higher types of functions are excluded from the +extensional hierarchy, and also all lower types (if any). Some +complications arise here. Suppose we start from \(\text{Indiv}\). +Then if \(\phi!\hat{z}\) is any predicative function of individuals, +\(\hat{z}(\phi!z) = \phi!\hat{z}\). But identity between a function +and a class does not have the usual properties of identity; in +fact, though every function is identical with some class, and +vice versa, the number of functions is likely to be greater than +the number of classes. This is due to the fact that we may have +\(\hat{z}(\phi!z)=\psi!\hat{z}.\hat{z}(\phi!z)=\chi!\hat{z}\) without +having \(\psi!\hat{z}=\chi!\hat{z}\).</p> + +<p>In the extensional hierarchy, we prove the extension from classes to +classes of classes, and so on, without fresh primitive propositions +(*20, *21). The primitive propositions involved are those concerning +the <i>functional</i> hierarchy.</p> + +<p>From all these various modes of extension we "see" that whatever <i>can +be proved</i> for lower types, whether functional or extensional, +can also be proved for higher types<a id="FNanchor_1" href="#Footnote_1" class="fnanchor">[1]</a>. Hence we assume that it is +unnecessary to know the types of our variables, though they must always +be confined within some one definite type.</p> + +<p>Now although everything that can be proved for lower types can be +proved for higher types, the converse does not hold. In Vol. I. only +two propositions occur which can be proved for higher but not for +lower types. These are \(\exists !2\) and \(\exists !2_{r}\). These +can be proved for any type except that of individuals. It is to be +observed that we do <i>not</i> state that whatever is <i>true</i> for +lower types is <i>true</i> for higher types, but only that whatever +<i>can be proved</i> for lower types <i>can be proved</i> for higher types. +If, for example, \(\text{Nc}ʻ{\text{Indiv}} = \nu\), then this +proposition is false for any higher type; but this proposition, +\(\text{Nc}ʻ{\text{Indiv}} = \nu\), is one which cannot be proved +logically; in fact, it is only ascertainable by a census, not by logic. +Thus among the propositions which can be proved by logic, there are +some which can only be proved for higher types, but none which can only +be proved for lower types.</p> + +<p>The propositions which can be proved in some types but not in others +all are or depend upon existence-theorems for cardinals. We can prove +\[\begin{array}{l} +\exists !0,\, \exists !1,\, \text{universally},\\ +\exists !2,\, \text{except for}\,\, \text{Indiv},\\ +\exists !3,\, \exists !4,\, \text{except for}\,\, \text{Indiv},\, \text{Cl}ʻ\text{Indiv},\, \text{Rl}ʻ\text{Indiv};\,\, \text{and so on}. +\end{array} +\] +Exactly similar remarks would apply to the functional hierarchy. In +both cases, the possibility of proving these propositions depends upon +the axiom of reducibility and the definition of identity. Suppose there +is only one individual, \(x\). Then \(\hat{y}=x\), \(\hat{y}\neq x\) +are two different functions, which, by the<span class="pagenum" id="Page_xiii">[Pg xiii]</span> axiom of reducibility, are +equivalent to two different predicative functions. Hence there are at +least two predicative functions of \(x\), and at least two classes +\(\iotaʻx\), \(\Lambda_{x}\). This argument fails both for classes and +functions if either we deny the axiom of reducibility or we suppose +that there may be two different individuals which agree in all their +predicates, <i>i.e.</i> that the definition of identity is misleading.</p> + +<p>The statement that what can be proved for lower types can be proved +for higher types requires certain limitations, or rather, a more exact +formulation. Taking \(\text{Indiv}\) as a primitive idea, put +\[ +\text{Kl} = \text{Cl}ʻ\text{Indiv} \quad\text{Df},\,\, \text{Kl}^{2} = \text{Cl}ʻ\text{Kl} \quad\text{Df}, \quad\text{etc}. +\] +Then consider the proposition \(\text{Nc}ʻ\text{Kl} = \Lambda\) . We +can prove +\[ +\text{Nc}ʻ\text{Kl} \cap tʻ\text{Indiv} = \Lambda . \exists ! \text{Nc}ʻ\text{Kl} \cap tʻ\text{Kl} . \exists ! \text{Nc}ʻ\text{Kl} \cap tʻ\text{Kl}^{2}. \quad\text{etc}. +\] +Thus \(\text{Nc}ʻ\text{Kl} = \Lambda\) can be proved in the lowest type +in which it is significant, and disproved in any other. The difficulty, +however, is avoided if Indiv is replaced by a variable \(\alpha\), and +\(\text{Kl}\) by \(\text{Cl}ʻt_{0}ʻ\alpha\). Then we have +\[ +\text{Nc}ʻ\text{Cl}ʻt_{0}ʻ\alpha \cap tʻ\alpha = \Lambda, +\] +and this holds whatever the type of \(\alpha\) may be. Thus in order +that our principle about lower and higher types may be true, it is +necessary that any relation there may be between two types occurring in +a proposition should be preserved; in other words, when one constant +type is <i>defined</i> in terms of another (as \(\text{Kl}\) and +\(\text{Indiv}\)), the definition must be restored before the type is +varied, so that when one type is varied, so is the other. With this +proviso, our principle about higher and lower types holds.</p> + +<p>With the above proviso, the truth of our statement is manifest. For +we have shown that the same primitive propositions, symbolically, +which hold for the lowest type concerned in our reasoning, hold also +for subsequent types; and therefore all our proofs can be repeated +symbolically unchanged.</p> + +<p>The importance of this lies in the fact that, when we have proved a +proposition for the lowest significant type, we "see" that it holds +in any other assigned significant type. Hence every proposition which +is proved without the mention of any type is to be regarded as proved +for the lowest significant type, and extended by analogy to any other +significant type.</p> + +<p>By exactly similar considerations we "see" that a proposition which +can be proved for some type other than the lowest significant type +must hold for any type in the direct descent from this. <i>E.g.</i> +suppose we can prove a proposition (such as \(\exists ! 2\)) for the +type \(\text{Kl}\) (where \(\text{Kl} = \text{Cl}ʻ\text{Indiv}\)); then +merely writing \(\text{Cl}ʻ\text{Indiv}\) for \(\text{Kl}\), we have +a proposition which is proved concerning \(\text{Indiv}\), namely +\(\exists!2 \cap tʻ\text{Cl}ʻ\text{Indiv}\), and here, by what was said +before, \(\text{Indiv}\)) may be replaced by any higher type.</p> + +<p>Thus given a typically ambiguous relation \(R\), such that, +if \(\tau\) is a type, \(Rʻ\tau\) is a type (\(\text{Cl}\) or +\(\text{Rl}\) is such a relation), we "see" that, if we can prove<span class="pagenum" id="Page_xiv">[Pg xiv]</span> +\(\phi (Rʻ\text{Indiv})\), we can also prove \(\phi (Rʻ\tau)\), where +\(\tau\) is any type, and \(\phi\) is composed of typically ambiguous +symbols. Similarly if we can prove \(\phi (\text{Indiv},Rʻ\text{Indiv})\), we can +prove \(\phi (\tau ,Rʻ\tau)\), where \(\tau\) is any type. But we +cannot in general prove \(\phi (\text{Indiv},Rʻ\tau)\) or +\(\phi (\tau,Rʻ\text{Indiv})\), and these may be in fact untrue. <i>e.g.</i> +we have \(\exists !\text{Nc}(\text{Kl})ʻ\text{Indiv}.{\sim}\exists!\text{Nc}(\text{Kl})ʻ\text{Kl}^{2}\).</p> + +<p>Thus more generally, when a proposition containing several ambiguities +can be proved for the types \(Rʻ\text{Indiv}\), \(Sʻ\text{Indiv}\), +..., but not for lower types, it is to be regarded as a function of +\(\text{Indiv}\), and then it becomes true for <i>any</i> type; that is, given +\[ +\phi (Rʻ\text{Indiv},Sʻ\text{Indiv}, \ldots), +\] +we shall also have +\[ +\phi (Rʻ\tau,Sʻ\tau, \ldots), +\] +where \(\tau\) is any type. In this way, <i>all</i> demonstrable +propositions are in the first instance about \(\text{Indiv}\), and +when so expressed remain true if any other type is substituted for +\(\text{Indiv}\).</p> + +<p>When a proposition containing typically ambiguous symbols <i>can be +proved</i> to be true in the lowest significant type, and we can "see" +that symbolically the same proof holds in any other assigned type, +we say that the proposition has "permanent truth." (We may also say, +loosely, that it is "true in all types.") When a proposition containing +typically ambiguous symbols <i>can be proved</i> to be false in the +lowest significant type, and we can "see" that it is false in any +other assigned type, we say that it has "permanent falsehood." Any +other proposition containing typically ambiguous symbols is said to +be "fluctuating," or to have "fluctuating truth-value," as opposed to +"permanent truth-value," which belongs to propositions that have either +permanent truth or permanent falsehood.</p> + +<p>In what follows, ambiguities concerned with the propositional hierarchy +will be ignored, since they never lead to fluctuating propositions. +Thus disjunction and negation and their derivatives will not receive +explicit typical determination, but only such typical determination +as results from assigning the types of the other typically ambiguous +symbols involved.</p> + +<p><span class="pagenum" id="Page_xv">[Pg xv]</span></p> + +<p>It is convenient to call the symbolic form of a propositional function +simply a "<i>symbolic form</i>." Thus, if a symbolic form contains +symbols of ambiguous type it represents different propositional +functions according as the types of its ambiguous symbols are +differently adjusted. The adjustment is of course always limited by +the necessity for the preservation of meaning. It is evident that the +ideas of "permanent truth-value" and "fluctuating truth-value" apply +in reality to symbolic forms and not to propositions or propositional +functions. Ambiguity of type can only exist in the process of +determination of meaning. When the meaning has been assigned to a +symbolic form and a propositional function thereby obtained, all +ambiguity of type has vanished.</p> + +<p>To "assert a symbolic form" is to assert each of the propositional +functions arising for the set of possible typical determinations which +are somewhere enumerated. We have in fact enumerated a very limited +number of types starting from that of individuals, and we "see" that +this process can be indefinitely continued by analogy. The form is +always asserted so far as the enumeration has arrived; and this is +sufficient for all purposes, since it is essentially impossible to use +a type which has not been arrived at by successive enumeration from the +lower types.</p> + +<p>The only difficulties which arise in Cardinal Arithmetic in connection +with the ambiguities of type of the symbols are those which enter +through the use of the symbol \(\text{ sm }\), or of the symbol \(\text{Nc}\), which +is \(\overrightarrow{\text{ sm }}\). For it may happen that a class +in one type has no class similar to it in some lower type (cf. +<a href="#*102·72">*102·72</a><a href="#*102·73">·73</a>). All fallacious reasoning in cardinal or ordinal arithmetic +in connection with types, apart from that due to the mere absence +of meaning in symbols, is due to this fact—in other words to the +fact that in some types \(\exists!\text{Nc}ʻ\alpha\) is true, and +in other types \(\exists !\text{Nc}ʻ\alpha\) may not be true. The +fallacy consists in neglecting this latter possibility of the failure +of \(\exists !\text{Nc}ʻ\alpha\) for a limited number of types, that +is, in taking the "fluctuating" form \(\exists !\text{Nc}ʻ\alpha\) as +though it possessed a "permanent" truth-value.</p> + +<p>A fluctuating form however often possesses what is here termed +a "stable" truth-value, which is as important as the permanent +truth-value of other forms. For example, anticipating our definitions +of elementary arithmetic, consider \(2 +_{c} 3 = 5\). There is no +abstract logical proof that there are two individuals; so suppose 2 and +3 refer to classes of individuals, but 5 refers to classes of a high +enough type, then with these determinations \(2 +_{c} 3 = 5\) cannot be +proved. But \(2 +_{c} 3 = 5\) has a <i>stable</i> truth-value, since +it can always be proved when all the types are high enough. In this +case the fact that our empirical census of individuals (at least of the +"relative" individuals of ordinary life) has outrun the capacity of +logical proof, makes the fluctuation in the truth-value of the form to +be entirely unimportant.</p> + +<p>In order to make this idea precise, it is necessary to have a +convention as to the order in which the types of symbols in a symbolic +form are assigned. The rule we adopt is that the types of the <i>real +variables</i> are to be first assigned, and then those of the +<i>constant symbols</i>. The types of the apparent variables, if any, +will then be completely determinate.</p> + +<p><span class="pagenum" id="Page_xvi">[Pg xvi]</span></p> + +<p>A symbolic form has a <i>stable</i> truth-value if, after any +assignment of types to the real variables, types can be assigned to +the constant symbols so that the truth-value of the proposition thus +obtained is the same as the truth-value of any proposition obtained by +modifying it by the assignment of higher types to some or all of the +constant symbols. This truth-value is the <i>stable</i> truth-value.</p> + + +<p class="nindc space-above2">II. <i>Formal Numbers.</i></p> + +<p>The conventions, which we shall give below as to the assignment of +types, practically restrict our interpretation of fluctuating symbolic +forms to types in which the forms possess their stable truth-value. +The assumption that these truth-values are stable never enters into +the reasoning. But we judge a truth-value to be stable when any method +of raising the types of the constant symbols by one step leaves it +unaltered.</p> + +<p>In practice the fluctuation of truth-values only enters into our +consideration through a limited number of symbols called "formal +numbers."</p> + +<p>Formal numbers may be "constant" or "functional."</p> + +<p>A <i>constant</i> formal number is any constant symbol for which +there is a constant \(\alpha\) such that, in whatever type the +constant symbol is determined, it is, in that type, identical with +\(\text{Nc}ʻ\alpha\). In other words if \(\sigma\) be a constant +symbol, then \(\sigma\) is a formal number provided that "truth" is +the permanent truth-value of \(\sigma =\text{Nc}ʻ\alpha\), for some +constant \(\alpha\).</p> + +<p>The <i>functional</i> formal numbers are defined by enumeration; they +are +\[ +\text{Nc}ʻ\alpha,\,\, \Sigma \text{Nc}ʻ\kappa,\,\, \Pi \text{Nc}ʻ\kappa,\,\, \text{ sm }ʻʻ\mu,\,\, \mu+_{c}\nu,\,\, \mu-_{c}\nu,\,\, \mu \times_{c}\nu,\,\, + \mu^\nu, +\] +where in each formal number the symbols \(\alpha\), \(\kappa\), +\(\mu\), \(\nu\) occurring in it are called the arguments of the +functional form even when they are complex symbols. The argument of +\(\text{Nc}ʻ(\alpha +\beta)\) is \(\alpha +\beta\), and those of +\(\mu+_{c} (\nu +_{c} \varpi)\) are \(\mu\) and \(\nu+_{c} \varpi\), +and those of \(1 +_{c} 2\) are 1 and 2.</p> + +<p>Thus among the constant formal numbers are +\[ +0,\,\, 1,\,\, 2,\,\, \ldots,\,\, \aleph_{0},\,\, 1 +_{c} 2,\,\, 2 \times _{c} \aleph_{0},\,\, 2^{2}. +\] +The references which support this statement are +</p><p class="nindc"> +<a href="#*101·11">*101·11</a><a href="#*101·21">·21</a><a href="#*101·32">·32</a>.<a href="#*123·36">*123·36</a>.<a href="#*110·42">*110·42</a>.<a href="#*113·23">*113·23</a>.<a href="#*116·23">*116·23</a>. +</p> +<p class="nind"> +Among the functional formal numbers are +\[ +\text{Nc}ʻ(\alpha +\beta),\,\, \mu +_{c} (\nu +_{c} \varpi),\,\,(\mu +_{c} \nu) \times _{c} \varpi,\,\, (\mu +_{c} \nu )^\varpi. +\] +It will be observed that <i>e.g.</i> \(1 +_{c} 2\) is both a constant +and a functional formal number, so that the two classes are not +mutually exclusive. In fact they possess an indefinite number of +members in common.</p> + +<p>All the formal numbers, with the exception of \(\text{ sm }ʻʻ\mu\) and +\(\mu -_{c} \nu\), are members of \(\text{NC}\) without any hypothesis +[cf. <a href="#*100·41">*100·41</a><a href="#*100·01">·01</a><a href="#*100·52"></a>.<a href="#*110·42">*110·42</a>.<a href="#*112·101">*112·101</a>.<a href="#*113·23">*113·23</a>.<a href="#*114·1">*114·1</a>.<a href="#*116·23">*116·23</a>, note to +<a href="#*119·12">*119·12</a>, and <a href="#*120·411">*120·411</a>].</p> + +<p>A functional formal number consists of two parts, namely, its argument +or arguments, and the constant "form." An argument of a functional +formal number may be a complex symbol, and may be constant or variable. +Thus \(\mu +_{c} \nu\) is an argument of \((\mu +_{c} \nu) +_{c}\rho\), +<span class="pagenum" id="Page_xvii">[Pg xvii]</span>and of (\(\mu +_{c} \nu) \times _{c} 1\) and of (\(\mu +_{c}\nu)^\rho\); +also \(2 +_{c} 3\) is an argument of (\(2 +_{c} 3)\times_{c} 1\). +The constant form is constituted by the other symbols which +are constants. Two occurrences of functional formal numbers are only +occurrences of the same formal number if the arguments and also the +constant forms are identical in symbolism. Thus two occurrences of +\(\text{Nc}ʻ\alpha\) are occurrences of the same formal number, even if +they are determined to be in different types; but \(\text{Nc}ʻ\alpha\) +and \(\text{Nc}ʻ\beta\) are different formal numbers. Also \(\mu^{1}\) +and \(\mu \times _{c} 1\) are different formal numbers because their +"forms" are different, though the arguments \(\mu\) and 1 are the +same and (in the same type) the entity denoted is the same. Thus the +distinction between formal numbers depends on the symbolism and not +on the entity denoted, and in considering them it is symbolic analogy +and not denotation which is to be taken into account. For example +two different occurrences of the same formal number will not denote +the same entity, if in the two occurrences the ambiguity of type is +determined differently.</p> + +<p>The functional formal numbers are divided into three sets: (i) the +<i>primary</i> set consisting of the forms \(\text{Nc}ʻ\alpha\), +\(\Sigma \text{Nc}ʻ\kappa \), \(\Pi \text{Nc}ʻ\kappa\), (ii) the +argumental set consisting only of \(\text{ sm }ʻʻ\mu\), (iii) the +arithmetical set consisting of \(\mu +_{c} \nu\), +\(\mu \times _{c}\nu\), \(\mu^\nu\), and \(\mu -_{c} \nu\).</p> + +<p>A functional formal number has at most two arguments. But an argument +of a functional formal number may itself be a functional formal +number, and will accordingly possess either one or two arguments, +which in their turn may be functional formal numbers, and so on. The +whole set of arguments and of arguments of arguments, thus obtained, +is called the set of <i>components</i> of the original formal +number. Thus \(\mu\), \(\nu\), \(\rho\), and \(\mu +_{c} \nu\) are +components of (\(\mu +_{c} \nu) +_{c} \rho\); and \(\mu\), \(\nu\) and +\(\text{ sm }ʻʻ\mu\) are components of \(\nu +_{c} \text{ sm }ʻʻ\mu\); and +\(\mu\), \(\alpha\) and \(\text{Nc}ʻ\alpha\) are components of +\(\mu +_{c} \text{Nc}ʻ\alpha\). The two arguments of (\(\mu +_{c} \nu) +_{c}\rho\) +are \(\mu +_{c} \nu\) and \(\rho\), and those of +\(\nu +_{c}\text{ sm }ʻʻ\mu\) are \(\nu\) and \(\text{ sm }ʻʻ\mu\), +and those of \(\mu+_{c} \text{Nc}ʻ\alpha\) are \(\mu\) and +\(\text{Nc}ʻ\alpha\).</p> + +<p><span class="pagenum" id="Page_xviii">[Pg xviii]</span></p> + +<p>Addition, multiplication, exponentiation, and subtraction will be +called the arithmetical operations; and in \(\mu +_{c} \nu\), +\(\mu \times _{c} \nu\), \(\mu ^\nu\), \(\mu -_{c} \nu\), \(\mu\) and \(\nu\) +will each be said to be subjected to these respective operations. +The <i>arithmetical components</i> of an arithmetical formal number +(<i>i.e.</i> one belonging to the arithmetical set) consist of those of +its components which do not appear in the capacity of components of a +component which does not belong to the arithmetical set. Thus \(\mu\), +\(\nu\), \(\rho\), \(\mu +_{c} \nu\) are arithmetical components of +\((\mu +_{c} \nu) +_{c} \rho\); and \(\nu\) and \(\text{ sm }ʻʻ\mu\) +are arithmetical components of \(\nu +_{c} \text{ sm }ʻʻ\mu\), but +\(\mu\) is not one; and \(\mu\) and \(\text{Nc}ʻ\alpha\) are +arithmetical components of \(\mu +_{c} \text{Nc}ʻ\alpha\), but +\(\alpha\) is not one; and \(\mu\) and \(\text{ sm }ʻʻ(\nu +_{c} +\rho)\) are arithmetical components of \(\mu +_{c} \text{ sm }ʻʻ(\nu ++_{c} \rho)\), but \(\nu +_{c} \rho\) and \(\nu\) and \(\rho\) are +components of \(\text{ sm }ʻʻ(\nu +_{c} \rho)\) and are therefore +not <i>arithmetical</i> components of \(\mu +_{c} \text{ sm }ʻʻ(\nu ++_{c} \rho)\). Only arithmetical formal numbers possess arithmetical +components.</p> + +<p>A formal number of the arithmetical set having no components +which are formal numbers of the argumental set is called a +<i>pure</i> arithmetical formal number. For example +\(\mu +_{c}(\nu +_{c} \rho)\) and \(\mu +_{c} \text{Nc}ʻ\alpha\) +are pure, but \(\mu +_{c} \text{ sm }ʻʻ(\nu +_{c} \rho)\) and +\(\mu +_{c}\text{ sm }ʻʻ\text{Nc}ʻ\alpha\) are not pure.</p> + +<p>There are many types involved in the consideration of a formal +number. For example, in \(\text{Nc}ʻ\alpha\) there is the type of +\(\text{Nc}ʻ\alpha\) and of \(\alpha\); in \(\mu +_{c} \nu\) there +is the type of \(\mu +_{c} \nu\), the type of \(\mu\), and the type +of \(\nu\); and so on for more complex formal numbers. The type of a +formal number as a whole in any occurrence is called its <i>actual +type</i>. This is the type of the entity which it then represents.</p> + +<p>The other types involved in a formal number in any occurrence are +called its subordinate types.</p> + +<p>The actual types are not indicated in the symbolism for the various +formal numbers as stated above. They can be indicated relatively +to the type of the variable \(\xi\) by writing \(\text{Nc}(\xi)ʻ\alpha\), +\(\text{ sm }_{\xi }ʻʻ\mu\), (\(\mu \times _{c} \nu)_{\xi}\), +(\(\mu^\nu)_{\xi}\), (\(\mu -_{c} \nu)_{\xi}\), by the notation of *65. +Even when the actual type of a complex formal number, such as +\(\mu +_{c}(\nu +_{c} \varpi)\), is settled—so for instance that we have +\(\{\mu +_{c} (\nu +_{c} \varpi)\}_{\xi}\)—the meaning of the symbol +is not completely determined, for the type of \(\nu +_{c} \varpi\) +remains ambiguous. It follows, however, from</p> +<p class="nindc"> +<a href="#*100·511">*100·511</a>.<a href="#*110·23">*110·23</a>.<a href="#*113·26">*113·26</a>.<a href="#*119·61"></a><a href="#*119·62">·62</a>, +</p><p class="nind"> +that the subordinate types make no difference to the value of a formal +number, so long as the components are not null.</p> + +<p>We can therefore make a formal number definite as soon as its actual +type is definite by securing that its components are not null. This is +done by the convention II T (below) combined with the definitions +</p><p class="nindc"> +<a href="#*110·03">*110·03</a><a href="#*110·04">·04</a>.<a href="#*113·04">*113·04</a><a href="#*113·05">·05</a>.<a href="#*116·03">*116·03</a><a href="#*116·04">·04</a>. +</p><p class="nind"> +When the subordinate types are adjusted in accordance with these +definitions and conventions, they will be said to be <i>normally +adjusted</i>.</p> + +<p><span class="pagenum" id="Page_xix">[Pg xix]</span></p> + +<p>But in order to state this convention \(\text{IIT}\) we require a definition +of what is here called the <i>adequacy</i> of the actual type of a +formal number. The general idea of adequacy is simple enough, namely +that, given the subordinate types of \(\sigma\), the actual type of +\(\sigma\) should be high enough to enable us logically to prove +\(\exists!\sigma\) when such a proof is possible for types which are +not too low. For example, all types except the lowest for which it has +meaning are adequate for the constant formal number 2. It is rather +difficult however to state the meaning of adequacy with precision in a +manner adapted to all formal numbers. Fortunately the definition of the +lowest type which corresponds to this general idea of adequacy is not +important for our purposes. It will be sufficient to define as adequate +some types which certainly do have the property in question.</p> + +<p>The method of definition which we adopt is to replace the formal +number \(\sigma\) by another one \(\sigma'\) so related to \(\sigma\) +that with the same actual type for both we can prove +\(\exists!\sigma'.\supset.\exists!\sigma\), whenever \(\sigma\) +is not equal to \(\Lambda\) in all types. If \(\sigma\) be functional, +we need only consider its argument, or its two arguments, and can +dismiss from consideration the other components; then we replace these +arguments by others so that the \(\sigma'\) has the required property. +Thus:</p> + +<p>(i) The actual types of \(\text{Nc}ʻ\alpha\), \(\Sigma\text{Nc}ʻ\kappa\), +\(\Pi \text{Nc}ʻ\kappa\), and \(\text{ sm }ʻʻ\mu\) +are adequate when we can logically prove +\[ +\exists !Ncʻt_{0}ʻ\alpha,\,\, \exists !\Sigma Ncʻt_{0}ʻ\kappa,\,\, \exists !\Pi Ncʻt_{0}ʻ\kappa,\,\, \text{and}\,\, \exists !\text{ sm }ʻʻt_{0}ʻ\mu; +\]</p> + +<p>(ii) The actual types of \(\mu +_{c} \nu\), \(\mu -_{c} \nu\), +\(\mu\times _{c} \nu\), and \(\mu^\nu\) are adequate when we can +logically prove +\[ +\begin{aligned} +&\exists!\text{N}_{0}cʻt_{1}ʻ\mu +_{c} \text{N}_{0}cʻt_{1}ʻ\nu,\quad \exists !\text{N}_{0}cʻt_{1}ʻ\mu -_{c} + 0\cap t_{0}ʻ\nu,\\ +&\exists!\text{N}_{0}cʻt_{1}ʻ\mu \times _{c} \text{N}_{0}cʻt_{1}ʻ\nu,\quad + \text{and}\quad \exists!\text{N}_{0}cʻt_{1}ʻ\mu^{{\text{N}_{0}cʻt_{1}ʻ\nu}}. +\end{aligned} +\]</p> + +<p>It will be noticed that \(t_{0}ʻ\alpha\), \(t_{0}ʻ\kappa\), +and \(t_0ʻ\mu\) are the greatest classes of the same type as +\(\alpha\), \(\kappa\), and \(\mu\) respectively, and that +\(\text{N}_{0}cʻt_{1}ʻ\mu\) and \(\text{N}_0cʻt_1ʻ\nu\) are the +greatest cardinal numbers of the same type as \(\mu\) and \(\nu\) +respectively. These definitions hold even when any of \(\alpha\), +\(\kappa\), \(\mu\), \(\nu\) are complex symbols.</p> + +<p>The remaining formal numbers which are not functional must certainly +be constant. The difficulty which arises here is that if \(\sigma\) +be such a formal number and \(\aleph_{0}\) occurs in its symbolism, +we have no logical method of deciding as to the truth or falsehood of +\(\exists !\aleph_{0}\) in any type. But we replace \(\aleph_{0}\) +by \(\text{N}_{0}cʻt_{1}ʻ\aleph_{0}\) which is the greatest existent +cardinal of the same type as \(\aleph_{0}\) in that occurrence. Thus:</p> + +<p>(iii) If \(\sigma\) be a formal number which is not functional, an +adequate actual type of \(\sigma\) is one for which we can logically +prove \(\exists !\sigma'\), where \(\sigma'\) is derived from +\(\sigma\) by replacing any occurrence of \(\aleph_{0}\) in \(\sigma\) +by \(\text{N}_{0}cʻt_{1}ʻ\aleph_{0}\). Accordingly if \(\aleph_{0}\) +does not occur in \(\sigma\), an adequate type is any actual type for +which we can logically prove \(\exists !\sigma\).</p> + +<p>In the case of members of the primary and argumental groups we have +substituted the \(\text{V}\) of the appropriate type in the place of +each variable. When the actual type is adequate we have +\[ +(\alpha ).\exists !\text{Nc}ʻ\alpha,\quad (\kappa ).\exists !\Sigma \text{Nc}ʻ\kappa,\quad (\kappa ).\exists !\Pi \text{Nc}ʻ\kappa,\quad (\mu).\exists !\text{ sm }ʻʻ\mu. +\]</p> + +<p>In the case of members of the arithmetical group (except in the +case of \(\mu -_c \nu)\), we have substituted for each argument the +largest cardinal number which can be obtained in the type of that +argument, namely the \(\text{N}_{0}cʻ\text{V}\) for the \(\text{V}\) of +the appropriate type. Accordingly we are sure (except in the case of +\(\mu -_c \nu)\) that for all other values of the arguments which are +existent cardinal numbers the formal number is not null.</p> + +<p>It will be noticed that normal adjustment only concerns the subordinate +types. For example <a href="#*110·03">*110·03</a> secures that in \(\text{Nc}ʻ\alpha +_{c}\mu\) +the actual type of<span class="pagenum" id="Page_xx">[Pg xx]</span> \(\text{Nc}ʻ\alpha\) is adequate, and <a href="#*110·23">*110·23</a> +shows that any adequate actual type of \(\text{Nc}ʻ\alpha\) will do. +But nothing is said about the actual type of \(\text{Nc}ʻ\alpha +_{c}\mu\). +We make the following definition: When the subordinate types of +a formal number are normally adjusted, and the actual type is adequate, +the types of the formal number are said to be <i>arithmetically</i> +adjusted.</p> + +<p>We notice that for the primary set, the arithmetical adjustment of +types means the same thing as the adequate adjustment of the actual +type. Also if the arguments of a formal number of the arithmetical set +are simple symbols, the two ideas come to the same thing.</p> + +<p>In the case of variable formal numbers of the primary set, it follows +from <a href="#*117·22">*117·22</a><a href="#*117·32">·32</a> that when their types are arithmetically adjusted they +are not equal to \(\Lambda\) for any values of their variables.</p> + +<p>Also in the case of those variable formal numbers which are of the +pure arithmetical set (excluding \(\mu -_{c} \nu\)) it follows +from <a href="#*100·4">*100·4</a><a href="#*100·52">·52</a><a href="#*100·42">·42</a>.<a href="#*113·23">*113·23</a>.<a href="#*116·23">*116·23</a> that, working from the ultimate +components reached by successive analysis upwards, for all values +of such ultimate components which are members of \(\text{NC} - \iotaʻ\Lambda\) +they can be reduced to the case of the formal numbers of the primary +group; and that therefore they are not equal to \(\Lambda\) when their +types are arithmetically adjusted. For example in \(\mu +_{c} \{\nu+_{c} (\rho +_{c} \sigma)\}\), +\(\mu\), \(\nu\), \(\rho\), \(\sigma\) are these ultimate components; +let them be existent cardinal numbers. Hence when the types are +arithmetically adjusted, the actual type of \(\rho +_{c} \sigma\) is +adequate and \(\rho +_{c} \sigma\) is an existent cardinal; we can +therefore substitute \(\text{N}_{0}cʻ\alpha\) for it. By the same +reasoning we can substitute \(\text{N}_{0}cʻ\beta\) for +\(\nu +_c\text{N}_0cʻ\alpha\), and again \(\text{N}_{0}\text{c}ʻ\gamma\) +for \(\mu +_{c} \text{N}_{0}cʻ\beta\).</p> + +<p>A definite standard arithmetical adjustment of types for any formal +number can always be found by making every use of \(\text{ sm }\), +whether explicit or concealed in \(\text{Nc}\) or in some other symbol, +to be homogeneous. Proofs which apply to any arithmetical adjustment +of types start by dealing with this standard type, and then by the use +of <a href="#*104·21">*104·21</a>.<a href="#*106·21">*106·21</a><a href="#*106·211">·211</a><a href="#*106·212">·212</a><a href="#*106·213">·213</a> the extension is made to the adjacent +higher classical and relational types. We then "see" that by the +analogy of symbolism this extension can always be formally proved at +each stage, so that we are dealing with the stable truth-value. For +some constant formal numbers a lower existential type can be found than +that indicated by this method.</p> + + +<p class="nindc space-above2"> +III. <i>Classification of Occurrences of Formal Numbers.</i></p> + +<p>A symbolic form of any of the kinds [cf. <a href="#*117·01">*117·01</a><a href="#*117·04">·04</a><a href="#*117·05">·05</a><a href="#*117·06">·06</a>] +\[ +\mu > \nu,\quad \mu < \nu,\quad \mu \geq \nu,\quad \mu \leq \nu, +\] +is called an <i>arithmetical inequality</i>.</p> + +<p><span class="pagenum" id="Page_xxi">[Pg xxi]</span></p> + +<p>These forms only arise when we are comparing cardinal numbers in +respect to the relation of being "greater than" or "less than." It +might seem natural to include equations among these arithmetical +inequalities. Their use however, even as between cardinal numbers, is +not so exclusively arithmetical, and it is convenient to consider them +separately under another heading during our preliminary investigations.</p> + +<p>In the arithmetical inequalities as above written, \(\mu\) and \(\nu\), +or any symbols replacing \(\mu\) and \(\nu\), are called the <i>opposed +sides</i> of the inequality, and either of \(\mu\) or \(\nu\) is called +a <i>side</i> of the inequality.</p> + +<p>Symbolic forms of the kinds \(\sigma =\kappa\) and \(\sigma \neq +\kappa\), where either \(\sigma\) or \(\kappa\) is a formal number, +will be called <i>equations</i> and <i>inequations</i> respectively; and +\(\sigma\) and \(\kappa\) are called the <i>opposed sides</i> of the +equation or inequation, and either of them is simply a <i>side</i> of +the equation or inequation.</p> + +<p>When we reach the exclusively arithmetical point of view, it will be +convenient to put together equations, inequations and arithmetical +inequalities as one sort of symbolic form. Their separation here is +for the sake of investigations into the exceptions due to the failure +of existence theorems in low types. It is unnecessary to consider +arithmetical inequalities in this connection.</p> + +<p>The ways in which a symbol \(\sigma\) can occur in a symbolic form are +named as follows:</p> + +<p>The occurrence of \(\sigma\) in \(\text{ sm }ʻʻ\sigma\) is called an +<i>argumental</i> occurrence,</p> + +<p>The occurrence of \(\sigma\) as an argument of an arithmetical formal +number (which may be a component of another formal number) or as one +side of an arithmetical inequality is called an <i>arithmetical</i> +occurrence,</p> + +<p>The occurrence of \(\sigma\) as one side of an equation is called an +<i>equational</i> occurrence,</p> + +<p>The occurrence of \(\sigma\) in "\(\xi \in \sigma\)" is called an +<i>attributive</i> occurrence,</p> + +<p>Any other occurrence of \(\sigma\) is called a <i>logical</i> +occurrence, so also is \(\sigma =\Lambda\).</p> + +<p>It is obvious that a pair of opposed sides of an equation or inequation +must be of the same type. Furthermore, if \(\sigma\) be a formal +number, and *20·18 is applied so as to give +\[ +\vdash \colon\ldotp \sigma =\kappa .\supset :f(\sigma).\equiv .f(\kappa), +\] +<span class="pagenum" id="Page_xxii">[Pg xxii]</span>the equational occurrence of \(\sigma\) must be of the same type as its occurrence in +\(f(\sigma\)), otherwise the inference is fallacious. Accordingly substitution in +arithmetical formulae can only be undertaken when the conventions as to the +relations of ambiguous types secure this identity. This question is considered +later in this prefatory statement, and the result appears in the text as +<a href="#*118·01">*118·01</a>.</p> + +<p>At this point some examples will be useful; they will also be referred +to subsequently in connection with the conventions limiting ambiguities +of type.</p> + +<p class="nind"> +<b>*100·35.</b> \[\begin{aligned}\vdash \colon\ldotp \exists !\text{Nc}ʻ\alpha .\lor.&\exists !\text{Nc}ʻ\beta :\supset:\\ +&\text{Nc}ʻ\alpha =\text{Nc}ʻ\beta .\equiv .\alpha \in \text{Nc}ʻ\beta .\equiv .\beta \in \text{Nc}ʻ\alpha .\equiv .\alpha\,\text{ sm }\,\beta\end{aligned}\]</p> + +<p>Here the formal numbers are \(\text{Nc}ʻ\alpha\) and +\(\text{Nc}ʻ\beta\), each of which has three occurrences. The +first occurrence of \(\text{Nc}ʻ\alpha\) is logical, its second is +equational, and its third is attributive.</p> + +<p class="nind"> +<b>*100·42</b> (in the demonstration).</p> + +<p>\(\vdash :\mu ,\nu \in \text{NC}.\exists !\mu \cap \nu .\supset .(\exists \alpha,\beta).\mu =\text{Nc}ʻ\alpha .\nu =\text{Nc}ʻ\beta .\text{Nc}ʻ\alpha =\text{Nc}ʻ\beta \)</p> + +<p>Here \(\text{Nc}ʻ\alpha\) and \(\text{Nc}ʻ\beta\) are the only formal +numbers, and all their occurrences are equational.</p> + +<p class="nind"> +<b>*100·44</b> (in the demonstration).<br> +\[ +\vdash :\mu \in \text{NC}.\exists !\text{Nc}ʻ\alpha .\alpha \in \mu .\supset .(\exists \beta ).\mu =\text{Nc}ʻ\beta .\text{Nc}ʻ\alpha =\text{Nc}ʻ\beta +\]</p> + +<p>Here \(\text{Nc}ʻ\alpha\) and \(\text{Nc}ʻ\beta\) are the only formal +numbers; the first occurrence of \(\text{Nc}ʻ\alpha\) is logical, its +second is equational; both the occurrences of \(\text{Nc}ʻ\beta\) are +equational.</p> + +<p class="nind"> +<b>*100·511.</b> \(\vdash :\exists !\text{Nc}ʻ\beta .\supset +.\text{ sm }ʻʻ\text{Nc}ʻ\beta =\text{Nc}ʻ\beta\)</p> + +<p>Here the formal numbers are \(\text{Nc}ʻ\beta\) and +\(\text{ sm }ʻʻ\text{Nc}ʻ\beta\). The first occurrence of +\(\text{Nc}ʻ\beta\) is logical, the second is argumental, the third is +equational; the only occurrence of \(\text{ sm }ʻʻ\text{Nc}ʻ\beta\) is +equational.</p> + +<p class="nind"> +<b>*100·521.</b> \(\vdash :\mu \in \text{NC}.\exists !\,\text{ sm }ʻʻ\mu .\supset .\text{ sm }ʻʻ\text{ sm }ʻʻ\mu =\mu\)</p> + +<p>Here \(\text{ sm }ʻʻ\mu\) and \(\text{ sm }ʻʻ\text{ sm }ʻʻ\mu\) are the only +formal numbers; \(\text{ sm }ʻʻ\mu\) has two occurrences, the first +logical, the second argumental; \(\text{ sm }ʻʻ\text{ sm }ʻʻ\mu\) has one +occurrence, which is equational.</p> + +<p class="nind"><b>*101·28</b> (in the demonstration).<br> +\[ +\vdash :\gamma\in \text{ sm }ʻʻ1.\equiv .(\exists \alpha ).\alpha \in 1.\gamma\, \text{ sm }\,\alpha +\]</p> + +<p>Here the formal numbers are 1 and \(\text{ sm }ʻʻ1\). The first occurrence +of 1 is argumental, the second is attributive; the occurrence of +\(\text{ sm }ʻʻ1\) is attributive.</p> + +<p class="nind"> +<b>*101·38.</b> \(\vdash :\exists !2.\supset .sʻ\text{Cl}ʻʻ2=0\cup 1\cup 2\)</p> + +<p>Here the formal numbers are 0, 1, and 2, and their occurrences are all +logical.</p> + +<p class="nind"> +<b>*110·54.</b> \(\vdash .(\text{Nc}ʻ\alpha +_{c} \text{Nc}ʻ\beta) +_{0} \text{Nc}ʻ\gamma = \text{Nc}ʻ(\alpha +\beta +\gamma)\)</p> + +<p>Here the formal numbers are +\[ +\text{Nc}ʻ\alpha,\,\, \text{Nc}ʻ\beta,\,\, \text{Nc}ʻ\gamma,\,\, \text{Nc}ʻ(\alpha +\beta +\gamma),\,\, \text{Nc}ʻ\alpha +_{c} \text{Nc}ʻ\beta,\,\, (\text{Nc}ʻ\alpha +_{c} + \text{Nc}ʻ\beta) +_{c} \text{Nc}ʻ\gamma. +\] +The occurrence of \(\text{Nc}ʻ(\alpha +\beta +\gamma\)) and that of +(\(\text{Nc}ʻ\alpha +_{c} \text{Nc}ʻ\beta) +_{c} \text{Nc}ʻ\gamma\) +are both equational, and they must be of the same type since they are +opposed sides of the same equation. The occurrences of the other formal +numbers<span class="pagenum" id="Page_xxiii">[Pg xxiii]</span> are as arithmetical components of a more complex arithmetical +formal number and are therefore arithmetical.</p> + +<p class="nind"> +<b>*116·63.</b> \(\vdash .\mu^{\nu \times _{c} \varpi} = (\mu^{\nu})^{\varpi}\)</p> + +<p>The formal numbers are \(\nu \times _{c} \varpi\), \(\mu^{\nu }\), +\(\mu ^{\nu \times _{c} \varpi}\), and (\(\mu^{\nu})^{\varpi}\). Each +formal number occurs once only. The occurrences of \(\nu \times _{c} +\varpi\) and \(\mu^{\nu}\) are arithmetical, and those of the other two +are equational.</p> + +<p class="nind"> +<b>*117·108.</b> \(\vdash \colon\ldotp \text{Nc}ʻ\alpha \geq \text{Nc}ʻ\beta . \equiv : \text{Nc}ʻ\alpha > \text{Nc}ʻ\beta . \lor . \text{Nc}ʻ\alpha = \text{Nc}ʻ\beta\)</p> + +<p>The formal numbers are \(\text{Nc}ʻ\alpha\) and \(\text{Nc}ʻ\beta\), +each with three occurrences. The first two occurrences of each formal +number are arithmetical, the last occurrence of each is equational.</p> + +<p class="nind"><b>*120·53</b> (in the demonstration).<br> +\[ +\vdash : \beta = \gamma +_{c} \delta . \exists ! \beta . \supset . \alpha ^\beta = \alpha^\gamma \times _{c} \alpha^\delta +\]</p> + +<p>Here the formal numbers are \(\gamma +_{c} \delta\), \(\alpha ^\beta\), +\(\alpha ^\gamma\), \(\alpha ^\delta\), \(\alpha ^\gamma \times _{c}\alpha ^\delta\). +Each formal number has one occurrence. Those of \(\gamma +_{c}\delta\), +\(\alpha ^\beta\) and \(\alpha ^\gamma \times _{c} \alpha^\delta\) +are equational, and those of \(\alpha ^\gamma\) and \(\alpha^\delta\) +are arithmetical.</p> + +<p class="nind"><b>*120·53</b> (in the demonstration).<br> + +\[ +\vdash : \alpha^\beta = \alpha^\gamma . \beta = \gamma +_{c} \delta . \exists ! \alpha^\beta . \supset . \alpha ^\gamma = \alpha ^\gamma + \times _{c} \alpha ^\delta +\]</p> + +<p>Here the formal numbers are \(\alpha ^\beta\), \(\alpha ^\gamma\), +\(\alpha ^\delta\), \(\alpha ^\gamma \times _{c} \alpha ^\delta\), +\(\gamma +_{c} \delta\). The first occurrence of \(\alpha ^\beta\) +is equational, its second occurrence is logical; the first two +occurrences of \(\alpha ^\gamma\) are equational, its third occurrence +is arithmetical; the only occurrence of \(\alpha ^\delta\) is +arithmetical; the only occurrences of \(\alpha ^\gamma \times _{c}\alpha ^\delta\) +and of \(\gamma +_{c} \delta\) are equational.</p> + + +<p class="nindc space-above2">IV. <i>The Conventions</i> \(\text{IT}\) <i>and</i> \(\text{IIT}\).</p> + +<p>Two occurrences of a formal number with the same actual type are said +to be <i>bound</i> to each other.</p> + +<p>The choice of types for formal numbers, when they are not made definite +in terms of variables by the notation of *65, is limited by the +following conventions, which enable us to dispense largely with the +elaboration produced by the definition of types.</p> + +<p>\(\text{IT}\). <i>All logical occurrences of the same formal number +are in the same type; argumental occurrences are bound to logical and +attributive occurrences; and, if there are no argumental occurrences, +equational occurrences are bound to logical occurrences.</i></p> + +<p>This rule only applies, so far as meaning permits, to those types which +remain ambiguous after the assignment of types to the real variables.</p> + +<p>It will be noticed that if there are no argumental or logical +occurrences of a formal number, \(\text{IT}\) does not in any way apply +to the assignment of types to the occurrences in the form of that +formal number.</p> + +<p><span class="pagenum" id="Page_xxiv">[Pg xxiv]</span></p> + +<p>The identification of types in argumental and attributive occurrences +by \(\text{IT}\) is rendered necessary to secure the use of the +equivalence +\[ +\gamma \in \text{ sm }ʻʻ\sigma .\equiv .(\exists \alpha ).\alpha \in \sigma .\gamma\,\text{ sm }\,\alpha, +\] +where \(\sigma\) is a formal number. Without the convention, this +application of *37·1 would be fallacious. The only one of our +examples to which this part of the convention applies is <a href="#*101·28">*101·28</a> +(demonstration), where it secures that the two occurrences of 1 are +in the same type. It is relevant however to the symbolism in the +demonstration of <a href="#*100·521">*100·521</a>.</p> + +<p>It will be found in practice that this convention relates the types of +occurrences in the same way as would naturally be done by anyone who was +not thinking of the convention at all. To see how the convention works, +we will run through the examples which have already been given above.</p> + +<p>In <a href="#*100·35">*100·35</a>, \(\text{IT}\) directs the logical and equational +occurrences of \(\text{Nc}ʻ\alpha\) to be in the same type, and +similarly for \(\text{Nc}ʻ\beta\). Also "meaning" secures that the +equational types of \(\text{Nc}ʻ\alpha\) and \(\text{Nc}ʻ\beta\) are +the same. Thus these four occurrences are all in one type, which has +no necessary relation to the types of the attributive occurrences of +\(\text{Nc}ʻ\alpha\) and \(\text{Nc}ʻ\beta\). Thus, using the notation +of *65·04 to secure typical definiteness, *100·35 is to mean +\[\begin{aligned} +\vdash \colon\ldotp \exists !&\text{Nc}(\xi )ʻ\alpha .\lor.\exists !\text{Nc}(\xi)ʻ\beta :\supset :\\ +&\text{Nc}(\xi)ʻ\alpha =\text{Nc}(\xi)ʻ\beta .\equiv .\alpha \in \text{Nc}(\alpha)ʻ\beta .\equiv .\beta \in \text{Nc}(\beta)ʻ\alpha .\equiv .\alpha\,\text{ sm }\,\beta.\end{aligned}\]</p> + +<p>The types of these attributive occurrences are settled by the necessity +of "meaning."</p> + +<p>In <a href="#*100·42">*100·42</a> (demonstration), since all the occurrences of formal numbers +are equational, \(\text{IT}\) produces no limitation of types.</p> + +<p>In <a href="#*100·44">*100·44</a> (demonstration), \(\text{IT}\) secures that the two +occurrences of \(\text{Nc}ʻ\alpha\) are in the same type. Also we +notice that the first occurrence of \(\text{Nc}ʻ\beta\) is really (cf. +*65·04) \(\text{Nc}(\alpha)ʻ\beta\), since "\(\alpha \in \mu\)" occurs, +and thus "meaning" requires this relation of types, and the second +occurrence of \(\text{Nc}ʻ\beta\) is in the type of the occurrences of +\(\text{Nc}ʻ\alpha\).</p> + +<p>In <a href="#*100·511">*100·511</a>, \(\text{IT}\) directs that the logical and argumental +occurrences are to have the same type. In <a href="#*100·521">*100·521</a>, \(\text{IT}\) +directs that the two occurrences of \(\text{ sm }ʻʻ\mu\) are to have the +same type. In <a href="#*101·28">*101·28</a> both occurrences of 1 are to be in the same type. +In <a href="#*101·38">*101·38</a>, \(\text{IT}\) directs that all the occurrences of 2 are to +have the same type.</p> + +<p>The convention \(\text{IT}\) in no way limits the types in <a href="#*110·54">*110·54</a>, nor +in <a href="#*116·63">*116·63</a>, nor in <a href="#*117·108">*117·108</a>.</p> + +<p>In the first example from <a href="#*120·53">*120·53</a> (in the demonstration) convention +\(\text{IT}\) has no application.</p> + +<p>In the second example from *120·53 (in the demonstration) convention +\(\text{IT}\) directs that the two occurrences of \(\alpha ^\beta\) +shall be in the same type; and the<span class="pagenum" id="Page_xxv">[Pg xxv]</span> necessity of "meaning" secures that +the first occurrence of \(\alpha ^\gamma\) shall also be in this type. +The same necessity secures that \(\gamma +_{c} \delta\) shall be in the +same type as \(\beta\); and it also secures that in +"\(\alpha ^\gamma = \alpha ^\gamma \times _{c} \alpha ^\delta\)" +the first occurrence of \(\alpha ^\gamma\) and that of +\(\alpha ^\gamma\times_c \alpha^\delta\) shall have a common type, +which is otherwise unfettered; also nothing has been decided as to the +types of \(\alpha ^\gamma\) and \(\alpha ^\delta\) in +\(\alpha ^\gamma\times _{c} \alpha ^\delta\).</p> + +<p>We now come to conventions embodying the outcome of arithmetical ideas. +The term "arithmetical" is here used to denote investigations in which +the interest lies in the comparison of formal numbers in respect to +equality or inequality, excluding the exceptional cases—whenever the +cases are exceptional—due to the failure of existence in low types. +The thorough-going arithmetical point of view, which we adopt later +in the investigation on Ratio and Quantity and also in this volume +in <a href="#*117">*117</a> and <a href="#*126">*126</a> and some earlier propositions, would sweep aside +as uninteresting all investigation of the exact ways in which the +failure of existence theorems is relevant to the truth of propositions, +thus concentrating attention exclusively on stable truth-values. +But the logical investigation has its own intrinsic interest among +the principles of the subject. It is obvious however that it should +be restrained to a consideration of the theorems of purely logical +interest. In practice this extrusion of uninteresting cases of the +failure of arithmetical theorems, even amid the logical investigations +of the first part of this volume, is effected by securing that <i>all +arithmetical occurrences of formal numbers have their actual types +adequate</i>.</p> + +<p>As far as formal numbers of the primary group, <i>i.e.</i> +\(\text{Nc}ʻ\alpha\), \(\Sigma \text{Nc}ʻ\kappa\), \(\Pi\text{Nc}ʻ\kappa\), +are concerned, the arithmetical adjustment of types +is secured formally in the symbolism by the definitions <a href="#*110·03">*110·03</a><a href="#*110·04">·04</a> +for addition, and <a href="#*113·04">*113·04</a><a href="#*113·05">·05</a> for multiplication, and <a href="#*116·03">*116·03</a><a href="#*116·04">·04</a> for +exponentiation, and <a href="#*117·02">*117·02</a><a href="#*117·03">·03</a> for arithmetical inequalities, and +<a href="#*119·02">*119·02</a><a href="#*119·03">·03</a> for subtraction.</p> + +<p>We save the symbolic elaboration which would arise from the extension +of similar definitions to other formal numbers by the following +convention:</p> + +<p>\(\text{IIT}\). <i>Whenever a formal number \(\sigma\) occurs, so that, +if it were replaced by</i> \(\text{Nc}ʻ\alpha\), <i>the actual type +of</i> \(\text{Nc}ʻ\alpha\) <i>would by definition have to be adequate, +then the actual type of \(\sigma\) is also to be adequate.</i></p> + +<p>For example in \(\mu +_{c} (\nu +_{c} \varpi)\), if \(\nu +_{c}\varpi\) +were replaced by \(\text{Nc}ʻ\alpha\), then by <a href="#*110·04">*110·04</a> the actual +type of \(\text{Nc}ʻ\alpha\) is adequate. Hence by \(\text{IIT}\) the +actual type of \(\nu +_{c} \varpi\) is to be adequate: accordingly so +long as \(\nu\) and \(\varpi\) are simple variables and members of +\(\text{NC}-\iotaʻ\Lambda\), we can always assume \(\exists !(\nu +_{c}\varpi)\) +for the type of the occurrence of \(\nu +_{c} \varpi \in \mu +_{c} (\nu +_{c} \varpi)\).</p> + +<p>It is essential to notice that so long as the argument of an argumental +formal number, or the arguments of an arithmetical formal number, are +adjusted arithmetically, the exact types chosen make no difference. +This follows for argumental formal numbers from <a href="#*102·862">*102·862</a><a href="#*102·87">·87</a><a href="#*102·88">·88</a>, +for addition from<span class="pagenum" id="Page_xxvi">[Pg xxvi]</span> <a href="#*110·25">*110·25</a>, for multiplication from <a href="#*113·26">*113·26</a>, for +exponentiation from <a href="#*116·26">*116·26</a>, for subtraction from <a href="#*119·61">*119·61</a><a href="#*119·62">·62</a>. Thus +(remembering also <a href="#*100·511">*100·511</a>) in any definite type a formal number has +one definite meaning provided that any subordinate formal number which +occurs in its symbolism is determined existentially. The convention +\(\text{IIT}\) directs us always to take this definite meaning for any +pure arithmetical formal number.</p> + +<p>The convention does not determine completely the meaning of an +arithmetical formal number which is not pure. For example, \(\mu +_{c}(\nu +_{c} \rho )\) +is a pure arithmetical formal number when \(\mu\), +\(\nu\), \(\rho\) are determined in type; and convention \(\text{IIT}\) +directs that the type of \((\nu +_{c} \rho)\) is to be adequate. But +\(\mu +_{c} \text{ sm }ʻʻ(\nu +_{c} \rho)\) is an arithmetical formal +number which is not pure, and convention \(\text{IIT}\) directs that +the type of the domain of \(\text{ sm }\) is to be adequate, but does +not affect the type of \(\nu +_{c} \rho\). Thus it is easy to see +that \(\text{IIT}\) secures the adequacy of the actual types of all +arithmetical components of any <i>arithmetical</i> formal numbers which +occur, but does not affect the actual type of a formal number which +occurs as the argument of an argumental formal number. But in this +case convention \(\text{IT}\) will bind the actual type of this occurrence +of the argument to any logical or attributive occurrence of the same +formal number. For example, if \(\exists ! \nu +_{c} \rho\) and +\(\mu +_c\text{ sm }ʻʻ(\nu +_c \rho)\) occur in the same form, then +these two occurrences of \(\nu +_{c} \rho\) must have the same actual +type. In practice argumental formal numbers are useful as components +of arithmetical formal numbers for the very purpose of avoiding the +automatic adjustment of types directed by \(\text{IIT}\).</p> + +<p>The meaning of \(\text{IIT}\) is best explained by examples. Among our +previous examples we need only consider those in which arithmetical +formal numbers occur.</p> + +<p>In <a href="#*110·54">*110·54</a> the convention or definitions direct us to determine the +types of \(\text{Nc}ʻ\alpha\) and \(\text{Nc}ʻ\beta\) adequately when +forming \(\text{Nc}ʻ\alpha +_{c} \text{Nc}ʻ\beta\), also to determine +\(\text{Nc}ʻ\alpha +_{c} \text{Nc}ʻ\beta\) and \(\text{Nc}ʻ\gamma\) +adequately when forming \((\text{Nc}ʻ\alpha +_{c} \text{Nc}ʻ\beta)+_{c} \text{Nc}ʻ\gamma\). +The convention does not apply to the types of +\((\text{Nc}ʻ\alpha +_{c} \text{Nc}ʻ\beta) +_{c} \text{Nc}ʻ\gamma\) and +\(\text{Nc}ʻ(\alpha + \beta + \gamma)\). These types must be identical +in order to secure meaning.</p> + +<p>In <a href="#*116·63">*116·63</a> the convention directs us to adjust the types of +\(\nu\times _c \varpi\) and \(\mu ^\nu\) adequately; it does not +affect the types of \(\mu ^{\nu \times _{c} \varpi}\) and +\((\mu^\nu)^\varpi\), which must be identical to secure meaning. If we +replace \(\mu\), \(\nu\), \(\varpi\) by formal numbers, by 2, +\(\aleph_{0}\), and 1 for example, we get "\(\vdash . 2^{\aleph_{0}\times \text{c1}} = (2^{\aleph _{0}})^{1}\)." +The convention now directs that 1 is to be determined adequately. It +so happens that any type is adequate for it, since \(\exists ! 1\) can +be proved in any type. Then adequate types for \(\aleph _{0} \times_{c} 1\) and \(2^{\aleph _{0}}\) +are types for which we can prove \(\exists! (\text{N}_{0} cʻt_{1}ʻ\aleph _{0}) \times _{c} 1\) +and \(\exists! 2^{\text{N}_{0} cʻt_{1}ʻ\aleph _{0}}\). Thus if \(\tau\) is the +type of \(\aleph _{0}\) in both cases, an adequate type for +\(\aleph_{0} \times _{c} 1\) is \(\tau\), and for \(2^{\aleph _{0}}\) is +\(\text{Cl}ʻ\tau\).</p> + +<p>In <a href="#*117·108">*117·108</a> we find arithmetical occurrences in arithmetical +inequalities. Thus \(\text{IIT}\) directs us to take the first +two occurrences of \(\text{Nc}ʻ\alpha\) and the first<span class="pagenum" id="Page_xxvii">[Pg xxvii]</span> two of +\(\text{Nc}ʻ\beta\) with adequate actual types. The type of +\(\text{Nc}ʻ\alpha\) and \(\text{Nc}ʻ\beta\) in \(\text{Nc}ʻ\alpha= \text{Nc}ʻ\beta\) +is not affected by it. It is evident that the +conventions \(\text{IT}\), \(\text{IIT}\) are not sufficient to secure +the truth of this proposition as thus symbolized. It is essential +that in the equation the type be adjusted adequately for both formal +numbers. In fact the general arithmetical convention, that types +of equational as well as of arithmetical occurrences are adjusted +arithmetically, is here used.</p> + + +<p class="nindc space-above2">V. <i>Some Important Principles.</i></p> + +<p><i>Principle of Arithmetical Substitution</i>. In <a href="#*120·53">*120·53</a>, the +application of \(\text{IIT}\) needs a consideration of the whole +question of arithmetical substitution. Consider the first of the two +examples. We have +\[ +\vdash : \beta = \gamma +_c \delta . \exists ! \beta . \supset . \alpha ^\beta = \alpha ^\gamma \times _c \alpha ^\delta. +\]</p> + +<p>It is obvious that unless we can pass with practical immediateness +from "\(\beta = \gamma +_c \delta. \alpha ^\beta = \alpha ^\beta\)" to +"\(\alpha ^\beta = \alpha^{\gamma +_c \delta}\)" by *20·18, arithmetic +is made practically impossible by the theory of types. But a difficulty +arises from the application of \(\text{IIT}\). Suppose we assign the +types of our real variables first. Then the types of \(\alpha\), +\(\beta\), \(\gamma\), \(\delta\) can be arbitrarily assigned, and +there is no necessary connection between them which arises from the +preservation of meaning. Thus \(\beta\) may be in a type which is +not an adequate type for \(\gamma +_c \delta\). Assume that this is +the case. But the equational use of \(\gamma +_c \delta\) is in the +same type as \(\beta\), and by \(\text{IIT}\) the arithmetical use +of \(\gamma +_c \delta\) in \(\alpha^{\gamma +_c \delta}\) is in an +adequate type. Thus, on the face of it, the reasoning, appealing to +*20·18, by which the substitution was justified, is fallacious; for the +two occurrences of \(\gamma +_c \delta\) in fact mean different things.</p> + +<p>In order to generalize our solution of this difficulty it is convenient +to define the term "arithmetical equation." An <i>arithmetical +equation</i> is an equation between purely arithmetical formal numbers +whose actual types are both determined adequately. Then it is evident +that from "\(\sigma = \tau.f(\tau)\)," where \(\sigma\) and \(\tau\) +are formal numbers and \(\tau\) occurs arithmetically in \(f(\tau)\), +we cannot infer \(f(\sigma)\) unless the equation \(\sigma = \tau\) +is arithmetical. For otherwise the \(\tau\) in the equation cannot be +identified with the \(\tau\) in \(f(\tau)\).</p> + +<p>When we have "\(\beta = \tau . f(\tau)\)," where \(\tau\) is a formal +number and \(\beta\) is a number in a definite type, and wish to pass +to "\(f(\beta)\)," or "\(\beta = \tau . f(\beta)\)" and wish to pass +to "\(f(\tau)\)," the occurrence of \(\tau\) in \(f(\tau)\) being +arithmetical, the type of \(\beta\) may not be an adequate type for +\(\tau\). Accordingly the \(\tau\) in "\(\beta = \tau\)" cannot be +identified with the \(\tau\) in \(f(\tau)\). The type of the \(\tau\) +in the equation ought to be freed from dependence on that of \(\beta\). +Accordingly the transition is only legitimate when we can write instead +\[ +\unicode{x201c}\beta +_c 0 = \tau . f(\tau )\unicode{x201d}\quad \text{or}\quad \unicode{x201c}\beta +_c 0 = \tau . f(\beta ),\unicode{x201d} +\]<span class="pagenum" id="Page_xxviii">[Pg xxviii]</span> +where in both cases the equation is arithmetical. For now all the +symbols are subject to the same rules.</p> + +<p>If this modification can be made without altering the truth-value of +the asserted propositions, the substitution is legitimate, otherwise it +is not.</p> + +<p>It is obvious that in the above our immediate passage is to or from +\(f(\beta +_{c}0)\). But is easy to see that, the occurrence of +\(\beta+_{c}0\) being arithmetical, we always have +\[ +f(\beta ).\equiv .f(\beta +_{c}0). +\] +In order to prove this, we have only to prove +\[ +\begin{array}{l} +\alpha +_{c}(\beta +_{c}0)&=\alpha +_{c}\beta,\\ +\alpha \times _{c}(\beta +_{c}0)&=\alpha \times _{c}\beta,\\ +(\alpha +_{c}0)^\beta &=\alpha ^\beta, +\alpha ^{\beta +_{c}0}&=\alpha ^\beta, +\end{array} +\] +\[ +\text{and}\qquad \alpha >\beta +_{c}0.\equiv .\alpha >\beta .\equiv .\alpha +_{c}0>\beta. +\]</p> + +<p>The demonstration of the first of these propositions runs as follows: +\[ +\begin{array}{l} +\vdash .*110·4.\supset \vdash \colon\ldotp \beta {\sim}\in \text{NC}.\lor.\beta =\Lambda :&\supset .\beta +_{c}0=\Lambda .\alpha +_{c}\beta =\Lambda.\\ +[*110·4] &\supset .\alpha +_{c}(\beta +_{c}0)=\Lambda =\alpha +_{c}\beta &\qquad \text{(1)}\\ +\vdash .*110·4.&\supset \vdash \colon\ldotp \alpha {\sim}\in \text{NC}.\lor.\alpha =\Lambda :\supset .\alpha +_{c}(\beta +_{c}0)=\Lambda =\alpha +_{c}\beta + &\qquad \text{(2)}\\ +\vdash .*110·6.\supset \vdash :\alpha ,\beta \in \text{NC}-{℩}ʻ\Lambda .\supset .\alpha +_{c}(\beta +_{c}0)&=\alpha +_{c}\text{ sm }ʻʻ\beta \\ +&=\alpha +_{c}\beta &\qquad \text{(3)}\\ +\vdash .(1).(2).(3).\supset \vdash :\alpha +_{c}(\beta +_{c}0)=\alpha +_{c}\beta +\end{array} +\]</p> + +<p>In the above demonstration the step to (3) is legitimate since by the +hypothesis \(\beta\) is a determination of \(\text{ sm }ʻʻ\beta\) in an +adequate type.</p> + +<p>Similar proofs hold for the other propositions, using <a href="#*113·204">*113·204</a> and +<a href="#*116·204">*116·204</a> and <a href="#*117·12">*117·12</a> and <a href="#*103·13">*103·13</a>.</p> + +<p>We must also consider the circumstances under which we can pass from +"\(\beta =\tau\)" to "\(\beta +_{c}0=\tau\)," where the latter equation +is arithmetical. In other words, using *65·01 we require the hypothesis +necessary for +\[ +\exists !\tau _{\eta }.\beta =\tau _{\xi }.\supset .\beta +_{c}0=\tau _{\eta }. +\]</p> + +<p>We have +\[ +\begin{array}{l} +\vdash .*20·18. &\supset \vdash :\beta =\tau_{\xi }.\supset .\beta +_{c}0=\tau_{\xi }+_{c}0 &\qquad \text{(1)}\\ +\vdash .*110·35. &\supset \vdash :\exists !\tau_{\xi }.\exists !\tau_{\eta }.\supset .\tau_{\xi }+_{c}0=\tau_{\eta }+_{c}0 &\qquad \text{(2)}\\ +\vdash .(1).(2). &\supset \vdash \colon\ldotp \exists !\tau_{\xi }.\exists !\tau_{\eta }.\supset :\beta =\tau_{\xi }.\supset + .\beta +_{c}0=\tau_{\eta }+_{c}0 &\qquad \text{(3)}\\ +\vdash .(3). &\supset \vdash \colon\ldotp \exists !\beta .\exists !\tau_{\eta }.\supset :\beta =\tau_{\xi }.\supset .\beta +_{c}0=\tau_{\eta }+_{c}0 + &\qquad \text{(4)} +\end{array} +\]</p> + +<p>Now in (4) the occurrences of \(\beta +_{c}0\) and \(\tau_{\eta}+_{c}0\), +which are in the same type, may be chosen to be in any type we like. +Hence we deduce +\[ +\begin{array}{l} +\vdash .(4).*110·6.\supset \vdash \colon\ldotp \exists !\beta .\exists !\tau _{\eta}.\supset :\beta =\tau _{\xi}.&\supset + .(\beta +_{c}0)_{\zeta }=\text{ sm }_{\zeta}ʻʻ\tau _{\eta}.\\ +[*100·511] &\supset .(\beta +_{c}0)_{\zeta}=\tau_{\zeta} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_xxix">[Pg xxix]</span></p> + +<p>Hence \(\exists !\beta\) is the requisite condition. Now since +\(\zeta\) can be in any type, we can also choose it in any existential +type for \(\tau\). Thus with \(\text{IIT}\) applying to the +arithmetical occurrence of \(\tau\) in \(f(\tau)\), we have, where +\(\tau\) is a formal number and \(\beta\) is a number in a definite +type, +\[ +\begin{aligned} +\vdash :\exists !\beta .\beta &=\tau .f(\tau).\supset .f(\beta),\\ +\vdash :\exists !\beta .\beta &=\tau .f(\beta).\supset .f(\tau),\\ +\vdash :\exists !\sigma .\sigma &=\tau .f(\tau).\supset .f(\sigma). +\end{aligned} +\]</p> + +<p>In the last proposition by \(\text{IT}\) the equation \(\sigma =\tau\) +is arithmetical. These equations are summed up in <a href="#*118·01">*118·01</a>.</p> + +<p>These three fundamental theorems embody the principle of arithmetical +substitution. The hypothesis \(\exists !\beta\) is really less than is +assumed in ordinary life, the usual tacit assumption being +\(\beta \in\text{NC}-\iota ʻ\Lambda\). In fact unless +\(\beta \in \text{NC}\), \(\beta =\tau\) is necessarily false.</p> + +<p><i>Principle of Identification of Types</i>. Suppose we have proved +"\(\vdash :\text{Hp}.\supset .\phi \sigma\)" and "\(\vdash :\phi(\sigma _{\xi}).\supset .p\)," +where \(\sigma\) is a formal number +whose occurrence in "\(\vdash :\text{Hp}.\supset .\phi \sigma\)" is +in an entirely ambiguous type, and \(\phi _{\xi}\) is the same formal +number \(\sigma\) with its type related to that of \(\xi\) by *65·01. +Then since the type of the \(\sigma\) in "\(\vdash :\text{Hp}.\supset.\phi \sigma\)" +is ambiguous, we can write "\(\vdash :\text{Hp}.\supset.\phi (\sigma _{\xi})\)" +and thence infer "\(\vdash .p\)."</p> + +<p>The principle is: An entirely undetermined type in an asserted symbolic +form can be identified with any type ambiguous or otherwise in any +other asserted symbolic form or in the same symbolic form.</p> + +<p>For example in <a href="#*100·42">*100·42</a> (demonstration) considered above, since +\(\exists !\mu \cap \nu\) occurs, the first occurrences of +\(\text{Nc}ʻ\alpha\) and \(\text{Nc}ʻ\beta\) are of the same +type, and so are their second occurrences in \(\text{Nc}ʻ\alpha=\text{Nc}ʻ\beta\). +But the two types are not determined by our +conventions to have any necessary connection. In fact the type +in \(\text{Nc}ʻ\alpha =\text{Nc}ʻ\beta\) is entirely arbitrary. +Accordingly it can be identified with the other type, and thus the +inference to the next line, <i>viz.</i> to "\(\vdash :\text{Hp}.\supset.\mu =\nu\)," +is justified.</p> + +<p>In the case of arithmetical equations, it is important to notice that +we have +\[ +\vdash .*100·321·33.\supset \vdash \colon\ldotp \exists !\text{Nc}(\xi)ʻ\alpha .\supset :\text{Nc}(\xi)ʻ\alpha =\text{Nc}(\xi)ʻ\beta .\supset .\text{Nc}ʻ\alpha =\text{Nc}ʻ\beta. +\] +Hence if \(\sigma\) and \(\tau\) are formal numbers, +\[ +\vdash \colon\ldotp \exists !\sigma_{\xi}.\supset :\sigma _{\xi}=\tau _{\xi}.\supset .\sigma =\tau. +\] +Thus if we have "\(\vdash :\text{Hp}.\exists !\sigma .\supset .\sigma=\tau\)" +and "\(\vdash :\text{Hp}ʻ.\sigma _{\eta}=\tau _{\eta}.\supset.p\)," +we can infer from the former proposition +"\(\vdash:\text{Hp}.\exists !\sigma .\supset .\sigma _{\eta}=\tau_{\eta}\)" +and from this and the latter proposition, we infer +"\(\vdash:\text{Hp}'.\text{Hp}.\exists !\sigma .\supset .p\)," +so the general principle of identification can be employed when the +\(\phi (\sigma)\) in the first proposition is an arithmetical equation.</p> + +<p>For example, in an example given above, <a href="#*100·44">*100·44</a> (demonstration), +<i>viz.</i> +\[ +\vdash :\mu \in \text{NC}.\exists !\text{Nc}ʻ\alpha .\alpha \in \mu .\supset .(\exists \beta).\mu =\text{Nc}ʻ\beta .\text{Nc}ʻ\alpha =\text{Nc}ʻ\beta, +\]<span class="pagenum" id="Page_xxx">[Pg xxx]</span> +the equation \(\text{Nc}ʻ\alpha =\text{Nc}ʻ\beta\) is arithmetical. +Accordingly we are justified in asserting the propositional function +\[ +\vdash :\mu \in \text{NC}.\exists !\text{Nc}ʻ\alpha .\alpha \in \mu .\supset .(\exists \beta).\mu =\text{Nc}(\alpha )ʻ\beta .\text{Nc}(\alpha)ʻ\alpha =\text{Nc}(\alpha)ʻ\beta, +\] +where \(\text{Nc}(\alpha )ʻ\beta\) in +"\(\mu =\text{Nc}(\alpha)ʻ\beta\)" has all along been +presupposed by the necessity of meaning. +Thus the inference follows, +\[\begin{aligned} +\vdash :\mu \in \text{NC}.\exists !\text{Nc}ʻ\alpha .\alpha \in \mu .&\supset .\text{Nc}(\alpha )ʻ\alpha =\mu .\\ +&\supset .\text{Nc}ʻ\alpha =\mu . +\end{aligned}\]</p> + +<p>This proof loses its point when \(\mu\) is looked on as a variable +with necessarily the same type throughout. For then the proposition +collapses into +\[ +\vdash \colon\ldotp \mu \in \text{NC}.\supset :\alpha \in \mu .\equiv .\text{Nc}(\alpha )ʻ\alpha =\mu . +\]</p> + +<p>But if \(\mu\) be a formal number necessarily a member of +\(\text{NC}\), the proposition is really +\[ +\vdash \colon\ldotp \exists !\text{Nc}ʻ\alpha .\supset :\alpha \in \mu .\equiv .\text{Nc}ʻ\alpha =\mu . +\]</p> + +<p>With this presupposition we should have in the first line of the +demonstration +\[ +\unicode{x201c}\vdash :\exists !\text{Nc}ʻ\alpha .\text{Nc}ʻ\alpha =\mu .\supset .\alpha \in \mu,\unicode{x201d} +\] +though with "\(\mu\)" a single variable, the line is formally correct +as it stands in the text.</p> + +<p><i>Recognition of Particular Cases</i>. It is important to notice +the conditions under which \(\phi \sigma\) can be recognized as a +particular case of \(\phi \xi\), where \(\xi\) is a real variable and +\(\sigma\) is a formal number. In the first place obviously we must +substitute \(\sigma \cap t_{0}ʻ\xi\) for \(\sigma\), wherever it occurs +in \(\phi \sigma\), and thus obtain \(\phi (\sigma \cap t_{0}ʻ\xi)\). +Then we may find that by the application of our conventions, we can +replace this by \(\phi \sigma\). For example we have</p> + +<p class="nind"> +<b>*100·42.</b> \(\vdash :\mu ,\nu \in \text{NC}.\exists !\mu \cap \nu .\supset .\mu =\nu \)</p> + +<p>Now put \(\text{Nc}ʻ\alpha \cap t_{0}ʻ\mu\) for \(\mu\), we obtain +\[ +\begin{array}{l} +\vdash :\text{Nc}ʻ\alpha \cap t_{0}ʻ\mu ,\nu \in \text{NC}.\exists !(\text{Nc}ʻ\alpha \cap t_{0}ʻ\mu )\cap \nu .\supset .\text{Nc}ʻ\alpha \cap t_{0}\mu + =\nu &\qquad \text{(1)}\\ +\vdash .(1).*100·41.\supset \vdash :\nu \in \text{NC}.\exists !\text{Nc}ʻ\alpha \cap t_{0}ʻ\mu \cap \nu .\supset .\text{Nc}ʻ\alpha \cap t_{0}ʻ\mu =\nu &\qquad \text{(2)} +\end{array} +\]</p> + +<p>Now by \(\text{IT}\), even when \(\nu\) is a formal number, the +identity of types of the two occurrences of \(\text{Nc}ʻ\alpha\) is +equally secured in +\[ +\vdash :\nu \in \text{NC}.\exists !\text{Nc}ʻ\alpha \cap \nu .\supset .\text{Nc}ʻ\alpha =\nu. +\]</p> + +<p>Thus this is a particular case of <a href="#*100·42">*100·42</a>. Such deductions can be made +in general without any explicit formal statement.</p> + +<p><span class="pagenum" id="Page_xxxi">[Pg xxxi]</span></p> + +<p><i>Ambiguity of</i> \(\text{NC}\). It follows (cf. <a href="#*100·02">*100·02</a> and <a href="#*103·02">*103·02</a>) +from the typical ambiguity of \(\text{Nc}\) that \(\text{NC}\) +is also typically ambiguous. Hence "\(\mu \in \text{NC}.\nu \in +\text{NC}\)" according to our methods of interpretation would not +necessitate that \(\mu\) and \(\nu\) should be of the same type. +We shall always interpret "\(\mu ,\nu \in \text{NC}\)" as standing +for "\(\mu \in \text{NC}.\nu \in \text{NC}\)" and therefore as not +necessarily identifying the types of \(\mu\) and \(\nu\). Similarly for +\(\text{N}_{0}\text{C}\), \(\text{NC induct}\), and \(\text{NC ind}\). +For example</p> + +<p class="nind"> +<b>*110·402.</b> \(\vdash :\mu,\nu \in \text{N}_{0}\text{C}.\supset .\exists !(\mu +_{c} \nu )\cap tʻtʻ(\mu \uparrow \nu)\)</p> + +<p>Here the \(\mu\) and \(\nu\) need not be of the same type. Again</p> + +<p class="nind"> +<b>*110·41.</b> \(\vdash :\mu,\nu \in \text{N}_{0}\text{C}.tʻ\mu = tʻ\nu .\supset .\exists !(\mu +_{c} \nu)\cap tʻ\mu\)</p> + +<p>Here the identification of the types of \(\mu\) and \(\nu\) requires +the hypothesis "\(tʻ\mu = tʻ\nu\)."</p> + + +<p class="nindc space-above2"> +VI. <i>Conventions</i> \(\text{AT}\) <i>and</i> \(\text{Infin T}\).</p> + +<p><i>General Arithmetical Convention</i>. Conventions \(\text{IT}\) and +\(\text{IIT}\) are always applied, but the following convention is not +used at first. This convention limits the remaining ambiguity of type +by sweeping away the exceptional cases in low types, due to the failure +of existence theorems. The convention will be cited as \(\text{AT}\).</p> + +<p>\(\text{AT}\). <i>All equations involving pure arithmetical formal +numbers are to be arithmetical.</i></p> + +<p>We have seen that from an arithmetical equation the analogous equation +in any other type can be deduced. Thus with \(\text{AT}\) all +equations between formal numbers are so determined in type that their +truth in "any type" is deducible. Thus in the few early propositions +where \(\text{AT}\) is introduced, the fact is noted by stating that +the equations hold "in <i>any</i> type." These propositions are <a href="#*103·16">*103·16</a>, +<a href="#*110·71">*110·71</a><a href="#*110·72">·72</a>.</p> + +<p>The effect of applying \(\text{AT}\) to other propositions in <a href="#*100">*100</a> +is to render some of the hypotheses (usually logical forms affirming +existence) unnecessary, but also materially to limit the scope of the +propositions. Take for example</p> + +<p class="nind"> +<b>*100·35.</b> \[\begin{aligned}\vdash :\exists !\text{Nc}ʻ\alpha .\lor.&\exists !\text{Nc}ʻ\beta :\supset:\\ +&\text{Nc}ʻ\alpha = \text{Nc}ʻ\beta .\equiv .\alpha \in \text{Nc}ʻ\beta .\equiv .\beta \in \text{Nc}ʻ\alpha .\equiv .\alpha\,\text{ sm }\,\beta\end{aligned}\]</p> + +<p>If we apply \(\text{AT}\) to this, we can write +\[ +\vdash :\text{Nc}ʻ\alpha = \text{Nc}ʻ\beta .\equiv .\alpha \in \text{Nc}ʻ\beta .\equiv .\beta \in \text{Nc}ʻ\alpha .\equiv .\alpha\,\text{ sm }\,\beta. +\]</p> + +<p>For the equational occurrences of \(\text{Nc}ʻ\alpha\) and +\(\text{Nc}ʻ\beta\) are by \(\text{AT}\) and \(\text{IIT}\) to be with +adequate actual types. But if \(\alpha\) is a small class in a high +type, an adequate actual type for \(\text{Nc}ʻ\alpha\) will be a high +type, whereas \(\exists !\text{Nc}ʻ\alpha\) may hold in a low type. +Thus with \(\text{AT}\), for the sake of simplicity we abandon the +statement of the minimum of hypothesis necessary for our propositions. +The enunciation of no other proposition in <a href="#*100">*100</a> is affected.</p> + +<p>The enunciation of no proposition in <a href="#*101">*101</a> is affected by \(\text{AT}\), +though it would unduly limit the scope of <a href="#*101·34">*101·34</a>. In <a href="#*110">*110</a>, +\(\text{AT}\) would unduly limit the scope of such propositions as +</p><p class="nindc"> +<a href="#*110·22">*110·22</a><a href="#*110·23">·23</a><a href="#*110·24">·24</a><a href="#*110·25">·25</a><a href="#*110·251">·251</a><a href="#*110·252">·252</a><a href="#*110·3">·3</a><a href="#*110·31">·31</a><a href="#*110·32">·32</a><a href="#*110·331">·331</a><a href="#*110·34">·34</a><a href="#*110·35">·35</a><a href="#*110·351">·351</a><a href="#*110·44">·44</a><a href="#*110·51">·51</a><a href="#*110·54">·54</a> +</p><p class="nind"> +and of many others, without altering their enunciations. There is no +proposition in <a href="#*110">*110</a> whose enunciation it would alter. \(\text{AT}\) is +already<span class="pagenum" id="Page_xxxii">[Pg xxxii]</span> applied to <a href="#*110·71">*110·71</a><a href="#*110·72">·72</a>; if \(\text{AT}\) is removed from these +propositions, then \(\exists !\text{Nc}ʻ\alpha\) must be added as an +hypothesis to both of them. The effect of \(\text{AT}\) on <a href="#*113">*113</a> and +<a href="#*116">*116</a> is entirely analogous to that on *110; in neither of these two +numbers is there any proposition to which \(\text{AT}\) is applied in +the text.</p> + +<p>As regards <a href="#*117">*117</a>, \(\text{AT}\) is applied throughout, so that +the propositions are all in the form suitable for subsequent +investigations in which the interest is purely arithmetical. It is +important however to analyse the effect of AT on the enunciations +for the sake of logical investigations, especially in connection +with <a href="#*120">*120</a>. First, \(\text{AT}\) can only affect propositions in +which equations or inequations occur, and among such propositions +it does not affect the enunciations of those in which both sides +of the equations are not formal numbers, so that the equations are +not arithmetical after the application of AT. These propositions +are <a href="#*117·104">*117·104</a><a href="#*117·14">·14</a><a href="#*117·24">·24</a><a href="#*117·241">·241</a><a href="#*117·243">·243</a><a href="#*117·31">·31</a><a href="#*117·551">·551</a>. These propositions, which are +characterized by the presence of a single letter on one side of any +equation involved, can be recognized at a glance. The propositions +involving arithmetical equations whose enunciations are unaltered +by the removal of \(\text{AT}\) are <a href="#*117·21">*117·21</a><a href="#*117·54">·54</a><a href="#*117·592">·592</a>. Propositions +involving inequations whose enunciations are unaltered by the removal +of \(\text{AT}\) are <a href="#*117·26">*117·26</a><a href="#*117·27">·27</a>. Finally the only propositions of *117 +whose enunciations are altered by the removal of \(\text{AT}\) are +<a href="#*117·108">*117·108</a><a href="#*117·211">·211</a><a href="#*117·23">·23</a><a href="#*117·25">·25</a><a href="#*117·3"></a>.</p> + +<p>In <a href="#*118">*118</a> and <a href="#*119">*119</a> \(\text{AT}\) is not used.</p> + +<p>In <a href="#*120">*120</a>, which is devoted to those properties of inductive cardinals +which are of logical interest, \(\text{AT}\) is never used. None of +the propositions <a href="#*117·108">*117·108</a><a href="#*117·211">·211</a><a href="#*117·23">·23</a><a href="#*117·25">·25</a><a href="#*117">·3</a> are cited in it, except <a href="#*117·25">*117·25</a> +in the demonstration of <a href="#*120·435">*120·435</a> for a use where \(\text{AT}\) is not +relevant. The application of AT to *120 would simplify the hypotheses +of <a href="#*120·31">*120·31</a><a href="#*120·41">·41</a><a href="#*120·451">·451</a><a href="#*120·53">·53</a><a href="#*120·55">·55</a>, and limit the scopes of the propositions.</p> + +<p>One other convention, which we will call "\(\text{Infin T}\)," is +required in certain propositions where the hypothesis implies that +there are types in which every inductive cardinal exists, <i>i.e.</i> +in which \(\text{V}\) is not an inductive class. Among such hypotheses +are \(\text{Infin ax}\), \(\exists !\text{Prog}\), \(\exists!\aleph_{0}\) +(or typically definite forms of these hypotheses), or +\(R\in \text{Prog}\) or \(\alpha \in \aleph_{0}\). When such hypotheses +occur, we shall assume that \(\text{NC}\) induct is, whenever +significance permits, to be determined in a type in which every +inductive cardinal exists, <i>i.e.</i> in which the axiom of infinity +holds (cf. <a href="#*120·03">*120·03</a><a href="#*120·04">·04</a>). The statement of this convention is as follows:</p> + +<p>\(\text{Infin T}\). <i>When the hypothesis of a proposition implies +that there is a type in which every inductive cardinal exists, every +occurrence of</i> "\(\text{NC induct}\)" <i>in this proposition is to +be taken (if conditions of significance permit) in a sufficiently high +type to insure the existence of every inductive cardinal.</i></p> + +<p>It is to be observed that this convention would be unnecessary if we +confined ourselves to one extensional hierarchy, for in any one such +hierarchy<span class="pagenum" id="Page_xxxiii">[Pg xxxiii]</span> all types are inductive or all are non-inductive, so that +if every inductive cardinal exists in one type in the hierarchy, the +same holds for any other type in the hierarchy. But when we no longer +confine ourselves to one extensional hierarchy, this result may not +follow. For example, it may be the case that the number of individuals +is inductive, but the number of predicative functions of individuals is +not inductive; at any rate, no <i>logical</i> reason can be given against this +possibility, which can only be rejected on empirical grounds, if at all.</p> + +<p>The way in which this convention is used may be illustrated by the +demonstration of <a href="#*122·33">*122·33</a>. In the second line of this demonstration, we +show that the hypothesis implies +\[ +\begin{array}{l} +\qquad\qquad\qquad\qquad\qquad \text{E}!\nu_{R} . \supset . \text{E}!(\nu +_{c}1)_{R} &\qquad \text{(1)}\\ +\text{where by} *121·04\qquad \nu_{R} = \breve{R}_{\nu -_{c}1}ʻBʻR &\qquad\text{Df},\\ +\text{and by} *121·02\qquad R_{\nu} = \hat{x} \hat{y}\{\text{N}_{0}\text{c}ʻR(x \vdash\dashv y) = \nu +_{c}1\} &\qquad\text{Df}. +\end{array} +\] +It will be seen that these definitions do not suffice to determine +the type of \(\nu\). Hence in (1), the \(\nu\) on the left may not +be of the same type as the \(\nu +_{c}1\) on the right. Now the use +of *122·473, which occurs in the next line of the demonstration of +<a href="#*122·33">*122·33</a>, requires that the \(\nu\) on the left and the \(\nu +_{c}1\) +on the right should be of the same type. This requires that the +\(\nu\) should not be taken in a type in which we have +\(\exists !\nu. \nu +_c 1 = \Lambda\). Hence in order to apply +<a href="#*120·473">*120·473</a>, we must choose a type in which all inductive cardinals +exist. Since "\(R \in \text{Prog}\)" occurs in the hypothesis, we know +that all inductive cardinals exist in the type of \(CʻR\). But it is +unnecessary to restrict ourselves to the type of \(CʻR\), since any +other type in which all inductive cardinals exist will equally secure +the validity of the demonstration. Thus the convention \(\text{Infin T}\) +secures the restriction required, and no more.</p> + +<p>The convention \(\text{Infin T}\) is often relevant when "\(\text{Infin ax}\)" +without any typical determination occurs in the hypothesis. +Whenever this is the case, if "\(\text{NC induct}\)" occurs in the +proposition in a way which leaves its type undetermined so far as +conditions of significance are concerned, it is to be taken in a type +in which all its members exist.</p> + + +<p class="nindc space-above2">VII. <i>Final Working Rule in Arithmetic.</i></p> + +<p>It is now (whenever \(\text{AT}\) is used, together with \(\text{Infin T}\) +when necessary) possible finally to sweep aside all consideration +of types in connection with inductive numbers. For by combining +<a href="#*126·121">*126·121</a> *126·122 and <a href="#*120·4232">*120·4232</a><a href="#*120·4622">·4622</a>, we see that it is always possible to +take the type high enough so that no definitely determined inductive +number shall be null (\(\Lambda)\), and that all the inductive +reasoning can take place within this type. Furthermore we have already +seen that the arithmetical operations are independent of the types +of the components, so long as they are existential. Thus, as far as +the ordinary<span class="pagenum" id="Page_xxxiv">[Pg xxxiv]</span> arithmetic of finite numbers is concerned, all the +conventions (including AT), and the necessity for hypotheses as to the +existence of inductive numbers, are finally superseded by the following +single rule:</p> + +<p><span class="allsmcap">Rule of Indefinite Numbers</span>. <i>The type assigned to any symbol +which represents an inductive number is such that the symbol is not +equal to</i> \(\Lambda\).</p> + +<p>We make the definition</p> + +<p class="nind"> +<b>*126·01.</b> \(\text{Nc ind} = \text{Nc induct} - \iota ʻ\Lambda \quad \text{Df}\)</p> + +<p>Wherever this symbol "\(\text{Nc ind}\)" for the class of "indefinite +inductive cardinal numbers" is used, the above rule is adhered to. +In other words, "\(\mu \in \text{Nc ind}\)" can always be replaced +by "\(\mu = \text{Nc}ʻ\alpha .\alpha \in \text{Cls induct}\)," where +\(\text{Nc}ʻ\alpha\) is a homogeneous or ascending cardinal, and +\(\alpha\) is the appropriate constant, or is a variable, as the case +may be. In the latter case, a symbolic form such as +\[ +(\mu).f(\mu \in \text{Nc ind},\mu) +\] +can be replaced by +\[ +(\mu,\alpha).f(\mu = \text{Nc}ʻ\alpha .\alpha \in \text{Cls induct},\mu). +\]</p> + +<p>Furthermore by <a href="#*120·4622">*120·4622</a> it follows that with this rule the result of +proceeding by induction in one type and then transforming to another +type is the same as that of proceeding by induction in the latter type. +Thus for example there is no advantage to be gained by discriminating +between \(2_{\xi }\) and \(2_{\eta }\); for +\(\text{ sm }_{\eta }ʻʻ2_{\xi} = 2_{\eta }\), \(\text{ sm }_{\xi }ʻʻ2_{\eta } = 2_{\xi }\), +\(\mu +_{c}2_{\xi } = \mu +_{c} 2_{\eta }\), +\(\mu \times _{c} 2_{\xi } = \mu \times _{c} 2_{\eta }\), +\(\mu ^{2_\xi } = \mu ^{2_\eta }\), \(2_{\xi^\mu } = 2_{\eta ^\mu }\), +and \(\mu \geq 2_{\xi }.\equiv .\mu \geq 2_{\eta }\), and so on.</p> + +<p><span class="pagenum" id="Page_1">[Pg 1]</span></p> + +<p>Hence all discrimination of the types of indefinite inductive numbers +may be dropped; and the types are entirely indefinite and irrelevant.</p> + + +<div class="footnotes"><h3>FOOTNOTES:</h3> + +<div class="footnote"> + +<p class="nind"> +<a id="Footnote_1" href="#FNanchor_1" class="label">[1]</a> +But cf. next page for a more exact statement of this +principle.</p> +</div> +</div> + + +<hr class="chap x-ebookmaker-drop"> +<p><span class="pagenum" id="Page_3">[Pg 3]</span></p> +<div class="chapter"> +<h2 class="nobreak" id="PART_III">PART III.<br> +<br> +CARDINAL ARITHMETIC.</h2> +</div> + + +<p class="nindc">SUMMARY OF PART III.</p> + +<p>IN this Part, we shall be concerned, +first, with the definition and general logical properties of +cardinal numbers (Section A); then with the operations of addition, +multiplication and exponentiation, of which the definitions and +formal laws do not require any restriction to finite numbers (Section +B); then with the theory of finite and infinite, which is rendered +somewhat complicated by the fact that there are two different senses +of "finite," which cannot (so far as is known) be identified without +assuming the multiplicative axiom. The theory of finite and infinite +will be resumed, in connection with series, in Part V, Section E.</p> + +<p>It is in this Part that the theory of types first becomes practically +relevant. It will be found that contradictions concerning the maximum +cardinal are solved by this theory. We have therefore devoted our +first section in this Part (with the exception of two numbers giving +the most elementary properties of cardinals in general, and of 0 and 1 +and 2, respectively) to the application of types to cardinals. Every +cardinal is typically ambiguous, and we confer typical definiteness +by the notations of *63, *64, and *65. It is especially where +existence-theorems are concerned that the theory of types is essential. +The chief importance of the propositions of the present part lies, not +only, as throughout the book, in the hypotheses necessary to secure the +conclusions, but also in the typical ambiguity which can be allowed to +the symbols consistently with the truth of the propositions in all the +cases thereby included.</p> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_4">[Pg 4]</span></p> +<h2 class="nobreak" id="SECTION_A">SECTION A.<br> +<span class="allsmcap">DEFINITION AND LOGICAL PROPERTIES OF CARDINAL NUMBERS.</span></h2> +</div> + + +<p class="nind"><i>Summary of Section A.</i></p> + +<p>The Cardinal Number of a class \(\alpha\), which we will denote by +"\(\text{Nc}ʻ\alpha\)," is defined as the class of all classes similar +to \(\alpha\), <i>i.e.</i> as \(\hat{\beta} (\beta\,\text{ sm }\,\alpha)\). +This definition is due to Frege, and was first published in his +<i>Grundlagen der Arithmetik</i><a id="FNanchor_2" href="#Footnote_2" class="fnanchor">[2]</a>; its symbolic expression and use +are to be found in his <i>Grundgesetze der Arithmetik</i><a id="FNanchor_3" href="#Footnote_3" class="fnanchor">[3]</a>. The chief +merits of this definition are (1) that the formal properties which +we expect cardinal numbers to have result from it; (2) that unless +we adopt this definition or some more complicated and practically +equivalent definition, it is necessary to regard the cardinal number of +a class as an indefinable. Hence the above definition avoids a useless +indefinable with its attendant primitive propositions.</p> + +<p>It will be observed that, if \(x\) is any object, 1 is not the cardinal +number of \(x\), but that of \(\iota ʻx\). This obviates a confusion +which otherwise is liable to arise in dealing with classes. Suppose we +have a class \(\alpha\) consisting of many terms; we say, nevertheless, +that it is <i>one</i> class. Thus it seems to be at once one and many. +But in fact it is \(\alpha\) that is many, and \(\iotaʻ\alpha\) that is +one. In regard to zero, the analogous point is still clearer. Suppose +we say "there are no Kings of France." This is equivalent to "the class +of Kings of France has no members," or, in our language, "the class +of Kings of France is a member of the class 0." It is obvious that we +cannot say "the King of France is a member of the class 0," because +there is no King of France. Thus in the case of 0 and 1, as more +evidently in all other cases, a cardinal number appertains to a class, +not to the members of the class.</p> + +<p>For the purposes of formal definition, we subject the formula +\[ +\text{Nc}ʻ\alpha = \hat{\beta} (\beta\,\text{ sm }\,\alpha) +\] +to some simplification. It will be seen that, according to this +formula, "\(\text{Nc}\)" is a relation, namely the relation of a +cardinal number to any class of which it is the number. Thus for +example 1 has to \(\iota ʻx\) the relation \(\text{Nc}\); so has<span class="pagenum" id="Page_5">[Pg 5]</span> 2 +to \(\iota ʻx \cup \iota ʻy\), provided \(x\neq y\). The relation +\(\text{Nc}\) is, in fact, the relation \(\overrightarrow{\text{ sm }}\); +for \(\overrightarrow{\text{ sm }}ʻ\alpha = \hat{\beta} (\beta\,\text{ sm }\, +\alpha)\). Hence for formal purposes of definition we put +\[ +\text{Nc} = \overrightarrow{\text{ sm }} \quad\text{Df}. +\]</p> + +<p>The class of cardinal numbers is the class of objects which are the +cardinal numbers of something or other, <i>i.e.</i> of objects which, +for some \(\alpha\), are equal to \(\text{Nc}ʻ\alpha\). We call the +class of cardinal numbers \(\text{NC}\); thus we have +\[ +\text{NC} = \hat{\mu}\{(\exists \alpha) . \mu = \text{Nc}ʻ\alpha\}. +\]</p> + +<p>For purposes of formal definition, we replace this by the simpler +formula +\[ +\text{NC} = \text{D}ʻ\text{Nc} \quad\text{Df}. +\]</p> + +<p>In the present section, we shall be concerned with what we may call the +purely logical properties of cardinal numbers, namely those which do +not depend upon the arithmetical operations of addition, multiplication +and exponentiation, nor upon the distinction of finite and infinite<a id="FNanchor_4" href="#Footnote_4" class="fnanchor">[4]</a>. +The chief point to be dealt with, as regards both importance and +difficulty, is the relation of a cardinal number in one type to the +same or an associated cardinal number in another type. When a symbol is +ambiguous as to type, we will call it <i>typically ambiguous</i>; when, +either always or in a given context, it is unambiguous as to type, we +will call it <i>typically definite</i>. Now the symbol "\(\text{ sm }\)" +is typically ambiguous; the only limitation on its type is that its +domain and converse domain must both consist of classes. When we have +\(\alpha\,\text{sm}\,\beta\), \(\alpha\) and \(\beta\) need not be of the +same type, in fact, in any type of classes, there are classes similar +to some of the classes of any other type of classes. For example, we +have \(\iota ʻx \text{ sm } \iota ʻy\), whatever types \(x\) and \(y\) +may belong to. This ambiguity of "\(\text{ sm }\)" is derived from that +of \(1 \rightarrow 1\), which in turn is derived from that of 1. We +denote (cf. *65·01) by "\(1_{\alpha}\)" all the unit classes which +are of the same type as \(\alpha\). Then (according to the definition +*70·01) \(1_{\alpha} \rightarrow 1_{\beta}\) will be the class of +those one-one relations whose domain is of the same type as \(\alpha\) +and whose converse domain is of the same type as \(\beta\). Thus +"\(1_{\alpha} \rightarrow 1_{\beta}\)" is typically definite as soon +as \(\alpha\) and \(\beta\) are given. Suppose now, instead of having +merely \(\gamma \text{ sm } \delta\), we have +\[ +(\exists R). R\in 1_{\alpha } \rightarrow 1_{\beta } . \text{D}ʻR = \gamma . \text{ᗡ}ʻR = \delta ; +\] +then we know not only that \(\gamma\,\text{ sm }\,\delta\), but also that +\(\gamma\) belongs to the same type as \(\alpha\), and \(\delta\) +belongs to the same type as \(\beta\). When the ambiguous symbol +"\(\text{ sm }\)" is rendered typically definite by having its domain +defined as being of the same type as \(\alpha\), and its converse +domain defined as being of the same type as \(\beta\), we write it +"\(\text{ sm }_{(\alpha ,\beta)}\)," because generally, in accordance +with *65·1, if \(R\) is a typically ambiguous relation, we write +<span class="pagenum" id="Page_6">[Pg 6]</span>\(R_{(\alpha ,\beta)}\) for the typically definite relation that +results when the domain of \(R\) is to consist of terms of the same +type as \(\alpha\), and the converse domain is to consist of terms of +the same type as \(\beta\). Thus we have +\[ +\gamma\,\text{ sm }_{(\alpha, \beta)}\,\delta . \equiv . (\exists R) . R \in 1_{\alpha} \rightarrow 1_{\beta} . \gamma = \text{D}ʻR . \delta = \text{ᗡ}ʻR. +\] +Here everything is typically definite if \(\alpha\) and \(\beta\) (or +their types) are given.</p> + +<p>Passing now to the relation "\(\text{Nc}\)," it will be seen that +it shares the typical ambiguity of "\(\text{sm}\)." In order to +render it typically definite, we must derive it from a typically +definite "\(\text{sm}\)." So long as nothing is added to give typical +definiteness, "\(\text{Nc}ʻ\gamma\)" will mean all the classes +belonging to some one (unspecified) type and similar to \(\gamma\). If +\(\alpha\) is a member of the type to which these classes are to belong, +then \(\text{Nc}ʻ\gamma\) is contained in the type of \(\alpha\). For +this case, it is convenient to introduce the following two notations, +already defined in *65. When a typically ambiguous relation \(R\) is to +be rendered typically definite as to its domain only, by deciding that +every member of the domain is to be <i>contained in</i> the type of +\(\alpha\), we write "\(R(\alpha)\)" in place of \(R\). When we further +wish to determine \(R\) as having members of the converse domain +<i>contained in</i> the type of \(\beta\), we write "\(R(\alpha, \beta)\)" in +place of \(R\); and when we wish members of the converse domain to be +<i>members</i> of the type of \(\beta\), we write "\(R(\alpha_{\beta})\)" +in place of \(R\). Thus +\[ +\text{sg}ʻ{R_{(\alpha, \beta)}} = \{\text{sg}ʻR\}(\alpha_{\beta}) +\] +(cf. *65·2), and in particular, since \(\text{Nc} =\overrightarrow{\text{ sm }}\), +\[ +\text{Nc}(\alpha_{\beta}) = \text{sg}ʻ\text{ sm }_{(\alpha, \beta)}. +\] +Thus "\(\text{Nc}(\alpha_{\beta})ʻ\gamma\)" is only significant when +\(\gamma\) is of the same type as \(\beta\), and then it means "classes +of the same type as \(\alpha\) and similar to \(\gamma\) (which is of +the same type as \(\beta\))."</p> + +<p>"\(\text{Nc}(\alpha)ʻ\gamma\)" will mean "classes of the same type as +\(\alpha\) and similar to \(\gamma\)." As soon as the types of \(\alpha\) +and \(\gamma\) are known, this is a typically definite symbol, being +in fact equal to \(\text{Nc}(\alpha_{\gamma})ʻ\gamma\). Hence so +long as we only wish to consider "\(\text{Nc}ʻ\gamma\)," typical +definiteness is secured by writing "\(\text{Nc}(\alpha)\)" in place of +"\(\text{Nc}\)."</p> + +<p>When we come to the consideration of \(\text{NC}\), +"\(\text{Nc}(\alpha)\)" is no longer a sufficient determination, +although it suffices to determine the type. Suppose we put +\[ +\text{NC}^{\beta}(\alpha) = \text{D}ʻ\text{Nc}(\alpha_{\beta}) \quad\text{Df}; +\] +we have also, in virtue of the definitions in *65, +\[ +\text{NC}(\alpha) = \text{NC} \cap t^{2}ʻ\alpha = \text{D}ʻ\text{Nc}(\alpha). +\] +Thus \(\text{NC}(\alpha)\) is definite as to type, but is the domain of +a relation whose converse domain is ambiguous as to type; and it will +appear that there are some propositions about \(\text{NC}(\alpha)\) +whose truth or falsehood depends upon the determination chosen +for the converse domain of \(\text{Nc}(\alpha)\). Hence if we +wish to have a symbol which is completely definite, we must write +<span class="pagenum" id="Page_7">[Pg 7]</span>"\(\text{NC}^{\beta}(\alpha)\)."</p> + +<p>This point is important in connection with the contradictions as to the +maximum cardinal. The following remarks will illustrate it further.</p> + +<p>Cantor has shown that, if \(\beta\) is any class, no class contained +in \(\beta\) is similar to \(\text{Cl}ʻ\beta\). Hence in particular +if \(\beta\) is a type, no class contained in \(\beta\) is similar +to \(\text{Cl}ʻ\beta\), which is the next type above \(\beta\). +Consequently, if \(\beta = \alpha \cup - \alpha\), where \(\alpha\) is +any class, we have +\[ +{\sim}(\exists \gamma) . \gamma \subset \alpha \cup - \alpha . \gamma \text{ sm } \text{Cl}ʻ(\alpha \cup - \alpha). +\]</p> + +<p>Now (cf. *63) we put +\[ +t_{0}ʻ\alpha = \alpha \cup - \alpha \quad\text{Df}, +\] +and we have \(tʻ\alpha = \text{Cl}ʻ(\alpha \cup - \alpha)\). Thus we +find +\[ +\begin{aligned} +{\sim}&(\exists \gamma) . \gamma \subset t_{0}ʻ\alpha . \gamma\, \text{ sm }\, tʻ\alpha.\\ +\text{Hence}\qquad &\text{Nc}(\alpha_{tʻ\alpha})ʻtʻ\alpha = \Lambda . +\end{aligned} +\] +That is to say, no class of the same type as \(\alpha\) has as many +members as \(tʻ\alpha\) has. Hence also +\[ +\begin{aligned} +&\Lambda \in \text{NC}^{tʻ\alpha}(\alpha).\\ +\text{But}\qquad \gamma \subset t_{0}ʻ\alpha . \supset . \gamma \in &\text{Nc}(\alpha_{\alpha })ʻ\gamma . \supset . \exists !\text{Nc}(\alpha_{\alpha })ʻ\gamma , +\end{aligned} +\] +and "\(\text{Nc}(\alpha_{\alpha})ʻ\gamma\)" is only significant when +\(\gamma \subset t_{0}ʻ\alpha\); hence +\[ +\begin{aligned} +\mu \in &\text{NC}^{\alpha}(\alpha) . \supset_{\mu } . \exists !\mu\\ +\text{and}\qquad\qquad &\Lambda {\sim}\in \text{NC}^{\alpha}(\alpha). +\end{aligned} +\]</p> + +<p>Now the notation "\(\text{NC}(\alpha)\)" will apply with +equal justice to \(\text{NC}^{\alpha }(\alpha)\) or to +\(\text{NC}^{tʻ\alpha}(\alpha)\); but we have just seen that in the +first case we shall have \(\Lambda {\sim}\in \text{NC}(\alpha)\), +and in the second we shall have \(\Lambda \in \text{NC}(\alpha)\). +Consequently "\(\text{NC}(\alpha)\)" has not sufficient definiteness +to prevent practically important differences between the various +determinations of which it is capable.</p> + +<p>A converse procedure to the above yields similar results. Let +\(\alpha\) be a class of classes; then \(sʻ\alpha\) is of lower type +than \(\alpha\). Let us consider \(\text{NC}^{sʻ\alpha}(\alpha)\). In +accordance with *63, we write \(t_{1}ʻ\alpha\) for the type containing +\(sʻ\alpha\), <i>i.e.</i> for \(sʻ\alpha \cup - sʻ\alpha\). Then +the greatest number in the class \(\text{NC}^{sʻ\alpha}(\alpha)\) +will be \(\text{Nc}(\alpha)ʻt_{1}ʻ\alpha\); but neither this nor +any lesser member of the class will be equal to +\(\text{Nc}(\alpha)ʻt_{0}ʻ\alpha\), because, as before, +\[ +{\sim}(\exists \gamma ) . \gamma \subset t_{1}ʻ\alpha . \gamma \text{ sm } t_{0}ʻ\alpha . +\] +Hence \(\text{Nc}(\alpha )ʻt_{0}ʻ\alpha\), which is a member +of \(\text{NC}^{\alpha }(\alpha)\), is not a member of +\(\text{NC}^{sʻ\alpha }(\alpha)\); but \(\text{NC}^{\alpha}(\alpha)\) +and \(\text{NC}^{sʻ\alpha}(\alpha)\) have an equal right to be called +\(\text{NC}(\alpha)\). Hence again "\(\text{NC}(\alpha)\)" is a symbol +not sufficiently definite for many of our purposes.</p> + +<p>The solution of the paradox concerning the maximum cardinal is evident +in view of what has been said. This paradox is as follows: It results +from a theorem of Cantor's that there is no maximum cardinal, since, +for all values of \(\alpha\), +\[ +\text{Nc}ʻ\text{Cl}ʻ\alpha \gt \text{Nc}ʻ\alpha . +\] +<span class="pagenum" id="Page_8">[Pg 8]</span> +But at first sight it would seem that the class which contains +everything must be the greatest possible class, and must therefore +contain the greatest possible number of terms. We have seen, however, +that a class \(\alpha\) must always be contained within some one type; +hence all that is proved is that there are greater classes in the next +type, which is that of \(\text{Cl}ʻ\alpha\). Since there is always a +next higher type, we thus have a maximum cardinal in each type, without +having any absolutely maximum cardinal. The maximum cardinal in the +type of \(\alpha\) is +\[ +\text{Nc}(\alpha)ʻ(\alpha \cup - \alpha). +\] +But if we take the corresponding cardinal in the next type, <i>i.e.</i> +\[ +\text{Nc}(\text{Cl}ʻ\alpha )ʻ(\alpha \cup - \alpha), +\] +this is not as great as \(\text{Nc}(\text{Cl}ʻ\alpha)ʻ\text{Cl}ʻ(\alpha\cup - \alpha)\), +and is therefore not the maximum cardinal of its +type. This gives the complete solution of the paradox.</p> + +<p>For most purposes, what we wish to know in order to have a sufficient +amount of typical definiteness is not the absolute types of +\(\alpha\) and \(\beta\), as above, but merely what we may call their +<i>relative</i> types. Thus, for example, \(\alpha\) and \(\beta\) +may be of the same type; in that case, \(\text{Nc}(\alpha _{\beta})\) +and \(\text{NC}^{\beta }(\alpha)\) are respectively equal to +\(\text{Nc}(\alpha _{\alpha })\) and \(\text{NC}^{\alpha }(\alpha)\). +We will call cardinals which, for some \(\alpha\), are members +of the class \(\text{NC}^{\alpha }(\alpha)\), <i>homogeneous</i> +cardinals, because the "\(\text{sm}\)" from which they are derived +is a homogeneous relation. We shall denote the homogeneous cardinal +of \(\alpha\) by "\(\text{N}_{0}\text{C}ʻ\alpha\)," and we shall +denote the class of homogeneous cardinals (in an unspecified type) by +"\(\text{N}_{0}\text{C}\)"; thus we put +\[ +\begin{aligned} +\text{N}_{0}\text{c}ʻ\alpha &= \text{Nc}ʻ\alpha \cap tʻ\alpha &\quad\text{Df},\\ +\text{N}_{0}\text{C} &= \text{D}ʻ\text{N}_{0}\text{c} &\quad\text{Df}. +\end{aligned} +\] +Almost all the properties of \(\text{N}_{0}\text{C}\) are the same +in different types. When further typical definiteness is required, +it can be secured by writing \(\text{N}_{0}\text{c}(\alpha)\), +\(\text{N}_{0}\text{C}(\alpha)\) in place of \(\text{N}_{0}\text{c}\), +\(\text{N}_{0}\text{C}\). For although \(\text{Nc}(\alpha)\) +and \(\text{NC}(\alpha)\) were not wholly definite, +\(\text{N}_{0}\text{c}(\alpha)\) and \(\text{N}_{0}\text{C}(\alpha)\) +are wholly definite. Apart from the fact of being of different +types, the only property in which \(\text{N}_{0}\text{C}(\alpha)\) +and \(\text{N}_{0}\text{C}(\beta)\) differ when \(\alpha\) and +\(\beta\) are of different types is in regard to the magnitude +of the cardinals belonging to them. Thus suppose the whole +universe consisted (as monists aver) of a single individual. Let +us call the type of this individual "\(\text{Indiv}\)." Then +\(\text{N}_{0}\text{C}(\text{Indiv})\) will consist of 0 and 1, +<i>i.e.</i> +\[ +\text{N}_{0}\text{C}(\text{Indiv}) = \iotaʻ0 \cup \iota ʻ1. +\] +But in the next higher type, there will be two members, namely +\(\Lambda\) and \(\text{Indiv}\). Thus +\[ +\begin{aligned} +&\text{N}_{0}\text{C}(tʻ\text{Indiv}) = \iotaʻ0 \cup \iotaʻ1 \cup \iotaʻ2.\\ +\text{Similarly}\qquad \text{N}_{0}\text{C}&(tʻtʻ\text{Indiv}) = \iotaʻ0 \cup \iotaʻ1 \cup \iotaʻ2 \cup \iotaʻ3 \cup \iotaʻ4, +\end{aligned} +\]<span class="pagenum" id="Page_9">[Pg 9]</span> +the members of \(tʻtʻ\text{Indiv}\) being \(\Lambda \cap +tʻ\text{Indiv}, \iotaʻ\Lambda\), \(\iotaʻ\text{Indiv}\), +\(\iotaʻ\Lambda \cup \iotaʻ\text{Indiv}\); and so on. (The greatest +cardinal in any except the lowest type is always a power of 2.)</p> + +<p>The maximum of \(\text{N}_{0}\text{C}(\alpha)\) is +\(\text{N}_{0}\text{c}ʻt_{0}ʻ\alpha\); but apart from this difference +of maximum and its consequences, \(\text{N}_{0}\text{C}(\alpha)\) +and \(\text{N}_{0}\text{C}(\beta)\) do not differ in any important +properties. Hence for most purposes \(\text{N}_{0}\text{C}\) and +\(\text{N}_{0}\text{c}\) have as much typical definiteness as is +necessary.</p> + +<p>Among cardinals which are not homogeneous we shall consider three +kinds. The first of these we shall call <i>ascending</i> cardinals. A +cardinal \(\text{NC}^{\beta}(\alpha)\) is called an <i>ascending</i> +cardinal if the type of \(\beta\) is \(tʻ\alpha\) or \(tʻtʻ\alpha\) or +\(tʻtʻtʻ\alpha\) or etc. We write \(t^{2}ʻ\alpha\) for \(tʻtʻ\alpha\), +\(t^{3}ʻ\alpha\) for \(tʻtʻtʻ\alpha\), and so on. We put +\[ +\begin{aligned} +\text{N}^{1}\text{c}ʻ\alpha &= \text{Nc}ʻ\alpha \cap tʻtʻ\alpha &&\quad\text{Df}\\ +\text{N}^{2}\text{c}ʻ\alpha &= \text{Nc}ʻ\alpha \cap tʻt^{2}ʻ\alpha &&\quad\text{Df}\\ +\text{N}^{3}\text{c}ʻ\alpha &= \text{Nc}ʻ\alpha \cap tʻt^{3}ʻ\alpha &&\quad\text{Df and so on},\\ +\text{and}\qquad \text{N}^{1}\text{C} &= \text{D}ʻ\text{N}^{1}\text{c} &&\quad\text{Df}\\ +\text{N}^{2}\text{C} &= \text{D}ʻ\text{N}^{2}\text{c} &&\quad\text{Df}\\ +\text{N}^{3}\text{C} &= \text{D}ʻ\text{N}^{3}\text{c} &&\quad\text{Df and so on}. +\end{aligned} +\] +We then have obviously +\[ +\text{N}^{1}\text{C}(tʻ\alpha) \subset \text{N}_{0}\text{C}(tʻ\alpha). +\] +We also have (by what was said earlier) +\[ +\begin{aligned} +&\text{N}_{0}\text{c}ʻtʻ\alpha {\sim}\in \text{N}^{1}\text{C}(tʻ\alpha).\\ +\text{Hence}\qquad &\exists !\text{N}_{0}\text{C}(tʻ\alpha) - \text{N}^{1}\text{C}(tʻ\alpha). +\end{aligned} +\] +The members of \(\text{N}_{0}\text{C}(tʻ\alpha) - +\text{N}^{1}\text{C}(tʻ\alpha)\) will be all cardinals which exceed +\(\text{Nc}ʻt_{0}ʻ\alpha\) but do not exceed \(\text{Nc}ʻtʻ\alpha\).</p> + +<p>Let us recur in illustration to our previous hypothesis +of the universe consisting of a single individual. Then +\(\text{N}^{1}\text{c}\,\text{Indiv}\) will consist of those classes +which are similar to "\(\text{Indiv}\)" but of the next higher type. +These are \(\iota ʻ\Lambda\) and \(\iota ʻ\text{Indiv}\). In our case +we had \(\text{N}_{0}\text{c}(\text{Indiv}) = 1\). This leads to +\[ +\text{N}^{1}\text{c}\,\text{Indiv} = 1 . \text{N}^{2}\text{c}\,\text{Indiv} = 1\, \quad\text{etc}. +\] +or, introducing typical definiteness, +\[ +\text{N}^{1}\text{c}\,\text{Indiv} = 1(tʻ\text{Indiv}) . \text{N}^{2}\text{c}\,\text{Indiv} = 1(t^{2}ʻ\text{Indiv})\, \quad\text{etc}. +\] +We have then \(1(tʻ\text{Indiv}) \in\text{N}^{1}\text{C}(tʻtʻ\text{Indiv})\). +Also +\[ +1(tʻ\text{Indiv}) \in \text{N}_{0}\text{C}(tʻtʻ\text{Indiv}). +\] +And in the case supposed, \(1(tʻ\text{Indiv})\) is the maximum of +\(\text{N}^{1}\text{C}(tʻtʻ\text{Indiv})\), but +\(2(tʻ\text{Indiv}) \in\text{N}_{0}\text{C}(tʻtʻ\text{Indiv})\). +Hence +\[ +\text{N}_{0}\text{C}(tʻtʻ\text{Indiv}) - \text{N}^{1}\text{C}(tʻtʻ\text{Indiv}) = \iota ʻ2. +\] +Generalizing, we see that \(\text{N}^{1}\text{C}(tʻ\alpha)\) +consists of the same numbers as \(\text{N}_{0}\text{C}(\alpha)\) +each raised one degree in type. Similar propositions +hold of \(\text{N}^{2}\text{C}(t^{2}ʻ\alpha)\), +<span class="pagenum" id="Page_10">[Pg 10]</span>\(\text{N}^{3}\text{C}(t^{3}ʻ\alpha)\) etc.</p> + +<p>It is often useful to have a notation for what we may call "the same +cardinal in another type." Suppose \(\mu\) is a typically definite +cardinal; then we will denote by \(\mu ^{(1)}\) the same cardinal in +the next type, <i>i.e.</i> +\[ +\text{ sm }ʻʻ\mu \cap tʻ\mu . +\] +Note that, if \(\mu\) is a cardinal, \(\text{ sm }ʻʻ\mu \cap \mu = \mu\); +and whether \(\mu\) is a typically definite cardinal or not, +\[ +\text{ sm }ʻʻ\mu \cap tʻ\alpha +\] +is a cardinal in a definite type. If \(\mu\) is typically definite, +then \(\text{ sm }ʻʻ\mu \cap tʻ\alpha\) is wholly definite; if \(\mu\) +is typically ambiguous, \(\text{ sm }ʻʻ\mu \cap tʻ\alpha\) has the same +kind of indefiniteness as belongs to \(\text{NC}(\alpha)\). The most +important case is when \(\mu\) is typically definite and \(\alpha\) has +an assigned relation of type to \(\mu\). We then put, as observed above, +\[ +\begin{aligned} +\mu ^{(1)} &= \text{ sm }ʻʻ\mu \cap tʻ\mu &&\quad\text{Df}\\ +\mu ^{(2)} &= \text{ sm }ʻʻ\mu \cap t^{2}ʻ\mu &&\quad\text{Df etc}. +\end{aligned} +\] +If \(\mu\) is an \(\text{N}_{0}\text{C}\), \(\mu ^{(1)}\) +is an \(\text{N}^{1}\text{C}\) and \(\mu ^{(2)}\) is an +\(\text{N}^{2}\text{C}\) and so on. \(\text{N}^{1}\text{C}(tʻ\alpha)\) +will consist of all numbers which are of the form \(\mu^{(1)}\) for +some \(\mu\) which is a member of \(\text{N}_{0}\text{C}(\alpha)\); +<i>i.e.</i> +\[ +\text{N}^{1}\text{C}(tʻ\alpha) = \hat{\nu}\{(\exists \mu). \mu \in \text{N}_{0}\text{C}(\alpha) . \nu = \mu^{(1)}\}. +\]</p> + +<p>The second kind of non-homogeneous cardinals to be considered is called +the class of "descending cardinals." These are such as go into a lower +type; <i>i.e.</i> \(\text{Nc}(\alpha)ʻ\beta\) is a descending cardinal +if \(\alpha\) is of a lower type than \(\beta\). We put +\[ +\begin{aligned} +\text{N}_{1}\text{c}ʻ\alpha &= \text{Nc}ʻ\alpha \cap tʻt_{1}ʻ\alpha &&\quad\text{Df}\\ +\text{N}_{2}\text{c}ʻ\alpha &= \text{Nc}ʻ\alpha \cap tʻt_{2}ʻ\alpha &&\quad\text{Df etc}.\\ +\text{N}_{1}\text{C} &= \text{D}ʻ\text{N}_{1}\text{C} &&\quad\text{Df}\\ +\text{N}_{2}\text{C} &= \text{D}ʻ\text{N}_{2}\text{C} &&\quad\text{Df etc}.\\ +\mu_{(1)} &= \text{ sm }ʻʻ\mu \cap t_{1}ʻ\mu &&\quad\text{Df}\\ +\mu_{(2)} &= \text{ sm }ʻʻ\mu \cap t_{2}ʻ\mu &&\quad\text{Df etc}. +\end{aligned} +\]</p> + +<p>We have obviously \(\text{N}_{0}\text{c}ʻ\alpha = \text{N}_{1}\text{c}ʻ\iotaʻʻ\alpha\).</p> + +<p>Hence \(\text{N}_{0}\text{C}(\alpha) \subset\text{N}_{1}\text{C}(\alpha)\).</p> + +<p class="nind"> +Also \(\gamma \in \text{N}_{1}\text{c}ʻ\delta . \supset .\text{N}_{1}\text{c}ʻ\delta = \text{N}_{0}\text{c}ʻ\gamma\), +whence \(\exists !\text{N}_{1}\text{c}ʻ\delta . \supset .\text{N}_{1}\text{c}ʻ\delta \in \text{N}_{0}\text{C}\), +whence +\(\text{N}_{1}\text{C} - \iota ʻ\Lambda \subset \text{N}_{0}\text{C}\).</p> + +<p class="nind"> +Since also \(\Lambda {\sim} \in\text{N}_{0}\text{C}\text{N}_{0}\text{C}(\alpha)\), we find +\[ +\text{N}_{0}\text{C} = \text{N}_{1}\text{C} - \iota ʻ\Lambda, +\] +this proposition not requiring any further typical definiteness, since +it holds however such definiteness may be introduced, remembering +that such definiteness is necessarily so introduced as to secure +significance. Further, in virtue of the fact that no class contained in +\(t_{0}ʻ\alpha\) is similar to \(tʻ\alpha\), we have +\[ +\Lambda \in \text{N}_{1}\text{C}(\alpha). +\]<span class="pagenum" id="Page_11">[Pg 11]</span> +Consequently \(\text{N}_{1}\text{C} = \text{N}_{0}\text{C} \cup\iotaʻ\Lambda\).</p> + +<p>We can prove in just the same way +\[ +\begin{aligned} +\text{N}_{2}\text{C} &= \text{N}_{0}\text{C} \cup \iotaʻ\Lambda.\\ +\text{Hence}\qquad \text{N}_{1}\text{C} &= \text{N}_{2}\text{C}, +\end{aligned} +\] +and this result can obviously be extended to all descending cardinals.</p> + +<p>The third kind of non-homogeneous cardinals to be considered may be +called "relational cardinals." They are those applicable to classes of +relations having a given relation of type to a given class. Consider +for example \(\text{Nc}ʻ\in_{\Delta}ʻ\kappa\). (We shall take this +as the definition of the product of the numbers of the members of +\(\kappa\).) Suppose now that \(\kappa\) consists of a single term: we +want to be able to say +\[ +\text{Nc}ʻ{\in}_{\Delta}ʻ\kappa = \text{Nc}ʻ\breve{\iota}ʻ\kappa . +\] +We have in this case, if \(\kappa = \iota ʻ\alpha\), +\[ +\in_{\Delta }ʻ\kappa = \downarrow \alphaʻʻ\alpha, +\] +and we know that \(\downarrow \alphaʻʻ\alpha\,\text{ sm }\,\alpha\). But if +we put simply +\[ +\text{Nc}ʻ\downarrow \alphaʻʻ\alpha = \text{Nc}ʻ\alpha, +\] +our proposition, though not mistaken, requires care in interpretation. +Just as we put \(\iota ʻʻ\alpha \in \text{N}^{1}\text{c}ʻ\alpha\), +so we want a notation giving typical definiteness to the proposition +\(\downarrow \alpha ʻʻ\alpha \in \text{Nc}ʻ\alpha\). This is provided +as follows.</p> + +<p>Using the notation of *64, put +\[ +\begin{aligned} +\text{N}_{00}\text{c}ʻ\alpha &= \text{Nc}ʻ\alpha \cap tʻt_{0}ʻ\alpha &&\quad\text{Df}\\ +\text{N}_{0}^{1}\text{c}ʻ\alpha &= \text{Nc}ʻ\alpha \cap tʻt_{0}^{1}ʻ\alpha &&\quad\text{Df etc}.\\ +\text{N}_{00}\text{C} &= \text{D}ʻ\text{N}_{00}c &&\quad\text{Df}\\ +\text{N}_{0}^{1}\text{C} &= \text{D}ʻ\text{N}_{0}^{1}c &&\quad\text{Df etc}.\\ +\mu_{00} &= \text{ sm }ʻʻ\mu \cap tʻt_{00}ʻt_{1}ʻ\mu &&\quad\text{Df etc}. +\end{aligned} +\] +Then we have, for example, +\[ +\downarrow \alphaʻʻ\alpha \subset t_{0}^{1}ʻ\alpha,\quad\textit{i.e.}\quad \downarrow \alphaʻʻ\alpha \in tʻt_{0}^{1}ʻ\alpha. +\] +Hence \(\downarrow \alphaʻʻ\alpha \in\text{N}_{0}^{1}\text{c}ʻ\alpha\), +where \(\text{N}_{0}^{1}ʻ\alpha =\text{Nc}ʻ\alpha \cap t_{0}^{1}ʻ\alpha\).</p> + +<p class="nind"> +Similarly \(x \in tʻ\alpha . \supset . \downarrow xʻʻ\alpha \in\text{N}_{00}\text{c}ʻ\alpha\).</p> + +<p class="nind"> +Thus the above definitions give us what is required.</p> + +<p>In order to complete our notation for types, we should need to be able +to express the type of the domain or converse domain of \(R\), or of +any relation whose domain and converse domain have respectively given +relations of type to the domain and converse domain of \(R\). Thus we +might put +\[ +\begin{aligned} +d_{0}ʻR &= t_{0}ʻ\text{D}ʻR &&\quad\text{Df}\\ +b_{0}ʻR &= t_{0}ʻ\text{ᗡ}ʻR &&\quad\text{Df} +\end{aligned} +\] +("b" appears here as "d" written backwards) +\[ +\begin{aligned} +d_{00}ʻR &= tʻ(d_{0}ʻR \uparrow b_{0}ʻR) &&\quad\text{Df}\\ +&= tʻR\\ +d^{mn}ʻR &= tʻ(t^{m}ʻd_{0}ʻR \uparrow t^{n}ʻb_{0}ʻR) &&\quad\text{Df and so on}. +\end{aligned} +\]</p> + +<p><span class="pagenum" id="Page_12">[Pg 12]</span></p> + +<p>This notation would enable us to deal with descending relational +cardinals. But it is not required in the present work, and is therefore +not introduced among the numbered propositions.</p> + +<p>When a typically ambiguous symbol, such as "\(\text{sm}\)" or +"\(\text{Nc}\)," occurs more than once in a given context, it must not +be assumed, unless required by the conditions of significance, that +it is to receive the same typical determination in each case. Thus +<i>e.g.</i> we shall write "\(\alpha\,\text{ sm }\, \beta . \supset .\beta\, \text{ sm }\, \alpha\)," +although, if \(\alpha\) and \(\beta\) are of different types, the two +symbols "\(\text{ sm }\)" must receive different typical determinations.</p> + +<p>Formulae which are typically ambiguous, or only partially definite as +to type, must not be admitted unless every significant interpretation +is true. Thus for example we may admit +\[ +\unicode{x201c}\vdash . \alpha \in \text{Nc}ʻ\alpha\unicode{x201d} +\] +because here "\(\text{Nc}\)" must mean "\(\text{Nc}(\alpha_{\alpha})\)," so that the only ambiguity remaining is as to the type of +\(\alpha\), and the formula holds whatever type \(\alpha\) may belong +to, provided "\(\text{Nc}ʻ\alpha\)" is significant, <i>i.e.</i> +provided \(\alpha\) is a class. But we must not, from "\(\alpha \in\text{Nc}ʻ\alpha\)," +allow ourselves to infer +\[ +\unicode{x201c}\exists !\text{Nc}ʻ\alpha.\unicode{x201d} +\] +For here the conditions of significance no longer demand that +"\(\text{Nc}\)" should mean "\(\text{Nc}(\alpha_{\alpha})\)": it +might just as well mean "\(\text{Nc}(\beta_{\alpha })\)." And as we +saw, if \(\beta\) is a lower type than \(\alpha\), and \(\alpha\) is +sufficiently large of its type, we may have +\[ +\text{Nc}(\beta_{\alpha})ʻ\alpha = \Lambda, +\] +so that "\(\exists !\text{Nc}ʻ\alpha\)" is not admissible without +qualification. Nevertheless, as we shall see in <a href="#*100">*100</a>, there are a +certain number of propositions to be made about a wholly ambiguous +\(\text{Nc}\) or \(\text{NC}\).</p> + + +<div class="footnotes"><h3>FOOTNOTES:</h3> + +<div class="footnote"> + +<p class="nind"> +<a id="Footnote_2" href="#FNanchor_2" class="label">[2]</a> +Breslau, 1884. Cf. especially pp. 79, 80.</p> + +</div> + +<div class="footnote"> + +<p class="nind"> +<a id="Footnote_3" href="#FNanchor_3" class="label">[3]</a> +Jena, Vol. I. 1893, Vol. II. 1903. Cf. Vol. I. §§ 40-42, +pp. 57, 58. The grounds in favour of this definition will be found at +length in <i>Principles of Mathematics</i>, Part II.</p> + +</div> + +<div class="footnote"> + +<p class="nind"> +<a id="Footnote_4" href="#FNanchor_4" class="label">[4]</a> +The definitions of the arithmetical operations, and of +finite and infinite, are really just as purely logical as what precedes +them; but if we are to draw a line between logic and arithmetic +somewhere, the arithmetical operations seem the natural point at which +to place the beginning of arithmetic.</p> + +</div> +</div> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<span class="pagenum" id="Page_13">[Pg 13]</span> +<h2 class="nobreak" id="*100"> +*100. DEFINITION AND ELEMENTARY PROPERTIES +OF CARDINAL NUMBERS.</h2> +</div> + +<p class="nind"><i>Summary of</i> *100.</p> + +<p>In this number we shall be concerned only with such immediate +consequences of the definition of cardinal numbers as do not require +typical definiteness, beyond what the inherent conditions of +significance may bestow. We introduce here the fundamental definitions:</p> + +<p class="nind"><b><a id="*100·01">*100·01</a>.</b> \(\text{Nc} = \overrightarrow{\text{ sm }} \quad\text{Df}\)</p> + +<p class="nind"><b><a id="*100·02">*100·02</a>.</b> \(\text{NC} = \text{D}ʻ\text{Nc} \quad\text{Df}\)</p> + +<p>The definition "\(\text{Nc}\)" is required chiefly for the sake of the +descriptive function \(\text{Nc}ʻ\alpha\). We have</p> + +<p class="nind"><b><a id="*100·1">*100·1</a>.</b> \(\vdash .\text{Nc}ʻ\alpha = \hat{\beta}(\beta\,\text{ sm }\,\alpha) = \hat{\beta}(\alpha\,\text{ sm }\,\beta)\)</p> + +<p>This may be stated in various equivalent forms, which are given at the +beginning of this number (<a href="#*100·1">*100·1</a>—<a href="#*100·16">·16</a>). After a few propositions on +\(\text{Nc}\) as a relation, we proceed to the elementary properties of +\(\text{Nc}ʻ\alpha\). We have</p> + +<p class="nind"><b>*100·3.</b> \(\vdash .\alpha \in \text{Nc}ʻ\alpha\)</p> + +<p class="nind"><b>*100·31.</b> \(\vdash :\alpha \in \text{Nc}ʻ\beta . \equiv . \beta \in \text{Nc}ʻ\alpha . \equiv . \alpha \text{ sm } \beta\)</p> + +<p class="nind"><b>*100·321.</b> \(\vdash : \alpha\,\text{ sm }\,\beta . \supset . \text{Nc}ʻ\alpha = \text{Nc}ʻ\beta\)</p> + +<p class="nind"><b>*100·33.</b> \(\vdash : \exists !\text{Nc}ʻ\alpha \cap \text{Nc}ʻ\beta . \supset . \alpha\,\text{ sm }\,\beta\)</p> + +<p>We proceed next to the elementary properties of \(\text{NC}\). We have</p> + +<p class="nind"><b>*100·4.</b> \(\vdash : \mu \in \text{NC} . \equiv . (\exists \alpha ) . \mu = \text{Nc}ʻ\alpha\)</p> + +<p class="nind"><b>*100·42.</b> \(\vdash : \mu ,\nu \in \text{NC} . \exists !\mu \cap \nu . \supset . \mu =\nu\)</p> + +<p class="nind"><b>*100·45.</b> \(\vdash : \mu \in \text{NC} . \alpha \in \mu . \supset . \text{Nc}ʻ\alpha = \mu \)</p> + +<p class="nind"><b>*100·51.</b> \(\vdash : \mu \in \text{NC} . \alpha \in \mu . \supset . \text{ sm }ʻʻ\mu = \text{Nc}ʻ\alpha\)</p> + +<p>Observe that when we have such a hypothesis as "\(\mu \in \text{NC}\)," +the \(\mu\), though it may be of any type, must be of <i>some</i> type; +hence the \(\mu\) cannot have the typical ambiguity which belongs +to \(\text{Nc}ʻ\alpha\). If we put \(\mu = \text{Nc}ʻ\alpha\), this +will hold only in the type of \(\mu\); but "\(\text{ sm }ʻʻ\mu\)" is +a typically ambiguous symbol, which<span class="pagenum" id="Page_14">[Pg 14]</span> will represent in any type the +"same" number as \(\mu\). Thus "\(\text{ sm }ʻʻ\mu = \text{Nc}ʻ\alpha\)" +is an equation which is applicable to all possible typical +determinations of "\(\text{ sm }\)" and "\(\text{Nc}\)."</p> + +<p class="nind"><b>*100·52.</b> \(\vdash :\mu \in \text{NC}.\exists !\mu .\supset .\text{ sm }ʻʻ\mu \in \text{NC}\)</p> + +<p>The hypothesis \(\exists !\mu\) is unnecessary, but we cannot prove +this till later (<a href="#*102">*102</a>).</p> + +<p>We end the number with some propositions (<a href="#*100·6">*100·6</a>—<a href="#*100·64">·64</a>) stating that +various classes (such as \(\iota ʻʻ\alpha\)), which have already been +proved to be similar to \(\alpha\), have \(\text{Nc}ʻ\alpha\) members.</p> + +<hr class="tb"> + +<p class="nind"><b>*100·01.</b> \(\text{Nc}=\overrightarrow{\text{ sm }} \quad\text{Df}\)</p> + +<p class="nind"><b>*100·02.</b> \(\text{NC}=\text{D}ʻ\text{Nc} \quad \text{Df}\)</p> + +<p class="nind"><b>*100·1.</b> \(\vdash .\text{Nc}ʻ\alpha =\hat{\beta}(\beta\, \text{ sm }\,\alpha)=\hat{\beta}(\alpha\, \text{ sm }\,\beta) \quad[*32·13.*73·31.(*100·01)]\)</p> + +<p class="nind"><b>*100·11.</b> \(\vdash .\text{Nc}ʻ\alpha =\hat{\beta}\{(\exists R).R\in 1\rightarrow 1.\text{D}ʻR=\alpha .\text{ᗡ}ʻR=\beta\} \quad[*100·1.*73·1]\)</p> + +<p class="nind"><b>*100·12.</b> \(\vdash . \text{Nc}ʻ\alpha =\hat{\beta}\{(\exists R).R\in 1\rightarrow 1.\alpha \subset \text{D}ʻR.\beta =\breve{R}ʻʻ\alpha\} \quad[*100·1.*73·11]\)</p> + +<p class="nind"><b>*100·13.</b> \(\vdash .\text{Nc}ʻ\alpha =\text{ᗡ}ʻʻ(1\rightarrow 1\cap \overleftarrow{\text{D}}ʻ\alpha )=\text{D}ʻʻ(1\rightarrow 1\cap \overleftarrow{\text{ᗡ}}ʻ\alpha)\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*100·11.*33·6. \supset \vdash .\text{Nc}ʻ\alpha &=\hat{\beta} \{(\exists R).R\in 1\rightarrow 1.R\in \overleftarrow{\text{D}}ʻ\alpha .\text{ᗡ}ʻR=\beta\}\\ +[*22·33.*37·6] &=\text{ᗡ}ʻʻ(1\rightarrow 1\cap \overleftarrow{\text{D}}ʻ\alpha) &\qquad \text{(1)}\\ +\vdash .*100·1.*73·1.*33·61. \supset \vdash .\text{Nc}ʻ\alpha &=\hat{\beta} \{(\exists R).R\in 1\rightarrow 1.R\in \overleftarrow{\text{ᗡ}}ʻ\alpha .\text{D}ʻR=\beta\}\\ +[*22·33.*37·6] &=\text{D}ʻʻ(1\rightarrow 1\cap \overleftarrow{\text{ᗡ}}ʻ\alpha) &\qquad \text{(2)}\\ +\vdash .(1).(2).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*100·14.</b> \(\vdash .\text{Nc}ʻ\alpha =\hat{\beta} \{(\exists R).\alpha \subset \text{ᗡ}ʻR.R\upharpoonright \alpha \in 1\rightarrow 1.\beta = Rʻʻ\alpha\} \quad [*73·15.*100·1]\)</p> + +<p class="nind"><b>*100·15.</b> \(\vdash .\text{Nc}ʻ\alpha =\hat{\beta} \{(\exists R):\text{E}!!Rʻʻ\alpha : +x,y\in \alpha .Rʻx=Rʻy.\supset _{x,y}.x=y:\beta =Rʻʻ\alpha\}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*74·1·11.\supset \\ +\vdash \colon\ldotp \text{E}!!Rʻʻ\alpha :x,y\in \alpha .&Rʻx=Rʻy.\supset _{x,y}.x=y:\beta =Rʻʻ\alpha :\equiv :\\ +&R\upharpoonright \alpha \in 1\rightarrow \text{Cls}.\alpha \subset \text{ᗡ}ʻR.R\upharpoonright \alpha \in 1\rightarrow 1.\beta = Rʻʻ\alpha \qquad \text{(1)}\\ +\vdash .(1).*4·71.*100·14.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*100·16">*100·16</a>.</b> \(\vdash .\text{Nc}ʻ\alpha =\hat{\beta} \{(\exists R)\colon\ldotp x,y\in \alpha .\supset _{x,y}:Rʻx=Rʻy.\equiv .x=y\colon\ldotp \beta =Rʻʻ\alpha\}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*71·59.\supset \\ +\vdash \colon\colon x,y\in \alpha .\supset_{x,y}:Rʻx=Rʻy.\equiv .x=y\colon\ldotp \equiv .R\upharpoonright \alpha \in 1\rightarrow 1.\alpha \subset \text{ᗡ}ʻR &\qquad \text{(1)}\\ +\vdash .(1).*100·14.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_15">[Pg 15]</span></p> + +<p class="nind"><b><a id="*100·2">*100·2</a>.</b> \(\vdash .\text{E}!\text{Nc}ʻ\alpha \quad[*32·12.(*100·01)]\)</p> + +<p class="nind"><b>*100·21.</b> \(\vdash .\text{ᗡ}ʻ\text{Nc} = \text{Cls}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +&\vdash .*37·76.(*100·01).\supset \vdash .\text{ᗡ}ʻ\text{Nc}\subset \text{Cls} &\qquad \text{(1)}\\ +&\vdash .*33·431.*100·2. \supset \vdash .\text{Cls}\subset \text{ᗡ}ʻ\text{Nc} &\qquad \text{(2)}\\ +&\vdash .(1).(2).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*100·22.</b> \(\vdash .\text{Nc}\in 1\rightarrow \text{Cls} \quad[*72·12.(*100·01)]\)</p> + +<p class="nind"><b><a id="*100·3">*100·3</a>.</b> \(\vdash .\alpha \in \text{Nc}ʻ\alpha \quad[*73·3.*100·1]\)</p> + +<p>Note that it is fallacious to infer \(\exists !\text{Nc}ʻ\alpha\), for +reasons explained in the introduction to the present section.</p> + +<p class="nind"><b>*100·31.</b> \(\vdash :\alpha \in \text{Nc}ʻ\beta .\equiv .\beta \in \text{Nc}ʻ\alpha .\equiv .\alpha\,\text{ sm }\,\beta \quad[*32·18.*73·31.(*100·01)]\)</p> + +<p class="nind"><b>*100·32.</b> \(\vdash :\alpha \in \text{Nc}ʻ\beta .\beta \in \text{Nc}ʻ\gamma .\supset .\alpha \in \text{Nc}ʻ\gamma \quad[*100·31.*73·32]\)</p> + +<p class="nind"><b><a id="*100·321">*100·321</a>.</b> \(\vdash :\alpha\,\text{ sm }\,\beta .\supset .\text{Nc}ʻ\alpha =\text{Nc}ʻ\beta\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*73·37.\supset \vdash \colon\ldotp \text{Hp}.&\supset :\gamma\,\text{ sm }\,\alpha .\equiv _{\gamma }.\gamma \text{ sm }\beta :\\ +[*100·1] &\supset :\text{Nc}ʻ\alpha =\text{Nc}ʻ\beta \colon\ldotp \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p>Note that \(\text{Nc}ʻ\alpha =\text{Nc}ʻ\beta .\supset .\alpha\,\text{ sm }\,\beta\) +is not always true. We might be tempted to prove it as follows: +\[ +\begin{array}{l} +\vdash .*100·1.\supset \vdash \colon\ldotp \text{Nc}ʻ\alpha =\text{Nc}ʻ\beta .\equiv :\gamma \,\text{ sm }\,\alpha .&\equiv _{\gamma }.\gamma \,\text{ sm }\,\beta :\\ +[*10·1] &\supset :\alpha \,\text{ sm }\,\alpha .\equiv .\alpha \,\text{ sm }\,\beta :\\ +[*73·3] &\supset :\alpha \,\text{ sm }\,\beta +\end{array} +\]</p> + +<p>But the use of *10·1 here is only legitimate when the "\(\text{sm}\)" +concerned is a homogeneous relation. If \(\text{Nc}ʻ\alpha\), +\(\text{Nc}ʻ\beta\) are descending cardinals, we may have +\(\text{Nc}ʻ\alpha =\Lambda = \text{Nc}ʻ\beta\) without having \(\alpha\text{ sm }\beta\).</p> + +<p class="nind"><b><a id="*100·33">*100·33</a>.</b> \(\vdash :\exists !\text{Nc}ʻ\alpha \cap \text{Nc}ʻ\beta .\supset .\alpha \,\text{ sm }\,\beta\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*100·1.\supset \vdash :\text{Hp}.&\supset .(\exists \gamma ).\gamma\,\text{ sm }\, \alpha .\gamma \,\text{ sm }\, \beta.\\ +[*73·31] &\supset .(\exists \gamma ).\alpha\, \text{ sm }\, \gamma .\gamma \text{ sm } \beta.\\ +[*73·32] &\supset .\alpha\,\text{ sm }\, \beta :\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p>Note that we do not always have +\[ +\alpha \,\text{ sm }\, \beta .\supset .\exists !\text{Nc}ʻ\alpha \cap \text{Nc}ʻ\beta. +\]</p> + +<p>For if the \(\text{Nc}\) concerned is a descending \(\text{Nc}\), and +\(\alpha\) and \(\beta\) are sufficiently great, \(\text{Nc}ʻ\alpha\) +and \(\text{Nc}ʻ\beta\) may both be \(\Lambda\). For example, we have +\[ +\text{Cl}ʻ(\alpha \cup - \alpha)\,\text{ sm }\, \text{Cl}ʻ(\alpha \cup - \alpha). +\]</p> + +<p>But \(\text{Nc}(\alpha)ʻ\text{Cl}ʻ(\alpha \cup - \alpha) = \Lambda\), +so that +\[ +{\sim}\exists !\text{Nc}(\alpha )ʻ\text{Cl}ʻ(\alpha \cup - \alpha ) \cap \text{Nc}(\alpha )ʻ\text{Cl}ʻ(\alpha \cup - \alpha). +\]</p> + +<p><span class="pagenum" id="Page_16">[Pg 16]</span></p> + +<p>Thus "\(\alpha\, \text{ sm }\,\beta . \supset . \exists ! +\text{Nc}ʻ\alpha \cap \text{Nc}ʻ\beta\)" is not always true when it is +significant.</p> + +<p class="nind"><b><a id="*100·34">*100·34</a>.</b> \(\vdash : \exists ! \text{Nc}ʻ\alpha \cap \text{Nc}ʻ\beta . \supset . \text{Nc}ʻ\alpha = \text{Nc}ʻ\beta \quad[*100·33·321]\)</p> + +<p class="nind"><b><a id="*100·35">*100·35</a>.</b> \[\begin{aligned}\vdash \colon\ldotp \exists ! \text{Nc}ʻ\alpha . \lor . &\exists ! \text{Nc}ʻ\beta : \supset:\\ +&\text{Nc}ʻ\alpha = \text{Nc}ʻ\beta . \equiv . \alpha \in \text{Nc}ʻ\beta . \equiv . \beta \in \text{Nc}ʻ\alpha . \equiv . \alpha \,\text{ sm }\, \beta \end{aligned}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*22·5. \supset \vdash \colon\ldotp \text{Hp} . \supset : \text{Nc}ʻ\alpha = \text{Nc}ʻ\beta . &\supset . \exists ! \text{Nc}ʻ\alpha \cap \text{Nc}ʻ\beta.\\ +[*100·33] &\supset . \alpha \, \text{ sm }\, \beta &\qquad \text{(1)}\\ +\vdash .(1).*100·321. &\supset \vdash \colon\ldotp \text{Hp} . \supset : \text{Nc}ʻ\alpha = \text{Nc}ʻ\beta . \equiv . \alpha \,\text{ sm }\, \beta &\qquad\text{(2)}\\ +\vdash .(2).*100·31. &\supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p>Thus the only case in which the implications in <a href="#*100·321">*100·321</a><a href="#*100·33">·33</a><a href="#*100·34">·34</a> cannot +be turned into equivalences is the case in which \(\text{Nc}ʻ\alpha\) +and \(\text{Nc}ʻ\beta\) are both \(\Lambda\).</p> + +<p class="nind"><b>*100·36.</b> \(\vdash \colon\ldotp \beta \in \text{Nc}ʻ\alpha . \supset : \exists ! \alpha . \equiv . \exists !\beta \quad[*100·31.*73·36]\)</p> + +<p class="nind"><b><a id="*100·4">*100·4</a>.</b> \(\vdash : \mu \in \text{NC} . \equiv . (\exists \alpha) . \mu = \text{Nc}ʻ\alpha \quad[*37·78·79.(*100·02·01)]\)</p> + +<p class="nind"><b><a id="*100·41">*100·41</a>.</b> \(\vdash . \text{Nc}ʻ\alpha \in \text{NC} \quad[*100·4·2.*14·204]\)</p> + +<p class="nind"><b><a id="*100·42">*100·42</a>.</b> \(\vdash : \mu,\nu \in \text{NC} . \exists !\mu \cap \nu . \supset . \mu = \nu\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *100·4 . \supset \vdash : \text{Hp} . &\supset . (\exists \alpha ,\beta ) . \mu = \text{Nc}ʻ\alpha . \nu = \text{Nc}ʻ\beta . \exists ! \text{Nc}ʻ\alpha \cap \text{Nc}ʻ\beta .\\ +[*100·34] &\supset . (\exists \alpha ,\beta) . \mu = \text{Nc}ʻ\alpha . \nu = \text{Nc}ʻ\beta . \text{Nc}ʻ\alpha = \text{Nc}ʻ\beta.\\ +[*14·15] &\supset . \mu = \nu : \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*100·43.</b> \(\vdash . \text{NC} \in \text{Cls}^{2} \text{excl} \quad[*100·42.*84·11]\)</p> + +<p class="nind"><b><a id="*100·44">*100·44</a>.</b> \(\vdash \colon\ldotp \mu \in \text{NC}. \exists ! \text{Nc}ʻ\alpha . \supset : \alpha \in \mu . \equiv . \text{Nc}ʻ\alpha = \mu\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *100·3. &\supset \vdash : \text{Nc}ʻ\alpha = \mu . \supset . \alpha \in \mu &\qquad\text{(1)}\\ +\vdash . *10·24. \supset \vdash : &\mu \in \text{NC} . \exists ! \text{Nc}ʻ\alpha . \alpha \in \mu . \supset.\\ +&\mu \in \text{NC} . \exists !\mu . \exists ! \text{Nc}ʻ\alpha . \alpha \in \mu.\\ +[*100·4] &\supset . (\exists \beta ) . \mu = \text{Nc}ʻ\beta . \exists ! \text{Nc}ʻ\beta. \exists ! \text{Nc}ʻ\alpha . \alpha \in \text{Nc}ʻ\beta.\\ +[*100·35] &\supset . (\exists \beta). \mu = \text{Nc}ʻ\beta . \text{Nc}ʻ\alpha = \text{Nc}ʻ\beta.\\ +[*14·15] &\supset . \text{Nc}ʻ\alpha = \mu &\qquad \text{(2)}\\ +\vdash . (1).(2). &\supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*100·45.</b> \(\vdash : \mu \in \text{NC} . \alpha \in \mu . \supset . \text{Nc}ʻ\alpha = \mu \quad[*100·4·31·321]\)</p> + +<p class="nind"><b>*100·5.</b> \(\vdash : \mu \in \text{NC} . \alpha ,\beta \in \mu . \supset . \alpha \,\text{ sm }\, \beta\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *100·4. \supset \vdash : \text{Hp} . &\supset . (\exists \gamma ) . \mu = \text{Nc}ʻ\gamma . \alpha ,\beta \in \text{Nc}ʻ\gamma.\\ +[*100·31] &\supset .(\exists \gamma). \alpha\,\text{sm}\,\gamma . \beta\,\text{ sm }\,\gamma.\\ +[*73·31·32] &\supset . \alpha\,\text{ sm }\,\beta : \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_17">[Pg 17]</span></p> + +<p class="nind"><b>*100·51.</b> \(\vdash : \mu \in \text{NC} . \alpha \in \mu . \supset . \text{ sm }ʻʻ\mu = \text{Nc}ʻ\alpha\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash. *100·5 . \text{Fact}. \supset \vdash \colon\ldotp \text{Hp} . \supset : \beta \in \mu . \gamma \,\text{ sm }\, \beta . &\supset . \alpha \text{ sm } \beta . \gamma \text{ sm } \beta.\\ +[*73·31·32] &\supset . \alpha \,\text{ sm }\, \gamma.\\ +[*100·31] &\supset . \gamma \in \text{Nc}ʻ\alpha &\qquad\text{(1)}\\ +\vdash .(1).*10·11·21·23.*37·1. \supset \vdash : \text{Hp} . &\supset . \text{ sm }ʻʻ\mu \subset \text{Nc}ʻ\alpha &\qquad \text{(2)}\\ +\vdash .*100·31. \supset \vdash \colon\ldotp \text{Hp} . &\supset : \gamma \in \text{Nc}ʻ\alpha . \supset . \gamma \,\text{ sm }\, \alpha . \alpha \in \mu.\\ +[*37·1] &\supset . \gamma \in \text{ sm }ʻʻ\mu &\qquad \text{(3)}\\ +\vdash .(2).(3). \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*100·511">*100·511</a>.</b> \(\vdash : \exists ! \text{Nc}ʻ\beta . \supset . \text{ sm }ʻʻ\text{Nc}ʻ\beta = \text{Nc}ʻ\beta\)</p> + +<p>Here the last "\(\text{Nc}ʻ\beta\)" may be of a different type from the +others: the proposition holds however its type is determined.</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*100·51·41. \supset \vdash : \alpha \in \text{Nc}ʻ\beta . \supset . \text{ sm }ʻʻ\text{Nc}ʻ\beta &= \text{Nc}ʻ\alpha \\ +[*100·31·321] &= \text{Nc}ʻ\beta &\qquad \text{(1)}\\ +\vdash .(1).*10·11·23. \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*100·52">*100·52</a>.</b> \(\vdash : \mu \in \text{NC} . \exists ! \mu . \supset . \text{ sm }ʻʻ\mu \in \text{NC} \quad[*100·51·4]\)</p> + +<p>This proposition still holds when \(\mu = \Lambda\), but the proof is +more difficult, since it depends upon the proof that every null-class +of classes is an \(\text{NC}\), which in turn depends upon the proof +that \(\text{Cl}ʻ\alpha\) is not similar to \(\alpha\) or to any class +contained in \(\alpha\).</p> + +<p class="nind"><b><a id="*100·521">*100·521</a>.</b> \(\vdash : \mu \in \text{NC} . \exists ! \text{ sm }ʻʻ\mu . \supset . \text{ sm }ʻʻ\text{ sm }ʻʻ\mu = \mu\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*37·29.\text{Transp}. \supset \vdash \colon\ldotp \text{Hp} . &\supset : \exists !\mu:\\ +[*100·52] &\supset : \text{ sm }ʻʻ\mu \in \text{NC}:\\ +[*100·51.\text{Hp}] &\supset : \gamma \in \text{ sm }ʻʻ\mu . \supset . \text{ sm }ʻʻ\text{ sm }ʻʻ\mu = \text{Nc}ʻ\gamma &\qquad \text{(1)}\\ +\vdash .*37·1.\text{Fact}. \supset \vdash : \text{Hp} . \gamma \in \text{ sm }ʻʻ\mu . &\supset . (\exists \alpha) . \alpha \in \mu . \mu \in \text{NC} . \gamma \text{ sm } \alpha.\\ +[*100·45·321] &\supset . (\exists \alpha) . \text{Nc}ʻ\alpha = \mu . \text{Nc}ʻ\gamma = \text{Nc}ʻ\alpha.\\ +[*13·17] &\supset . \text{Nc}ʻ\gamma = \mu &\qquad \text{(2)}\\ +\vdash .(1).(2). &\supset \vdash \colon\ldotp \text{Hp} . \gamma \in \text{ sm }ʻʻ\mu . \supset . \text{ sm }ʻʻ\text{ sm }ʻʻ\mu = \mu &\qquad \text{(3)}\\ +\vdash .(3).*10·11·23·35 . \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*100·53.</b> \(\vdash \colon\ldotp \exists !\mu . \exists !\nu . \supset : \mu \in \text{NC} . \nu = \text{ sm }ʻʻ\mu . \equiv . \nu \in \text{NC} . \mu = \text{ sm }ʻʻ\nu\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*100·52. &\supset \vdash \colon\ldotp \text{Hp} . \supset : \mu \in \text{NC} . \nu = \text{ sm }ʻʻ\mu . \supset . \nu \in \text{NC} &\qquad \text{(1)}\\ +\vdash .*100·521. &\supset \vdash \colon\ldotp \text{Hp} . \supset : \mu \in \text{NC} . \nu = \text{ sm }ʻʻ\mu . \supset . \mu = \text{ sm }ʻʻ\nu &\qquad \text{(2)}\\ +\vdash .(1).(2). &\supset \vdash \colon\ldotp \text{Hp} . \supset : \mu \in \text{NC} . \nu = \text{ sm }ʻʻ\mu . \supset . \nu \in \text{NC} . \mu = \text{ sm }ʻʻ\nu &\qquad \text{(3)}\\ +\vdash .(3).(3)\dfrac{\nu,\,\mu}{\mu,\,\nu}. \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_18">[Pg 18]</span></p> + +<p class="nind"><b><a id="*100·6">*100·6</a>.</b> \(\vdash .\iotaʻʻ\alpha \in \text{Nc}ʻ\alpha \quad[*73·41.*100·31]\)</p> + +<p class="nind"><b>*100·61.</b> \(\vdash .\hat{\beta}\{(\exists y).y \in \alpha .\beta = \iota ʻx \cup \iota ʻy\} \in \text{Nc}ʻ\alpha \quad [*73·27.*54·21.*100·31]\)</p> + +<p class="nind"><b>*100·62.</b> \(\vdash .x\downarrow ʻʻ\alpha \in \text{Nc}ʻ\alpha \quad[*73·61.*100·31]\)</p> + +<p class="nind"><b>*100·621.</b> \(\vdash .\downarrow xʻʻ\alpha \in \text{Nc}ʻ\alpha \quad[*73·611.*100·31]\)</p> + +<p class="nind"><b>*100·63.</b> \(\vdash .\in_{\Delta }ʻ\iota ʻ\alpha \in \text{Nc}ʻ\alpha \quad[*83·41.*100·31]\)</p> + +<p class="nind"><b>*100·631.</b> \(\vdash .\text{D}ʻʻ{\in}_{\Delta}ʻ\iota ʻ\alpha \in \text{Nc}ʻ\alpha \quad[*83·7.*100·6]\)</p> + +<p class="nind"><b><a id="*100·64">*100·64</a>.</b> \(\vdash :\kappa \in \text{Cls}^{2} \text{excl} .\supset . \text{D}ʻʻ\in_{\Delta }ʻ\kappa \subset \text{Nc}ʻ\kappa\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*84·3.*80·14.\supset \vdash :\text{Hp}.R\in {\in}_{\Delta}ʻ\kappa .&\supset .R \in 1\rightarrow 1.\kappa = \text{ᗡ}ʻR.\\ +[*73·2.*100·31] &\supset .\text{D}ʻR \in \text{Nc}ʻ\kappa :\supset \vdash . \text{Prop} +\end{array} +\]</p> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_19">[Pg 19]</span></p> + +<h2 class="nobreak" id="*101"> +*101. ON 0 AND 1 AND 2.</h2> +</div> + + +<p class="nind"><i>Summary of</i> *101.</p> + +<p>In the present number, we have to show that 0 and 1 and 2 as previously +defined are cardinal numbers in the sense defined in <a href="#*100">*100</a>, and to add a +few elementary propositions to those already given concerning them. We +prove (<a href="#*101·12">*101·12</a><a href="#*101·241">·241</a>) that 0 and 1 are not null, which cannot be proved, +with our axioms, for any other cardinal, except (in the case of finite +cardinals) when the type is specified as a sufficiently high one. Thus +we prove (<a href="#*101·42">*101·42</a><a href="#*101·43">·43</a>) that \(2_{\text{Cls}}\) and \(2_{\text{Rel}}\) +exist; this follows from \(\Lambda \neq \text{V}\) and \(\dot{\Lambda}\neq \dot{\text{V}}\). +We prove (<a href="#*101·22">*101·22</a><a href="#*101·34">·34</a>) that 0 and 1 and 2 are all different from +each other. We prove (<a href="#*101·15">*101·15</a><a href="#*101·28">·28</a>) that \(\text{ sm }ʻʻ0 = 0\) and +\(\text{ sm }ʻʻ1 = 1\), but we cannot prove \(\text{ sm }ʻʻ2 = 2\) unless +we assume the existence of at least two individuals, or define the +first 2 in "\(\text{ sm }ʻʻ2 = 2\)" as a 2 of some type other than +\(2_{\text{Indiv}}\), where "\(\text{Indiv}\)" stands for the type of +individuals.</p> + +<p>It should be observed that, since 0 and 1 and 2 are typically +ambiguous, their properties are analogous to those of +"\(\text{Nc}ʻ\alpha\)" rather than to those of \(\mu\), where \(\mu \in\text{NC}\). +For example, we have</p> + +<p class="nind"><b>*100·511.</b> \(\vdash : \exists ! \text{Nc}ʻ\beta . \supset . \text{ sm }ʻʻ\text{Nc}ʻ\beta = \text{Nc}ʻ\beta\)</p> + +<p>but we shall not have \(\mu \in \text{NC} . \exists ! \mu . \supset. \text{ sm }ʻʻ\mu =\mu\) +unless the "\(\text{ sm }\)" concerned is homogeneous, since in other +cases the symbols do not express a significant proposition. But in +<a href="#*100·511">*100·511</a> we may substitute 0 or 1 or 2, and the proposition remains +significant and true. In fact we have (<a href="#*101·1">*101·1</a><a href="#*101·2">·2</a><a href="#*101·31">·31</a>) +\[ +\vdash . 0 = \text{Nc}ʻ\Lambda . 1 = \text{Nc}ʻ\iota ʻx . 2 = \text{Nc}ʻ(\iota ʻ\iota ʻx \cup \iota ʻ\Lambda ), +\] +where 0 and 1 and 2 have an ambiguity corresponding to that of +"\(\text{Nc}\)."</p> + +<hr class="tb"> + +<p class="nind"><b><a id="*101·1">*101·1</a>.</b> \(\vdash . 0 = \text{Nc}ʻ\Lambda \quad[*73·48 . *100·1]\)</p> + +<p class="nind"><b><a id="*101·11">*101·11</a>.</b> \(\vdash . 0 \in \text{NC} \quad[*101·1 .*100·4]\)</p> + +<p class="nind"><b><a id="*101·12">*101·12</a>.</b> \(\vdash . \exists ! 0 \quad[*51·161. (*54·01)]\)</p> + +<p class="nind"><b>*101·13.</b> \(\vdash . \exists ! 0 \cap \text{Cl}ʻ\alpha . \Lambda \in 0 \cap \text{Cl}ʻ\alpha \quad[*51·16 . *60·3]\)</p> + +<p><span class="pagenum" id="Page_20">[Pg 20]</span></p> + +<p class="nind"><b>*101·14.</b> \(\vdash : \text{Nc}ʻ\gamma = 0 . \equiv . \gamma = \Lambda\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*101·1·12 . \supset \vdash : \text{Nc}ʻ\gamma = 0 . &\equiv . \text{Nc}ʻ\gamma = \text{Nc}ʻ\Lambda . \exists ! \text{Nc}ʻ\Lambda.\\ +[*13·194] &\equiv . \text{Nc}ʻ\gamma = \text{Nc}ʻ\Lambda . \exists ! \text{Nc}ʻ\Lambda . \exists ! \text{Nc}ʻ\gamma.\\ +[*100·35] &\equiv . \gamma \in \text{Nc}ʻ\Lambda . \exists ! \text{Nc}ʻ\Lambda . \exists ! \text{Nc}ʻ\gamma.\\ +[*101·1.*54·102] &\equiv . \gamma = \Lambda . \exists ! \text{Nc}ʻ\Lambda . \exists ! \text{Nc}ʻ\gamma .\\ +[*101·1·12.*13·194] &\equiv . \gamma = \Lambda : \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*101·15">*101·15</a>.</b> \(\vdash . \text{ sm }ʻʻ0 = 0\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*37·1 . \supset \vdash : \gamma \in \text{ sm }ʻʻ0 . &\equiv . (\exists \alpha ) . \alpha \in 0 . \gamma \text{ sm } \alpha.\\ +[*54·102] &\equiv . \gamma \text{ sm } \Lambda .\\ +[*73·48] &\equiv . \gamma \in 0 : \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*101·16.</b> \(\vdash \colon\ldotp \mu \in \text{NC}-\iota ʻ0 . \supset : \alpha \in \mu . \supset_{\alpha} . \exists !\alpha\)</p> + + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*100·45 . \supset \vdash : \mu \in \text{NC} . \Lambda \in \mu . &\supset . \mu = \text{Nc}ʻ\Lambda\\ +[*101·1] & =0 &\qquad \text{(1)}\\ +\vdash .(1).\text{Transp}. &\supset \vdash \colon\ldotp \mu \in \text{NC}-\iota ʻ0 . \supset : \Lambda {\sim}\in \mu :\\ +[*24·63] &\supset : \alpha \in \mu . \supset_{\alpha} . \exists !\alpha \colon\ldotp \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*101·17.</b> \(\vdash : \Lambda \in \text{Nc}ʻ\alpha . \equiv . \text{Nc}ʻ\alpha = 0 . \equiv . \text{Nc}ʻ\alpha = \text{Nc}ʻ\Lambda . \equiv . \alpha = \Lambda\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*100·31·321. \supset \vdash : \Lambda \in \text{Nc}ʻ\alpha . &\supset . \text{Nc}ʻ\alpha = \text{Nc}ʻ\Lambda.\\ +[*101·1] &\supset . \text{Nc}ʻ\alpha = 0 &\qquad \text{(1)}\\ +\vdash .*101·13. &\supset \vdash : \text{Nc}ʻ\alpha = 0 . \supset . \Lambda \in \text{Nc}ʻ\alpha &\qquad\text{(2)}\\ +\vdash .(1).(2). \supset \vdash : \Lambda \in \text{Nc}ʻ\alpha . &\equiv . \text{Nc}ʻ\alpha = 0. &\qquad\text{(3)}\\ +[*101·1] &\equiv . \text{Nc}ʻ\alpha = \text{Nc}ʻ\Lambda . &\qquad\text{(4)}\\ +[*101·14] &\equiv . \alpha = \Lambda &\qquad \text{(5)}\\ +\vdash .(3).(4).(5). \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*101·2">*101·2</a>.</b> \(\vdash . 1 = \text{Nc}ʻ\iota ʻx \quad[*73·45.*100·1]\)</p> + +<p class="nind"><b><a id="*101·21">*101·21</a>.</b> \(\vdash . 1 \in \text{NC} \quad[*101·2.*100·4]\)</p> + +<p class="nind"><b><a id="*101·22">*101·22</a>.</b> \(\vdash . 1 \neq 0\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*52·21.*101·13. &\supset \vdash . \Lambda {\sim}\in 1 . \Lambda \in 0.\\ +[*13·14] &\supset \vdash . 1\neq 0 +\end{array} +\]</p> + +<p class="nind"><b>*101·23.</b> \(\vdash . 1 \cap 0 = \Lambda\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*52·21. \supset \vdash : \alpha \in 1 . &\supset . \alpha \neq \Lambda.\\ +[*54·102] &\supset . \alpha {\sim}\in 0 &\qquad\text{(1)}\\ +\vdash .(1).*24·39. &\supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_21">[Pg 21]</span></p> + +<p class="nind"><b>*101·24.</b> \(\vdash : \exists !\alpha . \supset . \exists ! 1 \cap \text{Cl}ʻ\alpha\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *52·22.*60·6 . \supset \vdash : x \in \alpha . \supset . \iota ʻx \in 1 \cap \text{Cl}ʻ\alpha \qquad \text{(1)}\\ +\vdash .(1).*10·11·28. \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*101·241">*101·241</a>.</b> \(\vdash . \exists !1 \quad[*52·23]\)</p> + +<p class="nind"><b>*101·25.</b> \(\vdash : \alpha \in 1 . \beta \subset \alpha . \beta \neq \alpha . \supset . \beta \in 0\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*52·64.*22·621. &\supset \vdash : \alpha \in 1 . \beta \subset \alpha . \supset . \beta \in 1 \cup 0 &\qquad\text{(1)}\\ +\vdash .*52·46. &\supset \vdash : \alpha,\beta \in 1 . \beta \subset \alpha . \supset . \beta =\alpha:\\ +[\text{Transp}] &\supset \vdash : \alpha \in 1 . \beta \subset \alpha . \beta \neq \alpha . \supset . \beta {\sim}\in 1 &\qquad\text{(2)}\\ +\vdash .(1).(2). \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*101·26.</b> \(\vdash . sʻ\text{Cl}ʻʻ1 = 0 \cup 1\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*60·371.*40·43. &\supset \vdash . sʻ\text{Cl}ʻʻ1 \subset 0 \cup 1 &\qquad\text{(1)}\\ +\vdash .*60·3·34. &\supset \vdash . \Lambda \in \text{Cl}ʻ\iota ʻx . \iota ʻx \in \text{Cl}ʻ\iota ʻx.\\ +[*52·22.*40·4] &\supset \vdash . \Lambda \in sʻ\text{Cl}ʻʻ1 . \iota ʻx \in sʻ\text{Cl}ʻʻ1.\\ +[*51·2.*52·1] &\supset \vdash . 0 \subset sʻ\text{Cl}ʻʻ1 . 1 \subset sʻ\text{Cl}ʻʻ1 &\qquad\text{(2)}\\ +\vdash .(1).(2). \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*101·27.</b> \(\vdash . 1 = \hat{\alpha}\{(\exists x) . x \in \alpha . \alpha - \iota ʻx \in 0\}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*54·102. \supset \vdash : (\exists x) . x \in \alpha . \alpha - \iota ʻx \in 0 . &\equiv . (\exists x) . x \in \alpha . \alpha - \iota ʻx = \Lambda .\\ +[*24·3] &\equiv . (\exists x). x \in \alpha . \alpha \subset \iota ʻx.\\ +[*51·2] &\equiv . (\exists x) . \alpha = \iota ʻx.\\ +[*52·1] &\equiv . \alpha \in 1 : \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*101·28">*101·28</a>.</b> \(\vdash . \text{ sm }ʻʻ1 = 1\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*37·1. \supset \vdash : \gamma \in \text{ sm }ʻʻ1 . &\equiv . (\exists \alpha ) . \alpha \in 1 . \gamma\,\text{ sm }\,\alpha.\\ +[*52·1] &\equiv . (\exists x) . \gamma\,\text{ sm }\,\iota ʻx.\\ +[*73·45] &\equiv . \gamma \in 1 : \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*101·29.</b> \(\vdash : \iota ʻx\in \text{Nc}ʻ\alpha . \equiv . \text{Nc}ʻ\alpha = 1. \equiv . \text{Nc}ʻ\alpha = \text{Nc}ʻ\iota ʻx. = . \alpha \in 1\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*100·31·321. \supset \vdash : \iota ʻx \in \text{Nc}ʻ\alpha . &\supset . \text{Nc}ʻ\alpha = \text{Nc}ʻ\iota ʻx.\\ +[*101·2] &\supset . \text{Nc}ʻ\alpha = 1 &\qquad\text{(1)}\\ +\vdash .*52·22. &\supset \vdash : \text{Nc}ʻ\alpha = 1 . \supset . \iota ʻx \in \text{Nc}ʻ\alpha &\qquad\text{(2)}\\ +\vdash .(1).(2). \supset \vdash : \iota ʻx \in \text{Nc}ʻ\alpha . &\equiv . \text{Nc}ʻ\alpha = 1. &\qquad\text{(3)}\\ +[*101·2] &\equiv . \text{Nc}ʻ\alpha = \text{Nc}ʻ\iota ʻx &\qquad\text{(4)}\\ +\vdash .*101·2.*52·1. &\supset \vdash : \alpha \in 1 . \supset . \text{Nc}ʻ\alpha = 1 &\qquad\text{(5)}\\ +\vdash .*100·3 . &\supset \vdash : \text{Nc}ʻ\alpha = 1 . \supset . \alpha \in 1 &\qquad\text{(6)}\\ +\vdash .(3).(4).(5).(6). \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_22">[Pg 22]</span></p> + +<p class="nind"><b><a id="*101·3">*101·3</a>.</b> \(\vdash :x\neq y.\supset .2=\text{Nc}ʻ(\iota ʻx\cup \iota ʻy)\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*73·71·43.*51·231.\supset \vdash \colon\ldotp \text{Hp}.&\supset :z\neq w.\supset .(\iota ʻz\cup \iota ʻw)\text{ sm }(\iota ʻx\cup \iota ʻy):\\ +[*54·101]&\supset :\beta \in 2.\supset .\beta \text{ sm }(\iota ʻx\cup \iota ʻy):\\ +[*100·1] &\supset :2\subset \text{Nc}ʻ(\iota ʻx\cup \iota ʻy) &\qquad \text{(1)}\\ +\vdash .*53·32.*71·163.\supset \vdash :R\in 1\rightarrow 1.x,y&\in \text{ᗡ}ʻR.\supset .\\ +&Rʻʻ(\iota ʻx\cup \iota ʻy)=\iota ʻRʻx\cup \iota ʻRʻy &\qquad \text{(2)}\\ +\vdash .*71·56.\text{Transp}. &\supset \vdash :\text{Hp}.R\in 1\rightarrow 1.x,y\in \text{ᗡ}ʻR.\supset .Rʻx\neq Rʻy &\qquad \text{(3)}\\ +\vdash .(2).(3).*54·26.&\supset \\ +\vdash \colon\ldotp \text{Hp}.&\supset :R\in 1\rightarrow 1.x,y\in \text{ᗡ}ʻR.\beta =Rʻʻ(\iota ʻx\cup \iota ʻy).\supset .\beta \in 2:\\ +[*10·11·21·23.*51·234]&\supset :(\exists R).R\in 1\rightarrow 1.\iota ʻx\cup \iota ʻy\subset \text{ᗡ}ʻR.\beta =Rʻʻ(\iota ʻx\cup \iota ʻy).\\ +&\supset .\beta \in 2:\\ +[*73·12.*100·1] &\supset :\text{Nc}ʻ(\iota ʻx\cup \iota ʻy)\subset 2 &\qquad \text{(4)}\\ +\vdash .(1).(4).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*101·301.</b> \(\vdash.2=\hat{\alpha}\{(\exists x).x\in \alpha .\alpha -\iota ʻx\in 1\} \quad[*54·3]\)</p> + +<p>In comparing <a href="#*101·31">*101·31</a> with <a href="#*101·1">*101·1</a><a href="#*101·2">·2</a><a href="#*101·3">·3</a>, it should be observed that +\(\iota ʻx\) and \(\Lambda\) are both <i>classes</i>, whereas in *101·1·2·3 +there was no typical limitation beyond what was imposed by the +conditions of significance.</p> + +<p class="nind"><b><a id="*101·31">*101·31</a>.</b> \(\vdash .2=\text{Nc}ʻ(\iota ʻ\iota ʻx\cup \iota ʻ\Lambda)\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*51·161. &\supset \vdash .\iota ʻx\neq \Lambda &\qquad \text{(1)}\\ +\vdash .(1).*101·3.&\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*101·32">*101·32</a>.</b> \(\vdash .2\in \text{NC} \quad[*101·31.*100·4]\)</p> + +<p class="nind"><b>*101·33.</b> \(\vdash :\alpha,\beta \in 1.\alpha \cap \beta =\Lambda .\supset .\alpha \cup \beta \in 2 \quad[*54·43]\)</p> + +<p class="nind"><b><a id="*101·34">*101·34</a>.</b> \(\vdash .2\neq 0.2\neq 1\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*101·13.&\supset \vdash .\Lambda \in 0 &\qquad \text{(1)}\\ +\vdash .*101·301. &\supset \vdash :\alpha \in 2.\supset .\exists !\alpha :\\ +[*24·63] &\supset \vdash .\Lambda {\sim}\in 2 &\qquad \text{(2)}\\ +\vdash .(1).(2).*13·14. &\supset \vdash .2\neq 0 &\qquad \text{(3)}\\ +\vdash .*52·22.*54·26.*22·56. &\supset \vdash .\iota ʻy\in 1.\iota ʻy{\sim}\in 2.\\ +[*13·14]&\supset \vdash .1\neq 2 &\qquad \text{(4)}\\ +\vdash .(3).(4).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*101·35.</b> \(\vdash .2\cap 0=\Lambda .2\cap 1=\Lambda \quad[*100·42.\text{Transp}.*101·11·21·32·34]\)</p> + +<p class="nind"><b>*101·36.</b> \(\vdash :\alpha \in 2.\beta \subset \alpha .\beta \neq \alpha .\supset .\beta \in 0\cup 1\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*54·42. &\supset \vdash :\alpha \in 2.\beta \subset \alpha .\exists !\beta .\beta \neq \alpha .\supset .\beta \in 1 &\qquad \text{(1)}\\ +\vdash .*54·102. &\supset \vdash :{\sim}\exists !\beta .\supset .\beta \in 0 &\qquad \text{(2)}\\ +\vdash .(1).(2).&\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_23">[Pg 23]</span></p> + +<p class="nind"><b>*101·37.</b> \(\vdash . sʻ\text{Cl}ʻʻ2 \subset 0\cup 1\cup 2 \quad[*54·411]\)</p> + +<p class="nind"><b><a id="*101·38">*101·38</a>.</b> \(\vdash : \exists ! 2 . \supset . sʻ\text{Cl}ʻʻ2 = 0\cup 1\cup 2\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*60·3. \supset \vdash : \text{Hp} . &\supset . (\exists \alpha) . \alpha \in 2 . \Lambda \in \text{Cl}ʻ\alpha.\\ +[*40·4] &\supset . \Lambda \in sʻ\text{Cl}ʻʻ2.\\ +[*51·2] &\supset . 0 \subset sʻ\text{Cl}ʻʻ2 &\qquad\text{(1)}\\ +\vdash .*60·34.&\supset \vdash . 2 \subset sʻ\text{Cl}ʻʻ2 &\qquad\text{(2)}\\ +\vdash .*54·101. \supset \vdash \colon\colon \text{Hp} . &\supset \colon\ldotp (\exists x,y) . x \neq y \colon\ldotp \\ +[*13·171.\text{Transp}] &\supset \colon\ldotp (\exists x,y) \colon\ldotp (z) : z\neq x . \lor . z\neq y\colon\ldotp\\ +[*54·26] &\supset \colon\ldotp (\exists x,y) \colon\ldotp (z) : \iota ʻz \cup \iota ʻx \in 2 . \lor . \iota ʻz \cup \iota ʻy \in 2\colon\ldotp\\ +[*11·26.*22·58] &\supset \colon\ldotp (z) \colon\ldotp (\exists \alpha ,\beta ) : \alpha \in 2 . \iota ʻz \in \text{Cl}ʻ\alpha . \lor . \beta \in 2 . \iota ʻz \in \text{Cl}ʻ\beta \colon\ldotp \\ +[*40·4] &\supset \colon\ldotp (z) . \iota ʻz \in sʻ\text{Cl}ʻʻ2\colon\ldotp \\ +[*52·1] &\supset \colon\ldotp 1 \subset sʻ\text{Cl}ʻʻ2 &\qquad \text{(3)}\\ +\vdash .(1).(2).(3).*101·37 . \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*101·4.</b> \(\vdash : (\exists x,y) . x\neq y . \equiv . \exists !2\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *54·26 . &\supset \vdash : x\neq y . \supset . \exists !2:\\ +[*11·11·35] &\supset \vdash : (\exists x,y) . x\neq y . \supset . \exists !2 &\qquad \text{(1)}\\ +\vdash .*54·101. &\supset \vdash : \alpha \in 2 . \supset . (\exists x,y) . x\neq y:\\ +[*10·11·23] &\supset \vdash : \exists !2 . \supset . (\exists x,y) . x\neq y &\qquad \text{(2)}\\ +\vdash .(1).(2). &\supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p>When we are considering the lowest type occurring in a context, our +premisses do not suffice to prove \((\exists x,y).x\neq y\). For every +other type, this can be proved. Thus \(\Lambda \neq \text{V}\) and +\(\dot{\Lambda} \neq \dot{\text{V}}\) give the required result for +classes and relations respectively.</p> + +<p class="nind"><b>*101·41.</b> \(\vdash : (\exists x) . \iota ʻx\neq \text{V}. \equiv . \exists !2\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*24·14.\text{Transp}. \supset \\ +\vdash \colon\ldotp (\exists x) . \iota ʻx\neq \text{V} . &\equiv : (\exists x) : (\exists y) . y {\sim}\in \iota ʻx :\\ +[*51·15] &\equiv : (\exists x,y) . x\neq y:\\ +[*101·4] &\equiv : \exists !2 \colon\ldotp \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*101·42">*101·42</a>.</b> \(\vdash . \exists ! 2_{\text{Cls}} . \iota ʻ\Lambda \cup \iota ʻ\text{V} \in 2_{\text{Cls}}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*20·41.*24·1. &\supset \vdash . \Lambda , \text{V} \in \text{Cls} . \Lambda \neq \text{V} &\qquad \text{(1)}\\ +\vdash .(1).*54·26. &\supset \vdash . \iota ʻ\Lambda \cup \iota ʻ\text{V} \in 2 . \iota ʻ\Lambda \cup \iota ʻ\text{V} \subset \text{Cls}.\\ +[*63·371·105] &\supset \vdash . \iota ʻ\Lambda \cup \iota ʻ\text{V} \in 2 \cap tʻ\text{Cls}.\\ +[(*65·01)] &\supset \vdash . \iota ʻ\Lambda \cup \iota ʻ\text{V} \in 2_{\text{Cls}} . \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*101·43">*101·43</a>.</b> \(\vdash . \exists ! 2_{\text{Rel}} \quad[\text{Proof as in *101·42}]\)</p> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_24">[Pg 24]</span></p> +<h2 class="nobreak" id="*102"> +*102. ON CARDINAL NUMBERS OF ASSIGNED TYPES.</h2> +</div> + + +<p class="nind"><i>Summary of</i> *102.</p> + +<p>In this number, we shall consider a typically definite relation +"\(\text{Nc}\)," <i>i.e.</i> we shall consider the relation, to a +class \(\delta\) which is given as of the same type as \(\beta\), of +the class \(\mu\) of those classes \(\gamma\) which are similar to +\(\delta\) and of the same type as \(\alpha\). We shall then put +\[ +\begin{aligned} +\mu &= \text{Nc}(\alpha _{\beta })ʻ\delta ,\\ +\gamma &\in \text{Nc}(\alpha _{\beta })ʻ\delta ,\\ +\gamma &\,\text{ sm }_{(\alpha ,\beta )}\delta , +\end{aligned} +\] +and the class of all such numbers as \(\mu\) for a given \(\alpha\) and +\(\beta\) we shall call \(\text{NC}^{\beta }(\alpha)\), so that +\[ +\text{NC}^{\beta }(\alpha ) = \text{D}ʻ\text{Nc}(\alpha _{\beta }). +\]</p> + +<p>The notations here introduced for giving typical definiteness to +"\(\text{sm}\)" and "\(\text{Nc}\)" are those defined in *65 for any +typically ambiguous relation.</p> + +<p>By *63·01·02 we have, if \(\alpha\) is a typically ambiguous symbol, +\[ +\begin{aligned} +&\vdash .\alpha _{x} = \alpha \cap tʻx,\\ +&\vdash .\alpha (x) = \alpha \cap tʻ\iota ʻx. +\end{aligned} +\]</p> + +<p>Thus \(\vdash .\alpha (x) = \alpha _{\iota ʻx}\). If we apply the +definitions to 1, "\(1_{x}\)" is meaningless unless \(x\) is a class; +we therefore write a Greek letter in place of \(x\), and we have +\[ +\vdash .1_{\beta } = 1\cap tʻ\beta = 1\cap (\iota ʻ\beta \cup -\iota ʻ\beta ). +\]</p> + +<p>If \(x\in \beta\), we shall have \(\iota ʻx = \beta .\lor.\iota ʻx\neq\beta\). Hence +\[ +\begin{aligned} +&\vdash :x\in \beta .\supset .\iota ʻx\in 1_{\beta }.\\ +\text{Similarly}\qquad &\vdash :x{\sim}\in \beta .\supset .\iota ʻx\in 1_{\beta }.\\ +\text{Thus}\qquad &\vdash :x\in t_{0}ʻ\beta .\supset .\iota ʻx\in 1_{\beta }. +\end{aligned} +\]</p> + +<p>The converse implication also holds, so that +\[ +\vdash :x\in t_{0}ʻ\beta .\equiv .\iota ʻx\in 1_{\beta }. +\]</p> + +<p>Thus \(1_{\beta}\) consists of all unit classes whose sole members +\(x\) either are or are not members of \(\beta\), <i>i.e.</i> for which +"\(x\in \beta\)" is significant.</p> + +<p>In "\(x\in t_{0}ʻ\beta .\supset .\iota ʻx\in 1_{\beta }\)," the +hypothesis renders explicit the condition of significance; thus +"\(\iota ʻx\in 1_{\beta }\)" is always true when significant, and +always significant when \(x\in t_{0}ʻ\beta\). On the interpretation of +negative statements concerning types, see the note at the end of this +number.</p> + +<p>It should be noted that all the constant relations introduced in this +work are typically ambiguous. Consider <i>e.g.</i> \(\dot{\Lambda}\), +\(\text{sg}\), \(\text{D}\), \(s\), \(\dot{s}\), \(I\), \(\iota\), +\(\in\), \(\text{Cl}\), \(\text{Rl}\). These<span class="pagenum" id="Page_25">[Pg 25]</span> all have more or +less typical ambiguity, though all of them have what we will call +<i>relative</i> typical definiteness, <i>i.e.</i> when the type +of the relatum is given, that of the referent is given also. (In +regard to \(\text{D}\), it is not true that, conversely, when the +type of the referent is given, that of the relatum is also given.) +But "\(\text{sm}\)" and "\(\text{Nc}\)" have not even relative +definiteness. When the type of the relatum is given, that of the +referent becomes no more definite than before; the only restrictions +are that the relatum for "\(\text{sm}\)" or "\(\text{Nc}\)" must be +a class, that the referent for "\(\text{sm}\)" must be a class, and +that the referent for "\(\text{Nc}\)" must be a class of classes. When +a relation \(R\) has relative definiteness, it is enough to fix the +type of the relatum; and if further \(R\in 1\rightarrow \text{Cls}\), +so that \(R\) leads to a descriptive function, "\(Rʻy\)" has complete +typical definiteness as soon as the type of \(y\) is given. Now +the constant relations hitherto introduced, with the exception of +"\(\text{sm}\)" and "\(\dot{\text{V}}\)," have all been one-many +relations, and have been used almost exclusively in the form of +descriptive functions. Hence no special notation has been required to +give typical definiteness, since "\(Rʻy\)" in these circumstances, +has typical definiteness as soon as \(y\) is assigned. But with the +consideration of "\(\text{sm}\)" and "\(\text{Nc}\)," which do not +have even relative definiteness, an explicit means of giving typical +definiteness becomes necessary. It should be observed, however, that +"\(\text{Nc}ʻ\delta\)" has typical definiteness, when \(\delta\) is +known, as soon as the <i>domain</i> of "\(\text{Nc}\)" has typical +definiteness, since \(\delta\) must belong to the converse domain. +It is for the sake of this and similar cases that we introduced the +two definitions in *65, which only give typical definiteness to the +<i>domain</i>.</p> + +<p>In virtue of the definitions in *65, if \(R\) is a typically ambiguous +relation, and \(x\) is a referent, \(R\) becomes \(R_{x}\); if, +further, \(y\) is a relatum, \(R\) becomes \(R_{(x,y)}\). If \(x\) is a +referent for \(R\), we have \((\exists y).x\in \overrightarrow{R}ʻy\), +and \(\overrightarrow{R}ʻy\in \text{D}ʻ\overrightarrow{R}\). Thus +\(\text{D}ʻ\overrightarrow{R}\) has a member of the type next above +that of \(x\), <i>i.e.</i> of the type of \(\iota ʻx\). Thus +\[ +\begin{aligned} +&\vdash .\text{sg}ʻ(R_{x}) = (\overrightarrow{R})(x)\\ +\text{and}\qquad &\vdash .\text{sg}ʻ{R_{(x,y)}} = (\overrightarrow{R})(x_{y}) +\end{aligned} +\] +as was proved in *65. Hence in particular +\[ +\vdash .\text{sg}ʻ{\text{ sm }_{(\alpha ,\beta )}} = \text{Nc}(\alpha _{\beta }). +\]</p> + +<p>It is chiefly for this reason that it is worth while to introduce the +definition of \(R(x_{y})\).</p> + +<p>We have, in virtue of the above, as will be proved in <a href="#*102·46">*102·46</a>, +\[ +\vdash : \gamma \in tʻ\alpha .\delta \in tʻ\beta .\gamma\,\text{ sm }\,\delta . \equiv .\gamma \in \text{Nc}(\alpha _{\beta })ʻ\delta . +\]</p> + +<p>With regard to "\(\text{Nc}(\alpha)\)," which is to be interpreted by +*65·04, some caution is necessary. This will mean <i>some one</i> of +those typically different relations called "\(\text{Nc}\)" which have +their domains composed of terms of the same type as \(\alpha\). But +it will not mean the logical sum of all such relations, because these +relations are of different types according as their converse<span class="pagenum" id="Page_26">[Pg 26]</span> domains +differ in type, and therefore their logical sum is meaningless. Thus +for example if the type of \(\beta\) is lower than or equal to that of +\(\alpha\), we shall have +\[ +\vdash .\exists !\text{Nc}(\alpha )ʻ\beta , +\] +whence, if "\(\text{Nc}(\alpha)\)" has its converse domain composed of +terms of the same type as \(\beta\), +\[ +\vdash .\Lambda {\sim}\in \text{D}ʻ\text{Nc}(\alpha ). +\] +But if \(\beta\) is of higher type than \(\alpha\), we shall find +\[ +\vdash .\Lambda \in \text{D}ʻ\text{Nc}(\alpha). +\] +Thus "\(\text{Nc}(\alpha)\)" is indeterminate in a way that makes a +practical difference.</p> + +<p>Exactly similar remarks apply to \(\text{NC}(\alpha)\). We have +\[ +\vdash .\text{NC}(\alpha ) = \text{D}ʻ\text{Nc}(\alpha ); +\] +thus "\(\text{NC}(\alpha)\)" shares the ambiguity of +"\(\text{Nc}(\alpha)\)." The question whether \(\Lambda \in\text{NC}(\alpha)\) +depends upon the decision of this ambiguity. The +difficulty is that "\(\text{NC}(\alpha)\)" stands for the domain of +any one determination of "\(\text{Nc}\)" which has its domain composed +of objects of the type of \(\iota ʻ\alpha\); but it is the domain +of <i>only one</i> such determination of "\(\text{Nc}\)," because +different determinations are of different types, and therefore cannot +be taken together, even when their domains are all of the same type. +In consequence of this ambiguity, "\(\text{NC}(\alpha)\)" is a symbol +which is as a rule better avoided, and "\(\text{Nc}(\alpha)\)" is +not often useful except as a descriptive function, in which case the +relatum supplies the requisite typical definiteness.</p> + +<p>The peculiarity of "\(\text{NC}(\alpha)\)" is that it is +<i>typically</i> definite, and yet is capable of different meanings: +it is not <i>wholly</i> definite, being defined as the domain of a +relation whose converse domain is typically ambiguous. It results +that we cannot profitably make "\(\text{NC}\)" half-definite, as +"\(\text{NC}(\alpha)\)" does, but must make it completely definite, +as we do by taking \(\text{D}ʻ\text{Nc}(\alpha_{\beta})\). For this +we adopt the notation \(\text{NC}^\beta (\alpha)\). We cannot adopt +the notation \(\text{NC}(\alpha_{\beta})\), because that would +conflict with *65·11, nor \(\text{NC}(\alpha)_{\beta}\), because that +would conflict with *65·01, nor \(\text{NC}_{\beta}(\alpha)\), for +the same reason. But \(\text{NC}^\beta(\alpha)\) has no previously +defined meaning. We may if we like regard "\(\text{NC}^\beta\)" as +\(\text{D}ʻ(\text{Nc}\upharpoonright tʻ\beta)\). Then the required +meaning of "\(\text{NC}^\beta(\alpha)\)" would result from *65·04. But +as "\(\text{NC}^\beta\)" so defined is not required, it is simpler to +regard "\(\text{NC}^\beta(\alpha)\)" as a single symbol. We therefore +put</p> + +<p class="nind"><b>*102·01.</b> \(\text{NC}^\beta (\alpha ) = \text{D}ʻ\text{Nc}(\alpha _{\beta }) \quad\text{Df}\)</p> + +<p>The present number begins with various propositions (<a href="#*102·2">*102·2</a>—<a href="#*102·27">·27</a>) +on a typically definite relation of similarity, <i>i.e.</i> +\(\text{ sm }_{(\alpha,\beta)}\). We then have a set of propositions +(<a href="#*102·3">*102·3</a>-<a href="#*102">·46</a>) on "\(\text{Nc}(\alpha_{\beta})ʻ\delta\)." This is only +significant if \(\beta\) and \(\delta\) are of the same type; it then +denotes the class of those classes which are similar to \(\delta\) and +of the same type as \(\alpha\). We then have a set of propositions +(<a href="#*102·5">*102·5</a>-<a href="#*102·64">·64</a>) on \(\text{NC}^\beta(\alpha)\), <i>i.e.</i> on cardinals +consisting of classes of the same type as \(\alpha\) which are similar +to classes of the same type as \(\beta\). We next prove<span class="pagenum" id="Page_27">[Pg 27]</span> (<a href="#*102·71">*102·71</a>—<a href="#*102·75">·75</a>) +that no sub-class of \alpha is similar to \(\text{Cl}ʻ\alpha\), and +therefore (substituting \(t_{0}ʻ\alpha\) for \(\alpha\)) no class of +the same type as \(\alpha\) is similar to \(tʻ\alpha\), and therefore</p> + +<p class="nind"><b>*102*74.</b> \(\vdash . \Lambda \in \text{NC}^{tʻ\alpha }(\alpha)\)</p> + +<p>This proves that \(\Lambda\) is a cardinal, which is a proposition +constantly required. The remaining propositions of <a href="#*102">*102</a> are concerned +with \(\text{ sm }ʻʻ\mu\) where \(\mu\) is a typically definite cardinal.</p> + +<p>The most useful propositions in this number (apart from <a href="#*102·74">*102·74</a>) are</p> + +<p class="nind"><b>*102·3.</b> \(\vdash :\gamma\,\text{ sm }_{(\alpha,\beta)}\delta . \equiv .\gamma \in \text{Nc}(\alpha_{\beta})ʻ\delta\)</p> + +<p class="nind"><b>*102·46.</b> \(\vdash :\gamma \in \text{Nc}(\alpha _{\beta })ʻ\delta .\equiv .\delta \in \text{Nc}(\beta _{\alpha })ʻ\gamma + . \equiv .\gamma\,\text{ sm }\,\delta .\gamma \in tʻ\alpha .\delta \in tʻ\beta \)</p> + +<p class="nind"><b>*102·5.</b> \(\vdash :\mu \in \text{NC}^\beta (\alpha ). \equiv .(\exists \delta ).\mu = \text{Nc}(\alpha _{\beta })ʻ\delta \)</p> + +<p class="nind"><b>*102·6.</b> \(\vdash .\text{Nc}(\alpha )ʻ\beta = \text{Nc}(\alpha _{\beta })ʻ\beta + = \hat{\gamma} (\gamma\,\text{ sm }\,\beta .\gamma \in tʻ\alpha ) = \text{Nc}ʻ\beta \cap tʻ\alpha \)</p> + +<p class="nind"><b>*102·72.</b> \(\vdash :\beta \subset \alpha .\supset .{\sim}(\beta \text{ sm } \text{Cl}ʻ\alpha)\)</p> + +<p>This is used in proving \(\mu \in \text{NC}.\supset .2^\mu >\mu\), +which is the proposition from which Cantor deduced that there is +no greatest cardinal. (If \(\mu = \text{Nc}ʻ\alpha\), \(2^\mu =\text{Nc}ʻ\text{Cl}ʻ\alpha\), +and thus there is a rise of type.)</p> + +<p class="nind"><b>*102·84.</b> \(\vdash :(\exists \gamma ).\gamma \in tʻ\alpha .\gamma \text{ sm } \alpha .\delta \text{ sm } \gamma . \equiv .\delta \text{ sm } \alpha\)</p> + +<p class="nind"><b>*102·85.</b> \(\vdash .\text{ sm }ʻʻ\mu \cap tʻ\beta = \text{ sm }_{\beta }ʻʻ\mu \)</p> + +<hr class="tb"> + +<p class="nind"><b>*102·01.</b> \(\text{NC}^\beta (\alpha ) = \text{D}ʻ\text{Nc}(\alpha _{\beta }) \quad\text{Df}\)</p> + +<p class="nind"><b>*102·11.</b> \(\vdash :R \in 1\rightarrow 1.\supset .R_{(x,y)} \in 1(x)\rightarrow 1(y)\)</p> + +<p>Here, if \(R\) is a real variable, the conditions of significance +require \(R = R_{(x,y)}\). But if \(R\) is a typically ambiguous +constant, such as \(\iota\) or \(\dot{\Lambda}\) or \(\text{sg}\), +\(R_{(x,y)}\) is a typically definite constant. It is chiefly for such +cases that propositions such as the above are useful.</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*37·402.(*65·1).&\supset \vdash .\text{D}ʻR_{(x,y)}\subset tʻx.\\ +[*33·15]&\supset \vdash .\{\text{sg}ʻR_{(x,y)}\}ʻz\subset tʻx.\\ +[*63·5]&\supset \vdash .\{\text{sg}ʻR_{(x,y)}\}ʻz \in tʻtʻx &\qquad \text{(1)}\\ +\vdash .(1).*71·102.\supset \vdash :\text{Hp}.z \in \text{ᗡ}ʻR_{(x,y)}.&\supset .\{\text{sg}ʻR_{(x,y)}\}ʻz \in 1\cap tʻtʻx.\\ +[(*65·02)] &\supset .\{\text{sg}ʻR_{(x,y)}\}ʻz \in 1(x) &\qquad \text{(2)}\\ +\text{Similarly}\qquad &\vdash : \text{Hp}.\omega \in \text{D}ʻR_{(x,y)}.\supset .\{gsʻR_{(x,y)}\}ʻ\omega \in 1(y) &\qquad \text{(3)}\\ +\vdash .(2).(3).*70·1.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*102·13.</b> \(\vdash :R \in 1\rightarrow 1.\supset .R_{x} \in 1(x)\rightarrow 1 \quad[\text{Proof as in *102·11}]\)</p> + +<p class="nind"><b><a id="*102·2">*102·2</a>.</b> \(\vdash :\gamma \text{ sm }_{(\alpha ,\beta )}\delta . \equiv .\gamma \text{ sm } \delta .\gamma \in tʻ\alpha .\delta \in tʻ\beta \quad[*35·102.(*65·1)]\)</p> + +<p class="nind"><b>*102·21.</b> \[\begin{aligned}\vdash :\gamma \text{ sm }_{(\alpha ,\beta )}&\delta . \equiv .(\exists R).R \in 1\rightarrow 1.\text{D}ʻR \in tʻ\alpha .\\ +&\text{ᗡ}ʻR \in tʻ\beta .\text{D}ʻR = \gamma .\text{ᗡ}ʻR = \delta \quad[*102·2.*73·1]\end{aligned}\]</p> + +<p><span class="pagenum" id="Page_28">[Pg 28]</span></p> + +<p class="nind"><b>*102·22.</b> \(\vdash :\gamma \text{ sm }(x,y)\delta . \equiv .\gamma \text{ sm } \delta .\gamma \subset tʻx.\delta \subset tʻy \quad[*63·5.(*65·12)]\)</p> + +<p><b>102·23.</b> \[\begin{aligned}\vdash :\gamma \text{ sm }(x,y)&\delta . \equiv .(\exists R).R \in 1\rightarrow 1.\text{D}ʻR\subset tʻx.\\ +&\text{ᗡ}ʻR\subset tʻy.\text{D}ʻR = \gamma .\text{ᗡ}ʻR = \delta \quad[*102·22.*73·1]\end{aligned}\]</p> + +<p class="nind"><b>*102·24.</b> \(\vdash :\gamma \text{ sm } (x,y) \delta . \equiv .(\exists R).R \in 1(x)\rightarrow 1(y).\text{D}ʻR = \gamma .\text{ᗡ}ʻR = \delta \)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*102·23.*40·5·52·43.*37·25.\supset \\ +\vdash \colon\ldotp \gamma \text{ sm }v(x,y)\delta . &\equiv :(\exists R):R \in 1\rightarrow 1 : w \in \text{ᗡ}ʻR.\supset _{w}\overrightarrow{R}ʻw\subset tʻx:\\ +&z \in \text{D}ʻR.\supset _{z}.\overleftarrow{R}ʻz\subset tʻy : \text{D}ʻR = \gamma .\text{ᗡ}ʻR = \delta :\\ +[*63·5] &\equiv :(\exists R):R \in 1\rightarrow 1 . \overleftarrow{R}ʻʻ\text{ᗡ}ʻR\subset tʻtʻx.\overleftarrow{R}ʻʻ\text{D}ʻR\subset tʻtʻy.\\ +&\text{D}ʻR = \gamma .\text{ᗡ}ʻR = \delta :\\ +[*71·102.(*65·02)] &\equiv :(\exists R).\overrightarrow{R}ʻʻ\text{ᗡ}ʻR\subset 1(x).\overleftarrow{R}ʻʻ\text{D}ʻR\subset 1(y).\text{D}ʻR = \gamma .\text{ᗡ}ʻR = \delta :\\ +[*70·1] &\equiv :(\exists R).R \in 1(x)\rightarrow 1(y).\text{D}ʻR = \gamma .\text{ᗡ}ʻR = \delta \colon\ldotp \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*102·25.</b> \[\begin{aligned}&\vdash :\gamma \text{ sm }_{(\alpha ,\beta )} \delta . \equiv .(\exists R).R \in 1_{\alpha }\rightarrow 1_{\beta }.\text{D}ʻR + = \gamma .\text{ᗡ}ʻR = \delta\\ +&[\text{Proof as in *102·24}]\end{aligned}\]</p> + +<p class="nind"><b>*102·26.</b> \(\vdash :\gamma \text{ sm }_{(\alpha ,\beta )} \delta .\gamma'\text{ sm }_{(\alpha ,\beta )} + \delta .\supset .\gamma \text{ sm }_{(\alpha ,\alpha )}\gamma'\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*102·2. \supset \vdash : \text{Hp}.&\supset .\gamma \text{ sm } \delta .\gamma'\text{ sm } \delta .\gamma ,\gamma'\in \tau ʻ\alpha .\\ +[*73·32] &\supset .\gamma \text{ sm } \gamma'.\gamma ,\gamma'\in \tau ʻ\alpha .\\ +[*102·2] &\supset .\gamma \text{ sm }_{(\alpha ,\alpha )}\gamma':\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*102·27">*102·27</a>.</b> \(\vdash :\gamma \text{ sm }_{(\alpha ,\beta )} \delta .\gamma' \text{ sm }_{(\alpha ʻ,\beta )} \delta .\supset .\gamma \text{ sm }_{(\alpha ,\alpha ʻ)}\gamma' + \quad[\text{Proof as in *102·26}]\)</p> + +<p class="nind"><b><a id="*102·3">*102·3</a>.</b> \(\vdash :\gamma \text{ sm }_{(\alpha ,\beta )} \delta . \equiv .\gamma \in \text{Nc}{(\alpha _\beta )}ʻ\delta \)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*32·18.\supset \\ +\vdash :\gamma \text{ sm }_{(\alpha _\beta )} \delta . &\equiv .\gamma \in {\text{sg}ʻ\text{ sm }_(\alpha _{\beta })}ʻ\delta .\\ +[*65·2] &\equiv .\gamma \in {(\text{sg}ʻ\text{ sm })(\alpha _{\beta })}ʻ\delta .\\ +[(*100·01)] &\equiv \gamma \in \text{Nc}(\alpha _{\beta })ʻ\delta :\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*102·31.</b> \(\vdash .\text{Nc} (\alpha _{\beta })ʻ\delta = \text{D}ʻʻ\{1\rightarrow 1\cap \hat{R} (\text{D}ʻR \in tʻ\alpha .\text{ᗡ}ʻR \in tʻ\beta .\text{ᗡ}ʻR = \delta )\}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*102·3·21.\supset\\ +\vdash :\gamma \in \text{Nc}(\alpha _{\beta })ʻ\delta + . &\equiv .(\exists R).R \in 1\rightarrow 1.\text{D}ʻR \in tʻ\alpha .\text{ᗡ}ʻR \in tʻ\beta .\text{D}ʻR = \gamma .\text{ᗡ}ʻR = \delta .\\ +[*33·123.*37·1] &\equiv .\gamma \in \text{D}ʻʻ\{1\rightarrow 1\cap \hat{R} (\text{D}ʻR \in tʻ\alpha . \text{ᗡ}ʻR \in tʻ\beta .\text{ᗡ}ʻR = \delta )\}:\\ +\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_29">[Pg 29]</span></p> + +<p class="nind"><b>*102·32.</b> \(\vdash . \text{Nc}(\alpha _{\beta })ʻ\delta = \text{D}ʻʻ\{(1_{\alpha }\rightarrow 1_{\beta }) \cap \overleftarrow{\text{ᗡ}}ʻ\delta\}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*102·3·25. \supset \\ +\vdash : \gamma \in \text{Nc}(\alpha _{\beta })ʻ\delta . &\equiv . (\exists R) . R \in 1_{\alpha }\rightarrow 1_{\beta } + . \text{D}ʻR = \gamma . \text{ᗡ}ʻR = \delta .\\ +[*33·61] &\equiv . (\exists R) . R \in 1_{\alpha }\rightarrow 1_{\beta } . R \in \overleftarrow{\text{ᗡ}}ʻ\delta . \text{D}ʻR = \gamma .\\ +[*33·123.*37·1] &\equiv . \gamma \in \text{D}ʻʻ{(1_{\alpha }\rightarrow 1_{\beta }) \cap \overleftarrow{\text{ᗡ}}ʻ\delta } : \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*102·34.</b> + \[\begin{aligned}\vdash . \text{Nc}(\alpha ,\beta )ʻ\delta = &\text{D}ʻʻ\{1 \rightarrow 1 \cap \hat{R} (\text{D}ʻR \in tʻ\alpha . \text{ᗡ}ʻR \subset tʻ\beta . \text{ᗡ}ʻR = \delta )\}\\ +&[\text{Proof as in *102·31}]\end{aligned}\]</p> + +<p class="nind"><b>*102·35.</b> \(\vdash . \text{Nc}(\alpha ,\beta )ʻ\delta = \text{D}ʻʻ[{1_{\alpha } + \rightarrow 1(\beta )} \cap \overleftarrow{\text{ᗡ}}ʻ\delta ] \quad[\text{Proof as in *102·32}]\)</p> + +<p class="nind"><b>*102·36.</b> \(\vdash . \text{E}! \text{Nc}(\alpha _{\beta })ʻ\delta \quad[*102·31.*14·21]\)</p> + +<p>This proposition is true whenever it is significant, and is significant +whenever \(\delta \in tʻ\beta\). When \(\delta\) belongs to some other +type, the above proposition is not significant.</p> + +<p class="nind"><b>*102·361.</b> \(\vdash . \text{E}! \text{Nc}(\alpha , \beta )ʻ\delta \quad[*102·34.*14·21]\)</p> + +<p class="nind"><b>*102·37.</b> \(\vdash . \text{ᗡ}ʻ\text{Nc}(\alpha _{\beta }) = tʻ\beta\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*37·402.(*65·11). &\supset \vdash . \text{ᗡ}ʻ\text{Nc}(\alpha _{\beta }) \subset tʻ\beta &&\qquad \text{(1)}\\ +\vdash .*102·36.*33·43 . &\supset \vdash . (\delta ) . \delta \in \text{ᗡ}ʻ\text{Nc}(\alpha _{\beta }). +[*63·14] &\supset \vdash . t_{0}ʻ\text{ᗡ}ʻ\text{Nc}(\alpha _{\beta }) = \text{ᗡ}ʻ\text{Nc}(\alpha _{\beta }) &\qquad \text{(2)}\\ +\vdash .(1).*63·21. &\supset \vdash .t_{0}ʻ\text{ᗡ}ʻ\text{Nc}(\alpha _{\beta }) = tʻ\beta &&\qquad \text{(3)}\\ +\vdash .(2).(3). \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*102·4.</b> \(\vdash : \gamma \in \text{Nc}(\alpha _{\beta })ʻ\delta . \gamma' \in \text{Nc}(\alpha _{\beta })ʻ\delta + . \supset . \gamma \in \text{Nc}(\alpha _{\alpha })ʻ\gamma'\quad[*102·3·26]\)</p> + +<p class="nind"><b>*102·41.</b> \(\vdash : \gamma \in \text{Nc}(\alpha _{\beta })ʻ\delta . \gamma'\in \text{Nc}(\alpha _{\beta }ʻ)ʻ\delta + . \supset . \gamma \in \text{Nc}(\alpha _{\alpha ʻ})ʻ\gamma' \quad[*102·3·27]\)</p> + +<p class="nind"><b>*102·42.</b> \(\vdash . \alpha \in \text{Nc}(\alpha _{\alpha })ʻ\alpha \quad[*102·3·2.*73·3.*63·103]\)</p> + +<p class="nind"><b>*102·43.</b> \(\vdash . \exists ! \text{Nc}(\alpha _{\alpha })ʻ\alpha \quad[*102·42]\)</p> + +<p>This inference is legitimate because, when \(\alpha\) is given, +"\(\text{Nc}(\alpha _{\alpha })ʻ\alpha\)" is typically definite. The +inference from "\(\alpha \in \text{Nc}ʻ\alpha\)" (which is true) to +"\(\exists !\text{Nc}ʻ\alpha\)" is not valid, because "\(\exists!\text{Nc}ʻ\alpha\)" +may hold only for <i>some</i> of the possible +determinations of the ambiguity of "\(\text{Nc}\)."</p> + +<p class="nind"><b>*102·44.</b> \(\vdash : \alpha \text{ sm } \beta . \equiv . \alpha \in \text{Nc}(\alpha _{\beta })ʻ\beta . \equiv . \beta \in \text{Nc}(\beta _{\alpha })ʻ\alpha\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +&\vdash .*63·102 . \supset\\ +&\vdash : \alpha \text{ sm } \beta . \equiv . \alpha \text{ sm } \beta . \alpha \in tʻ\alpha . \beta \in tʻ\beta &\qquad \text{(1)}\\ +&\vdash .(1).*102·2·3 . \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_30">[Pg 30]</span></p> + +<p class="nind"><b>*102·45.</b> \(\vdash :\gamma \in \text{Nc}(\alpha _{\beta })ʻ\delta .\supset .\gamma \in \text{Nc}(\alpha _{\alpha })ʻ\gamma \)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +&\vdash .*102·3·2. \supset \vdash :\text{Hp}.\supset .\gamma \in tʻ\alpha &\qquad \text{(1)}\\ +&\vdash .*73·3. \supset \vdash .\gamma \text{ sm } \gamma &\qquad \text{(2)}\\ +&\vdash .(1).(2).*102·3·2.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*102·46">*102·46</a>.</b> \[\begin{aligned}&\vdash :\gamma \in \text{Nc}(\alpha _{\beta })ʻ\delta . \equiv .\delta \in \text{Nc}(\beta _{\alpha }ʻ\gamma + . \equiv .\gamma \text{ sm } \delta .\gamma \in tʻ\alpha.\delta \in tʻ\beta \\ +&[*102·2·3.*73·31]\end{aligned}\]</p> + +<p class="nind"><b><a id="*102·5">*102·5</a>.</b> \(\vdash :\mu \in \text{NC}^\beta (\alpha ). \colon\ldotp .(\exists \delta ).\mu = \text{Nc}(\alpha _{\beta })ʻ\delta \quad[*100·22.*71·41.(*102*01)]\)</p> + +<p>In using propositions, such as those of <a href="#*100">*100</a>, in which we have a +typically ambiguous "\(\text{Nc}\)" or "\(\text{NC}\)," any significant +typical definiteness may be added, since, when a typically ambiguous +proposition is asserted, that includes the assertion of every possible +proposition resulting from determining the ambiguity.</p> + +<p class="nind"><b>*102·501.</b> \(\vdash .\text{Nc}(\alpha _{\beta }ʻ\delta \in \text{NC}^\beta (\alpha ) \quad[*102·5·36]\)</p> + +<p class="nind"><b>*102·51.</b> \[\begin{aligned}\vdash :\gamma \in \text{Nc}(\alpha _{\beta })ʻ\delta .\supset .&\text{Nc}((\alpha _{\beta })ʻ\delta = \text{Nc}(\alpha _{\alpha })ʻ\gamma + .\\ +&\text{Nc}(\alpha _{\beta })ʻ\delta \in \text{NC}^\beta (\alpha ).\text{Nc}(\alpha _{\alpha })ʻ\gamma \in \text{NC}^\alpha (\alpha )\end{aligned}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*102·3·2.\supset \\ +\vdash \colon\ldotp \text{Hp}.&\supset :\gamma \text{ sm } \delta .\gamma \in tʻ\alpha .\delta \in tʻ\beta :\\ +[*73·37.*4·73] \supset :\xi \text{ sm } \delta . \equiv .&\xi \text{ sm } \gamma :\xi \text{ sm } \delta . \equiv .\xi \text{ sm } \delta .\delta \in tʻ\beta :\\ +&\xi \text{ sm } \gamma . \equiv .\xi \text{ sm } \gamma .\gamma \in tʻ\alpha :\\ +[*4·22] &\supset :\xi \text{ sm } \delta .\delta \in tʻ\beta . \equiv .\xi \text{ sm } \gamma .\gamma \in tʻ\alpha :\\ +[\text{Fact}] &\supset :\xi \text{ sm } \delta .\xi \in tʻ\alpha .\delta \in tʻ\beta . \equiv .\xi \text{ sm } \gamma .\xi \in tʻ\alpha .\gamma \in tʻ\alpha :\\ +[*102·2·3] &\supset :\text{Nc}(\alpha _{\beta })ʻ\delta = \text{Nc}(\alpha _{\alpha })ʻ\gamma &\qquad \text{(1)}\\ +\vdash .(1).*102·501 .\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*102·52.</b> \(\vdash :\exists ! \text{Nc}(\alpha _{\beta })ʻ\delta .\supset .\text{Nc}(\alpha _{\beta })ʻ\delta \in \text{NC}^\alpha (\alpha ) \quad[*102·51]\)</p> + +<p class="nind"><b>*102·53.</b> \(\vdash .\text{NC}^\beta (\alpha )-\iota ʻ\Lambda \subset \text{NC}^\alpha (\alpha )\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*102·52.\supset \vdash :\mu = \text{Nc}(\alpha _{\beta })ʻ\delta .\exists !\mu .\supset .\mu \in \text{NC}^\alpha (\alpha ) &\qquad \text{(1)}\\ +\vdash .(1).*102*5.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*102·54.</b> \(\vdash :\delta \in \text{Nc}(\beta _{\alpha })ʻ\gamma .\supset .\text{Nc}(\alpha _{\beta })ʻ\delta = \text{Nc}(\alpha _{\alpha })ʻ\gamma + \quad[*102·51·46]\)</p> + +<p class="nind"><b>*102·541.</b> \(\vdash :\exists ! \text{Nc}(\beta _{\alpha })ʻ\gamma .\supset .\text{Nc}(\alpha _{\alpha })ʻ\gamma \in \text{NC}^\beta (\alpha )-tʻ\Lambda\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *102·54·501.&\supset \vdash :\delta \in \text{Nc}(\beta_{\alpha}ʻ\gamma .\supset .\text{Nc}(\alpha _{\alpha })ʻ\gamma \in \text{NC}^\beta + (\alpha ) &\qquad \text{(1)}\\ +\vdash . *102·46·45. \supset :\vdash \delta \in \text{Nc}{(\beta_{\alpha})}ʻ\gamma .&\supset .\gamma \in \text{Nc}(\alpha _{\alpha })ʻ\gamma .\\ +[*10·24] &\supset .\exists !\text{Nc}(\alpha_{\alpha})ʻ\gamma &\qquad \text{(2)}\\ +\vdash .(1).(2).\supset \\ +\vdash :\delta \in \text{Nc}(\beta_{\alpha})ʻ\gamma .&\supset .\text{Nc}(\alpha_{\alpha})ʻ\gamma \in \text{NC}^\beta (\alpha )-tʻ\Lambda :\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_31">[Pg 31]</span></p> + +<p class="nind"><b>*102·55.</b> \(\vdash : \Lambda {\sim}\in \text{NC}^\alpha (\beta ) . \supset . \text{NC}^\beta (\alpha ) - \iota ʻ\Lambda = \text{NC}^\alpha (\alpha )\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*102·5 . \supset \\ +\vdash \colon\ldotp \text{Hp} . &\supset : \mu = \text{Nc}(\beta _{\alpha })ʻ\gamma . \supset _{\mu ,\gamma } . \exists !\mu .\\ +[*102·541] &\supset _{\mu ,\gamma } . \text{Nc}(\alpha _{\alpha })ʻ\gamma \in \text{NC}^\beta (\alpha ) - \iota ʻ\Lambda :\\ +[*10·23] &\supset : (\exists \mu ) . \mu = \text{Nc}(\beta _{\alpha })ʻ\gamma . \supset _{\gamma } . \text{Nc}(\alpha _{\alpha })ʻ\gamma \in \text{NC}^\beta + (\alpha ) - \iota ʻ\Lambda :\\ +[*102·36] &\supset : (\gamma ) . \text{Nc}(\alpha _{\alpha })ʻ\gamma \in \text{NC}^\beta (\alpha ) - \iota ʻ\Lambda :\\ +[*13·191] &\supset : \nu = \text{Nc}(\alpha _{\alpha })ʻ\gamma . \supset _{\nu ,\gamma } . \nu \in \text{NC}^\beta (\alpha ) - \iota ʻ\Lambda :\\ +[*102·5] &\supset : \nu \in \text{NC}^\alpha (\alpha ) . \supset _{\nu } . \nu \in \text{NC}^\beta (\alpha ) - \iota ʻ\Lambda &\qquad \text{(1)}\\ +\vdash .(1).*102·53. \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p>The above proposition shows that, if every class of the same type as +\(\beta\) is similar to some class of the same type as \(\alpha\), +then, given a class \(\gamma\) of the same type as \(\alpha\), there +is a class \(\delta\), of the same type as \(\beta\), such that the +classes similar to \(\delta\) and of the same type as \(\alpha\) are +the same as the classes similar to \(\gamma\) and of the same type +as \(\alpha\); and conversely, given any class \(\delta\), of the +same type as \(\beta\), and similar to some class of the same type +as \(\alpha\), then there is a class \(\gamma\), of the same type +as \(\alpha\), such that the classes similar to \(\gamma\) and of +the same type as \(\alpha\) are the same as the classes similar to +\(\delta\) and of the same type as \(\alpha\). We may express this by +saying that, if the cardinals which go from the type of \(\alpha\) +to the type of \(\beta\) are never null, then those that go from the +type of \(\beta\) to the type of \(\alpha\), with the exception of +\(\Lambda\) (if \(\Lambda\) is one of them), are the same as those +that begin and end within the type of \(\alpha\). The latter are +what we call "homogeneous" cardinals. Thus our proposition is a step +towards reducing the general study of cardinals to that of homogeneous +cardinals.</p> + +<p class="nind"><b>*102·6.</b> \(\vdash . \text{Nc}(\alpha )ʻ\beta = \text{Nc}(\alpha _{\beta })ʻ\beta + = \hat{\gamma} (\gamma \text{ sm } \beta . \gamma \in tʻ\alpha ) = \text{Nc}ʻ\beta \cap tʻ\alpha\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*35·1.(*65·04). \supset \\ +\vdash : \mu = \text{Nc}(\alpha )ʻ\beta . &\equiv . \mu = \text{Nc}ʻ\beta . \mu \in t^{2}ʻ\alpha .\\ +[*63·5] &\equiv . \mu = \text{Nc}ʻ\beta . \mu \subset tʻ\alpha .\\ +[*65·13] &\equiv . \mu = \text{Nc}ʻ\beta \cap tʻ\alpha . &&\qquad \text{(1)}\\ +[*100·1] &\equiv . \mu = \hat{\gamma} (\gamma \text{ sm } \beta . \gamma \in tʻ\alpha ). &&\qquad \text{(2)}\\ +[*63·103] &\equiv . \mu = \hat{\gamma} (\gamma \text{ sm } \beta . \gamma \in tʻ\alpha . \beta \in tʻ\beta ). +[*102·46] &\equiv . \mu = \text{Nc}(\alpha _{\beta })ʻ\beta &\qquad \text{(3)}\\ +\vdash .(1).(2).(3).*20·2.*100·1. \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*102·61.</b> \(\vdash : \delta \in tʻ\beta . \supset . \text{Nc}(\alpha )ʻ\delta = \text{Nc}(\alpha _{\beta })ʻ\delta \)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*4·73 . \supset \vdash : \text{Hp} . \supset . \hat{\gamma} (\gamma \text{ sm } \delta . \gamma \in tʻ\alpha ) &= \hat{\gamma} (\gamma \text{ sm } \delta . \gamma \in tʻ\alpha . \delta \in tʻ\beta )\\ +[*102·46] &= \text{Nc}(\alpha _{\beta })ʻ\delta &\qquad \text{(1)}\\ +\vdash .(1).*102·6. \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_32">[Pg 32]</span></p> + +<p class="nind"><b>*102·62.</b> \(\vdash . \text{NC}^\beta (\alpha ) = \text{Nc}(\alpha )ʻʻtʻ\beta\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*37·7.(*100·01). \supset \\ +\vdash . \text{Nc}(\alpha )ʻʻtʻ\beta &= \hat{\mu} \{(\exists \delta ) . \delta \in tʻ\beta . \mu = \text{Nc}(\alpha )ʻ\delta\}\\ +[*102·61] &= \hat{\mu} \{(\exists \delta ) . \delta \in tʻ\beta . \mu = \text{Nc}(\alpha _{\beta})ʻ\delta\}\\ +[*102·37] &= \text{D}ʻ\text{Nc}(\alpha _{\beta })\\ +[(*102·01)] &= \text{NC}^\beta (\alpha ) . \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*102·63.</b> \(\vdash : \mu = \text{Nc}ʻ\gamma . \alpha \in \mu . \supset . \mu = \text{Nc}(\alpha )ʻ\gamma\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *63·5 . \supset \vdash : \text{Hp} . &\supset . \mu = \text{Nc}ʻ\gamma . \mu \subset tʻ\alpha .\\ +[*65·13] &\supset . \mu = \text{Nc}ʻ\gamma \cap tʻ\alpha .\\ +[*102·6] &\supset . \mu = \text{Nc}(\alpha )ʻ\gamma : \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*102·64">*102·64</a>.</b> \(\vdash : \mu \in \text{NC} . \exists !\mu . \supset . (\exists \alpha ,\gamma ) . \mu = \text{Nc}(\alpha )ʻ\gamma \quad[*102·63.*100·4]\)</p> + +<p>The following propositions are part of Cantor's proof that there +is no greatest cardinal. They are inserted here in order to enable +us to prove that \(\Lambda\) is a cardinal, namely what we call +a "descending" cardinal, <i>i.e.</i> one whose corresponding +"\(\text{ sm }\)" goes from a higher to a lower type.</p> + +<p class="nind"><b><a id="*102·71">*102·71</a>.</b> \(\vdash : R \in \text{Cls} \rightarrow 1 . \text{D}ʻR\subset \alpha . \text{ᗡ}ʻR\subset \text{Cl}ʻ\alpha . \supset . \exists ! \text{Cl}ʻ\alpha - \text{ᗡ}ʻR\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*20·33.*4·73 . \supset \\ +\vdash \colon\colon \text{Hp} . \varpi = &\hat{x} (x \in \text{D}ʻR . x {\sim}\in \breve{R}ʻx) . \supset \colon\ldotp \\ +x \in \text{D}ʻR . &\supset _{x} : x \in \varpi . \equiv . x {\sim} \in \breve{R}ʻx:\\ +[*5·18] &\supset _{x} : {\sim}\{x \in \varpi . \equiv . x \in \breve{R}ʻx\}:\\ +[*20·43.\text{Transp}.*71·164] &\supset _{x} : \varpi \neq \breve{R}ʻx\colon\ldotp \\ +[*71·411.\text{Transp}] &\supset \colon\ldotp \varpi {\sim}\in \text{ᗡ}ʻR &\qquad \text{(1)}\\ +\vdash .*20·33.*3·26 . \supset \vdash : \text{Hp}(1) . &\supset . \varpi \subset \text{D}ʻR.\\ +[\text{Hp}] &\supset . \varpi \subset \alpha &\qquad \text{(2)}\\ +\vdash .(1).(2).*13·191 . \supset \\ +\vdash : \text{Hp} . \supset . \hat{x} (x \in \text{D}ʻR . x {\sim}\in \breve{R}ʻx) \in \text{Cl}ʻ\alpha - \text{ᗡ}ʻR : \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*102·72">*102·72</a>.</b> \(\vdash : \beta \subset \alpha . \supset .{\sim}(\beta \text{ sm } \text{Cl}ʻ\alpha )\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*102·71. \supset \vdash \colon\ldotp \text{Hp} . &\supset : R \in 1 \rightarrow 1 . \text{D}ʻR = \beta . \text{ᗡ}ʻR \subset \text{Cl}ʻ\alpha . \supset _{R} + . \exists ! \text{Cl}ʻ\alpha - \text{ᗡ}ʻR:\\ +[*24·55.*22·41] &\supset : R \in 1 \rightarrow 1 . \text{D}ʻR = \beta . \supset _{R}. \text{ᗡ}ʻR \neq \text{Cl}ʻ\alpha :\\ +[*10·51] &\supset : {\sim}(\exists R) . R \in 1 \rightarrow 1 . \text{D}ʻR = \beta . \text{ᗡ}ʻR = \text{Cl}ʻ\alpha :\\ +[*73·1] &\supset : {\sim}(\beta \text{ sm } \text{Cl}ʻ\alpha ) \colon\ldotp \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_33">[Pg 33]</span></p> + +<p class="nind"><b><a id="*102·73">*102·73</a>.</b> \(\vdash . \text{Nc}(\alpha )ʻtʻ\alpha = \Lambda\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*102·6 . \supset \vdash . \text{Nc}(\alpha )ʻtʻ\alpha &= \hat{\gamma} (\gamma \text{ sm } tʻ\alpha . \gamma \in tʻ\alpha )\\ +[*63·65] &= \hat{\gamma} (\gamma \text{ sm } \text{Cl}ʻt_{0}ʻ\alpha . \gamma \subset t_{0}ʻ\alpha )\\ +[*102·72] &= \Lambda . \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p>This proposition proves that no class of the same type as \(\alpha\) +is similar to \(tʻ\alpha\). Now \(tʻ\alpha\) is the greatest class +of its type; thus there are classes of the type next above that of +\(\alpha\) which are too great to be similar to any class of the type +of \(\alpha\). Thus (as will be explicitly proved later) the maximum +cardinal in one type is less than that in the next higher type. +Cantor's proposition that there is no maximum cardinal only holds when +we are allowed to rise to continually higher types: in each type, there +is a maximum for that type, namely the number of members of the type.</p> + +<p class="nind"><b><a id="*102·74">*102·74</a>.</b> \(\vdash . \Lambda \in \text{NC}^{tʻ\alpha }(\alpha )\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*102·6·501. &\supset \vdash . \text{Nc}(\alpha )ʻtʻ\alpha \in \text{NC}^{tʻ\alpha }(\alpha ) &\qquad \text{(1)}\\ +\vdash .(1).*102·73. &\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*102·75">*102·75</a>.</b> \(\vdash . \text{NC}^{tʻ\alpha }(\alpha ) = \text{NC}^\alpha (\alpha ) \cup \iota ʻ\Lambda \)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*100·6. &\supset \vdash : \gamma \in tʻ\alpha . \supset . \gamma \in tʻ\alpha . \iota ʻʻ\gamma \in \text{Nc}ʻ\gamma &\qquad \text{(1)}\\ +\vdash .*63·64·5. &\supset \vdash : \gamma \in tʻ\alpha . \supset . \iota ʻʻ\gamma \in t^{2}ʻ\alpha &\qquad \text{(2)}\\ +\vdash .(1).(2).*102·46. \supset \vdash : \gamma \in tʻ\alpha . &\supset . \iota ʻʻ\alpha \in \text{Nc}{(tʻ\alpha )_{\alpha }}ʻ\gamma .\\ +[*10·24] &\supset . \exists ! \text{Nc}{(tʻ\alpha )_{\alpha }}ʻ\gamma &\qquad \text{(3)}\\ +\vdash .(3).*102·55. &\supset \vdash . \text{NC}^{tʻ\alpha }(\alpha ) - \iota ʻ\Lambda = \text{NC}^\alpha (\alpha ) &\qquad \text{(4)}\\ +\vdash .(4).*102·74. &\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*102·8.</b> \(\vdash : \gamma \in \text{Nc}(\alpha _{\beta })ʻ\delta . \gamma \text{ sm } \zeta . \zeta \in tʻ\xi . \supset . \zeta \in \text{Nc}(\xi _{\beta })ʻ\delta + . \zeta \in \text{Nc}(\xi _{\alpha })ʻ\gamma\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *102·46 . \supset \\ +\vdash : \text{Hp}. &\supset . \gamma \text{ sm } \delta . \gamma \in tʻ\alpha . \delta \in tʻ\beta . \gamma \text{ sm } \zeta . \zeta \in tʻ\xi .\\ +[*73·31·32] &\supset . \zeta \text{ sm } \delta . \zeta \in tʻ\xi . \delta \in tʻ\beta . \zeta \text{ sm } \gamma . \zeta \in tʻ\xi . \gamma \in tʻ\alpha .\\ +[*102·46] &\supset . \zeta \in \text{Nc}(\xi _{\beta })ʻ\delta . \zeta \in \text{Nc}(\xi _{\alpha })ʻ\gamma : \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*102·81.</b> \(\vdash : \gamma \in \text{Nc}(\alpha _{\beta })ʻ\delta . \supset . \text{ sm }ʻʻ\text{Nc}(\alpha _{\beta })ʻ\delta \cap tʻ\xi = \text{Nc}(\xi _{\beta })ʻ\delta + = \text{Nc}(\xi _{\alpha })ʻ\gamma \)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*102·8.*73·31·32 . \supset \\ +\vdash : \gamma , \gamma ʻ &\in \text{Nc}(\alpha _{\beta })ʻ\delta . \zeta \text{ sm } \gamma ʻ . \zeta \in tʻ\xi . \supset . \zeta \in \text{Nc}(\xi _{\beta })ʻ\delta + . \zeta \in \text{Nc}(\xi _{\alpha })ʻ\gamma &\qquad \text{(1)}\\ +\vdash .(1).*37·1. \supset \vdash : \text{Hp} . \supset . &\text{ sm }ʻʻ\text{Nc}(\alpha _{\beta })ʻ\delta \cap tʻ\xi \subset \text{Nc}(\xi _{\beta })ʻ\delta .\\ +&\text{ sm }ʻʻ\text{Nc}(\alpha _{\beta })ʻ\delta \cap tʻ\xi \subset \text{Nc}(\xi _{\alpha })ʻ\gamma &\qquad \text{(2)}\\ +\vdash .*102·46.*73·31·32 . \supset \\ +\vdash : \text{Hp} . \zeta \in \text{Nc}(\xi _{\beta })ʻ\delta . &\supset . \zeta \in tʻ\xi . \zeta \text{ sm } \gamma . \gamma \in \text{Nc}(\alpha _{\beta })ʻ\delta + .\\ +[*37·1] &\supset . \zeta \in \text{ sm }ʻʻ\text{Nc}(\alpha _{\beta })ʻ\delta \cap tʻ\xi &\qquad \text{(3)}\\ +\text{Similarly}\\ +\vdash : \text{Hp} . \zeta \in \text{Nc}(\xi _{\alpha })ʻ\gamma . &\supset . \zeta \in \text{ sm }ʻʻ\text{Nc}(\alpha _{\beta })ʻ\delta \cap tʻ\xi &\qquad \text{(4)}\\ +\vdash .(2).(3).(4). \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_34">[Pg 34]</span></p> + +<p class="nind"><b>*102·82.</b> \(\vdash : \mu \in \text{NC}^\beta (\alpha ) . \exists !\mu . \supset . \text{ sm }ʻʻ\mu \cap tʻ\xi \in \text{NC}^\beta (\xi ) \quad[*102·81·5]\)</p> + +<p class="nind"><b>*102·83.</b> \[\begin{aligned}\vdash : \mu \in \text{NC}^\beta (\alpha ) . &\exists !\mu . \nu = \text{ sm }ʻʻ\mu \cap tʻ\xi . \exists !\nu . \supset .\\ +&\text{ sm }ʻʻ\mu \cap tʻ\zeta = \text{ sm }ʻʻ\nu \cap tʻ\zeta . \mu = \text{ sm }ʻʻ\nu \cap tʻ\alpha\end{aligned}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *102·81 . \supset \\ +\vdash : \text{Hp} . \gamma \in \mu . &\gamma' \in \nu . \mu = \text{Nc}(\alpha _{\beta })ʻ\delta . \supset .\\ +&\nu = \text{Nc}(\xi _{\beta })ʻ\delta = \text{Nc}(\xi _{\alpha })ʻ\gamma . \text{ sm }ʻʻ\mu \cap tʻ\zeta = \text{Nc}(\zeta _{\alpha })ʻ\gamma .\\ +[*102·81] \supset . &\text{ sm }ʻʻ\nu \cap tʻ\zeta = \text{Nc} (\zeta _{\alpha })ʻ\gamma = \text{ sm }ʻʻ\mu \cap tʻ\zeta .\\ +&\text{ sm }ʻʻ\nu \cap tʻ\alpha = \text{Nc}(\alpha _{\beta })ʻ\delta = \mu &\qquad \text{(1)}\\ +\vdash . (1) . *102·5 . \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*102·84.</b> \(\vdash : (\exists \gamma ) . \gamma \text{ sm } \alpha . \gamma \in tʻ\alpha . \delta \text{ sm } \gamma . \equiv . \delta \text{ sm } \alpha\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *73·32 . &\supset \vdash : (\exists \gamma ) . \gamma \text{ sm } \alpha . \gamma \in tʻ\alpha . \delta \text{ sm } \gamma . \supset . \delta \text{ sm } \alpha &\qquad \text{(1)}\\ +\vdash . *73·3 . *63·103 . \supset \\ +\vdash : \delta \text{ sm } \alpha . &\supset . \alpha \text{ sm } \alpha . \alpha \in tʻ\alpha . \delta \text{ sm } \alpha .\\ +[*10·24] &\supset . (\exists \gamma ) . \gamma \text{ sm } \alpha . \gamma \in tʻ\alpha . \delta \text{ sm } \gamma &\qquad \text{(2)}\\ +\vdash . (1) . (2) . \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*102·85.</b> \(\vdash . \text{ sm }ʻʻ\mu \cap tʻ\beta = \text{ sm }_{\beta }ʻʻ\mu \quad[*65·3]\)</p> + +<p class="nind"><b>*102·86.</b> \(\vdash : \mu = \text{Nc} (\alpha )ʻ\delta . \exists !\mu . \supset . \text{ sm }_{\xi }ʻʻ\mu = \text{Nc}(\xi )ʻ\delta\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *102·6·81 . \supset \\ +\vdash : \gamma \in \text{Nc}(\alpha )ʻ\delta . &\supset . \text{ sm }ʻʻ\text{Nc}(\alpha )ʻ\delta \cap tʻ\xi = \text{Nc}(\xi )ʻ\delta .\\ +[*102·85] &\supset . \text{ sm }_{\xi }ʻʻ\text{Nc}(\alpha )ʻ\delta = \text{Nc}(\xi )ʻ\delta &\qquad \text{(1)}\\ +\vdash . (1) . *13·12 . \supset \\ +\vdash : \mu = \text{Nc} (\alpha )ʻ\delta . \gamma \in \mu . &\supset . \text{ sm }_{\xi }ʻʻ\mu = \text{Nc}(\xi )ʻ\delta : \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*102·861.</b> \(\vdash . \text{ sm }_{\alpha }ʻʻ\text{ sm }_{\xi }ʻʻ \mu \subset \text{ sm }_{\alpha }ʻʻ\mu \)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *37·1 . \supset \vdash : \gamma \in \text{ sm }_{\alpha }ʻʻ\text{ sm }_{\xi }ʻʻ\mu + . &\supset . (\exists \zeta ,\eta ) . \eta \in \mu . \zeta \text{ sm } \eta . \zeta \in tʻ\xi . \gamma \text{ sm } \zeta . \gamma \in tʻ\alpha .\\ +[*73·32.*10·5] &\supset . (\exists \eta ) . \eta \in \mu . \gamma \text{ sm } \eta . \gamma \in tʻ\alpha .\\ +[*37·1] &\supset . \gamma \in \text{ sm }_{\alpha }ʻʻ\mu : \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*102·862">*102·862</a>.</b> \(\vdash \colon\ldotp \eta \in \mu . \supset _{\eta } . \exists !\text{Nc}(\xi )ʻ\eta : \supset . \text{ sm }_{\alpha }ʻʻ\mu + = \text{ sm }_{\alpha }ʻʻ\text{ sm }_{\xi }ʻʻ\mu \)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *102·6. \supset \vdash \colon\ldotp \text{Hp} . &\supset : \eta \in \mu . \supset . (\exists \zeta ) . \zeta \text{ sm } \eta . \zeta \in tʻ\xi :\\ +[\text{Fact}.*10·35] &\supset : \eta \in \mu . \gamma \text{ sm } \eta . \gamma \in tʻ\alpha . \supset .\\ +&(\exists \zeta ) . \zeta \text{ sm } \eta . \zeta \in tʻ \xi . \gamma \text{ sm } \eta . \gamma \in tʻ\alpha .\\ +[*73·37] &\supset . (\exists \zeta ) . \zeta \text{ sm } \eta . \zeta \in tʻ \xi . \gamma \text{ sm } \zeta . \zeta \in tʻ\alpha .\\ +[*37·1] &\supset . \gamma \in \text{ sm }_{\alpha }ʻʻ\text{ sm }_{\xi }ʻʻ\mu &\qquad \text{(1)}\\ +\vdash . (1) . *10·11·23 . *37·1 . &\supset \vdash : \text{Hp}. \supset . \text{ sm }_{\alpha }ʻʻ\mu \subset \text{ sm }_{\alpha }ʻʻ\text{ sm }_{\xi }ʻʻ\mu + &\qquad \text{(2)}\\ +\vdash . (2) . *102·861 . \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_35">[Pg 35]</span></p> + +<p class="nind"><b>*102·863.</b> \(\vdash \colon\ldotp \mu = \text{Nc}(\beta )ʻ\delta . \exists !\text{Nc}(\xi )ʻ\delta . \supset : \eta \in \mu . \supset _{\eta } + . \exists !\text{Nc}(\xi )ʻ\eta \)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *100·31·321. \supset \vdash : \text{Hp} . \eta \in \mu . &\supset . \text{Nc}(\xi )ʻ\eta = \text{Nc}(\xi )ʻ\delta .\\ +[\text{Hp}] &\supset . \exists !\text{Nc}(\xi )ʻ\eta : \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*102·87">*102·87</a>.</b> \[\begin{aligned}&\vdash : \mu = \text{Nc}(\beta )ʻ\delta . \exists !\text{Nc}(\xi )ʻ\delta . \supset . \text{ sm }_{\alpha }ʻʻ\mu + = \text{ sm }_{\alpha }ʻʻ\text{ sm }_{\xi }ʻʻ\mu \\ +&[*102·862·863]\end{aligned}\]</p> + +<p class="nind"><b><a id="*102·88">*102·88</a>.</b> \[\begin{aligned}\vdash : \mu = \text{Nc}(\beta )ʻ\delta . \exists !\text{ sm }_{\xi }ʻʻ\mu . \supset . &\text{ sm }_{\xi }ʻʻ\mu + = \text{Nc}(\xi )ʻ\delta . \text{ sm }_{\alpha }ʻʻ\mu = \text{Nc}(\alpha )ʻ\delta .\\ +&\text{ sm }_{\alpha }ʻʻ\mu = \text{ sm }_{\alpha }ʻʻ\text{ sm }_{\xi }ʻʻ\mu = \text{ sm }_{\alpha }ʻʻ\text{Nc}(\xi )ʻ\delta\end{aligned}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *37·29 . \text{Transp}. \supset \vdash : \text{Hp} . \supset . \exists !\mu .\\ +[*102·86] &\supset . \text{ sm }_{\xi }ʻʻ\mu = \text{Nc}(\xi )ʻ\delta . \text{ sm }_{\alpha }ʻʻ\mu = \text{Nc}(\alpha )ʻ\delta . &\qquad \text{(1)}\\ +[\text{Hp}] &\supset . \exists !\text{Nc}(\xi )ʻ\delta .\\ +[*102·87] \supset . \text{ sm }_{\alpha }ʻʻ\mu &= \text{ sm }_{\alpha }ʻʻ\text{ sm }_{\xi }ʻʻ\mu &\qquad \text{(2)}\\ +[(1)] &= \text{ sm }_{\alpha }ʻʻ\text{Nc}(\xi )ʻ\delta &\qquad \text{(3)}\\ +\vdash . (1) . (2) . (3) . \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p><i>Note on negative statements concerning types</i>. Statements +such as "\(x {\sim} \in tʻy\)" or "\(x {\sim} \in t_{0}ʻ\alpha\)" +are always false when they are significant. Hence when an object +belongs to one type, there is no significant way of expressing what +we mean when we say that it does not belong to some other type. The +reason is that, when, for example, \(tʻ\alpha\) and \( t_{0}ʻ\alpha\) +are said to be different, the statement is only significant if +interpreted as applying to the symbols, <i>i.e.</i> as meaning to +deny that the two symbols denote the same class. We cannot assert +that they denote <i>different</i> classes, since "\(tʻ\alpha \neq t_{0}ʻ\alpha\)" +is not significant, but we can deny that they denote +the same class. Owing to this peculiarity, propositions dealing +with types acquire their importance largely from the fact that they +can be interpreted as dealing with the symbols rather than directly +with the objects denoted by the symbols. Another reason for the +importance of typically definite propositions is that, when they +are implications of which the hypothesis can be asserted, they can +be used for <i>inference</i>, <i>i.e.</i> for the assertion of the +conclusion. Where typically ambiguous symbols occur in implications, +on the contrary, the conditions of significance may be different +for the hypothesis and the conclusion, so that fallacies may arise +from the use of such implications in inference. <i>E.g.</i> it is +a fallacy to infer "\(\vdash . \exists !\text{Nc}ʻ\alpha\)" from +the (true) propositions "\(\vdash : \alpha \in \text{Nc}ʻ\alpha .\supset . \exists !\text{Nc}ʻ\alpha\)" +and "\(\vdash . \alpha \in \text{Nc}ʻ\alpha\)." (The truth of the first +of these two requires that "\(\text{Nc}ʻ\alpha\)" should receive the +same typical determination in both its occurrences.) For these two +reasons hypothetical concerning types are often useful, in spite of the +fact that their hypotheses are always true when they are significant.</p> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_36">[Pg 36]</span></p> +<h2 class="nobreak" id="*103"> +*103. HOMOGENEOUS CARDINALS.</h2> +</div> + + +<p class="nind"><i>Summary of</i> *103.</p> + +<p>In this number, we shall consider cardinals generated by a +homogeneous relation of similarity. A "homogeneous" cardinal is +to mean all the classes similar to some class \(\alpha\) and +of the same type as \(\alpha\). The "homogeneous cardinal of +\(\alpha\)" will be defined as \(\text{Nc}ʻ\alpha \cap tʻ\alpha\); +we shall denote it by "\(\text{N}_{0}\text{c}ʻ\alpha\)." +Then the class of homogeneous cardinals is the class of +all such cardinals as "\(\text{N}_{0}\text{c}ʻ\alpha\)," +<i>i.e.</i> it is \(\text{D}ʻ\text{N}_{0}\text{c}\); this +we shall denote by "\(\text{N}_{0}\text{C}\)." The symbol +"\(\text{N}_{0}\text{c}ʻ\alpha\)" is typically definite as soon +as \(\alpha\) is assigned; "\(\text{N}_{0}\text{C}\)," on the +contrary, is typically ambiguous: it must be a \(\text{Cls}^{3}\), but +otherwise its type may vary indefinitely. Homogeneous cardinals have, +however, many properties which do not require that the ambiguity of +"\(\text{N}_{0}\text{C}\)" should be determined, and few which do +require this. They are important also as being the simplest kind of +cardinals, and as being a kind to which other kinds can usually be +reduced.</p> + +<p>The chief advantage of homogeneous cardinals is that they are never +null (<a href="#*103·13">*103·13</a><a href="#*103·22">·22</a>). This enables us to avoid by their means the explicit +exclusion of exceptional cases; thus throughout Section B we shall use +homogeneous cardinals in defining the arithmetical operations: the +arithmetical sum of \(\text{Nc}ʻ\alpha\) and \(\text{Nc}ʻ\beta\), for +example, will be defined by means of \(\text{N}_{0}\text{c}ʻ\alpha\) +and \(\text{N}_{0}\text{c}ʻ\beta\), in order to exclude such a +determination of the typical ambiguity of \(\text{Nc}ʻ\alpha\) and +\(\text{Nc}ʻ\beta\) as would make either of them null. It is true that +not only homogeneous cardinals, but also ascending cardinals (cf. +<a href="#*104">*104</a>), are never null. But homogeneous cardinals are much the simplest +kind of cardinals that are never null, and are therefore the most +convenient.</p> + +<p>The fact that no homogeneous cardinal is null is derived from</p> + +<p class="nind"><b>*103·12.</b> \(\vdash . \alpha \in \text{N}_{0}\text{c}ʻ\alpha\)</p> + +<p>Other important propositions in this number are the following:</p> + +<p class="nind"><b>*103·2.</b> \(\vdash : \mu \in \text{N}_{0}\text{C} + . \equiv . (\exists \alpha ) . \mu = \text{Nc}ʻ\alpha \cap tʻ\alpha . \equiv . (\exists \alpha ) . \mu = \text{N}_{0}\text{c}ʻ\alpha\)</p> + +<p class="nind"><b>*103·26.</b> \(\vdash \colon\ldotp \mu \in \text{NC} . \supset : \alpha \in \mu . \equiv . \text{N}_{0}\text{c}ʻ\alpha = \mu\)</p> + +<p>The above proposition is used constantly.</p> + +<p><span class="pagenum" id="Page_37">[Pg 37]</span></p> + +<p class="nind"><b>*103·27.</b> \(\vdash :\mu =\text{N}_{0}\text{c}ʻ\alpha .\equiv .\mu \in \text{NC}.\alpha \in \mu \)</p> + +<p>Thus to say that \(\mu\) is the homogeneous cardinal of \(\alpha\) is +equivalent to saying that \(\mu\) is a cardinal of which \(\alpha\) is +a member.</p> + +<p class="nind"><b>*103·301.</b> \(\vdash .\text{NC}^\alpha (\alpha )=\text{N}_{0}\text{C}(\alpha )\)</p> + +<p class="nind"><b>*103·34.</b> \(\vdash .\text{NC} - \iota ʻ\Lambda \subset \text{N}_{0}\text{C}\)</p> + +<p class="nind"><b>*103·4.</b> \(\vdash .\text{ sm }ʻʻ\text{N}_{0}\text{c}ʻ\alpha =\text{Nc}ʻ\alpha\)</p> + +<p class="nind"><b>*103·41.</b> \(\vdash .\text{ sm }ʻʻ\text{N}_{0}\text{c}ʻ\alpha \cap tʻ\beta =\text{Nc}(\beta )ʻ\alpha \)</p> + +<hr class="tb"> + +<p class="nind"><b>*103·01.</b> \(\text{N}_{0}\text{c}ʻ\alpha =\text{Nc}ʻ\alpha \cap tʻ\alpha \quad\text{Df}\)</p> + +<p class="nind"><b><a id="*103·02">*103·02</a>.</b> \(\text{N}_{0}\text{C}=\text{D}ʻ\text{N}_{0}\text{c} \quad\text{Df}\)</p> + +<p class="nind"><b>*103·1.</b> \(\vdash .\text{N}_{0}\text{c}ʻ\alpha =(\text{Nc}ʻ\alpha )_{\alpha }=\text{Nc}(\alpha )ʻ\alpha =\text{Nc}(\alpha _{\alpha })ʻ\alpha + \quad[*102·6.(*103·01)]\)</p> + +<p class="nind"><b>*103·11.</b> \[\begin{aligned}&\vdash :\beta \in \text{N}_{0}\text{c}ʻ\alpha + .\equiv .\beta \text{ sm }\alpha .\beta \in tʻ\alpha .\equiv .\beta \in \text{Nc}ʻ\alpha .\beta \in tʻ\alpha\\ +&[*103·1.*102·6]\end{aligned}\]</p> + +<p class="nind"><b><a id="*103·12">*103·12</a>.</b> \(\vdash .\alpha \in \text{N}_{0}\text{c}ʻ\alpha \quad[*103·11.*73·3.*63·103]\)</p> + +<p class="nind"><b><a id="*103·13">*103·13</a>.</b> \(\vdash .\exists !\text{N}_{0}\text{c}ʻ\alpha \quad[*103·12.*10·24]\)</p> + +<p>This is a legitimate inference from <a href="#*103·12">*103·12</a> because, when \(\alpha\) is +given, \(\text{N}_{0}\text{c}ʻ\alpha\) is typically definite.</p> + +<p class="nind"><b>*103·14.</b> \(\vdash :\text{N}_{0}\text{c}ʻ\alpha =\text{N}_{0}\text{c}ʻ\beta .\equiv .\alpha \in \text{N}_{0}\text{c}ʻ\beta .\equiv .\beta \in \text{N}_{0}\text{c}ʻ\alpha + .\equiv .\alpha \text{ sm }\beta .\alpha \in tʻ\beta\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*103·11.\supset \\ +\vdash \colon\ldotp \text{N}_{0}\text{c}ʻ\alpha =\text{N}_{0}\text{c}ʻ\beta .\equiv :\gamma \text{ sm }\alpha .\gamma \in tʻ\alpha .\equiv _{\gamma }.\gamma + \text{ sm }\beta .\gamma \in tʻ\beta : &\qquad \text{(1)}\\ +[*10·1] \supset :\alpha \text{ sm }\alpha .\alpha \in tʻ\alpha .\equiv .\alpha \text{ sm }\beta .\alpha \in tʻ\beta :\\ +[*73·3.*63·103] \supset :\alpha \text{ sm }\beta .\alpha \in tʻ\beta &\qquad \text{(2)}\\ +\vdash .*73·32.*63·17\\ +\vdash :\alpha \text{ sm }\beta .\alpha \in tʻ\beta .\gamma \text{ sm }\alpha .\gamma \in tʻ\alpha .\supset .\gamma \text{ sm }\beta .\gamma \in tʻ\beta &\qquad \text{(3)}\\ +\vdash .(3)\frac{\beta,\,\alpha}{\alpha,\,\beta}.*73·31.*63·16.\supset \\ +\vdash :\alpha \text{ sm }\beta .\alpha \in tʻ\beta .\gamma \text{ sm }\beta .\gamma \in tʻ\beta .\supset .\gamma \text{ sm }\alpha .\gamma \in tʻ\alpha &\qquad \text{(4)}\\ +\vdash .(3).(4).(1).\supset \\ +\vdash :\alpha \text{ sm }\beta .\alpha \in tʻ\beta .\supset .\text{N}_{0}\text{c}ʻ\alpha =\text{N}_{0}\text{c}ʻ\beta &\qquad \text{(5)}\\ +\vdash .(2).(5).*103·11.*73·31.*63·16.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*103·15.</b> \(\vdash :\exists !\text{N}_{0}\text{c}ʻ\alpha \cap \text{N}_{0}\text{c}ʻ\beta .\equiv .\text{N}_{0}\text{c}ʻ\alpha + =\text{N}_{0}\text{c}ʻ\beta\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*103·13.&\supset \vdash :\text{N}_{0}\text{c}ʻ\alpha =\text{N}_{0}\text{c}ʻ\beta .\supset .\exists !\text{N}_{0}\text{c}ʻ\alpha \cap \text{N}_{0}\text{c}ʻ\beta + &\qquad \text{(1)}\\ +\vdash .*103·14.\supset \vdash :\gamma \in \text{N}_{0}\text{c}ʻ\alpha .\gamma \in \text{N}_{0}\text{c}ʻ\beta .&\supset .\text{N}_{0}\text{c}ʻ\alpha =\text{N}_{0}\text{c}ʻ\gamma + .\text{N}_{0}\text{c}ʻ\beta =\text{N}_{0}\text{c}ʻ\gamma .\\ +[*14·131·144] &\supset .\text{N}_{0}\text{c}ʻ\alpha =\text{N}_{0}\text{c}ʻ\beta :\\ +[*10·11·23] &\supset \vdash :\exists !\text{N}_{0}\text{c}ʻ\alpha \cap \text{N}_{0}\text{c}ʻ\beta .\supset .\text{N}_{0}\text{c}ʻ\alpha =\text{N}_{0}\text{c}ʻ\beta + &\qquad \text{(2)}\\ +\vdash .(1).(2).&\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_38">[Pg 38]</span></p> + +<p class="nind"><b><a id="*103·16">*103·16</a>.</b> \(\vdash : \text{N}_{0}\text{c}ʻ\alpha = \text{Nc}ʻ\beta . \equiv . \text{Nc}ʻ\alpha = \text{Nc}ʻ\beta \)</p> + +<p>In this proposition, the equation "\(\text{Nc}ʻ\alpha =\text{Nc}ʻ\beta\)" +must be supposed to hold in any type for which +it is significant. Otherwise, we might find a type for which +\(\text{Nc}ʻ\alpha = \Lambda = \text{Nc}ʻ\beta\), without having +\(\text{N}_{0}\text{c}ʻ\alpha = \text{Nc}ʻ\beta\).</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *103·12 . \supset \vdash : \text{N}_{0}\text{c}ʻ\alpha = \text{Nc}ʻ\beta . &\supset . \alpha \in \text{Nc}ʻ\beta .\\ +[*100·31·321] &\supset . \text{Nc}ʻ\alpha = \text{Nc}ʻ\beta &\qquad \text{(1)}\\ +\vdash . *22·481. \supset \vdash : \text{Nc}ʻ\alpha = \text{Nc}ʻ\beta . &\supset . \text{Nc}ʻ\alpha \cap tʻ\alpha = \text{Nc}ʻ\beta \cap tʻ\alpha .\\ +[*65·13.(*103·01)] &\supset . \text{N}_{0}\text{c}ʻ\alpha = \text{Nc}ʻ\beta &\qquad \text{(2)}\\ +\vdash .(1).(2). \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*103·2.</b> \[\begin{aligned}&\vdash : \mu \in \text{N}_{0}\text{C} + . \equiv . (\exists \alpha ) . \mu = \text{Nc}ʻ\alpha \cap tʻ\alpha . \equiv . (\exists \alpha ). \mu = \text{N}_{0}\text{c}ʻ\alpha\\ +&[*71·41.*100·22.(*103·01·02)]\end{aligned}\]</p> + +<p class="nind"><b>*103·21.</b> \(\vdash . \text{N}_{0}\text{c}ʻ\alpha \in \text{N}_{0}\text{C} . \text{N}_{0}\text{c}ʻ\alpha \in \text{NC} \quad[*103·2.*100·2·4.*14·28.*65·13]\)</p> + +<p>In adducing a proposition, such as <a href="#*100·2">*100·2</a>, which is concerned with an +"\(\text{Nc}\)" entirely undetermined in type, any degree of typical +determination may be added to our "\(\text{Nc}\)," since an asserted +proposition containing an ambiguous "\(\text{Nc}\)" is only legitimate +if it is true for every possible determination of the ambiguity.</p> + +<p class="nind"><b><a id="*103·22">*103·22</a>.</b> \(\vdash : \mu \in \text{N}_{0}\text{C} . \supset . \exists !\mu \quad[*103·13·2]\)</p> + +<p class="nind"><b>*103·23.</b> \(\vdash . \Lambda {\sim}\in \text{N}_{0}\text{C} \quad[*103·22]\)</p> + +<p class="nind"><b>*103·24.</b> \(\vdash . \text{N}_{0}\text{C} \in \text{Cls ex}^{2}\,\text{excl}\quad[*100·43.*103·23.*84·13]\)</p> + +<p class="nind"><b>*103·25.</b> \(\vdash \colon\ldotp \mu ,\nu \in \text{N}_{0}\text{C} . \supset : \exists !\mu \cap \nu . \equiv . \mu = \nu \quad[*103·24 . *84·135]\)</p> + +<p class="nind"><b>*103·26.</b> \(\vdash \colon\ldotp \mu \in \text{NC}. \supset : \alpha \in \mu . \equiv . \text{N}_{0}\text{c}ʻ\alpha = \mu\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*100·45. &\supset \vdash \colon\ldotp \text{Hp} . \supset : \alpha \in \mu . \supset . \text{Nc}ʻ\alpha = \mu &\qquad \text{(1)}\\ +\vdash .*63·22. &\supset \vdash : \alpha \in \mu . \supset . \mu \subset tʻ\alpha &\qquad \text{(2)}\\ +\vdash .(1).(2).*22·621. \supset \vdash \colon\ldotp \text{Hp} . &\supset : \alpha \in \mu . \supset . \text{Nc}ʻ\alpha \cap tʻ\alpha = \mu .\\ +[(*103·01)] &\supset . \text{N}_{0}\text{c}ʻ\alpha = \mu &\qquad \text{(3)}\\ +\vdash . *103·12. &\supset \vdash : \text{N}_{0}\text{c}ʻ\alpha = \mu . \supset . \alpha \in \mu &\qquad \text{(4)}\\ +\vdash .(3).(4). \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*103·27.</b> \(\vdash : \mu = \text{N}_{0}\text{c}ʻ\alpha . \equiv . \mu \in \text{NC} . \alpha \in \mu\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*103·26. \supset \vdash : \mu \in \text{NC} . \mu = \text{N}_{0}\text{c}ʻ\alpha . \equiv . \mu \in \text{NC} . \alpha \in \mu &\qquad \text{(1)}\\ +\vdash .(1).*103·21. \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*103·28.</b> \(\vdash : (\exists \alpha ) . \gamma \text{ sm } \alpha . \mu = \text{N}_{0}\text{c}ʻ\alpha . \equiv . \exists ! \mu . \mu = \text{Nc}ʻ\gamma\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *103·27 . \supset \\ +\vdash : (\exists \alpha ) . \gamma \text{ sm } \alpha . \mu = \text{N}_{0}\text{c}ʻ\alpha + . &\equiv . (\exists \alpha ) . \gamma \text{ sm } \alpha . \mu \in \text{NC} . \alpha \in \mu .\\ +[*100·31] &\equiv . \mu \in \text{NC} . \exists ! \mu \cap \text{Nc}ʻ\gamma .\\ +[*100·42·41] &\equiv . \mu \in \text{NC} . \exists ! \mu \cap \text{Nc}ʻ\gamma . \mu = \text{Nc}ʻ\gamma .\\ +[*100·41] &\equiv . \exists ! \mu . \mu = \text{Nc}ʻ\gamma : \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_39">[Pg 39]</span></p> + +<p class="nind"><b>*103·3.</b> \(\vdash :\beta \in tʻ\alpha .\supset .\text{N}_{0}\text{c}ʻ\beta =\text{Nc}(\alpha )ʻ\beta =\text{Nc}(\alpha _{\alpha })ʻ\beta =\text{Nc}ʻ\beta \cap tʻ\alpha\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*63·16.\supset \vdash :\text{Hp}.\supset .tʻ\beta =tʻ\alpha .\\ +[*22·481.(*103·01)] \supset .\text{N}_{0}\text{c}ʻ\beta &=\text{Nc}ʻ\beta \cap tʻ\alpha &\qquad \text{(1)}\\ +[*102·6]&=\text{Nc}(\alpha )ʻ\beta &\qquad \text{(2)}\\ +[*102·61] &=\text{Nc}(\alpha _{\alpha })ʻ\beta &\qquad \text{(3)}\\ +\vdash .(1).(2).(3).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*103·301.</b> \(\vdash .\text{NC}^\alpha (\alpha )=\text{N}_{0}\text{C}(\alpha )\)</p> + +<p>Note that although "\(\text{NC}(\alpha )\)" is not definite, +"\(\text{N}_{0}\text{C}(\alpha )\)" is absolutely definite as soon as +\(\alpha\) is assigned.</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*103·3.\supset \vdash :\beta \in tʻ\alpha .\mu =\text{N}_{0}\text{c}ʻ\beta .\equiv .\beta \in tʻ\alpha .\mu =\text{Nc}(\alpha _{\alpha })ʻ\beta .\\ +[*102·37] \equiv .\mu =\text{Nc}(\alpha _{\alpha })ʻ\beta &\qquad \text{(1)}\\ +\vdash .*63·5.(*103·01).\supset \\ +\vdash \colon\ldotp \mu =\text{N}_{0}\text{c}ʻ\beta .\supset :\beta \in tʻ\alpha .\equiv .\mu \in t^{2}ʻ\alpha &\qquad \text{(2)}\\ +\vdash .(1).(2).\supset \vdash :\mu \in t^{2}ʻ\alpha .\mu =\text{N}_{0}\text{c}ʻ\beta .\equiv .\mu =\text{Nc}(\alpha _{\alpha })ʻ\beta &\qquad \text{(3)}\\ +\vdash .(3).*10·11·281·35.\supset \\ +\vdash \colon\ldotp \mu \in t^{2}ʻ\alpha :(\exists \beta ).\mu =\text{N}_{0}\text{c}ʻ\beta :\equiv .(\exists \beta ).\mu =\text{Nc}(\alpha _{\alpha })ʻ\beta .\\ +[*102·5] \equiv .\mu \in \text{NC}^\alpha (\alpha ) &\qquad \text{(4)}\\ +\vdash .(4).*103·2.\supset \vdash :\mu \in t^{2}ʻ\alpha \cap \text{N}_{0}\text{C}.\equiv .\mu \in \text{NC}^\alpha (\alpha ) &\qquad \text{(5)}\\ +\vdash .(5).(*65·02).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*103·31.</b> \(\vdash :\exists !\text{Nc}(\alpha _{\beta })ʻ\delta .\supset .\text{Nc}(\alpha _{\beta })ʻ\delta \in \text{N}_{0}\text{C}(\alpha )\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*102·52.\supset \vdash :\text{Hp}.&\supset .\text{Nc}(\alpha _{\beta })ʻ\delta \in \text{NC}^\alpha (\alpha ).\\ +[*103·301] &\supset .\text{Nc}(\alpha _{\beta })ʻ\delta \in \text{N}_{0}\text{C}(\alpha ):\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*103·32.</b> \(\vdash .\text{NC}^\beta (\alpha )-\iota ʻ\Lambda \subset \text{N}_{0}\text{C}(\alpha )\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*103·31.\supset \vdash :\mu =\text{Nc}(\alpha _{\beta })ʻ\delta .\exists !\mu .\supset .\mu \in \text{N}_{0}\text{C}(\alpha ) &\qquad \text{(1)}\\ +\vdash .(1).*102·5.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p>In the above proposition, the "\(\beta\)" may be omitted, and we may +write (cf. <a href="#*103·33">*103·33</a>, below) +\[ +\vdash .\text{NC}(\alpha )-\iota ʻ\Lambda \subset \text{N}_{0}\text{C}(\alpha ). +\]</p> + +<p>For the \(\beta\) is wholly arbitrary, so that any possible +determination of \(\text{NC}(\alpha )\) makes the above proposition +true. We may proceed a step further, and write (<a href="#*103·34">*103·34</a>, below) +\[ +\vdash .\text{NC}-\iota ʻ\Lambda \subset \text{N}_{0}\text{C}. +\]</p> + +<p><span class="pagenum" id="Page_40">[Pg 40]</span></p> + +<p>But although we also have \(\text{N}_{0}\text{C}\subset \text{NC}-\iotaʻ\Lambda\), +provided the "\(\text{NC}\)" on the right is suitably determined, +we do not have this always. For example, if "\(\text{NC}\)" +is determined as \(\text{NC}^\alpha (tʻ\alpha )\), and +"\(\text{N}_{0}\text{C}\)" as \(\text{N}_{0}\text{C}(tʻ\alpha )\), then +\(\text{N}_{0}\text{c}ʻtʻ\alpha \in \text{N}_{0}\text{C}-\text{NC}\).</p> + +<p class="nind"><b><a id="*103·33">*103·33</a>.</b> \(\vdash . \text{NC}(\alpha ) - \iota ʻ\Lambda \subset \text{N}_{0}\text{C}(\alpha )\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *4·2 . (*65·02) . \supset \\ +\vdash \colon\ldotp \mu \in \text{NC}(\alpha ) - \iota ʻ\Lambda . \equiv : \mu \in \text{NC} . \mu \in t^{2}ʻ\alpha . \exists ! \mu :\\ +[*100·4 . *63·5] \equiv : (\exists \beta ) . \mu = \text{Nc}ʻ\beta : \mu \subset tʻ\alpha . \exists ! \mu :\\ +[*65·13] \equiv : (\exists \beta ) . \mu = \text{Nc}ʻ\beta \cap tʻ\alpha : \exists ! \mu :\\ +[*102·6] \equiv : (\exists \beta ) . \mu = \text{Nc}(\alpha _{\beta })ʻ\beta . \exists ! \mu :\\ +[*103·31] \supset : \mu \in \text{N}_{0}\text{C}(\alpha ) \colon\ldotp \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*103·34">*103·34</a>.</b> \(\vdash . \text{NC} - \iota ʻ\Lambda \subset \text{N}_{0}\text{C}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *100·31·321 . *63·5 . \supset \\ +\vdash : \mu = \text{Nc}ʻ\alpha . \beta \in \mu . \supset . \mu = \text{Nc}ʻ\beta \cap tʻ\beta \\ +[(*103·01)] = \text{N}_{0}\text{c}ʻ\beta .\\ +[*103·2] \supset . \mu \in \text{N}_{0}\text{C} &\qquad \text{(1)}\\ +\vdash . (1) . *100·4 . *11·11·35·54 . \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p>Thus every cardinal except \(\Lambda\) is a homogeneous cardinal +in the appropriate type. Note that although of course every +homogeneous cardinal is a cardinal, yet "\(\text{N}_{0}\text{C}\subset \text{NC}\)" +must not be asserted, because it is possible +to determine the ambiguity of "\(\text{NC}\)" in such a way as to +make this false. Hence we do not get \(\text{NC} - \iota ʻ\Lambda =\text{N}_{0}\text{C}\).</p> + +<p class="nind"><b>*103·35.</b> \(\vdash : \Lambda {\sim} \in \text{NC}^\alpha (\beta ) . \supset . \text{NC}^\beta (\alpha ) - \iota ʻ\Lambda = \text{N}_{0}\text{C}(\alpha + ) \quad[*102·55 . *103·301]\)</p> + +<p>The hypothesis of this proposition is satisfied, as will appear later, +if the type of \(\beta\) is in what we may call the direct ascent from +that of \(\alpha\), <i>i.e.</i> if it can be reached from \(\alpha\) by +a finite number of steps each of which takes us from a type \(\tau\) +to either \(\text{Cl}ʻ\tau\) or \(\text{Rl}ʻ(\tau \uparrow \tau)\). +Thus in such a case the cardinals (other than \(\Lambda\)) which go +from \(tʻ\beta\) to \(tʻ\alpha\) are the same as those which begin +and end within \(tʻ\alpha\). It will also appear that in such a case +\(\Lambda\) always is a member of \(\text{NC}^\beta (\alpha )\). +<i>If</i> two cardinals which are not equal must always be one greater +and the other less, then \(\Lambda \in \text{NC}^\beta (\alpha )\) is +the condition for \(\text{N}_{0}\text{c}ʻtʻ\beta \gt \text{Nc}(\beta +)ʻtʻ\alpha\). In that case, we shall have \(\Lambda \in \text{NC}^\beta +(\alpha ). \supset . \Lambda {\sim} \in \text{NC}^\alpha(\beta )\). But +there is no known proof that of two different cardinals one must be the +greater, except by assuming the multiplicative axiom and proving thence +(by Zermelo's theorem) that every class can be well-ordered (cf. *258).</p> + +<p class="nind"><b>*103·4.</b> \(\vdash . \text{ sm }ʻʻ\text{N}_{0}\text{c}ʻ\alpha = \text{Nc}ʻ\alpha\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *37·1 . \supset \\ +\vdash : \delta \in \text{ sm }ʻʻ\text{N}_{0}\text{c}ʻ\alpha . &\equiv . (\exists \gamma ) . \gamma \text{ sm } \alpha . \gamma \in tʻ\alpha . \delta \text{ sm } \gamma .\\ +[*102·84] &\equiv . \delta \text{ sm } \alpha : \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*103·41.</b> \(\vdash . \text{ sm }ʻʻ\text{N}_{0}\text{c}ʻ\alpha \cap tʻ\beta = \text{Nc}(\beta )ʻ\alpha \)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *103·4 . \supset \vdash .\\ \text{ sm }ʻʻ\text{N}_{0}\text{c}ʻ\alpha \cap tʻ\beta &= \text{Nc}ʻ\alpha \cap tʻ\beta \\ +[*102·6] &= \text{Nc}(\beta )ʻ\alpha . \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_41">[Pg 41]</span></p> + +<p class="nind"><b>*103·42.</b> \(\vdash :\beta \text{ sm } \alpha . \equiv .\text{Nc}(\beta )ʻ\alpha = \text{N}_{0}\text{c}ʻ\beta \)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*100·321.\supset \vdash :\beta \text{ sm } \alpha .&\supset .\text{Nc}ʻ\alpha = \text{Nc}ʻ\beta .\\ +[*22·481] &\supset .\text{Nc}ʻ\alpha \cap tʻ\beta = \text{Nc}ʻ\beta \cap tʻ\beta .\\ +[*102·6.(*103·01)] &\supset .\text{Nc}(\beta )ʻ\alpha = \text{N}_{0}\text{c}ʻ\beta &\qquad \text{(1)}\\ +\vdash .*103·12.\supset \vdash :\text{Nc}(\beta )ʻ\alpha = \text{N}_{0}\text{c}ʻ\beta .&\supset .\beta \in \text{Nc}(\beta )ʻ\alpha .\\ +[*100·31] &\supset .\beta \text{ sm } \alpha &\qquad \text{(2)}\\ +\vdash .(1).(2).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*103·43">*103·43</a>.</b> \(\vdash :\mu \in \text{NC}.\supset .\text{ sm }ʻʻ\mu \cap t_{0}ʻ\mu = \mu\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*37·29. &\supset \vdash :\mu = \Lambda .\supset .\text{ sm }ʻʻ\mu \cap t_{0}ʻ\mu = \Lambda &\qquad \text{(1)}\\ +\vdash .*103·27. \supset \vdash :\mu \in \text{NC}.\alpha \in \mu .&\supset .\mu = \text{N}_{0}\text{c}ʻ\alpha .t_{0}ʻ\mu = tʻ\alpha .\\ +[*103·41] &\supset .\text{ sm }ʻʻ\mu \cap t_{0}ʻ\mu = \text{Nc}(\alpha )ʻ\alpha \\ +[*103·3·27] &= \mu &\qquad \text{(2)}\\ +\vdash .(1).(2).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*103·44.</b> \(\vdash \colon\ldotp \mu ,\nu \in \text{N}_{0}\text{C}.\supset :\mu = \text{ sm }ʻʻ\nu .\equiv .\nu = \text{ sm }ʻʻ\mu\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*100·53. &\supset \vdash \colon\ldotp \exists !\mu .\exists !\nu .\mu ,\nu \in \text{NC}.\supset :\mu = \text{ sm }ʻʻ\nu .\equiv .\nu = \text{ sm }ʻʻ\mu &\qquad \text{(1)}\\ +\vdash .*103·27·2.&\supset \vdash :\text{Hp}.\supset .\exists !\mu .\exists !\nu .\mu ,\nu \in \text{NC} &\qquad \text{(2)}\\ +\vdash .(1).(2).&\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*103.5.</b> \(\vdash .0\in \text{N}_{0}\text{C}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*101·11·12.&\supset \vdash .0\in \text{NC}.\exists !0.\\ +[*103·34]&\supset \vdash .0\in \text{N}_{0}\text{C}.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*103·51.</b> \(\vdash .1\in \text{N}_{0}\text{C}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*101·21·241.&\supset \vdash .1\in \text{NC}.\exists !1.\\ +[*103·34] &\supset \vdash .1\in \text{N}_{0}\text{C}.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p>0 and 1 are the only cardinals of which the above property can be +proved universally with our assumptions. If (as is possible so far as +our assumptions go) the lowest type is a unit class, we shall have <i>in +that type</i> (though in no other) \(2 = \Lambda\), so that in that +type \(2 {\sim} \in \text{N}_{0}\text{C}\).</p> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_42">[Pg 42]</span></p> +<h2 class="nobreak" id="*104"> +*104. ASCENDING CARDINALS.</h2> +</div> + + +<p class="nind"><i>Summary of</i> *104.</p> + +<p>In this number we have to consider cardinals derived from a relation +of similarity which goes from the type of \(\alpha\) to that of +\(tʻ\alpha\), or to that of \(t^{2}ʻ\alpha\). The propositions to +be proved can be extended, by a mere repetition of the proofs, to +\(t^{3}ʻ\alpha\), \(t^{4}ʻ\alpha\), etc. This extension must, however, +be made afresh in each instance; we cannot prove that it can be made +generally, because mathematical induction cannot be applied to the +series +\[ +t_{0}ʻ\alpha,\quad tʻ\alpha,\quad t^{2}ʻ\alpha,\quad t^{3}ʻ\alpha ,\ldotp\ldots +\]</p> + +<p>Ascending cardinals, though less important than homogeneous +cardinals, yet have considerable importance in arithmetic, because +\(\text{Nc}ʻ\alpha \times \text{Nc}ʻ\beta\) and \((\text{Nc}ʻ\alpha)^{\text{Nc}ʻ\beta }\) +are defined as the cardinals of classes of higher types than those of +\(\alpha\) and \(\beta\), and the same applies to the product of the +cardinals of members of a class of classes. In these cases, however, we +also need cardinals of relational types, which will be dealt with in +<a href="#*106">*106</a>.</p> + +<p>We have to deal, in this number, with three different sets of notions, +namely</p> + +<p class="nind"><b>*104·01.</b> \(\text{N}^{1}\text{c}ʻ\alpha =\text{Nc}ʻ\alpha \cap tʻtʻ\alpha \quad\text{Df}\)</p> + +<p class="nind"><b>*104·02.</b> \(\text{N}^{1}\text{C}=\text{D}ʻ\text{N}^{1}\text{c} \quad\text{Df}\)</p> + +<p class="nind"><b>*104·03.</b> \(\mu ^{(1)}=\text{ sm }ʻʻ\mu \cap tʻ\mu \quad\text{Df}\)</p> + +<p class="nind"> +with similar definitions of \(\text{N}^{2}\text{c}ʻ\alpha\), etc. +Thus \(\text{N}^{1}\text{c}ʻ\alpha\) consists of all classes +similar to \(\alpha\) but of the next higher type, <i>i.e.</i> it +is the cardinal number of \(\alpha\) in the type next above that of +\(\text{N}_{0}\text{c}ʻ\alpha\); \(\text{N}^{1}\text{C}\) is the class +of all such cardinals as \(\text{N}^{1}\text{c}ʻ\alpha\), and is a +typically ambiguous symbol, though \(\text{N}^{1}\text{c}ʻ\alpha\) +is typically definite when \(\alpha\) is given; \(\mu ^{(1)}\) (if +\(\mu\) is a cardinal which is not null) is the "same" cardinal in the +next higher type, so that, <i>e.g.</i>, if \(\mu\) is 1 determined +as consisting of unit classes of <i>individuals</i>, \(\mu ^{(1)}\) +will be 1 determined as consisting of unit classes of classes of +individuals. (When \(\mu\) is not an existent cardinal, \(\mu ^{(1)}\) +is unimportant.)</p> + +<p>The following are the most useful propositions in the present number:</p> + +<p class="nind"><b>*104·12.</b> \(\vdash :\beta \in \text{N}^{1}\text{c}ʻ\alpha .\gamma \in \text{N}^{1}\text{c}ʻ\beta .\supset .\gamma \in \text{N}^{2}\text{c}ʻ\alpha\)</p> + +<p><span class="pagenum" id="Page_43">[Pg 43]</span></p> + +<p class="nind"><b>*104·2.</b> \(\vdash .ʻʻ\alpha \in \text{N}^{1}\text{c}ʻ\alpha\)</p> + +<p class="nind"><b>*104·21.</b> \(\vdash .\exists !\text{N}^{1}\text{c}ʻ\alpha\)</p> + +<p class="nind"><b>*104·24.</b> \(\vdash :\mu = \text{N}^{1}\text{c}ʻ\alpha .\supset .\mu = \text{N}_{0}\text{c}ʻ\iota ʻʻ\alpha = \text{N}_{0}\text{c}ʻ\hat{\beta} + \{(\exists y).y\in \alpha .\beta = \iota ʻx\cup \iota ʻy\}\)</p> + +<p class="nind"><b>*104·25.</b> \(\vdash .\text{N}^{1}\text{C}\subset \text{C}\text{N}_{0}\text{C}\)</p> + +<p class="nind"><b>*104·26.</b> \(\vdash :\mu = \text{N}_{0}\text{c}ʻ\alpha .\supset .\mu ^{(1)} = \text{N}_{0}\text{c}ʻ\iota ʻʻ\alpha = \text{N}^{1}\text{c}ʻ\alpha\)</p> + +<p class="nind"><b>*104·265.</b> \(\vdash .\mu ^{(1)} = \text{ sm }_{\mu }ʻʻ\mu\)</p> + +<p class="nind"><b>*104·27.</b> \(\vdash \colon\ldotp \mu \in \text{NC}.\supset :\mu = \text{N}_{0}\text{c}ʻ\alpha . \equiv .\mu ^{(1)} = \text{N}^{1}\text{c}ʻ\alpha\)</p> + +<p class="nind"><b>*104·35.</b> \(\vdash .\text{N}^{2}\text{C}\subset \text{N}^{1}\text{C}.\text{N}^{2}\text{C}\subset \text{N}_{0}\text{C}\)</p> + +<p class="nind"><b>*104·43.</b> \(\vdash :tʻ\alpha = tʻ\beta .\supset .(\exists \gamma ,\delta ).\gamma \in \text{N}^{1}\text{c}ʻ\alpha .\delta \in \text{N}^{1}\text{c}ʻ\beta + .\gamma \cap \delta = \Lambda\)</p> + +<hr class="tb"> + +<p class="nind"><b>*104·01.</b> \(\text{N}^{1}\text{c}ʻ\alpha = \text{Nc}ʻ\alpha \cap tʻtʻ\alpha \quad\text{Df}\)</p> + +<p>This defines the cardinal number of \(\alpha\) in the next +type above that of \(\text{N}_{0}\text{c}ʻ\alpha\); thus +\(\text{N}^{1}\text{c}ʻ\alpha\) consists of all classes similar to +\(\alpha\) and of the next type above that of \(\alpha\).</p> + +<p class="nind"><b>*104·011.</b> \(\text{N}^{2}\text{c}ʻ\alpha = \text{Nc}ʻ\alpha \cap tʻt^{2}ʻ\alpha \quad\text{Df}\)</p> + +<p>Similar definitions are to be assumed for +\(\text{N}^{3}\text{c}ʻ\alpha\), etc.</p> + +<p class="nind"><b>*104·02.</b> \(\text{N}^{1}\text{C} = \text{D}ʻ\text{N}^{1}\text{c} \quad\text{Df}\)</p> + +<p>\(\text{N}^{1}\text{C}\), like \(\text{N}_{0}\text{C}\), is typically ambiguous; but \(\text{N}^{1}\text{C}(\alpha )\) is typically definite.</p> + +<p class="nind"><b>*104·021.</b> \(\text{N}^{2}\text{C} = \text{D}ʻ\text{N}^{2}\text{c} \quad\text{Df}\)</p> + +<p>Similar definitions are to be assumed for \(\text{N}^{3}\text{C}\), etc.</p> + +<p class="nind"><b>*104·03.</b> \(\mu ^{(1)} = \text{ sm }ʻʻ\mu \cap tʻ\mu \quad\text{Df}\)</p> + +<p>Here, if \(\mu\) is a cardinal, \(\mu ^{(1)}\) is the same cardinal +in the next higher type. For example, if \(\mu\) is couples of +individuals, \(\mu ^{(1)}\) is couples of classes of individuals.</p> + +<p class="nind"><b>*104·031.</b> \(\mu ^{(2)} = \text{ sm }ʻʻ\mu \cap t^{2}ʻ\mu \quad\text{Df}\)</p> + +<p>Similar definitions are to be assumed for \(\mu ^{(3)}\), etc.</p> + +<p class="nind"><b>*104·1.</b> \[\begin{aligned}&\vdash :\beta \in \text{N}^{1}\text{c}ʻ\alpha + . \equiv .\beta \in \text{Nc}ʻ\alpha .\beta \in tʻtʻ\alpha . \equiv .\beta \in \text{Nc}ʻ\alpha .\beta \subset tʻ\alpha \\ +&[*63·5. (*104-01)]\end{aligned}\]</p> + +<p class="nind"><b>*104·101.</b> \(\vdash :\beta \in \text{N}^{1}\text{c}ʻ\alpha . \equiv .\beta \text{ sm } \alpha .\beta \subset tʻ\alpha \quad[*100·31.*104·1]\)</p> + +<p class="nind"><b>*104·102.</b> \(\vdash .\text{N}^{1}\text{c}ʻ\alpha = \text{Nc}(tʻ\alpha )ʻ\alpha = \text{Nc} {(tʻ\alpha )_{\alpha }}ʻ\alpha \quad[*102·6.(*104·01)]\)</p> + +<p class="nind"><b>*104·11.</b> \[\begin{aligned}&\vdash :\beta \in \text{N}^{2}\text{c}ʻ\alpha . \equiv .\beta \in \text{Nc}ʻ\alpha . \beta \in tʻt^{2}ʻ\alpha + . \equiv .\beta \in \text{Nc}ʻ\alpha .\beta \subset t^{2}ʻ\alpha\\ +&[*63·5.(*104·011)]\end{aligned}\]</p> + +<p class="nind"><b>*104·111.</b> \(\vdash :\beta \in \text{N}^{2}\text{c}ʻ\alpha . \equiv .\beta \text{ sm } \alpha .\beta \subset t^{2}ʻ\alpha \quad[*100·31.*104·11]\)</p> + +<p><span class="pagenum" id="Page_44">[Pg 44]</span></p> + +<p class="nind"><b>*104·112.</b> \(\vdash .\text{N}^{2}\text{c}ʻ\alpha = \text{Nc}(t^{2}ʻ\alpha )ʻ\alpha = \text{Nc}{(t^{2}ʻ\alpha )_{\alpha }}ʻ\alpha + \quad[*102·6.(*104·011)]\)</p> + +<p class="nind"><b>*104·12.</b> \(\vdash :\beta \in \text{N}^{1}\text{c}ʻ\alpha .\gamma \in \text{N}^{1}\text{c}ʻ\beta .\supset .\gamma \in \text{N}^{2}\text{c}ʻ\alpha\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*104·1.\supset \vdash :\text{Hp}.&\supset .\beta \in \text{Nc}ʻ\alpha .\beta \in tʻtʻ\alpha .\gamma \in \text{Nc}ʻ\beta .\gamma \in tʻtʻ\beta .\\ +[*100·32] &\supset .\gamma \in \text{Nc}ʻ\alpha .\beta \in tʻtʻ\alpha .\gamma \in tʻtʻ\beta .\\ +[*63·16] &\supset .\gamma \in \text{Nc}ʻ\alpha .tʻ\beta = tʻtʻ\alpha .\gamma \in tʻtʻ\beta .\\ +[*13·12] &\supset .\gamma \in \text{Nc}ʻ\alpha .\gamma \in tʻtʻtʻ\alpha .\\ +[*104·11] &\supset .\gamma \in \text{N}^{2}\text{c}ʻ\alpha :\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*104·121.</b> \(\vdash :\beta \in \text{N}^{1}\text{c}ʻ\alpha .\gamma \in \text{N}^{2}\text{c}ʻ\alpha .\supset .\gamma \in \text{N}^{1}\text{c}ʻ\beta \)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*104·102·112.\supset \vdash :\text{Hp}.&\supset .\beta \in \text{Nc}\{(tʻ\alpha)_{\alpha}\} +ʻ\alpha .\gamma \in \text{Nc}\{(t^{2}ʻ\alpha)_{\alpha}\}ʻ\alpha.\\ +[*102·41] &\supset .\gamma \in \text{Nc}\{(t^{2}ʻ\alpha )_{tʻ\alpha}\}ʻ\beta &\qquad \text{(1)}\\ +\vdash .*104·1.\supset \vdash :\text{Hp}.&\supset .\beta \in tʻtʻ\alpha .\\ +[*63*16] &\supset .tʻ\beta = tʻtʻ\alpha .\\ +[(*65·11)] &\supset .\text{Nc}\{(t^{2}ʻ\alpha )_{tʻ\alpha}\} = \text{Nc}\{(tʻ\beta)_{\beta}\} &\qquad \text{(2)}\\ +\vdash .(1).(2).*104·102.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*104·122.</b> \(\vdash :\beta \in \text{N}^{1}\text{c}ʻ\alpha .\supset .\text{N}^{1}\text{c}ʻ\beta = \text{N}^{2}\text{c}ʻ\alpha \quad[*104·12·121]\)</p> + +<p class="nind"><b>*104·123.</b> \(\vdash :\text{N}_{0}\text{c}ʻ\beta = \text{N}^{1}\text{c}ʻ\alpha .\supset . \text{N}^{1}\text{c}ʻ\beta = \text{N}^{2}\text{c}ʻ\alpha + \quad[*104·122.*103·26]\)</p> + +<p class="nind"><b>*104·13.</b> \(\vdash :\mu \in \text{N}^{1}\text{C}. \equiv .(\exists \alpha ).\mu = \text{N}^{1}\text{c}ʻ\alpha \quad[*100·122.*71·41.(*104·02)]\)</p> + +<p class="nind"><b>*104·14.</b> \[\begin{aligned}&\vdash :\delta \in \mu ^{(1)}. + \equiv .(\exists \gamma ).\gamma \in \mu .\delta \text{ sm } \gamma .\delta \in tʻ\mu . \equiv .(\exists \gamma ).\gamma \in \mu .\delta \text{ sm } \gamma .\delta \subset tʻ\gamma\\ +&[*37·1.*63·22.(*104·03)]\end{aligned}\]</p> + +<p class="nind"><b>*104·141.</b> \(\vdash :\mu \in \text{NC}.\exists !\mu .\supset .\mu ^{(1)}\in \text{NC} \quad[*100·52]\)</p> + +<p>When the hypothesis "\(\exists !\mu\)" is omitted, this proposition is +still true, but with a difference. <i>E.g.</i> let us put +\[ +\mu = \text{Nc}(\alpha )ʻtʻ\alpha . +\] +Then \(\mu = \Lambda .\mu ^{(1)} = \Lambda\). Thus \(\mu^{(1)} \neq +\text{Nc}(tʻ\alpha)ʻtʻ\alpha\). But we still have +\[ +\mu ^{(1)} = \text{Nc}(tʻ\alpha )ʻt^{2}ʻ\alpha . +\] +Thus \(\mu^{(1)}\in \text{NC}\), but \(\mu^{(1)}\) is not the same +cardinal as \(\mu\) in a higher type, <i>i.e.</i> there are classes +whose cardinal in one type is \(\mu\), but whose cardinal in the next +higher type is not \(\mu ^{(1)}\).</p> + +<p class="nind"><b>*104·142.</b> \(\vdash :\mu \in \text{NC}.\exists !\mu .\supset .\mu ^{(2)}\in \text{NC} \quad[*100·52]\)</p> + +<p class="nind"><b>*104·15.</b> \(\vdash :\mu \in \text{N}^{2}\text{C}. \equiv .(\exists \alpha ).\mu = \text{N}^{2}\text{c}ʻ\alpha \quad[*100·22.*71·41. (*104·021)]\)</p> + +<p class="nind"><b>*104·2.</b> \(\vdash .\iota ʻʻ\alpha \in \text{N}^{1}\text{c}ʻ\alpha\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*63·621.&\supset \vdash x\in \alpha .\supset _{x}.\iota ʻx\in tʻ\alpha :\\ +[*37·61] &\supset \vdash .\iotaʻʻ\alpha \subset tʻ\alpha &\qquad \text{(1)}\\ +\vdash .(1).*100·6.*104·1.\supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_45">[Pg 45]</span></p> + +<p class="nind"><b>*104·201.</b> \(\vdash :\beta \in \text{N}_{0}\text{c}ʻ\alpha .\supset .\iotaʻʻ\beta \in \text{N}^{1}\text{c}ʻ\alpha .\text{N}^{1}\text{c}ʻ\alpha + =\text{N}^{1}\text{c}ʻ\beta\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*100·31·321.\supset \vdash :\text{Hp}.&\supset .\text{Nc}ʻ\alpha =\text{Nc}ʻ\beta &\qquad \text{(1)}\\ +\vdash .*103·11. \supset \vdash :\text{Hp}.&\supset .\beta \in tʻ\alpha .\\ +[*63·16] &\supset .tʻ\alpha =tʻ\beta .\\ +[*30·37] &\supset .tʻtʻ\alpha =tʻtʻ\beta &\qquad \text{(2)}\\ +\vdash .(1).(2).(*104·01).&\supset \vdash .\text{N}^{1}\text{c}ʻ\alpha =\text{N}^{1}\text{c}ʻ\beta &\qquad \text{(3)}\\ +\vdash .(3).*104·2.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*104·21">*104·21</a>.</b> \(\vdash .\exists !\text{N}^{1}\text{c}ʻ\alpha \quad[*104·2]\)</p> + +<p>It follows from this proposition that <i>ascending</i> cardinals +are never null. The proof has to be made separately for each kind +of ascending cardinal, <i>i.e.</i> \(\text{N}^{1}\text{C}\), +\(\text{N}^{2}\text{C}\), etc.</p> + +<p class="nind"><b>*104·211.</b> \(\vdash .\exists !\text{N}^{1}\text{c}ʻ\alpha \cap \text{Cl}ʻ1 \quad[*104·2.*52·3]\)</p> + +<p class="nind"><b>*104·23.</b> \(\vdash .\hat{\beta} \{(\exists y).y\in \alpha .\beta =\iota ʻx\cup \iota ʻy\}\in \text{N}^{1}\text{c}ʻ\alpha\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*51·16. &\supset \vdash :y\in \alpha .\supset y\in \alpha \cap (\iota ʻx\cup \iota ʻy).\\ +[*63·16] &\supset .\iota ʻx\cup \iota ʻy\in tʻ\alpha &\qquad \text{(1)}\\ +\vdash .(1).*10·11·23.&\supset \vdash .\hat{\beta} \{(\exists y).y\in \alpha .\beta =\iota ʻx\cup \iota ʻy\}\subset tʻ\alpha &\qquad \text{(2)}\\ +\vdash .(2).*100·61.*104·1.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*104·231.</b> \(\vdash :\text{N}^{1}\text{c}ʻ\alpha =\text{N}^{1}\text{c}ʻ\beta .\supset .\text{N}_{0}\text{c}ʻ\alpha =\text{N}_{0}\text{c}ʻ\beta\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*104·2.\supset \vdash :\text{Hp}.&\supset .\iotaʻʻ\beta \in \text{N}_{1}\text{c}ʻ\alpha .\\ +[*104·101] &\supset .\iotaʻʻ\beta \text{ sm }\alpha .\iotaʻʻ\beta \subset tʻ\alpha .\\ +[*73·41.*63·21·64] &\supset .\beta \text{ sm }\alpha .tʻ\beta =tʻ\alpha .\\ +[*103·11.*63·16] &\supset .\beta \in \text{N}_{0}\text{c}ʻ\alpha .\\ +[*103·14] &\supset .\text{N}_{0}\text{c}ʻ\alpha =\text{N}_{0}\text{c}ʻ\beta :\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*104·232.</b> \[\begin{aligned}&\vdash :\text{N}^{1}\text{c}ʻ\alpha =\text{N}^{1}\text{c}ʻ\beta .\equiv .\text{N}_{0}\text{c}ʻ\alpha =\text{N}_{0}\text{c}ʻ\beta + .\equiv .\beta \in \text{N}_{0}\text{c}ʻ\alpha\\ +&[*104·231·201.*103·14]\end{aligned}\]</p> + +<p class="nind"><b>*104·24.</b> \[\begin{aligned}&\vdash :\mu =\text{N}^{1}\text{c}ʻ\alpha .\supset .\mu =\text{N}_{0}\text{c}ʻ\iotaʻʻ\alpha =\text{N}_{0}\text{c}ʻ\hat{\beta} + \{(\exists y).y\in \alpha .\beta =\iota ʻx\cup \iota ʻy\}\\ +&[*104·2·23.*103·26]\end{aligned}\]</p> + +<p class="nind"><b>*104·25.</b> \(\vdash .\text{N}^{1}\text{C}\subset \text{N}_{0}\text{C} \quad[*104·24·13]\)</p> + +<p>This proposition holds for each possible determination of the typical +ambiguities, <i>i.e.</i> for every \(\alpha\) we have +\[ +\begin{aligned} +\text{N}^{1}\text{C}(tʻ\alpha )&\subset \text{N}_{0}\text{C}(tʻ\alpha ).\\ +\text{We do not have}\qquad \text{N}^{1}\text{C}(tʻ\alpha )&=\text{N}_{0}\text{C}(tʻ\alpha ),\\ +\text{because}\qquad\qquad\qquad \text{N}_{0}\text{c}ʻtʻ\alpha &\in \text{N}_{0}\text{C}(tʻ\alpha )-\text{N}^{1}\text{C}(tʻ\alpha ). +\end{aligned} +\]</p> + +<p><span class="pagenum" id="Page_46">[Pg 46]</span></p> + +<p class="nind"><b>*104·251.</b> \(\vdash .\Lambda {\sim}\in \text{N}^{1}\text{C} \quad[*104·25.*103·23]\)</p> + +<p class="nind"><b>*104·252.</b> \(\vdash .\text{N}^{1}\text{C}\in \text{Cls ex}^{2}\,\text{excl} \quad[*104·25.*103·24.*84·26]\)</p> + +<p class="nind"><b>*104·26.</b> \(\vdash :\mu =\text{N}_{0}\text{c}ʻ\alpha .\supset .\mu ^{(1)}=\text{N}_{0}\text{c}ʻ\iotaʻʻ\alpha =\text{N}^{1}\text{c}ʻ\alpha\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*104·14.*103·11.\supset \\ +\vdash \colon\ldotp \text{Hp}.\supset :\delta \in \mu ^{(1)}.&\equiv + .(\exists \gamma ).\gamma \text{ sm }\alpha .\gamma \in tʻ\alpha .\delta \text{ sm }\gamma .\delta \subset tʻ\gamma . &\qquad \text{(1)}\\ +[*73·32.*63·16] &\supset .\delta \text{ sm }\alpha .\delta \subset tʻ\alpha .\\ +[*104·101] &\supset .\delta \in \text{N}^{1}\text{c}ʻ\alpha &\qquad \text{(2)}\\ +\vdash .*104·101.\supset \\ +\vdash :\delta \in \text{N}^{1}\text{c}ʻ\alpha . &\supset .\delta \text{ sm }\alpha .\delta \subset tʻ\alpha .\\ +[*73·3.*63·103]&\supset .\alpha \text{ sm }\alpha .\alpha \in tʻ\alpha .\delta \text{ sm }\alpha .\delta \subset tʻ\alpha .\\ +[*10·24] &\supset .(\exists \gamma ).\gamma \text{ sm }\alpha .\gamma \in tʻ\alpha .\delta \text{ sm }\gamma .\delta \subset tʻ\gamma &\qquad \text{(3)}\\ +\vdash .(3).(1).&\supset \vdash \colon\ldotp \text{Hp}.\supset :\delta \in \text{N}^{1}\text{c}ʻ\alpha .\supset .\delta \in \mu ^{(1)} &\qquad \text{(4)}\\ +\vdash .(2).(4).*104·24.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*104·261.</b> \(\vdash :\mu ^{(1)}=\text{N}^{1}\text{c}ʻ\alpha .\supset .\mu \subset \text{N}_{0}\text{c}ʻ\alpha\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*104·14·101.\supset \\ +\vdash \colon\ldotp \text{Hp}. \supset :(\exists \gamma ).\gamma \in \mu .\delta \text{ sm }\gamma .\delta &\subset tʻ\gamma .\equiv _{\delta }.\delta + \text{ sm }\alpha .\delta \subset tʻ\alpha :\\ +[*10·23] \supset :\gamma \in \mu .\delta \text{ sm }\gamma .\delta \subset tʻ\gamma .&\supset _{\gamma ,\delta }.\delta \text{ sm }\alpha .\delta \subset tʻ\alpha .\\ +[*4·7] &\supset _{\gamma ,\delta }.\delta \text{ sm }\alpha .\delta \text{ sm }\gamma .\delta \subset tʻ\alpha .\delta \subset tʻ\gamma .\\ +[*73·32.*63·13] &\supset _{\gamma ,\delta }.\gamma \text{ sm }\alpha .\gamma \in tʻ\alpha .\\ +[*103·11] &\supset _{\gamma ,\delta }.\gamma \in \text{N}_{0}\text{c}ʻ\alpha &\qquad \text{(1)}\\ +\vdash .(1).*10·23·35.*104·101.\supset \\ +\vdash \colon\ldotp \text{Hp}. \supset :\gamma \in \mu .\exists !\text{N}^{1}\text{c}ʻ\gamma .\supset _{\gamma }.\gamma \in \text{N}_{0}\text{c}ʻ\alpha :\\ +[*104·21] \supset :\gamma \in \mu .\supset _{\gamma }.\gamma \in \text{N}_{0}\text{c}ʻ\alpha \colon\ldotp \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*104·262.</b> \(\vdash :\mu \in \text{NC}.\mu ^{(1)}=\text{N}^{1}\text{c}ʻ\alpha .\supset .\mu =\text{N}_{0}\text{c}ʻ\alpha\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*104·21.\supset \vdash :\text{Hp}.&\supset .\exists !\mu ^{(1)}.\\ +[*37·29.\text{Transp}] &\supset .\exists !\mu &\qquad \text{(1)}\\ +\vdash .*103·26.&\supset \vdash :\text{Hp}.\gamma \in \mu .\supset .\mu =\text{N}_{0}\text{c}ʻ\gamma &\qquad \text{(2)}\\ +\vdash .(1).(2).\supset \vdash :\text{Hp}.&\supset .(\exists \gamma ).\mu =\text{N}_{0}\text{c}ʻ\gamma .\\ +[*104·26.\text{Hp}] &\supset .(\exists \gamma ).\mu =\text{N}_{0}\text{c}ʻ\gamma .\text{N}^{1}\text{c}ʻ\alpha =\text{N}^{1}\text{c}ʻ\gamma .\\ +[*104·231] &\supset .(\exists \gamma ).\mu =\text{N}_{0}\text{c}ʻ\gamma .\text{N}_{0}\text{c}ʻ\alpha =\text{N}_{0}\text{c}ʻ\gamma .\\ +[*13·172] &\supset .\mu =\text{N}_{0}\text{c}ʻ\alpha :\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*104·263.</b> \(\vdash :\alpha \in \mu .\supset .\iotaʻʻ\alpha \in \mu ^{(1)}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*73·41.*37·1.&\supset \vdash :\text{Hp}.\supset .\iotaʻʻ\alpha \in \text{ sm }ʻʻ\mu &\qquad \text{(1)}\\ +\vdash .*63·64. &\supset \vdash :\text{Hp}.\supset .\iotaʻʻ\alpha \in tʻ\mu &\qquad \text{(2)}\\ +\vdash .(1).(2).(*104·03).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_47">[Pg 47]</span></p> + +<p class="nind"><b>*104·264.</b> \(\vdash :\exists !\mu .\equiv .\exists !\mu ^{(1)}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*104·263. &\supset \vdash :\exists !\mu .\supset .\exists !\mu ^{(1)} &\qquad \text{(1)}\\ +\vdash .*37·29.\text{Transp}.(*104·03).&\supset \vdash :\exists !\mu ^{(1)}.\supset .\exists !\mu &\qquad \text{(2)}\\ +\vdash .(1).(2).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*104·265.</b> \(\vdash .\mu ^(1)=\text{ sm }_\muʻʻ\mu \quad[*102·85.(*104·03)]\)</p> + +<p class="nind"><b>*104·27.</b> \(\vdash \colon\ldotp \mu \in \text{NC}.\supset :\mu =\text{N}_{0}\text{c}ʻ\alpha .\equiv .\mu ^{(1)}=\text{N}^{1}\text{c}ʻ\alpha \quad[*104·26·262]\)</p> + +<p class="nind"><b>*104·28.</b> \(\vdash :\mu \in \text{NC}-\iota ʻ\Lambda .\supset .\mu ^{(1)}\in \text{N}^{1}\text{C} \quad[*104·26.*103·34]\)</p> + +<p class="nind"><b>*104·29.</b> \(\vdash :\nu \in \text{N}^{1}\text{C}.\equiv .(\exists \mu )·\mu \in \text{N}_{0}\text{C}.\nu =\mu ^{(1)}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*104·26. \supset \vdash :\mu =\text{N}_{0}\text{c}ʻ\alpha .\nu =\mu ^{(1)}.\supset .\nu =\text{N}^{1}\text{c}ʻ\alpha :\\ +[*10·11·28] \supset \vdash :(\exists \alpha ).\mu =\text{N}_{0}\text{c}ʻ\alpha .\nu =\mu ^{(1)}.\supset .(\exists \alpha ).\nu =\text{N}^{1}\text{c}ʻ\alpha :\\ +[*103·2.*104·13] \supset \vdash :\mu \in \text{N}_{0}\text{C}.\nu =\mu ^{(1)}.\supset .\nu \in \text{N}^{1}\text{C} &\qquad \text{(1)}\\ +\vdash .*104·26.*103·2.\supset \\ +\vdash :\nu =\text{N}^{1}\text{c}ʻ\alpha .\mu =\text{N}_{0}\text{c}ʻ\alpha .\supset .\nu =\mu ^{(1)}.\mu \in \text{N}_{0}\text{C} &\qquad \text{(2)}\\ +\vdash .(2).*10·11·28·35.\supset \\ +\vdash \colon\ldotp \nu =\text{N}^{1}\text{c}ʻ\alpha :(\exists \mu ).\mu =\text{N}_{0}\text{c}ʻ\alpha :\supset .(\exists \mu ).\mu \in \text{N}_{0}\text{C}.\nu =\mu ^{(1)} + &\qquad \text{(3)}\\ +\vdash .(3).*100·2.*14·204.\supset \\ +\vdash :\nu =\text{N}^{1}\text{c}ʻ\alpha .\supset .(\exists \mu ).\mu \in \text{N}_{0}\text{C}.\nu =\mu ^{(1)} &\qquad \text{(4)}\\ +\vdash .(4).*10·11·23.*104·13.\supset \\ +\vdash :\nu \in \text{N}^{1}\text{C}.\supset .(\exists \mu ).\mu \in \text{N}_{0}\text{C}.\nu =\mu ^{(1)} &\qquad \text{(5)}\\ +\vdash .(1).(5).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*104·3.</b> \(\vdash .\iotaʻʻ\iotaʻʻ\alpha \in \text{N}^{2}\text{c}ʻ\alpha\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*104·2.&\supset \vdash .\iotaʻʻ\alpha \in \text{N}^{1}\text{c}ʻ\alpha .\iotaʻʻ\iotaʻʻ\alpha \in \text{N}^{1}\text{c}ʻ\iotaʻʻ\alpha .\\ +[*104·12] &\supset \vdash .\iotaʻʻ\iotaʻʻ\alpha \in \text{N}^{2}\text{c}ʻ\alpha +\end{array} +\]</p> + +<p class="nind"><b>*104·31.</b> \(\vdash .\exists !\text{N}^{2}\text{c}ʻ\alpha \quad[*104·3]\)</p> + +<p class="nind"><b>*104·311.</b> \(\vdash .\text{N}^{2}\text{c}ʻ\alpha =\text{N}_{0}\text{c}ʻ\iotaʻʻ\iotaʻʻ\alpha =\text{N}^{1}\text{c}ʻ\iotaʻʻ\alpha \quad[*104·3·2.*103·26]\)</p> + +<p class="nind"><b>*104·32.</b> \(\vdash :\mu =\text{N}_{0}\text{c}ʻ\alpha .\supset .\mu ^{(2)}=\text{N}_{0}\text{c}ʻ\iotaʻʻ\iotaʻʻ\alpha =\text{N}^{1}\text{c}ʻ\iotaʻʻ\alpha + =\text{N}^{2}\text{c}ʻ\alpha =\{\mu ^{(1)}\}^{(1)}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*104·26.\supset \vdash :\text{Hp}.\supset .\{\mu ^{(1)}\}^{(1)}&=\text{N}_{0}\text{c}ʻ\iotaʻʻ\iotaʻʻ\alpha &\qquad \text{(1)}\\ +[*104·311] &=\text{N}^{2}\text{c}ʻ\alpha &\qquad \text{(2)}\\ +\vdash .*103·11.(*104·031).\supset \\ +\vdash \colon\ldotp \text{Hp}.\supset :\delta \in \mu ^{(2)}.&\equiv .(\exists \gamma ).\gamma \text{ sm }\alpha .\gamma \in tʻ\alpha .\delta \text{ sm }\gamma .\delta \in tʻt^{2}ʻ\gamma + .\\ +[*102·84.*63·16] &\equiv .\delta \text{ sm }\alpha .\delta \in tʻt^{2}ʻ\alpha .\\ +[*104·11] &\equiv .\delta \in \text{N}^{2}\text{c}ʻ\alpha &\qquad \text{(3)}\\ +\vdash .(1).(2).(3).*104·24.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_48">[Pg 48]</span></p> + +<p class="nind"><b>*104·33.</b> \(\vdash \colon\ldotp \mu \in \text{NC}.\supset :\mu =\text{N}_{0}\text{c}ʻ\alpha .\equiv .\mu ^{(2)}=\text{N}^{2}\text{c}ʻ\alpha\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*104·27.\supset \vdash \colon\ldotp \text{Hp}.\supset :\mu =\text{N}_{0}\text{c}ʻ\alpha .&\equiv .\mu ^{(1)}=\text{N}^{1}\text{c}ʻ\alpha .\\ +[*104·24] &\equiv .\mu ^{(1)}=\text{N}_{0}\text{c}ʻ\iotaʻʻ\alpha .\\ +[*104·27·141.*103·13] &\equiv .\{\mu ^{(1)}\}^{(1)}=\text{N}^{1}\text{c}ʻ\iotaʻʻ\alpha .\\ +[*104·32·24] &\equiv .\mu ^{(2)}=\text{N}^{2}\text{c}ʻ\alpha \colon\ldotp \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*104·34.</b> \(\vdash :\varpi \in \text{N}^{2}\text{C}.\equiv .(\exists \nu ).\nu \in \text{N}^{1}\text{C}.\varpi =\nu ^{(1)}.\equiv + .(\exists \mu ).\mu \in \text{N}_{0}\text{C}.\varpi =\mu ^{(2)}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*104·32.\supset \\ +\vdash :\varpi =\text{N}^{2}\text{c}ʻ\alpha .\mu =\text{N}_{0}\text{c}ʻ\alpha .\supset .\varpi =\mu ^{(2)}.\mu \in \text{N}_{0}\text{C} &\qquad \text{(1)}\\ +\vdash .(1).*100·2.*10·11·28·35.\supset\\ +\vdash :(\exists \alpha ).\varpi =\text{N}^{2}\text{c}ʻ\alpha .\supset .(\exists \mu ).\mu \in \text{N}_{0}\text{C}.\varpi =\mu ^{(2)} &\qquad \text{(2)}\\ +\vdash *104·32. \supset \vdash :\mu =\text{N}_{0}\text{C}ʻ\alpha .\varpi =\mu ^{(2)}.\supset .\varpi =\text{N}^{2}\text{c}ʻ\alpha .\\ +[*104·15.*103·2]\supset \vdash :\mu \in \text{N}_{0}\text{C}.\varpi =\mu ^{(2)}.\supset .\varpi \in \text{N}^{2}\text{C} &\qquad \text{(3)}\\ +\vdash (2).(3). \supset \vdash :\varpi \in \text{N}^{2}\text{C}.\equiv .(\exists \mu ).\mu \in \text{N}_{0}\text{C}.\varpi =\mu ^{(2)}. &\qquad \text{(4)}\\ +[*104·32] \equiv .(\exists \mu ).\mu \in \text{N}_{0}\text{C}.\varpi =\{\mu ^{(1)}\}^{(1)}.\\ +[*13·195] \equiv .(\exists \mu ,\nu ).\mu \in \text{N}_{0}\text{C}.\nu =\mu ^{(1)}.\varpi =\nu^{(1)}. +[*104·29] \equiv .(\exists \nu).\nu \in \text{N}^{1}\text{C}.\varpi =\nu ^{(1)} &\qquad \text{(5)}\\ +\vdash .(4).(5).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*104·35.</b> \(\vdash .\text{N}^{2}\text{C}\subset \text{N}^{1}\text{C}.\text{N}^{2}\text{C}\subset \text{N}_{0}\text{C} \quad[*104·311·13·15]\)</p> + +<p class="nind"><b>*104·36.</b> \(\vdash :\gamma \in \text{N}^{2}\text{c}ʻ\alpha .\gamma \in \text{N}^{1}\text{c}ʻ\beta .\supset .\beta \in \text{N}^{1}\text{c}ʻ\alpha .\text{N}^{1}\text{c}ʻ\alpha + =\text{N}_{0}\text{c}ʻ\beta\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*104·1·11.\supset \vdash :\text{Hp}.&\supset .\gamma \in \text{Nc}ʻ\alpha .\gamma \in tʻt^{2}ʻ\alpha .\gamma \in \text{Nc}ʻ\beta .\gamma \in tʻtʻ\beta .\\ +[*100·34.*63·16] & \supset .\text{Nc}ʻ\alpha =\text{Nc}ʻ\beta .tʻt^{2}ʻ\alpha =tʻtʻ\beta .\\ +[*63·35·15] &\supset .\text{Nc}ʻ\alpha =\text{Nc}ʻ\beta .t^{2}ʻ\alpha =tʻ\beta .\\ +[(*104·01.*103·01)] &\supset .\text{N}^{1}\text{c}ʻ\alpha =\text{N}_{0}\text{c}ʻ\beta &\qquad \text{(1)}\\ +\vdash .(1).*103·12.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*104·37.</b> \(\vdash :\text{N}^{2}\text{c}ʻ\alpha =\text{N}^{1}\text{c}ʻ\beta .\equiv .\text{N}^{1}\text{c}ʻ\alpha =\text{N}_{0}\text{c}ʻ\beta\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*104·21.\supset \vdash :\text{N}^{2}\text{c}ʻ\alpha =\text{N}^{1}\text{c}ʻ\beta .&\supset .\exists !\text{N}^{2}\text{c}ʻ\alpha \cap \text{N}^{1}\text{c}ʻ\beta.\\ +[*104·36] &\supset .\text{N}^{1}\text{c}ʻ\alpha =\text{N}_{0}\text{c}ʻ\beta &\qquad \text{(1)}\\ +\vdash .(1).*104·123.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p>The following propositions are concerned with the proof that, given +any two cardinals \(\mu\) and \(\nu\), of the same type, we can find +two mutually exclusive classes one of which has \(\mu\) terms while +the other has \(\nu\) terms. The proof requires that we should raise +the types of both \(\mu\) and \(\nu\) one degree above<span class="pagenum" id="Page_49">[Pg 49]</span> that in which +they were originally given, <i>i.e.</i> that we should turn \(\mu\) +and \(\nu\) into \(\mu ^{(1)}\) and \(\nu ^{(1)}\). Thus, for example, +suppose the total number of individuals in the universe were finite (a +supposition which is consistent with our primitive propositions), and +suppose \(\mu\) were this number. Then unless \(\nu =0\), a class of +\(\nu\) individuals will be an existent sub-class of the only class +which consists of \(\mu\) individuals, and therefore we shall have +\[ +\alpha \in \mu .\beta \in \nu .\supset _{\alpha ,\beta }.\exists !\alpha \cap \beta . +\]</p> + +<p>But if we consider classes of \(\mu\) classes and \(\nu\) classes, we +shall always be able to find a \(\gamma\) and a \(\delta\) such that +\[ +\gamma \in \mu ^{(1)}.\delta \in \nu^{(1)}.\gamma \cap \delta =\Lambda . +\]</p> + +<p>The existence of such a \(\gamma\) and \(\delta\) is important in +connection with the arithmetical operations, and is therefore proved +here.</p> + +<p class="nind"><b>*104·4.</b> \[\begin{aligned}\vdash \colon\ldotp x\in \alpha .x\neq y.x\neq z.y\neq z:&(w).w_{\iota }=\hat{\alpha}\hat{u}(\alpha =\iota ʻw\cup \iota ʻu):\supset .\\ +&x_{\iota }ʻʻ(\alpha -\iota ʻx)\cup \iota ʻy_{\iota }ʻz\in \text{N}^{1}\text{c}ʻ\alpha \cap \text{Cl}ʻ2\end{aligned}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*100·61. &\supset \vdash :\text{Hp}.\supset .x_{\iota }ʻʻ(\alpha -\iota ʻx)\text{ sm }(\alpha -\iota ʻx) &\qquad \text{(1)}\\ +\vdash .*73·43. &\supset \vdash :\text{Hp}.\supset .\iota ʻy_{\iota }ʻz\text{ sm }\iota ʻx &\qquad \text{(2)}\\ +\vdash .*51·232.\text{Transp}.&\supset \vdash :\text{Hp}.\supset .x{\sim}\in y_{\iota }ʻz &\qquad \text{(3)}\\ +\vdash .*51·232. &\supset \vdash :\text{Hp}.\gamma \in x_{\iota }ʻʻ(\alpha -\iota ʻx).\supset .x\in \gamma &\qquad \text{(4)}\\ +\vdash .(3).(4). \supset \vdash :\text{Hp}.&\supset .y_{\iota }ʻz{\sim}\in x_{\iota }ʻʻ(\alpha -\iota ʻx).\\ +[*51·211] &\supset .x_{\iota }ʻʻ(\alpha -\iota ʻx)\cap \iota ʻy_{\iota }ʻz=\Lambda &\qquad \text{(5)}\\ +\vdash .*51·21·211. &\supset \vdash .(\alpha -\iota ʻx)\cap \iota ʻx=\Lambda &\qquad \text{(6)}\\ +\vdash .(1).(2).(5).(6).*73·71.*51·221.\supset \\ +\vdash :\text{Hp}.&\supset .x_{\iota }ʻʻ(\alpha -\iota ʻx)\cup \iota ʻy_{\iota }ʻz\text{ sm }\alpha &\qquad \text{(7)}\\ +\vdash .*63·101·16.*51·232·16.\supset \\ +\vdash :\text{Hp}. &\supset .tʻx=tʻy.x\in \alpha .y\in y_{\iota }ʻz.y_{\iota }ʻz\in x_{\iota }ʻʻ(\alpha -\iota ʻx)\cup \iota ʻy_{\iota }ʻz.\\ +[*63·53·2] &\supset .t^{2}ʻx=tʻ\alpha .t^{2}ʻy=t_{0}ʻ\{x_{\iota }ʻʻ(\alpha -\iota ʻx)\cup \iota ʻy_{\iota}ʻz\}.t^{2}ʻx=t^{2}ʻy.\\ +[*13·17] &\supset .tʻ\alpha =t_{0}ʻ\{x_{\iota }ʻʻ(\alpha -\iota ʻx)\cup \iota ʻy_{\iota }ʻz\}.\\ +[*63·105] &\supset .x_{\iota }ʻʻ(\alpha -\iota ʻx)\cup \iota ʻy_{\iota }ʻz\subset tʻ\alpha &\qquad \text{(8)}\\ +\vdash .*54·26.&\supset \vdash :\text{Hp}.\supset .x_{\iota }ʻʻ(\alpha -\iota ʻx)\cup \iota ʻy_{\iota }ʻz\subset 2 &\qquad \text{(9)}\\ +\vdash .(7).(8).(9).*104·101.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*104·41</b> \[\begin{aligned}\vdash \colon\ldotp tʻ\alpha =tʻ\beta :(\exists x,y,z).x\in \alpha .&x\neq y.x\neq z.y\neq z:\supset .\\ +&(\exists \gamma ,\delta ).\gamma \in \text{N}^{1}\text{c}ʻ\alpha .\delta \in \text{N}^{1}\text{c}ʻ\beta .\gamma \cap \delta =\Lambda\end{aligned}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*104·4·2.*52·3.\supset \\ +\vdash :\text{Hp}.\text{Hp}*104·4.&\supset .(\exists x,y,z).x_{\iota }ʻʻ(\alpha -\iota ʻx)\cup \iota ʻy_{\iota }ʻz\in \text{N}^{1}\text{c}ʻ\alpha \cap \text{Cl}ʻ2.\\ +&\iotaʻʻ\beta \in \text{N}^{1}\text{c}ʻ\beta \cap \text{Cl}ʻ1.\\ +[*13·22] &\supset .(\exists x,y,z,\gamma ,\delta ).\gamma =x_{\iota }ʻʻ(\alpha -\iota ʻx)\cup \iota ʻy_{\iota }ʻz.\delta =\iotaʻʻ\beta .\\ +&\gamma \in \text{N}^{1}\text{c}ʻ\alpha \cap \text{Cl}ʻ2.\delta \in \text{N}^{1}\text{c}ʻ\beta \cap \text{Cl}ʻ1.\\ +[*11·55] &\supset .(\exists \gamma ,\delta ).\gamma \in \text{N}^{1}\text{c}ʻ\alpha \cap \text{Cl}ʻ2.\delta \in \text{N}^{1}\text{c}ʻ\beta + \cap \text{Cl}ʻ1 &\qquad \text{(1)}\\ +\vdash .(1).*101·35.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_50">[Pg 50]</span></p> + +<p>This proposition proves the desired conclusions provided \(\exists!\alpha\), +and \(t_{0}ʻ\alpha\) consists of at least three terms. The +following propositions deal with the cases in which this hypothesis is +not verified.</p> + +<p class="nind"><b>*104·411.</b> \(\vdash :tʻ\alpha =tʻ\beta .\alpha \in 0.\gamma =\Lambda _{\alpha }.\delta =\iotaʻʻ\beta .\supset .\gamma \in \text{N}^{1}\text{c}ʻ\alpha + .\delta \in \text{N}^{1}\text{c}ʻ\beta .\gamma \cap \delta =\Lambda\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*73·47. &\supset \vdash :\text{Hp}.\supset .\gamma \text{ sm }\alpha &\qquad \text{(1)}\\ +*22·43.(*65·01). &\supset \vdash :\text{Hp}.\supset .\gamma \subset tʻ\alpha &\qquad \text{(2)}\\ +\vdash .(1).(2).*104·101.&\supset \vdash :\text{Hp}.\supset .\gamma \in \text{N}^{1}\text{c}ʻ\alpha &\qquad \text{(3)}\\ +\vdash .(3).*104·2.*24·23.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*104·412.</b> \[\begin{aligned}\vdash :tʻ\alpha =tʻ\beta .\alpha =\iota ʻx.\gamma =\iota ʻ\Lambda _{x}.&\delta =\iotaʻʻ\beta .\supset .\\ +&\gamma \in \text{N}^{1}\text{c}ʻ\alpha .\delta \in \text{N}^{1}\text{c}ʻ\beta .\gamma \cap \delta =\Lambda\end{aligned}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*73·43. &\supset \vdash :\text{Hp}.\supset .\gamma \text{ sm }\alpha &\qquad \text{(1)}\\ +\vdash .*63·61·103. &\supset \vdash :\text{Hp}.\supset .\alpha \in t^{2}ʻx &\qquad \text{(2)}\\ +\vdash .*22·43.(*65·01). \supset \vdash :\text{Hp}.\xi \in \gamma .&\supset _{\xi }.\xi \subset tʻx.\\ +[*63·5] &\supset _{\xi }.\xi \in t^{2}ʻx.\\ +[(2).*63·13] &\supset _\xi .\xi \in tʻ\alpha . &\qquad \text{(3)}\\ +\vdash .(1).(3).*104·101.&\supset \vdash :\text{Hp}.\supset .\gamma \in \text{N}^{1}\text{c}ʻ\alpha &\qquad \text{(4)}\\ +\vdash .*101·23. &\supset \vdash :\text{Hp}.\supset .\gamma \cap \delta =\Lambda &\qquad \text{(5)}\\ +\vdash .(4).(5).*104·2.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*104·413.</b> + \[\begin{aligned}\vdash :tʻ\alpha =tʻ\beta .\alpha =\iota ʻx\cup \iota ʻy.&x \neq y.\gamma =\iota ʻ\Lambda \cup \iota ʻ(\iota ʻx\cup \iota ʻy).\delta =\iotaʻʻ\beta .\supset .\\ +&\gamma \in \text{N}^{1}\text{c}ʻ\alpha .\delta \in \text{N}^{1}\text{c}ʻ\beta .\gamma \cap \delta =\Lambda\end{aligned}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*54·26. \supset \vdash :\text{Hp}.&\supset .\iota ʻx\cup \iota ʻy\in 2. &\qquad \text{(1)}\\ +[*101·35] &\supset .\Lambda \neq \iota ʻx\cup \iota ʻy.\\ +[*54·26] &\supset .\iota ʻ\Lambda \cup \iota ʻ(\iota ʻx\cup \iota ʻy)\in 2.\\ +[*101·3] &\supset .\iota ʻ\Lambda \cup \iota ʻ(\iota ʻx\cup \iota ʻy)\in \text{Nc}ʻ(\iota ʻx\cup \iota ʻy) &\qquad \text{(2)}\\ +\vdash .*51·16. \supset \vdash :\text{Hp}.&\supset .\alpha \in \gamma .\\ +[*63·5] &\supset .\gamma \subset tʻ\alpha &\qquad \text{(3)}\\ +\vdash .(2).(3).*104·1. &\supset \vdash :\text{Hp}.\supset .\gamma \in \text{N}^{1}\text{c}ʻ\alpha &\qquad \text{(4)}\\ +\vdash .*52·21·3. &\supset \vdash .\Lambda {\sim}\in \iotaʻʻ\beta &\qquad \text{(5)}\\ +\vdash .(1).*52·3.*54·25. &\supset \vdash :\text{Hp}.\supset .\iota ʻx\cup \iota ʻy{\sim}\in \iotaʻʻ\beta &\qquad \text{(6)}\\ +\vdash .(5).(6). &\supset \vdash :\text{Hp}.\supset .\gamma \cap \delta =\Lambda &\qquad \text{(7)}\\ +\vdash .(4).(7).*104·2.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*104·42.</b> \[\begin{aligned}&\vdash :tʻ\alpha =tʻ\beta .\alpha \in 0\cup 1\cup 2.\supset .(\exists \gamma ,\delta ).\gamma \in \text{N}^{1}\text{c}ʻ\alpha + .\delta \in \text{N}^{1}\text{c}ʻ\beta .\gamma \cap \delta =\Lambda \\ +&[*104·411·412·413.*52·1.*54·101]\end{aligned}\]</p> + +<p><span class="pagenum" id="Page_51">[Pg 51]</span></p> + +<p class="nind"><b>*104·43.</b> \(\vdash :tʻ\alpha =tʻ\beta .\supset .(\exists \gamma ,\delta ).\gamma \in \text{N}^{1}\text{c}ʻ\alpha .\delta \in \text{N}^{1}\text{c}ʻ\beta + .\gamma \cap \delta =\Lambda \)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*54·56.\supset \\ +\vdash :\text{Hp}.\alpha {\sim}\in 0\cup 1\cup 2.&\supset .(\exists x,y,z).x,y,z \in \alpha .x\neq y.x\neq z.y\neq z.\\ +[*104·41] &\supset .(\exists \gamma ,\delta ).\gamma \in \text{N}^{1}\text{c}ʻ\alpha .\delta \in \text{N}^{1}\text{c}ʻ\beta .\gamma \cap \delta =\Lambda &\qquad \text{(1)}\\ +\vdash .(1).*104·42.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p>The above proposition gives the desired result. The following +propositions re-state this result in other forms.</p> + +<p class="nind"><b>*104·44.</b> \[\begin{aligned}&\vdash :\mu ,\nu \in \text{N}^{1}\text{C}.tʻ\mu + =tʻ\nu .\supset .(\exists \gamma ,\delta ).\gamma \in \mu .\delta \in \nu .\gamma \cap \delta =\Lambda\\ +&[*104·13·43]\end{aligned}\]</p> + +<p class="nind"><b>*104·45.</b> \[\begin{aligned}&\vdash :\mu ,\nu \in \text{N}_{0}\text{C}.tʻ\mu =tʻ\nu .\supset .(\exists \gamma ,\delta ).\gamma \in \mu ^{(1)}.\delta \in \nu ^{(1)}.\gamma + \cap \delta =\Lambda\\ +&[*104·29·44]\end{aligned}\]</p> + +<p class="nind"><b>*104·46.</b> \[\begin{aligned}&\vdash :\mu ,\nu \in \text{NC} - \iota ʻ\Lambda .tʻ\mu =tʻ\nu .\supset .(\exists \gamma ,\delta ).\gamma \in \mu ^{(1)}.\delta \in \nu ^{(1)}.\gamma + \cap \delta =\Lambda \\ +&[*104·28·44]\end{aligned}\]</p> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_52">[Pg 52]</span></p> + +<h2 class="nobreak" id="*105"> +*105. DESCENDING CARDINALS.</h2> +</div> + + +<p class="nind"><i>Summary of</i> *105.</p> + +<p>In this number, we consider cardinals generated by a relation of +similarity which goes from a higher to a lower type, <i>i.e.</i> given +any class of classes \(\kappa\), we consider \(\text{Nc}ʻ\kappa\) +in the type of members of \(\kappa\) (which we shall call +\(\text{N}_{1}\text{c}ʻ\kappa\)) or in some lower type. Thus +<i>e.g.</i> we shall have +\[ +\kappa =\iotaʻʻ\alpha .\supset .\alpha \in \text{N}_{1}\text{c}ʻ\kappa , +\] +where "\(\text{N}_{1}\text{c}ʻ\kappa\)" means "classes similar to +\(\kappa\) but of the next lower type." Similarly +\[ +\kappa =\iotaʻʻ\iotaʻʻ\alpha .\supset .\alpha \in \text{N}_{2}\text{c}ʻ\kappa , +\] +and so on. We shall have generally +\[ +\begin{aligned} +\beta \in \text{N}_{1}\text{c}ʻ\alpha .&\equiv \alpha \in \text{N}^{1}\text{c}ʻ\beta ,\\ +\beta \in \text{N}_{2}\text{c}ʻ\alpha .&\equiv \alpha \in \text{N}^{2}\text{c}ʻ\beta , +\end{aligned} +\] +and so on. The chief difference between ascending and descending +cardinals is that \(\Lambda\) is one of the latter, but not one of the +former. Otherwise the propositions of the present number are mostly +analogous to corresponding propositions of <a href="#*104">*104</a>.</p> + +<p>On the analogy of the definitions in *104, we put +\[ +\begin{aligned} +\text{N}_{1}\text{C}=\text{D}ʻ\text{N}_{1}\text{c} \quad\text{Df},\\ +\mu _{(1)}=\text{ sm }ʻʻ\mu \cap t_{1}ʻ\mu \quad\text{Df}, +\end{aligned} +\] +with similar definitions for \(\text{N}_{2}\text{C}\) and \(\mu_(2)\).</p> + +<p>No proposition of the present number is ever referred to in the sequel, +and the reader who is not interested in the subject may therefore omit +it without detriment to what follows. The principal propositions proved +are the following:</p> + +<p class="nind"><b>*105·25.</b> \(\vdash .\text{N}_{0}\text{C}=\text{N}_{1}\text{C}-\iota ʻ\Lambda\)</p> + +<p class="nind"><b>*105·251.</b> \(\vdash .\text{N}_{0}\text{C}=\text{N}_{2}\text{C}-\iota ʻ\Lambda\)</p> + +<p class="nind"><b>*105·26.</b> \(\vdash .\text{N}_{1}\text{c}ʻtʻ\alpha =\Lambda\)</p> + +<p><span class="pagenum" id="Page_53">[Pg 53]</span></p> + +<p>Thus \(\text{N}_{1}\text{C}\) or \(\text{N}_{2}\text{C}\), in any given +type, only differs from \(\text{N}_{0}\text{C}\) in that type by the +addition of \(\Lambda\).</p> + +<p class="nind"><b>*105·3.</b> \(\vdash :\mu =\text{N}_{0}\text{c}ʻ\alpha .\supset .\mu _{(1)}=\text{N}_{1}\text{c}ʻ\alpha\)</p> + +<p class="nind"><b>*105·322.</b> \(\vdash \colon\ldotp \exists !\text{N}_{1}\text{c}ʻ\alpha .\supset :\text{N}_{1}\text{c}ʻ\alpha =\text{N}_{1}\text{c}ʻ\beta .\equiv .\text{N}_{0}\text{c}ʻ\alpha + =\text{N}_{0}\text{c}ʻ\beta\)</p> + +<p class="nind"><b>*105·34.</b> \(\vdash \colon\ldotp \mu \in \text{NC}.\exists !\mu _{(1)}.\supset :\mu _{(1)}=\text{N}_{1}\text{c}ʻ\alpha + .\equiv .\mu =\text{N}_{0}\text{c}ʻ\alpha\)</p> + +<p class="nind"><b>*105·35.</b> \(\vdash \colon\ldotp \mu \in \text{NC}.\nu \in \text{N}_{0}\text{C}.\supset :\mu =\nu ^{(1)}.\equiv .\mu _{(1)}=\nu\)</p> + +<p class="nind"><b>*105·38.</b> \(\vdash .\{\mu _{(1)}\}_{(1)}=\mu _{(2)}\)</p> + +<hr class="tb"> + +<p class="nind"><b><a id="*105·01">*105·01</a>.</b> \(\text{N}_{1}\text{c}ʻ\alpha =\text{Nc}ʻ\alpha \cap tʻt_{1}ʻ\alpha \quad\text{Df}\)</p> + +<p>We might write +\[ +\text{N}_{1}\text{c}ʻ\alpha =\text{Nc}ʻ\alpha \cap t_{0}ʻ\alpha \quad\text{Df}, +\] +which would be equivalent to the above. But we choose the above form +for the sake of uniformity. If \(s\) is any suffix, we put, provided +\(t_{s}ʻ\alpha\) has been defined, +\[ +\text{N}_{s}\text{c}ʻ\alpha =\text{Nc}ʻ\alpha \cap tʻt_{s}ʻ\alpha \quad\text{Df}, +\] +and if \(i\) is any index for which \(t^{i}ʻ\alpha\) has been defined, we +put +\[ +\text{N}^{i}\text{c}ʻ\alpha =\text{Nc}ʻ\alpha \cap tʻt^{i}ʻ\alpha \quad\text{Df}. +\]</p> + +<p>Thus for the sake of uniformity it is better, in the above definition +<a href="#*105·01">*105·01</a>, to write "\(tʻt_{1}ʻ\alpha\)" rather than "\(t_{0}ʻ\alpha\)."</p> + +<p class="nind"><b>*105·011.</b> \(\text{N}_{2}\text{c}ʻ\alpha =\text{Nc}ʻ\alpha \cap tʻt_{2}ʻ\alpha \quad\text{Df}\)</p> + +<p class="nind"><b>*105·02.</b> \(\text{N}_{1}\text{C}=\text{D}ʻ\text{N}_{1}\text{c} \quad\text{Df}\)</p> + +<p class="nind"><b>*105·021.</b> \(\text{N}_{2}\text{C}=\text{D}ʻ\text{N}_{2}\text{c} \quad\text{Df}\)</p> + +<p class="nind"><b>*105·03.</b> \(\mu _{(1)}=\text{ sm }ʻʻ\mu \cap t_{1}ʻ\mu \quad\text{Df}\)</p> + +<p class="nind"><b>*105·031.</b> \(\mu _{(2)}=\text{ sm }ʻʻ\mu \cap t_{2}ʻ\mu \quad\text{Df}\)</p> + +<p class="nind"><b>*105·1.</b> \(\vdash .\text{N}_{1}\text{c}ʻ\alpha =\text{Nc}ʻ\alpha \cap t_{0}ʻ\alpha \quad[*63·383.(*105·01)]\)</p> + +<p class="nind"><b>*105·101.</b> \(\vdash .\text{N}_{2}\text{c}ʻ\alpha =\text{Nc}ʻ\alpha \cap t_{1}ʻ\alpha \quad[*63·41.(*105·011)]\)</p> + +<p class="nind"><b>*105·11.</b> \[\begin{aligned}&\vdash :\beta \in \text{N}_{1}\text{c}ʻ\alpha .\equiv .\beta \in \text{Nc}ʻ\alpha .\beta \in t_{0}ʻ\alpha + .\equiv .\beta \text{ sm }\alpha .\beta \in t_{0}ʻ\alpha .\equiv .\beta \text{ sm }\alpha .\beta \subset t_{1}ʻ\alpha \\ +&[*105·1.*100·31.*63·51]\end{aligned}\]</p> + +<p class="nind"><b>*105·111.</b> \[\begin{aligned}&\vdash :\beta \in \text{N}_{2}\text{c}ʻ\alpha .\equiv .\beta \in \text{Nc}ʻ\alpha .\beta \in t_{1}ʻ\alpha + .\equiv .\beta \text{ sm }\alpha .\beta \in t_{1}ʻ\alpha .\equiv .\beta \text{ sm }\alpha .\beta \subset t_{2}ʻ\alpha\\ +&[*105·101.*100·31.*63·52]\end{aligned}\]</p> + +<p class="nind"><b>*105·12.</b> \[\begin{aligned}&\vdash :\beta \in \text{N}_{1}\text{c}ʻ\alpha + .\equiv .\beta \in \text{Nc}ʻ\alpha .\alpha \subset tʻ\beta .\equiv .\beta \text{ sm }\alpha .\alpha \subset tʻ\beta .\equiv .\alpha \in \text{N}^{1}\text{c}ʻ\beta \\ +&[*105·11.*63·51.*104·1]\end{aligned}\]</p> + +<p class="nind"><b>*105·121.</b> \[\begin{aligned}&\vdash :\beta \in \text{N}_{2}\text{c}ʻ\alpha .\equiv .\beta \in \text{Nc}ʻ\alpha .\alpha \subset t^{2}ʻ\beta + .\equiv .\beta \text{ sm }\alpha .\alpha \subset t^{2}ʻ\beta .\equiv .\alpha \in \text{N}^{2}\text{c}ʻ\beta \\ +&[*105·111.*63·52.*104·11]\end{aligned}\]</p> + +<p class="nind"><b>*105·13.</b> \(\vdash .\text{N}_{1}\text{c}ʻ\alpha =\text{Nc}(t_{1}ʻ\alpha )ʻ\alpha =\text{Nc}\{(t_{1}ʻ\alpha )_{\alpha }\}ʻ\alpha + \quad[*102·6.(*105·.01)]\)</p> + +<p><span class="pagenum" id="Page_54">[Pg 54]</span></p> + +<p class="nind"><b>*105·131.</b> \(\vdash .\text{N}_{2}\text{c}ʻ\alpha =\text{Nc}(t_{2}ʻ\alpha )ʻ\alpha =\text{Nc}\{(t_{2}ʻ\alpha )_{\alpha }\}ʻ\alpha + \quad[*102·6.(*105·011)]\)</p> + +<p class="nind"><b>*105·14.</b> \(\vdash :\alpha \in t_{0}ʻ\beta .\supset .\text{N}_{1}\text{c}ʻ\beta =\text{Nc}(a)ʻ\beta =\text{Nc}(a_{\beta })ʻ\beta\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*63·22.\supset \vdash :\text{Hp}.&\supset .tʻ\alpha =t_{0}ʻ\beta .\\ +[*105·1] &\supset .\text{N}_{1}\text{c}ʻ\beta =\text{Nc}ʻ\beta \cap tʻ\alpha &\qquad \text{(1)}\\ +\vdash .(1).*102·6.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*105·141.</b> \(\vdash :\alpha \in t_{1}ʻ\beta .\supset .\text{N}_{2}\text{c}ʻ\beta =\text{Nc}(\alpha )ʻ\beta =\text{Nc}(\alpha _{\beta })ʻ\beta + \quad[\text{Proof as in}\, *105·14]\)</p> + +<p class="nind"><b>*105·142.</b> \(\vdash :\beta \subset tʻ\alpha .\supset .\text{N}_{1}\text{c}ʻ\beta =\text{Nc}(\alpha )ʻ\beta =\text{Nc}(\alpha _{\beta })ʻ\beta \quad[*105·14.*63·51]\)</p> + +<p class="nind"><b>*105·143.</b> \(\vdash :\beta \subset t^{2}ʻ\alpha .\supset .\text{N}_{2}\text{c}ʻ\beta =\text{Nc}(\alpha )ʻ\beta =\text{Nc}(\alpha _{\beta })ʻ\beta + \quad[*105·141.*63·52]\)</p> + +<p class="nind"><b>*105·15.</b> \(\vdash :\mu \in \text{N}_{1}\text{C}.\equiv .(\exists \alpha ).\mu =\text{N}_{1}\text{c}ʻ\alpha \quad[*100·22.*71·41.(*105·02)]\)</p> + +<p class="nind"><b>*105·151.</b> \(\vdash :\mu \in \text{N}_{2}\text{C}.\equiv .(\exists \alpha ).\mu =\text{N}_{2}\text{c}ʻ\alpha\)</p> + +<p class="nind"><b>*105·16.</b> \[\begin{aligned}\vdash :\delta \in \mu _{(1)}.&\equiv .(\exists \gamma ).\gamma \in \mu .\delta \text{ sm }\gamma .\delta \in t_{1}ʻ\mu .\\ +&\equiv .(\exists \gamma ).\gamma \in \mu .\delta \text{ sm }\gamma .\delta \in t_{0}ʻ\gamma .\\ +&\equiv .(\exists \gamma ).\gamma \in \mu .\delta \text{ sm }\gamma .\gamma \subset tʻ\delta \quad[*37·1.*63·51·54]\end{aligned}\]</p> + +<p class="nind"><b>*105·161.</b> \[\begin{aligned}\vdash :\delta \in \mu _{(2)}.&\equiv .(\exists \gamma ).\gamma \in \mu .\delta \text{ sm }\gamma .\delta \in t_{2}ʻ\mu .\\ +&\equiv .(\exists \gamma ).\gamma \in \mu .\delta \text{ sm }\gamma .\delta \in t_{1}ʻ\gamma .\\ +&\equiv .(\exists \gamma ).\gamma \in \mu .\delta \text{ sm }\gamma .\gamma \subset t^{2}ʻ\delta \quad[*37·1.*63·52·55]\end{aligned}\]</p> + +<p>In what follows, propositions concerning \(\text{N}_{2}\text{c}\) or +\(\text{N}_{2}\text{C}\) have proofs exactly analogous to those of +the corresponding propositions concerning \(\text{N}_{1}\text{c}\) or +\(\text{N}_{1}\text{C}\).</p> + +<p class="nind"><b>*105·2.</b> \(\vdash .\text{N}_{0}\text{c}ʻ\alpha =\text{N}_{1}\text{c}ʻ\iotaʻʻ\alpha\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*105·12.*104·2.&\supset \vdash .\alpha \in \text{N}_{1}\text{c}ʻ\iotaʻʻ\alpha .\\ +[*103·26]&\supset \vdash .\text{N}_{0}\text{c}ʻ\alpha =\text{N}_{1}\text{c}ʻ\iotaʻʻ\alpha +\end{array} +\]</p> + +<p class="nind"><b>*105·201.</b> \(\vdash .\text{N}_{0}\text{c}ʻ\alpha =\text{N}_{2}\text{c}ʻ\iotaʻʻ\iotaʻʻ\alpha\)</p> + +<p class="nind"><b>*105·21.</b> \(\vdash .\text{N}_{0}\text{C}\subset \text{N}_{1}\text{C} \quad[*105·2·15]\)</p> + +<p class="nind"><b>*105·211.</b> \(\vdash .\text{N}_{0}\text{C}\subset \text{N}_{2}\text{C}\)</p> + +<p class="nind"><b>*105·22.</b> \(\vdash :\gamma \in \text{N}_{1}\text{c}ʻ\delta .\supset .\text{N}_{1}\text{c}ʻ\delta =\text{N}_{0}\text{c}ʻ\gamma \quad[*103·26]\)</p> + +<p class="nind"><b>*105·221.</b> \(\vdash :\gamma \in \text{N}_{2}\text{c}ʻ\delta .\supset .\text{N}_{2}\text{c}ʻ\delta =\text{N}_{0}\text{c}ʻ\gamma\)</p> + +<p class="nind"><b>*105·23.</b> \(\vdash :\exists !\text{N}_{1}\text{c}ʻ\delta .\supset .\text{N}_{1}\text{c}ʻ\delta \in \text{N}_{0}\text{C} \quad[*105·22]\)</p> + +<p class="nind"><b>*105·231.</b> \(\vdash :\exists !\text{N}_{2}\text{c}ʻ\delta .\supset .\text{N}_{2}\text{c}ʻ\delta \in \text{N}_{0}\text{C}\)</p> + +<p class="nind"><b>*105·24.</b> \(\vdash .\text{N}_{1}\text{C}-\iota ʻ\Lambda \subset \text{N}_{0}\text{C} \quad[*105·23]\)</p> + +<p class="nind"><b>*105·241.</b> \(\vdash .\text{N}_{2}\text{C}-\iota ʻ\Lambda \subset \text{N}_{0}\text{C}\)</p> + +<p class="nind"><b>*105·25.</b> \(\vdash .\text{N}_{0}\text{C}=\text{N}_{1}\text{C}-\iota ʻ\Lambda \quad[*105·21·24.*103·23]\)</p> + +<p><span class="pagenum" id="Page_55">[Pg 55]</span></p> + +<p class="nind"><b>*105·251.</b> \(\vdash .\text{N}_{0}\text{C}=\text{N}_{2}\text{C}-\iota ʻ\Lambda\)</p> + +<p class="nind"><b>*105·252.</b> \(\vdash .\text{N}_{1}\text{c}ʻ\beta =\text{N}_{2}\text{c}ʻ\iotaʻʻ\beta\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*105·111.\supset \vdash :\alpha \in \text{N}_{2}\text{c}ʻ\iotaʻʻ\beta .&\equiv .\alpha \text{ sm }\iotaʻʻ\beta .\alpha \in t_{1}ʻ\iotaʻʻ\beta .\\ +[*73·41.*63·64·54] &\equiv .\alpha \text{ sm }\beta .\alpha \in t_{0}ʻ\beta .\\ +[*105·11] &\equiv .\alpha \in \text{N}_{1}\text{c}ʻ\beta :\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*105·26.</b> \(\vdash .\text{N}_{1}\text{c}ʻtʻ\alpha =\Lambda\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*105·142.\supset \vdash .\text{N}_{1}\text{c}ʻtʻ\alpha =\text{Nc}(\alpha )ʻtʻ\alpha &\qquad \text{(1)}\\ +\vdash .(1).*102·73.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*105·261.</b> \(\vdash .\text{N}_{2}\text{c}ʻ\iotaʻʻtʻ\alpha =\Lambda \quad[*105·26·252]\)</p> + +<p class="nind"><b>*105·27.</b> \(\vdash .\Lambda \in \text{N}_{1}\text{C} \quad[*105·26]\)</p> + +<p class="nind"><b>*105·271.</b> \(\vdash .\Lambda \in \text{N}_{2}\text{C}\)</p> + +<p class="nind"><b>*105·28.</b> \(\vdash .\text{N}_{1}\text{C}=\text{N}_{0}\text{C}\cup \iota ʻ\Lambda \quad[*105·25·27]\)</p> + +<p class="nind"><b>*105·281.</b> \(\vdash .\text{N}_{2}\text{C}=\text{N}_{1}\text{C}=\text{N}_{0}\text{C}\cup \iota ʻ\Lambda\)</p> + +<p class="nind"><b>*105·29.</b> \(\vdash .\text{NC}\subset \text{N}_{1}\text{C}.\text{NC}\subset \text{N}_{2}\text{C} \quad[*105·281.*103·34]\)</p> + +<p class="nind"><b>*105·3.</b> \(\vdash :\mu =\text{N}_{0}\text{c}ʻ\alpha .\supset .\mu _{(1)}=\text{N}_{1}\text{c}ʻ\alpha\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*103·4.(*105·03).\supset \vdash :\mu =\text{N}_{0}\text{c}ʻ\alpha .&\supset .\mu _{(1)}=\text{Nc}ʻ\alpha \cap t_{1}ʻ\mu &\qquad \text{(1)}\\ +\vdash .*103·12. &\supset \vdash :\mu =\text{N}_{0}\text{c}ʻ\alpha .\supset .\alpha \in \mu .\\ +[*63·105] &\supset .\alpha \in t_{0}ʻ\mu .\\ +[*63·54] &\supset .t_{0}ʻ\alpha =t_{1}ʻ\mu &\qquad \text{(2)}\\ +\vdash .(1).(2). \supset \vdash :\mu =\text{N}_{0}\text{c}ʻ\alpha .&\supset .\mu _{(1)}=\text{Nc}ʻ\alpha \cap t_{0}ʻ\alpha .\\ +[*105·1] &\supset .\mu _{(1)}=\text{N}_{1}\text{c}ʻ\alpha :\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*105·301.</b> \(\vdash :\mu =\text{N}_{0}\text{c}ʻ\alpha .\supset .\mu _{(2)}=\text{N}_{2}\text{c}ʻ\alpha\)</p> + +<p class="nind"><b>*105·31.</b> \(\vdash :\mu \in \text{N}_{0}\text{C}.\supset .\mu {(1)}\in \text{N}_{1}\text{C} \quad[*105·3·15.*103·2]\)</p> + +<p class="nind"><b>*105·311.</b> \(\vdash :\mu \in \text{N}_{0}\text{C}.\supset .\mu _{(2)}\in \text{N}_{2}\text{C}\)</p> + +<p class="nind"><b>*105·312.</b> \(\vdash :\gamma \in \text{N}_{1}\text{c}ʻ\alpha .\supset .\alpha \in \text{N}^{1}\text{c}ʻ\gamma .\text{N}^{1}\text{c}ʻ\gamma =\text{N}_{0}\text{c}ʻ\alpha + \quad[*105·12.*103·26]\)</p> + +<p class="nind"><b>*105·313.</b> \(\vdash :\gamma \in \text{N}_{2}\text{c}ʻ\alpha .\supset .\alpha \in \text{N}^{2}\text{c}ʻ\gamma .\text{N}^{2}\text{c}ʻ\gamma + =\text{N}_{0}\text{c}ʻ\alpha\)</p> + +<p class="nind"><b>*105·314.</b> \(\vdash :\text{N}_{1}\text{c}ʻ\alpha =\text{N}_{0}\text{c}ʻ\gamma .\supset .\text{N}_{0}\text{c}ʻ\alpha =\text{N}^{1}\text{c}ʻ\gamma + \quad[*105·312.*103·12]\)</p> + +<p class="nind"><b>*105·315.</b> \(\vdash :\text{N}_{2}\text{c}ʻ\alpha =\text{N}_{0}\text{c}ʻ\gamma .\supset .\text{N}_{0}\text{c}ʻ\alpha =\text{N}^{2}\text{c}ʻ\gamma\)</p> + +<p class="nind"><b>*105·316.</b> \(\vdash :\exists !\text{N}_{1}\text{c}ʻ\alpha .\text{N}_{1}\text{c}ʻ\alpha =\text{N}_{1}\text{c}ʻ\beta .\supset .\text{N}_{0}\text{c}ʻ\alpha + =\text{N}_{0}\text{c}ʻ\beta\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*105·312.&\supset \vdash :\gamma \in \text{N}_{1}\text{c}ʻ\alpha \\.\text{N}_{1}\text{c}ʻ\alpha =\text{N}_{1}\text{c}ʻ\beta .&\supset .\text{N}^{1}\text{c}ʻ\gamma + =\text{N}_{0}\text{c}ʻ\alpha .\text{N}^{1}\text{c}ʻ\gamma =\text{N}_{0}\text{c}ʻ\beta .\\ +[*13·171] &\supset .\text{N}_{0}\text{c}ʻ\alpha =\text{N}_{0}\text{c}ʻ\beta &\qquad \text{(1)}\\ +\vdash .(1).*10·11·23·35.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_56">[Pg 56]</span></p> + +<p class="nind"><b>*105·317.</b> \(\vdash :\exists !\text{N}_{2}\text{c}ʻ\alpha .\text{N}_{2}\text{c}ʻ\alpha =\text{N}_{2}\text{c}ʻ\beta .\supset .\text{N}_{0}\text{c}ʻ\alpha + =\text{N}_{0}\text{c}ʻ\beta\)</p> + +<p class="nind"><b>*105·32.</b> \(\vdash :\text{N}_{0}\text{c}ʻ\alpha =\text{N}_{0}\text{c}ʻ\beta .\supset .\text{N}_{1}\text{c}ʻ\alpha =\text{N}_{1}\text{c}ʻ\beta\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*103·41.\supset \vdash :\text{Hp}.&\supset .\text{Nc}(t_{1}ʻ\alpha )ʻ\alpha =\text{Nc}(t_{1}ʻ\alpha )ʻ\beta &\qquad \text{(1)}\\ +\vdash .*103·14.\supset \vdash :\text{Hp}.&\supset .\beta \in tʻ\alpha .\\ +[*63·16·36] &\supset .t_{1}ʻ\alpha =t_{1}ʻ\beta &\qquad \text{(2)}\\ +\vdash .(1).(2).\supset \vdash :\text{Hp}.&\supset .\text{Nc}(t_{1}ʻ\alpha )ʻ\alpha =\text{Nc}(t_{1}ʻ\beta )ʻ\beta .\\ +[*105·13] &\supset .\text{N}_{1}\text{c}ʻ\alpha =\text{N}_{1}\text{c}ʻ\beta :\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*105·321.</b> \(\vdash :\text{N}_{0}\text{c}ʻ\alpha =\text{N}_{0}\text{c}ʻ\beta .\supset .\text{N}_{2}\text{c}ʻ\alpha =\text{N}_{2}\text{c}ʻ\beta\)</p> + +<p class="nind"><b>*105·322.</b> \(\vdash \colon\ldotp \exists !\text{N}_{1}\text{c}ʻ\alpha .\supset :\text{N}_{1}\text{c}ʻ\alpha =\text{N}_{1}\text{c}ʻ\beta .\equiv .\text{N}_{0}\text{c}ʻ\alpha + =\text{N}_{0}\text{c}ʻ\beta \quad[*105·316·32]\)</p> + +<p class="nind"><b>*105·323.</b> \(\vdash \colon\ldotp \exists !\text{N}_{2}\text{c}ʻ\alpha .\supset :\text{N}_{2}\text{c}ʻ\alpha =\text{N}_{2}\text{c}ʻ\beta .\equiv .\text{N}_{0}\text{c}ʻ\alpha + =\text{N}_{0}\text{c}ʻ\beta\)</p> + +<p class="nind"><b>*105·324.</b> \(\vdash :\exists !\mu _{(1)}.\supset .\exists !\mu \quad[*37·29.(*105·03)]\)</p> + +<p class="nind"><b>*105·325.</b> \(\vdash :\exists !\mu _{(2)}.\supset .\exists !\mu \)</p> + +<p class="nind"><b>*105·326.</b> \(\vdash :\mu \in \text{NC}.\mu _{(1)}=\text{N}_{0}\text{c}ʻ\gamma .\supset .\mu =\text{N}^{1}\text{c}ʻ\gamma\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*103·26. \supset \vdash :\text{Hp}.\alpha \in \mu .&\supset .\mu =\text{N}_{0}\text{c}ʻ\alpha . &\qquad \text{(1)}\\ +[*105·3] & \supset .\mu _{(1)}=\text{N}_{1}\text{c}ʻ\alpha .\\ +[\text{Hp}] &\supset .\text{N}_{1}\text{c}ʻ\alpha =\text{N}_{0}\text{c}ʻ\gamma .\\ +[*105·314] & \supset .\text{N}_{0}\text{c}ʻ\alpha =\text{N}^{1}\text{c}ʻ\gamma .\\ +[(1)] &\supset .\mu =\text{N}^{1}\text{c}ʻ\gamma &\qquad \text{(2)}\\ +\vdash .(2).*10·11·23·35.\supset \vdash :\text{Hp}.\exists !\mu .&\supset .\mu =\text{N}^{1}\text{c}ʻ\gamma &\qquad \text{(3)}\\ +\vdash .(3).*105·324.*103·13.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*105·327.</b> \(\vdash :\mu \in \text{NC}.\mu _{(2)}=\text{N}_{0}\text{c}ʻ\gamma .\supset .\mu =\text{N}^{2}\text{c}ʻ\gamma\)</p> + +<p class="nind"><b>*105·33.</b> \(\vdash :\mu \in \text{NC}.\exists !\mu _{(1)}.\mu _{(1)}=\text{N}_{1}\text{c}ʻ\alpha .\supset .\mu =\text{N}_{0}\text{c}ʻ\alpha\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*103·26.\supset \vdash :\gamma \in \mu _{(1)}.\mu _{(1)}=\text{N}_{1}\text{c}ʻ\alpha .\supset .\text{N}_{1}\text{c}ʻ\alpha =\text{N}_{0}\text{c}ʻ\gamma + .\\ +[*105·314] \supset .\text{N}_{0}\text{c}ʻ\alpha =\text{N}^{1}\text{c}ʻ\gamma &\qquad \text{(1)}\\ +\vdash .(1).*105·326.\supset \\ +\vdash :\gamma \in \mu _{(1)}.\mu _{(1)}=\text{N}_{1}\text{c}ʻ\alpha .\mu \in \text{NC}.\supset .\mu =\text{N}_{0}\text{c}ʻ\alpha &\qquad \text{(2)}\\ +\vdash .(2).*10·11·23·35.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*105·331.</b> \(\vdash :\mu \in \text{NC}.\exists !\mu _{(2)}.\mu _{(2)}=\text{N}_{2}\text{c}ʻ\alpha .\supset .\mu =\text{N}_{0}\text{c}ʻ\alpha\)</p> + +<p class="nind"><b>*105·34.</b> \(\vdash \colon\ldotp \mu \in \text{NC}.\exists !\mu _{(1)}.\supset :\mu _{(1)}=\text{N}_{1}\text{c}ʻ\alpha .\equiv .\mu =\text{N}_{0}\text{c}ʻ\alpha + \quad[*105·33·3]\)</p> + +<p class="nind"><b>*105·341.</b> \(\vdash \colon\ldotp \mu \in \text{NC}.\exists !\mu _{(2)}.\supset :\mu _{(2)}=\text{N}_{2}\text{c}ʻ\alpha + .\equiv .\mu =\text{N}_{0}\text{c}ʻ\alpha\)</p> + +<p class="nind"><b>*105·342.</b> \(\vdash .\mu \in \text{NC}.\supset .\mu _{(1)}\in \text{N}_{1}\text{C}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*103·34. \supset \vdash :\text{Hp}.\exists !\mu .&\supset .\mu \in \text{N}_{0}\text{C}.\\ +[*105·31] &\supset .\mu _{(1)}\in \text{N}_{1}\text{C} &\qquad \text{(1)}\\ +\vdash .*105·324.\supset \vdash :\text{Hp}.{\sim}\exists !\mu .&\supset .{\sim}\exists !\mu _{(1)}.\\ +[*105·27] &\supset .\mu _{(1)}\in \text{N}_{1}\text{C} &\qquad \text{(2)}\\ +\vdash .(1).(2). \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_57">[Pg 57]</span></p> + +<p class="nind"><b>*105·343.</b> \(\vdash :\mu \in \text{NC}.\supset .\mu _{(2)}\in \text{N}_{2}\text{C}\)</p> + +<p class="nind"><b>*105·344.</b> \(\vdash :\mu =\text{N}^{1}\text{c}ʻ\gamma .\supset .\mu _{(1)}=\text{N}_{0}\text{c}ʻ\gamma\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *104·24.\supset \vdash :\text{Hp}.&\supset .\mu =\text{N}_{0}\text{c}ʻ\iotaʻʻ\gamma .\\ +[*105·3] &\supset .\mu _{(1)}=\text{N}_{1}\text{c}ʻ\iotaʻʻ\gamma .\\ +[*105·2] &\supset .\mu _{(1)}=\text{N}_{0}\text{c}ʻ\gamma :\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*105·345.</b> \(\vdash :\mu =\text{N}^{2}\text{c}ʻ\gamma .\supset .\mu _{(2)}=\text{N}_{0}\text{c}ʻ\gamma\)</p> + +<p class="nind"><b>*105·35.</b> \(\vdash \colon\ldotp \mu \in \text{NC}.\nu \in \text{N}_{0}\text{C}.\supset :\mu =\nu ^{(1)}.\equiv .\mu _{(1)}=\nu\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*105·326.*104·26.\supset \\ +\vdash :\mu \in \text{NC}.\nu =\text{N}_{0}\text{c}ʻ\gamma .\mu _{(1)}=\nu .\supset .\mu =\text{N}^{1}\text{c}ʻ\gamma . \nu ^{(1)}=\text{N}_{1}\text{c}ʻ\gamma + .\\ +[*13·172] \supset .\mu =\nu ^{(1)} &\qquad \text{(1)}\\ +\vdash .*104·26.\text{Fact}.\supset \\ +\vdash :\mu \in \text{NC}.\nu =\text{N}_{0}\text{c}ʻ\gamma .\mu =\nu ^{(1)}.\supset .\mu =\text{N}^{1}\text{c}ʻ\gamma .\nu =\text{N}_{0}\text{c}ʻ\gamma .\\ +[*105·344] \supset .\mu _{(1)}=\text{N}_{0}\text{c}ʻ\gamma .\nu =\text{N}_{0}\text{c}ʻ\gamma .\\ +[*13·172] \supset .\mu _{(1)}=\nu &\qquad \text{(2)}\\ +\vdash .(1).(2).\supset \vdash \colon\ldotp \mu \in \text{NC}.\nu =\text{N}_{0}\text{c}ʻ\gamma .\supset :\mu =\nu ^{(1)}.\equiv .\mu _{(1)}=\nu &\qquad \text{(3)}\\ +\vdash .(3).*103·2.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*105·351.</b> \(\vdash \colon\ldotp \mu \in \text{NC}.\nu \in \text{N}_{0}\text{C}.\supset :\mu =\nu ^{(2)}.\equiv .\mu _{(2)}=\nu\)</p> + +<p class="nind"><b>*105·352.</b> \(\vdash \colon\ldotp \mu ,\nu \in \text{NC}.\exists !\nu .\supset :\mu =\nu ^{(1)}.\equiv .\mu _{(1)}=\nu \quad[*105·35.*103·34]\)</p> + +<p class="nind"><b>*105·353.</b> \(\vdash \colon\ldotp \mu ,\nu \in \text{NC}.\exists !\nu .\supset :\mu =\nu ^{(2)}.\equiv .\mu _{(2)}=\nu\)</p> + +<p class="nind"><b>*105·354.</b> \(\vdash :\nu \in \text{NC}.\exists !\nu .\supset .{\nu ^{(1)}}_{(1)}=\nu \quad[*105·352]\)</p> + +<p class="nind"><b>*105·355.</b> \(\vdash :\nu \in \text{NC}.\exists !\nu .\supset .\{\nu ^{(2)}\}_{(2)}=\nu\)</p> + +<p class="nind"><b>*105·356.</b> \(\vdash :\mu \in \text{NC}.\exists !\mu _{(1)}.\supset .\{\mu _{(1)}\}^{(1)}=\mu \quad[*105·352]\)</p> + +<p class="nind"><b>*105·357.</b> \(\vdash :\mu \in \text{NC}.\exists !\mu _{(2)}.\supset .\{\mu _{(2)}\}^{(2)}=\mu\)</p> + +<p class="nind"><b>*105·36.</b> \(\vdash :\beta \in \text{N}_{1}\text{c}ʻ\alpha .\gamma \in \text{N}_{1}\text{c}ʻ\beta .\supset .\gamma \in \text{N}_{2}\text{c}ʻ\alpha\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*105·11.\supset \vdash :\text{Hp}.&\supset .\beta \text{ sm }\alpha .\beta \in t_{0}ʻ\alpha .\gamma \text{ sm }\beta .\gamma \in t_{0}ʻ\beta .\\ +[*73·32.*63·38] &\supset .\gamma \text{ sm }\alpha .\gamma \in t_{1}ʻ\alpha .\\ +[*105·111] &\supset .\gamma \in \text{N}_{2}\text{c}ʻ\alpha :\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*105·361.</b> \(\vdash :\beta \in \text{N}_{1}\text{c}ʻ\alpha .\gamma \in \text{N}_{2}\text{c}ʻ\alpha .\supset .\gamma \in \text{N}_{1}\text{c}ʻ\beta\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*105·11·111 .\supset \vdash :\text{Hp}.&\supset .\beta \text{ sm }\alpha .\beta \in t_{0}ʻ\alpha .\gamma \text{ sm }\alpha .\gamma \in t_{1}ʻ\alpha .\\ +[*73·31·32] &\supset .\gamma \text{ sm }\beta .\beta \in t_{0}ʻ\alpha .\gamma \in t_{1}ʻ\alpha &\qquad \text{(1)}\\ +\vdash .*63·54. &\supset \vdash :\beta \in t_{0}ʻ\alpha .\supset .t_{0}ʻ\beta =t_{1}ʻ\alpha &\qquad \text{(2)}\\ +\vdash .(1).(2). &\supset \vdash :\text{Hp}.\supset .\gamma \text{ sm }\beta .\gamma \in t_{0}ʻ\beta .\\ +[*105·11] \supset .\gamma \in \text{N}_{1}\text{c}ʻ\beta :\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_58">[Pg 58]</span></p> + +<p class="nind"><b>*105·362.</b> \(\vdash :\beta \in \text{N}_{1}\text{c}ʻ\alpha .\supset .\text{N}_{1}\text{c}ʻ\beta =\text{N}_{2}\text{c}ʻ\alpha \quad[*105·36·361]\)</p> + +<p class="nind"><b>*105·37.</b> \(\vdash :\text{N}_{0}\text{c}ʻ\beta =\text{N}_{1}\text{c}ʻ\alpha .\supset .\text{N}_{1}\text{c}ʻ\beta =\text{N}_{2}\text{c}ʻ\alpha + \quad[*105·362.*103·12]\)</p> + +<p class="nind"><b>*105·371.</b> \(\vdash :\exists !\mu _{(2)}.\supset .\exists !\mu _{(1)}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*63·381.(*63·05).\supset \\ +\vdash :\gamma \text{ sm }\alpha .\alpha \in \mu .\gamma \in t_{2}ʻ\mu .&\supset .\gamma \text{ sm }\alpha .\alpha \in \mu .tʻ\gamma =t_{2}ʻ\mu .\\ +[*73·41.*63·64] &\supset .\iotaʻʻ\gamma \text{ sm }\alpha .\alpha \in \mu .t_{0}ʻ\iotaʻʻ\gamma =t_{2}ʻ\mu .\\ +[*63·57] &\supset .\iotaʻʻ\gamma \text{ sm }\alpha .\alpha \in \mu .tʻ\iotaʻʻ\gamma =t_{1}ʻ\mu .\\ +[*63·103] &\supset .\iotaʻʻ\gamma \text{ sm }\alpha .\alpha \in \mu .\iotaʻʻ\gamma \in t_{1}ʻ\mu .\\ +[*105·16] &\supset .\iotaʻʻ\gamma \in \mu _{(1)}.\\ +[*10·24] &\supset .\exists !\mu _{(1)} &\qquad \text{(1)}\\ +\vdash .(1).*10·11·23.\supset \\ +\vdash :(\exists \alpha ).\gamma \text{ sm }\alpha .\alpha \in \mu .\gamma \in t_{2}ʻ\mu .&\supset .\exists !\mu _{(1)} &\qquad \text{(2)}\\ +\vdash .(2).*105·161.\supset \vdash :\gamma \in \mu _{(2)}.&\supset .\exists !\mu _{(1)} &\qquad \text{(3)}\\ +\vdash .(3).*10·11·23.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*105·372.</b> \(\vdash :\mu _{(1)}=\Lambda .\supset .\mu _{(2)}=\Lambda \quad[*105·371.\text{Transp}]\)</p> + +<p class="nind"><b>*105·38.</b> \(\vdash .\{\mu _{(1)}\}_{(1)}=\mu _{(2)}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*105·16.\supset \vdash :\gamma \in \{\mu _{(1)}\}_{(1)}.&\equiv .(\exists \beta ).\beta \in \mu _{(1)}.\gamma \text{ sm }\beta .\gamma \in t_{0}ʻ\beta .\\ +[*105·16] &\equiv .(\exists \alpha ,\beta ).\alpha \in \mu .\beta \text{ sm }\alpha .\beta \in t_{0}ʻ\alpha .\gamma \text{ sm }\beta .\gamma \in t_{0}ʻ\beta . &\qquad \text{(1)}\\ +[*73·32.*63·38] &\supset .(\exists \alpha ).\alpha \in \mu .\gamma \text{ sm }\alpha .\gamma \in t_{1}ʻ\alpha &\qquad \text{(2)}\\ +\vdash .*73·41.*63·64·53·57.\supset \\ +\vdash :\alpha \in \mu .\gamma \text{ sm }\alpha .\gamma \in t_{1}ʻ\alpha .&\supset .\alpha \in \mu .\iotaʻʻ\gamma \text{ sm }\alpha .\gamma \text{ sm }\iotaʻʻ\gamma .\gamma \in t_{0}ʻ\iotaʻʻ\gamma + .\iotaʻʻ\gamma \in t_{0}ʻ\alpha .\\ +[(1)] &\supset .\gamma \in \{\mu _{(1)}\}_{(1)} &\qquad \text{(3)}\\ +\vdash .(2).(3).\supset \vdash :\gamma \in \{\mu _{(1)}\}_{(1)}.&\equiv .(\exists \alpha ).\alpha \in \mu .\gamma \text{ sm }\alpha .\gamma \in t_{1}ʻ\alpha .\\ +[*105·161] &\equiv .\gamma \in \mu _{(2)}:\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*105·4.</b> \(\vdash :\gamma \in \text{N}_{2}\text{c}ʻ\alpha .\supset .\iotaʻʻ\gamma \in \text{N}_{1}\text{c}ʻ\alpha\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*105·111.*73·41.*63·64.\supset \vdash :\text{Hp}.&\supset .\iotaʻʻ\gamma \text{ sm }\alpha .\gamma \in t_{1}ʻ\alpha .\gamma \in t_{0}ʻ\iotaʻʻ\gamma .\\ +[*63·41·383·16·55] &\supset .\iotaʻʻ\gamma \text{ sm }\alpha .t_{1}ʻ\alpha =t_{0}ʻ\iotaʻʻ\gamma .\\ +[*63·54] &\supset .\iotaʻʻ\gamma \text{ sm }\alpha .\iotaʻʻ\gamma \in t_{0}ʻ\alpha .\\ +[*105·11] &\supset .\iotaʻʻ\gamma \in \text{N}_{1}\text{c}ʻ\alpha :\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*105·41.</b> \(\vdash :\exists !\text{N}_{2}\text{c}ʻ\alpha .\supset .\exists !\text{N}_{1}\text{c}ʻ\alpha \quad[*105·4]\)</p> + +<p><span class="pagenum" id="Page_59">[Pg 59]</span></p> + +<p class="nind"><b>*105·42.</b> \(\vdash :\text{N}_{1}\text{c}ʻ\alpha =\Lambda .\supset .\text{N}_{2}\text{c}ʻ\alpha =\Lambda \quad[*105·41]\)</p> + +<p class="nind"><b>*105·43.</b> \(\vdash :\mu _{(1)}=\text{N}_{1}\text{c}ʻ\alpha .\supset .\mu _{(2)}=\text{N}_{2}\text{c}ʻ\alpha \)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*105·11. \supset \vdash :\text{Hp}.\beta \in \mu _{(1)}.&\supset .\beta \in \text{Nc}ʻ\alpha \cap t_{0}ʻ\alpha .\\ +[*63·54.*100·31·321] &\supset .\text{Nc}ʻ\beta =\text{Nc}ʻ\alpha .t_{0}ʻ\beta =t_{1}ʻ\alpha .\\ +[*105·1·101] &\supset .\text{N}_{1}\text{c}ʻ\beta =\text{N}_{2}\text{c}ʻ\alpha &\qquad \text{(1)}\\ +\vdash .*105·3.*103·26.\supset \vdash :\text{Hp}.\beta \in \mu _{(1)}.\supset .\text{N}_{1}\text{c}ʻ\beta &=\{\mu _{(1)}\}_{(1)}\\ +[*105·38]&=\mu _{(2)} &\qquad \text{(2)}\\ +\vdash .(1).(2). &\supset \vdash :\text{Hp}.\exists !\mu _{(1)}.\supset .\mu _{(2)}=\text{N}_{2}\text{c}ʻ\alpha &\qquad \text{(3)}\\ +\vdash .*105·372·42. &\supset \vdash :\text{Hp}.\mu _{(1)}=\Lambda .\supset .\mu _{(2)}=\Lambda .\text{N}_{2}\text{c}ʻ\alpha =\Lambda &\qquad \text{(4)}\\ +\vdash .(3).(4).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*105·44.</b> \(\vdash .\text{N}_{2}\text{c}ʻt^{2}ʻ\alpha =\Lambda\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*105·26.&\supset \vdash .\text{N}_{1}\text{c}ʻtʻtʻ\alpha =\Lambda .\\ +[*105·42] &\supset \vdash .\text{N}_{2}\text{c}ʻtʻtʻ\alpha =\Lambda .\supset \vdash .\text{Prop} +\end{array} +\]</p> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_60">[Pg 60]</span></p> +<h2 class="nobreak" id="*106"> +*106. CARDINALS OF RELATIONAL TYPES.</h2> +</div> + + +<p class="nind"><i>Summary of</i> *106.</p> + +<p>In this number we have to consider the cardinals whose members are +classes of relations which have a given relation of type to some given +class. For example, we have \(\downarrow xʻʻ\alpha \text{ sm }\alpha\), +and \(\downarrow xʻʻ\alpha\) has a given relation of type to \(\alpha\) +when \(x\) is given. Thus we want a notation for +\[ +\text{Nc}ʻ\alpha \cap tʻ\downarrow xʻʻ\alpha +\] +and all the associated ideas. In this number, we shall deal only with +relations in which the referent and relatum have a relation, as to +type, which can be expressed by the notations of *63, <i>i.e.</i> +roughly speaking, when, for suitable values of \(\alpha\), \(m\), +\(n\), our relations are contained in +\[ +t^{m}ʻ\alpha \uparrow t^{n}ʻ\alpha\,\,\, \text{or}\,\,\, t_{m}ʻ\alpha \uparrow t_{n}ʻ\alpha\,\,\, \text{or}\,\,\, t^{m}ʻ\alpha \uparrow t_{n}ʻ\alpha\,\,\, + \text{or}\,\,\, t_{m}ʻ\alpha \uparrow t^{n}ʻ\alpha. +\]</p> + +<p>Thus if \(t_{\mu \nu }ʻ\alpha\) has been defined, we shall put +\[ +\begin{aligned} +\text{N}_{\mu \nu }\text{c}ʻ\alpha &=\text{Nc}ʻ\alpha \cap tʻt_{\mu \nu }ʻ\alpha &\quad\text{Df},\\ +\text{N}_{\mu \nu }\text{C}&=\text{D}ʻ\text{N}_{\mu \nu }\text{c} &\quad\text{Df},\\ +\xi _{\mu \nu }&=\text{ sm }ʻʻ\xi \cap tʻt_{\mu \nu }ʻt_{1}ʻ\xi &\quad\text{Df}, +\end{aligned} +\] +with analogous definitions for \(t^{\mu \nu }ʻ\alpha\), \(t^\mu _{\nu}ʻ\alpha\) +and \(^\mu t_{\nu }ʻ\alpha\).</p> + +<p>Much the most important case is that of \(t_{00}ʻ\alpha\). For this +case we have</p> + +<p class="nind"><b>*106·1.</b> \[\begin{aligned}\vdash :\beta \in \text{N}_{00}\text{c}ʻ\alpha .\equiv .\beta \in \text{Nc}ʻ\alpha .\beta \in tʻt_{00}ʻ\alpha + .&\equiv .\beta \text{ sm }\alpha .\beta \in tʻtʻ(t_{0}ʻ\alpha \uparrow t_{0}ʻ\alpha ).\\ +&\equiv .\beta \text{ sm }\alpha .\beta \subset tʻ(\alpha \uparrow \alpha )\end{aligned}\]</p> + +<p>Thus \(\text{N}_{00}\text{c}ʻ\alpha\) will be the number of a class of +relations whose fields are of the same type as \(\alpha\), provided +this class of relations is similar to \(\alpha\). <i>E.g.</i> the +number of terms such as \(x\downarrow x\), where \(x\in \alpha\), will +be \(\text{N}_{00}\text{c}ʻ\alpha\).</p> + +<p>We have</p> + +<p class="nind"><b>*106·21.</b> \(\vdash .\exists !\text{N}_{00}\text{c}ʻ\alpha .\text{N}_{00}\text{c}ʻ\alpha \in \text{N}_{0}\text{C}\)</p> + +<p class="nind"><b>*106·22.</b>\(\vdash :\lambda \in \text{N}_{0}^{1}\text{c}ʻ\alpha .\equiv .\text{Cnv}ʻʻ\lambda \in ^{1}\text{N}_{0}\text{c}ʻ\alpha \)</p> + +<p class="nind"><b>*106·23.</b> \(\vdash :\beta \in \text{N}^{1}\text{c}ʻ\alpha .\supset .\text{N}^{11}\text{c}ʻ\alpha =\text{N}_{00}\text{c}ʻ\beta\)</p> + +<p class="nind"><b>*106·32.</b> \(\vdash :t_{0}ʻ\alpha =t_{0}ʻ\beta .\supset .(\exists \gamma ,\delta ).\gamma \in \text{N}_{00}\text{c}ʻ\alpha .\delta \in \text{N}_{00}\text{c}ʻ\beta + .\gamma \cap \delta =\Lambda\)</p> + +<p><span class="pagenum" id="Page_61">[Pg 61]</span></p> + +<p class="nind"><b>*106·4·41·411.</b> \(\vdash :\mu =\text{N}_{0}\text{c}ʻ\alpha .\supset .\mu _{(00)}=\text{N}_{00}\text{c}ʻ\alpha .\mu ^{(11)}=\text{N}^{11}\text{c}ʻ\alpha + .\mu _{(11)}=\text{N}_{11}\text{c}ʻ\alpha\)</p> + +<p class="nind"><b>*106·53.</b> \(\vdash .\text{Nc}(\alpha )ʻt_{00}ʻ\alpha = \Lambda\)</p> + +<p class="nind">whence it follows that</p> + +<p class="nind"><b>*106·54.</b> \(\vdash .\text{N}_{0}\text{c}ʻt_{00}ʻ\alpha {\sim} \in \text{N}_{00}\text{C}\)</p> + +<p>The propositions of this number, except <a href="#*106·21">*106·21</a>, are never referred +to again (except in <a href="#*154·25">*154·25</a><a href="#*154·251">·251</a><a href="#*154·262">·262</a>, which are themselves never +used again), but they have a somewhat greater importance than the +propositions of <a href="#*105">*105</a>, owing to the fact that the arithmetical +operations are defined by means of classes of relations, <i>i.e.</i> +the sum of two cardinals (for instance) is defined as the cardinal +number of a certain class of relations (cf. <a href="#*110">*110</a>).</p> + +<hr class="tb"> + +<p class="nind"><b>*106·01</b> \(\text{N}_{00}\text{c}ʻ\alpha = \text{Nc}ʻ\alpha \cap tʻt_{00}ʻ\alpha \quad\text{Df}\)</p> + +<p class="nind"><b>*106·011.</b> \(\text{N}^{11}\text{c}ʻ\alpha = \text{Nc}ʻ\alpha \cap tʻt^{11}ʻ\alpha \quad\text{Df}\)</p> + +<p class="nind"><b>*106·012.</b> \(\text{N}_{01}\text{c}ʻ\alpha = \text{Nc}ʻ\alpha \cap tʻt_{01}ʻ\alpha \quad\text{Df etc.}\)</p> + +<p class="nind"><b>*106·02.</b> \(\text{N}^{1}\text{c}ʻ\alpha = \text{Nc}ʻ\alpha \cap tʻt_{0}^{1}ʻ\alpha \quad\text{Df etc.}\)</p> + +<p class="nind"><b>*106·021.</b> \(^{1}\text{N}_{0}\text{c}ʻ\alpha = \text{Nc}ʻ\alpha \cap tʻ^{1}t_{0}ʻ\alpha \quad\text{Df etc.}\)</p> + +<p class="nind"><b>*106·03.</b> \(\text{N}_{00}\text{C} = \text{D}ʻ\text{N}_{00}\text{c} \quad\text{Df etc.}\)</p> + +<p class="nind"><b>*106·04.</b> \(\mu _{(00)} = \text{ sm }ʻʻ\mu \cap tʻt_{00}ʻt_{1}ʻ\mu \quad\text{Df}\)</p> + +<p class="nind"><b>*106·041.</b> \(\mu ^{(11)} = \text{ sm }ʻʻ\mu \cap tʻt^{11}ʻt_{1}ʻ\mu \quad\text{Df etc.}\)</p> + +<p class="nind"><b>*106·1.</b> \[\begin{aligned}\vdash :\beta \in \text{N}_{00}\text{c}ʻ\alpha . &\equiv .\beta \in \text{Nc}ʻ\alpha .\beta \in tʻt_{00}ʻ\alpha .\\ +&\equiv .\beta \text{ sm } \alpha .\beta \in tʻtʻ(t_{0}ʻ\alpha \uparrow t_{0}ʻ\alpha ).\\ +&\equiv .\beta \text{ sm } \alpha .\beta \subset tʻ(\alpha \uparrow \alpha )\\ +&[*100·1.(*106·01.*64·01).*64·11]\end{aligned}\]</p> + +<p class="nind"><b>*106·101.</b> \[\begin{aligned}\vdash :\beta \in \text{N}^{11}\text{c}ʻ\alpha . &\equiv .\beta \in \text{Nc}ʻ\alpha .\beta \in tʻt^{11}ʻ\alpha .\\ +&\equiv .\beta \text{ sm } \alpha .\beta \in tʻtʻ(tʻ\alpha \uparrow tʻ\alpha ).\\ +&\equiv .\beta \text{ sm } \alpha .\beta \subset tʻ(tʻ\alpha \uparrow tʻ\alpha )\end{aligned}\]</p> + +<p>Similar propositions hold for any other double index \(mn\) for which +\(t^{mn}ʻ\alpha\) has been defined.</p> + +<p class="nind"><b>*106·11.</b> \[\begin{aligned}\vdash :\beta \in \text{N}_{01}\text{c}ʻ\alpha . &\equiv .\beta \in \text{Nc}ʻ\alpha .\beta tʻt_{01}ʻ\alpha .\\ +&\equiv .\beta \text{ sm } \alpha .\beta \in tʻtʻ(t_{0}ʻ\alpha \uparrow t_{1}ʻ\alpha ).\\ +&\equiv .\beta \text{ sm } \alpha .\beta \subset tʻ(t_{0}ʻ\alpha \uparrow t_{1}ʻ\alpha )\end{aligned}\]</p> + +<p>Similar propositions hold for any other double suffix \(mn\) for which +\(t_{mn}ʻ\alpha\) has been defined.</p> + +<p class="nind"><b><a id="*106·12">*106·12</a>.</b> \[\begin{aligned}\vdash :\beta \in \text{N}^{1}\text{c}ʻ\alpha . &\equiv .\beta \in \text{Nc}ʻ\alpha .\beta \in tʻt_{0}^{1}ʻ\alpha .\\ +&\equiv .\beta \text{ sm } \alpha .\beta \in tʻtʻ(t_{0}ʻ\alpha \uparrow tʻ\alpha ).\\ +&\equiv .\beta \text{ sm } \alpha .\beta \subset tʻ(t_{0}ʻ\alpha \uparrow tʻ\alpha )\end{aligned}\]</p> + +<p class="nind"><b>*106·121.</b> \[\begin{aligned}\vdash :\beta \in ^{1}\text{N}_{0}\text{c}ʻ\alpha . &\equiv .\beta \in \text{Nc}ʻ\alpha .\beta \in tʻ^{1}t_{0}ʻ\alpha .\\ +&\equiv .\beta \text{ sm } \alpha .\beta \in tʻtʻ(tʻ\alpha \uparrow t_{0}ʻ\alpha ).\\ +&\equiv .\beta \text{ sm } \alpha .\beta \subset tʻ(tʻ\alpha \uparrow t_{0}ʻ\alpha )\end{aligned}\]</p> + +<p><span class="pagenum" id="Page_62">[Pg 62]</span></p> + +<p>Similar propositions hold for any other index and suffix for which +\(t_{m}^{n}ʻ\alpha\) or \(^{n}t_{m}ʻ\alpha\) has been defined.</p> + +<p class="nind"><b>*106·13.</b> \(\vdash :\mu \in \text{N}_{00}\text{C}.\equiv .(\exists \alpha ).\mu =\text{N}_{00}\text{c}ʻ\alpha \quad[*100·22.*71·41]\)</p> + +<p>Similar propositions hold for \(\text{N}^{11}\text{C}ʻ\alpha\) etc.</p> + +<p class="nind"><b>*106·14.</b> \[\begin{aligned}\vdash :\beta \in \mu _{(00)}.&\equiv .(\exists \alpha ).\alpha \in \mu .\beta \text{ sm }\alpha .\beta \in tʻtʻ(t_{1}ʻ\mu \uparrow t_{1}ʻ\mu + ).\\ +&\equiv .(\exists \alpha ).\alpha \in \mu .\beta \text{ sm }\alpha .\beta \in tʻt_{00}ʻ\alpha .\\ +&\equiv .(\exists \alpha ).\alpha \in \mu .\beta \text{ sm }\alpha .\beta \subset tʻ(\alpha \uparrow \alpha ) \quad[*64·33·11]\end{aligned}\]</p> + +<p class="nind"><b>*106·141.</b> \[\begin{aligned}\vdash :\beta \in \mu _{0}^{1}.&\equiv .(\exists \alpha ).\alpha \in \mu .\beta \text{ sm }\alpha .\beta \in tʻtʻ(t_{1}ʻ\mu + \uparrow t_{0}ʻ\mu ).\\ +&\equiv .(\exists \alpha ).\alpha \in \mu .\beta \text{ sm }\alpha .\beta \in tʻt_{0}^{1}ʻ\alpha .\\ +&\equiv .(\exists \alpha ).\alpha \in \mu .\beta \text{ sm }\alpha .\beta \subset tʻ(\alpha \uparrow tʻ\alpha )\end{aligned}\]</p> + +<p>Similar propositions hold for \(^{1}\mu _{0}\), \(\mu ^{11}\), \(\mu_{11}\) etc.</p> + +<p class="nind"><b>*106·2</b> \(\vdash :x\in t_{0}ʻ\alpha .\supset .\downarrow xʻʻ\alpha \in \text{N}_{00}\text{c}ʻ\alpha .\downarrow xʻʻ\alpha \in \text{N}_{0}\text{c}ʻ\downarrow + xʻʻ\alpha\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*55·15.&\supset \vdash :R\in \downarrow xʻʻ\alpha .\supset .\text{D}ʻR\subset \alpha .\text{ᗡ}ʻR=\iota ʻx:\\ +[*63·105] &\supset \vdash \colon\ldotp x\in t_{0}ʻ\alpha .\supset :R\in \downarrow xʻʻ\alpha .\supset _{R}.\text{D}ʻR\subset t_{0}ʻ\alpha .\text{ᗡ}ʻR\subset t_{0}ʻ\alpha + .\\ +[*35·83] &\supset _{R}.R\unicode{x2abd} t_{0}ʻ\alpha \uparrow t_{0}ʻ\alpha .\\ +[*64·16·13] &\supset _{R}.R\in tʻ(\alpha \uparrow \alpha ):\\ +[*22·1] &\supset :\downarrow xʻʻ\alpha \subset tʻ(\alpha \uparrow \alpha ) &\qquad \text{(1)}\\ +\vdash .(1).*73·611.*106·1.*103·12.&\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*106·201.</b> \(\vdash :\beta \in tʻ\alpha .\supset .\downarrow \beta ʻʻ\alpha \in \text{N}_{0}^{1}\text{c}ʻ\alpha \)</p> + +<p class="nind"><b>*106·202.</b> \(\vdash :\beta \in t^{2}ʻ\alpha .\supset .\downarrow \beta ʻʻ\alpha \in \text{N}_{0}^{2}\text{c}ʻ\alpha\)</p> + +<p class="nind"><b>*106·203.</b> \(\vdash .\downarrow \alpha ʻʻ\alpha \in \text{N}_{0}^{1}\text{c}ʻ\alpha \quad[*106·201]\)</p> + +<p class="nind"><b>*106·204.</b> \(\vdash .\downarrow (\iotaʻʻ\alpha )ʻʻ\alpha \in \text{N}_{0}^{2}\text{c}ʻ\alpha \quad[*106·202]\)</p> + +<p class="nind"><b><a id="*106·21">*106·21</a>.</b> \(\vdash .\exists !\text{N}_{00}\text{c}ʻ\alpha .\text{N}_{00}\text{c}ʻ\alpha \in \text{N}_{0}\text{C} \quad[*106·2.*63·18]\)</p> + +<p class="nind"><b><a id="*106·211">*106·211</a>.</b> \(\vdash .\Lambda {\sim}\in \text{N}_{00}\text{C}.\text{N}_{00}\text{C}\subset \text{N}_{0}\text{C}.\text{N}_{00}\text{C}\in \text{Cls ex}^{2}\,\text{excl} + \quad[*106·21.*103·24]\)</p> + +<p class="nind"><b><a id="*106·212">*106·212</a>.</b> \(\vdash .\Lambda {\sim}\in \text{N}_{0}^{1}\text{C}.\text{N}_{0}^{1}\text{C}\subset \text{N}_{0}\text{C}.\text{N}_{0}^{1}\text{C}\in + \text{Cls ex}^{2}\,\text{excl} \quad[*106·203]\)</p> + +<p class="nind"><b><a id="*106·213">*106·213</a>.</b> \(\vdash .\Lambda {\sim}\in \text{N}_{0}^{2}\text{C}.\text{N}_{0}^{2}\text{C}\subset \text{N}_{0}\text{C}.\text{N}_{0}^{2}\text{C}\in + \text{Cls ex}^{2}\,\text{excl} \quad[*106·204]\)</p> + +<p class="nind"><b>*106·22.</b> \(\vdash :\lambda \in \text{N}_{0}^{1}\text{c}ʻ\alpha .\equiv .\text{Cnv}ʻʻ\lambda \in ^{1}\text{N}_{0}\text{c}ʻ\alpha\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*73·4. \supset \vdash :\lambda \text{ sm }\alpha .&\equiv .\text{Cnv}ʻʻ\lambda \text{ sm }\alpha &\qquad \text{(1)}\\ +\vdash .*64·16. \supset \vdash \colon\ldotp \lambda \subset tʻ(t_{0}ʻ\alpha \uparrow tʻ\alpha ).&\equiv :R\in \lambda .\supset _{R}.R\,\unicode{x2abd}\,t_{0}ʻ\alpha + \uparrow tʻ\alpha :\\ +[*35·84] &\equiv :R\in \lambda .\supset _{R}.\breve{R}\,\unicode{x2abd}\, tʻ\alpha \uparrow t_{0}ʻ\alpha :\\ +[*37·63] &\equiv :S\in \text{Cnv}ʻʻ\lambda .\supset _{S}.S\,\unicode{x2abd}\, tʻ\alpha \uparrow t_{0}ʻ\alpha :\\ +[*64·16] &\equiv :\text{Cnv}ʻʻ\lambda \subset tʻ(tʻ\alpha \uparrow t_{0}ʻ\alpha ) &\qquad \text{(2)}\\ +\vdash .(1).(2).*106·12.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_63">[Pg 63]</span></p> + +<p>The proof requires, in addition to <a href="#*106·12">*106·12</a>, its analogue for +\(^{1}\text{N}_{0}\text{c}ʻ\alpha\). Such analogues will be assumed as +required.</p> + +<p class="nind"><b>*106·221.</b> \(\vdash :\lambda \in \text{N}_{0}^{2}\text{c}ʻ\alpha .\equiv .\text{Cnv}ʻʻ\lambda \in ^{2}\text{N}_{0}\text{c}ʻ\alpha\)</p> + +<p class="nind"><b>*106·222.</b> \(\vdash .\Lambda {\sim}\in ^{1}\text{N}_{0}\text{C}.^{1}\text{N}_{0}\text{C}\subset \text{N}_{0}\text{C}.^{1}\text{N}_{0}\text{C}\in + \text{Cls ex}^{2}\,\text{excl} \quad[*106·22·212]\)</p> + +<p class="nind"><b>*106·223.</b> \(\vdash .\Lambda {\sim}\in ^{2}\text{N}_{0}\text{C}.^{2}\text{N}_{0}\text{C}\subset \text{N}_{0}\text{C}.^{2}\text{N}_{0}\text{C}\in + \text{Cls ex}^{2}\,\text{excl}\)</p> + +<p>Other propositions of the same kind as the above may be proved by +observing that, if \(m\) and \(n\) are indices for which \(t^{m}ʻ\alpha\) +and \(t^{n}ʻ\alpha\) have been defined, we have +\[ +\gamma \subset t^{n}ʻ\alpha .\beta \in \text{N}^{m}\text{c}ʻ\alpha .\supset .\downarrow \beta ʻʻ\gamma \in \text{N}^{m}\text{c}ʻ\alpha , +\] +of which the proof is direct and simple. Hence, since we always have +\(\exists !\text{N}^{m}\text{c}ʻ\alpha\), we also always have +\[ +\begin{aligned} +&\exists !\text{N}^{m}\text{c}ʻ\alpha ,\\ +\text{whence}\qquad \text{N}^{mm}\text{C}\subset \text{N}_{0}\text{C}.&\text{N}^{mm}\text{C}\in \text{Cls ex}^{2}\,\text{excl}. +\end{aligned} +\] +We have in like manner +\[ +\exists !\text{N}_{0}^{m}\text{c}ʻ\alpha .\exists !^{m}\text{N}_{0}\text{c}ʻ\alpha. +\] +But we do not always have +\[ +\exists !\text{N}_{mn}\text{c}ʻ\alpha \,\,\text{or}\,\, \exists !\text{N}_{n}^{m}\text{c}ʻ\alpha \,\,\text{or}\,\, \exists !^{m}\text{N}_{n}\text{c}ʻ\alpha . +\]</p> + +<p class="nind"><b>*106·23.</b> \(\vdash :\beta \in \text{N}^{1}\text{c}ʻ\alpha .\supset .\text{N}^{11}\text{c}ʻ\alpha =\text{N}_{00}\text{c}ʻ\beta\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +&\vdash .*64·33.*104·1.*63·5 .\supset \vdash :\text{Hp}.\supset .t^{11}ʻ\alpha =t_{00}ʻ\beta \\ +&\vdash .(1).(*106·01·011).*100·321.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*106·231.</b> \(\vdash :\beta \in \text{N}_{1}\text{c}ʻ\alpha .\supset .\text{N}_{11}\text{c}ʻ\alpha =\text{N}_{00}\text{c}ʻ\beta + \quad[\text{Proof as in}\, *106·23]\)</p> + +<p class="nind"><b>*106·24.</b> \(\vdash :\text{N}^{1}\text{c}ʻ\alpha =\text{N}_{0}\text{c}ʻ\beta .\supset .\text{N}^{11}\text{c}ʻ\alpha =\text{N}_{00}\text{c}ʻ\beta + \quad[*106·23]\)</p> + +<p class="nind"><b>*106·241.</b> \(\vdash :\text{N}_{1}\text{c}ʻ\alpha =\text{N}_{0}\text{c}ʻ\beta .\supset .\text{N}_{11}\text{c}ʻ\alpha =\text{N}_{00}\text{c}ʻ\beta\)</p> + +<p>The analogues of the above propositions for other indices or suffixes +are similarly proved.</p> + +<p class="nind"><b>*106·25.</b> \(\vdash .\text{N}^{11}\text{c}ʻ\alpha =\text{N}_{00}\text{c}ʻ\iotaʻʻ\alpha \quad[*106·23.*104·2]\)</p> + +<p class="nind"><b>*106·251.</b> \(\vdash .\text{N}_{00}\text{c}ʻ\alpha =\text{N}_{11}\text{c}ʻ\iotaʻʻ\alpha\)</p> + +<p class="nind"><b>*106·31.</b> \[\begin{aligned}\vdash :x,y\in t_{0}ʻ\alpha .t_{0}ʻ\alpha &=t_{0}ʻ\beta .x\neq y.\supset .\\ +&\downarrow xʻʻ\alpha \in \text{N}_{00}\text{c}ʻ\alpha .\downarrow yʻʻ\beta \in \text{N}_{00}\text{c}ʻ\beta .\downarrow xʻʻ\alpha \cap \downarrow xʻʻ\beta =\Lambda \\ +\quad[*106·2.*55·233]\end{aligned}\]</p> + +<p class="nind"><b>*106·311.</b> \[\begin{aligned}\vdash \colon\ldotp x\in t_{0}ʻ\alpha .t_{0}ʻ\alpha &=t_{0}ʻ\beta :\alpha =\Lambda .\lor.\beta =\Lambda :\supset .\\ +&\downarrow xʻʻ\alpha \in \text{N}_{00}\text{c}ʻ\alpha .\downarrow xʻʻ\beta \in \text{N}_{00}\text{c}ʻ\beta .\downarrow xʻʻ\alpha \cap \downarrow xʻʻ\beta =\Lambda \\ +\quad[*106·2.*55·232.\text{Transp}]\end{aligned}\]</p> + +<p class="nind"><b>*106·312.</b> \[\begin{aligned}\vdash :&t_{0}ʻ\alpha =\iota ʻx.\alpha =\beta =\iota ʻx.\supset .\\ +&\iota ʻ(\iota ʻx\uparrow \iota ʻx)\in \text{N}_{00}\text{c}ʻ\alpha .\iota ʻ(\Lambda \uparrow \iota ʻx)\in \text{N}_{00}\text{c}ʻ\beta + .\iota ʻ(\iota ʻx\uparrow \iota ʻx)\cap \iota ʻ(\Lambda \uparrow \iota ʻx)=\Lambda \end{aligned}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*73·43.&\supset \vdash .\iota ʻ(\iota ʻx\uparrow \iota ʻx)\text{ sm }\iota ʻx.\iota ʻ(\Lambda \uparrow \iota ʻx)\text{ sm }\iota ʻx.\\ +[*13·12] &\supset \vdash :\text{Hp}.\supset .\iota ʻ(\iota ʻx\uparrow \iota ʻx)\text{ sm }\alpha .\iota ʻ(\Lambda \uparrow \iota ʻx)\text{ sm }\beta &\qquad \text{(1)}\\ +\vdash .*64·16.&\supset \vdash :\text{Hp}.\supset .\iota ʻx\uparrow \iota ʻx\in t_{00}ʻ\alpha .\Lambda \uparrow \iota ʻx\in t_{00}ʻ\alpha &\qquad \text{(2)}\\ +\vdash .(1).(2).*106·1.*51·161.*24·54.*55·202.&\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_64">[Pg 64]</span></p> + +<p class="nind"><b>*106·32.</b> \(\vdash :t_{0}ʻ\alpha =t_{0}ʻ\beta .\supset .(\exists \gamma ,\delta ).\gamma \in \text{N}_{00}\text{c}ʻ\alpha .\delta \in \text{N}_{00}\text{c}ʻ\beta + .\gamma \cap \delta =\Lambda\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*106·31.\supset \vdash \colon\ldotp \text{Hp}:(\exists x,y).x,y\in t_{0}ʻ\alpha .x\neq y:\supset .\\ +(\exists \gamma ,\delta ).\gamma \in \text{N}_{00}\text{c}ʻ\alpha .\delta \in \text{N}_{00}\text{c}ʻ\beta .\gamma \cap \delta =\Lambda &\qquad \text{(1)}\\ +\vdash .*52·4. \supset \vdash :{\sim}(\exists x,y).x,y\in t_{0}ʻ\alpha .x\neq y.\supset .t_{0}ʻ\alpha \in 1\cup \iota ʻ\Lambda .\\ +[*63·18] \supset .t_{0}\alpha \in 1 &\qquad \text{(2)}\\ +\vdash .(2).*60·38.*63·105.*52·46.\supset \vdash :{\sim}(\exists x,y).x,y\in t_{0}ʻ\alpha .x\neq y.\exists !\alpha .\exists !\beta .\supset .\\ +\alpha =\beta =t_{0}ʻ\alpha .t_{0}ʻ\alpha \in 1.\\ +[*106·312] \supset .(\exists \gamma ,\delta ).\gamma \in \text{N}_{00}\text{c}ʻ\alpha .\delta \in \text{N}_{00}\text{c}ʻ\beta + .\gamma \cap \delta =\Lambda &\qquad \text{(3)}\\ +\vdash .*106·311.*63·18.\supset \\ +\vdash \colon\ldotp \text{Hp}:{\sim}(\exists !\alpha .\exists !\beta ):\supset .(\exists \gamma ,\delta ).\gamma \in \text{N}_{00}\text{c}ʻ\alpha .\delta \in \text{N}_{00}\text{c}ʻ\beta + .\gamma \cap \delta =\Lambda &\qquad \text{(4)}\\ +\vdash .(1).(3).(4).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*106·4.</b> \(\vdash :\mu =\text{N}_{0}\text{c}ʻ\alpha .\supset .\mu _{(00)}=\text{N}_{00}\text{c}ʻ\alpha\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*106·14.\supset \vdash \colon\colon \text{Hp}.\supset \colon\ldotp \beta \in \mu _{00}.&\equiv :(\exists \gamma ).\gamma \in \text{N}_{0}\text{c}ʻ\alpha + .\beta \text{ sm }\gamma .\beta \in tʻt_{00}ʻ\gamma :\\ +[*64·3] &\equiv :(\exists \gamma ).\gamma \in \text{N}_{0}\text{c}ʻ\alpha .\beta \text{ sm }\gamma :\beta \in tʻt_{00}ʻ\alpha :\\ +[*102·84] & \equiv :\beta \text{ sm }\alpha .\beta \in tʻt_{00}ʻ\alpha :\\ +[*106·1] &\equiv :\beta \in \text{N}_{00}\text{c}ʻ\alpha \colon\colon \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*106·401.</b> \(\vdash :\mu =\text{N}^{1}\text{c}ʻ\alpha .\supset .\mu _{(00)}=\text{N}^{11}\text{c}ʻ\alpha\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*104·24.*106·4.\supset \vdash :\text{Hp}.\supset .\mu _{(00)}&=\text{N}_{00}\text{c}ʻ\iotaʻʻ\alpha \\ +[*106·25] &=\text{N}^{11}\text{c}ʻ\alpha :\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*106·402.</b> \(\vdash :\mu =\text{N}_{1}\text{c}ʻ\alpha .\exists !\mu .\supset .\mu _{(00)}=\text{N}_{11}\text{c}ʻ\alpha\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*106·231. \supset \vdash :\text{Hp}.\beta \in \mu .\supset .\text{N}_{11}\text{c}ʻ\alpha &=\text{N}_{00}\text{c}ʻ\beta \\ +[*106·4.*103·26] & =\mu _{(00)} &\qquad \text{(1)}\\ +\vdash .(1).*10·11·23·35.&\supset \vdash :\text{Hp}.\exists !\mu .\supset .\mu _{(00)}=\text{N}_{11}\text{c}ʻ\alpha :\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*106·41.</b> \(\vdash :\mu =\text{N}_{0}\text{c}ʻ\alpha .\supset .\mu ^{(11)}=\text{N}^{11}\text{c}ʻ\alpha\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*63·54.(*106·041).*103·27.\supset \\ +\vdash \colon\colon \text{Hp}.\supset \colon\ldotp \beta \in \mu ^{(11)}.&\equiv :(\exists \gamma ).\gamma \in \text{N}_{0}\text{c}ʻ\alpha + .\beta \text{ sm }\gamma :\beta \in tʻt^{11}ʻt_{0}ʻ\alpha :\\ +[*102·84.*64·32] &\equiv :\beta \text{ sm }\alpha .\beta \in tʻt^{11}ʻ\alpha :\\ +[(*106·011)] &\equiv :\beta \in \text{N}^{11}\text{c}ʻ\alpha \colon\colon \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*106·411.</b> \(\vdash :\mu =\text{N}_{0}\text{c}ʻ\alpha .\supset .\mu _{(11)}=\text{N}_{11}\text{c}ʻ\alpha \quad[\text{Proof as in}\, *106·41]\)</p> + +<p class="nind"><b>*106·43.</b> \(\vdash :\mu ,\nu \in \text{N}_{0}\text{C}.tʻ\mu =tʻ\nu .\supset .(\exists \gamma ,\delta ).\gamma \in \mu _{(00)}.\delta \in \nu _{(00)}.\gamma + \cap \delta =\Lambda\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*103·2.\supset \vdash :\text{Hp}.&\supset .(\exists \alpha ,\beta ).\mu =\text{N}_{0}\text{c}ʻ\alpha .\nu =\text{N}_{0}\text{c}ʻ\beta .\\ +[*106·4] &\supset .(\exists \alpha ,\beta ).\mu _{(00)}=\text{N}_{00}\text{c}ʻ\alpha .\nu _{(00)}=\text{N}_{00}\text{c}ʻ\beta .\\ +[*106·32] &\supset .(\exists \gamma ,\delta ).\gamma \in \mu _{(00)}.\delta \in \nu _{(00)}.\gamma \cap \delta =\Lambda :\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_65">[Pg 65]</span></p> + +<p class="nind"><b>*106·44.</b> \(\vdash :\mu ,\nu \in \text{N}_{00}\text{C}.tʻ\mu + =tʻ\nu .\supset .(\exists \gamma ,\delta ).\gamma \in \mu .\delta \in \nu .\gamma \cap \delta =\Lambda \quad[*106·32]\)</p> + +<p>The following propositions are analogous to <a href="#*102·71">*102·71</a> ff., and similar +remarks apply to them.</p> + +<p class="nind"><b>*106·5.</b> \[\begin{aligned}\vdash :R\in \text{Cls}\rightarrow &1.\text{D}ʻR\subset \alpha .\text{ᗡ}ʻR\subset \text{ᗡ}ʻʻʻ(\alpha \uparrow \alpha ).\\ +&W=\hat{x} \hat{y} \{x,y\in \alpha .{\sim}x(\breve{R} ʻx)y\}.\supset .W{\sim}\in \text{ᗡ}ʻR.W\unicode{x2abd} \alpha \uparrow \alpha\end{aligned}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*4·73.\supset \vdash \colon\colon \text{Hp}.\supset \colon\ldotp x,y\in \alpha .&\supset _{x,y}:xWy.\equiv .{\sim}x(\breve{R} ʻx)y:\\ +[*5·18] &\supset _{x,y}:{\sim}\{xWy.\equiv .x(\breve{R} ʻx)y\}\colon\ldotp \\ +[*10·1] \supset \colon\ldotp x\in \alpha .&\supset _{x}.{\sim}\{xWx.\equiv .x(\breve{R} ʻx)x\}.\\ +[*21·43.\text{Transp}] &\supset _{x}.W\neq \breve{R} ʻx\colon\ldotp +[\text{Hp}] \supset \colon\ldotp x\in \text{D}ʻR.\supset _{x}.W\neq \breve{R} ʻx\colon\ldotp \\ +[*71·411.\text{Transp}] &\supset \colon\ldotp W{\sim}\in \text{ᗡ}ʻR &\qquad \text{(1)}\\ +\vdash .*21·33.(*35·04).&\supset \vdash :\text{Hp}.\supset .W\,\unicode{x2abd}\, \alpha \uparrow \alpha &\qquad \text{(2)}\\ +\vdash .(1).(2).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*106·51.</b> \(\vdash :\beta \subset \alpha .\supset .{\sim}\{\beta \text{ sm }\text{Rl}ʻ(\alpha \uparrow \alpha )\}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*106·5. \supset \vdash :\text{Hp}.R\in 1\rightarrow 1.\text{D}ʻR=&\beta .\text{ᗡ}ʻR\subset \text{ᗡ}ʻʻʻ(\alpha \uparrow \alpha ).\supset .\\ +&(\exists W).W\in \text{ᗡ}ʻʻʻ(\alpha \uparrow \alpha ).W{\sim}\in \text{ᗡ}ʻR.\\ +[*13·14] &\supset .\text{ᗡ}ʻR\neq \text{ᗡ}ʻʻʻ(\alpha \uparrow \alpha ) &\qquad \text{(1)}\\ +\vdash .(1).*22·41.\supset \vdash \colon\ldotp \text{Hp}.&\supset :R\in 1\rightarrow 1.\text{D}ʻR=\beta .\supset _{R}.\text{ᗡ}ʻR \neq \text{ᗡ}ʻʻʻ(\alpha \uparrow \alpha ):\\ +[*10·51.*73·1] &\supset :{\sim}{\beta \text{ sm }\text{ᗡ}ʻʻʻ(\alpha \uparrow \alpha )}\colon\ldotp \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*106·52.</b> \(\vdash :\beta \subset t_{0}ʻ\alpha .\supset .\beta {\sim}\in \text{Nc}ʻt_{00}ʻ\alpha\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*106·51.\supset \vdash :\text{Hp}.&\supset .{\sim}\{\beta \text{ sm }\text{Rl}ʻ(t_{0}ʻ\alpha \uparrow t_{0}ʻ\alpha)\}.\\ +[*64·54] &\supset .{\sim}\{\beta \text{ sm }t_{00}ʻ\alpha\}.\\ +[*100·1] &\supset .\beta {\sim}\in \text{Nc}ʻt_{00}ʻ\alpha :\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*106·53.</b> \(\vdash .\text{Nc}(\alpha )ʻt_{00}ʻ\alpha =\Lambda \quad[*106·52.*102·6.*63·371]\)</p> + +<p class="nind"><b>*106·54.</b> \(\vdash .\text{N}_{0}\text{c}ʻt_{00}ʻ\alpha {\sim}\in \text{N}_{00}\text{C}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*100·33.*103·15.&\supset \\ +\vdash :\text{N}_{00}\text{c}ʻ\beta =\text{N}_{0}\text{c}ʻt_{00}ʻ\alpha .&\supset .\beta \text{ sm }t_{00}ʻ\alpha &\qquad \text{(1)}\\ +\vdash .*103·12.(*106·01).&\supset \\ +\vdash :\text{N}_{00}\text{c}ʻ\beta =\text{N}_{0}\text{c}ʻt_{00}ʻ\alpha .&\supset .t_{00}ʻ\alpha \in tʻt_{00}ʻ\beta .\\ +[*63·16.(*64·01)] &\supset .tʻtʻ(t_{0}ʻ\alpha \uparrow t_{0}ʻ\alpha )=tʻtʻ(t_{0}ʻ\beta \uparrow t_{0}ʻ\beta ).\\ +[*63·391] &\supset .tʻ(t_{0}ʻ\alpha \uparrow t_{0}ʻ\alpha )=tʻ(t_{0}ʻ\beta \uparrow t_{0}ʻ\beta ) .\\ +[*64·3.(*64·01)] &\supset .t_{0}ʻ\alpha =t_{0}ʻ\beta .\\ +[*63·105] &\supset .\beta \subset t_{0}ʻ\alpha &\qquad \text{(2)}\\ +\vdash .(1).(2).&\supset \vdash :\text{N}_{00}\text{c}ʻ\beta =\text{N}_{0}\text{c}ʻt_{00}ʻ\alpha .\supset .\beta \in \text{Nc}ʻt_{00}ʻ\alpha .\beta \subset t_{0}ʻ\alpha + &\qquad \text{(3)}\\ +\vdash .(3).\text{Transp}.*106·52.&\supset \vdash .(\beta ).\text{N}_{00}\text{c}ʻ\beta \neq \text{N}_{0}\text{c}ʻt_{00}ʻ\alpha .\\ +[*106·13.\text{Transp}] &\supset \vdash .\text{N}_{0}\text{c}ʻt_{00}ʻ\alpha {\sim}\in \text{N}_{00}\text{C}.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*106·55.</b> \(\vdash .\exists !\text{N}_{0}\text{C}-\text{N}_{00}\text{C} \quad[*106·54]\)</p> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_66">[Pg 66]</span></p> +<h2 class="nobreak" id="SECTION_B">SECTION B.<br> +<span class="allsmcap">ADDITION, MULTIPLICATION AND EXPONENTIATION.</span></h2> +</div> + + +<p class="nind"><i>Summary of Section B.</i></p> + +<p>In the present section, we have to consider the arithmetical operations +as applied to cardinals, as well as the relation of greater and less +between cardinals. Thus the topics to be dealt with in this section are +the first that can properly be said to belong to Arithmetic.</p> + +<p>The treatment of addition, multiplication and exponentiation to be +given in what follows is guided by the desire to secure the greatest +possible generality. In the first place, everything to be said +generally about the arithmetical operations must apply equally to +finite and infinite classes or cardinals. In the second place, we +desire such definitions as shall allow the number of summands in a sum +or of factors in a product to be infinite. In the third place, we wish +to be able to add or multiply two numbers which are not necessarily +of the same type. In the fourth place, we wish our definitions to +be such that the sum of the cardinal numbers of two or more classes +shall depend only upon the cardinal numbers of those classes, and +shall be the same when the classes overlap as when they are mutually +exclusive; with similar conditions for the product. The desire to +obtain definitions fulfilling all these conditions leads to somewhat +more complicated definitions than would otherwise be required; but in +the outcome, the result is simpler than if we started with simpler +definitions, since we avoid vexatious exceptions.</p> + +<p>The above observations will become clearer through their applications. +Let us begin with the case of arithmetical addition of two classes.</p> + +<p>If \(\alpha\) and \(\beta\) are mutually exclusive classes, the sum +of their cardinal numbers will be the cardinal number of \(\alpha +\cup \beta\). But in order that \(\alpha\) and \(\beta\) may be +mutually exclusive, they must have no common members, and this is +only significant when they are of the same type. Hence, given two +perfectly general classes \(\alpha\) and \(\beta\), we require to +find two classes which are mutually exclusive and are respectively +similar to \(\alpha\) and \(\beta\); if these two classes are called +\(\alpha'\) and \(\beta'\), then \(\text{Nc}ʻ(\alpha' \cup \beta')\) +will be the sum of the cardinal numbers of \(\alpha\) and \(\beta\). We +note that \(\Lambda \cap \alpha\) and \(\Lambda \cap \beta\) indicate +respectively the \(\Lambda\)'s of the same types as \(\alpha\) and +\(\beta\), and accordingly we take as \(\alpha'\) and \(\beta'\) the +two classes +\[ +\downarrow (\Lambda \cap \beta )ʻʻ\iotaʻʻ\alpha\,\,\text{and}\,\,(\Lambda \cap \alpha )\downarrow ʻʻ\iotaʻʻ\beta ; +\]<span class="pagenum" id="Page_67">[Pg 67]</span> +these two classes are always of the same type, always mutually +exclusive, and always similar to \(\alpha\) and \(\beta\) respectively. +Hence we define +\[ +\alpha +\beta =\downarrow (\Lambda \cap \beta )ʻʻ\iotaʻʻ\alpha \cup (\Lambda \cap \alpha )\downarrow ʻʻ\iotaʻʻ\beta \quad\text{Df}. +\]</p> + +<p>The sum of the cardinal numbers of \(\alpha\) and \(\beta\) will +then be the cardinal number of \(\alpha + \beta\); hence we may call +\(\alpha + \beta\) the <i>arithmetical</i> class-sum of two classes, in +contradistinction to \(\alpha \cup \beta\), which is the <i>logical</i> +sum. It will be noted that \(\alpha + \beta\), unlike \(\alpha \cup +\beta\), does not require that \(\alpha\) and \(\beta\) should be +of the same type. Also \(\alpha + \alpha\) is not identical with +\(\alpha\), but when \(\alpha = \Lambda\), \(\alpha + \alpha\) is also +\(\Lambda\), though in a different type. Thus the law of tautology does +not hold of the arithmetical class-sum of two classes.</p> + +<p>If \(\mu\) and \(\nu\) are two cardinals of assigned types, we denote +their arithmetical sum by \(\mu +_{\text{c}} \nu\). (As many kinds +of arithmetical addition occur in our work, and as it is essential +to our purpose to distinguish them, we effect the distinction by +suffixes to the sign of addition. It is, of course, only in dealing +with principles that these different symbols are needed: we do not +wish to suggest that they should be adopted in ordinary mathematics.) +Now if \(\mu +_{\text{c}} \nu\) is to have the properties which we +commonly associate with the sum of two cardinals, it must be typically +ambiguous, and must be the cardinal number of any class which can +be divided into two mutually exclusive parts having \(\mu\) terms +and \(\nu\) terms respectively. Hence we are led to the following +definition: +\[ +\mu +_{\text{c}}\nu =\hat{\xi} \{(\exists \alpha ,\beta ).\mu =\text{N}_{0}\text{c}ʻ\alpha .\nu =\text{N}_{0}\text{c}ʻ\beta .\xi \text{ sm }(\alpha +\beta )\} \quad\text{Df}. +\]</p> + +<p>In this definition, various points should be noted. In the first +place, it does not require that \(\mu\) and \(\nu\) should be of the +same type; \(\mu +_{\text{c}} \nu\) is significant whenever \(\mu\) +and \(\nu\) are classes of classes. Thus it is not necessary for +significance that \(\mu\) and \(\nu\) should be cardinals, though if +they are not both cardinals, \(\mu +_{\text{c}} \nu = \Lambda\). If +they are both cardinals, we find +\[ +\mu +_{\text{c}}\nu =\hat{\xi} \{(\exists \alpha ,\beta ).\alpha \in \mu .\beta \in \nu .\xi \text{ sm } (\alpha +\beta )\}. +\]</p> + +<p>Thus in this case \(\alpha \in \mu .\beta \in \nu .\supset .\alpha+\beta \in \mu +_{\text{c}}\nu\).</p> + +<p>Hence if neither \(\mu\) nor \(\nu\) is null, and if \(\alpha\) has +\(\mu\) terms and \(\beta\) has \(\nu\) terms, \(\alpha + \beta\) is a +member of \(\mu +_{\text{c}} \nu\). It easily follows that +\[ +\vdash :\mu =\text{N}_{0}\text{c}ʻ\alpha .\nu =\text{N}_{0}\text{c}ʻ\beta .\supset .\mu +_{\text{c}}\nu =\text{Nc}ʻ(\alpha +\beta ). +\]</p> + +<p>Hence when \(\mu\) and \(\nu\) are homogeneous cardinals (<i>i.e.</i> +when they are cardinals other than \(\Lambda\)), their sum is the +number of the arithmetical class-sum of any two classes having \(\mu\) +terms and \(\nu\) terms respectively.</p> + +<p>A few words are necessary to explain why, in the definition, +we put \(\mu = \text{N}_{0}\text{c}ʻ\alpha . \nu =\text{N}_{0}\text{c}ʻ\beta\) +rather than \(\mu = \text{Nc}ʻ\alpha .\nu = \text{Nc}ʻ\beta\). +The reason is this. Suppose either \(\mu\) +or \(\nu\), say \(\mu\), is \(\Lambda\). Then, by <a href="#*102·73">*102·73</a>, +\(\mu = \text{Nc}(\zeta)ʻtʻ\zeta\), if \(\zeta\) is of the appropriate +type. Hence if we had put +\[ +\mu +_{\text{c}}\nu =\hat{\xi} \{(\exists \alpha , \beta ).\mu =\text{Nc}ʻ\alpha .\nu =\text{Nc}ʻ\beta .\xi \text{ sm }(\alpha +\beta )\} \quad\text{Df}, +\]<span class="pagenum" id="Page_68">[Pg 68]</span> +where the ambiguities of type involved in \(\text{Nc}ʻ\alpha\) and +\(\text{Nc}ʻ\beta\) may be determined as we please, we should have +\[ +\begin{aligned} +\nu &=\text{Nc}ʻ\beta .\supset .tʻ\zeta +\beta \in \mu +_{\text{c}}\nu ,\\ +\textit{i.e.}\qquad \nu &=\text{Nc}ʻ\beta .\supset .tʻ\zeta +\beta \in \Lambda +_{\text{c}} \nu . +\end{aligned} +\] +We should also have \(tʻtʻ\zeta +\beta \in \Lambda +_{\text{c}}\nu\) +and so on. Thus \(\Lambda +_{\text{c}}\nu\) would not have a definite +value, <i>i.e.</i> it would not merely have typical ambiguity, which +it ought to have, but it would not have a definite value even when +its type was assigned. Thus such a definition would be unsuitable. +For the above reasons, we put \(\mu =\text{N}_{0}\text{c}ʻ\alpha . +\nu =\text{N}_{0}\text{c}ʻ\beta\) in the definition, and obtain the +typical ambiguity which we desire by means of the typical ambiguity +of the "\(\text{sm}\)" in "\(\xi \text{ sm }(\alpha +\beta)\)." It is +always essential to right symbolism that the values of typically +ambiguous symbols should be unique as soon as their type is assigned. +The scope of these definitions and of the corresponding definitions for +multiplication and exponentiation (<a href="#*113·04">*113·04</a><a href="#*113·05">·05</a>.<a href="#*116·03">*116·03</a><a href="#*116·04">·04</a>) is extended +by convention \(\text{IIT}\) of the prefatory statement.</p> + +<p>The above definition of \(\mu +_{\text{c}}\nu\) is designed +for the case in which \(\mu\) and \(\nu\) are typically +definite. But we must be able to speak of "\(\text{Nc}ʻ\gamma+_{\text{c}}\text{Nc}ʻ\delta\)," +and this must be a definite cardinal, namely \(\text{Nc}ʻ(\gamma +\delta)\). +If we simply write \(\text{Nc}ʻ\gamma , \text{Nc}ʻ\delta\) +in place of \(\mu\), \(\nu\) in the definition of \(\mu +_{\text{c}}\nu\), +we find +\[ +\text{Nc}ʻ\gamma +_{\text{c}}\text{Nc}ʻ\delta =\hat{\xi} \{(\exists \alpha ,\beta ).\text{Nc}ʻ\gamma =\text{N}_{0}\text{c}ʻ\alpha .\text{Nc}ʻ\delta =\text{N}_{0}\text{c}ʻ\beta + .\xi \text{ sm }(\alpha +\beta )\}. +\] +But this will not always have a definite value when the type of +\(\text{Nc}ʻ\gamma +_{\text{c}}\text{Nc}ʻ\delta\) is assigned. To +take a simple case, write \(tʻ\zeta\) for \(\gamma\) and \(tʻy\) for +\(\delta\). Then +\[ +\text{Nc}ʻtʻ\zeta +_{\text{c}}\text{Nc}ʻtʻy=\hat{\xi} \{(\exists \alpha ,\beta ).\text{Nc}ʻtʻ\zeta =\text{N}_{0}\text{c}ʻ\alpha .\text{Nc}ʻtʻy=\text{N}_{0}\text{c}ʻ\beta + .\xi \text{ sm }(\alpha +\beta )\}. +\] +whence we easily obtain +\[ +\text{Nc}ʻtʻ\zeta +_{\text{c}}\text{Nc}ʻtʻy=\hat{\xi} \{(\exists \alpha ).\text{Nc}ʻtʻ\zeta =\text{N}_{0}\text{c}ʻ\alpha .\xi \text{ sm }(\alpha +\iota ʻy)\}. +\] +If we determine the ambiguity of \(\text{Nc}ʻtʻ\zeta\) to be +\(\text{N}_{1}\text{c}ʻtʻ\zeta\), we find +\[ +\text{Nc}ʻtʻ\zeta +_{\text{c}}\text{Nc}ʻ\iota ʻy=\Lambda +\] +in all types; but if we determine the ambiguity to be +\(\text{N}_{0}\text{c}ʻtʻ\zeta\), we have +\[ +\text{Nc}ʻtʻ\zeta +_{\text{c}}\text{Nc}ʻ\iota ʻy=\text{Nc}ʻ(tʻ\zeta +\iota ʻy), +\] +and this exists in the type of \(tʻ\zeta +\iota ʻy\), if +not in lower types. Hence the value of \(\text{Nc}ʻtʻ\zeta+_{\text{c}}\text{Nc}ʻ\iota ʻy\) +depends upon the determination of the ambiguity of +\(\text{Nc}ʻtʻ\zeta\). It is obvious that we want our definition to +yield +\[ +\text{Nc}ʻ\gamma +_{\text{c}}\text{Nc}ʻ\delta =\text{Nc}ʻ(\gamma +\delta ) +\] +in all types; but in order to insure that this shall hold even when, +for some values of \(\zeta\), \(\text{Nc}(\zeta )ʻ\gamma =\Lambda\), we +must introduce two new definitions, namely +\[ +\begin{aligned} +\text{Nc}ʻ\alpha +_{\text{c}}\mu &=\text{N}_{0}\text{c}ʻ\alpha +_{\text{c}}\mu \quad\text{Df},\\ +\mu +_{\text{c}}\text{Nc}ʻ\alpha &=\mu +_{\text{c}}\text{N}_{0}\text{c}ʻ\alpha \quad\text{Df},\\ +\text{whence}\qquad \vdash :\text{Nc}ʻ\alpha +_{\text{c}}\text{Nc}ʻ\beta &=\text{N}_{0}\text{c}ʻ\alpha +_{\text{c}}\text{N}_{0}\text{c}ʻ\beta + =\text{Nc}ʻ(\alpha +\beta ). +\end{aligned} +\] +This definition is to be applied when "\(\text{Nc}ʻ\gamma\)" and +"\(\text{Nc}ʻ\delta\)" occur without any<span class="pagenum" id="Page_69">[Pg 69]</span> determination of type. On the +other hand, if we have \(\text{Nc}(\zeta)ʻ\gamma\) and \(\text{Nc}(\eta)ʻ\delta\), +we apply the definition of \(\mu +_{\text{c}}\nu\). +We shall find that whenever \(\text{Nc}(\zeta )ʻ\gamma\) and +\(\text{Nc}(\eta )ʻ\delta\) both exist, +\[ +\text{Nc}(\zeta )ʻ\gamma +_{\text{c}} \text{Nc}(\eta )ʻ\delta =\text{N}_{0}\text{c}ʻ\gamma +_{\text{c}} \text{N}_{0}\text{c}ʻ\delta . +\] +Thus the above definition is only required in order to exclude values +of \(\zeta\) or \(\eta\) for which either \(\text{Nc}(\zeta )ʻ\gamma\) +or \(\text{Nc}(\eta )ʻ\delta\) is \(\Lambda\).</p> + +<p>The commutative and associative laws of arithmetical addition are easily +deduced from the definition of \(\alpha +\beta\). We shall have +\[ +\begin{aligned} +\vdash .\alpha &+\beta =\text{Cnv}ʻʻ(\beta +a),\\ +\text{whence}\qquad \vdash .\text{Nc}ʻ\alpha &+_{\text{c}}\text{Nc}ʻ\beta =\text{Nc}ʻ\beta +_{\text{c}}\text{Nc}ʻ\alpha , +\end{aligned} +\] +because each = \(\text{Nc}ʻ(\alpha +\beta)\). A similar though slightly +longer proof shows that +\[ +\begin{aligned} +\vdash .(\alpha +\beta )&+\gamma \text{ sm }\alpha +(\beta +\gamma ),\\ +\text{whence}\qquad \vdash .(\text{Nc}ʻ\alpha +_{\text{c}}\text{Nc}ʻ\beta )&+_{\text{c}}\text{Nc}ʻ\gamma =\text{Nc}ʻ\alpha +_{\text{c}}(\text{Nc}ʻ\beta + +_{\text{c}}\text{Nc}ʻ\gamma). +\end{aligned} +\]</p> + +<p>The above definition of \(\alpha +\beta\) enables us to proceed to the +sum of any finite number of classes, and allows any one class to recur +in the summation. But it does not enable us to define the sum of an +infinite number of classes. For this we need a new definition. Since +an infinite number of classes cannot be given by enumeration, but only +by intension, we shall have to take a class of classes \(\kappa\), and +define the arithmetical sum of the members of \(\kappa\). Thus now the +classes which are the summands must all be of the same type (since +they are all members of \(\kappa\)), and no one class can occur more +than once, since each member of \(\kappa\) only counts once. (In order +to deal with repetition, we must advance to multiplication, which will +be explained shortly.) Thus in removing the limitation to a finite +number of summands, we introduce certain other limitations. This is the +reason which makes it worth while to introduce the above definition of +\(\alpha +\beta\) in addition to the definition now to be given.</p> + +<p>If \(\kappa\) is a class of classes, the sum of the cardinal numbers of +the members of \(\kappa\) will evidently be obtained by constructing +a class of mutually exclusive classes whose members have a one-one +relation to the members of corresponding members of \(\kappa\). Suppose +\(\alpha\), \(\beta\) are two different members of \(\kappa\), and +suppose \(x\) is a member both of \(\alpha\) and of \(\beta\). Then we +wish to count \(x\) twice over, once as a member of \(\alpha\) and once +as a member of \(\beta\). The simplest way to do this is to form the +ordinal couples \(x\downarrow \alpha\) and \(x\downarrow \beta\), which +are not identical except when \(\alpha\) and \(\beta\) are identical. +Thus if we take all such ordinal couples, <i>i.e.</i> if we take the +class +\[ +\hat{R} \{(\exists x).x\in \alpha .R=x\downarrow \alpha\}, +\] +for every \(\alpha\) which is a member of \(\kappa\), we get a class of +mutually exclusive classes, namely the classes of the form \(\downarrow\alpha ʻʻ\alpha\), +where \(\alpha \in \kappa\), each of these is +similar to the corresponding member of \(\kappa\). Hence the logical sum +of this class of classes, <i>i.e.</i> +\[ +\hat{R} \{(\exists \alpha ,x).\alpha \in \kappa .x\in \alpha .R=x\downarrow \alpha\}, +\]<span class="pagenum" id="Page_70">[Pg 70]</span> +has the required number of terms. Now, by *85·601, +\[ +\downarrow \alpha ʻʻ\alpha =\in \unicode{x21A7}ʻ\alpha . +\] +Hence the class whose logical sum we are taking is +\(\in\unicode{x21A7}ʻʻ\kappa\). Hence we put +\[ +\Sigma ʻ\kappa =sʻ\in \unicode{x21A7}ʻʻ\kappa \quad\text{Df}. +\] +\(\Sigma ʻ\kappa\) may be called the <i>arithmetical</i> sum of +\(\kappa\), in contradistinction to \(sʻ\kappa\), which is the logical +sum. Thus \(\Sigma ʻ\kappa\) bears to \(sʻ\kappa\) a relation analogous +to that which \(\alpha +\beta\) bears to \(\alpha \cup \beta\).</p> + +<p>We put further \(\Sigma \text{Nc}ʻ\kappa =\text{Nc}ʻsʻ\in\unicode{x21A7}ʻʻ\kappa \quad\text{Df}.\)</p> + +<p>Thus \(\Sigma \text{Nc}ʻ\kappa\) is the sum of the numbers of members +of \(\kappa\).</p> + +<p>It is to be observed that \(\Sigma \text{Nc}ʻ\kappa\) is not in +general a function of \(\text{Nc}ʻʻ\kappa\). For, if two members +of \(\kappa\) have the same cardinal number, this will only count +once in \(\text{Nc}ʻʻ\kappa\), whereas it counts twice in \(\Sigma +\text{Nc}ʻ\kappa\).</p> + +<p>We shall find that, provided \(\alpha \neq \beta\), +\[ +\Sigma \text{Nc}ʻ(\iota ʻ\alpha \cup \iota ʻ\beta )=\text{Nc}ʻ\alpha +_{\text{c}}\text{Nc}ʻ\beta . +\] +Thus where a finite number of summands are concerned, the two +definitions of addition agree, except that the first allows one class +to count several times over, while the second does not.</p> + +<p>In dealing with multiplication, our procedure is closely analogous +to the procedure for addition. We first define the <i>arithmetical +class-product</i> of two classes \(\alpha\) and \(\beta\), which is +a certain class whose cardinal number is the product of the cardinal +numbers of \(\alpha\) and \(\beta\). We write \(\beta \times \alpha\) +for the arithmetical class-product of \(\beta\) and \(\alpha\), and +define it as the class of all ordinal couples of which the referent +is a member of \(\alpha\) and the relatum a member of \(\beta\), +<i>i.e.</i> as +\[ +\hat{R} \{(\exists x,y).x\in \alpha .y\in \beta .R=x\downarrow y\}. +\] +By *40·7, this class is \(sʻ\alpha \downarrow_{,,}ʻʻ\beta\). Hence we +put +\[ +\beta \times \alpha =sʻ\alpha \downarrow_{,,}ʻʻ\beta \quad\text{Df}. +\] +The class \(\alpha \downarrow_{,,}ʻʻ\beta\) is similar to +\(\beta\), and each member of it is similar to \(\alpha\); hence if +\(\text{N}_{0}\text{c}ʻ\alpha =\mu\) and \(\text{N}_{0}\text{c}ʻ\beta=\nu\), +\(sʻ\alpha \downarrow_{,,}ʻʻ\beta\) consists of \(\nu\) +classes having \(\mu\) members each. The class +\(\alpha\downarrow_{,,}ʻʻ\beta\) is important also in connection with +exponentiation.</p> + +<p>The product of two cardinals is defined as follows: +\[ +\mu \times _{\text{c}}\nu =\hat{\xi} \{(\exists \alpha ,\beta ).\mu =\text{N}_{0}\text{c}ʻ\alpha .\nu =\text{N}_{0}\text{c}ʻ\beta + .\xi \text{ sm }(\alpha \times \beta )\} \quad\text{Df}. +\] +In regard to types, this definition calls for analogous remarks to +those which were made on \(\mu +_{\text{c}}\nu\). Also, as before, we +need definitions of \(\mu \times _{\text{c}}\text{Nc}ʻ\alpha\) and +\(\text{Nc}ʻ\alpha \times _{\text{c}}\mu\), whence we obtain +\[ +\text{Nc}ʻ\alpha \times _{\text{c}}\text{Nc}ʻ\beta =\text{N}_{0}\text{c}ʻ\alpha \times _{\text{c}}\text{N}_{0}\text{c}ʻ\beta \quad\text{Df}. +\] +<span class="pagenum" id="Page_71">[Pg 71]</span>By means of these definitions, we can define the product of any finite +number of cardinals; but in order to define products which have an +infinite number of factors, we need a new definition.</p> + +<p>If \(\kappa\) is a class of classes, we take \({\in}_{\Delta}ʻ\kappa\) +as its arithmetical product. In simple cases, it is easy to see the +justification of this decision. <i>E.g.</i> let \(\kappa\) consist of +the three classes \(\alpha _{1}\), \(\alpha _{2}\), \(\alpha _{3}\), +and let the members of \(\alpha _{1}\) be \(x_{1}\), \(x_{2}\); those +of \(\alpha _{2}\), \(y_{1}\), \(y_{2}\); those of \(\alpha _{3}\), +\(z_{1}\), \(z_{2}\). Then the members of \({\in}_{\Delta}ʻ\kappa\) are +\[ +\begin{aligned} +&x_{1}\downarrow \alpha _{1}\unicode{x228d} y_{1}\downarrow \alpha _{2}\unicode{x228d} z_{1}\downarrow \alpha _{3},\\ +&x_{2}\downarrow \alpha _{1}\unicode{x228d} y_{1}\downarrow \alpha _{2}\unicode{x228d} z_{1}\downarrow \alpha _{3},\\ +&x_{1}\downarrow \alpha _{1}\unicode{x228d} y_{2}\downarrow \alpha _{2}\unicode{x228d} z_{1}\downarrow \alpha _{3},\\ +&x_{2}\downarrow \alpha _{1}\unicode{x228d} y_{2}\downarrow \alpha _{2}\unicode{x228d} z_{1}\downarrow \alpha _{3},\\ +\end{aligned} +\] +with four more obtained by substituting \(z_{2}\) for \(z_{1}\) in the +above. Thus +\(\text{Nc}ʻ{\in}_{\Delta}ʻ\kappa = 8 = \text{Nc}ʻ\alpha_{1} \times _{\text{c}} \text{Nc}ʻ\alpha _{2} \times _{\text{c}}\text{Nc}ʻ\alpha _{3}\). +In general, however, the existence of \({\in}_{\Delta}ʻ\kappa\) is +doubtful, owing to the doubt as to the validity of the multiplicative +axiom. (We shall return to this point shortly.) Hence there is no proof +that the product of an infinite number of factors cannot be zero unless +one of the factors is zero.</p> + +<p>When \(\kappa\) is a class of mutually exclusive +classes, \({\in}_{\Delta}ʻ\kappa\) is similar to +\(\text{D}ʻʻ{\in}_{\Delta}ʻ\kappa\). On account of its lower type, +\(\text{D}ʻʻ{\in}_{\Delta}ʻ\kappa\) is often more convenient than +\({\in}_{\Delta}ʻ\kappa\). Hence we put +\[ +\text{Prod}ʻ\kappa =\text{D}ʻʻ{\in}_{\Delta}ʻ\kappa \quad\text{Df}, +\] +or (what comes to the same thing) +\[ +\text{Prod}=\text{D}_{\in }\mid {\in}_{\Delta} \quad\text{Df}. +\] +For the product of the cardinal numbers of the members of \(\kappa\), we +put +\[ +\Pi \text{Nc}ʻ\kappa =\text{Nc}ʻ{\in}_{\Delta}ʻ\kappa \quad\text{Df}. +\] +As in the case of \(\Sigma \text{Nc}ʻ\kappa\), \(\Pi \text{Nc}ʻ\kappa\) +is not in general a function of \(\text{Nc}ʻʻ\kappa\). We shall have +\[ +\vdash :\alpha \neq \beta .\supset .\Pi \text{Nc}ʻ(\iota ʻ\alpha \cup \iota ʻ\beta )=\text{Nc}ʻ\alpha \times _{\text{c}}\text{Nc}ʻ\beta . +\] +Thus for products of a finite number of different factors, the two +definitions of multiplication agree.</p> + +<p>It remains to define exponentiation. Since this is not a commutative +operation, it essentially involves an order as between the base and the +exponent; hence we do not obtain a definition of the exponentiation of +a class \(\kappa\), analogous to \(\Sigma \text{Nc}ʻ\kappa\) or +\(\Pi\text{Nc}ʻ\kappa\), but only a definition of \(\mu ^\nu\), +which may be extended to any finite number of exponentiations. We put +\[ +\alpha\,\text{exp}\,\beta =\text{Prod}ʻ\alpha \downarrow_{,,}ʻʻ\beta \quad\text{Df}. +\] +where \(\alpha \downarrow_{,,}ʻʻ\beta\) has the meaning explained +above, resulting from *38·03. It will be observed that, if +\(\text{N}_{0}\text{c}ʻ\alpha = \mu\) and \(\text{N}_{0}\text{c}ʻ\beta= \nu\), +\(\alpha \downarrow_{,,}ʻʻ\beta\) is a class of \(\nu\) +mutually exclusive classes each of which has \(\mu\) terms; hence +\(\alpha \text{exp}\beta\) may suitably be used to define \(\mu ^\nu\). +Hence we put +\[ +\mu ^\nu =\hat{\xi} \{(\exists \alpha ,\beta ).\mu =\text{N}_{0}\text{c}ʻ\alpha .\nu =\text{N}_{0}\text{c}ʻ\beta .\xi \text{ sm }(\alpha \,\text{exp}\,\,\beta )\} \quad\text{Df}, +\] +and for the same reasons as before, we put +\[ +\text{Nc}ʻ\alpha ^{\text{N}_{c}\text{c}ʻ\beta} = \text{N}_{0}\text{c}ʻ\alpha ^{\text{N}_{0}\text{c}ʻ\beta} \quad\text{Df}. +\]</p> + +<p><span class="pagenum" id="Page_72">[Pg 72]</span></p> + +<p>The above definition of exponentiation gives the same value of +\(\mu^\nu\) as results from Cantor's definition by means of +"Belegungen." The class of Cantor's "Belegungen" is +\[ +\begin{aligned} +\hat{R} \{R\in 1\rightarrow &\text{Cls}.\text{D}ʻR\subset \alpha .\text{ᗡ}ʻR=\beta \},\\ +\textit{i.e.}\qquad &(\alpha \uparrow \beta )_{\Delta }ʻ\beta, +\end{aligned} +\] +and it is easily proved that this is similar to \(\alpha \,\text{exp}\,\beta\).</p> + +<p>The usual formal properties of exponentiation result without much +difficulty from the above definitions.</p> + +<p>The above definition of exponentiation is so framed as to make +propositions on exponentiation independent of the multiplicative axiom, +except when exponentiation is to be connected with multiplication, +<i>i.e.</i> when it is to be shown that the product of \(\nu\) factors, +each of which is \(\mu\), is \(\mu ^\nu\). This proposition cannot be +proved generally without the multiplicative axiom. Similarly, in the +theory of multiplication, the proposition that the sum of \(\nu \mu\)'s +is \(\mu \times _{\text{c}} \nu\) requires the multiplicative axiom (as +does also the proposition that a product is zero when and only when +one of its factors is zero). Otherwise, the theory of multiplication +proceeds without the need for employing the multiplicative axiom.</p> + +<figure class="figcenter width500" id="i_072" style="width: 1234px;"> +<img src="images/i_072.jpg" width="1234" height="1870" +alt="Mathematical diagram showing a 4x2 grid with rows labeled κ₁-κ₄ +(left) and λ₁-λ₄ (right). Each row contains arrows s₁-s₄ connecting +dots. Curved arrows labeled S at top and bottom suggest cyclic +relationships, typical of formal logic or category theory diagrams."> +</figure> + +<p>To take first the connection of addition and multiplication: this +connection, in the form in which we naturally suppose it to hold, is +affirmed in the proposition: +\[ +\begin{aligned} +\mu ,\nu \in \text{NC}.\kappa \in \nu \cap \text{Cls excl}ʻ\mu .&\supset .sʻ\kappa \in \mu \times _{\text{c}}\nu &\qquad \text{(A)}\\ +\textit{or}\qquad \mu ,\nu \in \text{NC}.\kappa \in \nu \cap \text{Cl}ʻ\mu .&\supset .\Sigma ʻ\kappa \in \mu \times _{\text{c}}\nu . +\end{aligned} +\] +We will take the first of these as being simpler. It affirms that +the sum of \(\nu \mu\)'s is \(\mu \times _{\text{c}}\nu\). This +can be proved when \(\nu\) is finite, whether \(\mu\) is finite or +not; but when \(\nu\) is infinite, it cannot be proved without the +multiplicative axiom. This may be seen as follows. We know that +\[ +\begin{aligned} +\vdash :&\mu ,\nu \in \text{NC}.\alpha \in \mu .\beta \in \nu .\supset .\\ +&\alpha \downarrow_{,,}ʻʻ\beta \in \nu \cap \text{Cls excl}ʻ\mu .sʻ\alpha \downarrow_{,,}ʻʻ\beta \in \mu \times _{\text{c}}\nu &\qquad \text{(A)}\\ +\end{aligned} +\] +Thus (A) above will result if we can prove +\[ +\kappa ,\lambda \in \nu \cap \text{Cls excl}ʻ\mu .\supset .sʻ\kappa \text{ sm }sʻ\lambda , +\] +since we shall put \(\alpha \downarrow_{,,}ʻʻ\beta\) for \(\lambda\), +and use (B).</p> + +<p>Since \(\kappa\), \(\lambda \in \nu\), we have \(\kappa\text{ sm }\lambda\). +Assume +\[ +S\in 1\rightarrow 1.\text{D}ʻS=\kappa .\text{ᗡ}ʻS=\lambda . +\] +Let \(\kappa _{1}\), \(\kappa _{2}\), ... be members of \(\kappa\), +and let \(\lambda _{1}\), \(\lambda _{2}\), ... be the members of +\(\lambda\) which are correlated with \(\kappa _{1}\), \(\kappa_{2}\), +... by \(S\), <i>i.e.</i> \(\lambda _{1}=\breve{S}ʻ\kappa _{1} . \lambda _{2}=\breve{S} ʻ\kappa _{2}.\text{etc}\). +We have, since \(\kappa\), \(\lambda \in \text{Cl}ʻ\mu\), +<span class="pagenum" id="Page_73">[Pg 73]</span>\(\kappa _{1}\text{ sm }\lambda _{1}.\kappa _{2}\text{ sm }\lambda_{2}.\text{etc}\). +Thus \(\alpha S\beta .\supset _{\alpha ,\beta}.\alpha \text{ sm }\beta\), +<i>i.e.</i> \(S\,\unicode{x2abd}\, \text{ sm }\). +If \(\kappa\) and \(\lambda\) are finite, we can pick out arbitrarily +a correlation \(S_{1}\) for \(\kappa _{1}\) and \(\lambda _{1}\), +another \(S_{2}\) for \(\kappa _{2}\) and \(\lambda _{2}\), and so +on; then \(S_{1}\unicode{x228d} S_{2}\unicode{x228d}\) ... correlates +\(sʻ\kappa\) and \(sʻ\lambda\), and therefore \(sʻ\kappa \text{ sm }sʻ\lambda\). +But when \(\kappa\) and \(\lambda\) are infinite, this +method is impracticable. In this case, we proceed as follows.</p> + +<p>By *73·01, \(\alpha \overline{\text{ sm }} \beta =(1\rightarrow 1)\cap\overleftarrow{\text{D}}ʻ\alpha \cap \overleftarrow{\text{ᗡ}}ʻ\beta\quad\text{Df}\).</p> + +<p>Thus "\(\alpha \overline{\text{ sm }} \alpha\)" will stand for all the +permutations of a class into itself; "\(\alpha \overline{\text{ sm }}\beta\)" +stands for all the permutations of \(\alpha\) into \(\beta\), +<i>i.e.</i> all the \(1\rightarrow 1\)'s whose domain is \(\alpha\) and +whose converse domain is \(\beta\). It is obvious that +\[ +\vdash :\exists !\alpha \overline{\text{ sm }} \beta \cap \gamma \overline{\text{ sm }} \delta .\supset .\alpha =\gamma .\beta =\delta . +\] +In the case of the \(\kappa\) and \(\lambda\) above, we know that +\(\alpha \text{ sm }\beta\) when \(\alpha S\beta\); thus +\[ +\begin{aligned} +\alpha \in \kappa .\supset _{\alpha }.\exists !\alpha \overline{\text{ sm }} (\breve{S} ʻ\alpha )\\ +\text{or}\qquad\qquad\quad \beta \in \lambda .\supset _{\beta }.\exists !(Sʻ\beta )\overline{\text{ sm }} \beta .\\ +\text{Put}\qquad \text{Crp}(S)ʻ\beta =(Sʻ\beta )\overline{\text{ sm }} \beta \qquad\text{Df}, +\end{aligned} +\] +where "\(\text{Crp}\)" stands for "correspondence." Thus +\(\text{Crp}(S)ʻ\beta\) is the class of all correspondences of +\(Sʻ\beta\) and \(\beta\); \(\text{Crp}(S)ʻʻ\lambda\) is the class of all +such classes of correspondences. If we extract one member out of each +of these classes of correspondences, we get a class of relations whose +sum is a correlator of \(sʻ\kappa\) and \(sʻ\lambda\); <i>i.e.</i> +\[ +\varpi \in \text{D}ʻʻ{\in}_{\Delta}ʻ\text{Crp}(S)ʻʻ\lambda .\supset .\dot{s} ʻ\varpi \in (sʻ\kappa )\overline{\text{ sm }} (sʻ\lambda ). +\] +Thus the desired result follows whenever +\[ +\exists !{\in}_{\Delta }ʻ\text{Crp}(S)ʻʻ\lambda . +\] +Now we have \(S\in 1\rightarrow 1.S\,\unicode{x2abd}\, \text{ sm }.\supset .\text{Crp}(S)ʻʻ\lambda \in \text{Cls}^{2} \text{excl}\).</p> + +<p>Consequently +\[ +\text{Mult ax}.\supset :S\in 1\rightarrow 1.S\,\unicode{x2abd}\, \text{ sm }.\text{D}ʻS=\kappa .\text{ᗡ}ʻS=\lambda .\kappa ,\lambda \in \text{Cls}^{2} \text{excl}. +\supset .sʻ\kappa \text{ sm }sʻ\lambda , +\] +whence, by what was said previously, +\[ +\text{Mult ax}.\supset :\kappa \in \nu \cap \text{Cls excl}ʻ\mu .\supset .sʻ\kappa \in \mu \times _{\text{c}}\nu .\Sigma \text{Nc}ʻ\kappa =\mu \times _{\text{c}}\nu . +\]</p> + +<p>The consideration of \({\in}_{\Delta }ʻ\text{Crp}(S)ʻʻ\lambda\) leads +similarly to the proposition +\[ +\vdash \colon\ldotp \text{Mult ax}.\supset :\mu ,\nu \in \text{NC}.\kappa \in \nu \cap \text{Cl}ʻ\mu .\supset .{\in}_{\Delta}ʻ\kappa \in \mu ^\nu .\Pi \text{Nc}ʻ\kappa =\mu ^\nu. +\] +The proof is closely analogous to that for the connection of addition +and multiplication.</p> + +<p>It will be seen that, in the above use of the multiplicative axiom, we +have two classes of classes \(\kappa\) and \(\lambda\) concerning which +we assume +\[ +(\exists S).S\in 1\rightarrow 1.S\,\unicode{x2abd}\, \text{ sm }.\text{D}ʻS=\kappa .\text{ᗡ}ʻS=\lambda , +\] +<i>i.e.</i> we assume that \(\kappa\) and \(\lambda\) are similar +classes of similar classes. A slightly modified hypothesis concerning +\(\kappa\) and \(\lambda\) will enable us to obtain many results, +without the multiplicative axiom, which otherwise might be expected to +require this axiom. This is effected as follows.</p> + +<p><span class="pagenum" id="Page_74">[Pg 74]</span></p> + +<p>Put \(\kappa \text{ sm } \text{ sm } \lambda .\equiv .(\exists T).T\in 1\rightarrow 1.\text{ᗡ}ʻT=sʻ\lambda .\kappa =T_{\in }ʻʻ\lambda\), +where "\(\text{ sm }\text{ sm }\)" is a single symbol representing a relation.</p> + +<p>When this relation holds between \(\kappa\) and \(\lambda\), we shall +say that \(\kappa\) and \(\lambda\) have "double similarity." In this +case, \(T\) correlates \(sʻ\kappa\) and \(sʻ\lambda\), while \(T_{\in}\) +correlates \(\kappa\) and \(\lambda\), so that if \(\beta\) is a +member of \(\lambda\), \(T_{\in }ʻ\beta\), <i>i.e.</i> \(Tʻʻ\beta\), is +its correlate in \(\kappa\). We shall then have +\[ +\begin{aligned} +&\vdash :\kappa \text{ sm } \text{ sm } \lambda .\supset .sʻ\kappa \text{ sm } sʻ\lambda ,\\ +&\vdash :\kappa \text{ sm } \text{ sm } \lambda .\supset .\Sigma \text{Nc}ʻ\kappa =\Sigma \text{Nc}ʻ\lambda ,\\ +&\vdash :\kappa \text{ sm } \text{ sm } \lambda .\supset .\Pi \text{Nc}ʻ\kappa =\Pi \text{Nc}ʻ\lambda . +\end{aligned} +\]</p> + +<p>Also we have +\[ +\vdash :\kappa \text{ sm } \text{ sm } \lambda .\supset .(\exists S).S\in 1\rightarrow 1.S\,\unicode{x2abd}\, \text{ sm }.\text{D}ʻS=\kappa .\text{ᗡ}ʻS=\lambda . +\] +Conversely, +\[ +\begin{aligned} +&\vdash :\kappa ,\lambda \in \text{Cls}^{2} \text{excl}.S\in 1\rightarrow 1.S\,\unicode{x2abd}\, \text{ sm }.\text{D}ʻS=\kappa .\text{ᗡ}ʻS=\lambda .\\ +&\varpi \in \text{D}ʻʻ{\in}_{\Delta}ʻ\text{Crp}(S)ʻʻ\lambda .T=\dot{s} ʻ\varpi .\supset .T\in 1\rightarrow 1.\text{ᗡ}ʻT=sʻ\lambda .\kappa =T_{\in }ʻʻ\lambda , +\end{aligned} +\] +whence +\[ +\begin{aligned} +\vdash \colon\colon \text{Mult ax}.\supset \colon\ldotp \kappa ,\lambda \in \text{Cls}^{2} + \text{excl}:&(\exists S).S\in 1\rightarrow 1.S\,\unicode{x2abd}\, \text{ sm }.\text{D}ʻS=\kappa .\text{ᗡ}ʻS=\lambda :\\ +&\supset .\kappa \text{ sm } \text{ sm } \lambda . +\end{aligned} +\] +Hence the multiplicative axiom is only required in order to pass from +\[ +(\exists S).S\in 1\rightarrow 1.S\,\unicode{x2abd}\, \text{ sm }.\text{D}ʻS=\kappa .\text{ᗡ}ʻS=\lambda +\] +to \(\kappa \text{ sm } \text{ sm } \lambda\). It is this fact, and the +consequent possibility of diminishing the use of the multiplicative +axiom, which has led us to the employment of "\(\text{sm} \,\text{sm}\)" +in the present section.</p> + +<p>We treat also, in this section, the relation of greater and less +between cardinals. We say that \(\text{Nc}ʻ\alpha \gt \text{Nc}ʻ\beta\) +when there is a part of \(\alpha\) which is similar to \(\beta\), but +no part of \(\beta\) is similar to \(\alpha\). The principal proposition +in this subject is the Schröder-Bernstein theorem, <i>i.e.</i> +\[ +\vdash :\mu \geq \nu .\nu \geq \mu .\supset .\mu =\nu . +\] +This is an immediate consequence of *73·88. It cannot be shown, without +assuming the multiplicative axiom, that of any two cardinals one must +be the greater, <i>i.e.</i> +\[ +\mu ,\nu \in \text{NC}.\mu \neq \nu .\supset :\mu >\nu .\lor.\nu >\mu. +\] +If we assume the multiplicative axiom, this results from Zermelo's +proof that on that assumption, every class can be well-ordered, +together with Cantor's proof that of any two well-ordered series which +are not similar, one must be similar to a part of the other. But these +propositions cannot be proved till a much later stage (*258).</p> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_75">[Pg 75]</span></p> +<h2 class="nobreak" id="*110"> +*110. THE ARITHMETICAL SUM OF TWO CLASSES AND OF +TWO CARDINALS.</h2> +</div> + + +<p class="nind"><i>Summary of</i> *110.</p> + +<p>In this number, we start from the definition:</p> + +<p class="nind"><b>*110·01.</b> \(\alpha +\beta =\downarrow (\Lambda \cap \beta )ʻʻ\iotaʻʻ\alpha \cup (\Lambda \cap \alpha )\downarrow ʻʻ\iotaʻʻ\beta \quad\text{Df}\)</p> + +<p>\(\alpha +\beta\) is called the "arithmetical class-sum" of \(\alpha\) +and \(\beta\). The definition is framed so as to give two mutually +exclusive classes respectively similar to \(\alpha\) and \(\beta\), so +that the number of terms in the logical sum of these two classes is the +arithmetical sum of the numbers of terms in \(\alpha\) and \(\beta\) +respectively. \(\alpha +\beta\) is significant whenever \(\alpha\) and +\(\beta\) are classes, whatever their types may be.</p> + +<p>By means of \(\alpha +\beta\), we define the arithmetical sum of two +cardinals as follows:</p> + +<p class="nind"><b>*110·02.</b> \(\mu +_{\text{c}} \nu =\hat{\xi} \{(\exists \alpha ,\beta ).\mu =\text{N}_{0}\text{c}ʻ\alpha .\nu =\text{N}_{0}\text{c}ʻ\beta + .\xi \text{ sm }(\alpha +\beta )\} \quad\text{Df}\)</p> + +<p>This defines the "arithmetical sum of two cardinals." (It is not +necessary to <i>significance</i> that \(\mu\) and \(\nu\) should +be cardinals, but only that they should be classes of classes. If, +however, either is not a cardinal, \(\mu +_{\text{c}}\nu =\Lambda\)). +It will be observed that, when \(\mu\) and \(\nu\) are typically +definite, so are \(\alpha\) and \(\beta\) in the above definition; +but \(\xi\) is typically ambiguous, on account of the ambiguity of +"\(\text{ sm }\)." Hence \(\mu +_{\text{c}}\nu\) is also typically +ambiguous.</p> + +<p>It will be shown that \(\mu +_{\text{c}}\nu\) is always a cardinal, and +that, if +\[ +\mu =\text{N}_{0}\text{c}ʻ\alpha .\nu =\text{N}_{0}\text{c}ʻ\beta, \quad\text{then}\quad \mu +_{\text{c}}\nu =\text{Nc}ʻ(\alpha +\beta ). +\] +Hence whenever \(\mu\) and \(\nu\) are cardinals other than +\(\Lambda\), \(\mu +_{\text{c}}\nu\) is an existent cardinal in some +types, though it may be \(\Lambda\) in others.</p> + +<p>Two more definitions are required in this number, namely:</p> + +<p class="nind"><b>*110·03.</b> \(\text{Nc}ʻ\alpha +_{\text{c}}\mu =\text{N}_{0}\text{c}ʻ\alpha +_{\text{c}}\mu \quad\text{Df}\)</p> + +<p class="nind"><b>*110·04.</b> \(\mu +_{\text{c}} \text{Nc}ʻ\alpha =\mu +_{\text{c}}\text{N}_{0}\text{c}ʻ\alpha \quad\text{Df}\)</p> + +<p>These definitions are needed in order to apply the definition of +\(\mu +_{\text{c}}\nu\) to the case in which \(\mu\) and \(\nu\) are +replaced by typically ambiguous symbols \(\text{Nc}ʻ\alpha\) and +\(\text{Nc}ʻ\beta\). It does not make any difference to the value of +\(\text{Nc}ʻ\alpha +_{\text{c}}\text{Nc}ʻ\beta\) how the ambiguities of +\(\text{Nc}ʻ\alpha\) and \(\text{Nc}ʻ\beta\) are determined, so long as +they are determined in a way that insures \(\exists !\text{Nc}ʻ\alpha.\exists !\text{Nc}ʻ\beta\); +but if there are types in which either \(\text{Nc}ʻ\alpha\) or +\(\text{Nc}ʻ\beta\) is \(\Lambda\), we get \(\text{Nc}ʻ\alpha +_{\text{c}}\text{Nc}ʻ\beta =\Lambda\) +in all types if we determine the ambiguities so that +\(\text{Nc}ʻ\alpha=\Lambda\) or \(\text{Nc}ʻ\beta =\Lambda\). It is in +order to<span class="pagenum" id="Page_76">[Pg 76]</span> exclude such determinations of the ambiguity that the above +definitions are required. Also in connection with these definitions and +the corresponding definitions <a href="#*113·04">*113·04</a><a href="#*113·05">·05</a> and <a href="#*116·03">*116·03</a><a href="#*116·04">·04</a> and <a href="#*117·02">*117·02</a><a href="#*117·03">·03</a>, +the convention \(\text{IIT}\) of the prefatory statement must be noted.</p> + +<p>The propositions of the present number begin with the properties of +\(\alpha + \beta\). We show (<a href="#*110·11">*110·11</a><a href="#*110·12">·12</a>) that \(\alpha + \beta\) +consists of two mutually exclusive parts, which are respectively +similar to \(\alpha\) and \(\beta\); we show (<a href="#*110·14">*110·14</a>) that if +\(\alpha\) and \(\beta\) are mutually exclusive, \(\alpha \cup \beta\) +is similar to \(\alpha + \beta\), and (<a href="#*110·15">*110·15</a>) that if \(\gamma\) and +\(\delta\) are respectively similar to \(\alpha\) and \(\beta\), then +\(\gamma + \delta\) is similar to \(\alpha + \beta\). We show (<a href="#*110·16">*110·16</a>) +that \(\text{Nc}ʻ(\alpha + \beta)\) consists of all classes which can +be divided into two mutually exclusive parts which are respectively +similar to \(\alpha\) and \(\beta\).</p> + +<p>We then proceed (<a href="#*110·2">*110·2</a>—<a href="#*110·252">·252</a>) to the consideration of \(\mu +_{\text{c}} \nu\). +Here \(\mu\) and \(\nu\) are typically definite, and the definition +<a href="#*110·02">*110·02</a> applies to any typically definite symbols, such as +\(\text{N}_{0}\text{c}ʻ\alpha\) or \(\text{Nc}(\eta )ʻ\alpha\). We +prove (<a href="#*110·21">*110·21</a>) that if \(\mu\) and \(\nu\) are cardinals, their +sum consists of all classes similar to some class of the form +\(\alpha + \beta\), where \(\alpha \in \mu .\beta \in \nu\); we +prove (<a href="#*110·22">*110·22</a>) that the sum of \(\text{N}_{0}\text{c}ʻ\alpha\) and +\(\text{N}_{0}\text{c}ʻ\beta\) is \(\text{Nc}ʻ(\alpha + \beta)\), and +(<a href="#*110·25">*110·25</a>) that if \(\mu\) and \(\nu\) are cardinals, their sum is equal +to the sum of the "same" cardinals in any other types in which they are +not null, <i>i.e.</i></p> + +<p class="nind"><b>*110·25.</b> \(\vdash :\mu ,\nu \in \text{NC}.\exists !\text{ sm }_{\eta }ʻʻ\mu .\exists !\text{ sm }_{\zeta }ʻʻ\nu .\supset .\mu +_{\text{c}}\nu = \text{ sm }_{\eta }ʻʻ\mu + +_{\text{c}}\text{ sm }_{\zeta }ʻʻ\nu \)</p> + +<p>We then (<a href="#*110·3">*110·3</a>—<a href="#*110·351">·351</a>) consider \(\text{Nc}ʻ\alpha +_{\text{c}}\text{Nc}ʻ\beta\), +to which we apply the definitions *110·03·04. We have</p> + +<p class="nind"><b>*110·3.</b> \(\vdash .\text{Nc}ʻ\alpha +_{\text{c}}\text{Nc}ʻ\beta = \text{N}_{0}\text{c}ʻ\alpha +_{\text{c}}\text{N}_{0}\text{c}ʻ\beta + = \text{Nc}ʻ(\alpha + \beta)\) +whence the other properties of \(\text{Nc}ʻ\alpha +_{\text{c}}\text{Nc}ʻ\beta\) +follow from previous propositions.</p> + +<p>We then have (<a href="#*110·4">*110·4</a>—<a href="#*110·44">·44</a>) various propositions on the type of \(\mu +_{\text{c}}\nu\) and +its existence and kindred matters. The chief of these are</p> + +<p class="nind"><b>*110·4.</b> \(\vdash :\exists !\mu +_{\text{c}}\nu .\supset .\mu ,\nu \in \text{NC} - \iota ʻ\Lambda .\mu ,\nu \in \text{N}_{0}\text{C}\)</p> + +<p class="nind"><b>*110·42.</b> \(\vdash .\mu +_{\text{c}}\nu \in \text{NC}\)</p> + +<p>This proposition requires no hypothesis, because, if \(\mu\) and +\(\nu\) are not both cardinals, \(\mu +_{\text{c}}\nu = \Lambda\), and +\(\Lambda\) is a cardinal, by <a href="#*102·74">*102·74</a>.</p> + +<p>Our next set of propositions (<a href="#*110·5">*110·5</a>—<a href="#*110·57">·57</a>) are concerned with the +permutative and associative laws, which are <a href="#*110·51">*110·51</a> and <a href="#*110·56">*110·56</a> +respectively.</p> + +<p>We then (<a href="#*110·6">*110·6</a>—<a href="#*110·643">·643</a>) consider the addition of 0 or 1, proving +(<a href="#*110·61">*110·61</a>) that a cardinal is unchanged by the addition of 0, and +(<a href="#*110·643">*110·643</a>) that \(1 +_{\text{c}}1 = 2\).</p> + +<hr class="tb"> + +<p class="nind"><b><a id="*110·01">*110·01</a>.</b> \(\alpha +\beta = \downarrow (\Lambda \cap \beta )ʻʻ\iota ʻʻ\alpha \cup (\Lambda \cap \alpha ) \downarrow ʻʻ\iota ʻʻ\beta \quad\text{Df}\)</p> + +<p class="nind"><b><a id="*110·02">*110·02</a>.</b> \(\mu +_{\text{c}}\nu = \hat{\xi} \{(\exists \alpha ,\beta ).\mu = \text{N}_{0}\text{c}ʻ\alpha .\nu = \text{N}_{0}\text{c}ʻ\beta + .\xi \text{ sm }(\alpha +\beta )\} \quad\text{Df}\)</p> + +<p class="nind"><b><a id="*110·03">*110·03</a>.</b> \(\text{Nc}ʻ\alpha +_{\text{c}}\mu = \text{N}_{0}\text{c}ʻ\alpha +_{\text{c}}\mu \quad\text{Df}\)</p> + +<p class="nind"><b><a id="*110·04">*110·04</a>.</b> \(\mu +_{\text{c}}\text{Nc}ʻ\alpha = \mu +_{\text{c}}\text{N}_{0}\text{c}ʻ\alpha \quad\text{Df}\)</p> + +<p><span class="pagenum" id="Page_77">[Pg 77]</span></p> + +<p>These definitions are extended by \(\text{IIT}\) of the prefatory +statement.</p> + +<p class="nind"><b>*110·1.</b> \[\begin{aligned}\vdash \colon\ldotp R\in \alpha +\beta .\equiv :&(\exists x).x\in \alpha .R = (\iota ʻx)\downarrow (\Lambda \cap \beta ).\lor.\\ +&(\exists y).y\in \beta .R = (\Lambda \cap \alpha )\downarrow (\iota ʻy)\\ +&[*38·13·131 .(*110·01)]\end{aligned}\]</p> + +<p class="nind"><b>*110·101.</b> \(\vdash .(\iota ʻx)\downarrow (\Lambda \cap \beta ) \neq (\Lambda \cap \alpha )\downarrow (\iota ʻy)\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*55·15. &\supset \vdash .\text{D}ʻ(\iota ʻx)\downarrow (\Lambda \cap \beta ) = \iota ʻ\iota ʻx.\text{D}ʻ(\Lambda \cap \alpha )\downarrow (\iota ʻy) = \iota ʻ(\Lambda \cap \alpha ) &\qquad \text{(1)}\\ +\vdash . *51·161. &\supset \vdash .\iota ʻx \neq (\Lambda \cap \alpha ).\\ +[*51·23] &\supset \vdash .\iota ʻ\iota ʻx \neq \iota ʻ(\Lambda \cap \alpha ) &\qquad \text{(2)}\\ +\vdash .(1).(2).&\supset \vdash .\text{D}ʻ(\iota ʻx)\downarrow (\Lambda \cap \beta ) \neq \text{D}ʻ(\Lambda \cap \alpha )\downarrow (\iota ʻy).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*110·11">*110·11</a>.</b> \(\vdash .\downarrow (\Lambda \cap \beta )ʻʻ\iotaʻʻ\alpha \cap (\Lambda \cap \alpha )\downarrow ʻʻ\iotaʻʻ\beta = \Lambda\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*110·101.&\supset \vdash :x\in \alpha .R = \downarrow (\Lambda \cap \beta )ʻ\iota ʻx.y\in \beta .S = (\Lambda \cap \alpha )\downarrow ʻ\iota ʻy.\supset .R \neq S:\\ +[*37·67] &\supset \vdash :R\in \downarrow (\Lambda \cap \beta )ʻʻ\iotaʻʻ\alpha .S\in (\Lambda \cap \alpha )\downarrow ʻʻ\iotaʻʻ\beta .\supset .R \neq S &\qquad \text{(1)}\\ +\vdash .(1).*24·37.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*110·12">*110·12</a>.</b> \(\vdash .\downarrow (\Lambda \cap \beta )ʻʻ\iotaʻʻ\alpha \text{ sm }\alpha .(\Lambda \cap \alpha )\downarrow ʻʻ\iotaʻʻ\beta \text{ sm }\beta \quad[*73·41·61·611]\)</p> + +<p><a href="#*110·11">*110·11</a><a href="#*110·12">·12</a> give the justification for the use of \(\alpha + \beta\) in +defining arithmetical addition, since they show that \(\alpha + \beta\) +consists of two mutually exclusive parts which are respectively similar +to \(\alpha\) and \(\beta\).</p> + +<p class="nind"><b>*110·13.</b> \(\vdash :\gamma \text{ sm }\alpha .\delta \text{ sm }\beta .\gamma \cap \delta = \Lambda .\supset .\gamma \cup \delta \text{ sm }(\alpha +\beta)\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*110·12.\supset \vdash :\text{Hp}.\supset .\gamma \text{ sm }\downarrow (\Lambda \cap \beta )ʻʻ\iotaʻʻ\alpha .\delta \text{ sm }(\Lambda \cap \alpha )\downarrow ʻʻ\iotaʻʻ\beta &\qquad \text{(1)}\\ +\vdash .(1).*110·11.*73·71.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*110·14">*110·14</a>.</b> \(\vdash :\alpha \cap \beta = \Lambda .\supset .\alpha \cup \beta \text{ sm }(\alpha +\beta ) \quad[*110·13.*73·3]\)</p> + +<p>Thus whenever \(\alpha\) and \(\beta\) are mutually exclusive, their +logical sum may replace their arithmetical sum in defining the sum of +their cardinal numbers.</p> + +<p class="nind"><b><a id="*110·15">*110·15</a>.</b> \(\vdash :\gamma \text{ sm }\alpha .\delta \text{ sm }\beta .\supset .\gamma +\delta \text{ sm }\alpha +\beta\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*110·12.\supset \vdash :\text{Hp}.\supset .\downarrow (\Lambda \cap \delta )ʻʻ\iotaʻʻ\gamma \text{ sm }\alpha .(\Lambda \cap \gamma )\downarrow ʻʻ\iotaʻʻ\text{ sm }\beta &\qquad \text{(1)}\\ +\vdash .*110·11.\supset \vdash .\downarrow (\Lambda \cap \delta )ʻʻ\iotaʻʻ\gamma \cap (\Lambda \cap \gamma )\downarrow ʻʻ\iotaʻʻ\delta = \Lambda &\qquad \text{(2)}\\ +\vdash .(1).(2).*110·13.\supset \\ +\vdash :\text{Hp}.\supset .\downarrow (\Lambda \cap \delta )ʻʻ\iotaʻʻ\gamma \cup (\Lambda \cap \gamma )\downarrow ʻʻ\iotaʻʻ\delta \text{ sm }\alpha +\beta :\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*110·151.</b> + \(\vdash \colon\ldotp \alpha \cap \beta = \Lambda .\supset :\xi \text{ sm }(\alpha \cup \beta ).\equiv .(\exists \gamma ,\delta ).\gamma \text{ sm }\alpha .\delta \text{ sm }\beta .\gamma \cap \delta = \Lambda .\xi = \gamma \cup \delta\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*73·71.&\supset \vdash \colon\ldotp \text{Hp}.\supset :\\ +&(\exists \gamma ,\delta ).\gamma \text{ sm }\alpha .\delta \text{ sm }\beta .\gamma \cap \delta = \Lambda .\xi = \gamma \cup \delta .\supset .\xi \text{ sm }(\alpha \cup \beta ) &\qquad \text{(1)}\\ +\vdash .*72·411.*37·25·22.*73·22.\supset \\ +\vdash :S\in &1\rightarrow 1.\text{D}ʻS = \xi .\text{ᗡ}ʻS = \alpha \cup \beta .\alpha \cap \beta = \Lambda .\supset .\\ +&Sʻʻ\alpha \cap Sʻʻ\beta = \Lambda .\xi = Sʻʻ\alpha \cup Sʻʻ\beta .Sʻʻ\alpha \text{ sm }\alpha .Sʻʻ\beta \text{ sm }\beta .\\ +[*11·36] &\supset .(\exists \gamma ,\delta ).\gamma \text{ sm }\alpha .\delta \text{ sm }\beta .\gamma \cap \delta = \Lambda .\xi = \gamma \cup \delta &\qquad \text{(2)}\\ +\vdash .(2).*10·11·23·35.*73·1.\supset \\ +\vdash \colon\ldotp \text{Hp}.&\supset :\xi \text{ sm }(\alpha \cup \beta ).\supset .(\exists \gamma ,\delta ).\gamma \text{ sm }\alpha .\delta \text{ sm }\beta .\gamma \cap \delta = \Lambda .\xi = \gamma \cup \delta &\qquad \text{(3)}\\ +\vdash .(1).(3).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_78">[Pg 78]</span></p> + +<p class="nind"><b>*110·152.</b> + \(\vdash :\xi \text{ sm } (\alpha +\beta ). \equiv .(\exists \gamma ,\delta ).\gamma \text{ sm } \alpha .\delta \text{ sm } \beta .\gamma \cap \delta = \Lambda .\xi = \gamma \cup \delta\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*110·151·11.\supset \\ +\vdash :\xi \text{ sm } (\alpha +\beta ). &\equiv .(\exists \gamma ,\delta ).\gamma \text{ sm } \downarrow (\Lambda \cap \beta )ʻʻ\iota ʻʻ\alpha .\delta \text{ sm } (\Lambda \cap \alpha )\downarrow ʻʻ\iota ʻʻ\beta .\\ +&\gamma \cap \delta = \Lambda .\xi = \gamma \cup \delta .\\ +[*73·37.*110·12] &\equiv .(\exists \gamma ,\delta ).\gamma \text{ sm } \alpha .\delta \text{ sm } \beta .\gamma \cap \delta = \Lambda .\xi = \gamma \cup \delta :\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*110·16">*110·16</a>.</b> + \[\begin{aligned}&\vdash .\text{Nc}ʻ(\alpha +\beta ) = \hat{\xi} \{(\exists \gamma ,\delta ).\gamma \text{ sm } \alpha .\delta \text{ sm } \beta .\gamma \cap \delta = \Lambda .\xi = \gamma \cup \delta \}\\ +&[*110·152.*100·1]\end{aligned}\]</p> + +<p class="nind"><b>*110·17.</b> \(\vdash :\alpha \in tʻ\beta .\supset .\exists !\text{Nc}(tʻ\alpha )ʻ(\alpha +\beta )\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *104·43.\supset \\ +\vdash :\text{Hp}.&\supset .(\exists \gamma ,\delta ).\gamma \text{ sm } \alpha .\gamma \subset tʻ\alpha .\delta \text{ sm } \beta .\delta \subset tʻ\alpha .\gamma \cap \delta = \Lambda .\\ +[*22·59] & \supset .(\exists \gamma ,\delta ).\gamma \text{ sm } \alpha .\delta \text{ sm } \beta .\gamma \cap \delta = \Lambda .\gamma \cup \delta \subset tʻ\alpha .\\ +[*110·16] &\supset .(\exists \xi ).\xi \subset tʻ\alpha .\xi \in \text{Nc}ʻ(\alpha +\beta ).\\ +[*102·6.*63·5] &\supset .\exists !\text{Nc}(tʻ\alpha )ʻ(\alpha +\beta ):\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p>Thus when \(\alpha\) and \(\beta\) are of the same type, +\(\text{Nc}ʻ(\alpha +\beta )\) exists at least in the type next above +that of \(\alpha\) and \(\beta\). We cannot prove that it exists in +the type of \(\alpha\) and \(\beta\). <i>E.g.</i> suppose the lowest +type contained only one member; then if \(x\) were that one member, +\(\text{Nc}ʻ(\iota ʻx+\iota ʻx)\) would not exist in the type to which +\(\iota ʻx\) belongs, but would exist in the next type, <i>i.e.</i> +there would not be two individuals, but there would be two classes, +namely \(\Lambda\) and \(\iota ʻx\), so that \(\iota ʻ\Lambda \cup +\iota ʻ\iota ʻx\in \text{Nc}ʻ(\iota ʻx+\iota ʻx)\).</p> + +<p class="nind"><b>*110·18.</b> \(\vdash .\alpha +\beta \in tʻtʻ(tʻ\alpha \uparrow tʻ\beta)\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*64·53. &\supset \vdash :x\in \alpha .\supset .\downarrow (\Lambda \cap \beta )ʻ\iota ʻx\in tʻ(tʻ\alpha \uparrow tʻ\beta ) &\qquad \text{(1)}\\ +\vdash .(1).*37·61.&\supset \vdash .\downarrow (\Lambda \cap \beta )ʻʻ\iota ʻʻ\alpha \subset tʻ(tʻ\alpha \uparrow tʻ\beta ) &\qquad \text{(2)}\\ +\text{Similarly}\quad &\vdash .(\Lambda \cap \alpha )\downarrow ʻʻ\iota ʻʻ\beta \subset tʻ(tʻ\alpha \uparrow tʻ\beta ) &\qquad \text{(3)}\\ +\vdash .(2).(3). &\supset \vdash .\alpha +\beta \subset tʻ(tʻ\alpha \uparrow tʻ\beta ).\\ +[*63·5] & \supset \vdash .\alpha +\beta \in tʻtʻ(tʻ\alpha \uparrow tʻ\beta ).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*110·2">*110·2</a>.</b> \[\begin{aligned}&\vdash :\xi \in \mu +_{\text{c}}\nu . \equiv .(\exists \alpha ,\beta ).\mu = \text{N}_{0}\text{c}ʻ\alpha .\nu = \text{N}_{0}\text{c}ʻ\alpha + .\xi \text{ sm } (\alpha +\beta )\\ +&[(*110·02)]\end{aligned}\]</p> + +<p class="nind"><b>*110·201.</b> \[\begin{aligned}&\vdash \colon\ldotp \xi \in \mu +_{\text{c}}\nu + . \equiv :\mu ,\nu \in \text{NC}:(\exists \alpha ,\beta ).\alpha \in \mu .\beta \in \nu .\xi \text{ sm } (\alpha +\beta )\\ +&[*103·27.*110·2]\end{aligned}\]</p> + +<p class="nind"><b>*110·202.</b> \[\begin{aligned}\vdash \colon\ldotp \xi \in &\mu +_{\text{c}}\nu . \equiv :\\ +&\exists !\mu .\exists !\nu :(\exists \gamma ,\delta ).\mu = \text{Nc}ʻ\gamma .\nu = \text{Nc}ʻ\delta .\gamma \cap \delta = \Lambda .\xi = \gamma \cup \delta\end{aligned}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*110·2·152 .&\supset \vdash \colon\ldotp \xi \in \mu +_{\text{c}}\nu . \equiv :\\ +(\exists \alpha ,\beta ,\gamma ,\delta ).&\mu = \text{N}_{0}\text{c}ʻ\alpha .\nu = \text{N}_{0}\text{c}ʻ\beta + .\gamma \text{ sm } \alpha .\delta \text{ sm } \beta .\gamma \cap \delta = \Lambda .\xi = \gamma \cup \delta :\\ +[*103·28] &\equiv :(\exists \gamma ,\delta ).\exists !\mu .\exists !\nu .\mu = \text{Nc}ʻ\gamma .\nu = \text{Nc}ʻ\delta .\gamma \cap \delta = \Lambda .\xi = \gamma \cup \delta \colon\ldotp \\ +&\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_79">[Pg 79]</span></p> + +<p class="nind"><b><a id="*110·21">*110·21</a>. </b> \[\begin{aligned}&\vdash \colon\ldotp \mu ,\nu \in \text{NC}.\supset :\xi \in \mu +_{\text{c}}\nu + .\equiv .(\exists \alpha ,\beta ).\alpha \in \mu .\beta \in \nu .\xi \text{ sm }(\alpha +\beta )\\ +&[*110·201]\end{aligned}\]</p> + +<p class="nind"><b>*110·211.</b> \[\begin{aligned}\vdash \colon\ldotp \mu ,\nu \in \text{NC}.\supset :\xi &\in \mu +_{\text{c}}\nu .\equiv .\\ +&(\exists \gamma ,\delta ).\gamma \in \text{ sm }ʻʻ\mu .\delta \in \text{ sm }ʻʻ\nu .\gamma \cap \delta =\Lambda .\xi =\gamma \cup \delta\end{aligned}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*110·21·152.&\supset \vdash \colon\ldotp \text{Hp}.\supset :\xi \in \mu +_{\text{c}}\nu .\equiv .\\ +&(\exists \alpha ,\beta ,\gamma ,\delta ).\alpha \in \mu .\beta \in \nu .\gamma \text{ sm }\alpha .\delta \text{ sm }\beta .\gamma \cap \delta =\Lambda .\xi =\gamma \cup \delta .\\ +[*37·1] &\equiv .(\exists \gamma ,\delta ).\gamma \in \text{ sm }ʻʻ\mu .\delta \in \text{ sm }ʻʻ\nu .\gamma \cap \delta =\Lambda .\xi =\gamma \cup \delta \colon\ldotp \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*110·212.</b> \(\vdash \colon\ldotp \mu ,\nu \in \text{NC}.\supset :\xi \in \mu +_{\text{c}}\nu + .\equiv .(\exists \gamma ).\gamma \in \text{ sm }ʻʻ\mu .\gamma \subset \xi .\xi -\gamma \in \text{ sm }ʻʻ\nu\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*110·211.*24·47.\supset \\ +\vdash \colon\ldotp \text{Hp}.\supset :\xi \in \mu +_{\text{c}}\nu + .&\equiv .(\exists \gamma ,\delta ).\gamma \in \text{ sm }ʻʻ\mu .\delta \in \text{ sm }ʻʻ\nu .\gamma \subset \xi .\delta =\xi -\gamma .\\ +[*13·195] &\equiv .(\exists \gamma ).\gamma \in \text{ sm }ʻʻ\mu .\gamma \subset \xi .\xi -\gamma \in \text{ sm }ʻʻ\nu \colon\ldotp \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*110·22">*110·22</a>.</b> \(\vdash .\text{N}_{0}\text{c}ʻ\alpha +_{\text{c}}\text{N}_{0}\text{c}ʻ\beta = \text{Nc}ʻ(\alpha + \beta )\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*103·4.*110·211.\supset \\ +\vdash :\xi \in \text{N}_{0}\text{c}ʻ\alpha +_{\text{c}}\text{N}_{0}\text{c}ʻ\beta + .&\equiv .(\exists \gamma ,\delta ).\gamma \in \text{Nc}ʻ\alpha .\delta \in \text{Nc}ʻ\beta .\gamma \cap \delta =\Lambda .\xi =\gamma \cup \delta .\\ +[*100·31] &\equiv .(\exists \gamma ,\delta ).\gamma \text{ sm }\alpha .\delta \text{ sm }\beta .\gamma \cap \delta =\Lambda .\xi =\gamma \cup \delta .\\ +[*110·16] &\equiv .\xi \in \text{Nc}ʻ(\alpha + \beta ):\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*110·221.</b> \(\vdash :\xi \in \text{Nc}(\eta )ʻ\alpha +_{\text{c}}\text{Nc}(\zeta + )ʻ\beta .\equiv .\exists !\text{Nc}(\eta )ʻ\alpha .\exists !\text{Nc}(\zeta )ʻ\beta .\xi \in \text{Nc}ʻ(\alpha +\beta )\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*110·202.&\supset \vdash \colon\ldotp \xi \in \text{Nc}(\eta )ʻ\alpha +_{\text{c}}\text{Nc}(\zeta )ʻ\beta .\\ +&\equiv :\exists !\text{Nc}(\eta )ʻ\alpha .\exists !\text{Nc}(\zeta )ʻ\beta :(\exists \gamma ,\delta ).\text{Nc}(\eta )ʻ\alpha =\text{Nc}ʻ\gamma .\\ +&\text{Nc}(\zeta )ʻ\beta =\text{Nc}ʻ\delta .\gamma \cap \delta =\Lambda .\xi =\gamma \cup \delta :\\ +[*100·35]&\equiv :\exists !\text{Nc}(\eta )ʻ\alpha .\exists !\text{Nc}(\zeta )ʻ\beta :(\exists \gamma ,\delta ).\gamma \text{ sm }\alpha .\delta \text{ sm }\beta .\gamma \cap \delta =\Lambda .\xi =\gamma \cup \delta :\\ +[*110·16]&\equiv :\exists !\text{Nc}(\eta )ʻ\alpha .\exists !\text{Nc}(\zeta )ʻ\beta .\xi \in \text{Nc}ʻ(\alpha +\beta )\colon\ldotp \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*110·23">*110·23</a>.</b> \[\begin{aligned}\vdash :&\exists !\text{Nc}(\eta )ʻ\alpha .\exists !\text{Nc}(\zeta )ʻ\beta .\supset .\\ +&\text{Nc}(\eta )ʻ\alpha +_{\text{c}}\text{Nc}(\zeta )ʻ\beta =\text{Nc}ʻ(\alpha +\beta )=\text{N}_{0}\text{c}ʻ\alpha +_{\text{c}}\text{N}_{0}\text{c}ʻ\beta + \quad[*110·221·22]\end{aligned}\]</p> + +<p>Thus \(\text{Nc}(\eta )ʻ\alpha +_{\text{c}}\text{Nc}(\zeta )ʻ\beta\) is +independent of \(\eta\) and \(\zeta\) so long as \(\text{Nc}ʻ\alpha\) +and \(\text{Nc}ʻ\beta\) exist in the types of \(\eta\) and \(\zeta\) +respectively.</p> + +<p class="nind"><b>*110·231.</b> \[\begin{aligned}&\vdash \colon\ldotp \text{Nc}(\eta )ʻ\alpha =\Lambda .\lor.\text{Nc}(\zeta )ʻ\beta =\Lambda :\supset .\text{Nc}(\eta )ʻ\alpha +_{\text{c}}\text{Nc}(\zeta + )ʻ\beta =\Lambda\\ +&[*110·221]\end{aligned}\]</p> + +<p class="nind"><b><a id="*110·24">*110·24</a>.</b> \(\vdash :\eta \text{ sm }\alpha .\zeta \text{ sm }\beta .\supset .\text{N}_{0}\text{c}ʻ\eta +_{\text{c}}\text{N}_{0}\text{c}ʻ\zeta =\text{N}_{0}\text{c}ʻ\alpha + +_{\text{c}}\text{N}_{0}\text{c}ʻ\beta\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*103·42. \supset \vdash :\text{Hp}.&\supset.\text{N}_{0}\text{c}ʻ\eta =\text{Nc}(\eta )ʻ\alpha .\text{N}_{0}\text{c}ʻ\zeta + =\text{Nc}(\zeta )ʻ\beta &\qquad \text{(1)}\\ +\vdash .(1).*103·13.\supset \vdash :\text{Hp}.&\supset .\exists !\text{Nc}(\eta )ʻ\alpha .\exists !\text{Nc}(\zeta )ʻ\beta .\\ +[*110·23] &\supset .\text{Nc}(\eta )ʻ\alpha +_{\text{c}}\text{Nc}(\zeta )ʻ\beta =\text{N}_{0}\text{c}ʻ\alpha +_{\text{c}}\text{N}_{0}\text{c}ʻ\beta + &\qquad \text{(2)}\\ +\vdash .(1).(2).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_80">[Pg 80]</span></p> + +<p class="nind"><b><a id="*110·25">*110·25</a>.</b> \(\vdash :\mu ,\nu \in \text{NC}.\exists ! \text{ sm }_{\eta }ʻʻ\mu .\exists ! \text{ sm }_{\zeta }ʻʻ\nu .\supset .\mu +_{\text{c}}\nu = \text{ sm }_{\eta }ʻʻ\mu + +_{\text{c}} \text{ sm }_{\zeta }ʻʻ\nu\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*103·27.\supset \vdash :\mu ,\nu \in \text{NC}.\alpha \in \mu .\beta \in \nu .\exists ! \text{ sm }_{\eta }ʻʻ\mu .\exists ! \text{ sm }_{\zeta }ʻʻ\nu .\\ +\supset .\mu = \text{N}_{0}\text{c}ʻ\alpha .\nu = \text{N}_{0}\text{c}ʻ\beta .\exists ! \text{ sm }_{\eta }ʻʻ\mu .\exists ! \text{ sm }_{\zeta }ʻʻ\nu .\\ +[*103·41.*102·85]\supset .\mu = \text{N}_{0}\text{c}ʻ\alpha .\nu = \text{N}_{0}\text{c}ʻ\beta . \text{ sm }_{\eta }ʻʻ\mu = \text{Nc}(\eta )ʻ\alpha .\\ +\text{ sm }_{\zeta }ʻʻ\nu = \text{Nc}(\zeta )ʻ\beta .\exists !\text{Nc}(\eta )ʻ\alpha .\exists !\text{Nc}(\zeta )ʻ\beta .\\ +[*110·23] \supset .\mu +_{\text{c}}\nu = \text{N}_{0}\text{c}ʻ\alpha +_{\text{c}}\text{N}_{0}\text{c}ʻ\beta = \text{ sm }_{\eta }ʻʻ\mu +_{\text{c}} + \text{ sm }_{\zeta }ʻʻ\nu &\qquad \text{(1)}\\ +\vdash .(1).*10·11·23·35.\supset \\ +\vdash :\mu ,\nu \in \text{NC}.\exists !\mu .\exists !\nu .\exists ! \text{ sm }_{\eta }ʻʻ\mu .\exists ! \text{ sm }_{\zeta }ʻʻ\nu .\supset .\mu +_{\text{c}}\nu + = \text{ sm }_{\eta }ʻʻ\mu +_{\text{c}} \text{ sm }_{\zeta }ʻʻ\nu &\qquad \text{(2)}\\ +\vdash .*37·29.\text{Transp}.\supset \vdash :\exists ! \text{ sm }_{\eta }ʻʻ\mu .\exists ! \text{ sm }_{\zeta }ʻʻ\nu .\supset .\exists !\mu .\exists !\nu &\qquad \text{(3)}\\ +\vdash .(2).(3).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*110·251">*110·251</a>.</b> \(\vdash :\mu ,\nu \in \text{NC}.\supset .\mu ^{(1)}+_{\text{c}}\nu ^{(1)} = \mu +_{\text{c}}\nu\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash *110·25.*104·265.\supset \\ +\vdash :\text{Hp}.\exists !\mu ^{(1)}.\exists !\nu ^{(1)}.&\supset .\mu ^{(1)}+_{\text{c}}\nu ^{(1)} = \mu +_{\text{c}}\nu &\qquad \text{(1)}\\ +\vdash .*110·202.&\supset \vdash :{\sim}(\exists !\mu ^{(1)}.\exists !\nu ^{(1)}).\supset .\mu ^{(1)}+_{\text{c}}\nu ^{(1)} + = \Lambda &\qquad \text{(2)}\\ +\vdash .*104·264.\supset \vdash :\text{Hp}(2).&\supset .{\sim}(\exists !\mu .\exists !\nu ).\\ +[*110·202] &\supset .\mu +_{\text{c}}\nu = \Lambda .\\ +[(2)] &\supset .\mu ^{(1)}+_{\text{c}}\nu ^{(1)} = \mu +_{\text{c}}\nu &\qquad \text{(3)}\\ +\vdash .(1).(3).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*110·252">*110·252</a>.</b> \(\vdash :\mu ,\nu \in \text{NC}.\supset .\mu _{(00)}+_{\text{c}}\nu _{(00)} = \mu +_{\text{c}}\nu \quad[\text{Proof as in *110·251}]\)</p> + +<p>A similar proof applies to \(\mu ^{(2)}\), \(\nu ^{(2)}\), etc., +and to any such derived cardinals whose existence follows from that +of \(\mu\) and \(\nu\). The proposition does not hold generally for +\(\mu _{(1)}\), \(\nu _{(1)}\) and other descending derived cardinals, +because they may be null when \(\mu\) and \(\nu\) exist.</p> + +<p>The following proposition (*110·3) is more often used than any other in +this number except <a href="#*110·4">*110·4</a>.</p> + +<p class="nind"><b><a id="*110·3">*110·3</a>.</b> \(\vdash .\text{Nc}ʻ\alpha +_{\text{c}}\text{Nc}ʻ\beta = \text{N}_{0}\text{c}ʻ\alpha +_{\text{c}}\text{N}_{0}\text{c}ʻ\beta + = \text{Nc}ʻ(\alpha +\beta ) \quad[*110·22.(*110·03·04)]\)</p> + +<p class="nind"><b><a id="*110·31">*110·31</a>.</b> \(\vdash :\gamma \text{ sm } \alpha .\delta \text{ sm } \beta .\supset .\text{Nc}ʻ\gamma +_{\text{c}}\text{Nc}ʻ\delta = \text{Nc}ʻ\alpha +_{\text{c}}\text{Nc}ʻ\beta + \quad[*110·24·3]\)</p> + +<p>The following proposition is frequently used.</p> + +<p class="nind"><b><a id="*110·32">*110·32</a>.</b> \(\vdash :\alpha \cap \beta = \Lambda .\supset .\text{Nc}ʻ\alpha +_{\text{c}}\text{Nc}ʻ\beta = \text{Nc}ʻ(\alpha \cup \beta ) \quad[*110·3·14]\)</p> + +<p class="nind"><b>*110·33.</b> \[\begin{aligned}&\vdash :\xi \in \text{Nc}ʻ\alpha +_{\text{c}}\text{Nc}ʻ\beta + . \equiv .(\exists \gamma ,\delta ).\gamma \text{ sm } \alpha .\delta \text{ sm } \beta .\gamma \cap \delta = \Lambda .\xi = \gamma \cup \delta \\ +&[*110·3·16]\end{aligned}\]</p> + +<p>The above proposition is used in <a href="#*110·63">*110·63</a>. We might have used the above +to define arithmetical addition, but this method would have been less +convenient than the method adopted in this number, both because there +would<span class="pagenum" id="Page_81">[Pg 81]</span> have been more difficulty in dealing with types, and because the +existence of \(\text{Nc}ʻ\alpha +_{\text{c}} \text{Nc}ʻ\beta\) (in the +types in which it does exist) is less evident with the above definition +than with the definitions given in <a href="#*110·01">*110·01</a><a href="#*110·02">·02</a><a href="#*110·03">·03</a><a href="#*110·04">·04</a>.</p> + +<p class="nind"><b><a id="*110·331">*110·331</a>.</b> \(\vdash .\text{Nc}ʻ\alpha +_{\text{c}}\text{Nc}ʻ\beta =\hat{\xi} \{(\exists \gamma ).\gamma \text{ sm }\alpha .\xi -\gamma \text{ sm }\beta .\gamma \subset \xi\}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *110·33.*24·47.&\supset \\ +\vdash :\xi \in \text{Nc}ʻ\alpha +_{\text{c}}\text{Nc}ʻ\beta .&\equiv .(\exists \gamma ,\delta ).\gamma \text{ sm }\alpha .\delta \text{ sm }\beta .\gamma \subset \xi .\delta =\xi -\gamma .\\ +[*13·195] &\equiv .(\exists \gamma ).\gamma \text{ sm }\alpha .\xi -\gamma \text{ sm }\beta .\gamma \subset \xi :\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*110·34">*110·34</a>.</b> \[\begin{aligned}&\vdash :\exists !\text{Nc}(\eta )ʻ\alpha .\exists !\text{Nc}(\zeta )ʻ\beta .\supset .\text{Nc}(\eta )ʻ\alpha +_{\text{c}} + \text{Nc}(\zeta )ʻ\beta =\text{Nc}ʻ\alpha +_{\text{c}} \text{Nc}ʻ\beta \\ +&[*110·23·3]\end{aligned}\]</p> + +<p class="nind"><b><a id="*110·35">*110·35</a>.</b> \(\vdash .\text{N}^{1}\text{c}ʻ\alpha +_{\text{c}} \text{N}^{1}\text{c}ʻ\beta =\text{Nc}ʻ\alpha +_{\text{c}} + \text{Nc}ʻ\beta \quad[*104·102·21.*110·34]\)</p> + +<p class="nind"><b><a id="*110·351">*110·351</a>.</b> \(\vdash .\text{N}_{00}\text{c}ʻ\alpha +_{\text{c}} \text{N}_{00}\text{c}ʻ\beta =\text{Nc}ʻ\alpha +_{\text{c}} + \text{Nc}ʻ\beta \quad[*106·21.*110·34]\)</p> + +<p>Similar propositions will hold generally for <i>ascending</i> cardinals.</p> + +<p>The following proposition (*110·4) is the most used of the propositions +in this number. It is useful both in the form given, and in the form +resulting from transposition, in which it shows that \(\mu +_{\text{c}}\nu =\Lambda\) +unless both \(\mu\) and \(\nu\) are existent cardinals. +It is chiefly useful in avoiding the necessity of the hypothesis +\(\mu\), \(\nu \in \text{NC}\) in such propositions as the commutative +and associative laws.</p> + +<p class="nind"><b><a id="*110·4">*110·4</a>.</b> \(\vdash :\exists !\mu +_{\text{c}} \nu .\supset .\mu ,\nu \in \text{NC}-\iota ʻ\Lambda .\mu ,\nu \in \text{N}_{0}\text{C} \quad[*110·201·202·2]\)</p> + +<p>The following propositions, down to <a href="#*110·411">*110·411</a> inclusive, are concerned +with types. They are not referred to in the sequel.</p> + +<p class="nind"><b>*110·401.</b> \(\vdash :\mu =\text{N}_{0}\text{c}ʻ\alpha .\nu =\text{N}_{0}\text{c}ʻ\beta .\supset .\alpha + \beta \in tʻtʻ(\mu \uparrow \nu )\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*110·18.*103·12.\supset \vdash :\text{Hp}.&\supset .\alpha + \beta \in tʻtʻ(tʻ\alpha \uparrow tʻ\beta ).\alpha \in \mu .\beta \in \nu .\\ +[*63·11] &\supset .\alpha + \beta \in tʻtʻ(tʻ\alpha \uparrow tʻ\beta ).tʻ\alpha =t_{0}ʻ\mu .tʻ\beta =t_{0}ʻ\nu .\\ +[*13·12] &\supset .\alpha + \beta \in tʻtʻ(t_{0}ʻ\mu \uparrow t_{0}ʻ\nu ).\\ +[*64·13] &\supset .\alpha + \beta \in tʻtʻ(\mu \uparrow \nu ):\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*110·402.</b> \(\vdash :\mu ,\nu \in \text{N}_{0}\text{C}.\supset .\exists !(\mu +_{\text{c}} \nu )\cap tʻtʻ(\mu \uparrow \nu )\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*110·22.*100·3.\supset \\ +\vdash :\mu =\text{N}_{0}\text{c}ʻ\alpha .\nu =\text{N}_{0}\text{c}ʻ\beta .&\supset .\alpha + \beta \in \mu +_{\text{c}} \nu .\\ +[*110·401] &\supset .\alpha + \beta \in (\mu +_{\text{c}} \nu )\cap tʻtʻ(\mu \uparrow \nu ).\\ +[*10·24] &\supset .\exists !(\mu +_{\text{c}} \nu )\cap tʻtʻ(\mu \uparrow \nu ) (1) +\vdash .(1).*103·2.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*110·403.</b> \(\vdash :\mu ,\nu \in \text{N}_{0}\text{C}.\equiv .\exists !(\mu +_{\text{c}} \nu )\cap tʻtʻ(\mu \uparrow \nu ) \quad[*110·402·4]\)</p> + +<p><span class="pagenum" id="Page_82">[Pg 82]</span></p> + +<p class="nind"><b>*110·404.</b> \(\vdash .\exists !(\text{Nc}ʻ\alpha +_{\text{c}} \text{Nc}ʻ\beta )\cap tʻtʻ(tʻ\alpha \uparrow tʻ\beta ) \quad[*110·18·3.*100·3]\)</p> + +<p class="nind"><b>*110·41.</b> \(\vdash :\mu ,\nu \in \text{N}_{0}\text{C}.tʻ\mu = tʻ\nu .\supset .\exists !(\mu +_{\text{c}}\nu )\cap tʻ\mu \)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*103·11.\supset \vdash :&\mu = \text{N}_{0}\text{c}ʻ\alpha .\nu = \text{N}_{0}\text{c}ʻ\beta .tʻ\mu = tʻ\nu .\supset .\\ +&\mu \subset tʻ\alpha .\nu \subset tʻ\beta .tʻ\mu = tʻ\nu .\\ +[*63·21·35] &\supset .t_{0}ʻ\mu = tʻ\alpha .t_{0}ʻ\nu = tʻ\beta .t_{0}ʻ\mu = t_{0}ʻ\nu .\\ +[*13·16·17] &\supset .tʻ\alpha = tʻ\beta = t_{0}ʻ\mu .\\ +[*110·17] &\supset .\exists !\text{Nc}ʻ(\alpha +\beta )\cap tʻt_{0}ʻ\mu .\\ +[*110·22.*63·19] &\supset .\exists !(\mu +_{\text{c}}\nu )\cap tʻ\mu :\supset .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*110·411">*110·411</a>.</b> \[\begin{aligned}&\vdash :tʻ\alpha = tʻ\beta .\supset .\exists !(\text{Nc}ʻ\alpha +_{\text{c}}\text{Nc}ʻ\beta + )\cap tʻtʻ\alpha .\exists !\text{Nc}(tʻ\alpha )ʻ(\alpha +\beta )\\ +&[*110·17·3]\end{aligned}\]</p> + +<p>It will be observed that the following proposition (*110·42) requires +no hypothesis. This is owing to <a href="#*110·4">*110·4</a> and <a href="#*102·74">*102·74</a>.</p> + +<p class="nind"><b><a id="*110·42">*110·42</a>.</b> \(\vdash .\mu +_{\text{c}}\nu \in \text{NC}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*110·22. \supset \vdash :\mu = \text{N}_{0}\text{c}ʻ\alpha .\nu = \text{N}_{0}\text{c}ʻ\beta .&\supset .\mu +_{\text{c}}\nu = \text{Nc}ʻ(\alpha +\beta ).\\ +[*100·41] &\supset .\mu +_{\text{c}}\nu \in \text{NC}&\qquad \text{(1)}\\ +\vdash .(1).*103·2. &\supset \vdash :\mu ,\nu \in \text{N}_{0}\text{C}.\supset .\mu +_{\text{c}}\nu \in \text{NC}&\qquad \text{(2)}\\ +\vdash .*110·4.\text{Transp}.\supset .\vdash :{\sim}(\mu ,\nu \in \text{N}_{0}\text{C}).&\supset .\mu +_{\text{c}}\nu = \Lambda .\\ +[*102·74] &\supset .\mu +_{\text{c}}\nu \in \text{NC} &\qquad \text{(3)}\\ +\vdash .(2).(3).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*110·43.</b> \(\vdash :\mu +_{\text{c}}\nu = \text{N}_{0}\text{c}ʻ\eta . \equiv .\eta \in \mu +_{\text{c}}\nu \quad[*110·42.*103·26]\)</p> + +<p class="nind"><b><a id="*110·44">*110·44</a>.</b> \(\vdash .\text{ sm }ʻʻ(\mu +_{\text{c}}\nu ) = \mu +_{\text{c}}\nu\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*37·1.*110·2.\supset \\ +\vdash :\xi \in \text{ sm }ʻʻ(\mu +_{\text{c}}\nu ). &\equiv .(\exists \eta ,\alpha ,\beta ).\mu = \text{N}_{0}\text{c}ʻ\alpha .\nu = \text{N}_{0}\text{c}ʻ\beta + .\eta \text{ sm }(\alpha +\beta ).\xi \text{ sm }\eta .\\ +[*73·3·32] &\equiv .(\exists \alpha ,\beta ).\mu = \text{N}_{0}\text{c}ʻ\alpha .\nu = \text{N}_{0}\text{c}ʻ\beta .\xi \text{ sm }(\alpha +\beta ).\\ +[*110·2] &\equiv .\xi \in \mu +_{\text{c}}\nu :\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p>The above proposition depends upon the fact that \(\mu +_{\text{c}}\nu\) +is typically ambiguous, even when \(\mu\) and \(\nu\) are typically +definite. It is used in the theory of inductive cardinals +(<a href="#*120·32">*120·32</a><a href="#*120·41">·41</a><a href="#*120·424">·424</a>).</p> + +<p>The following propositions are concerned with the commutative and +associative laws for arithmetical addition of cardinals.</p> + +<p class="nind"><b><a id="*110·5">*110·5</a>.</b> \(\vdash .\beta +\alpha = \text{Cnv}ʻʻ(\alpha +\beta)\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*55·14.\supset \vdash .\text{Cnv}ʻʻ(\alpha +\beta ) &= \Lambda _{\beta }\downarrow ʻʻ\iota ʻʻ\alpha \cup \downarrow \Lambda _{\alpha }ʻʻ\iota ʻʻ\beta \\ +[(*110·01)] &= \beta +\alpha .\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_83">[Pg 83]</span></p> + +<p class="nind"><b>*110·501.</b> \(\vdash .\beta + \alpha \text{ sm }\alpha + \beta \quad[*110·5.*73·4]\)</p> + +<p class="nind"><b><a id="*110·51">*110·51</a>.</b> \(\vdash .\mu +_{\text{c}} \nu =\nu +_{\text{c}} \mu \quad[*110·2·501.*73·37]\)</p> + +<p>It is not necessary to the truth of the above proposition that \(\mu\) +and \(\nu\) should be cardinals. If either is not a cardinal, \(\mu +_{\text{c}} \nu\) +and \(\nu +_{\text{c}} \mu\) are both \(\Lambda\).</p> + +<p>The following propositions lead to the associative law (<a href="#*110·56">*110·56</a>).</p> + +<p class="nind"><b>*110·52.</b> \[\begin{aligned}\vdash :\xi \text{ sm }(\alpha + \beta ) + \gamma .&\equiv .(\exists \pi ,\rho ,\sigma ).\pi \text{ sm }\alpha .\rho \text{ sm }\beta .\sigma \text{ sm }\gamma .\\ +&\pi \cap \rho =\Lambda .\pi \cap \sigma =\Lambda .\rho \cap \sigma =\Lambda .\xi =\pi \cup \rho \cup \sigma\end{aligned}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*110·152.\supset \vdash \colon\ldotp \xi \text{ sm }(\alpha + \beta ) + \gamma .\equiv :(\exists \eta ,\sigma ).&\eta \text{ sm }(\alpha + \beta ).\\ +&\sigma \text{ sm }\gamma .\eta \cap \sigma =\Lambda .\xi =\eta \cup \sigma :\\ +[*110·152] \equiv :(\exists \pi ,\rho ,\eta ,\sigma ).\pi \text{ sm }\alpha .&\rho \text{ sm }\beta .\pi \cap \rho =\Lambda .\eta =\pi \cup \rho .\\ +&\sigma \text{ sm }\gamma .\eta \cap \sigma =\Lambda .\xi =\eta \cup \sigma :\\ +[*13·195.*22·68.*24·32] &\equiv :(\exists \pi ,\rho ,\sigma ).\pi \text{ sm }\alpha .\rho \text{ sm }\beta .\sigma \text{ sm }\gamma .\pi \cap \rho =\Lambda .\\ +&\pi \cap \sigma =\Lambda .\rho \cap \sigma =\Lambda .\xi =\pi \cup \rho \cup \sigma \colon\ldotp \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*110·521.</b> \[\begin{aligned}\vdash :&\xi \text{ sm }\alpha + (\beta + \gamma ).\equiv .(\exists \pi ,\rho ,\sigma ).\pi \text{ sm }\alpha .\rho \text{ sm }\beta .\sigma \text{ sm }\gamma .\\ +&\pi \cap \rho =\Lambda .\pi \cap \sigma =\Lambda .\rho \cap \sigma =\Lambda .\xi =\pi \cup \rho \cup \sigma \quad[*110·501·52]\end{aligned}\]</p> + +<p class="nind"><b>*110·53.</b> \(\vdash .(\alpha + \beta )+ \gamma \text{ sm }\alpha + (\beta + \gamma ) \quad[*110·52·521]\)</p> + +<p class="nind"><b>*110·531.</b> \(\alpha + \beta + \gamma =(\alpha + \beta ) + \gamma \quad\text{Df}\)</p> + +<p class="nind"><b><a id="*110·54">*110·54</a>.</b> \(\vdash .(\text{Nc}ʻ\alpha +_{\text{c}} \text{Nc}ʻ\beta ) +_{\text{c}} \text{Nc}ʻ\gamma =\text{Nc}ʻ(\alpha + \beta + \gamma )\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*110.3.\supset \vdash .(\text{Nc}ʻ\alpha +_{\text{c}} \text{Nc}ʻ\beta ) +_{\text{c}} \text{Nc}ʻ\gamma &=\text{Nc}ʻ(\alpha + \beta ) +_{\text{c}} + \text{Nc}ʻ\gamma \\ +[*110·3.(*110·531)] & =\text{Nc}ʻ(\alpha + \beta + \gamma ).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*110·541.</b> \(\vdash .\text{Nc}ʻ\alpha +_{\text{c}} (\text{Nc}ʻ\beta +_{\text{c}} \text{Nc}ʻ\gamma )=\text{Nc}ʻ(\alpha + \beta + \gamma )\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*110·3.\supset \vdash .\text{Nc}ʻ\alpha +_{\text{c}} (\text{Nc}ʻ\beta +_{\text{c}} \text{Nc}ʻ\gamma )&=\text{Nc}ʻ{\alpha + (\beta + \gamma )}\\ +[*110·53.(*110·531)] & =\text{Nc}ʻ(\alpha + \beta + \gamma ).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*110·55.</b> \(\vdash .(\text{Nc}ʻ\alpha +_{\text{c}} \text{Nc}ʻ\beta ) +_{\text{c}} \text{Nc}ʻ\gamma =\text{Nc}ʻ\alpha +_{\text{c}} (\text{Nc}ʻ\beta +_{\text{c}} + \text{Nc}ʻ\gamma ) \quad[*110·54·541]\)</p> + +<p class="nind"><b>*110·551.</b> \[\begin{aligned}&\vdash .(\text{N}_{0}\text{c}ʻ\alpha +_{\text{c}} \text{N}_{0}\text{c}ʻ\beta ) +_{\text{c}} \text{N}_{0}\text{c}ʻ\gamma + =\text{N}_{0}\text{c}ʻ\alpha +_{\text{c}} (\text{N}_{0}\text{c}ʻ\beta +_{\text{c}} \text{N}_{0}\text{c}ʻ\gamma )\\ +&[*110·55.(*110·03·04)]\end{aligned}\]</p> + +<p class="nind"><b><a id="*110·56">*110·56</a>.</b> \(\vdash .(\mu +_{\text{c}} \nu ) +_{\text{c}} \varpi =\mu +_{\text{c}} (\nu +_{\text{c}} \varpi )\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*110·551.*103·2.\supset \\ +\vdash :\mu ,\nu ,\varpi \in \text{N}_{0}\text{C}.&\supset .(\mu +_{\text{c}} \nu ) +_{\text{c}} \varpi =\mu +_{\text{c}} (\nu +_{\text{c}} + \varpi ) &\qquad \text{(1)}\\ +\vdash .*110·4.\text{Transp}.&\supset \\ +\vdash :{\sim}(\mu ,\nu ,\varpi \in \text{N}_{0}\text{C}).&\supset .(\mu +_{\text{c}} \nu ) +_{\text{c}} \varpi =\Lambda .\mu +_{\text{c}} (\nu +_{\text{c}} + \varpi )=\Lambda .\\ +[*13.171] &\supset .(\mu +_{\text{c}} \nu ) +_{\text{c}} \varpi =\mu +_{\text{c}} (\nu +_{\text{c}} \varpi )&\qquad \text{(2)}\\ +\vdash .(1).(2).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_84">[Pg 84]</span></p> + +<p>This is the associative law for arithmetical addition. It will be seen +that, like the commutative law, it does not require that \(\mu\), +\(\nu\), \(\varpi\) should be cardinals.</p> + +<p class="nind"><b>*110·561.</b> \(\mu +_{\text{c}} \nu +_{\text{c}} \varpi = (\mu +_{\text{c}} \nu ) +_{\text{c}} \varpi \quad\text{Df}\)</p> + +<p class="nind"><b><a id="*110·57">*110·57</a>.</b> \(\vdash .(\mu +_{\text{c}} \nu ) +_{\text{c}} (\varpi +_{\text{c}} \rho ) = \mu +_{\text{c}} \nu +_{\text{c}} \varpi +_{\text{c}} + \rho \quad[*110·56.(*110·561)]\)</p> + +<p>The following propositions, concerning the addition of 0 or 1, are used +frequently in dealing with inductive cardinals (<a href="#*120">*120</a>).</p> + +<p class="nind"><b><a id="*110·6">*110·6</a>.</b> \(\vdash :\mu \in \text{NC}.\supset .\mu +_{\text{c}} 0 = \text{ sm }ʻʻ\mu\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*101·11.*110·21.\supset \\ +\vdash \colon\ldotp \text{Hp}.\supset :\xi \in \mu +_{\text{c}} 0.&\equiv .(\exists \alpha ,\beta ).\alpha \in \mu .\beta \in 0.\xi \text{ sm }(\alpha + \beta ).\\ +[*54·102] &\equiv .(\exists \alpha ).\alpha \in \mu .\xi \text{ sm }(\alpha + \Lambda ).\\ +[*110·152] & \equiv .(\exists \alpha ,\gamma ,\delta ).\alpha \in \mu .\gamma \text{ sm }\alpha .\delta \text{ sm }\Lambda .\gamma \cap \delta =\Lambda .\xi =\gamma \cup \delta .\\ +[*73·47] &\equiv .(\exists \alpha ,\gamma ).\alpha \in \mu .\gamma \text{ sm }\alpha .\xi =\gamma .\\ +[*13·195] &\equiv .(\exists \alpha ).\alpha \in \mu .\xi \text{ sm }\alpha .\\ +[*37·1] &\equiv .\xi \in \text{ sm }ʻʻ\mu \colon\ldotp \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p>When \(\mu\) is a typically definite cardinal, \(\text{ sm }ʻʻ\mu\) is +the same cardinal rendered typically ambiguous; when \(\mu\) is a +typically ambiguous cardinal, \(\text{ sm }ʻʻ\mu\), is \(\mu\). In place +of the above proposition, we might write \(\mu \in \text{NC}.\supset.\mu +_{\text{c}} 0 = \mu\); +this would be true whenever the ambiguity of \(\mu +_{\text{c}} 0\) was +so determined as to make it significant. But the above form gives more +information.</p> + +<p class="nind"><b><a id="*110·61">*110·61</a>.</b> \(\vdash .\text{Nc}ʻ\alpha +_{\text{c}} 0=\text{Nc}ʻ\alpha\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*101·1.\supset \vdash .\text{Nc}ʻ\alpha +_{\text{c}} 0&=\text{Nc}ʻ\alpha +_{\text{c}} \text{Nc}ʻ\Lambda \\ +[*110·32] &= \text{Nc}ʻ(\alpha \cup \Lambda )\\ +[*24·24] &= \text{Nc}ʻ\alpha .\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p>In this proposition, \(\text{Nc}ʻ\alpha\) is typically ambiguous; hence +we escape the necessity of putting \(\text{ sm }ʻʻ\text{Nc}ʻ\alpha\) +on the right, as we should have to do if \(\text{Nc}ʻ\alpha\) were +typically definite. We can deduce <a href="#*110·61">*110·61</a> from <a href="#*110·6">*110·6</a> as follows: +\[ +\begin{array}{l} +\vdash .*110·3.\supset \vdash .\text{Nc}ʻ\alpha +_{\text{c}} 0&=\text{N}_{0}\text{c}ʻ\alpha +_{\text{c}} 0\\ +[*110·6] &=\text{ sm }ʻʻ\text{N}_{0}\text{c}ʻ\alpha \\ +[*103·4] &=\text{Nc}ʻ\alpha +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_85">[Pg 85]</span></p> + +<p>We have to travel via \(\text{N}_{0}\text{c}ʻ\alpha\) in this +proof, in order to avoid the possibility of a typical determination +of \(\text{Nc}ʻ\alpha\) which would make \(\text{Nc}ʻ\alpha=\Lambda\). +It is for the same reason that we cannot put +"\(\text{ sm }ʻʻ\text{Nc}ʻ\alpha =\text{Nc}ʻ\alpha\)"; for if the +first \(\text{Nc}ʻ\alpha\) is determined to a type in which +\(\text{Nc}ʻ\alpha =\Lambda\), while the second is not, this equation +becomes false.</p> + +<p class="nind"><b>*110·62.</b> \(\vdash :\mu +_{\text{c}} \nu =0.\equiv .\mu =0.\nu =0\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*103·27.*101·11·13.&\supset \vdash .0=\text{N}_{0}\text{c}ʻ\Lambda &\quad \text{(1)}\\ +\vdash .(1).*110·43.\supset \\ +\vdash \colon\ldotp \mu +_{\text{c}} \nu =0.&\equiv :\Lambda \in \mu +_{\text{c}} \nu :\\ +[*110·202] &\equiv :\exists !\mu .\exists !\nu :(\exists \gamma ,\delta ).\mu =\text{Nc}ʻ\gamma .\nu =\text{Nc}ʻ\delta .\gamma \cap \delta =\Lambda .\gamma \cup \delta =\Lambda :\\ +[*24·32.*13·22] &\equiv :\exists !\mu .\exists !\nu :\mu =\text{Nc}ʻ\Lambda .\nu =\text{Nc}ʻ\Lambda :\\ +[*101·1·12] &\equiv :\mu =0.\nu =0\colon\ldotp \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*110·63">*110·63</a>.</b> \(\vdash .\text{Nc}ʻ\alpha +_{\text{c}} 1=\hat{\xi} \{(\exists \gamma ,y).\gamma \text{ sm }\alpha .y{\sim}\in \gamma .\xi =\gamma \cup \iota ʻy\}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*101·2.\supset \\ +\vdash .\text{Nc}ʻ\alpha +_{\text{c}} 1&=\text{Nc}ʻ\alpha +_{\text{c}} \text{Nc}ʻ\iota ʻx\\ +[*110·33] &=\hat{\xi} \{(\exists \gamma ,\delta ).\gamma \text{ sm }\alpha .\delta \text{ sm }\iota ʻx.\gamma \cap \delta =\Lambda .\xi =\gamma \cup \delta\}\\ +[*73·45] &=\hat{\xi} \{(\exists \gamma ,\delta ).\gamma \text{ sm }\alpha .\delta \in 1.\gamma \cap \delta =\Lambda .\xi =\gamma \cup \delta\}\\ +[*52·1] &=\hat{\xi} \{(\exists \gamma ,\delta ,y).\gamma \text{ sm }\alpha .\delta =\iota ʻy.\gamma \cap \delta =\Lambda .\xi =\gamma \cup \delta\}\\ +[*13·195.*51·211] &=\hat{\xi} \{(\exists \gamma ,y).\gamma \text{ sm }\alpha .y{\sim}\in \gamma .\xi =\gamma \cup \iota ʻy\}.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p>The above proposition is much used in the theory of finite and +infinite, both cardinal and ordinal. It connects mathematical induction +for inductive cardinals with mathematical induction for inductive +classes (cf. <a href="#*120">*120</a>).</p> + +<p class="nind"><b>*110·631.</b> \(\vdash :\mu \in \text{NC}.\supset \mu +_{\text{c}}1=\hat{\xi} \{(\exists \gamma ,y).\gamma \in \text{ sm }ʻʻ\mu .y{\sim}\in \gamma .\xi =\gamma \cup \iota ʻy\}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*110·211.*101·21.\supset \\ +\vdash :\text{Hp}.\supset .\mu +_{\text{c}} + 1&=\hat{\xi} \{(\exists \gamma ,\delta ).\gamma \in \text{ sm }ʻʻ\mu .\delta \in \text{ sm }ʻʻ1.\gamma \cap \delta =\Lambda .\xi =\gamma \cup \delta\}\\ +[*101·28] &=\hat{\xi} \{(\exists \gamma ,\delta ).\gamma \in \text{ sm }ʻʻ\mu .\delta \in 1.\gamma \cap \delta =\Lambda .\xi =\gamma \cup \delta\}\\ +[*52·1.*51·211] &=\hat{\xi} \{(\exists \gamma ,y).\gamma \in \text{ sm }ʻʻ\mu .y{\sim}\in \gamma .\xi =\gamma \cup \iota ʻy\}:\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p>The proposition +\[ +\mu \in \text{NC}.\supset .\mu +_{\text{c}} 1=\hat{\xi} \{(\exists \gamma ,y).\gamma \in \mu .y{\sim}\in \gamma .\xi \text{ sm }\gamma \cup \iota ʻy\} +\] +which might at first sight seem demonstrable, will only be true +universally if the total number of objects in any one type +is not finite. For suppose \(\alpha\) is a type, and \(\mu=\text{N}_{0}\text{c}ʻ\alpha\). +Then if \(\alpha\) is a finite class, \(\mu =\iota ʻ\alpha\). Hence +\(\gamma \in \mu . \supset _{\gamma ,y}. y\in \gamma\). Hence +\(\hat{\xi} \{(\exists \gamma ,y).\gamma \in \mu .y{\sim}\in \gamma.\xi \text{ sm }(\gamma \cup \iota ʻy)\}=\Lambda\) +in all types. But \(\mu +_{\text{c}} 1\) will exist in all types higher +than that of \(\gamma\). If on the other hand the number of entities in +\(\alpha\) is infinite, we shall have +\[ +y\in \alpha .\supset .\alpha -\iota ʻy\in \text{Nc}ʻ\alpha .y{\sim}\in \alpha -\iota ʻy. +\] +Hence in this case the above proposition will be true universally.</p> + +<p><span class="pagenum" id="Page_86">[Pg 86]</span></p> + +<p class="nind"><b>*110·632.</b> \(\vdash :\mu \in \text{NC}.\supset .\mu +_{\text{c}} 1=\hat{\xi} \{(\exists y).y\in \xi .\xi -\iota ʻy\in \text{ sm }ʻʻ\mu \}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*110·631.*51·211·22.\supset \\ +\vdash :\text{Hp}.&\supset .\mu +_{\text{c}} 1=\hat{\xi} \{(\exists \gamma ,y).\gamma \in \text{ sm }ʻʻ\mu .y\in \xi .\gamma =\xi -\iota ʻy\}\\ +[*13·195] &=\hat{\xi} \{(\exists y).y\in \xi .\xi -\iota ʻy\in \text{ sm }ʻʻ\mu\}:\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*110·64.</b> \(\vdash .0 +_{\text{c}} 0=0 \quad[*110·62]\)</p> + +<p class="nind"><b>*110·641.</b> \(\vdash .1 +_{\text{c}} 0=0 +_{\text{c}} 1=1 \quad[*110·51·61.*101·2]\)</p> + +<p class="nind"><b>*110·642.</b> \(\vdash .2 +_{\text{c}} 0=0 +_{\text{c}} 2=2 \quad[*110·51·61.*101·31]\)</p> + +<p class="nind"><b><a id="*110·643">*110·643</a>.</b> \(\vdash .1 +_{\text{c}} 1=2\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*110·632.*101·21·28.\supset \\ +\vdash .1 +_{\text{c}} 1=\hat{\xi} \{(\exists y).y\in \xi .\xi -\iota ʻy\in 1\}\\ +[*54·3] =2.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p>The above proposition is occasionally useful. It is used at least three +times, in <a href="#*113·66">*113·66</a> and <a href="#*120·123">*120·123</a><a href="#*120·472">·472</a>.</p> + +<p><a href="#*110·7">*110·7</a><a href="#*110·71">·71</a> are required for proving <a href="#*110·72">*110·72</a>, and *110·72 is used in +<a href="#*117·3">*117·3</a>, which is a fundamental proposition in the theory of greater and +less.</p> + +<p class="nind"><b><a id="*110·7">*110·7</a>.</b> \(\vdash :\beta \subset \alpha .\supset .(\exists \mu ).\mu \in \text{NC}.\text{Nc}ʻ\alpha =\text{Nc}ʻ\beta +_{\text{c}} \mu\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*24·411·21.\supset \vdash :\text{Hp}.&\supset .\alpha =\beta \cup (\alpha -\beta ).\beta \cap (\alpha -\beta )=\Lambda .\\ +[*110·32] &\supset .\text{Nc}ʻ\alpha =\text{Nc}ʻ\beta +_{\text{c}} \text{Nc}ʻ(\alpha -\beta ):\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*110·71">*110·71</a>.</b> \(\vdash :(\exists \mu ).\text{Nc}ʻ\alpha =\text{Nc}ʻ\beta +_{\text{c}} \mu .\supset .(\exists \delta ).\delta \text{ sm }\beta .\delta \subset \alpha\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*100·3.*110·4.\supset \\ +\vdash :\text{Nc}ʻ\alpha =\text{Nc}ʻ\beta +_{\text{c}} \mu .&\supset .\mu \in \text{NC}-\iota ʻ\Lambda &\qquad \text{(1)}\\ +\vdash .*110·3.\supset \vdash :\text{Nc}ʻ\alpha =\text{Nc}ʻ\beta +_{\text{c}} &\text{Nc}ʻ\gamma . \equiv .\text{Nc}ʻ\alpha =\text{Nc}ʻ(\beta + \gamma ).\\ +[*100·3·31] &\supset .\alpha \text{ sm }(\beta + \gamma ).\\ +[*73·1] &\supset .(\exists R).R\in 1\rightarrow 1.\text{D}ʻR=\alpha .\text{ᗡ}ʻR=\downarrow \Lambda _{\gamma }ʻʻ\iotaʻʻ\beta \cup \Lambda _{\beta }\downarrow ʻʻ\iotaʻʻ\gamma .\\ +[*37·15] &\supset .(\exists R).R\in 1 \rightarrow 1.\downarrow \Lambda _{\gamma }ʻʻ\iotaʻʻ\beta \subset \text{ᗡ}ʻR.Rʻʻ\downarrow \Lambda _{\gamma }ʻʻ\iotaʻʻ\beta + \subset \alpha .\\ +[*110·12.*73·22] &\supset .(\exists \delta ).\delta \subset \alpha .\delta \text{ sm }\beta &\qquad \text{(2)}\\ +\vdash .(1).(2).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_87">[Pg 87]</span></p> + +<p>The above proof depends upon the fact that "\(\text{Nc}ʻ\alpha\)" +and "\(\text{Nc}ʻ\beta +_{\text{c}} \mu\)" are typically ambiguous, +and therefore, when they are asserted to be equal, this must +hold in any type, and therefore, in particular, in that type for +which we have \(\alpha \in \text{Nc}ʻ\alpha\), <i>i.e.</i> for +\(\text{N}_{0}\text{c}ʻ\alpha\). This is why the use of <a href="#*100·3">*100·3</a> is +legitimate.</p> + +<p class="nind"><b><a id="*110·72">*110·72</a>.</b> \(\vdash :(\exists \delta ).\delta \text{ sm }\beta .\delta \subset \alpha .\equiv .(\exists \mu ).\mu \in \text{NC}.\text{Nc}ʻ\alpha =\text{Nc}ʻ\beta +_{\text{c}} \mu\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*100·321.*110·7.\supset \\ +\vdash \colon\ldotp \delta \text{ sm } \beta .\delta \subset \alpha .&\supset :\text{Nc}ʻ\delta =\text{Nc}ʻ\beta :(\exists \mu ).\mu \in \text{NC}.\text{Nc}ʻ\alpha =\text{Nc}ʻ\delta +_{\text{c}} + \mu :\\ +[*13·12] &\supset :(\exists \mu ).\mu \in \text{NC}.\text{Nc}ʻ\alpha =\text{Nc}ʻ\beta +_{\text{c}} \mu &\qquad \text{(1)}\\ +\vdash .(1).*110·71.\supset \vdash .\text{Prop} +\end{array} +\]</p> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_88">[Pg 88]</span></p> +<h2 class="nobreak" id="*111"> +*111. DOUBLE SIMILARITY.</h2> +</div> + + +<p class="nind"><i>Summary of</i> *111.</p> + +<p>The arithmetical properties of a class, so far as these do not require +or assume that it is a class of classes, are the same for any similar +class. But a class of classes has many arithmetical properties +which it does not share with all similar classes of classes. For +example, if \(\kappa\) is a class of classes, the number of members +of \(sʻ\kappa\) is an arithmetical property of \(\kappa\), but it +is obvious that this is not determined by the number of members of +\(\kappa\), but requires also a knowledge of the numbers of members +of members of \(\kappa\). For example, let \(\kappa\) consist of the +two members \(\alpha\) and \(\beta\), and let \(\lambda\) consist +of \(\gamma\) and \(\delta\). Then \(\kappa \text{ sm } \lambda\); +but in order to be able to infer \(sʻ\kappa \text{ sm } sʻ\lambda\), +we require \(\kappa\), \(\lambda \in \text{Cls}^{2} \text{excl}\) and +\(\alpha \text{ sm } \gamma .\beta \text{ sm } \delta\) or \(\alpha +\text{ sm } \delta .\beta \text{ sm } \gamma\) or some such further +datum. The relation of "double similarity," to be defined in the +present number, is a relation between classes of classes, which, +when it holds between \(\kappa\) and \(\lambda\), insures that all +the arithmetical properties of \(\kappa\) and \(\lambda\) are the +same, <i>e.g.</i> we have (in particular) \(\text{Nc}ʻsʻ\kappa =\text{Nc}ʻsʻ\lambda\) +and \(\text{Nc}ʻ{\in}_{\Delta }ʻ\kappa=\text{Nc}ʻ{\in}_{\Delta }ʻ\lambda\). +This relation we denote by "\(\text{sm}\, \text{sm}\)," which is to be +read as one symbol. It is defined as follows: We define first the class +of "double correlators" of \(\kappa\) and \(\lambda\), which we denote +by "\(\kappa \overline{\text{ sm }}\,\overline{\text{ sm }}\, \lambda\)," +and of which the definition is</p> + +<p class="nind"><b>*111·01.</b> \(\kappa \overline{\text{ sm }}\, \overline{\text{ sm }} \lambda =(1\rightarrow 1)\cap \overleftarrow{\text{ᗡ}}ʻsʻ\lambda \cap \hat{T} (\kappa =T_{\in }ʻʻ\lambda + ) \quad\text{Df}\)</p> + +<p>so that +\[ +\vdash :T\in \kappa \overline{\text{ sm }}\, \overline{\text{ sm }} \lambda .\equiv .T\in 1\rightarrow 1.\text{ᗡ}ʻT=sʻ\lambda .\kappa =T_{\in }ʻʻ\lambda . +\]</p> + +<p>We then define "\(\kappa \text{ sm } \text{ sm } \lambda\)" as meaning +that \(\kappa \overline{\text{ sm }}\, \overline{\text{ sm }} \lambda\) is +not null, <i>i.e.</i> that there is at least one double correlator of +\(\kappa\) and \(\lambda\).</p> + +<p>To illustrate the nature of a double correlator, let us suppose that +\(\kappa\) consists of the two classes \(\alpha _{1}\) and \(\alpha_{2}\), +and that \(\alpha _{1}\) consists of \(x_{11}\), \(x_{12}\), +while \(\alpha _{2}\) consists of \(x_{21}\), \(x_{22}\), \(x_{23}\). +Similarly let \(\lambda\) consist of \(\beta _{1}\) and \(\beta _{2}\) +while \(\beta _{1}\) consists of \(y_{11}\), \(y_{12}\) and \(\beta_{2}\) +consists of \(y_{21}\), \(y_{22}\), \(y_{23}\). Now let \(T\) +correlate each \(x\) with the \(y\) having the same two suffixes. +Then \(T\) is a one-one, and its converse domain is \(sʻ\lambda\). +Moreover \(T_{\in }ʻ\beta _{1}\) (which is \(Tʻʻ\beta _{1}\)) = \(\alpha_{1}\), +and \(T_{\in }ʻ\beta _{2}=\alpha _{2}\), so that \(T_{\in}ʻʻ\lambda =\kappa\). +Thus \(T\) is a double correlator according to +the definition.</p> + +<p>The essential characteristic of a double correlator \(T\) is that +(1) \(T\) is a correlator of \(sʻ\kappa\) and \(sʻ\lambda\), (2) +\(T_{\in }\upharpoonright \lambda\) is a correlator of \(\kappa\) and +<span class="pagenum" id="Page_89">[Pg 89]</span>\(\lambda\). If we write \(S\) in place of \(T_{\in }\upharpoonright +\lambda\), then if \(\beta \in \lambda\), we have \(Sʻ\beta \in +\kappa\); moreover \(T\upharpoonright \beta\) is a correlator of +\(Sʻ\beta\) and \(\beta\). Thus \(\kappa\) and \(\lambda\) are similar +classes of similar classes. They are not merely this, however, for +we not only know that \(Sʻ\beta\) is similar to \(\beta\), but we +know a particular correlator of \(Sʻ\beta\) and \(\beta\), namely +\(T\upharpoonright \beta\). This is essential to the use of double +similarity, as will appear shortly.</p> + +<p>Let us consider the relation between \(\kappa\) and \(\lambda\) which +consists in their being similar classes of similar classes. This means +that there is a correlator \(S\) of \(\kappa\) and \(\lambda\), such +that, if \(\beta \in \lambda\), \(Sʻ\beta\) is similar to \(\beta\). +That is to say, we are to consider the hypothesis +\[ +(\exists S).S\in 1\rightarrow 1.\text{D}ʻS=\kappa .\text{ᗡ}ʻS=\lambda .S \,\unicode{x2abd}\, \text{ sm } +\] +or, as it may be more briefly expressed, +\[ +\exists !\kappa \overline{\text{ sm }} \lambda \cap \text{Rl}ʻ\text{ sm }. +\] +Let us assume \(S\in \kappa \overline{\text{ sm }} \lambda \cap\text{ᗡ}ʻʻʻ\text{ sm }\). +If we attempt to prove (say) that \(sʻ\kappa\) is similar to +\(sʻ\lambda\), we find that we are forced to assume the multiplicative +axiom, unless \(\kappa\) and \(\lambda\) are finite. This necessity +arises as follows. Let us put +\[ +\text{Crp}(S)ʻ\beta =(Sʻ\beta )\,\overline{\text{ sm }}\, \beta , +\] +where "\(\text{Crp}\)" stands for "correspondence." Then we know that +whenever \(\beta \in \lambda\), \(\text{Crp}(S)ʻ\beta\) is not null. +Further it is easy to prove that, if \(\kappa\) and \(\lambda\) are +classes of mutually exclusive classes, and if we can pick out one +representative member of \(\text{Crp}(S)ʻ\beta\) for each value of +\(\beta\) which is a member of \(\lambda\), then the relational sum +of all these representative correlations gives us a correlator of +\(sʻ\kappa\) and \(sʻ\lambda\). That is, we have +\[ +\vdash :\kappa ,\lambda \in \text{Cls}^{2} \text{excl}.S\in \kappa \,\overline{\text{ sm }}\, \lambda \cap \text{ᗡ}ʻʻʻ\text{ sm }.R\in {\in}_{\Delta}ʻ\text{Crp}(S)ʻʻ\lambda + .\supset .\dot{s} ʻ\text{D}ʻR\in (sʻ\kappa )\,\overline{\text{ sm }}\, (sʻ\lambda ). +\]</p> + +<p>But in order to infer hence \(sʻ\kappa \text{ sm } sʻ\lambda\), we need +\(\exists !{\in}_{\Delta}ʻ\text{Crp}(S)ʻʻ\lambda\), <i>i.e.</i> we need +to be able to pick out a particular correlator for each pair of similar +classes \(Sʻ\beta\) and \(\beta\). This, however, cannot be done in +general without assuming the multiplicative axiom. It follows that we +must not define two classes as having double similarity when +\(\exists!\kappa \,\overline{\text{ sm }}\, \lambda \cap \text{Rl}ʻ\text{ sm }\), +but must give a definition which enables us to specify a particular +correlator for each pair of similar classes. This is what is effected +by the above definition of double correlators, where our \(S\) is given +as of the form \(T_{\in }\upharpoonright \lambda\), where +\(T\in 1\rightarrow 1.\text{ᗡ}ʻT =sʻ\lambda\). If the multiplicative +axiom is assumed, but in general not otherwise, we have (<a href="#*111·5">*111·5</a>) +\[ +\kappa ,\lambda \in \text{Cls}^{2} \text{excl}.\supset :\kappa \text{ sm } \text{ sm } \lambda .\equiv .\exists !\kappa \,\overline{\text{ sm }}\, \lambda \cap \text{Rl}ʻ\text{ sm }. +\]</p> + +<p>In the present number, we shall begin with various properties of +double correlators. We prove (<a href="#*111·11">*111·11</a>) that \(T\) is a double +correlator of \(\kappa\) and \(\lambda\) when, and only when, \(T\) +is a correlator of \(sʻ\kappa\) and \(sʻ\lambda\), and \(T_{\in}\upharpoonright \lambda\) +is a correlator of \(\kappa\) and \(\lambda\). We prove (<a href="#*111·112">*111·112</a>) +that in the same hypothesis, \(T_{\in }\upharpoonright \lambda \in\kappa \,\overline{\text{ sm }}\, \lambda \cap \text{Rl}ʻ\text{ sm }\). +We prove (<a href="#*111·13">*111·13</a>) that \(1\upharpoonright sʻ\lambda\) is a double +correlator of \(\lambda\) with itself; that (<a href="#*111·131">*111·131</a>) if \(T\) is a +double correlator of<span class="pagenum" id="Page_90">[Pg 90]</span> \(\kappa\) and \(\lambda\), \(\breve{T}\) is a +double correlator of \(\lambda\) and \(\kappa\); that (<a href="#*111·132">*111·132</a>) if +\(S\), \(T\) are double correlators of \(\kappa\) with \(\lambda\) +and of \(\lambda\) with \(\mu\) respectively, \(S\mid T\) is a +double correlator of \(\kappa\) with \(\mu\). Hence it follows +(<a href="#*111·45">*111·45</a><a href="#*111·451">·451</a><a href="#*111·452">·452</a>) that double similarity is reflexive, symmetrical, and +transitive.</p> + +<p>We then proceed (<a href="#*111·2">*111·2</a>—<a href="#*111·34">·34</a>) to consider \(\text{Crp}(S)ʻʻ\lambda\), +where it is to be supposed that \(S\) is a correlator of \(Sʻʻ\lambda\) +and \(\lambda\), and that \(Sʻ\beta\) is similar to \(\beta\) if +\(\beta \in \lambda\). We prove</p> + +<p class="nind"><b>*111·32.</b> \[\begin{aligned}\vdash :\lambda ,Sʻʻ\lambda \in \text{Cls}^{2} \text{excl}.&S\in 1\rightarrow 1.R\in {\in}_{\Delta }ʻ\text{Crp}(S)ʻʻ\lambda + .M = \dot{s} ʻ\text{D}ʻR.\supset .\\ +&M\in 1\rightarrow 1.\text{ᗡ}ʻM = sʻ\lambda .Sʻʻ\lambda = M_{\in }ʻʻ\lambda .S\upharpoonright \lambda = M_{\in }\upharpoonright \lambda\end{aligned}\]</p> + +<p>Thus in the case supposed, \(M\) is a double correlator of +\(Sʻʻ\lambda\) and \(\lambda\). Thus</p> + +<p class="nind"><b>*111·322.</b> \[\begin{aligned}\vdash :\kappa ,\lambda \in \text{Cls}^{2} \text{excl}.S\in \kappa \,\overline{\text{ sm }}\, \lambda .R\in {\in}_{\Delta}ʻ\text{Crp}(S)ʻʻ\lambda + .&M = \dot{s} ʻ\text{D}ʻR.\supset .\\ +&M\in \kappa \,\overline{\text{ sm }}\, \,\overline{\text{ sm }}\, \lambda .S = M_{\in }\upharpoonright \lambda\end{aligned}\]</p> + +<p>We then proceed (<a href="#*111·4">*111·4</a>—<a href="#*111·47">·47</a>) to various propositions on "\(\text{ sm }\,\text{ sm }\)," +and finally (<a href="#*111·5">*111·5</a><a href="#*111·51">·51</a><a href="#*111·53">·53</a>) state three propositions which +assume the multiplicative axiom, namely</p> + +<p class="nind"><b>*111·5.</b> If \(\kappa\), \(\lambda \in \text{Cls}^{2} \text{excl}\), +then \(\kappa \text{ sm } \text{ sm } \lambda . \equiv .\exists !\kappa \,\overline{\text{ sm }}\, \lambda \cap \text{ᗡ}ʻʻʻ\text{ sm }\).</p> + +<p class="nind"><b>*111·51.</b> In the same case, \(\exists !\kappa\,\overline{\text{ sm }}\, \lambda \cap \text{Rl}ʻ\text{ sm }.\supset .sʻ\kappa\text{ sm } sʻ\lambda\), +<i>i.e.</i> if \(\kappa\) and \(\lambda\) are similar classes of +mutually exclusive similar classes, their sums are similar.</p> + +<p class="nind"><b>*111·53.</b> In the same case, if \(\kappa\), \(\lambda \in \text{Cls}^{2} \text{excl}\), +\(\kappa \text{ sm } \text{ sm } \lambda\). Hence the multiplicative axiom +implies that two classes of \(\mu\) mutually exclusive classes each of +which has \(\nu\) terms, have the same number of terms in their sum.</p> + +<hr class="tb"> + +<p class="nind"><b><a id="*111·01">*111·01</a>.</b> \(\kappa \,\overline{\text{ sm }}\, \,\overline{\text{ sm }}\, \lambda = (1\rightarrow 1)\cap \overleftarrow{\text{ᗡ}}ʻsʻ\lambda \cap \hat{T} (\kappa = T_{\in }ʻʻ\lambda + ) \quad\text{Df}\)</p> + +<p class="nind"><b>*111·02.</b> \(\text{Crp}(S)ʻ\beta = (Sʻ\beta ) \,\overline{\text{ sm }}\, \beta \quad \text{Df}\)</p> + +<p class="nind"><b>*111·03.</b> \(\text{ sm } \text{ sm } = \hat{\kappa} \hat{\lambda} (\exists ! \kappa \,\overline{\text{ sm }}\, \,\overline{\text{ sm }}\, \lambda ) \quad\text{Df}\)</p> + +<p class="nind"><b>*111·1.</b> \(\vdash :T\in \kappa \,\overline{\text{ sm }}\, \,\overline{\text{ sm }}\, \lambda . \equiv .T\in 1\rightarrow 1.\text{ᗡ}ʻT= sʻ\lambda .\kappa = T_{\in }ʻʻ\lambda + \quad [(*111·01)]\)</p> + +<p class="nind"><b><a id="*111·11">*111·11</a>.</b> \(\vdash :T\in \kappa \,\overline{\text{ sm }}\, \,\overline{\text{ sm }}\, \lambda . \equiv .T\in (sʻ\kappa ).\,\overline{\text{ sm }}\, (sʻ\lambda ).T_{\in }\upharpoonright + \lambda \in \kappa \,\overline{\text{ sm }}\, \lambda \)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*37·25.\text{Fact}.\supset \vdash :\text{ᗡ}ʻT = sʻ\lambda .\kappa = T_{\in }ʻʻ\lambda .&\supset .\text{D}ʻT = Tʻʻsʻ\lambda .\kappa = T_{\in }ʻʻ\lambda .\\ +[*40·38] & \supset .\text{D}ʻT = sʻTʻʻʻ\lambda .\kappa = T_{\in }ʻʻ\lambda .\\ +[(*37·04)] & \supset .\text{D}ʻT=sʻ\kappa &\qquad \text{(1)}\\ +\vdash .*72·451.*60·57.*35·65.\supset \\ +\vdash :T\in 1\rightarrow 1.\text{ᗡ}ʻT = sʻ\lambda .&\supset .T_{\in }\upharpoonright \lambda \in 1\rightarrow 1.\lambda = \text{ᗡ}ʻ(T_{\in }\upharpoonright + \lambda ) &\qquad \text{(2)}\\ +\vdash .*37·401. &\supset \vdash :\kappa = T_{\in }ʻʻ\lambda . \equiv .\kappa = \text{D}ʻ(T_{\in }\upharpoonright \lambda ) &\qquad \text{(3)}\\ +\vdash .(1).(2).(3).*4·71.&\supset \vdash :T\in 1\rightarrow 1.\text{ᗡ}ʻT = sʻ\lambda .\kappa = T_{\in }ʻʻ\lambda . \equiv .\\ +&T\in 1\rightarrow 1.\text{D}ʻT = sʻ\kappa .\text{ᗡ}ʻT = sʻ\lambda .T_{\in }\upharpoonright \lambda \in 1\rightarrow 1.\text{D}ʻ(T_{\in }\upharpoonright + \lambda ) = \kappa .\text{ᗡ}ʻ(T_{\in }\upharpoonright \lambda ) = \lambda &\qquad \text{(4)}\\ +\vdash .(4).*111·1.*73·03.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_91">[Pg 91]</span></p> + +<p class="nind"><b>*111·111.</b> \(\vdash :T\in \kappa \,\overline{\text{ sm }}\, \,\overline{\text{ sm }}\, \lambda .\supset .T_{\in }\upharpoonright \lambda \,\unicode{x2abd}\, \text{ sm }\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *111·1. *60·57.\supset \vdash :\text{Hp}.&\supset .T\in 1\rightarrow 1.\lambda \subset \text{Cl}ʻ\text{ᗡ}ʻT.\\ +[*73·5] &\supset .T_{\in }\upharpoonright \lambda \,\unicode{x2abd}\, \text{ sm }:\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*111·112">*111·112</a>.</b> \(\vdash :T\in \kappa \,\overline{\text{ sm }}\, \,\overline{\text{ sm }}\, \lambda .\supset .T_{\in }\upharpoonright + \lambda \in \kappa \,\overline{\text{ sm }}\, \lambda \cap \text{ᗡ}ʻʻʻ\text{ sm } \quad[*111·11·111]\)</p> + +<p>The two following propositions are useful lemmas for the case when +\(T\) is replaced (as it often is) by \(T\upharpoonright \alpha\).</p> + +<p class="nind"><b>*111·12.</b> \(\vdash :sʻ\lambda \subset \alpha .\supset .(T\upharpoonright \alpha )_{\in }ʻʻ\lambda = T_{\in }ʻʻ\lambda .(T\upharpoonright \alpha )_{\in }\upharpoonright + \lambda = T_{\in }\upharpoonright \lambda\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *37·101·421. &\supset \vdash :\beta \subset \alpha .\supset .(T\upharpoonright \alpha )_{\in }ʻ\beta = T_{\in }ʻ\beta &\qquad \text{(1)}\\ +\vdash . *40·13. &\supset \vdash \colon\ldotp \text{Hp}.\supset :\beta \in \lambda .\supset .\beta \subset \alpha &\qquad \text{(2)}\\ +\vdash .(1).(2). \supset \vdash \colon\ldotp \text{Hp}.&\supset :\beta \in \lambda .\supset .(T\upharpoonright \alpha )_{\in }ʻ\beta = T_{\in }ʻ\beta :\\ +[*37·69.*35·71] &\supset :(T\upharpoonright \alpha )_{\in }ʻʻ\lambda = T_{\in }ʻʻ\lambda .(T\upharpoonright \alpha )_{\in }\upharpoonright \lambda = T_{\in }\upharpoonright + \lambda \colon\ldotp \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*111·121.</b> \(\vdash .(T\upharpoonright sʻ\lambda )_{\in }ʻʻ\lambda = T_{\in }ʻʻ\lambda = (T_{\in }\upharpoonright + \lambda )ʻʻ\lambda .(T\upharpoonright sʻ\lambda )_{\in }\upharpoonright \lambda = T_{\in }\upharpoonright \lambda\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*37·421.\supset \vdash .T_{\in }ʻʻ\lambda = (T_{\in }\upharpoonright \lambda )ʻʻ\lambda &\qquad \text{(1)}\\ +\vdash .(1).*111·12 \frac{sʻ\lambda}{\alpha}.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*111·13">*111·13</a>.</b> \(\vdash .I\upharpoonright sʻ\lambda \in \lambda \,\overline{\text{ sm }}\, \,\overline{\text{ sm }}\, \lambda\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*72·17.*50·5·52.&\supset \vdash .I\upharpoonright sʻ\lambda \in 1\rightarrow 1.\text{ᗡ}ʻ(I\upharpoonright sʻ\lambda ) = sʻ\lambda &\qquad \text{(1)}\\ +\vdash .*111·121. \supset \vdash .(I\upharpoonright sʻ\lambda )_{\in }ʻʻ\lambda &= I_{\in }ʻʻ\lambda \\ +[*50·16·17] &= \lambda &\qquad \text{(2)}\\ +\vdash .(1).(2).*111·1.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*111·131">*111·131</a>.</b> \(\vdash :T\in \kappa \,\overline{\text{ sm }}\, \,\overline{\text{ sm }}\, \lambda . \equiv .\breve{T} \in \lambda \,\overline{\text{ sm }}\, \,\overline{\text{ sm }}\, \kappa\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*71·212.&\supset \vdash :T\in 1\rightarrow 1. \equiv .\breve{T} \in 1\rightarrow 1 &\qquad \text{(1)}\\ +\vdash .*111·11.&\supset \vdash :T\in \kappa \,\overline{\text{ sm }}\, \,\overline{\text{ sm }}\, \lambda .\supset .\text{D}ʻT = sʻ\kappa &\qquad \text{(2)}\\ +\vdash .*111·1.(2).*60·57.\supset \\ +\vdash :T\in \kappa \,\overline{\text{ sm }}\, \,\overline{\text{ sm }}\, \lambda .&\supset .T\in 1\rightarrow 1.\kappa \subset \text{Cl}ʻ\text{D}ʻT.\lambda \subset \text{Cl}ʻ\text{ᗡ}ʻT.\kappa = T_{\in }ʻʻ\lambda + .\\ +[*74·6] &\supset .\lambda = (\breve{T} )_{\in }ʻʻ\kappa &\qquad \text{(3)}\\ +\vdash .(1).(2).(3).*111·1.&\supset \vdash :T\in \kappa \,\overline{\text{ sm }}\, \,\overline{\text{ sm }}\, \lambda .\supset .\breve{T} \in \lambda \,\overline{\text{ sm }}\, \,\overline{\text{ sm }}\, \kappa &\qquad \text{(4)}\\ +\vdash .(4)\frac{\breve{T}}{T}. &\supset \vdash :\breve{T} \in \lambda \,\overline{\text{ sm }}\, \,\overline{\text{ sm }}\, \kappa .\supset .T\in \kappa \,\overline{\text{ sm }}\, \,\overline{\text{ sm }}\, \lambda &\qquad \text{(5)}\\ +\vdash .(4).(5).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_92">[Pg 92]</span></p> + +<p class="nind"><b><a id="*111·132">*111·132</a>.</b> + \(\vdash : S \in \kappa \,\overline{\text{ sm }}\, \,\overline{\text{ sm }}\, \lambda . T \in \lambda \,\overline{\text{ sm }}\, \,\overline{\text{ sm }}\, \mu .\supset . S \mid T \in \kappa \,\overline{\text{ sm }}\, \,\overline{\text{ sm }}\, \mu\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *111·11 . *73·311 . \supset \\ +\vdash : \text{Hp} . &\supset . S \mid T e (sʻ\kappa ) \,\overline{\text{ sm }}\, (sʻ\mu ) . (S_{\in } \upharpoonright \lambda ) \mid (T_{\in } + \upharpoonright \mu ) \in \kappa \,\overline{\text{ sm }}\, \mu &\qquad \text{(1)}\\ +\vdash . *35·354 . &\supset \vdash . (S_{\in } \upharpoonright \lambda ) \mid (T_{\in } \upharpoonright \mu ) = S_{\in } \mid (\lambda \upharpoonleft T_{\in } + \upharpoonright \mu ) &\qquad \text{(2)}\\ +\vdash . *74·251 . *111·1 . &\supset \vdash : \text{Hp} . \supset . S_{\in } \mid (\lambda \upharpoonleft T_{\in } \upharpoonright \mu ) = S_{\in } + \mid (T_{\in } \upharpoonright \mu )\\ +[*35·23] &= (S_{\in } \mid T_{\in }) \upharpoonright \mu \\ +[*37·34] &= (S \mid T)_{\in } \upharpoonright \mu &\qquad \text{(3)}\\ +\vdash . (1) . (2) . (3) . \supset \vdash : \text{Hp} . &\supset . S \mid T_{\in } (sʻ\kappa ) \,\overline{\text{ sm }}\, (sʻ\mu ) . (S \mid T)_{\in } + \upharpoonright \mu \in \kappa \,\overline{\text{ sm }}\, \mu .\\ +[*111·11] &\supset . S \mid T \in \kappa \,\overline{\text{ sm }}\, \,\overline{\text{ sm }}\, \mu : \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*111·14.</b> + \(\vdash : T \upharpoonright sʻ\lambda \in \kappa \,\overline{\text{ sm }}\, \,\overline{\text{ sm }}\, \lambda . \equiv . T \upharpoonright sʻ\lambda \in 1 \rightarrow 1 . sʻ\lambda \subset \text{ᗡ}ʻT . \kappa = T_{\in }ʻʻ\lambda\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *111·1·121 . \supset \\ +\vdash : T \upharpoonright sʻ\lambda \in \kappa \,\overline{\text{ sm }}\, \,\overline{\text{ sm }}\, \lambda . &\equiv . T \upharpoonright sʻ\lambda \in 1 \rightarrow 1 . \text{ᗡ}ʻ(T \upharpoonright sʻ\lambda ) = sʻ\lambda . \kappa = T_{\in }ʻʻ\lambda .\\ +[*35·65] &\equiv . T \upharpoonright sʻ\lambda \in 1 \rightarrow 1 . sʻ\lambda \subset \text{ᗡ}ʻT . \kappa = T_{\in }ʻʻ\lambda : \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*111·15.</b> + \(\vdash : T \upharpoonright sʻ\lambda \in \kappa \,\overline{\text{ sm }}\, \,\overline{\text{ sm }}\, \lambda . \equiv . T \upharpoonright sʻ\lambda \in (sʻ\kappa ) \,\overline{\text{ sm }}\, (sʻ\lambda ) . T_{\in } \upharpoonright \lambda \in \kappa \,\overline{\text{ sm }}\, \lambda\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *111·11 . \supset \\ +\vdash : T \upharpoonright sʻ\lambda \in \kappa \,\overline{\text{ sm }}\, \,\overline{\text{ sm }}\, \lambda . \equiv . T \upharpoonright sʻ\lambda \in (sʻ\kappa ) \,\overline{\text{ sm }}\, (sʻ\lambda ) . (T \upharpoonright sʻ\lambda )_{\in } \upharpoonright \lambda \in \kappa \,\overline{\text{ sm }}\, \lambda &\qquad \text{(1)}\\ +\vdash . (1) . *111·121 . \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*111·16.</b> \(\vdash : \exists ! \alpha \,\overline{\text{ sm }}\, \beta \cap \gamma \,\overline{\text{ sm }}\, \delta . \supset . \alpha = \gamma . \beta = \delta\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *73·03 . \supset \vdash : \text{Hp} . &\supset . (\exists R) . \text{D}ʻR = \alpha . \text{ᗡ}ʻR = \beta . \text{D}ʻR = \gamma . \text{ᗡ}ʻR = \delta .\\ +[*13·171] &\supset . \alpha = \gamma . \beta = \delta : \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*111·18.</b> \(\vdash . \alpha \,\overline{\text{ sm }}\, \beta \subset (\alpha \uparrow \beta )_{\Delta }ʻ\beta\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *35·83 . *73·03 . &\supset \vdash : R \in \alpha \,\overline{\text{ sm }}\, \beta . \supset . R \,\unicode{x2abd}\, \alpha \uparrow \beta &\qquad \text{(1)}\\ +\vdash . *73·03 . &\supset \vdash : R \in \alpha \,\overline{\text{ sm }}\, \beta . \supset . R \in 1 \rightarrow \text{Cls} . \text{ᗡ}ʻR = \beta &\qquad \text{(2)}\\ +\vdash . (1) . (2) . *80·14 . &\supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p>The class \((\alpha \uparrow \beta )_{\Delta }ʻ\beta\) is important, +being the class of Cantor's "Belegungen," used by him to define +exponentiation; we have in fact +\[ +\text{Nc}ʻ(\alpha \uparrow \beta )_{\Delta }ʻ\beta = (\text{Nc}ʻ\alpha )^{\text{Nc}ʻ\beta }. +\] +Thus the above proposition shows that \(\text{Nc}ʻ(\alpha\,\overline{\text{ sm }}\, \beta )\) +is less than or equal to \((\text{Nc}ʻ\alpha )^{\text{Nc}ʻ\beta}\); and +since, whenever it is not zero, \(\text{Nc}ʻ\alpha = \text{Nc}ʻ\beta\), +it is less than or equal to +\[ +(\text{Nc}ʻ\alpha )^{\text{Nc}ʻ\alpha }. +\]</p> + +<p><span class="pagenum" id="Page_93">[Pg 93]</span></p> + +<p>The following propositions lead up to <a href="#*111·32">*111·32</a><a href="#*111·33">·33</a><a href="#*111·34">·34</a>:</p> + +<p class="nind"><b><a id="*111·2">*111·2</a>.</b> \(\vdash :\text{E}!Sʻ\beta .\supset .\text{Crp}(S)ʻ\beta = (Sʻ\beta )\,\overline{\text{ sm }}\, \beta \quad[*14·28.(*111·02)]\)</p> + +<p class="nind"><b>*111·201.</b> \(\vdash :f\{\text{Crp}(S)ʻ\beta\}. \equiv .f\{(Sʻ\beta )\,\overline{\text{ sm }}\, \beta\} \quad[*4·2.(*111·02)]\)</p> + +<p class="nind"><b>*111·202.</b> \[\begin{aligned}\vdash :R\in \text{Crp}(S)ʻ\beta . \equiv .R\in 1\rightarrow 1.\text{D}ʻR = &Sʻ\beta .\text{ᗡ}ʻR = \beta \\ +&[*111·201.*73·03]\end{aligned}\]</p> + +<p class="nind"><b>*111·21.</b> \(\vdash :\exists !\text{Crp}(S)ʻ\beta . \equiv .Sʻ\beta \text{ sm }\beta \quad[*111·201.*73·04]\)</p> + +<p class="nind"><b>*111·211.</b> \(\vdash :\exists !\text{Crp}(S)ʻ\beta .\supset .\text{E}!Sʻ\beta .\beta \in \text{ᗡ}ʻS \quad[*111·21.*14·21.*33·43]\)</p> + +<p class="nind"><b>*111·22.</b> \(\vdash \colon\ldotp \beta \in \text{ᗡ}ʻS.\supset _{\beta }.\exists !\text{Crp}(S)ʻ\beta : \equiv .S\in 1\rightarrow \text{Cls}.S\,\unicode{x2abd}\, \text{ sm }\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*111·21.\supset \vdash \colon\ldotp \beta \in \text{ᗡ}ʻS.\supset _{\beta }.\exists !\text{Crp}(S)ʻ\beta : &\equiv :\beta \in \text{ᗡ}ʻS.\supset _{\beta }.Sʻ\beta + \text{ sm }\beta :\\ +[*72·93] &\equiv :S\in 1\rightarrow \text{Cls}.S\,\unicode{x2abd}\, \text{ sm }\colon\ldotp \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*111·221.</b> \(\vdash \colon\ldotp S\in 1\rightarrow \text{Cls}.S\,\unicode{x2abd}\, \text{ sm }.\supset :\exists !\text{Crp}(S)ʻ\beta . \equiv .\beta \in \text{ᗡ}ʻS\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*111·22 .\supset \vdash \colon\ldotp \text{Hp}.\supset :\beta \in \text{ᗡ}ʻS.\supset .\exists !\text{Crp}(S)ʻ\beta &\qquad \text{(1)}\\ +\vdash .(1).*111·211.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*111·23.</b> \(\vdash :S\in 1\rightarrow 1.\beta \in \text{ᗡ}ʻS.\supset .\text{Crp}(S)ʻ\beta = \text{Cnv}ʻʻ\text{Crp}(\breve{S} )ʻSʻ\beta\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*111·2.*71·163.\supset \\ +\vdash \colon\ldotp \text{Hp}.\supset :\text{Crp}(S)ʻ\beta & = (Sʻ\beta )\,\overline{\text{ sm }}\, \beta \\ +[*73·301] &= \text{Cnv}ʻʻ(\beta \,\overline{\text{ sm }}\, Sʻ\beta )\\ +[*72·241] &= \text{Cnv}ʻʻ(\breve{S} ʻSʻ\beta \,\overline{\text{ sm }}\, Sʻ\beta ) &\qquad \text{(1)}\\ +\vdash .(1).*111·201 \frac{\breve{S},Sʻ\beta}{S,\, \beta}.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*111·24.</b> \(\vdash :S\in 1\rightarrow \text{Cls}.\lambda \subset \text{ᗡ}ʻS.\supset .\text{Crp}(S)ʻʻ\lambda \in \text{Cls}^{2} \text{excl}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*111·2.*71·163.&\supset \\ +\vdash \colon\ldotp \text{Hp}.&\supset :\beta ,\gamma \in \lambda .\supset _{\beta , \gamma }.\text{Crp}(S)ʻ\beta + = (Sʻ\beta )\,\overline{\text{ sm }}\, \beta .\text{Crp}(S)ʻ\gamma = (Sʻ\gamma )\,\overline{\text{ sm }}\, \gamma .&\qquad \text{(1)}\\ +[*111·16] &\supset _{\beta , \gamma }.\exists !\text{Crp}(S)ʻ\beta \cap \text{Crp}(S)ʻ\gamma .\supset .\beta = \gamma .\\ +[(1).*30·37] &\supset .\text{Crp}(S)ʻ\beta = \text{Crp}(S)ʻ\gamma &\qquad \text{(2)}\\ +\vdash .(2).*37·63.&\supset \vdash \colon\ldotp \text{Hp}.\supset :\rho ,\sigma \in \text{Crp} (S)ʻʻ\lambda .\exists !\rho \cap \sigma .\supset _{\rho ,\sigma }.\rho + = \sigma \colon\ldotp \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*111·25.</b> + \[\begin{aligned}&\vdash :S\in 1\rightarrow \text{Cls}.S\,\unicode{x2abd}\, \text{ sm }.\lambda \subset \text{ᗡ}ʻS.\supset .\text{Crp}(S)ʻʻ\lambda \in \text{Cls ex}^{2}\,\text{excl}\\ +&[*111·24·22]\end{aligned}\]</p> + +<p><span class="pagenum" id="Page_94">[Pg 94]</span></p> + +<p class="nind"><b>*111·3.</b> \(\vdash : \lambda \in \text{Cls ex}^{2}\,\text{excl} . \supset . \dot{s} ʻʻ\text{D}ʻʻ {\in}_{\Delta} + ʻ\alpha \,\overline{\text{ sm }}\, ʻʻ\lambda \subset (\alpha \uparrow sʻ\lambda )_{\Delta }ʻsʻ\lambda\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *37·29 . *24·12 . \supset \\ +\vdash : {\in}_{\Delta}ʻ\alpha \,\overline{\text{ sm }}\, ʻʻ\lambda = \Lambda . &\supset . \dot{s} ʻʻ\text{D}ʻʻ{\in}_{\Delta}ʻ\alpha + \,\overline{\text{ sm }}\, ʻʻ\lambda \subset (\alpha \uparrow sʻ\lambda )_{\Delta }ʻsʻ\lambda &\qquad \text{(1)}\\ +\vdash . *83·1 . \supset \\ +\vdash \colon\ldotp \text{Hp} . \exists ! {\in}_{\Delta}ʻ\alpha \,\overline{\text{ sm }}\, ʻʻ\lambda . \supset : \beta \in \lambda . &\supset _{\beta } + . \exists ! \alpha \,\overline{\text{ sm }}\, ʻ\beta .\\ +[*111·18] &\supset _{\beta } . \exists ! (\alpha \uparrow \beta )_{\Delta }ʻ\beta .\\ +[*80·15] &\supset _{\beta } . \exists ! (\alpha \uparrow sʻ\lambda )_{\Delta }ʻ\beta :\\ +[*80·83] &\supset : \{(\alpha \uparrow sʻ\lambda )_{\Delta }ʻʻ\lambda\} \upharpoonleft (\alpha \uparrow sʻ\lambda )_{\Delta } \in 1 \rightarrow 1 &\qquad \text{(2)}\\ +\vdash . (2) . *111·18 . *85·72 \frac{\alpha \,\overline{\text{ sm }}\, , (\alpha \uparrow sʻ\lambda )_{\Delta }}{R,\, S}. \supset \\ +\vdash : \text{Hp} . \exists ! {\in}_{\Delta}ʻ\alpha \,\overline{\text{ sm }}\, ʻʻ\lambda . &\supset . \text{D}ʻʻ{\in}_{\Delta}ʻ\alpha + \,\overline{\text{ sm }}\, ʻʻ\lambda \subset \text{D}ʻʻ{\in}_{\Delta}ʻ(\alpha \uparrow sʻ\lambda )_{\Delta }ʻʻ\lambda .\\ +[*37·2] \supset . \dot{s} ʻʻ\text{D}ʻʻ{\in}_{\Delta}ʻ\alpha \,\overline{\text{ sm }}\, ʻʻ\lambda &\subset \dot{s} ʻʻ\text{D}ʻʻ{\in}_{\Delta}ʻ(\alpha + \uparrow sʻ\lambda )_{\Delta }ʻʻ\lambda \\ +[*85·27] &\subset (\alpha \uparrow sʻ\lambda )_{\Delta }ʻsʻ\lambda &\qquad \text{(3)}\\ +\vdash . (1) . (3) . \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*111·31.</b> \[\begin{aligned}\vdash : \lambda , Sʻʻ\lambda \in \text{Cls}^{2} \text{excl} . S \in 1 \rightarrow 1 . R \in {\in}_{\Delta}ʻ\text{Crp} + &(S)ʻʻ\lambda . \supset .\\ +&\dot{s} ʻ\text{D}ʻR \in (sʻSʻʻ\lambda ) \,\overline{\text{ sm }}\, (sʻ\lambda )\end{aligned}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *83·2 . \supset \\ +\vdash \colon\ldotp \text{Hp} . \supset : \beta \in \lambda . &\equiv . Rʻ\text{Crp} (S)ʻ\beta \in \text{Crp} (S)ʻ\beta .\\ +[*111·202] &\equiv . Rʻ\text{Crp} (S)ʻ\beta \in 1 \rightarrow 1 . \text{D}ʻRʻ\text{Crp} (S)ʻ\beta = Sʻ\beta .\\ +&\text{ᗡ}ʻRʻ\text{Crp} (S)ʻ\beta = \beta &\qquad \text{(1)}\\ +\vdash . (1) . *72·322 . &\supset \vdash : \text{Hp} . \supset . \dot{s} ʻRʻʻ\text{Crp} (S)ʻʻ\lambda \in 1 \rightarrow 1 .\\ +[*80·34] &\supset . \dot{s} ʻ\text{D}ʻR \in 1 \rightarrow 1 &\qquad \text{(2)}\\ +\vdash . (1) . *37·68 . *50·17 . \supset \vdash : \text{Hp} . &\supset . \text{D}ʻʻRʻʻ\text{Crp} (S)ʻʻ\lambda = Sʻʻ\lambda .\\ +&\text{ᗡ}ʻʻRʻʻ\text{Crp} (S)ʻʻ\lambda = \lambda .\\ +[*80·34] &\supset . \text{D}ʻʻ\text{D}ʻR = Sʻʻ\lambda . \text{ᗡ}ʻʻ\text{D}ʻR = \lambda .\\ +[*41·43·44] &\supset . \text{D}ʻ\dot{s} ʻ\text{D}ʻR = sʻSʻʻ\lambda . \text{ᗡ}ʻ\dot{s} ʻ\text{D}ʻR = sʻ\lambda &\qquad \text{(3)}\\ +\vdash . (2) . (3) . *73·03 . \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*111·311.</b> \[\begin{aligned}&\vdash : \lambda , Sʻʻ\lambda \in \text{Cls}^{2} \text{excl} . S \in 1 \rightarrow 1 . \exists ! {\in}_{\Delta}ʻ\text{Crp} + (S)ʻʻ\lambda . \supset · sʻSʻʻ\lambda \text{ sm } sʻ\lambda\\ +&[*111·31 . *73·04]\end{aligned}\]</p> + +<p class="nind"><b>*111·313.</b> \[\begin{aligned}\vdash : \lambda \in \text{Cls}^{2} \text{excl} . R \in &{\in}_{\Delta }ʻ\text{Crp} + (S)ʻʻ\lambda . \beta \in \lambda . M = \dot{s} ʻ\text{D}ʻR . \supset .\\ +&M \upharpoonright \beta = Rʻ\text{Crp} (S)ʻ\beta . M \upharpoonright \beta \in \text{Crp} (S)ʻ\beta\end{aligned}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *83·2 . \supset \vdash \colon\colon \text{Hp} . \supset \colon\ldotp \alpha \in \lambda . &\supset _{\alpha } + : Rʻ\text{Crp} (S)ʻ\alpha \in \text{Crp} (S)ʻ\alpha : &\qquad \text{(1)}\\ +[*111·202] &\supset _{\alpha } : \text{ᗡ}ʻRʻ\text{Crp} (S)ʻ\alpha = \alpha :\\ +[*33·14 . *4·71] &\supset _{\alpha } + : x {Rʻ\text{Crp} (S)ʻ\alpha } y . \equiv . x {Rʻ\text{Crp} (S)ʻ\alpha } y . y \in \alpha &\qquad \text{(2)}\\ +\vdash .*35·101.*83·23.*41·11.\supset \\ +\vdash \colon\ldotp \text{Hp}.\supset :x(M\upharpoonright \beta )y.&\equiv .(\exists \alpha ).\alpha \in \lambda .x\{Rʻ\text{Crp}(S)ʻ\alpha \}y.y\in \beta \\ +[(2)] & \equiv .(\exists \alpha ).\alpha \in \lambda .x\{Rʻ\text{Crp}(S)ʻ\alpha\}y.y\in \alpha \cap \beta .\\ +[*84·11.*22·5] &\equiv .(\exists \alpha ).\alpha \in \lambda .x\{Rʻ\text{Crp}(S)ʻ\alpha\}y.y\in \beta .\alpha = \beta .\\ +[*13·195] &\equiv .\beta \in \lambda .x\{Rʻ\text{Crp}(S)ʻ\beta \}y.y\in \beta .\\ +[\text{Hp}.*4·73.(2)] &\equiv .x\{Rʻ\text{Crp}(S)ʻ\beta\}y &\qquad \text{(3)}\\ +\vdash .(1).(3).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_95">[Pg 95]</span></p> + +<p class="nind"><b><a id="*111·32">*111·32</a>.</b> \[\begin{aligned}\vdash :\lambda ,Sʻʻ\lambda \in &\text{Cls}^{2} \text{excl}.S\in 1\rightarrow 1.R\in {\in}_{\Delta}ʻ\text{Crp}(S)ʻʻ\lambda + .M = \dot{s} ʻ\text{D}ʻR.\supset .\\ +&M\in 1\rightarrow 1.\text{ᗡ}ʻM = sʻ\lambda .Sʻʻ\lambda = M_{\in }ʻʻ\lambda .S\upharpoonright \lambda = M_{\in }\upharpoonright \lambda\end{aligned}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*111·31.*73·03.\supset \vdash :\text{Hp}.\supset .M\in 1\rightarrow 1.\text{ᗡ}ʻM = sʻ\lambda &\qquad \text{(1)}\\ +\vdash .*111·313·202. \supset \vdash \colon\ldotp \text{Hp}.\supset :\beta \in \lambda .\supset .\text{D}ʻ(M\upharpoonright \beta ) = Sʻ\beta .\text{ᗡ}ʻ(M\upharpoonright \beta ) = \beta .\\ +[*37·25] \supset .(M\upharpoonright \beta )ʻʻ\beta = Sʻ\beta .\\ +[*37·421·11] \supset .M_{\in }ʻ\beta = Sʻ\beta :\\ +[*35·71.*37·69] \supset :M_{\in }\upharpoonright \lambda = S\upharpoonright \lambda .M_{\in }ʻʻ\lambda = Sʻʻ\lambda &\qquad \text{(2)}\\ +\vdash .(1).(2).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*111·321.</b> \[\begin{aligned}\vdash :\lambda ,&Sʻʻ\lambda \in \text{Cls}^{2} \text{excl}.S\in 1\rightarrow 1.\exists !{\in}_{\Delta}ʻ\text{Crp}(S)ʻʻ\lambda .\supset .\\ +&(\exists M).M\in 1\rightarrow 1.\text{ᗡ}ʻM = sʻ\lambda .Sʻʻ\lambda = M_{\in }ʻʻ\lambda .S\upharpoonright \lambda = M_{\in }\upharpoonright \lambda\\ +&[*111·32]\end{aligned}\]</p> + +<p class="nind"><b>*111·322.</b> \[\begin{aligned}\vdash :\kappa ,\lambda \in \text{Cls}^{2} \text{excl}.&S\in \kappa \,\overline{\text{ sm }}\, \lambda .R\in {\in}_{\Delta}ʻ\text{Crp}(S)ʻʻ\lambda + .M = \dot{s} ʻ\text{D}ʻR.\supset .\\ +&M\in \kappa \,\overline{\text{ sm }}\, \,\overline{\text{ sm }}\, \lambda .S = M_{\in }\upharpoonright \lambda \quad[*111·32·1.*35·66.*73·03]\end{aligned}\]</p> + +<p class="nind"><b><a id="*111·33">*111·33</a>.</b> \[\begin{aligned}\vdash \colon\ldotp \text{Mult ax}.\supset :S\in 1\rightarrow 1.S\,\unicode{x2abd}\, \text{ sm }.&\kappa ,\lambda \in \text{Cls}^{2} + \text{excl}.\kappa =Sʻʻ\lambda .\lambda \subset \text{ᗡ}ʻS.\supset .\\ +&sʻ\kappa \text{ sm }sʻ\lambda\end{aligned}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*111·221.\supset \\ +\vdash \colon\ldotp S\in 1\rightarrow 1.&S\,\unicode{x2abd}\, \text{ sm }.\kappa ,\lambda \in \text{Cls}^{2} \text{excl}.\kappa =Sʻʻ\lambda .\lambda \subset \text{ᗡ}ʻS.\supset :\\ +&\beta \in \lambda .\supset _{\beta }.\exists !\text{Crp}(S)ʻ\beta :\\ +[*88·37] \supset :\text{Mult ax}.&\supset .\exists !{\in}_{\Delta}ʻ\text{Crp}(S)ʻʻ\lambda .\\ +[*111·311] &\supset .sʻ\kappa \text{ sm }sʻ\lambda \colon\ldotp \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*111·34">*111·34</a>.</b> \[\begin{aligned}\vdash \colon\ldotp &\text{Mult ax}.\supset :\\ +&(\exists S).S\in 1\rightarrow 1.S\,\unicode{x2abd}\, \text{ sm }.\text{D}ʻS=\kappa .\text{ᗡ}ʻS = \lambda .\kappa ,\lambda \in \text{Cls}^{2} \text{excl}.\supset .\\ +&(\exists M).M\in 1\rightarrow 1.\text{ᗡ}ʻM = sʻ\lambda .\kappa = M_{\in }ʻʻ\lambda\end{aligned}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*111·25.\supset \\ +\vdash \colon\ldotp S\in 1\rightarrow 1.&S\,\unicode{x2abd}\, \text{ sm }.\text{D}ʻS = \kappa .\text{ᗡ}ʻS = \lambda .\kappa ,\lambda \in \text{Cls}^{2} \text{excl}.\supset :\\ +&\text{Crp}(S)ʻʻ\lambda \in \text{Cls ex}^{2}\,\text{excl}:\\ +[*88·32] &\supset :\text{Mult ax}.\supset .\exists !{\in}_{\Delta}ʻ\text{Crp}(S)ʻʻ\lambda .\\ +[*111·321] &\supset .(\exists M).M\in 1\rightarrow 1.\text{ᗡ}ʻM = sʻ\lambda .\kappa = M_{\in }ʻʻ\lambda &\qquad \text{(1)}\\ +\vdash .(1).*10·11·23.\text{Comm}.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_96">[Pg 96]</span></p> + +<p>The following propositions are concerned with the elementary properties +of "\(\text{sm} \,\text{sm}\)." It will be seen that they are closely +analogous to those of "\(\text{sm}\)."</p> + +<p class="nind"><b><a id="*111·4">*111·4</a>.</b> \[\begin{aligned}&\vdash :\kappa \text{ sm } \text{ sm }\lambda . \equiv .(\exists T).T\in 1\rightarrow 1.\text{ᗡ}ʻT = sʻ\lambda .\kappa = T_{\in }ʻʻ\lambda + . \equiv .\exists !\kappa \,\overline{\text{ sm }}\, \,\overline{\text{ sm }}\, \lambda \\ +&[*111·1.(*111·03)]\end{aligned}\]</p> + +<p class="nind"><b>*111·401.</b> \(\vdash :\kappa \text{ sm } \text{ sm } \lambda . \equiv .(\exists T).T\in 1\rightarrow 1.sʻ\lambda \subset \text{ᗡ}ʻT.\kappa = T_{\in }ʻʻ\lambda\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *22·42 . *111·4.\supset \vdash :\kappa \text{ sm } \text{ sm } \lambda .\supset .(\exists T).T\in 1\rightarrow 1.sʻ\lambda \subset \text{ᗡ}ʻT.\kappa = T_{\in }ʻʻ\lambda + &\qquad \text{(1)}\\ +\vdash .(1).*111·14.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*111·402.</b> + \[\begin{aligned}&\vdash :\kappa \text{ sm } \text{ sm } \lambda . \equiv .(\exists T).T\upharpoonright sʻ\lambda \in 1\rightarrow 1.sʻ\lambda \subset \text{ᗡ}ʻT.\kappa = T_{\in }ʻʻ\lambda\\ +&[*111·14·1·121]\end{aligned}\]</p> + +<p class="nind"><b>*111·43.</b> \[\begin{aligned}&\vdash :\kappa \text{ sm } \text{ sm } \lambda .\supset .(\exists S).S\in 1\rightarrow 1.S\,\unicode{x2abd}\, \text{ sm }.\text{D}ʻS = \kappa .\text{ᗡ}ʻS = \lambda \\ +&[*111·11·111]\end{aligned}\]</p> + +<p class="nind"><b>*111·44.</b> \(\vdash :\kappa \text{ sm } \text{ sm } \lambda .\supset .\kappa \text{ sm } \lambda .sʻ\kappa \text{ sm } sʻ\lambda \quad[*111·11·4.*73·03]\)</p> + +<p class="nind"><b><a id="*111·45">*111·45</a>.</b> \(\vdash .\lambda \text{ sm } \text{ sm } \lambda \quad[*111·13·4]\)</p> + +<p class="nind"><b><a id="*111·451">*111·451</a>.</b> \(\vdash :\kappa \text{ sm } \text{ sm } \lambda . \equiv .\lambda \text{ sm } \text{ sm } \kappa \quad[*111·131·4]\)</p> + +<p class="nind"><b><a id="*111·452">*111·452</a>.</b> \(\vdash :\kappa \text{ sm } \text{ sm } \lambda .\lambda \text{ sm } \text{ sm } \mu .\supset .\kappa \text{ sm } \text{ sm } \mu \quad[*111·132·4]\)</p> + +<p class="nind"><b>*111·46.</b> \[\begin{aligned}&\vdash :\lambda ,Sʻʻ\lambda \in \text{Cls}^{2} \text{excl}.S\in 1\rightarrow 1.\exists !{\in}_{\Delta}ʻ\text{Crp}(S)ʻʻ\lambda + .\supset .Sʻʻ\lambda \text{ sm } \text{ sm } \lambda \\ +&[*111·32·4]\end{aligned}\]</p> + +<p class="nind"><b><a id="*111·47">*111·47</a>.</b> \(\vdash \colon\ldotp \kappa \text{ sm } \text{ sm } \lambda .\supset :\kappa \in \text{Cls}^{2} \text{excl}. \equiv .\lambda \in \text{Cls}^{2} \text{excl}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *111·4. &\supset \vdash \colon\ldotp \text{Hp}.\supset :(\exists T).T\in 1\rightarrow 1.\text{ᗡ}ʻT = sʻ\lambda .\kappa = Tʻʻʻ\lambda :\\ +[*84·53] &\supset :\lambda \in \text{Cls}^{2} \text{excl}.\supset .\kappa \in \text{Cls}^{2} \text{excl} &\qquad \text{(1)}\\ +\vdash .(1).*111·451.&\supset \vdash \colon\ldotp \text{Hp}.\supset :\kappa \in \text{Cls}^{2} \text{excl}.\supset .\lambda \in \text{Cls}^{2} + \text{excl} &\qquad \text{(2)}\\ +\vdash .(1).(2).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*111·5">*111·5</a>.</b> \[\begin{aligned}&\vdash \colon\colon \text{Mult ax}.\supset \colon\ldotp \kappa ,\lambda \in \text{Cls}^{2} \text{excl}.\supset :\\ +&\kappa \text{ sm } \text{ sm } \lambda . \equiv .(\exists S).S\in 1\rightarrow 1.S\,\unicode{x2abd}\, \text{ sm }.\text{D}ʻS = \kappa .\text{ᗡ}ʻS = \lambda .\\ +&\equiv .\exists !\kappa \,\overline{\text{ sm }}\, \lambda \cap \text{ᗡ}ʻʻʻ\text{ sm } \quad[*111·34·43·4]\end{aligned}\]</p> + +<p class="nind"><b><a id="*111·51">*111·51</a>.</b> \[\begin{aligned}&\vdash \colon\ldotp \text{Mult ax}.\supset :\kappa ,\lambda \in \text{Cls}^{2} + \text{excl}.\exists !\kappa \,\overline{\text{ sm }}\, \lambda \cap \text{ᗡ}ʻʻʻ\text{ sm }.\supset .sʻ\kappa \text{ sm }sʻ\lambda \\ +&[*111·5·44]\end{aligned}\]</p> + +<p class="nind"><b>*111·52.</b>\(\vdash :\mu ,\nu \in \text{NC}.\kappa ,\lambda \in \mu \cap \text{Cl}ʻ\nu .\supset .\exists !\kappa \,\overline{\text{ sm }}\, \lambda \cap \text{Rl}ʻ\text{ sm }\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*100·5.*73·1.&\supset \vdash :\text{Hp}.\supset .(\exists S).S\in 1\rightarrow 1.\text{D}ʻS = \kappa .\text{ᗡ}ʻS = \lambda &\qquad \text{(1)}\\ +\vdash .*100·5. &\supset \vdash \colon\ldotp \text{Hp}.\supset :\alpha \in \kappa .\beta \in \lambda .\supset .\alpha \text{ sm } \beta &\qquad \text{(2)}\\ +\vdash .(1).(2).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*111·53">*111·53</a>.</b> + \[\begin{aligned}&\vdash \colon\ldotp \text{Mult ax}.\supset :\mu ,\nu \in \text{NC}.\kappa ,\lambda \in \mu \cap \text{Cl excl}ʻ\nu .\supset .\kappa \text{ sm } \text{ sm } \lambda\\ +&[*111·52·5]\end{aligned}\]</p> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_97">[Pg 97]</span></p> +<h2 class="nobreak" id="*112"> +*112. THE ARITHMETICAL SUM OF A CLASS OF CLASSES.</h2> +</div> + + +<p class="nind"><i>Summary of</i> *112.</p> + +<p>In this number, we return to the arithmetical operations. The +definition of addition in <a href="#*110">*110</a> was only applicable to a finite number +of summands, because the summands had to be enumerated. In the present +number, we define the arithmetical sum of a class of classes, so that +the summands are given as the members of a class, and do not require to +be enumerated. Hence the definition in this number is as applicable to +an infinite number of summands as to a finite number.</p> + +<p>If \(\kappa\) is a class of mutually exclusive classes, the number of +\(sʻ\kappa\) will be the sum of the numbers of members of \(\kappa\); +<i>i.e.</i> if we write "\(\Sigma \text{Nc}ʻ\kappa\)" for the sum of +the numbers of members of \(\kappa\), +\[ +\kappa \in \text{Cls}^{2} \text{excl}.\supset .\text{Nc}ʻsʻ\kappa =\Sigma \text{Nc}ʻ\kappa . +\] +But when the members of \(\kappa\) are not mutually exclusive, a +term \(x\) which is a member of two members (say \(\alpha\) and +\(\beta\)) of \(\kappa\) has to be counted twice over in obtaining +the arithmetical sum of \(\kappa\), whereas in the logical sum \(x\) is +only counted once. Thus we need a construction which shall duplicate +\(x\), taking it first as a member of \(\alpha\), and then as a +member of \(\beta\). This is effected if we replace \(x\) first by +\(x\downarrow \alpha\), and then by \(x\downarrow \beta\). In fact, +\(x\downarrow \alpha\) has the kind of arithmetical properties which +we mean to secure when we speak of "\(x\) considered as a member of +\(\alpha\)"—a phrase which, as it stands, does not serve our purpose, +for \(x\) is simply \(x\) however we may choose to consider it. Thus we +replace \(\alpha\) by \(\downarrow \alphaʻʻ\alpha\) and \(\beta\) by +\(\downarrow \betaʻʻ\beta\) and so on; <i>i.e.</i> (using *85·5), we +replace \(\alpha\) by \(\in \unicode{x21A7}\alpha\) and \(\beta\) by +\(\in \unicode{x21A7}\beta\) and so on. These new classes are similar +to \(\alpha\) and \(\beta\) and so on, and are mutually exclusive. +Hence their <i>logical</i> sum has the number of terms which is wanted +for the <i>arithmetical</i> sum of the members of \(\kappa\). Thus we +put +\[ +\begin{aligned} +\Sigma ʻ\kappa &=sʻ\in \unicode{x21A7}ʻʻ\kappa &\quad\text{Df},\\ +\Sigma \text{Nc}ʻ\kappa &=\text{Nc}ʻ\Sigma ʻ\kappa &\quad\text{Df}. +\end{aligned} +\]</p> + +<p>With regard to the second of these definitions, it is to be +observed that \(\Sigma \text{Nc}ʻ\kappa\) is not a function of +\(\text{Nc}ʻʻ\kappa\), unless no two members of \(\kappa\) are similar; +for \(\text{Nc}ʻʻ\kappa\) cannot contain the same number twice over. +For the same reason, if \(\lambda\) is a class of cardinals, and we +define "\(\text{Sum}ʻ\lambda\)," we do not get what<span class="pagenum" id="Page_98">[Pg 98]</span> is wanted for +arithmetical addition, because our definition will not enable us to +deal with summations in which there are numbers that are repeated. +We could, if it were worth while, define "\(\text{Sum}ʻ\lambda\)" as +follows: Take a class of classes \(\kappa\), consisting of one class +having each number which is a member of \(\lambda\), <i>i.e.</i> let +\(\kappa\) be a selection from \(\lambda\); then \(\Sigma ʻ\kappa\) +will have the required number of terms. <i>i.e.</i> we might put +\[ +\text{Sum}ʻ\lambda = \hat{\xi} \{(\exists \kappa ).\kappa \in \text{D}ʻʻ{\in}_{\Delta}ʻ\lambda .\xi \text{ sm }\Sigma ʻ\kappa\} \quad\text{Df}. +\] +But since this definition is only available for sums in which no number +is repeated, it is not worth while to introduce it.</p> + +<p>In this number we prove the following propositions among others.</p> + +<p class="nind"><b>*112·15.</b> \(\vdash :\kappa \in \text{Cls}^{2} \text{excl}.\supset .sʻ\kappa \in \Sigma \text{Nc}ʻ\kappa\)</p> + +<p>This is an extension of <a href="#*110·32">*110·32</a>.</p> + +<p class="nind"><b>*112·17.</b> \(\vdash :\kappa \text{ sm }\text{ sm }\lambda .\supset .\Sigma \text{Nc}ʻ\kappa = \Sigma \text{Nc}ʻ\lambda .\Sigma ʻ\kappa \text{ sm }\Sigma ʻ\lambda\)</p> + +<p>The chief point in the above proposition is that it does not require +\(\kappa\), \(\lambda \in \text{Cls}^{2} \text{excl}\).</p> + +<p><a href="#*112·2">*112·2</a>—<a href="#*112·24">·24</a> are concerned with the use of the multiplicative axiom and +the propositions of <a href="#*111">*111</a> in which it appears as hypothesis. We have</p> + +<p class="nind"><b>*112·22.</b> + \(\vdash \colon\ldotp \text{Mult ax}.\supset :\exists !(\in \unicode{x21A7}ʻʻ\kappa )\,\overline{\text{ sm }}\, (\in \unicode{x21A7}ʻʻ\lambda )\cap \text{Rl}ʻ\text{ sm }.\supset .\Sigma \text{Nc}ʻ\kappa = \Sigma \text{Nc}ʻ\lambda\)</p> + +<p class="nind">whence we derive the proposition</p> + +<p class="nind"><b>*112·24.</b> \(\vdash \colon\ldotp \text{Mult ax}.\supset :\mu ,\nu \in \text{NC}.\kappa ,\lambda \in \mu \cap \text{Cl}ʻ\nu .\supset .\Sigma \text{Nc}ʻ\kappa = \Sigma \text{Nc}ʻ\lambda\)</p> + +<p><i>I.e.</i> assuming the multiplicative axiom, two classes which +each consist of \(\mu\) classes of \(\nu\) terms each have the same +number of terms in their sum. This number would naturally be defined +as \(\mu\) multiplied by \(\nu\), but owing to the necessity of the +multiplicative axiom in this proposition, we have selected a different +definition of multiplication (<a href="#*113">*113</a>) which does not depend upon the +multiplicative axiom. The reader should observe that the similarity +of two classes, each of which consists of \(\mu\) mutually exclusive +sets of \(\nu\) terms, cannot be proved in general without the +multiplicative axiom.</p> + +<p>The remaining propositions of this number give properties of \(\Sigma\) +in special cases. We prove that \(\Sigma ʻ\Lambda = \Lambda\) (<a href="#*112·3">*112·3</a>), +that \(\Sigma \text{Nc}ʻ\iota ʻ\alpha = \text{Nc}ʻ\alpha\) (<a href="#*112·321">*112·321</a>), +that \(\alpha \neq \beta .\supset .\Sigma \text{Nc}ʻ(\iota ʻ\alpha \cup\iota ʻ\beta ) = \text{Nc}ʻ\alpha +_{c}\text{Nc}ʻ\beta\) (<a href="#*112·34">*112·34</a>), +which connects the definition of addition in this number with that in +<a href="#*110">*110</a>. Finally we prove the general associative law for addition, in the +following two forms:</p> + +<p class="nind"><b>*112·41.</b> \(\vdash .sʻ\Sigma ʻʻ\lambda = \Sigma ʻsʻ\lambda\)</p> + +<p class="nind"><b>*112·43.</b> \(\vdash :\lambda \in \text{Cls}^{2} \text{excl}.\supset .\text{Nc}ʻ\Sigma ʻ\Sigma ʻʻ\lambda = \text{Nc}ʻ\Sigma ʻsʻ\lambda\)</p> + +<p><span class="pagenum" id="Page_99">[Pg 99]</span></p> + +<hr class="tb"> + +<p class="nind"><b><a id="*112·01">*112·01</a>.</b> \(\Sigma ʻ\kappa = sʻ\in \unicode{x21A7} ʻʻ\kappa \quad\text{Df}\)</p> + +<p class="nind"><b>*112·02.</b> \(\Sigma \text{Nc}ʻ\kappa = \text{Nc}ʻ\Sigma ʻ\kappa \quad\text{Df}\)</p> + +<p class="nind"><b>*112·1.</b> \(\vdash .\Sigma ʻ\kappa = sʻ\in \unicode{x21A7} ʻʻ\kappa \quad[*20·2. (*112·01)]\)</p> + +<p class="nind"><b><a id="*112·101">*112·101</a>.</b> \(\vdash .\Sigma \text{Nc}ʻ\kappa = \text{Nc}ʻ\Sigma ʻ\kappa = \text{Nc}ʻsʻ\in \unicode{x21A7}ʻʻ\kappa \quad[*20·2. *112·1. (*112*02)]\)</p> + +<p class="nind"><b>*112·102.</b> \(\vdash .\Sigma ʻ\kappa = \hat{R} \{(\exists \alpha ,x).\alpha \in \kappa .x\in \alpha .R=x\downarrow \alpha\}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *85·6. *40·11. *112·1.\supset \\ +\vdash . \Sigma ʻ\kappa &= \hat{R} \{(\exists \mu ,\alpha ).\alpha \in \kappa .\mu = \downarrow \alpha ʻʻ\alpha .R\in \mu\}\\ +[*13·195] &= \hat{R} \{(\exists \alpha ).\alpha \in \kappa .R\in \downarrow \alpha ʻʻ\alpha\}\\ +[*55·231] &= \hat{R} \{(\exists \alpha , x).\alpha \in \kappa .x\in \alpha .R=x\downarrow \alpha\}.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*112·103.</b> \(\vdash .\Sigma ʻ\kappa = sʻ\hat{\mu} \{(\exists \alpha ).\alpha \in \kappa .\mu =\downarrow \alpha ʻʻ\alpha\} \quad[*112·1. *85·6]\)</p> + +<p class="nind"><b>*112·11.</b> \(\vdash :\beta \in \Sigma \text{Nc}ʻ\kappa .\equiv . \beta \text{ sm } sʻ\in \unicode{x21A7}ʻʻ\kappa \quad[*112·101]\)</p> + +<p class="nind"><b>*112·12.</b> \(\vdash .sʻ\in \unicode{x21A7}ʻʻ\kappa \in \Sigma \text{Nc}ʻ\kappa \quad[*112·11]\)</p> + +<p class="nind"><b>*112·13.</b> \(\vdash :\lambda \text{ sm } \text{ sm } \in \unicode{x21A7}ʻʻ\kappa . \supset . sʻ\lambda \in \Sigma \text{Nc}ʻ\kappa \quad[*111·44. *112·11]\)</p> + +<p class="nind"><b><a id="*112·14">*112·14</a>.</b> \(\vdash :\kappa \in \text{Cls}^{2} \text{excl}. \supset . \in \unicode{x21A7} ʻʻ\kappa \text{ sm } \text{ sm } \kappa\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *21·33 .&\supset \vdash \colon\ldotp \text{Hp}.T= \hat{R} \hat{x} \{(\exists \alpha ).\alpha \in \kappa .x\in \alpha .R= x\downarrow \alpha\}.\supset :\\ +&xTR.yTR . \supset . (\exists \alpha ,\beta ).R = x\downarrow \alpha .R = y\downarrow \beta .\\ +[*55·31] &\supset .x=y:\\ +[*71·17] &\supset :T\in 1\rightarrow \text{Cls} &\qquad \text{(1)}\\ +\vdash . *21·33 .\supset\\ +\vdash :\text{Hp}(1).xTR.xTS. &\supset . (\exists \alpha ,\beta ).\alpha ,\beta \in \kappa .x\in \alpha \cap \beta .R = x\downarrow \alpha .S=x\downarrow \beta .\\ +[*84·11.\text{Hp}] &\supset . (\exists \alpha ,\beta ).\alpha =\beta .R = x\downarrow \alpha .S = x\downarrow \beta .\\ +[*13·195] &\supset .R=S:\\ +[*71·171] &\supset :T\in \text{Cls}\rightarrow 1 &\qquad \text{(2)}\\ +\vdash .*33·131.\supset \vdash \colon\ldotp \text{Hp}(1).\supset :x\in \text{ᗡ}ʻT. &\equiv . (\exists R,\alpha ).\alpha \in \kappa .x\in \alpha .R = x\downarrow \alpha .\\ +[*55·12] &\equiv .x\in sʻ\kappa &\qquad \text{(3)}\\ +\vdash .*37·1·11 .\supset \\ +\vdash \colon\colon \text{Hp} .\supset \colon\ldotp \alpha \in \kappa .\supset : R\in T_{\in }ʻ\alpha + . &\equiv . (\exists x,\beta ).x\in \alpha \cap \beta .\beta \in \kappa .R=x\downarrow \beta .\\ +[*84·11.\text{Hp}] &\equiv . (\exists x,\beta ).x\in \alpha \cap \beta .\beta \in \kappa .\alpha =\beta .R = x\downarrow \beta .\\ +[*13·195] &\equiv . (\exists x).x\in \alpha .R = x\downarrow \beta .\\ +[*85·601] &\equiv .R\in \in \unicode{x21A7}ʻ\alpha \colon\ldotp \\ +[*37·69] &\supset \colon\ldotp T_{\in }ʻʻ\kappa = \in \unicode{x21A7}ʻʻ\kappa &\qquad \text{(4)}\\ +\vdash .(1).(2).(3).(4).*111·4.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_100">[Pg 100]</span></p> + +<p class="nind"><b>*112·15.</b> \(\vdash :\kappa \in \text{Cls}^{2} \text{excl}. \supset . sʻ\kappa \in \Sigma \text{Nc}ʻ\kappa \quad[*112·14·11. *111·44]\)</p> + +<p class="nind"><b>*112·151.</b> + \(sʻ\in \unicode{x21A7}ʻʻ\lambda = \hat{R} \{(\exists \alpha ,x).\alpha \in \lambda .x\in \alpha .R = x\downarrow \alpha \}.\dot{s} ʻsʻ\in \unicode{x21A7}ʻʻ\lambda = \in \upharpoonright \lambda\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*40·11.(*85·5).\supset \\ +\vdash . sʻ\in \unicode{x21A7}ʻʻ\lambda &= \hat{R} \{(\exists \alpha ).\alpha \in \lambda .R\in \downarrow \alpha ʻʻ\alpha\}\\ +[*38·131] &= \hat{R} \{(\exists \alpha ,x).\alpha \in \lambda .x\in \alpha .R = x\downarrow \alpha\} &\qquad \text{(1)}\\ +\vdash .(1).*41·11.\supset \\ +\vdash .\dot{s} ʻsʻ\in \unicode{x21A7}ʻʻ\lambda &= \hat{y} \hat{\beta} \{(\exists R,\alpha ,x).\alpha \in \lambda .x\in \alpha .R = x\downarrow \alpha .yR\beta\}\\ +[*13·195.*55·13] &= \hat{y} \hat{\beta} \{(\exists \alpha ,x).\alpha \in \lambda .x\in \alpha .y = x.\beta = \alpha\}\\ +[*13·22] &= \hat{y} \hat{\beta} \{\beta \in \lambda .y\in \beta\}\\ +[*35·101] &= \in \upharpoonright \lambda &\qquad \text{(2)}\\ +\vdash .(1).(2).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p>The following proposition is a lemma for <a href="#*112·153">*112·153</a>, which is required +for <a href="#*112·16">*112·16</a>. *112·16 in turn is used in <a href="#*112·17">*112·17</a>, which is a fundamental +proposition in the theory of addition.</p> + +<p class="nind"><b><a id="*112·152">*112·152</a>.</b> \(\vdash :T\in 1\rightarrow Cls.\beta \subset \text{ᗡ}ʻT.\supset .(T\parallel \breve{T} _{\in })ʻʻ\in \unicode{x21A7}\beta = \in \unicode{x21A7}(Tʻʻ\beta )\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *37·6.*85·601.&\supset \vdash .(T\parallel \breve{T} _{\in })ʻʻ\in \unicode{x21A7}\beta = \hat{R} \{(\exists y).y\in \beta .R = (T\parallel \breve{T} _{\in })ʻ(y\downarrow + \beta )\} &\qquad \text{(1)}\\ +\vdash .(1).*55·61.\supset \\ +\vdash :\text{Hp}.\supset .(T\parallel \breve{T} _{\in })ʻʻ\in \unicode{x21A7}\beta &= \hat{R} \{(\exists y).y\in \beta .R = (Tʻy)\downarrow (T_{\in }ʻ\beta )\}\\ +[*37·11] &= \hat{R} \{(\exists y).y\in \beta .R = (Tʻy)\downarrow (Tʻʻ\beta )\}\\ +[*38·131] &= \downarrow (Tʻʻ\beta )ʻʻ(Tʻʻ\beta )\\ +[*85·601] &= \in \unicode{x21A7}(Tʻʻ\beta ):\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p>In the following proposition, we have a double correlator of a sort +which will frequently occur in cardinal arithmetic, namely \(T\parallel\breve{T}_{\in }\) +with its converse domain limited, where \(T\) is a given double +correlator (or single correlator, on other occasions). As appears from +the propositions used in the above proof of <a href="#*112·152">*112·152</a>, if \(T\) is a +correlator whose converse domain includes \(\beta\) and has \(y\) as +a member, \((T\parallel \breve{T} _{\in })ʻ(y\downarrow \beta ) =(Tʻy)\downarrow (Tʻʻ\beta )\). +Thus \(T\parallel \breve{T}_{\in }\) is an operation which, when +operating on suitable relations of individuals to classes (including +selectors), turns the individuals into their correlates and the classes +into the classes of their members' correlates. This is why it is a +useful relation.</p> + +<p class="nind"><b><a id="*112·153">*112·153</a>.</b> \(T\in \kappa \,\overline{\text{ sm }}\, \,\overline{\text{ sm }}\, \lambda .\supset .(T\parallel \breve{T} _{\in })\upharpoonright + sʻ\in \unicode{x21A7}ʻʻ\lambda \in (\in \unicode{x21A7}ʻʻ\kappa )\,\overline{\text{ sm }}\, \,\overline{\text{ sm }}\, (\in \unicode{x21A7}ʻʻ\lambda )\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*112·151.*41·43·44.&\supset \vdash .sʻ\text{D}ʻʻsʻ\in \unicode{x21A7}ʻʻ\lambda = \text{D}ʻ(\in \upharpoonright \lambda ).sʻ\text{ᗡ}ʻʻsʻ\in \unicode{x21A7}ʻʻ\lambda = \text{ᗡ}ʻ(\in \upharpoonright \lambda ).\\ +[*62·41·43] &\supset \vdash .sʻ\text{D}ʻʻsʻ\in \unicode{x21A7}ʻʻ\lambda = sʻ\lambda .sʻ\text{ᗡ}ʻʻsʻ\in \unicode{x21A7}ʻʻ\lambda = \lambda -\iota ʻ\Lambda &\qquad \text{(1)}\\ +\vdash . (1) . *111·1 . *37·231 . &\supset \vdash : \text{Hp} . \supset . sʻ\text{D}ʻʻsʻ\in \unicode{x21A7} ʻʻ\lambda \subset \text{ᗡ}ʻT . sʻ\text{ᗡ}ʻʻsʻ\in \unicode{x21A7} ʻʻ\lambda \subset \text{ᗡ}ʻT_{\in } &\qquad \text{(2)}\\ +\vdash . *111·1 .*71·29 . &\supset \vdash : \text{Hp} . \supset . T \upharpoonright sʻ\text{D}ʻʻsʻ\in \unicode{x21A7} ʻʻ\lambda \in 1 \rightarrow 1 &\qquad \text{(3)}\\ +\vdash . *111·11 . (1) . &\supset \vdash : \text{Hp} . \supset . T_{\in } \upharpoonright sʻ\text{ᗡ}ʻʻsʻ\in \unicode{x21A7} ʻʻ\lambda \in 1 \rightarrow 1 &\qquad \text{(4)}\\ +\vdash . (2) . (3) . (4) . *74·775 \frac{sʻ\in \unicode{x21A7} ʻʻ\lambda , T, T_{\in }}{\lambda,\, Q,\, R}. &\supset \vdash : \text{Hp} . \supset . (T \parallel \breve{T} _{\in }) + \upharpoonright sʻ\in \unicode{x21A7} ʻʻ\lambda \in 1 \rightarrow 1 &\qquad \text{(5)}\\ +\vdash . *43·302 . &\supset \vdash . sʻ\in \unicode{x21A7} ʻʻ\lambda \subset \text{ᗡ}ʻ(T \parallel \breve{T} _{\in }) &\qquad \text{(6)}\\ +\vdash . *112·152 . \supset \vdash : \text{Hp} . &\supset . (T \parallel \breve{T} _{\in })ʻʻʻ\in \unicode{x21A7} ʻʻ\lambda = \in \unicode{x21A7} ʻʻTʻʻʻ\lambda .\\ +[*37·11] \supset . (T \parallel \breve{T} _{\in })_{\in }ʻʻ\in \unicode{x21A7} ʻʻ\lambda &= \in \unicode{x21A7} ʻʻT_{\in }ʻʻ\lambda \\ +[*111·1 . \text{Hp}] &= \in \unicode{x21A7} ʻʻ\kappa &\qquad \text{(7)}\\ +\vdash . (5) . (6) . (7) . *111·14 . \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_101">[Pg 101]</span></p> + +<p class="nind"><b><a id="*112·16">*112·16</a>.</b> \(\vdash : \kappa \text{ sm } \text{ sm } \lambda . \supset . \in \unicode{x21A7} ʻʻ\kappa \text{ sm } \text{ sm } \in \unicode{x21A7} ʻʻ\lambda \quad[*112·153 . *111·4]\)</p> + +<p class="nind"><b><a id="*112·17">*112·17</a>.</b> \(\vdash : \kappa \text{ sm } \text{ sm } \lambda . \supset . \Sigma \text{Nc}ʻ\kappa . \Sigma \text{Nc}ʻ\lambda . \Sigma ʻ\kappa \text{ sm } \Sigma ʻ\lambda\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *112·16 . *111·44 . \supset \vdash : \text{Hp} . \supset . sʻ\in \unicode{x21A7} ʻʻ\kappa \text{ sm } sʻ\in \unicode{x21A7} ʻʻ\lambda &\qquad \text{(1)}\\ +\vdash . (1) . *112·1·101 . \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*112·18.</b> \(\vdash . \Sigma \text{Nc}ʻ\kappa = \Sigma \text{Nc}ʻ\in \unicode{x21A7} ʻʻ\kappa\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *85·61 . *112·15 . \supset \vdash . sʻ\in \unicode{x21A7} ʻʻ\kappa \in \Sigma \text{Nc}ʻ\in \unicode{x21A7} ʻʻ\kappa &\qquad \text{(1)}\\ +\vdash . (1) . *112·12 . *100·34 . \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*112·2">*112·2</a>.</b> \[\begin{aligned}\vdash : S \in 1 \rightarrow 1 . \text{D}ʻS = \in &\unicode{x21A7} ʻʻ\kappa . \text{ᗡ}ʻS = \in \unicode{x21A7} ʻʻ\lambda . \exists ! {\in}_{\Delta} + ʻ\text{Crp} (S)ʻʻ\lambda .\\ +&\supset . \Sigma \text{Nc}ʻ\kappa = \Sigma \text{Nc}ʻ\lambda . \Sigma ʻ\kappa \text{ sm } \Sigma ʻ\lambda\end{aligned}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *111·311 . *85·61 . \supset \vdash : \text{Hp} . \supset . sʻ\in \unicode{x21A7} ʻʻ\kappa \text{ sm } sʻ\in \unicode{x21A7} ʻʻ\lambda &\qquad \text{(1)}\\ +\vdash . (1) . *112·1·101 . \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*112·21.</b> + \[\begin{aligned}\vdash \colon\ldotp \text{Mult ax} . \supset : (\exists S) . S \in 1 \rightarrow 1 . &S \,\unicode{x2abd}\, \text{ sm } . \text{D}ʻS = \in \unicode{x21A7} ʻʻ\kappa . \text{ᗡ}ʻS = \in \unicode{x21A7} ʻʻ\lambda .\\ +&\equiv . \in \unicode{x21A7} ʻʻ\kappa \text{ sm } \text{ sm } \in \unicode{x21A7} ʻʻ\lambda \quad[*111·5 . *85·61]\end{aligned}\]</p> + +<p class="nind"><b>*112·22.</b> + \[\begin{aligned}\vdash \colon\ldotp \text{Mult ax} . \supset : \exists ! (\in \unicode{x21A7} ʻʻ\kappa ) \,\overline{\text{ sm }}\, (\in \unicode{x21A7} ʻʻ\lambda \cap \text{ᗡ}ʻʻ ʻ\text{ sm } . \supset .\\ +\Sigma \text{Nc}ʻ\kappa = \Sigma \text{Nc}ʻ\lambda \quad[*112·17·18·21]\end{aligned}\]</p> + +<p class="nind"><b>*112·23.</b> \[\begin{aligned}\vdash \colon\ldotp \text{Mult ax} . \supset : \kappa , \lambda \in \text{Cls}^{2} + \text{excl} . \exists ! \kappa &\,\overline{\text{ sm }}\, \lambda \cap \text{Rl}ʻ\text{ sm } . \supset .\\ +&sʻ\kappa , sʻ\lambda \in \Sigma \text{Nc}ʻ\kappa . \Sigma \text{Nc}ʻ\kappa = \Sigma \text{Nc}ʻ\lambda\end{aligned}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *112·15 . &\supset \vdash : \text{Hp} . \kappa , \lambda \in \text{Cls}^{2} + \text{excl} . \supset . sʻ\kappa \in \Sigma \text{Nc}ʻ\kappa . sʻ\lambda \in \Sigma \text{Nc}ʻ\lambda &\qquad \text{(1)}\\ +\vdash . *111·51 . &\supset \vdash : \text{Hp} (1) . \exists ! \kappa \,\overline{\text{ sm }}\, \lambda \cap \text{ᗡ}ʻʻ ʻ\text{ sm } . \supset . sʻ\kappa \text{ sm } sʻ\lambda &\qquad \text{(2)}\\ +\vdash . (1) . (2) . &\supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*112·231.</b> + \(\vdash : S \in \kappa \,\overline{\text{ sm }}\, \lambda \cap \text{ᗡ}ʻʻ ʻ\text{ sm } . \supset . \in \unicode{x21A7} \mid S\mid \text{Cnv}ʻ\in \unicode{x21A7} \in (\in \unicode{x21A7} ʻʻ\kappa ) \,\overline{\text{ sm }}\, ( \in \unicode{x21A7} ʻʻ\lambda ) \cap \text{Rl} ʻ\text{ sm }\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *73·63 . *85·601 . &\supset \vdash : S \in \kappa \,\overline{\text{ sm }}\, \lambda . \supset . \in \unicode{x21A7} \mid S\mid \text{Cnv}ʻ\in \unicode{x21A7} \in (\in \unicode{x21A7} ʻʻ\kappa ) \,\overline{\text{ sm }}\, (\in \unicode{x21A7} ʻʻ\lambda ) &\qquad \text{(1)}\\ +\vdash . *85·601 . *73·33·34 .& \supset \vdash : S \,\unicode{x2abd}\, \text{ sm } . \supset . \in \unicode{x21A7} \mid S\mid \text{Cnv}ʻ\in \unicode{x21A7} \,\unicode{x2abd}\, \text{ sm } &\qquad \text{(2)}\\ +\vdash . (1) . (2) . &\supset \vdash : S \in \kappa \,\overline{\text{ sm }}\, \lambda \cap \text{ᗡ}ʻʻ ʻ\text{ sm } . \supset . \in \unicode{x21A7} \mid S\mid \text{Cnv}ʻ\in \unicode{x21A7} \in (\in \unicode{x21A7} ʻʻ\kappa ) \,\overline{\text{ sm }}\, (\in \unicode{x21A7} ʻʻ\lambda ) \cap \text{Rl} ʻ\text{ sm }:\\ +&\supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_102">[Pg 102]</span></p> + +<p class="nind"><b><a id="*112·24">*112·24</a>.</b> \(\vdash \colon\ldotp \text{Mult ax}.\supset :\mu ,\nu \in \text{NC}.\kappa ,\lambda \in \mu \cap \text{Cl}ʻ\nu .\supset .\Sigma \text{Nc}ʻ\kappa = \Sigma \text{Nc}ʻ\lambda\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*111·52.\supset \vdash :\mu ,\nu \in \text{NC}.\kappa ,\lambda \in \mu \cap \text{Cl}ʻ\nu .&\supset .\exists !\kappa \,\overline{\text{ sm }}\, \lambda \cap \text{ᗡ}ʻʻʻ\text{ sm }.\\ +[*112·231] &\supset .\exists !(\in \unicode{x21A7}ʻʻ\kappa )\,\overline{\text{ sm }}\, (\in \unicode{x21A7}ʻʻ\lambda )\cap \text{ᗡ}ʻʻʻ\text{ sm } &\qquad \text{(1)}\\ +\vdash .(1).*111·51.*85·61.\supset \\ +\vdash \colon\ldotp \text{Mult ax}.\supset :\mu ,\nu \in \text{NC}.\kappa ,\lambda \in \mu \cap \text{Cl}ʻ\nu .&\supset .sʻ\in \unicode{x21A7}ʻʻ\kappa \text{ sm }sʻ\in \unicode{x21A7}ʻʻ\lambda .\\ +[*112·101] &\supset .\Sigma \text{Nc}ʻ\kappa = \Sigma \text{Nc}ʻ\lambda \colon\ldotp \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*112·3">*112·3</a>.</b> \(\vdash .\Sigma ʻ\Lambda = \Lambda \quad[*37·29.*40·21.*112·1]\)</p> + +<p class="nind"><b>*112·301.</b> \(\vdash .\Sigma ʻ\iota ʻ\Lambda = \Lambda\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*112·102.\supset \vdash .\Sigma ʻ\iota ʻ\Lambda &= \hat{R} \{(\exists \alpha ,x).\alpha \in \iota ʻ\Lambda .x\in \alpha .R = x\downarrow \alpha\}\\ +[*51·15] &= \hat{R} \{(\exists x).x\in \Lambda .R = x\downarrow \Lambda\}\\ +[*24·15] &= \Lambda .\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*112·302.</b> \(\vdash .\Sigma ʻ\kappa = \Sigma ʻ(\kappa -\iota ʻ\Lambda )\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*112·102.\supset \vdash .\Sigma ʻ\kappa &= \hat{R} \{(\exists \alpha ,x).\alpha \in \kappa .x\in \alpha .R = x\downarrow \alpha\}\\ +[*10·24] &= \hat{R} \{(\exists \alpha ,x).\alpha \in \kappa .\exists !\alpha .x\in \alpha .R = x\downarrow \alpha\}\\ +[*53·52] &= \hat{R} \{(\exists \alpha ,x).\alpha \in \kappa - \iota ʻ\Lambda .x\in \alpha .R = x\downarrow \alpha\}\\ +[*112·102] &= \Sigma ʻ(\kappa - \iota ʻ\Lambda ).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p>Thus if \(\Lambda\) is a member of a class of classes, it does not +affect the value of their arithmetical sum.</p> + +<p class="nind"><b>*112·303.</b> \(\vdash : \kappa \cap \Lambda = \Lambda . \supset . \Sigma ʻ\kappa \cap \Sigma ʻ\lambda = \Lambda\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*112·102.\supset \\ +\vdash :R\in \Sigma ʻ\kappa \cap \Sigma ʻ\lambda . &\equiv .(\exists \alpha ,\beta ,x,y).\alpha \in \kappa .\beta \in \lambda .x\in \alpha .y\in \beta .R = x\downarrow \alpha = y\downarrow \beta .\\ +[*55·202] &\supset .(\exists \alpha ,x).\alpha \in \kappa \cap \lambda .x\in \alpha .\\ +[*24·5] &\supset .\exists !\kappa \cap \lambda &\qquad \text{(1)}\\ +\vdash .(1).\text{Transp}.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*112·304.</b> \(\vdash :\Sigma ʻ\kappa = \Lambda . \equiv .sʻ\kappa = \Lambda\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*112·3·301.*53·24.&\supset \vdash :sʻ\kappa = \Lambda .\supset .\Sigma ʻ\kappa = \Lambda &\qquad \text{(1)}\\ +\vdash . *112·102. &\supset \vdash :\alpha \in \kappa .x\in \alpha .\supset .x\downarrow \alpha \in \Sigma ʻ\kappa :\\ +[*10·24.*40·11] &\supset \vdash :\exists !sʻ\kappa .\supset .\exists !\Sigma ʻ\kappa &\qquad \text{(2)}\\ +\vdash .(1).(2).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*112·31.</b> \(\vdash .\Sigma ʻ(\kappa \cup \lambda ) = \Sigma ʻ\kappa \cup \Sigma ʻ\lambda\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*112·1.\supset \vdash .\Sigma ʻ(\kappa \cup \lambda ) &= sʻ\in \unicode{x21A7}ʻʻ(\kappa \cup \lambda )\\ +[*40·31] &= sʻ\in \unicode{x21A7}ʻʻ\kappa \cup sʻ\in \unicode{x21A7}ʻʻ\lambda \\ +[*112·1] &= \Sigma ʻ\kappa \cup \Sigma ʻ\lambda .\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_103">[Pg 103]</span></p> + +<p class="nind"><b>*112·311.</b> \(\vdash : \kappa \cap \lambda = \Lambda . \supset . \Sigma \text{Nc}ʻ(\kappa \cup \lambda ) = \Sigma \text{Nc}ʻ\kappa +_{c} \Sigma \text{Nc}ʻ\lambda\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *112·303 . *110·32 . \supset \\ +\vdash : \text{Hp} . \supset . \text{Nc}ʻ(\Sigma ʻ\kappa \cup \Sigma ʻ\lambda ) &= \text{Nc}ʻ\Sigma ʻ\kappa +_{c} \text{Nc}ʻ\Sigma ʻ\lambda \\ +[*112·101] &= \Sigma \text{Nc}ʻ\kappa +_{c} \Sigma \text{Nc}ʻ\lambda &\qquad \text{(1)}\\ +\vdash . (1) . *112·31 . \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*112·32.</b> \(\vdash . \Sigma ʻ\iota ʻ\alpha = \in \unicode{x21A7} \alpha\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *53·31 .*112·1 . \supset \vdash . \Sigma ʻ\iota ʻ\alpha &= sʻ\iota ʻ\in \unicode{x21A7} \alpha \\ +[*53·02] &= \in \unicode{x21A7} \alpha . \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*112·321">*112·321</a>.</b> \(\vdash . \Sigma \text{Nc}ʻ\iota ʻ\alpha = \text{Nc}ʻ\alpha \quad[*112·32·101 . *85·601]\)</p> + +<p class="nind"><b>*112·33.</b> \(\vdash . \Sigma ʻ(\iota ʻ\alpha \cup \iota ʻ\beta ) = \in \unicode{x21A7} \alpha \cup \in \unicode{x21A7} \beta \quad[*112·32·31]\)</p> + +<p class="nind"><b>*112·331.</b> \(\vdash . \Sigma ʻ(\kappa \cup \iota ʻ\beta ) = \Sigma ʻ\kappa \cup \in \unicode{x21A7} \beta \quad[*112·31·32]\)</p> + +<p class="nind"><b><a id="*112·34">*112·34</a>.</b> \(\vdash : \alpha \neq \beta . \supset . \Sigma \text{Nc}ʻ(\iota ʻ\alpha \cup \iota ʻ\beta ) = \text{Nc}ʻ\alpha +_{c} \text{Nc}ʻ\beta\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *51·231 . *112·311 . \supset \\ +\vdash : \text{Hp} . \supset . \Sigma \text{Nc}ʻ(\iota ʻ\alpha \cup \iota ʻ\beta ) &= \Sigma \text{Nc}ʻ\iota ʻ\alpha +_{c} \Sigma \text{Nc}ʻ\iota ʻ\beta \\ +[*112·321] &= \text{Nc}ʻ\alpha +_{c} \text{Nc}ʻ\beta : \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p>This proposition establishes the agreement of the two definitions of +addition, namely that in <a href="#*110">*110</a> and that in <a href="#*112">*112</a>. It will be seen that +the definition of *112 is inapplicable to the addition of a class to +itself, if this is to give the double of the class, instead of (like +logical addition) simply reproducing the class. Hence the need of the +condition \(\alpha\neq \beta\) in the above proposition.</p> + +<p class="nind"><b>*112·341.</b> \(\vdash : \beta {\sim} \in \kappa . \supset . \Sigma \text{Nc}ʻ(\kappa \cup \iota ʻ\beta ) = \Sigma \text{Nc}ʻ\kappa +_{c} \text{Nc}ʻ\beta\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *51·211 . \supset \vdash : \text{Hp} . &\supset . \kappa \cap \iota ʻ\beta = \Lambda .\\ +[*112·311] \supset . \Sigma \text{Nc}ʻ(\kappa \cup \iota ʻ\beta ) &= \Sigma \text{Nc}ʻ\kappa +_{c} \Sigma \text{Nc}ʻ\iota ʻ\beta \\ +[*112·321] &= \Sigma \text{Nc}ʻ\kappa +_{c} \text{Nc}ʻ\beta : \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*112·35.</b> + \(\vdash : \alpha \neq \beta . \alpha \neq \gamma . \beta \neq \gamma . \supset . \Sigma \text{Nc}ʻ(\iota ʻ\alpha \cup \iota ʻ\beta \cup \iota ʻ\gamma ) = \text{Nc}ʻ\alpha +_{c} \text{Nc}ʻ\beta +_{c} \text{Nc}ʻ\gamma\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *51·231 . *112·311 . \supset \\ +\vdash : \text{Hp} . \supset . \Sigma \text{Nc}ʻ(\iota ʻ\alpha \cup \iota ʻ\beta \cup \iota ʻ\gamma ) &= \Sigma \text{Nc}ʻ(\iota ʻ\alpha \cup \iota ʻ\beta ) +_{c} + \Sigma \text{Nc}ʻ\iota ʻ\gamma \\ +[*112·34·321] &= \text{Nc}ʻ\alpha +_{c} \text{Nc}ʻ\beta +_{c} \text{Nc}ʻ\gamma : \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p>Similar propositions can obviously be proved for any finite number of +summands.</p> + +<p><span class="pagenum" id="Page_104">[Pg 104]</span></p> + +<p class="nind"><b>*112·4.</b> \(\vdash :sʻ\kappa ,sʻʻ\kappa \in \text{Cls}^{2} \text{excl}.\supset .\Sigma \text{Nc}ʻsʻ\kappa = \Sigma \text{Nc}ʻsʻʻ\kappa\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*112·15.\supset \vdash :\text{Hp}.\supset .\Sigma \text{Nc}ʻsʻ\kappa &= \text{Nc}ʻsʻsʻ\kappa \\ +[*42·1] &= \text{Nc}ʻsʻsʻʻ\kappa \\ +[*112·15] &= \Sigma \text{Nc}ʻsʻʻ\kappa :\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*112·41.</b> \(\vdash .sʻ\Sigma ʻʻ\lambda = \Sigma ʻsʻ\lambda\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*112·1.\supset \vdash .sʻ\Sigma ʻʻ\lambda &= sʻsʻʻ\in \unicode{x21A7}ʻʻʻ\lambda \\ +[*42·1] &= sʻsʻ\in \unicode{x21A7}ʻʻʻ\lambda \\ +[*40·38] &= sʻ\in \unicode{x21A7}ʻʻsʻ\lambda\\ +[*112·1] &= \Sigma ʻsʻ\lambda .\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*112·42.</b> \(\vdash :\lambda \in \text{Cls}^{2} \text{excl}.\supset .\Sigma ʻʻ\lambda \in \text{Cls}^{2} \text{excl}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*112·303.\supset \vdash \colon\ldotp \lambda \in \text{Cls}^{2} + \text{excl}.&\supset :\beta ,\gamma \in \lambda .\beta \neq \gamma .\supset _[\beta ,\gamma ].\Sigma ʻ\beta \cap \Sigma ʻ\gamma = \Lambda :\\ +[*30·37.\text{Transp}.*37·63] &\supset :\mu ,\nu \in \Sigma ʻʻ\lambda .\mu \neq \nu .\supset _{\mu ,\nu }.\mu \cap \nu = \Lambda :\\ +[*84·1] &\supset :\Sigma ʻʻ\lambda \in \text{Cls}^{2} \text{excl}\colon\ldotp \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*112·43.</b> \(\vdash :\lambda \in \text{Cls}^{2} \text{excl}.\supset .\text{Nc}ʻ\Sigma ʻ\Sigma ʻʻ\lambda = \text{Nc}ʻ\Sigma ʻsʻ\lambda\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*112·15·42.\supset \vdash :\text{Hp}.\supset .\text{Nc}ʻ\Sigma ʻ\Sigma ʻʻ\lambda &= \text{Nc}ʻsʻ\Sigma ʻʻ\lambda \\ +[*112·41] &= \text{Nc}ʻ\Sigma ʻsʻ\lambda :\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p>The above is the associative law for arithmetical addition.</p> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_105">[Pg 105]</span></p> +<h2 class="nobreak" id="*113"> +*113. ON THE ARITHMETICAL PRODUCT OF TWO CLASSES OR OF +TWO CARDINALS.</h2> +</div> + + +<p class="nind"><i>Summary of</i> *113.</p> + +<p>In this number, we give a definition of multiplication which can be +extended to any finite number of factors, but not to an infinite +number of factors. We define first the arithmetical class-product of +two classes \(\alpha\) and \(\beta\), and thence the product of two +cardinals \(\mu\) and \(\nu\) as the number of terms in the product +of \(\alpha\) and \(\beta\) when \(\alpha\) has \(\mu\) terms and +\(\beta\) has \(\nu\) terms. In <a href="#*114">*114</a>, we shall give a definition of +multiplication which is not restricted to a finite number of factors. +The advantages of the definition to be given in this number are, that +it does not require the factors to be of the same type, and that +it enables us to multiply a class by itself without (as in logical +addition and multiplication) simply reproducing the class in question. +The disadvantage of the definition in this number is the impossibility +of extending it to an infinite number of factors.</p> + +<p>The arithmetical class-product of two classes \(\alpha\) and \(\beta\), +which we denote by \(\beta \times \alpha\)<a id="FNanchor_5" href="#Footnote_5" class="fnanchor">[5]</a>, is the class of all +ordinal couples which take their referent from \(\alpha\) and their +relatum from \(\beta\), <i>i.e.</i> it is the class of all such +relations as \(x\downarrow y\), where \(x\in \alpha\) and +\(y\in\beta\). For a given \(y\), the class of couples we obtain is +\(\downarrow yʻʻ\alpha\), which is similar to \(\alpha\); and the +number of such classes, for varying \(y\), is \(\text{Nc}ʻ\beta\). Thus +we have \(\text{Nc}ʻ\beta\) classes of \(\text{Nc}ʻ\alpha\) couples, +and \(\beta \times \alpha\) is the logical sum of these classes of +couples. The class of such classes as \(\downarrow yʻʻ\alpha\), where +\(y\in \beta\), is important again in connection with exponentiation; +we have \(\downarrow yʻʻ\alpha =\alpha \downarrow_{,,}y\), whence the +class of such classes, when \(y\) is varied among the \(\beta\)'s, is +\(\alpha \downarrow_{,,}ʻʻ\beta\), and +\[ +\beta \times \alpha =sʻ\alpha \downarrow_{,,}ʻʻ\beta \quad(\text{cf}. *40·7), +\] +which we take as the definition of \(\beta \times \alpha\).</p> + +<p><span class="pagenum" id="Page_106">[Pg 106]</span></p> + +<p>We represent the arithmetical product of \(\mu\) and \(\nu\) by +\(\mu \times _{c}\nu\). This, as well as \(\text{Nc}ʻ\alpha \times_{c}\text{Nc}ʻ\beta\), +is defined in terms of \(\alpha \times \beta\) exactly as, in <a href="#*110">*110</a>, the +sum was defined in terms of \(\alpha +\beta\).</p> + +<p>The present number contains many propositions which belong to the +theory of \(\alpha \downarrow_{,,}ʻʻ\beta\) rather than (specially) +of \(\beta \times \alpha\); and many propositions are rather logical +than arithmetical in their nature, <i>i.e.</i> they might have +been given in *55. The line is, however, so hard to draw that it +has seemed better to deal simultaneously with all propositions on +\(\alpha \downarrow_{,,}ʻʻ\beta\) or on its sum, which is \(\beta\times \alpha\). +Thus in the present number, the early propositions, +down to <a href="#*113·118">*113·118</a>, deal mainly with logical properties of \(\alpha\downarrow_{,,}ʻʻ\beta\) +and \(\beta \times \alpha\); the following +propositions, down to <a href="#*113·13">*113·13</a>, deal mainly with arithmetical properties +of \(\alpha \downarrow_{,,}ʻʻ\beta\); the propositions <a href="#*113·14">*113·14</a>—<a href="#*113·191">·191</a> +are concerned mainly with arithmetical properties of \(\beta \times\alpha\); +<a href="#*113·2">*113·2</a>—<a href="#*113·27">·27</a> deal with the simpler properties of \(\mu \times_{c}\nu\); +<a href="#*113·3">*113·3</a>—<a href="#*113·34">·34</a> give propositions involving the multiplicative +axiom, and exhibiting the connection (assuming this axiom) of addition +and multiplication; <a href="#*113·4">*113·4</a>—<a href="#*113·491">·491</a> are concerned with various forms of +the distributive law; <a href="#*113·5">*113·5</a>—<a href="#*113·541">*113·541</a> deal with the associative law of +multiplication, and the remaining propositions deal with multiplication +by 0 or 1 or 2.</p> + +<p>The most important propositions in the present number are the following:</p> + +<p class="nind"><b>*113·101.</b> \(\vdash :R\in \beta \times \alpha .\equiv .(\exists x,y).x\in \alpha .y\in \beta .R=x\downarrow y\)</p> + +<p>This merely embodies the definition of \(\beta \times \alpha\).</p> + +<p class="nind"><b>*113·105.</b> \(\vdash :\exists !\alpha .\supset .\alpha \downarrow_{,,}.\in 1\rightarrow 1\)</p> + +<p>This proposition is especially useful in dealing with exponentiation +(<a href="#*116">*116</a>).</p> + +<p class="nind"><b>*113·114.</b> \(\vdash \colon\ldotp \alpha =\Lambda .\lor.\beta =\Lambda :\equiv .\beta \times \alpha =\Lambda\)</p> + +<p>It is in virtue of this proposition that a product of a finite number of +factors only vanishes when one of its factors vanishes.</p> + +<p class="nind"><b>*113·118.</b> \(\vdash .sʻ\text{D}ʻʻ(\beta \times \alpha )\subset \alpha .sʻ\text{ᗡ}ʻʻ(\beta \times \alpha )\subset \beta\)</p> + +<p>This proposition is chiefly useful in the analogous theory of ordinal +products (<a href="#*165">*165</a>, <a href="#*166">*166</a>), where it enables us to apply *74·773. Unless +\(\beta =\Lambda\), we have \(sʻ\text{D}ʻʻ(\beta \times \alpha)=\alpha\), +and unless \(\alpha =\Lambda ,sʻ\text{ᗡ}ʻʻ(\beta \times +\alpha )=\beta\) (<a href="#*113·116">*113·116</a>).</p> + +<p class="nind"><b>*113·12.</b> \(\vdash :\exists !\alpha .\supset .\alpha \downarrow_{,,}ʻʻ\beta \in \text{Nc}ʻ\beta \cap \text{Cl excl}ʻ\text{Nc}ʻ\alpha\)</p> + +<p><i>I.e.</i> unless \(\alpha\) is null, \(\alpha\downarrow_{,,}ʻʻ\beta\) +consists of \(\text{Nc}ʻ\beta\) mutually +exclusive classes each having \(\text{Nc}ʻ\alpha\) members.</p> + +<p class="nind"><b>*113·127.</b> \[\begin{aligned}\vdash :R\upharpoonright y\in \alpha \,\overline{\text{ sm }}\, \gamma .S\upharpoonright \delta \in \beta &\,\overline{\text{ sm }}\, \delta .\supset .\\ +&(R\parallel \breve{S} )\upharpoonright (\delta \times \gamma )\in (\alpha \downarrow_{,,}ʻʻ\beta )\,\overline{\text{ sm }}\, \,\overline{\text{ sm }}\, (\gamma \downarrow_{,,}ʻʻ\delta + )\end{aligned}\]</p> + +<p><span class="pagenum" id="Page_107">[Pg 107]</span></p> + +<p>This is an important proposition, since it gives a double +correlator of \(\alpha \downarrow_{,,}ʻʻ\beta\) with +\(\gamma\downarrow_{,,}ʻʻ\delta\) whenever simple correlators of +\(\alpha\) with \(\gamma\) and of \(\beta\) with \(\delta\) are given. +It leads at once to</p> + +<p class="nind"><b>*113·13.</b> \(\vdash : \alpha \text{ sm } \gamma . \beta \text{ sm } \delta . \supset . \alpha \downarrow_{,,} ʻʻ\beta \text{ sm } \text{ sm } \gamma \downarrow_{,,} + ʻʻ\delta . (\beta \times \alpha ) \text{ sm } (\delta \times \gamma )\)</p> + +<p>This proposition is fundamental in the theory of multiplication, since +it shows that the number of members of \(\beta \times \alpha\) depends +only upon the numbers of members of \(\alpha\) and \(\beta\). It is +also fundamental in the theory of exponentiation, as will appear in +<a href="#*116">*116</a>.</p> + +<p class="nind"><b>*113·141.</b> \(\vdash . \text{Nc}ʻ(\alpha \times \beta ) = \text{Nc}ʻ(\beta \times \alpha )\)</p> + +<p>This is the source of the commutative law of multiplication (<a href="#*113·27">*113·27</a>).</p> + +<p class="nind"><b>*113·146.</b> \(\vdash : \alpha \neq \beta . \supset . \alpha \times \beta \text{ sm } {\in}_{\Delta}ʻ(\iota ʻ\alpha \cup \iota ʻ\beta )\)</p> + +<p>This connects our present theory of multiplication with the theory of +selections.</p> + +<p>We come next to propositions concerning \(\mu \times _{c} \nu\). We have</p> + +<p class="nind"><b>*113·204.</b> \(\vdash \colon\ldotp \mu = \Lambda . \lor . \nu = \Lambda . \lor . {\sim} (\mu , \nu \in \text{NC}) : \supset . \mu \times _{c} \nu = \Lambda\)</p> + +<p>The use of this proposition, like that of <a href="#*110·4">*110·4</a>, is for avoiding +trivial exceptions.</p> + +<p class="nind"><b>*113·23.</b> \(\vdash . \mu \times _{c} \nu \in \text{NC}\)</p> + +<p class="nind"><b>*113·25.</b> \(\vdash . \text{Nc}ʻ\gamma \times _{c} \text{Nc}ʻ\delta = \text{Nc}ʻ(\gamma \times \delta )\)</p> + +<p>This proposition enables us to infer propositions on products of +cardinals from propositions on products of classes, and is therefore +constantly used.</p> + +<p class="nind"><b>*113·27.</b> \(\vdash . \mu \times _{c} \nu = \nu \times _{c} \mu\)</p> + +<p>This is the commutative law of cardinal multiplication.</p> + +<p>The chief proposition using the multiplicative axiom is</p> + +<p class="nind"><b>*113·31.</b> \(\vdash \colon\ldotp \text{Mult ax} . \supset : \mu , \nu \in \text{NC} . \kappa \in \nu \cap \text{Cl}ʻ\mu . \supset . \Sigma ʻ\kappa \in \mu \times _{c} + \nu\)</p> + +<p><i>I.e.</i> assuming the multiplicative axiom, the sum of the numbers +of members in \(\nu\) classes of \(\mu\) terms is \(\mu \times _{c}\nu\). +If we had taken this sum as <i>defining</i> \(\mu \times _{c}\nu\), +almost all propositions on multiplication would have required +the multiplicative axiom. The advantage of \(\alpha \downarrow_{,,}ʻʻ\beta\) +is that, given \(\alpha \text{ sm } \gamma\) and \(\beta\text{ sm } \delta\), +we can construct a double correlator of \(\alpha\downarrow_{,,} ʻʻ\beta\) +with \(\gamma \downarrow_{,,} ʻʻ\delta \), without using the +multiplicative axiom. This is proved in <a href="#*113·127">*113·127</a> (mentioned above).</p> + +<p>The distributive law, which is next considered, has various forms. We +have, to begin with,</p> + +<p class="nind"><b>*113·4.</b> \(\vdash . (\beta \cup \gamma ) \times \alpha = (\beta \times \alpha ) \cup (\gamma \times \alpha )\)</p> + +<p class="nind">whence, using also the commutative law, we easily deduce</p> + +<p class="nind"><b>*113·43.</b> \(\vdash . (\nu +_{c} \varpi ) \times _{c} \mu = \mu \times _{c} (\nu +_{c} \varpi ) = (\mu \times _{c} \nu ) +_{c} + (\mu \times _{c} \varpi )\)</p> + +<p><span class="pagenum" id="Page_108">[Pg 108]</span></p> + +<p>But the distributive law also holds when, instead of enumerated +summands \(\beta\), \(\gamma\) or \(\nu\), \(\varpi\), the summands are +given as the members of a class \(\kappa\), which may be infinite. We +have</p> + +<p class="nind"><b>*113·48.</b> \(\vdash . sʻ\alpha \times ʻʻ\kappa = \alpha \times sʻ\kappa = \text{Cnv}ʻʻ(sʻ\kappa \times \alpha )\)</p> + +<p class="nind">whence, using the definitions of <a href="#*112">*112</a>, we find</p> + +<p class="nind"><b>*113·491.</b> \(\vdash : \kappa \in \text{Cls}^{2} + \text{excl} . \supset . \Sigma \text{Nc}ʻ\alpha \times ʻʻ\kappa = \text{Nc}ʻ(\alpha \times \Sigma ʻ\kappa ) = \text{Nc}ʻ\alpha \times _{c} \Sigma \text{Nc}ʻ\kappa\)</p> + +<p>This is an extension of the distributive law to the case where the +number of summands may be infinite.</p> + +<p>The associative law</p> + +<p class="nind"><b>*113·54.</b> \(\vdash . (\mu \times _{c} \nu ) x_{c} \varpi = \mu \times _{c} (\nu \times _{c} \varpi )\)</p> + +<p class="nind">is proved without any difficulty.</p> + +<p>We prove next that \(\mu \times _{c} \nu = 0\) when, and only when, +\(\mu = 0\) or \(\nu = 0\), \(\mu\), \(\nu\) being existent cardinals +(<a href="#*113·602">*113·602</a>); that a cardinal is unchanged when it is multiplied by 1 +(<a href="#*113·62">*113·62</a><a href="#*113·621">·621</a>); that \(\mu \times _{c} 2 = \mu +_{c} \mu\) (<a href="#*113·66">*113·66</a>) and +that \(\mu \times _{c} (\nu + 1) = (\mu \times _{c} \nu ) +_{c} \mu\) (<a href="#*113·671">*113·671</a>).</p> + +<hr class="tb"> + +<p class="nind"><b>*113·02.</b> \(\beta \times \alpha = sʻ\alpha \downarrow_{,,} ʻʻ\beta \quad\text{Df}\)</p> + +<p class="nind"><b><a id="*113·03">*113·03</a>.</b> \(\mu \times _{c} \nu = \hat{\xi} \{(\exists \alpha , \beta ) . \mu = \text{N}_{0}\text{c}ʻ\alpha . \nu \text{N}_{0}\text{c}ʻ\beta + . \xi \text{ sm } (\alpha \times \beta )\} \quad\text{Df}\)</p> + +<p class="nind"><b><a id="*113·04">*113·04</a>.</b> \(\text{Nc}ʻ\beta \times _{c} \mu = \text{N}_{0}\text{c}ʻ\beta \times _{c} \mu \quad\text{Df}\)</p> + +<p class="nind"><b><a id="*113·05">*113·05</a>.</b> \(\mu \times _{c} \text{Nc}ʻ\alpha = \mu \times _{c} \text{N}_{0}\text{c}ʻ\alpha \quad\text{Df}\)</p> + +<p>In relation to types, <a href="#*113·03">*113·03</a><a href="#*113·04">·04</a><a href="#*113·05">·05</a> call for similar remarks to those +made in <a href="#*110">*110</a> for addition.</p> + +<p class="nind"><b>*113·1.</b> \(\vdash . \beta \times \alpha = sʻ\alpha \downarrow_{,,} ʻʻ\beta \quad[(*113·02)]\)</p> + +<p class="nind"><b>*113·101.</b> \(\vdash : R \in \beta \times \alpha . \equiv . (\exists x, y) . x \in \alpha . y \in \beta . R = x\downarrow y \quad[*40·7 . *113·1]\)</p> + +<p class="nind"><b>*113·102.</b> \(\vdash : y \in \beta . \supset . \alpha \downarrow_{,,} y = (\alpha \uparrow \beta )_{\Delta }ʻ\iota ʻy\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *35·103 . \supset \\ +\vdash \colon\ldotp \text{Hp} . \supset : x (\alpha \uparrow \beta ) y . \equiv . x \in \alpha :\\ +[*85·51] \supset : (\alpha \uparrow \beta )_{\Delta }ʻ\iota ʻy = \downarrow yʻʻ\alpha \\ +[(*38·03)] = \alpha \downarrow_{,,} y : \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*113·103">*113·103</a>.</b> \(\vdash . \alpha \downarrow_{,,} ʻʻ\beta = (\alpha \uparrow \beta )_{\Delta }ʻʻ\iotaʻʻ\beta + = (\alpha \uparrow \beta ) \unicode{x21A7} ʻʻ\beta \quad[*113·102 . *85·52]\)</p> + +<p><span class="pagenum" id="Page_109">[Pg 109]</span></p> + +<p class="nind"><b>*113·104.</b> \(\vdash . \text{E}! \alpha \downarrow_{,,} ʻy \quad[*38·12]\)</p> + +<p class="nind"><b>*113·105.</b> \(\vdash : \exists ! \alpha . \supset . \alpha \downarrow_{,,} \in 1 \rightarrow 1\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *113·104. *71·166 . &\supset \vdash . \alpha \downarrow_{,,} \in 1 \rightarrow \text{Cls} &\qquad \text{(1)}\\ +\vdash . *38·131. \supset \vdash : \alpha \downarrow_{,,} ʻy = \alpha \downarrow_{,,} ʻz . x \in \alpha . &\supset . x \downarrow y \in \alpha \downarrow_{,,} + ʻz .\\ +[*38·131] &\supset . (\exists xʻ) . xʻ \in \alpha . x \downarrow y = xʻ \downarrow z.\\ +[*55·202] &\supset . y = z &\qquad \text{(2)}\\ +\vdash .(2). *10·11·23·35. &\supset \vdash : \exists ! \alpha .\alpha \downarrow_{,,} ʻy = \alpha \downarrow_{,,} ʻz . \supset . y = z &\qquad \text{(3)}\\ +\vdash .(1).(3). *71·54. \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*113·106.</b> \(\vdash : x \in \alpha . y \in \beta . \supset . x \downarrow y \in \beta \times \alpha \quad[*113·101]\)</p> + +<p class="nind"><b>*113·107.</b> \(\vdash : \exists ! \alpha . \exists ! \beta . \supset . \exists ! \beta \times \alpha \quad [*113·106]\)</p> + +<p class="nind"><b>*113·11.</b> \(\vdash : \exists ! \alpha . \supset . \alpha \downarrow_{,,}ʻʻ\beta \in \text{Nc}ʻ\beta : (y) . \alpha \downarrow_{,,} y \in \text{Nc}ʻ\alpha\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *113·105·104 . *73·26 . &\supset \vdash : \exists ! \alpha . \supset . \alpha \downarrow_{,,} ʻʻ\beta \text{ sm } \beta &\qquad \text{(1)}\\ +\vdash . *38·2. *73·611. & \supset \vdash . \alpha \downarrow_{,,} y \text{ sm } \alpha &\qquad \text{(2)}\\ +\vdash . (1) . (2) . &\supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*113·111.</b> \(\vdash . \alpha \downarrow_{,,} ʻʻ\beta \in \text{Cls}^{2} \text{excl} \quad[*113·103 . *85·55]\)</p> + +<p class="nind"><b>*113·112.</b> \(\vdash : \alpha = \Lambda . \exists ! \beta . \supset . \alpha \downarrow_{,,} ʻʻ\beta = \iota ʻ\Lambda\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *38·3 . \supset \vdash : \text{Hp} . \supset . \alpha \downarrow_{,,} ʻʻ\beta &= \hat{\mu} \{ (\exists y).y \in \beta . \mu = \downarrow yʻʻ\Lambda \}\\ +[*37·29] &= \hat{\mu} \{ (\exists y).y \in \beta . \mu = \Lambda\}\\ +[\text{Hp}] & = \iota ʻ\Lambda +\end{array} +\]</p> + +<p class="nind"><b>*113·113.</b> \(\vdash : \beta = \Lambda . \supset . \alpha \downarrow_{,,} ʻʻ\beta = \Lambda \quad[*37·29]\)</p> + +<p class="nind"><b>*113·114.</b> \(\vdash \colon\ldotp \alpha = \Lambda .\lor.\beta = \Lambda : \equiv .\beta \times \alpha = \Lambda \quad[*113·1·112·113·107.*53·24]\)</p> + +<p class="nind"><b>*113·115.</b> \(\vdash . \dot{s} ʻ(\beta \times \alpha ) = \alpha \uparrow \beta\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *113·101.*41·11. \supset \\ +\vdash : u \{ \dot{s} ʻ(\beta \times \alpha )\} v. &\equiv . (\exists R, x, y) . x \in \alpha . y \in \beta . R = x \downarrow y . uRv .\\ +[*13·195.*55·13] &\equiv . (\exists x, y) . x \in \alpha . y \in \beta . u = x . v = y .\\ +[*13·22] &\equiv . u \in \alpha . v \in \beta .\\ +[*35·103] &\equiv . u (\alpha \uparrow \beta ) v : \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*113·116">*113·116</a>.</b> + \[\begin{aligned}&\vdash : \exists ! \beta . \supset . sʻ\text{D}ʻʻ(\beta \times \alpha ) = \alpha : \exists ! \alpha . \supset . sʻ\text{ᗡ}ʻʻ(\beta \times \alpha ) = \beta \\ +&[*113·115 . *41·43·44. *35·85·86]\end{aligned}\]</p> + +<p class="nind"><b>*113·117.</b> + \[\begin{aligned}&\vdash \colon\ldotp \alpha = \Lambda . \lor . \beta = \Lambda : \supset . sʻ\text{D}ʻʻ(\beta \times \alpha ) = \Lambda . sʻ\text{ᗡ}ʻʻ(\beta \times \alpha ) = \Lambda\\ +&[*113·115 . *41·43·44. *35·88]\end{aligned}\]</p> + +<p class="nind"><b><a id="*113·118">*113·118</a>.</b> \(\vdash . sʻ\text{D}ʻʻ(\beta \times \alpha ) \subset \alpha . sʻ\text{ᗡ}ʻʻ(\beta \times \alpha ) \subset \beta \quad[*113·116·117]\)</p> + +<p><span class="pagenum" id="Page_110">[Pg 110]</span></p> + +<p class="nind"><b>*113·12.</b> \(\vdash : \exists ! \alpha . \supset . \alpha \downarrow_{,,} ʻʻ\beta \in \text{Nc}ʻ\beta \cap \text{Cl excl}ʻ\text{Nc}ʻ\alpha \quad[*113·11·111]\)</p> + +<p class="nind"><b>*113·121.</b> \(\vdash . \Sigma ʻ\alpha \downarrow_{,,} ʻʻ\beta \text{ sm } \beta \times \alpha \quad[*112·15 . *113·111·1]\)</p> + +<p class="nind"><b>*113·122.</b> + \[\begin{aligned}&\vdash : R \upharpoonright \gamma , S \upharpoonright \delta \in \text{Cls} \rightarrow 1 . \gamma \subset \text{ᗡ}ʻR . \delta \subset \text{ᗡ}ʻS . \supset . (R \parallel \breve{S} ) \upharpoonright (\delta \times \gamma ) \in 1 \rightarrow 1\\ +&[*74·773. *113·118]\end{aligned}\]</p> + +<p class="nind"><b>*113·123.</b> + \[\begin{aligned}\vdash : R \upharpoonright \gamma , S \upharpoonright \delta \in 1 \rightarrow \text{Cls}. \gamma &\subset \text{ᗡ}ʻR . \delta \subset \text{ᗡ}ʻS . z \in \gamma . w \in \delta . \supset .\\ +&(R \parallel \breve{S} )ʻ(z \downarrow w) = (Rʻz) \downarrow (Sʻw) \quad[*55·61]\end{aligned}\]</p> + +<p class="nind"><b>*113·124.</b> + \[\begin{aligned}\vdash : R \upharpoonright \gamma , S \upharpoonright \delta \in 1 \rightarrow \text{Cls}. \gamma &\subset \text{ᗡ}ʻR . \delta \subset \text{ᗡ}ʻS . w \in \delta . \supset .\\ +&(R \parallel \breve{S} )ʻʻ\gamma \downarrow_{,,} w = (Rʻʻ\gamma ) \downarrow_{,,} (Sʻw)\end{aligned}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *113·123 . *38·131 . \supset \vdash : \text{Hp} . &\supset . (R \parallel \breve{S} )ʻʻ \downarrow wʻʻ\gamma = \downarrow (Sʻw)ʻʻRʻʻ\gamma .\\ +[*38·2] &\supset . (R \parallel \breve{S} )ʻʻ\gamma \downarrow_{,,} w = (Rʻʻ\gamma ) \downarrow_{,,} (Sʻw) : \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*113·125.</b> + \[\begin{aligned}\vdash : R \upharpoonright \gamma , S \upharpoonright \delta \in 1 \rightarrow &\text{Cls}. \gamma \subset \text{ᗡ}ʻR . \delta \subset \text{ᗡ}ʻS . \supset .\\ +&(R \parallel \breve{S} )_{\in }ʻʻ\gamma \downarrow_{,,} ʻʻ\delta = (Rʻʻ\gamma ) \downarrow_{,,} ʻʻ(Sʻʻ\delta ) \quad[*113·124]\end{aligned}\]</p> + +<p class="nind"><b>*113·126.</b> \(\vdash : \text{Hp} *113·125 . \supset . (R \parallel \breve{S} )ʻʻ(\delta \times \gamma ) = (Sʻʻ\delta ) \times (Rʻʻ\gamma )\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *113·1 . *40·38 . &\supset \vdash . (R \parallel \breve{S} )ʻʻ(\delta \times \gamma ) = sʻ(R \parallel \breve{S} )ʻʻʻ\gamma \downarrow_{,,} + ʻʻ\delta &\qquad \text{(1)}\\ +\vdash . (1) . *113·125 . \supset \vdash : \text{Hp} . \supset . (R \parallel \breve{S} )ʻʻ(\delta \times \gamma ) &= sʻ(Rʻʻ\gamma ) \downarrow_{,,} ʻʻ(Sʻʻ\delta )\\ +[*113·1] &= (Sʻʻ\delta ) \times (Rʻʻ\gamma ) : \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*113·127">*113·127</a>.</b> \[\begin{aligned}\vdash : R \upharpoonright \gamma \in \alpha &\,\overline{\text{ sm }}\, \gamma . S \upharpoonright \delta \in \beta \,\overline{\text{ sm }}\, \delta . \supset .\\ +&(R \parallel \breve{S} ) \upharpoonright (\delta \times \gamma ) \in (\alpha \downarrow_{,,} ʻʻ\beta ) \,\overline{\text{ sm }}\, \,\overline{\text{ sm }}\, (\gamma \downarrow_{,,} + ʻʻ\delta )\\ +&[*113·122·125 . *43·302 . *73·142 . *111·14]\end{aligned}\]</p> + +<p class="nind"><b>*113·128.</b> + \[\begin{aligned}\vdash : \text{Hp} &*113·127 . \supset . (R \parallel \breve{S} ) \upharpoonright (\delta \times \gamma ) \in (\beta \times \alpha ) \,\overline{\text{ sm }}\, (\delta \times \gamma ).\\ +&(R \parallel \breve{S} )_{\in } \upharpoonright (\gamma \downarrow_{,,} ʻʻ\delta ) \in (\alpha \downarrow_{,,} + ʻʻ\beta ) \,\overline{\text{ sm }}\, (\gamma \downarrow_{,,} ʻʻ\delta ) \quad[*113·127 . *111·15]\end{aligned}\]</p> + +<p class="nind"><b><a id="*113·13">*113·13</a>.</b> \[\begin{aligned}&\vdash : \alpha \text{ sm } \gamma . \beta \text{ sm } \delta . \supset . \alpha \downarrow_{,,} + ʻʻ\beta \text{ sm } \text{ sm } \gamma \downarrow_{,,} ʻʻ\delta . (\beta \times \alpha ) \text{ sm } (\delta \times \gamma )\\ +&[*113·127 . *111·4·44 . *113·1]\end{aligned}\]</p> + +<p class="nind"><b><a id="*113·14">*113·14</a>.</b> \(\vdash . \alpha \times \beta = \text{Cnv}ʻʻ(\beta \times \alpha ) \quad[*113·101 . *55·14]\)</p> + +<p class="nind"><b>*113·141.</b> \(\vdash . \text{Nc}ʻ(\alpha \times \beta ) = \text{Nc}ʻ(\beta \times \alpha ) \quad[*113·14 . *73·4]\)</p> + +<p class="nind"><b>*113·142.</b> + \(\vdash : \exists ! \beta . \supset . \text{D}ʻʻ(\beta \times \alpha ) = \iotaʻʻ\alpha : \exists ! \alpha . \supset . \text{ᗡ}ʻʻ(\beta \times \alpha ) = \iotaʻʻ\beta \)</p> + +<p><span class="pagenum" id="Page_111">[Pg 111]</span></p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *55·261 . *2·02 . &\supset \vdash : y \in \beta .\supset . \text{D}ʻʻ\alpha \downarrow_{,,} y = \iotaʻʻ\alpha \\ +[*37·63] &\supset \vdash : \gamma \in \text{D}ʻʻʻ \alpha \downarrow_{,,} + ʻʻ\beta . \supset . \gamma = \iotaʻʻ\alpha &\qquad \text{(1)}\\ +\vdash . *37·45. \supset \vdash :\exists !\beta .&\supset .\exists !\text{D}ʻʻʻ\alpha \downarrow_{,,}ʻʻ\beta &\qquad \text{(2)}\\ +\vdash .(1).(2).*51·141. \supset \vdash :\exists !\beta .&\supset .\text{D}ʻʻʻ\alpha \downarrow_{,,}ʻʻ\beta = \iota ʻ\iotaʻʻ\alpha .\\ +[*40·38.*53·02] &\supset .\text{D}ʻʻsʻ\alpha \downarrow_{,,}ʻʻ\beta = \iotaʻʻ\alpha &\qquad \text{(3)}\\ +\vdash . *55·251. \supset \vdash :\exists !\alpha .&\supset .\text{ᗡ}ʻʻ\alpha \downarrow_{,,}y = \iota ʻ\iota ʻy.\\ +[*37·355] &\supset .\text{ᗡ}ʻʻʻ\alpha \downarrow_{,,}ʻʻ\beta = \iotaʻʻ\iotaʻʻ\beta .\\ +[*40·38.*53·22] &\supset .\text{ᗡ}ʻʻsʻ\alpha \downarrow_{,,}ʻʻ\beta = \iotaʻʻ\beta &\qquad \text{(4)}\\ +\vdash .(3).(4).*113·1. \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*113·143">*113·143</a>.</b> \[\begin{aligned}\vdash :\alpha \neq \beta .P = x\downarrow y.&R = x\downarrow \alpha \unicode{x228d} y\downarrow \beta .\supset .\\ +&P = (Rʻ\alpha )\downarrow (Rʻ\beta ).R = \text{D}ʻP\uparrow \iota ʻ\alpha \unicode{x228d} \text{ᗡ}ʻP\uparrow \iota ʻ\beta\end{aligned}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*55·62.\supset \vdash :\text{Hp}.&\supset .Rʻ\alpha = x.Rʻ\beta = y .\\ +[*30·19.*13·15] &\supset .P = (Rʻ\alpha )\downarrow (Rʻ\beta ) &\qquad \text{(1)}\\ +\vdash .*55·15.\supset \vdash :\text{Hp}.&\supset .\text{D}ʻP = \iota ʻx.\text{ᗡ}ʻP = \iota ʻy.\\ +[*55·1] &\supset .R = \text{D}ʻP\uparrow \iota ʻ\alpha \unicode{x228d} \text{ᗡ}ʻP\uparrow \iota ʻ\beta &\qquad \text{(2)}\\ +\vdash .(1).(2).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*113·144">*113·144</a>.</b> + \[\begin{aligned}\vdash :\alpha &\neq \beta .T = \hat{P} \hat{R} \{(\exists x,y).x\in \alpha .y\in \beta .P = x\downarrow y.R = x\downarrow \alpha \unicode{x228d} y\downarrow \beta\}.\\ +&\supset .T\in 1\rightarrow 1.\text{D}ʻT = \beta \times \alpha .\text{ᗡ}ʻT = {\in}_{\Delta}ʻ(\iota ʻ\alpha \cup \iota ʻ\beta )\end{aligned}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*21·33.&\supset \vdash \colon\ldotp \text{Hp}.\supset :\\ +&PTR.QTR.\supset .(\exists x,y,z,w).x,z\in \alpha .y,w\in \beta .P = x\downarrow y.Q = z\downarrow w.\\ +&R = x\downarrow \alpha \unicode{x228d} y\downarrow \beta = z\downarrow \alpha \unicode{x228d} w\downarrow \beta .\\ +[*113·143]&\supset .P = (Rʻ\alpha )\downarrow (Rʻ\beta ).Q = (Rʻ\alpha )\downarrow (Rʻ\beta ).\\ +[*13·172] &\supset .P = Q &\qquad \text{(1)}\\ +\vdash .*21·33.&\supset \vdash \colon\ldotp \text{Hp}.\supset :PTQ.PTR.\supset .\\ +&(\exists x,y,z,w).x,z\in \alpha .y,w\in \beta .P = x\downarrow y = w\downarrow z.Q = x\downarrow \alpha \unicode{x228d} y\downarrow \beta .R=z\downarrow \alpha \unicode{x228d} w\downarrow \beta .\\ +[*113·143]&\supset .Q = DʻP\uparrow \iota ʻ\alpha \unicode{x228d} \text{ᗡ}ʻP\uparrow \iota ʻ\beta .R = \text{D}ʻP\uparrow \iota ʻ\alpha \unicode{x228d} \text{ᗡ}ʻP\uparrow \iota ʻ\beta .\\ +[*13·172] &\supset .Q = R &\qquad \text{(2)}\\ +\vdash .*33·13. &\supset \vdash :\text{Hp}.\supset .\\ +&\text{D}ʻT = \hat{P} \{(\exists R,x,y).x\in \alpha .y\in \beta .P = x\downarrow y.R = x\downarrow \alpha \unicode{x228d} y\downarrow \beta\}\\ +[*11·55.*13·19] &= \hat{P} \{(\exists x,y).x\in \alpha .y\in \beta .P = x\downarrow y\}\\ +[*113·101] &= \beta \times \alpha &\qquad \text{(3)}\\ +\vdash .*33·131.\supset \vdash :\text{Hp}.\supset .\\ +\text{ᗡ}ʻT &= \hat{R} \{(\exists P,x,y).x\in \alpha .y\in \beta .P = x\downarrow y.R = x\downarrow \alpha \unicode{x228d} y\downarrow \beta\}\\ +[*11·55.*13·19] &= \hat{R} \{(\exists x,y).x\in \alpha .y\in \beta .R = x\downarrow \alpha \unicode{x228d} y\downarrow \beta \}\\ +[*80·9] &= {\in}_{\Delta}ʻ(\iota ʻ\alpha \cup \iota ʻ\beta ) &\qquad \text{(4)}\\ +\vdash .(1).(2).(3).(4).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_112">[Pg 112]</span></p> + +<p><i>Note to</i> <a href="#*113·144">*113·144</a>. In virtue of <a href="#*113·143">*113·143</a> and *55·61 we have +\[ +\vdash \colon\ldotp \text{Hp} *113·144.\supset :PTR. \equiv .R\in {\in}_{\Delta}ʻ(\iota ʻ\alpha \cup \iota ʻ\beta ).P = (R\parallel \breve{R} )ʻ(\alpha \downarrow \beta ). +\]</p> + +<p>At a later stage (in <a href="#*150">*150</a>) we shall put +\[ +R\dagger S = (R\parallel \breve{R} )ʻS \quad\text{Df}. +\]</p> + +<p>Thus we shall have, anticipating this notation, +\[ +\vdash : \text{Hp} *113·144.\supset .T = \{\dagger(\alpha \downarrow \beta )\}\upharpoonright {\in}_{\Delta}ʻ(\iota ʻ\alpha \cup \iota ʻ\beta ). +\]</p> + +<p>Hence we have +\[ +\vdash :\alpha \neq \beta .\supset .\{\dagger(\alpha \downarrow \beta )\}\upharpoonright {\in}_{\Delta}ʻ(\iota + ʻ\alpha \cup \iota ʻ\beta )\in (\beta \times \alpha )\,\overline{\text{ sm }}\, {\in}_{\Delta}ʻ(\iota ʻ\alpha \cup \iota ʻ\beta ). +\]</p> + +<p class="nind"><b>*113·145.</b> \(\vdash :\alpha \neq \beta .\supset .\beta \times \alpha \text{ sm }{\in}_{\Delta}ʻ(\iota ʻ\alpha \cup \iota ʻ\beta ) \quad[*113·144]\)</p> + +<p class="nind"><b>*113·146.</b> \(\vdash :\alpha \neq \beta .\supset .\alpha \times \beta \text{ sm } {\in}_{\Delta}ʻ(\iota ʻ\alpha \cup \iota ʻ\beta ) \quad[*113·141·145]\)</p> + +<p class="nind"><b>*113·147.</b> \[\begin{aligned}\vdash :\text{Hp}*133·144.&\beta \times \alpha = \mu .\supset .\\ +&T = \hat{P} \hat{R} \{P\in \mu .R = \text{D}ʻP\uparrow \iota ʻsʻ\text{D}ʻʻ\mu \unicode{x228d} \text{ᗡ}ʻP\uparrow \iota ʻsʻ\text{ᗡ}ʻʻ\mu \}\end{aligned}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*113·114.\text{Transp}.\supset \vdash :\text{Hp}.P\in \mu .&\supset .\exists !\alpha .\exists !\beta .\\ +[*113·142.*53·22]&\supset .\alpha = sʻ\text{D}ʻʻ\mu .\beta = sʻ\text{ᗡ}ʻʻ\mu &\qquad \text{(1)}\\ +\vdash .*133·101·143.&\supset \vdash \colon\ldotp \text{Hp}.P\in \mu .\supset :PTR. \equiv .R = \text{D}ʻP\uparrow \iota ʻ\alpha \unicode{x228d} \text{ᗡ}ʻP\uparrow \iota ʻ\beta &\qquad \text{(2)}\\ +\vdash .*113·144.&\supset \vdash :\text{Hp}.PTR.\supset .P\in \mu &\qquad \text{(3)}\\ +\vdash .(1).(2).(3).*113·101.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p>The advantage of this proposition is that it exhibits the correlator of +\(\beta \times \alpha\) and \(({\in}_{\Delta}ʻ(\iota ʻ\alpha \cup \iotaʻ\beta)\) +as a function of \(\beta \times \alpha\).</p> + +<p class="nind"><b>*113·148.</b> \(\vdash :\alpha \cap \beta = \Lambda .\supset .C\upharpoonright (\alpha \times \beta )\in 1\rightarrow 1\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*113·101.*55·15.\supset \\ +&\vdash \colon\ldotp \text{Hp}.\supset :R,S\in \alpha \times \beta .CʻR = CʻS.\equiv .\\ +&(\exists x,x',y,y').x,x'\in \alpha .y,y'\in \beta .R = y\downarrow x,S = y'\downarrow x'.\iota ʻx\cup \iota ʻy = \iota ʻx'\cup \iota ʻy'.\\ +[*54·6] &\supset .(\exists x,x',y,y').x,x'\in \alpha .y,y'\in \beta .R = y\downarrow x.S = y'\downarrow x'.x = x'.y = y'.\\ +[*13·22·172] &\supset .R = S &\qquad \text{(1)}\\ +\vdash .(1).*71·55.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*113·15.</b> \(\vdash .Cʻʻ(\alpha \times \beta ) = Cʻʻ(\beta \times a) = \hat{\xi} \{(\exists x,y).x\in \alpha .y\in \beta .\xi = \iota ʻx\cup \iota ʻy\}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*113·1.*40·38.\supset \vdash .Cʻʻ(\beta \times \alpha ) &= sʻCʻʻʻ\alpha \downarrow_{,,}ʻʻ\beta \\ +[*40·4] &= \hat{\xi} \{(\exists y).y\in \beta .\xi \in Cʻʻ\alpha \downarrow_{,,}y\}\\ +[*55·27.*38·2] &= \hat{\xi} \{(\exists x,y).x\in \alpha .y\in \beta .\xi = \iota ʻx\cup \iota ʻy\} &\qquad \text{(1)}\\ +\vdash .(1)\frac{\beta,\,\alpha}{\alpha,\,\beta}. \supset \vdash .Cʻʻ(\alpha \times \beta ) &= \hat{\xi} \{(\exists x,y).x\in \alpha .y\in \beta .\xi = \iota ʻx\cup \iota ʻy\} &\qquad \text{(2)}\\ +\vdash .(1).(2).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_113">[Pg 113]</span></p> + +<p class="nind"><b>*113·151.</b> \(\vdash :\alpha \neq \beta .\supset .Cʻʻ(\alpha \times \beta ) = \text{D}ʻʻ{\in}_{\Delta}ʻ(\iota ʻ\alpha \cup \iota ʻ\beta ) \quad[*113·15.*80·92]\)</p> + +<p class="nind"><b>*113·152.</b> \(\vdash :\alpha \cap \beta = \Lambda .\supset .Cʻʻ(\alpha \times \beta )\text{ sm }(\alpha \times \beta ).\text{D}ʻʻ{\in}_{\Delta}ʻ(\iota + ʻ\alpha \cup \iota ʻ\beta )\text{ sm }(\alpha \times \beta )\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*84·41·62.&\supset \vdash :\text{Hp}.\alpha \neq \beta .\supset .\text{D}ʻʻ{\in}_{\Delta}ʻ(\iota ʻ\alpha \cup \iota ʻ\beta )\text{ sm }{\in}_{\Delta}ʻ(\iota + ʻ\alpha \cup \iota ʻ\beta ) &\qquad \text{(1)}\\ +\vdash .(1).*113·146·151.\supset \\ +\vdash :\text{Hp}.\alpha \neq \beta .&\supset .Cʻʻ(\alpha \times \beta )\text{ sm }(\alpha \times \beta ).\text{D}ʻʻ{\in}_{\Delta}ʻ(\iota + ʻ\alpha \cup \iota ʻ\beta )\text{ sm }\alpha \times \beta &\qquad \text{(2)}\\ +\vdash .*24·38.\supset \vdash :\text{Hp}.\alpha = \beta .&\supset .\alpha = \Lambda .\beta = \Lambda .\\ +[*113·114.*83·11.*37·29] &\supset .\alpha \times \beta = \Lambda .\text{D}ʻʻ{\in}_{\Delta}ʻ(\iota ʻ\alpha \cup \iota ʻ\beta ) = \Lambda .Cʻʻ(\alpha \times \beta )=\Lambda .\\ +[*73·47] &\supset .Cʻʻ(\alpha \times \beta )\text{ sm }(\alpha \times \beta ).\text{D}ʻʻ{\in}_{\Delta}ʻ(\iota + ʻ\alpha \cup \iota ʻ\beta )\text{ sm }(\alpha \times \beta ) &\qquad \text{(3)}\\ +\vdash .(2).(3).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p>The following proposition is only significant when \(\lambda\) and +\(\mu\) are classes of relations. It is used in relation-arithmetic +(<a href="#*172·34">*172·34</a>).</p> + +<p class="nind"><b>*113·153.</b> + \(\vdash :\dot{s} ʻ\lambda \dot{\cap} \dot{s} ʻ\mu = \dot{\Lambda} .\supset .\dot{s} \mid C\upharpoonright (\lambda \times \mu )\in (sʻ\lambda \unicode{x228d}_{,,}ʻʻ\mu )\,\overline{\text{ sm }}\, (\lambda \times \mu ).sʻ\lambda \unicode{x228d}_{,,}ʻʻ\mu \text{ sm }\lambda \times \mu\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*55·15.*53·13.&\supset \vdash :R = T\downarrow S.\supset .\dot{s} ʻCʻR = S\unicode{x228d} T &\qquad \text{(1)}\\ +\vdash .(1).*113·101.\supset \\ +&\vdash :R,R'\in \lambda \times \mu .\dot{s} ʻCʻR = \dot{s} ʻCʻR'.\supset.\\ +&(\exists S,S',T,T').S,S'\in \lambda .T,T'\in \mu .R = T\downarrow S.R'=T'\downarrow S'.S\unicode{x228d} T=S'\unicode{x228d} T' &\qquad \text{(2)}\\ +\vdash .(2).*25·48.*41·13.\supset \\ +&\vdash \colon\ldotp \text{Hp}.\supset :R,R'\in \lambda \times \mu .\dot{s} ʻCʻR = \dot{s} ʻCʻR'.\supset .R = R' &\qquad \text{(3)}\\ +\vdash .(1).*113·101.&\supset \vdash .\dot{s} ʻʻCʻʻ(\lambda \times \mu ) = \hat{M} \{(\exists S,T).S\in \lambda .T\in \mu .M = S\unicode{x228d} T\}\\ +[*40·7] & =sʻ\lambda \unicode{x228d}_{,,}ʻʻ\mu &\qquad \text{(4)}\\ +\vdash .(3).(4).*73·25.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*113·16.</b> \[\begin{aligned}\vdash :&tʻ\alpha = tʻ\beta .\supset .\text{Nc}ʻ(\alpha \times \beta ) =\\ +&\hat{\xi} \{(\exists \gamma ,\delta ).\gamma \in \text{N}^{1}\text{c}ʻ\alpha .\delta \in \text{N}^{1}\text{c}ʻ\beta + .\gamma \cap \delta = \Lambda .\xi \text{ sm }\text{D}ʻʻ{\in}_{\Delta}ʻ(\iota ʻ\gamma \cup \iota ʻ\delta )\}\end{aligned}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*113·152.\supset \vdash \colon\ldotp \gamma \in \text{N}^{1}\text{c}ʻ\alpha .&\delta \in \text{N}^{1}\text{c}ʻ\beta .\gamma \cap \delta = \Lambda .\supset :\\ +\xi \text{ sm }\text{D}ʻʻ{\in}_{\Delta}ʻ(\iota ʻ\gamma \cup \iota ʻ\delta ). &\equiv .\xi \text{ sm }(\gamma \times \delta ).\\ +[*113·13.*104·101] &\equiv .\xi \text{ sm }(\alpha \times \beta ).\\ +[*100·31] &\equiv .\xi \in \text{Nc}ʻ(\alpha \times \beta ) &\qquad \text{(1)}\\ +\vdash .(1).*5·32.*11·11·341.\supset \\ +\vdash \colon\ldotp (\exists \gamma ,\delta ).\gamma \in &\text{N}^{1}\text{c}ʻ\alpha .\delta \in \text{N}^{1}\text{c}ʻ\beta + .\gamma \cap \delta = \Lambda .\xi \text{ sm }\text{D}ʻʻ{\in}_{\Delta}ʻ(\iota ʻ\gamma \cup \iota ʻ\delta ). \equiv :\\ +&(\exists \gamma ,\delta ).\gamma \in \text{N}^{1}\text{c}ʻ\alpha .\delta \in \text{N}^{1}\text{c}ʻ\beta + .\gamma \cap \delta = \Lambda .\xi \in \text{Nc}ʻ(\alpha \times \beta ):\\ +[*11·45] &\equiv :(\exists \gamma ,\delta ).\gamma \in \text{N}^{1}\text{c}ʻ\alpha .\delta \in \text{N}^{1}\text{c}ʻ\beta + .\gamma \cap \delta = \Lambda :\xi \in \text{Nc}ʻ(\alpha \times \beta ) &\qquad \text{(2)}\\ +\vdash .(2). *104·43.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_114">[Pg 114]</span></p> + +<p class="nind"><b>*113·17.</b> \(\vdash .\beta \times \alpha \in tʻtʻ(\alpha \uparrow \beta)\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *113·115.*41·13.\supset \vdash :R\in \beta \times \alpha .&\supset .R\,\unicode{x2abd}\, \alpha \uparrow \beta .\\ +[*64·201] &\supset .R\in tʻ(\alpha \uparrow \beta ) &\qquad \text{(1)}\\ +\vdash .(1).*63·5.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*113·171.</b> \(\vdash :\alpha \cap \beta = \Lambda .\supset .\exists !\text{Nc}(tʻ\alpha )ʻ(\alpha \times \beta )\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*113·152·15.&\supset \vdash :\text{Hp}.\supset .\hat{\xi}\{(\exists x,y).x\in \alpha .y\in \beta .\xi = \iota ʻx\cup \iota ʻy\}\in \text{Nc}ʻ(\alpha \times \beta ) &\qquad \text{(1)}\\ +\vdash .*51·16. & \supset \vdash :x\in \alpha .y\in \beta .\xi = \iota ʻx\cup \iota ʻy.\supset .x\in \alpha .x\in \xi .\\ +[*63·13] &\supset .\xi \in tʻ\alpha &\qquad \text{(2)}\\ +\vdash .(2).*11·11·35.\supset \\ +&\vdash .\hat{\xi}\{(\exists x,y).x\in \alpha .y\in \beta .\xi = \iota ʻx\cup \iota ʻy\}\subset tʻ\alpha .\\ +[*63·5] &\supset \vdash .\hat{\xi} \{(\exists x,y).x\in \alpha .y\in \beta .\xi = \iota ʻx\cup \iota ʻy\}\in tʻtʻ\alpha &\qquad \text{(3)}\\ +\vdash .(1).(3).&\supset \vdash :\text{Hp}.\supset .\exists !\text{Nc}ʻ(\alpha \times \beta )\cap tʻtʻ\alpha &\qquad \text{(4)}\\ +\vdash .(4).*102·6.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p>Note that the hypothesis \(\alpha \cap \beta = \Lambda\) is only +significant when \(\alpha\) and \(\beta\) are of the same type.</p> + +<p class="nind"><b>*113·172.</b> \(\vdash :\alpha \in tʻ\beta .\supset .\exists !\text{Nc}(t^{2}ʻ\alpha )ʻ(\alpha \times \beta )\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*113·16. \supset \vdash \colon\ldotp \text{Hp}.\supset :\gamma \in \text{N}^{1}\text{c}ʻ\alpha .&\delta \in \text{N}^{1}\text{c}ʻ\beta + .\gamma \cap \delta = \Lambda .\supset .\\ +&\text{D}ʻʻ{\in}_{\Delta}ʻ(\iota ʻ\gamma \cup \iota ʻ\delta )\in \text{Nc}ʻ(\alpha \times \beta ) &\qquad \text{(1)}\\ +\vdash .(1).*104·43.&\supset \vdash :\text{Hp}.\supset .\\ +&(\exists \gamma ,\delta ).\gamma \in \text{N}^{1}\text{c}ʻ\alpha .\delta \in \text{N}^{1}\text{c}ʻ\beta .\text{D}ʻʻ{\in}_{\Delta}ʻ(\iota + ʻ\gamma \cup \iota ʻ\delta )\in \text{Nc}ʻ(\alpha \times \beta ) &\qquad \text{(2)}\\ +\vdash .*104·1.\supset \vdash :\gamma \in \text{N}^{1}\text{c}ʻ\alpha .&\supset .\gamma \in t^{2}ʻ\alpha .\\ +[*63·61·621] &\supset .\iota ʻ\gamma \cup \iota ʻ\delta \in tʻt^{2}ʻ\alpha .\\ +[*83·81] &\supset .\text{D}ʻʻ{\in}_{\Delta}ʻ(\iota ʻ\gamma \cup \iota ʻ\delta )\in tʻt^{2}ʻ\alpha &\qquad \text{(3)}\\ +\vdash .(2).(3).&\supset \vdash :\text{Hp}.\supset .\exists !\text{Nc}ʻ(\alpha \times \beta )\cap tʻt^{2}ʻ\alpha &\qquad \text{(4)}\\ +\vdash .(4).*102·6.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*113·18.</b> \(\vdash :\exists !\alpha .\exists !\beta .\alpha \times \beta = \alpha' \times \beta'.\supset .\alpha = \alpha'.\beta = \beta'\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*113·114.\supset \vdash :\text{Hp}.&\supset .\exists !\alpha' \times \beta'.\\ +[*113·114] &\supset .\exists !\alpha'.\exists !\beta' &\qquad \text{(1)}\\ +\vdash .*30·37. \supset \vdash :\text{Hp}.&\supset .sʻ\text{ᗡ}ʻʻ(\alpha \times \beta ) = sʻ\text{ᗡ}ʻʻ(\alpha' \times \beta').\\ +[*113·142.(1)] & \supset .sʻtʻʻ\alpha = sʻ\iota ʻʻ\alpha'.\\ +[*53·22] &\supset .\alpha = \alpha' &\qquad \text{(2)}\\ +\text{Similarly}\quad \vdash :\text{Hp}.&\supset .\beta = \beta' &\qquad \text{(3)}\\ +\vdash .(2).(3).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_115">[Pg 115]</span></p> + +<p class="nind"><b>*113·181.</b> \(\vdash :\exists !\alpha .\exists !\alpha'.\alpha \times \beta = \alpha' \times \beta'.\supset .\beta = \beta'\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *13·172.&\supset \vdash :\beta = \Lambda .\beta' = \Lambda .\supset .\beta = \beta' &\qquad \text{(1)}\\ +\vdash . *113·18.&\supset \vdash :\text{Hp}.{\sim}(\beta = \Lambda .\beta' = \Lambda ).\supset .\beta = \beta' &\qquad \text{(2)}\\ +\vdash .(1).(2).&\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*113·182.</b> \[\begin{aligned}\vdash :&\exists !\beta .\exists !\beta'.\alpha \times \beta = \alpha' \times \beta'.\supset .\alpha = \alpha'\\ +&[\text{Proof as in *113·181}]\end{aligned}\]</p> + +<p class="nind"><b>*113·183.</b> \(\vdash :\exists !\alpha .\exists !\beta .\supset .Fʻʻ(\alpha \times \beta ) = sʻCʻʻ(\alpha \times \beta ) = \alpha \cup \beta\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*40·57. \supset \vdash .sʻCʻʻ(\alpha \times \beta ) = sʻ\text{D}ʻʻ(\alpha \times \beta )\cup sʻ\text{ᗡ}ʻʻ(\alpha \times \beta ) &\qquad \text{(1)}\\ +\vdash .*40·56. \supset \vdash .Fʻʻ(\alpha \times \beta ) = sʻCʻʻ(\alpha \times \beta ) &\qquad \text{(2)}\\ +\vdash . *113·142. \supset \vdash :\text{Hp}.\supset .sʻ\text{ᗡ}ʻʻ(\alpha \times \beta ) = sʻ\iotaʻʻ\alpha\\ +[*53·22] = \alpha &\qquad \text{(3)}\\ +\vdash . *113·142. \supset \vdash :\text{Hp}.\supset .sʻ\text{D}ʻʻ(\alpha \times \beta ) = sʻ\iotaʻʻ\beta \\ +[*53·22] = \beta &\qquad \text{(4)}\\ +\vdash .(3).(4). \supset \vdash :\text{Hp}.\supset .sʻ\text{D}ʻʻ(\alpha \times \beta )\cup sʻ\text{ᗡ}ʻʻ(\alpha \times \beta ) = \alpha \cup \beta &\qquad \text{(5)}\\ +\vdash .(1).(2).(5).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*113·19.</b> \(\vdash :\exists !(\alpha \times \beta )\cap (\gamma \times \delta ). \equiv .\exists !\alpha \cap \gamma .\exists !\beta \cap \delta\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *113·101.&\supset \vdash \colon\ldotp \exists !(\alpha \times \beta )\cap (\gamma \times \delta ). \equiv :\\ +&(\exists x,y,z,w).x\in \alpha .y\in \beta .z\in \gamma .w\in \delta .x\downarrow y = w\downarrow z:\\ +[*55·202]&\equiv :(\exists x,y,z,w).x\in \alpha .y\in \beta .z\in \gamma .w\in \delta .x = z.y = w:\\ +[*13·22] &\equiv :(\exists x,y).x\in \alpha \cap \gamma .y\in \beta \cap \delta \colon\ldotp \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*113·191">*113·191</a>.</b> \(\vdash \colon\ldotp \exists !\alpha .\supset :\exists !\alpha \downarrow_{,,}ʻʻ\beta \cap \alpha \downarrow_{,,}ʻʻ\gamma + . \equiv .\exists !\beta \cap \gamma\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *37·6.&\supset \vdash :\exists !\alpha \downarrow_{,,}ʻʻ\beta \cap \alpha \downarrow_{,,}ʻʻ\gamma . \equiv .(\exists y,z).y\in \beta .z\in \gamma .a\downarrow_{,,}y + = \alpha \downarrow_{,,}z &\qquad \text{(1)}\\ +\vdash . *113·105.*71·57.\supset \vdash \colon\ldotp \text{Hp}.&\supset :\alpha \downarrow_{,,}y = \alpha \downarrow_{,,}z. \equiv .y = z:\\ +[(1)] \supset :\exists !\alpha \downarrow_{,,}ʻʻ\beta \cap \alpha \downarrow_{,,}ʻʻ\gamma . &\equiv .(\exists y,z).y\in \beta .z\in \gamma .y = z.\\ +[*13·195] &\equiv .\exists !\alpha \cap \beta \colon\ldotp \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*113·2">*113·2</a>.</b> \[\begin{aligned}&\vdash :\xi \in \mu \times _{c} \nu . \equiv .(\exists \alpha ,\beta ).\mu = \text{N}_{0}\text{c}ʻ\alpha .\nu = \text{N}_{0}\text{c}ʻ\beta + .\xi \text{ sm }(\alpha \times \beta )\\ +&[(*113·03)]\end{aligned}\]</p> + +<p class="nind"><b>*113·201.</b> \[\begin{aligned}&\vdash \colon\ldotp \xi \in \mu \times _{c} + \nu . \equiv :\mu ,\nu \in \text{NC}:(\exists \alpha ,\beta ).\alpha \in \mu .\beta \in \nu .\xi \text{ sm }(\alpha \times \beta )\\ +&[*113·2.*103·27]\end{aligned}\]</p> + +<p><span class="pagenum" id="Page_116">[Pg 116]</span></p> + +<p class="nind"><b>*113·202.</b> \(\vdash \colon\ldotp \xi \in \mu \times _{c} + \nu . \equiv :\exists !\mu .\exists !\nu :(\exists \gamma ,\delta ).\mu = \text{Nc}ʻ\gamma .\nu = \text{Nc}ʻ\delta .\xi \text{ sm }(\gamma \times \delta )\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*113·201.*100·4.\supset \\ +\vdash \colon\ldotp \xi \in \mu \times _{c} + \nu . &\equiv :(\exists \alpha ,\beta ,\gamma ,\delta ).\mu = \text{Nc}ʻ\gamma .\nu = \text{Nc}ʻ\delta .\alpha \in \mu .\beta \in \nu.\xi \text{ sm }(\alpha \times \beta ).\\ +[*100·31] &\equiv : (\exists \alpha ,\beta ,\gamma ,\delta ).\mu = \text{Nc}ʻ\gamma .\nu = \text{Nc}ʻ\delta .\alpha \text{ sm }\gamma .\beta \text{ sm }\delta .\xi \text{ sm }(\alpha \times \beta ).\\ +[*113·13.*73·37] &\equiv :(\exists \alpha ,\beta ,\gamma ,\delta ).\mu = \text{Nc}ʻ\gamma .\nu = \text{Nc}ʻ\delta .\alpha \text{ sm }\gamma .\beta \text{ sm }\delta .\xi \text{ sm }(\gamma \times \delta ).\\ +[*100·31] &\equiv :(\exists \alpha ,\beta ,\gamma ,\delta ).\mu = \text{Nc}ʻ\gamma .\nu = \text{Nc}ʻ\delta .\alpha \in \mu .\beta \in \nu .\xi \text{ sm }(\gamma \times \delta ).\\ +[*10·35] &\equiv :\exists !\mu .\exists !\nu :(\exists \gamma ,\delta ).\mu = \text{Nc}ʻ\gamma .\nu = \text{Nc}ʻ\delta .\xi \text{ sm }(\gamma \times \delta )\colon\ldotp \\ +&\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*113·203.</b> \(\vdash :\exists !\mu \times _{c} \nu .\supset .\mu ,\nu \in \text{NC} - \iota ʻ\Lambda .\mu ,\nu \in \text{N}_{0}\text{C} \quad[*113·201·202·2]\)</p> + +<p class="nind"><b><a id="*113·204">*113·204</a>.</b> \(\vdash \colon\ldotp \mu = \Lambda .\lor.\nu = \Lambda .\lor.{\sim}(\mu ,\nu \in \text{NC}):\supset .\mu \times _{c} \nu = \Lambda \quad[*113·203]\)</p> + +<p class="nind"><b>*113·205.</b> \(\vdash :{\sim}(\mu ,\nu \in \text{N}_{0}\text{C}).\supset .\mu \times _{c} \nu = \Lambda \quad[*113·203]\)</p> + +<p class="nind"><b>*113·21.</b> \[\begin{aligned}&\vdash \colon\ldotp \mu ,\nu \in \text{NC}.\supset :\xi \in \mu \times _{c} + \nu . \equiv .(\exists \alpha ,\beta ).\alpha \in \mu .\beta \in \nu .\xi \text{ sm }(\alpha \times \beta )\\ +&[*113·201]\end{aligned}\]</p> + +<p class="nind"><b>*113·22.</b> \(\vdash :\xi \in \text{Nc}(\eta )ʻ\gamma \times _{c} + \text{Nc}(\zeta )ʻ\delta . \equiv .\exists !\text{Nc}(\eta )ʻ\gamma .\exists !\text{Nc}(\zeta )ʻ\delta .\xi \text{ sm }(\gamma \times \delta )\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*113·21.*100·41.&\supset \vdash :\xi \in \text{Nc}(\eta )ʻ\gamma \times _{c} \text{Nc}(\zeta )ʻ\delta . \equiv .\\ +&(\exists \alpha ,\beta ).\alpha \in \text{Nc}(\eta )ʻ\gamma .\beta \in \text{Nc}(\zeta )ʻ\delta .\xi \text{ sm }(\alpha \times \beta ).\\ +[*102·6] &\equiv .(\exists \alpha ,\beta ).\alpha \in \text{Nc}(\eta )ʻ\gamma .\beta \in \text{Nc}(\zeta )ʻ\delta .\alpha \text{ sm }\gamma .\beta \text{ sm }\delta .\xi \text{ sm }(\alpha \times \beta ).\\ +[*113·13.*73·37] &\equiv .(\exists \alpha ,\beta ).\alpha \in \text{Nc}(\eta )ʻ\gamma .\beta \in \text{Nc}(\zeta )ʻ\delta .\alpha \text{ sm }\gamma .\beta \text{ sm }\delta .\xi \text{ sm }(\gamma \times \delta ).\\ +[*102·6] &\equiv .(\exists \alpha ,\beta ).\alpha \in \text{Nc}(\eta )ʻ\gamma .\beta \in \text{Nc}(\zeta )ʻ\delta .\xi \text{ sm }(\gamma \times \delta ).\\ +[*10·35] &\equiv .\exists !\text{Nc}(\eta )ʻ\gamma .\exists !\text{Nc}(\zeta )ʻ\delta .\xi \text{ sm }(\gamma \times \delta ):\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*113·221.</b> \[\begin{aligned}&\vdash :\exists !\text{Nc}(\eta )ʻ\gamma .\exists !\text{Nc}(\zeta )ʻ\delta .\supset .\text{Nc}(\eta )ʻ\gamma \times _{c} + \text{Nc}(\zeta )ʻ\delta = \text{Nc}ʻ(\gamma x \delta )\\ +&[*113·22]\end{aligned}\]</p> + +<p class="nind"><b>*113·222.</b> \(\vdash .\text{N}_{0}\text{c}ʻ\gamma \times _{c} \text{N}_{0}\text{c}ʻ\delta = \text{Nc}ʻ(\gamma \times \delta )\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*103·1·13.&\supset \vdash .\text{N}_{0}\text{c}ʻ\gamma = \text{Nc}(\gamma )ʻ\gamma .\text{N}_{0}\text{c}ʻ\delta = \text{Nc}(\delta )ʻ\delta .\exists !\text{N}_{0}\text{c}ʻ\gamma + .\exists !\text{N}_{0}\text{c}ʻ\delta .\\ +[*113·221] &\supset \vdash .\text{N}_{0}\text{c}ʻ\gamma \times _{c} \text{N}_{0}\text{c}ʻ\delta = \text{Nc}ʻ(\gamma x \delta ).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*113·23">*113·23</a>.</b> \(\vdash .\mu \times _{c} \nu \in \text{NC}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*113·222.*100·41.&\supset \vdash :\mu ,\nu \in \text{N}_{0}\text{C}.\supset .\mu \times _{c} \nu \in \text{NC} &\qquad \text{(1)}\\ +\vdash .*113·205.*102·74.&\supset \vdash :{\sim}(\mu ,\nu \in \text{N}_{0}\text{C}).\supset .\mu \times _{c} \nu \in \text{NC} &\qquad \text{(2)}\\ +\vdash .(1).(2).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*113·24.</b> \(\vdash .\text{Nc}ʻ\gamma \times _{c} \text{Nc}ʻ\delta = \text{N}_{0}\text{c}ʻ\gamma \times _{c} \text{N}_{0}\text{c}ʻ\delta + \quad[(*113·04·05)]\)</p> + +<p class="nind"><b>*113·25.</b> \(\vdash .\text{Nc}ʻ\gamma \times _{c} \text{Nc}ʻ\delta = \text{Nc}ʻ(\gamma \times \delta )\quad [*113·24·222]\)</p> + +<p><span class="pagenum" id="Page_117">[Pg 117]</span></p> + +<p>This proposition constitutes part of the reason for our definitions. It +is obvious that such definitions ought, if possible, to be chosen as +will yield this proposition.</p> + +<p class="nind"><b>*113·251.</b> \(\vdash .\gamma \times \delta \in \text{Nc}ʻ\gamma \times _{c} \text{Nc}ʻ\delta \quad[*113·25.*100·3]\)</p> + +<p class="nind"><b><a id="*113·26">*113·26</a>.</b> \(\vdash :\mu ,\nu \in \text{NC}.\exists !\text{ sm }_{\eta }ʻʻ\mu .\exists !\text{ sm }_{\zeta }ʻʻ\nu .\supset .\mu \times _{c} \nu = \text{ sm }_{\eta }ʻʻ\mu + \times _{c} \text{ sm }_{\zeta }ʻʻ\nu\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*37·29. \text{Transp}.\supset \vdash :\text{Hp}.&\supset .\exists !\mu .\exists !\nu .\\ +[*102·64] &\supset .(\exists \alpha ,\beta ,\gamma ,\delta ).\mu = \text{Nc}(\alpha )ʻ\gamma .\nu = \text{Nc}(\beta )ʻ\delta &\qquad \text{(1)}\\ +\vdash .*102·88.&\supset \vdash :\mu = \text{Nc}(\alpha )ʻ\gamma .\nu = \text{Nc}(\beta )ʻ\delta .\exists !\text{ sm }_{\eta }ʻʻ\mu .\exists !\text{ sm }_{\zeta }ʻʻ\nu .\supset .\\ +&\text{ sm }_{\eta }ʻʻ\mu = \text{Nc}(\eta )ʻ\gamma .\text{ sm }_{\zeta }ʻʻ\nu = \text{Nc}(\zeta )ʻ\delta .\exists !\text{Nc}(\eta )ʻ\gamma .\exists !\text{Nc}(\zeta )ʻ\delta .\\ +[*113·221]&\supset .\text{ sm }_{\eta }ʻʻ\mu \times _{c} \text{ sm }_{\zeta }ʻʻ\nu = \text{Nc}ʻ(\gamma \times \delta ) &\qquad \text{(2)}\\ +\vdash .*37·29.\text{Transp}.*113·221.\supset \\ +\vdash :\mu &= \text{Nc}(\alpha )ʻ\gamma .\nu = \text{Nc}(\beta )ʻ\delta .\exists !\text{ sm }_{\eta }ʻʻ\mu .\exists !\text{ sm }_{\zeta }ʻʻ\nu .\supset .\mu \times _{c} + \nu = \text{Nc}ʻ(\gamma \times \delta ) &\qquad \text{(3)}\\ +\vdash .(2).(3).\supset \vdash :\mu &= \text{Nc}(\alpha )ʻ\gamma .\nu = \text{Nc}(\beta )ʻ\delta .\exists !\text{ sm }_{\eta }ʻʻ\mu .\exists !\text{ sm }_{\varsigma }ʻʻ\nu + .\supset .\\&\mu \times _{c} \nu = \text{ sm }_{\eta }ʻʻ\mu \times _{c} \text{ sm }_{\varsigma }ʻʻ\nu &\qquad \text{(4)}\\ +\vdash .(4).*11·11·35·45.(1).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*113·261.</b> \(\vdash :\mu ,\nu \in \text{NC}.\supset .\mu \times _{c} \nu = \mu ^{(1)} \times _{c} \nu ^{(1)} = \mu _{(00)} \times _{c} + \nu _{(00}) = \text{etc}.\)</p> + +<p>Here "\(\text{etc}\)." includes all ascending derivatives of \(\mu\). +We shall only prove the result for \(\mu ^{(1)}\) and \(\nu ^{(1)}\), +since it is proved in just the same way for the other cases. +\(\mu^{(1)} \times _{c} \nu ^{(2)}\) or \(\mu ^{(1)} \times _{c} \nu_{(00)}\) +or etc. will serve equally well; <i>i.e.</i> it is not necessary to +take the same derivative of \(\mu\) as of \(\nu\).</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*104·264·265.\supset \\ +\vdash :\text{Hp}.\exists !\mu .\exists !\nu .\supset .\mu ^{(1)} = \text{ sm }_{\mu }ʻʻ\mu .\nu ^{(1)} = \text{ sm }_{\nu }ʻʻ\mu .\exists !\mu ^{(1)}.\exists + !\nu ^{(1)}.\\ +[*113·26] \supset :\mu \times _{c} \nu = \mu ^{(1)} \times _{c} \nu ^{(1)} &\qquad \text{(1)}\\ +\vdash .*104·264.*113·204.\supset\\ +\vdash :{\sim}(\exists !\mu .\exists !\nu ).\supset .\mu \times _{c} \nu = \Lambda .\mu ^{(1)} \times _{c} \nu ^{(1)} = \Lambda &\qquad \text{(2)}\\ +\vdash .(1).(2).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p>As appears in the above proof, if \(\mu^{i}\) and \(\iota^{j}\) are any +derivatives of \(\mu\) and \(\nu\), the above proposition holds +provided we have +\[ +\exists !\mu .\exists !\nu .\supset .\exists !\mu^{i}.\exists !\nu^{j} . +\]</p> + +<p>Thus it holds for all ascending derivatives, but not always for +descending derivatives.</p> + +<p class="nind"><b><a id="*113·27">*113·27</a>.</b> \(\vdash .\mu \times _{c} \nu = \nu \times _{c} \mu\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*113·2·141.\supset \\ +\vdash :\xi \in \mu \times _{c} \nu . &\equiv .(\exists \alpha ,\beta ).\mu = \text{N}_{0}\text{c}ʻ\alpha .\nu = \text{N}_{0}\text{c}ʻ\beta + .\xi \text{ sm }(\beta \times \alpha ).\\ +[*113·2] &\equiv .\xi \in \nu \times _{c} \mu :\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p>Note that this proposition is not confined to the case in which \(\mu\) +and \(\nu\) are cardinals. When either or both are not cardinals, +\[ +\mu \times _{c} \nu = \Lambda = \nu \times _{c} \mu . +\]</p> + +<p><span class="pagenum" id="Page_118">[Pg 118]</span></p> + +<p class="nind"><b><a id="*113·3">*113·3</a>.</b> \(\vdash \colon\ldotp \text{Mult ax}.\supset :\kappa \in \text{Nc}ʻ\beta \cap \text{Cl}ʻ\text{Nc}ʻ\alpha .\supset .\Sigma ʻ\kappa \in \text{Nc}ʻ\alpha \times _{c} + \text{Nc}ʻ\beta\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*112·24.*113·12.\supset \\ +\vdash \colon\ldotp \text{Mult ax}.\exists !\alpha .\supset :\kappa \in \text{Nc}ʻ\beta \cap \text{Cl}ʻ\text{Nc}ʻ\alpha .&\supset .\Sigma ʻ\kappa \text{ sm }\Sigma ʻ\alpha \downarrow_{,,}ʻʻ\beta + .\\ +[*113·121]& \supset .\Sigma ʻ\kappa \text{ sm } \beta \times \alpha .\\ +[*113·141·25] & \supset .\Sigma ʻ\kappa \in \text{Nc}ʻ\alpha \times _{c} \text{Nc}ʻ\beta &\qquad \text{(1)}\\ +\vdash .*113·114·25.&\supset \vdash :\alpha = \Lambda .\supset .\text{Nc}ʻ\alpha \times _{c} \text{Nc}ʻ\beta = 0 &\qquad \text{(2)}\\ +\vdash .*101·14.\supset \vdash \colon\ldotp \alpha = \Lambda .\kappa \in \text{Nc}ʻ\beta \cap \text{Cl}ʻ\text{Nc}ʻ\alpha .&\supset :\kappa \in \text{Cl}ʻ\iota ʻ\Lambda :\\ +[*60·362] & \supset :\kappa = \iota ʻ\Lambda .\lor.\kappa = \Lambda: \\ +[*112·3·301] &\supset :\Sigma ʻ\kappa = \Lambda &\qquad \text{(3)}\\ +\vdash .(2).(3).*54·102.&\supset \vdash :\alpha = \Lambda.\kappa \in \text{Nc}ʻ\beta \cap \text{Cl}ʻ\text{Nc}ʻ\alpha .\supset .\Sigma ʻ\kappa \in \text{Nc}ʻ\alpha \times _{c} + \text{Nc}ʻ\beta &\qquad \text{(4)}\\ +\vdash .(1).(4).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*113·31">*113·31</a>.</b> \(\vdash \colon\ldotp \text{Mult ax}.\supset :\mu ,\nu \in \text{NC}.\kappa \in \nu \cap \text{Cl}ʻ\mu .\supset .\Sigma ʻ\kappa \in \mu \times _{c} + \nu \quad[*113·3]\)</p> + +<p class="nind"><b>*113·32.</b> \[\begin{aligned}&\vdash \colon\ldotp \text{Mult ax}.\supset :\mu ,\nu \in \text{NC}.\kappa \in \nu \cap \text{Cl excl}ʻ\mu .\supset .sʻ\kappa \in \mu \times _{c} + \nu \\ +&[*112·15.*113·31·23]\end{aligned}\]</p> + +<p class="nind"><b>*113·33.</b> \[\begin{aligned}\vdash \colon\ldotp \text{Mult ax} .\supset :\mu ,\nu \in &\text{NC}.\kappa \in \nu \cap \text{Cl}ʻ\mu .\lambda\in \mu \cap \text{Cl}ʻ\nu .\supset .\\ +&\Sigma \text{Nc}ʻ\kappa = \Sigma \text{Nc}ʻ\lambda = \mu \times _{c} \nu \quad[*113·31·27·23]\end{aligned}\]</p> + +<p class="nind"><b><a id="*113·34">*113·34</a>.</b> + \[\begin{aligned}\vdash \colon\ldotp \text{Mult ax} .\supset :\mu ,\nu \in &\text{NC}.\kappa \in \nu \cap \text{Cl excl}ʻ\mu .\lambda \in \mu \cap \text{Cl excl}ʻ\nu .\supset .\\ +&\text{Nc}ʻsʻ\kappa = \text{Nc}ʻsʻ\lambda = \mu \times _{c} \nu \quad[*113·32·27]\end{aligned}\]</p> + +<p>The above propositions give the connection of addition and +multiplication.</p> + +<p>The following propositions are concerned with various forms of the +distributive law.</p> + +<p class="nind"><b><a id="*113·4">*113·4</a>.</b> \(\vdash .(\beta \cup \gamma ) \times \alpha = (\beta \times \alpha )\cup (\gamma \times \alpha )\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*113·1.\supset \vdash .(\beta \cup \gamma ) \times \alpha &= sʻ\alpha \downarrow_{,,}ʻʻ(\beta \cup \gamma )\\ +[*40·31] & = sʻ\alpha \downarrow_{,,}ʻʻ\beta \cup sʻ\alpha \downarrow_{,,}ʻʻ\gamma \\ +[*113·1] &= (\beta \times \alpha )\cup (\gamma \times \alpha ).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*113·401.</b> \(\vdash :\beta \cap \gamma = \Lambda .\supset .(\beta \times \alpha )\cap (\gamma \times \alpha ) = \Lambda \quad[*113·19.\text{Transp}]\)</p> + +<p class="nind"><b>*113·41.</b> \[\begin{aligned}\vdash .\text{Nc}ʻ(\beta + \gamma ) \times _{c} + \text{Nc}ʻ\alpha = \text{Nc}ʻ\{(\beta + \gamma ) \times \alpha\} &= \text{Nc}ʻ\{(\beta \times \alpha ) + (\gamma \times \alpha )\}\\ +&= \text{Nc}ʻ(\beta \times \alpha ) +_{c} \text{Nc}ʻ(\gamma \times \alpha )\end{aligned}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *113·25.*110·3. &\supset \vdash .\text{Nc}ʻ(\beta + \gamma ) \times _{c} \text{Nc}ʻ\alpha = \text{Nc}ʻ\{(\beta + \gamma ) \times \alpha\}.\\ +&\text{Nc}ʻ\{(\beta \times \alpha ) + (\gamma \times \alpha )\} = \text{Nc}ʻ(\beta \times \alpha ) +_{c} \text{Nc}ʻ(\gamma \times \alpha ) &\qquad \text{(1)}\\ +\vdash .*113·4.(*110·01). & \supset \vdash .(\beta + \gamma ) \times \alpha = (\downarrow \Lambda _{\gamma }ʻʻ\iotaʻʻ\beta \times \alpha )\cup (\Lambda _{\beta }\downarrow + ʻʻ\iotaʻʻ\gamma \times \alpha ) &\qquad \text{(2)}\\ +\vdash .*113·13.*110·12. &\supset \vdash .\downarrow \Lambda _{\gamma }ʻʻ\iotaʻʻ\beta \times \alpha \text{ sm }\beta \times \alpha .\Lambda _{\beta }\downarrow + ʻʻ\iotaʻʻ\gamma \times \alpha \text{ sm }\gamma \times \alpha &\qquad \text{(3)}\\ +\vdash .*113·401.*110·11. & \supset \vdash .(\downarrow \Lambda _{\gamma }ʻʻ\iotaʻʻ\beta \times \alpha )\cap (\Lambda _{\beta }\downarrow + ʻʻ\iotaʻʻ\gamma \times \alpha ) = \Lambda &\qquad \text{(4)}\\ +\vdash .*110·152.(2).(3).(4). &\supset \vdash .(\beta + \gamma ) \times \alpha \text{ sm }{(\beta \times \alpha ) + (\gamma \times \alpha )} &\qquad \text{(5)}\\ +\vdash .(1).(5).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_119">[Pg 119]</span></p> + +<p class="nind"><b>*113·42.</b> \[\begin{aligned}\vdash .(\text{Nc}ʻ\beta +_{c} \text{Nc}ʻ\gamma ) \times _{c} \text{Nc}ʻ\alpha &= \text{Nc}ʻ(\beta + \gamma ) \times _{c} + \text{Nc}ʻ\alpha \\ +&= (\text{Nc}ʻ\beta \times _{c} \text{Nc}ʻ\alpha ) +_{c} (\text{Nc}ʻ\gamma \times _{c} \text{Nc}ʻ\alpha )\\ +[*110·3.*113·25.*113·41]\end{aligned}\]</p> + +<p class="nind"><b>*113·421.</b> \[\begin{aligned}\vdash . \text{Nc}ʻ\alpha \times _{c} (\text{Nc}ʻ\beta &+_{c} \text{Nc}ʻ\gamma ) = \text{Nc}ʻ\alpha \times _{c} + \text{Nc}ʻ(\beta + \gamma )\\ +&= (\text{Nc}ʻ\alpha \times _{c} \text{Nc}ʻ\beta ) +_{c} (\text{Nc}ʻ\alpha \times _{c} \text{Nc}ʻ\gamma ) \quad[*113·42·27]\end{aligned}\]</p> + +<p class="nind"><b>*113·43.</b> \(\vdash .(\nu +_{c} \varpi ) \times _{c} \mu = \mu \times _{c} (\nu +_{c} \varpi ) = (\mu \times _{c} \nu ) +_{c} + (\mu \times _{c} \varpi )\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *113·27·421 .\supset \vdash :\mu ,\nu ,\varpi \in \text{NC}.\supset .(\nu +_{c} \varpi ) \times _{c} \mu &= \mu \times _{c} (\nu +_{c} \varpi )\\ +&= (\mu \times _{c} \nu ) +_{c} (\mu \times _{c} \varpi ) &\qquad \text{(1)}\\ +\vdash .*113·204.*110·4.\supset\\ +\vdash :{\sim}(\mu ,\nu ,\varpi \in \text{NC}).\supset .(\nu +_{c} \varpi ) \times _{c} \mu = \Lambda .\mu \times _{c} &(\nu +_{c} \varpi ) = \Lambda .\\ +&(\mu \times _{c} \nu ) +_{c} (\mu \times _{c} \varpi ) = \Lambda &\qquad \text{(2)}\\ +\vdash .(1).(2).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p>The following propositions are concerned with various forms of the +distributive law, when the summands are not enumerated, but given as +the members of a class.</p> + +<p>The first of them (<a href="#*113·44">*113·44</a>) gives the distributive law with regard to +arithmetical class-multiplication and logical addition of classes.</p> + +<p class="nind"><b><a id="*113·44">*113·44</a>.</b> \(\vdash .(sʻ\kappa ) \times \alpha = sʻ(x \alpha )ʻʻ\kappa\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*113·1 .\supset \vdash .sʻ(\times \alpha )ʻʻ\kappa &= sʻsʻʻ\alpha \downarrow_{,,}ʻʻʻ\kappa \\ +[*42·1] &= sʻsʻ\alpha \downarrow_{,,}ʻʻʻ\kappa \\ +[*40·38] &= sʻ\alpha \downarrow_{,,}ʻʻsʻ\kappa \\ +[*113·1] &= (sʻ\kappa ) \times \alpha .\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*113·45.</b> \(\vdash :\kappa \in \text{Cls}^{2} \text{excl}.\supset . \times \alpha ʻʻ\kappa \in \text{Cls}^{2} \text{excl}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *113·19. &\supset \vdash :\exists ! \times \alpha ʻ\beta \cap \times \alpha ʻ\gamma .\supset .\exists !\supset \cap \gamma &\qquad \text{(1)}\\ +\vdash .(1).*84·11.\supset \vdash \colon\ldotp \text{Hp}.\supset :\beta ,\gamma \in \kappa .\exists ! \times \alpha ʻ\beta \cap \times \alpha ʻ\gamma .&\supset_{\beta,\gamma}.\beta = \gamma .\\ +[*30·37] &\supset_{\beta,\gamma}. \times \alpha ʻ\beta = \times \alpha ʻ\gamma :\\ +[*37·63] &\supset :\rho ,\sigma \in \times \alpha ʻʻ\kappa .\exists !\rho \cap \sigma .\supset _{\rho ,\sigma}.\rho = \sigma &\qquad \text{(2)}\\ +\vdash .(2).*84·11.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*113·46.</b> \(\vdash :\kappa \in \text{Cls}^{2} \text{excl}.\supset .\Sigma ʻ \times \alpha ʻʻ\kappa \text{ sm }(\Sigma ʻ\kappa ) \times \alpha\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*112·15. \supset \vdash :\text{Hp}.&\supset .\Sigma ʻ\kappa \text{ sm }sʻ\kappa .\\ +[*113·13] &\supset .(\Sigma ʻ\kappa ) \times \alpha \text{ sm }(sʻ\kappa ) \times \alpha &\qquad \text{(1)}\\ +\vdash . *112·15 . *113·45. &\supset \vdash :\text{Hp}.\supset .\Sigma ʻ \times \alpha ʻʻ\kappa \text{ sm }sʻ \times \alpha ʻʻ\kappa &\qquad \text{(2)}\\ +\vdash .(1).(2).*113·44.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_120">[Pg 120]</span></p> + +<p class="nind"><b>*113·47.</b> \[\begin{aligned}&\vdash :\kappa \in \text{Cls}^{2} + \text{excl}.\supset .\Sigma \text{Nc}ʻ \times \alpha ʻʻ\kappa = \text{Nc}ʻ{(\Sigma ʻ\kappa ) \times \alpha } = \Sigma \text{Nc}ʻ\kappa \times _{c} \text{Nc}ʻ\alpha \\ +&[*113·46]\end{aligned}\]</p> + +<p>This is the distributive law for arithmetical multiplication and +arithmetical addition of the kind defined in <a href="#*112">*112</a>.</p> + +<p class="nind"><b>*113·48.</b> \(\vdash .sʻ\alpha \times ʻʻ\kappa = \alpha \times (sʻ\kappa ) = \text{Cnv}ʻʻ\{(sʻ\kappa ) \times \alpha\}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*113·14. \supset \vdash .sʻ\alpha \times ʻʻ\kappa & = sʻ\text{Cnv}ʻʻʻ \times \alpha ʻʻ\kappa \\ +[*40·38] &= \text{Cnv}ʻʻsʻ \times \alpha ʻʻ\kappa \\ +[*113·44] &= \text{Cnv}ʻʻ{(sʻ\kappa ) \times \alpha } &\qquad \text{(1)}\\ +[*113·14] &= \alpha \times sʻ\kappa &\qquad \text{(2)}\\ +\vdash .(1).(2).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*113·49.</b> \(\vdash :\kappa \in \text{Cls}^{2} \text{excl}.\supset .\Sigma ʻ\alpha \times ʻʻ\kappa \text{ sm }\alpha \times (\Sigma ʻ\kappa )\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *113·14.&\supset \vdash .\alpha \times ʻʻ\kappa = \text{Cnv}ʻʻʻ \times \alpha ʻʻ\kappa &\qquad \text{(1)}\\ +\vdash .(1).*113·45.*72·11.*84·53.\supset \\ +\vdash :\text{Hp}.&\supset .\alpha \times ʻʻ\kappa \in \text{Cls}^{2} \text{excl}.\\ +[*112·15] &\supset . \Sigma ʻ\alpha \times ʻʻ\kappa \text{ sm }sʻ\alpha \times ʻʻ\kappa .\\ +[*113·48] &\supset .\Sigma ʻ\alpha \times ʻʻ\kappa \text{ sm }\alpha \times (sʻ\kappa ).\\ +[*112·15.*113·13] &\supset .\Sigma ʻ\alpha \times ʻʻ\kappa \text{ sm }\alpha \times (\Sigma ʻ\kappa ):\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*113·491">*113·491</a>.</b> \[\begin{aligned}&\vdash :\kappa \in \text{Cls}^{2} + \text{excl}.\supset .\Sigma \text{Nc}ʻ\alpha \times ʻʻ\kappa = \text{Nc}ʻ{\alpha \times (\Sigma ʻ\kappa )} = \text{Nc}ʻ\alpha \times _{c} \Sigma \text{Nc}ʻ\kappa \\ +&[*113·49·25]\end{aligned}\]</p> + +<p>The following propositions are concerned with the associative law for +arithmetical multiplication.</p> + +<p class="nind"><b><a id="*113·5">*113·5</a>.</b> \(\vdash .(\gamma \times \beta ) \times \alpha = \hat{R} \{(\exists x,y,z).x\in \alpha .y\in \beta .z\in \gamma .R = x\downarrow (y\downarrow z)\}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*113·101.\supset \\ +\vdash .(\gamma \times \beta ) \times \alpha &= \hat{R} \{(\exists x,P).x\in \alpha .P\in (\gamma \times \beta ).R = x\downarrow P\}\\ +[*113·101] &= \hat{R} \{(\exists x,y,z).x\in \alpha .y\in \beta .z\in \gamma .R = x\downarrow (y\downarrow z)\}.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*113·51.</b> \(\vdash .(\alpha \times \beta ) \times \gamma \text{ sm }\alpha \times (\beta \times \gamma )\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*113·141.&\supset \vdash .\alpha \times (\beta \times \gamma )\text{ sm }(\beta \times \gamma ) x \alpha &\qquad \text{(1)}\\ +\vdash .*113·5. \supset \vdash .(\alpha \times \beta ) \times \gamma &= \hat{R} \{(\exists x,y,z).x\in \alpha .y\in \beta .z\in \gamma .R = z\downarrow (y\downarrow x)\} .\\ +(\beta \times \gamma ) \times \alpha &= \hat{P} \{(\exists x,y,z).x\in \alpha .y\in \beta .z\in \gamma .P = x\downarrow (z\downarrow y)\} &\qquad \text{(2)}\\ +\vdash .(2).\supset \vdash :T &= \hat{R} \hat{P} \{(\exists x,y,z).x\in \alpha .y\in \beta .z\in \gamma .R = z\downarrow (y\downarrow x).P =x\downarrow (y\downarrow z)\}.\supset .\\ +&\text{D}ʻT = (\alpha \times \beta ) \times \gamma .\text{ᗡ}ʻT = (\beta \times \gamma ) \times \alpha &\qquad \text{(3)}\\ +\vdash .*21·33.&\supset \vdash :\text{Hp}(3).RTP.RTQ.\supset\\ +&(\exists x,x',y,y',z,z').x,x'\in \alpha .y,y'\in \beta .z,z'\in \gamma .R = z\downarrow (y\downarrow x) = z'\downarrow (y'\downarrow x').\\ +&P = x\downarrow (z\downarrow y).Q = x'\downarrow (z'\downarrow y').\\ +[*55·202]&\supset .P = Q &\qquad \text{(4)}\\ +\text{Similarly}\qquad &\vdash :\text{Hp}(3).RTP.QTP.\supset .R = Q &\qquad \text{(5)}\\ +\vdash .(3).(4).(5).&\supset \vdash .(\alpha \times \beta ) \times \gamma \text{ sm }(\beta \times \gamma ) \times \alpha &\qquad \text{(6)}\\ +\vdash .(1).(6).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_121">[Pg 121]</span></p> + +<p class="nind"><b>*113·511.</b> \(\alpha \times \beta \times \gamma = (\alpha \times \beta ) \times \gamma \quad\text{Df}\)</p> + +<p class="nind"><b>*113·52.</b> \(\vdash . (\text{Nc}ʻ\alpha \times _{c} \text{Nc}ʻ\beta ) \times _{c} \text{Nc}ʻ\gamma = \text{Nc}ʻ(\alpha \times \beta \times \gamma ) \quad[*113·25]\)</p> + +<p class="nind"><b>*113·53.</b> \(\vdash . (\text{Nc}ʻ\alpha \times _{c} \text{Nc}ʻ\beta ) \times _{c} \text{Nc}ʻ\gamma = \text{Nc}ʻ\alpha \times _{c} (\text{Nc}ʻ\beta \times _{c} + \text{Nc}ʻ\gamma )\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *113·52·51 . \supset \\ +\vdash . (\text{Nc}ʻ\alpha \times _{c} \text{Nc}ʻ\beta ) \times _{c} \text{Nc}ʻ\gamma &= \text{Nc}ʻ\{\alpha \times (\beta \times \gamma )\}\\ +[*113·25] &= \text{Nc}ʻ\alpha \times _{c} (\text{Nc}ʻ\beta \times _{c} \text{Nc}ʻ\gamma ).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*113·531.</b> \[\begin{aligned}&\vdash . (\text{N}_{0}\text{c}ʻ\alpha \times _{c} \text{N}_{0}\text{c}ʻ\beta ) \times _{c} \text{N}_{0}\text{c}ʻ\gamma + = \text{N}_{0}\text{c}ʻ\alpha \times _{c} (\text{N}_{0}\text{c}ʻ\beta \times _{c} \text{N}_{0}\text{c}ʻ\gamma )\\ +&[*113·53.(*113·04·05)]\end{aligned}\]</p> + +<p class="nind"><b><a id="*113·54">*113·54</a>.</b> \(\vdash . (\mu \times _{c} \nu ) \times _{c} \varpi = \mu \times _{c} (\nu \times _{c} \varpi )\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*113·531.*103·2.\supset \\ +\vdash :\mu ,\nu ,\varpi \in \text{N}_{0}\text{C}.\supset .(\mu \times _{c} \nu ) \times _{c} \varpi = \mu \times _{c} (\nu \times _{c} + \varpi ) &\qquad \text{(1)}\\ +\vdash .*113·204.\supset \\ +\vdash :{\sim}(\mu ,\nu ,\varpi \in \text{N}_{0}\text{C}).\supset .(\mu \times _{c} \nu ) \times _{c} \varpi = \Lambda .\mu \times _{c} + (\nu \times \varpi ) = \Lambda &\qquad \text{(2)}\\ +\vdash .(1).(2).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*113·541">*113·541</a>.</b> \(\mu \times _{c} \nu \times _{c} \varpi = (\mu \times _{c} \nu ) \times _{c} \varpi \quad\text{Df}\)</p> + +<p class="nind"><b>*113·6.</b> \(\vdash . \text{Nc}ʻ\alpha \times _{c} 0 = 0\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*113·25.*101·1.\supset \vdash .\text{Nc}ʻ\alpha \times _{c} 0 &= \text{Nc}ʻ(\alpha \times \Lambda )\\ +[*113·114.*101·1] &= 0.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*113·601.</b> \(\vdash :\mu \in \text{NC} - \iota ʻ\Lambda .\supset .\mu \times _{c} 0 = 0\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*103·26.\supset \vdash :\text{Hp}.\supset .(\exists \alpha ).\mu = \text{N}_{0}\text{c}ʻ\alpha &\qquad \text{(1)}\\ +\vdash .*101·11·13.*103·27.\supset \vdash .0 = \text{N}_{0}\text{c}ʻ\Lambda &\qquad \text{(2)}\\ +\vdash .(1).(2).\supset \vdash :\text{Hp}.\supset .(\exists \alpha ).\mu \times _{c} 0 = \text{N}_{0}\text{c}ʻ\alpha \times _{c} \text{N}_{0}\text{c}ʻ\Lambda \\ +[*113·222] = \text{Nc}ʻ(\alpha x \Lambda )\\ +[*113·114.*101·1] = 0:\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_122">[Pg 122]</span></p> + +<p class="nind"><b><a id="*113·602">*113·602</a>.</b> \(\vdash \colon\ldotp \mu \times _{c} \nu = 0. \equiv :\mu ,\nu \in \text{NC} -{℩} ʻ\Lambda :\mu = 0.\lor.\nu = 0\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*113·203.*101·12.\supset \\ +\vdash :\mu \times _{c} \nu = 0.&\supset .\mu ,\nu \in \text{NC}-\iota ʻ\Lambda &\qquad \text{(1)}\\ +\vdash .(1).*113·201.\supset\\ +\vdash \colon\colon \mu \times _{c} \nu = 0.&\supset \colon\ldotp \xi \in 0. \equiv _{\xi } + :(\exists \alpha ,\beta ).\alpha \in \mu .\beta \in \nu .\xi \text{ sm }(\alpha \times \beta )\colon\ldotp \\ +[*54·102] &\supset \colon\ldotp \xi = \Lambda . \equiv _{\xi } :(\exists \alpha ,\beta ).\alpha \in \mu .\beta \in \nu .\xi \text{ sm }(\alpha \times \beta )\colon\ldotp \\ +[*10·1.*13·15] &\supset \colon\ldotp (\exists \alpha ,\beta ).\alpha \in \mu .\beta \in \nu .\Lambda \text{ sm }(\alpha \times \beta )\colon\ldotp \\ +[*73·47] &\supset \colon\ldotp (\exists \alpha ,\beta ).\alpha \in \mu .\beta \in \nu .\alpha \times \beta = \Lambda \colon\ldotp \\ +[*113·114] &\supset \colon\ldotp (\exists \alpha ,\beta ).\alpha \in \mu .\beta \in \nu :\alpha = \Lambda .\lor.\beta = \Lambda \colon\ldotp \\ +[*13·195] &\supset \colon\ldotp \Lambda \in \mu .\lor.\Lambda \in \nu \colon\ldotp \\ +[(1).*100·45] &\supset \colon\ldotp \mu = \text{Nc}ʻ\Lambda .\lor.\nu = \text{Nc}ʻ\Lambda \colon\ldotp \\ +[*101·1] &\supset \colon\ldotp \mu = 0.\lor.\nu = 0 &\qquad \text{(2)}\\ +\vdash . *113·601·27. &\supset \vdash \colon\ldotp \mu ,\nu \in \text{NC} - \iota ʻ\Lambda :\mu = 0.\lor.\nu = 0:\supset .\mu \times _{c} \nu = 0 &\qquad \text{(3)}\\ +\vdash .(2).(3).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p>The following propositions are concerned with multiplication by a unit +class or by 1 or 2.</p> + +<p class="nind"><b>*113·61.</b> \(\vdash .\iota ʻz \times \alpha = \downarrow zʻʻ\alpha\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*113·1.\supset \vdash .\iota ʻz \times \alpha &= sʻ\alpha \downarrow_{,,}ʻʻ\iota ʻz\\ +[*53·31·02] &= \alpha \downarrow_{,,}ʻz\\ +[*38·2] &= \downarrow zʻʻ\alpha .\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*113·611">*113·611</a>.</b> \(\vdash .\iota ʻz \times \alpha \text{ sm }\alpha \quad[*113·61.*73·611]\)</p> + +<p class="nind"><b>*113·612.</b> \(\vdash .\alpha \times \iota ʻz\text{ sm }\alpha \quad[*113·611·141]\)</p> + +<p class="nind"><b><a id="*113·62">*113·62</a>.</b> \(\vdash .\text{Nc}ʻ\alpha \times _{c} 1 = \text{Nc}ʻ\alpha\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*101·2.\supset \vdash .\text{Nc}ʻ\alpha \times _{c} 1 &= \text{Nc}ʻ\alpha \times _{c} \text{Nc}ʻ\iota ʻz\\ +[*113·25] &= \text{Nc}ʻ(\alpha \times \iota ʻz)\\ +[*113·612] &= \text{Nc}ʻ\alpha .\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*113·621">*113·621</a>.</b> \(\vdash :\mu \in \text{NC}.\supset .\mu \times _{c} 1 = \text{ sm }ʻʻ\mu\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*113·204.\supset \vdash :\mu = \Lambda .\supset .\mu \times _{c} 1 &= \Lambda \\ +[*37·29] &= \text{ sm }ʻʻ\mu &\qquad \text{(1)}\\ +\vdash .*103·26. \supset \vdash :\text{Hp}.\alpha \in \mu .&\supset .\mu = \text{N}_{0}\text{c}ʻ\alpha . &\qquad \text{(2)}\\ +[(*113·04)] \supset .\mu \times _{c} 1 &= \text{Nc}ʻ\alpha \times _{c} 1\\ +[*113·62] &= \text{Nc}ʻ\alpha \\ +[*103·4.(2)] &= \text{ sm }ʻʻ\mu &\qquad \text{(3)}\\ +\vdash .(2).*10·11·23·35.&\supset \vdash :\text{Hp}.\exists !\mu .\supset .\mu \times _{c} 1 = \text{ sm }ʻʻ\mu &\qquad \text{(4)}\\ +\vdash .(1).(4).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_123">[Pg 123]</span></p> + +<p>Observe that if \(\mu\) is a typically definite cardinal, +\(\text{ sm }ʻʻ\mu\) is the "same" cardinal rendered typically ambiguous; +while if \(\mu\) is typically ambiguous, \(\mu = \text{ sm }ʻʻ\mu\) in +every type.</p> + +<p class="nind"><b>*113·63.</b> \(\vdash :z{\sim}\in \alpha .\supset .\downarrow zʻʻ\alpha \text{ sm } \text{D}ʻʻ{\in}_{\Delta}ʻ(\iota ʻ\alpha \cup \iota ʻ\iota ʻz)\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*113·152.\supset \vdash :\text{Hp}.\supset .\text{D}ʻʻ{\in}_{\Delta}ʻ(\iota ʻ\alpha \cup \iota ʻ\iota ʻz)\text{ sm } \alpha \times \iota ʻz &\qquad \text{(1)}\\ +\vdash .(1).*113·61·141.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*113·64.</b> + \(\vdash .\downarrow zʻʻ\alpha \times \downarrow zʻʻ\beta \text{ sm } \alpha \times \beta .\downarrow zʻʻ\alpha \times \downarrow zʻʻ\beta \text{ sm }\downarrow zʻʻ(\alpha \times \beta )\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*73·611.*113·13.&\supset \vdash .\downarrow zʻʻ\alpha \times \downarrow zʻʻ\beta \text{ sm } \alpha \times \beta &\qquad \text{(1)}\\ +\vdash .(1).*73·611.&\supset \vdash .\downarrow zʻʻ\alpha \times \downarrow zʻʻ\beta \text{ sm }\downarrow zʻʻ(\alpha \times \beta ) &\qquad \text{(2)}\\ +\vdash .(1).(2).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*113·65">*113·65</a>.</b> \(\vdash .\downarrow zʻʻ\alpha \times \downarrow zʻʻ\beta = (\downarrow z\parallel \text{Cnv}ʻ\downarrow z)ʻʻ(\alpha \times \beta )\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*72·184.*55·21.&\supset \vdash .\downarrow z\in 1\rightarrow 1.\alpha \subset \text{ᗡ}ʻ\downarrow z.\beta \subset \text{ᗡ}ʻ\downarrow z.\\ +[*113·126] &\supset \vdash .\downarrow zʻʻ\alpha \times \downarrow zʻʻ\beta = (\downarrow z\parallel \text{Cnv}ʻ\downarrow z)ʻʻ(\alpha \times \beta ).\\ +&\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*113·66">*113·66</a>.</b> \(\vdash .\mu \times _{c} 2 = \mu +_{c} \mu\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*110·643.\supset \vdash .\mu \times _{c} 2 &= \mu \times _{c} (1 +_{c} 1)\\ +[*113·43] &= (\mu \times _{c} 1) +_{c} (\mu \times _{c} 1) &\qquad \text{(1)}\\ +\vdash .(1).\supset \vdash \colon\ldotp \mu = \text{N}_{0}\text{c}ʻ\alpha .\supset .\mu \times _{c} 2 &= (\text{N}_{0}\text{c}ʻ\alpha \times _{c} 1) +_{c} + (\text{N}_{0}\text{c}ʻ\alpha \times _{c} 1)\\ +[*113·62.(*113·04)] &= \text{Nc}ʻ\alpha +_{c} \text{Nc}ʻ\alpha \\ +[*110·3] &= \mu +_{c} \mu &\qquad \text{(2)}\\ +\vdash .(2).*103·2. &\supset \vdash :\mu \in \text{N}_{0}\text{C}.\supset .\mu \times _{c} 2 = \mu +_{c} \mu &\qquad \text{(3)}\\ +\vdash .*113·205.*110·4. &\supset \vdash :\mu {\sim}\in \text{N}_{0}\text{C}.\supset .\mu \times _{c} 2 = \Lambda .\mu +_{c} \mu = \Lambda &\qquad \text{(4)}\\ +\vdash .(3).(4).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*113·67.</b> \(\vdash .\text{Nc}ʻ\alpha \times _{c} \text{Nc}ʻ(\beta + \iota ʻy) = (\text{Nc}ʻ\alpha \times _{c} \text{Nc}ʻ\beta ) +_{c} \text{Nc}ʻ\alpha\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*113·421.*101·2.\supset \\ +\vdash .\text{Nc}ʻ\alpha \times _{c} \text{Nc}ʻ(\beta + \iota ʻy) &= (\text{Nc}ʻ\alpha \times _{c} \text{Nc}ʻ\beta ) +_{c} (\text{Nc}ʻ\alpha \times _{c} 1)\\ +[*113·62] &= (\text{Nc}ʻ\alpha \times _{c} \text{Nc}ʻ\beta ) +_{c} \text{Nc}ʻ\alpha .\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*113·671">*113·671</a>.</b> \(\vdash .\mu \times _{c} (\nu +_{c} 1) = (\mu \times _{c} \nu ) +_{c} \mu \quad[*113·67·205.*110·4]\)</p> + + +<div class="footnotes"><h3>FOOTNOTES:</h3> + +<div class="footnote"> + +<p class="nind"> +<a id="Footnote_5" href="#FNanchor_5" class="label">[5]</a> +We define this as \(\beta \times \alpha\), rather than +\(\alpha \times \beta\), for the sake of certain analogies with +products in relation-arithmetic. Cf. <a href="#*166">*166</a>.</p> + +</div> +</div> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_124">[Pg 124]</span></p> +<h2 class="nobreak" id="*114"> +*114. THE ARITHMETICAL PRODUCT OF A CLASS OF CLASSES.</h2> +</div> + + +<p class="nind"><i>Summary of</i> *114.</p> + +<p>The kind of multiplication defined in <a href="#*113">*113</a> cannot be extended beyond +a finite number of factors. We therefore, as in the case of addition, +introduce another definition, defining the product of the numbers of a +class of classes, and capable of being applied to an infinite number of +factors. We define the product of the numbers of members of \(\kappa\) +as \(\text{Nc}ʻ{\in}_{\Delta }ʻ\kappa\); thus we put +\[ +\Pi \text{Nc}ʻ\kappa = \text{Nc}ʻ{\in}_{\Delta}ʻ\kappa \quad\text{Df}. +\]</p> + +<p>It is to be observed that \(\Pi \text{Nc}ʻ\kappa\) is not a function of +\(\text{Nc}ʻʻ\kappa\), because, if two members of \(\kappa\) have the +same number, this will count only once in \(\text{Nc}ʻʻ\kappa\), but +will count twice in \(\Pi \text{Nc}ʻ\kappa\).</p> + +<p>It is very easy to see that, in case \(\kappa\) is finite, +\(\text{Nc}ʻ{\in}_{\Delta }ʻ\kappa\) will be what we should ordinarily +regard as the product of the numbers of members of \(\kappa\). For +suppose (<i>e.g.</i>) +\[ +\kappa = \iota ʻ\alpha \cup \iota ʻ\beta \cup \iota ʻ\gamma , +\] +where \(\alpha \neq \beta . \alpha \neq \gamma . \beta \neq \gamma\). Then +\[ +{\in}_{\Delta }ʻ\kappa = \hat{R} \{(\exists x,y,z).R = x\downarrow \alpha \unicode{x228d} y\downarrow \beta \unicode{x228d} z\downarrow \gamma .x\in \alpha .y\in \beta .z\in \gamma\}. +\]</p> + +<p>Thus if \(R\) is a member of \({\in}_{\Delta }ʻ\kappa\), \(R\) is +determinate when \(x\), \(y\), \(z\) are given, \(x\), \(y\), \(z\) +being the referents to \(\alpha\), \(\beta\), \(\gamma\). Whether +\(\alpha\), \(\beta\), \(\gamma\) overlap or not, the choice of any +one of \(x\), \(y\), \(z\) is entirely independent of the choice of +the other two, and therefore the total number of choices possible +is obviously the product of the numbers of \(\alpha\), \(\beta\), +\(\gamma\). Thus our definition will not conflict with what is commonly +understood by a product.</p> + +<p>The propositions of this number are less numerous and less important +than those of <a href="#*113">*113</a>. We shall deal first with products of a single +factor, and products in which one factor is null (<a href="#*114·2">*114·2</a>—<a href="#*114·27">·27</a>). We +shall then deal (<a href="#*114·3">*114·3</a>—<a href="#*114·36">·36</a>) with the relations between the sort of +multiplication here defined and the sort defined in *113. Then we have +a few propositions (<a href="#*114·4">*114·4</a>—<a href="#*114·43">*114·43</a>) showing that unit factors make no +difference to the value of a product. Then we prove (<a href="#*114·5">*114·5</a>—<a href="#*114·52">·52</a>) that +the value of the product is the same for two classes having double +similarity, and then (<a href="#*114·53">*114·53</a>—<a href="#*114·571">·571</a>) we give extensions of this result +which depend upon the multiplicative axiom. Finally, we give some new +forms of the associative law of multiplication.</p> + +<p><span class="pagenum" id="Page_125">[Pg 125]</span></p> + +<p>Among the more important propositions in this number are the following:</p> + +<p class="nind"><b>*114·21.</b> \(\vdash . \Pi \text{Nc}ʻ\iota ʻ\alpha = \text{Nc}ʻ\alpha\)</p> + +<p><i>I.e.</i> a product of one factor is equal to that factor.</p> + +<p class="nind"><b>*114·23.</b> \(\vdash :\Lambda \in \kappa .\supset .\Pi \text{Nc}ʻ\kappa = 0\)</p> + +<p><i>I.e.</i> a product vanishes if one of its factors is zero. The +converse requires the multiplicative axiom, as appears from the +proposition</p> + +<p class="nind"><b>*114·26.</b> \(\vdash \colon\ldotp \text{Mult ax}. \equiv :\Pi \text{Nc}ʻ\kappa = 0. \equiv _{\kappa } .\Lambda \in \kappa\)</p> + +<p><i>I.e.</i> the multiplicative axiom is equivalent to the assumption +that a product vanishes when, and only when, one of its factors is zero.</p> + +<p class="nind"><b>*114·301.</b> \(\vdash :\kappa \cap \lambda = \lambda .\supset .{\in}_{\Delta}ʻ(\kappa \cup \lambda )\text{ sm } {\in}_{\Delta}ʻ\kappa \times {\in}_{\Delta} + ʻ\lambda \)</p> + +<p class="nind">whence</p> + +<p class="nind"><b>*114·31.</b> \(\vdash :\kappa \cap \lambda = \Lambda .\supset .\Pi \text{Nc}ʻ\kappa \times _{c} \Pi \text{Nc}ʻ\lambda = \Pi \text{Nc}ʻ(\kappa \cup \lambda )\)</p> + +<p class="nind">which is a form of the associative law, and</p> + +<p class="nind"><b>*114·35.</b> \(\vdash :\alpha \neq \beta .\supset .\Pi \text{Nc}ʻ(\iota ʻ\alpha \cup \iota ʻ\beta ) = \text{Nc}ʻ\alpha \times _{c} \text{Nc}ʻ\beta\)</p> + +<p class="nind">which connects the two sorts of multiplication.</p> + +<p class="nind"><b>*114·41.</b> \(\vdash :\lambda \subset 1.\supset .\Pi \text{Nc}ʻ(\kappa \cup \lambda ) = \Pi \text{Nc}ʻ\kappa\)</p> + +<p><i>I.e.</i> unit factors make no difference to the value of a product.</p> + +<p class="nind"><b>*114·51.</b> \(\vdash :T\upharpoonright sʻ\lambda \in \kappa \,\overline{\text{ sm }}\, \,\overline{\text{ sm }}\, \lambda .\supset .(T\parallel \breve{T} _{\in })\upharpoonright {\in}_{\Delta}ʻ\lambda + \in ({\in}_{\Delta}ʻ\kappa )\,\overline{\text{ sm }}\, ({\in}_{\Delta}ʻ\lambda )\)</p> + +<p>This proposition gives a correlator of \({\in}_{\Delta }ʻ\kappa\) and +\({\in}_{\Delta }ʻ\lambda\) as a function of a double correlator of +\(\kappa\) and \(\lambda\), and thus leads to</p> + +<p class="nind"><b>*114·52.</b> \(\vdash :\kappa \text{ sm }\text{ sm }\lambda .\supset .\Pi \text{Nc}ʻ\kappa = \Pi \text{Nc}ʻ\lambda .{\in}_{\Delta}ʻ\kappa \text{ sm }{\in}_{\Delta}ʻ\lambda\)</p> + +<p>Hence, by the propositions of <a href="#*111">*111</a>, we infer</p> + +<p class="nind"><b>*114·571.</b> \(\vdash \colon\ldotp \text{Mult ax}.\supset :\mu ,\nu \in \text{NC}.\kappa ,\lambda \in \mu \cap \text{Cl}ʻ\nu .\supset .\Pi \text{Nc}ʻ\kappa = \Pi \text{Nc}ʻ\lambda\)</p> + +<p><i>I.e.</i> assuming the multiplicative axiom, if \(\kappa\) and +\(\lambda\) each consist of \(\mu\) classes of \(\nu\) terms each, +their products are equal.</p> + +<p>We have next various forms of the associative law, beginning with</p> + +<p class="nind"><b>*114·6.</b> \(\vdash : \kappa \in \text{Cls}^{2} \text{excl} . \supset . \Pi \text{Nc}ʻ{\in}_{\Delta }ʻʻ\kappa = \Pi \text{Nc}ʻsʻ\kappa \)</p> + +<p>which is an immediate consequence of *85·44. The other form is</p> + +<p class="nind"><b>*114·632.</b> \[\begin{aligned}\vdash :S\upharpoonright &\gamma \in 1\rightarrow 1.\gamma \subset \text{ᗡ}ʻS.\gamma \cap Sʻʻ\gamma = \Lambda .\supset .\\ +&{\in}_{\Delta }ʻ\hat{\mu}\{(\exists \alpha ).\alpha \in \gamma .\mu = \alpha \times Sʻ\alpha\}\text{ sm }{\in}_{\Delta}ʻ(\gamma \cup Sʻʻ\gamma )\end{aligned}\]</p> + +<p>As to the sense in which this is a form of the associative law, see the +observations following <a href="#*114·6">*114·6</a>.</p> + +<hr class="tb"> + +<p class="nind"><b>*114·01.</b> \(\Pi \text{Nc}ʻ\kappa = \text{Nc}ʻ{\in}_{\Delta }ʻ\kappa \quad\text{Df}\)</p> + +<p class="nind"><b><a id="*114·1">*114·1</a>.</b> \(\vdash . \Pi \text{Nc}ʻ\kappa = \text{Nc}ʻ{\in}_{\Delta}ʻ\kappa \quad[(*114·01)]\)</p> + +<p class="nind"><b>*114·11.</b> \(\vdash :\beta \in \Pi \text{Nc}ʻ\kappa . \equiv .\beta \text{ sm }{\in}_{\Delta}ʻ\kappa . \equiv .\beta \in \text{Nc}ʻ{\in}_{\Delta}ʻ\kappa + \quad[*114·1.*100·31]\)</p> + +<p class="nind"><b>*114·12.</b> \(\vdash .{\in}_{\Delta}ʻ\kappa \in \Pi \text{Nc}ʻ\kappa \quad[*100·3.*114·1]\)</p> + +<p class="nind"><b><a id="*114·2">*114·2</a>.</b> \(\vdash .\Pi \text{Nc}ʻ\Lambda = 1 \quad[*83·15 .*101·2]\)</p> + +<p><span class="pagenum" id="Page_126">[Pg 126]</span></p> + +<p>Thus a product of no factors is 1. This is the source of \(\mu^{0} =1\), +as we shall see later.</p> + +<p class="nind"><b>*114·21.</b> \(\vdash .\Pi \text{Nc}ʻ\iota ʻ\alpha = \text{Nc}ʻ\alpha \quad[*83·41]\)</p> + +<p class="nind"><b>*114·22.</b> \(\vdash . \Pi \text{Nc}ʻ\iota ʻ\Lambda = 0 \quad[*114·21.*101·1]\)</p> + +<p class="nind"><b>*114·23.</b> \(\vdash :\Lambda \in \kappa .\supset .\Pi \text{Nc}ʻ\kappa = 0 \quad[*83·11.*101·1]\)</p> + +<p>Thus an arithmetical product is zero if any of its factors is zero. To +prove the converse, we have to assume the multiplicative axiom, which, +in fact, is equivalent to the proposition that an arithmetical product +is only zero when at least one of its factors is zero.</p> + +<p class="nind"><b>*114·24.</b> \(\vdash :\Pi \text{Nc}ʻ\lambda \neq 0.\kappa \subset \lambda .\supset .\Pi \text{Nc}ʻ\kappa \neq 0\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*114·1.*101·1. &\supset \vdash :\Pi \text{Nc}ʻ\lambda \neq 0.\supset .\exists !{\in}_{\Delta}ʻ\lambda &\qquad \text{(1)}\\ +\vdash . (1) . *80·6. \supset \vdash :\Pi \text{Nc}ʻ\lambda \neq 0.\kappa \subset \lambda .&\supset .\exists !{\in}_{\Delta}ʻ\kappa .\\ +[*114·1.*101·1] &\supset .\Pi \text{Nc}ʻ\kappa \neq 0:\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*114·25.</b> \(\vdash \colon\ldotp \text{Mult ax}. \equiv :\Pi \text{Nc}ʻ\kappa = 0.\supset _{\kappa }.\Lambda \in \kappa\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*88·37.\text{Transp}.\supset \\ +\vdash \colon\ldotp \text{Mult ax}. &\equiv :{\in}_{\Delta}ʻ\kappa = \Lambda .\supset _{\kappa }.\Lambda \in \kappa :\\ +[*114·1.*101·1] &\equiv :\Pi \text{Nc}ʻ\kappa = 0.\supset _{\kappa }.\Lambda \in \kappa \colon\ldotp \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p>Note that \(\Lambda \in \kappa . \equiv .0\in \text{Nc}ʻʻ\kappa\).</p> + +<p class="nind"><b>*114·26.</b> \(\vdash \colon\ldotp \text{Mult ax}. \equiv :\Pi \text{Nc}ʻ\kappa = 0. \equiv _{\kappa } .\Lambda \in \kappa \quad[*88·372.*101·1]\)</p> + +<p class="nind"><b>*114·261.</b> \(\vdash \colon\ldotp \text{Mult ax}. \equiv :\Pi \text{Nc}ʻ\kappa = 0. \equiv _{\kappa } .0\in \text{Nc}ʻʻ\kappa \quad[*114·26.*101·1]\)</p> + +<p class="nind"><b><a id="*114·27">*114·27</a>.</b> \[\begin{aligned}&\vdash \colon\colon \text{Mult ax}. \equiv \colon\ldotp \alpha \in \kappa .\supset _{\alpha }.\exists !\alpha : \equiv _{\kappa } + .\Pi \text{Nc}ʻ\kappa \neq 0\\ +&[*114·26. \text{Transp}.*24·63]\end{aligned}\]</p> + +<p class="nind"><b><a id="*114·3">*114·3</a>.</b> \(\vdash :\kappa \neq \lambda .\supset .{\in}_{\Delta}ʻ(\iota ʻ{\in}_{\Delta}ʻ\kappa \cup \iota ʻ{\in}_{\Delta}ʻ\lambda )\text{ sm }{\in}_{\Delta}ʻ\kappa + \times {\in}_{\Delta}ʻ\lambda\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*113·146. \supset \vdash :{\in}_{\Delta}ʻ\kappa \neq {\in}_{\Delta}ʻ\lambda .\supset .{\in}_{\Delta}ʻ(\iota ʻ{\in}_{\Delta}ʻ\kappa + \cup \iota ʻ{\in}_{\Delta}ʻ\lambda )\text{ sm }{\in}_{\Delta}ʻ\kappa \times {\in}_{\Delta}ʻ\lambda &\qquad \text{(1)}\\ +\vdash .*80·81. \supset \vdash \colon\ldotp \exists !{\in}_{\Delta}ʻ\kappa .\lor.\exists !{\in}_{\Delta}ʻ\lambda :\kappa \neq \lambda :\supset .{\in}_{\Delta}ʻ\kappa + \neq {\in}_{\Delta}ʻ\lambda &\qquad \text{(2)}\\ +\vdash .*83·903.*113·114.\supset \\ +\vdash :{\in}_{\Delta}ʻ\kappa = \Lambda .{\in}_{\Delta}ʻ\lambda = \Lambda .\supset .{\in}_{\Delta}ʻ(\iota ʻ{\in}_{\Delta}ʻ\kappa \cup \iota ʻ{\in}_{\Delta}ʻ\lambda + ) = \Lambda .{\in}_{\Delta}ʻ\kappa \times {\in}_{\Delta}ʻ\lambda = \Lambda &\qquad \text{(3)}\\ +\vdash .(1).(2).(3).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*114·301">*114·301</a>.</b> \(\vdash :\kappa \cap \lambda = \Lambda .\supset .{\in}_{\Delta}ʻ(\kappa \cup \lambda )\text{ sm }{\in}_{\Delta}ʻ\kappa + \times {\in}_{\Delta}ʻ\lambda\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*85·45.*114·3.\supset\\ +\vdash :\kappa \cap \lambda = \Lambda .\kappa \neq \lambda .\supset .{\in}_{\Delta}ʻ(\kappa \cup \lambda )\text{ sm }{\in}_{\Delta}ʻ\kappa \times {\in}_{\Delta}ʻ\lambda + &\qquad \text{(1)}\\ +\vdash .*22·5. \supset \vdash :\kappa \cap \lambda = \Lambda .\kappa = \lambda .\supset .\kappa = \Lambda .\lambda = \Lambda .\\ +[*83·15] \supset .{\in}_{\Delta}ʻ(\kappa \cup \lambda ) = \iota ʻ\dot{\Lambda} .{\in}_{\Delta}ʻ\kappa = \iota ʻ\dot{\Lambda} .{\in}_{\Delta}ʻ\lambda + = \iota ʻ\dot{\Lambda} .\\ +[*113·611] \supset .{\in}_{\Delta}ʻ(\kappa \cup \lambda )\text{ sm }{\in}_{\Delta}ʻ\kappa \times {\in}_{\Delta}ʻ\lambda &\qquad \text{(2)}\\ +\vdash .(1).(2).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_127">[Pg 127]</span></p> + +<p class="nind"><b><a id="*114·31">*114·31</a>.</b> \[\begin{aligned}&\vdash :\kappa \cap \lambda = \Lambda .\supset .\Pi \text{Nc}ʻ\kappa \times _{c} \Pi \text{Nc}ʻ\lambda = \Pi \text{Nc}ʻ(\kappa \cup \lambda )\\ +&[*114·301·1.*113·25]\end{aligned}\]</p> + +<p>The above is one form of the associative law of multiplication.</p> + +<p class="nind"><b>*114·311.</b> \(\vdash .\Pi \text{Nc}ʻ(\kappa \cup \lambda ) = \Pi \text{Nc}ʻ\kappa \times _{c} \Pi \text{Nc}ʻ(\lambda - \kappa ) \quad[*114·31.*22·91]\)</p> + +<p class="nind"><b>*114·32.</b> \(\vdash :\Pi \text{Nc}ʻ(\kappa \cup \lambda ) \neq 0. \equiv .\Pi \text{Nc}ʻ\kappa \neq 0.\Pi \text{Nc}ʻ\lambda \neq 0\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*114·311.*113·602.\supset \\ +\vdash :\Pi \text{Nc}ʻ(\kappa \cup \lambda ) \neq 0.\supset .\Pi \text{Nc}ʻ\kappa \neq 0 &\qquad \text{(1)}\\ +\vdash .(1) \frac{\lambda ,\kappa}{\kappa ,\lambda}. \supset \vdash :\Pi \text{Nc}ʻ(\kappa \cup \lambda ) \neq 0.\supset .\Pi \text{Nc}ʻ\lambda \neq 0 &\qquad \text{(2)}\\ +\vdash .*114·24.\supset \vdash :\Pi \text{Nc}ʻ\lambda \neq 0.\supset .\Pi \text{Nc}ʻ(\lambda - \kappa ) \neq 0:\\ +[\text{Fact}] \supset \vdash :\Pi \text{Nc}ʻ\kappa \neq 0.\Pi \text{Nc}ʻ\lambda \neq 0.\supset .\Pi \text{Nc}ʻ\kappa \neq 0.\Pi \text{Nc}ʻ(\lambda -\kappa ) \neq 0.\\ +[*113·602.*114·311] \supset .\Pi \text{Nc}ʻ(\kappa \cup \lambda ) \neq 0 &\qquad \text{(3)}\\ +\vdash .(1).(2).(3).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*114·33.</b> \(\vdash :\alpha {\sim}\in \kappa .\supset .\Pi \text{Nc}ʻ(\kappa \cup \iota ʻ\alpha ) = \Pi \text{Nc}ʻ\kappa \times _{c} \text{Nc}ʻ\alpha \quad[*114·31·21]\)</p> + +<p class="nind"><b>*114·34.</b> \[\begin{aligned}&\vdash :\Pi \text{Nc}ʻ\kappa \neq 0.\exists !\alpha . \equiv .\Pi \text{Nc}ʻ(\kappa \cup \iota ʻ\alpha ) \neq 0\\ +&[*114·32·21.*101·14]\end{aligned}\]</p> + +<p class="nind"><b>*114·35.</b> \(\vdash :\alpha \neq \beta .\supset .\Pi \text{Nc}ʻ(\iota ʻ\alpha \cup \iota ʻ\beta ) = \text{Nc}ʻ\alpha \times _{c} \text{Nc}ʻ\beta \quad[*114·33·21]\)</p> + +<p class="nind"><b><a id="*114·36">*114·36</a>.</b> + \[\begin{aligned}&\vdash :\alpha \neq \beta .\alpha \neq \gamma .\beta \neq \gamma .\supset .\Pi \text{Nc}ʻ(\iota ʻ\alpha \cup \iota ʻ\beta \cup \iota ʻ\gamma ) = \text{Nc}ʻ\alpha \times _{c} \text{Nc}ʻ\beta \times _{c} \text{Nc}ʻ\gamma \\ +&[*114·33·35]\end{aligned}\]</p> + +<p class="nind"><b><a id="*114·4">*114·4</a>.</b> \(\vdash :\lambda \subset 1.\supset .\Pi \text{Nc}ʻ\lambda = 1 \quad[*83·44]\)</p> + +<p class="nind"><b>*114·41.</b> \(\vdash :\lambda \subset 1.\supset .\Pi \text{Nc}ʻ(\kappa \cup \lambda ) = \Pi \text{Nc}ʻ\kappa \quad[*83·57]\)</p> + +<p class="nind"><b>*114·42.</b> \(\vdash .\Pi \text{Nc}ʻ\kappa = \Pi \text{Nc}ʻ(\kappa -1)\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*24·41.\supset \vdash .\kappa = (\kappa - 1)\cup (\kappa \cap 1) &\qquad \text{(1)}\\ +\vdash .(1).*114·41.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*114·43">*114·43</a>.</b> \(\vdash .\Pi \text{Nc}ʻ(\kappa \cup \iotaʻʻ\alpha ) = \Pi \text{Nc}ʻ\kappa \quad[*114·41.*52·3]\)</p> + +<p class="nind"><b><a id="*114·5">*114·5</a>.</b> \(\vdash :T\in \kappa \,\overline{\text{ sm }}\, \,\overline{\text{ sm }}\, \lambda .\supset .(T\parallel \breve{T} _{\in })\upharpoonright {\in}_{\Delta}ʻ\lambda + \in ({\in}_{\Delta}ʻ\kappa )\,\overline{\text{ sm }}\, ({\in}_{\Delta}ʻ\lambda )\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*111·1·11. &\supset \vdash :\text{Hp}.\supset .T,T_{\in }\upharpoonright \lambda \in 1\rightarrow 1 &\qquad \text{(1)}\\ +\vdash .*80·14.*83·21.&\supset \vdash sʻ\text{D}ʻʻ{\in}_{\Delta}ʻ\lambda \subset sʻ\lambda .sʻ\text{ᗡ}ʻʻ{\in}_{\Delta}ʻ\lambda \subset \lambda &\qquad \text{(2)}\\ +\vdash .(1).(2).*74·773.\supset \\ +\vdash :\text{Hp}.&\supset .(T\parallel \breve{T} _{\in })\upharpoonright {\in}_{\Delta}ʻ\lambda \in \{(T\parallel \breve{T} _{\in })ʻʻ{\in}_{\Delta}ʻ\lambda + \},\overline{\text{ sm }}\, ({\in}_{\Delta}ʻ\lambda ) &\qquad \text{(3)}\\ +\vdash .*82.43 \frac{\in,T_{\in}}{P,\,Q}.*62·3.\supset \\ +\vdash :T,T_{\in }\upharpoonright \lambda \in 1\rightarrow 1.sʻ\lambda \subset \text{ᗡ}ʻT.\lambda \subset \text{ᗡ}ʻT_{\in }.\kappa &= T_{\in }ʻʻ\lambda .\supset .\\ +&(T\mid \in \upharpoonright \lambda \mid \breve{T} _{\in })_{\Delta }ʻ\kappa = (T\parallel \breve{T} _{\in })ʻʻ{\in}_{\Delta}ʻ\lambda &\qquad \text{(4)}\\ +\vdash.(4).(1).*111·1.*37·111.\supset \vdash :\text{Hp}.\supset .&(T\mid \in \upharpoonright \lambda \mid \breve{T} _{\in })_{\Delta }ʻ\kappa + = (T\parallel \breve{T} _{\in })ʻʻ{\in}_{\Delta}ʻ\lambda &\qquad \text{(5)}\\ +\vdash .*34·1.*37·101.\supset \\ +\vdash :x(T\mid \in \upharpoonright \lambda \mid \breve{T} _{\in }\alpha . &\equiv .(\exists y,\beta ).xTy.y\in \beta .\beta \in \lambda .\alpha = Tʻʻ\beta &\qquad \text{(6)}\\ +\vdash .(6).*72·52.*111·1.\supset \\ +\vdash \colon\ldotp \text{Hp}.\supset :x(T\mid \in \upharpoonright \lambda \mid \breve{T} _{\in })\alpha + . &\equiv .(\exists y,\beta ).xTy.y\in \beta .\beta \in \lambda .\beta = \breve{T} ʻʻ\alpha .\alpha \subset \text{D}ʻT.\\ +[*111·1·131.*13·195] &\equiv .(\exists y).xTy.y\in \breve{T} ʻʻ\alpha .\alpha \in \kappa .\\ +[*37·1] &\equiv .x\in Tʻʻ\breve{T} ʻʻ\alpha .a\in \kappa .\\ +[*72·502.*111·1] &\equiv .x(\in \upharpoonright \kappa )\alpha &\qquad \text{(7)}\\ +\vdash .(5).(7).\supset \vdash :\text{Hp}.&\supset .(\in \upharpoonright \kappa )_{\Delta }ʻ\kappa = (T\parallel \breve{T} _{\in })ʻʻ{\in}_{\Delta}ʻ\lambda .\\ +[*83·12] &\supset .{\in}_{\Delta}ʻ\kappa = (T \parallel ]\breve{T} _{\in })ʻʻ{\in}_{\Delta}ʻ\lambda &\qquad \text{(8)}\\ +\vdash .(3).(8).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_128">[Pg 128]</span></p> + +<p class="nind"><b>*114·501.</b> \(\vdash :S = T\upharpoonright sʻ\lambda .\supset .(S\parallel \breve{S} _\in )\upharpoonright {\in}_{\Delta}ʻ\lambda = (T\parallel \breve{T} _{\in })\upharpoonright + {\in}_{\Delta}ʻ\lambda\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*80·14.*83·21.\supset \\ +\vdash \colon\ldotp R\in {\in}_{\Delta}ʻ\lambda .\supset :yR\beta .&\supset .y\in sʻ\lambda .\beta \in \lambda .\\ +[*40·13] &\supset .y\in sʻ\lambda .\beta \subset sʻ\lambda : &\qquad \text{(1)}\\ +[*4·71 . \text{Fact}] &\supset :xTy.yR\beta .\beta \breve{T} _{\in }\alpha . \equiv .xTy.y\in sʻ\lambda .yR\beta .\beta \breve{T} _{\in }\alpha .\beta \subset sʻ\lambda .\\ +[*37·101.*22·621] &\equiv .x(T\upharpoonright sʻ\lambda )y.yR\beta .\alpha = Tʻʻ\beta .\beta = \beta \cap sʻ\lambda .\\ +[(1).*37·412] &\equiv .x(T\upharpoonright sʻ\lambda )y.yR\beta .\alpha = (T\upharpoonright sʻ\lambda )ʻʻ\beta &\qquad \text{(2)}\\ +\vdash .(2).\supset \vdash \colon\ldotp \text{Hp}.&\supset :R\in {\in}_{\Delta}ʻ\kappa .\supset .T\mid R\mid \breve{T} _{\in } = S\mid R\mid \breve{S} _{\in }:\\ +[*35·71] &\supset :(T\parallel \breve{T} _{\in })\upharpoonright {\in}_{\Delta}ʻ\kappa = (S\parallel \breve{S} _{\in })\upharpoonright {\in}_{\Delta}ʻ\kappa + \colon\ldotp \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*114·51">*114·51</a>.</b> \[\begin{aligned}&\vdash :T\upharpoonright sʻ\lambda \in \kappa \,\overline{\text{ sm }}\, \,\overline{\text{ sm }}\, \lambda .\supset .(T\parallel \breve{T} _{\in })\upharpoonright + {\in}_{\Delta}ʻ\lambda \in ({\in}_{\Delta}ʻ\kappa )\,\overline{\text{ sm }}\, ({\in}_{\Delta}ʻ\lambda )\\ +&[*114·5·501]\end{aligned}\]</p> + +<p class="nind"><b><a id="*114·52">*114·52</a>.</b> \(\vdash :\kappa \text{ sm }\text{ sm } \lambda .\supset .\Pi \text{Nc}ʻ\kappa = \Pi \text{Nc}ʻ\lambda .{\in}_{\Delta}ʻ\kappa \text{ sm }{\in}_{\Delta}ʻ\lambda + \quad[*114·51 .*111·4]\)</p> + +<p class="nind"><b><a id="*114·53">*114·53</a>.</b> \[\begin{aligned}\vdash \colon\colon &\text{Mult ax}.\supset \colon\ldotp \kappa ,\lambda \in \text{Cls}^{2} \text{excl}:\\ +&(\exists S).S\in 1\rightarrow 1.S\,\unicode{x2abd}\, \text{ sm }.\text{D}ʻS = \kappa .\text{ᗡ}ʻS = \lambda :\supset .\Pi \text{Nc}ʻ\kappa = \Pi \text{Nc}ʻ\lambda\\ +&[*114·52.*111·5]\end{aligned}\]</p> + +<p class="nind"><b><a id="*114·54">*114·54</a>.</b> + \[\begin{aligned}&\vdash \colon\ldotp \text{Mult ax}.\supset :\mu ,\nu \in \text{NC}.\kappa ,\lambda \in \mu \cap \text{Cl excl}ʻ\nu .\supset .\Pi \text{Nc}ʻ\kappa = \Pi \text{Nc}ʻ\lambda \\ +&[*114·52.*111·53]\end{aligned}\]</p> + +<p>The condition \(\kappa\), \(\lambda \in \text{Cls}^{2} \text{excl}\), +which is involved in the hypothesis of <a href="#*114·54">*114·54</a> (through \(\kappa\), +\(\lambda \in \text{Cl excl}ʻ\nu\)), is not necessary. The following +propositions enable us to remove it. We first prove +\[ +{\in}_{\Delta}ʻ\kappa \text{ sm }{\in}_{\Delta}ʻ\in \unicode{x21A7}ʻʻ\kappa +\] +and then we use *114·54 to take us from \({\in}_{\Delta}ʻ\in\unicode{x21A7}ʻʻ\kappa\) +to \({\in}_{\Delta}ʻ\in\unicode{x21A7}ʻʻ\lambda \). Thence we arrive at +\({\in}_{\Delta}ʻ\kappa \text{ sm } {\in}_{\Delta}ʻ\lambda\).</p> + +<p class="nind"><b>*114·56.</b> \(\vdash .{\in}_{\Delta}ʻ\kappa \text{ sm }{\in}_{\Delta}ʻ\in + \unicode{x21A7}ʻʻ\kappa .\Pi \text{Nc}ʻ\kappa = \Pi \text{Nc}ʻ\in \unicode{x21A7}ʻʻ\kappa \quad[*85·54]\)</p> + +<p class="nind"><b>*114·561.</b> + \[\begin{aligned}&\vdash :S\in \kappa \,\overline{\text{ sm }}\, \lambda \cap \text{ᗡ}ʻʻʻ\text{ sm }.\supset .\in \unicode{x21A7}\mid S\mid \text{Cnv}ʻ(\in \unicode{x21A7})\in (\in \unicode{x21A7}ʻʻ\kappa )\,\overline{\text{ sm }}\, (e\unicode{x21A7}ʻʻ\lambda )\cap \text{ᗡ}ʻʻʻ\text{ sm }\\ +&[*73·63.*85·601.*38·12.*33·432]\end{aligned}\]</p> + +<p><span class="pagenum" id="Page_129">[Pg 129]</span></p> + +<p class="nind"><b>*114·562.</b> \(\vdash \colon\ldotp \text{Mult ax}.\supset : +(\exists S).S\in 1\rightarrow 1.S\,\unicode{x2abd}\, \text{ sm }.\text{D}ʻS = \kappa .\text{ᗡ}ʻS = \lambda .\supset .\in \unicode{x21A7}ʻʻ\kappa \text{ sm }\text{ sm }\in \unicode{x21A7}ʻʻ\lambda\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*114·561.*85·61.\supset \\ +\vdash \colon\ldotp (\exists S).S\in 1\rightarrow 1.S\,\unicode{x2abd}\, \text{ sm }.\text{D}ʻS = \kappa .\text{ᗡ}ʻS = \lambda .\supset :\\ +\in \unicode{x21A7}ʻʻ\kappa ,\in \unicode{x21A7}ʻʻ\lambda \in \text{Cls}^{2} + \text{excl}:(\exists T).T\in 1\rightarrow 1.T\,\unicode{x2abd}\, \text{ sm }.\text{D}ʻT = \in \unicode{x21A7}ʻʻ\kappa .\text{ᗡ}ʻT = \in \unicode{x21A7}ʻʻ\lambda :\\ +[*111·5] \supset :\text{Mult ax}.\supset .\in \unicode{x21A7}ʻʻ\kappa \text{ sm } \text{ sm }\in \unicode{x21A7}ʻʻ\lambda \colon\ldotp \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*114·57.</b> \[\begin{aligned}\vdash \colon\ldotp &\text{Mult ax}.\supset :\\ +&(\exists S).S\in 1\rightarrow 1.S\,\unicode{x2abd}\, \text{ sm }.\text{D}ʻS = \kappa .\text{ᗡ}ʻS = \lambda .\supset .\Pi \text{Nc}ʻ\kappa = \Pi \text{Nc}ʻ\lambda \end{aligned}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*114·562·52.\supset \\ +\vdash \colon\ldotp \text{Mult ax}.\supset :(\exists S).&S\in 1\rightarrow 1.S\,\unicode{x2abd}\, \text{ sm }.\text{D}ʻS = \kappa .\text{ᗡ}ʻS = \lambda .\supset .\\ +&\Pi \text{Nc}ʻ\in \downarrow ʻʻ\kappa = \Pi \text{Nc}ʻ\in \downarrow ʻʻ\lambda .\\ +[*114·56] &\supset .\Pi \text{Nc}ʻ\kappa = \Pi \text{Nc}ʻ\lambda \colon\ldotp \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*114·571">*114·571</a>.</b> + \[\begin{aligned}&\vdash \colon\ldotp \text{Mult ax}.\supset :\mu ,\nu \in \text{NC}.\kappa ,\lambda \in \mu \cap \text{Cl}ʻ\nu .\supset .\Pi \text{Nc}ʻ\kappa = \Pi \text{Nc}ʻ\lambda \\ +&[*111·52 . *114·57]\end{aligned}\]</p> + +<p class="nind"><b><a id="*114·6">*114·6</a>.</b> \(\vdash :\kappa \in \text{Cls}^{2} \text{excl}.\supset .\Pi \text{Nc}ʻ{\in}_{\Delta}ʻʻ\kappa = \Pi \text{Nc}ʻsʻ\kappa \quad[*85·44]\)</p> + +<p>This is the most general form of the associative law for arithmetical +multiplication.</p> + +<p>Owing to the fact that we have two kinds of multiplication, namely +\(\alpha \times \beta\) and \({\in}_{\Delta}ʻ\kappa\), we have four +forms of the associative law of multiplication, namely:</p> + +<p>(1) *114·6, above,</p> + +<p>(2) <a href="#*113·54">*113·54</a>, <i>i.e.</i> \(\vdash .(\mu \times _{c} \nu ) \times _{c}\varpi = \mu \times _{c} (\nu \times _{c} \varpi )\),</p> + +<p>(3) <a href="#*114·31">*114·31</a>, <i>i.e.</i> \(\vdash :\kappa \cap \lambda = \Lambda.\supset .\Pi \text{Nc}ʻ\kappa \times _{c} \Pi \text{Nc}ʻ\lambda = \Pi +\text{Nc}ʻ(\kappa \cup \lambda )\),</p> + +<p>(4) a form of the associative law which has not yet been proved, which +may be explained as follows.</p> + +<p>Suppose we have a number of pairs of classes, <i>e.g.</i> \((\alpha_{1}, \beta _{1})\), +\((\alpha _{2}, \beta _{2})\), \((\alpha _{3},\beta _{3})\),.... +Suppose we form the products \(\alpha _{1} \times\beta _{1}\), +\(\alpha _{2} \times \beta _{2}\), \(\alpha _{3} \times\beta _{3}\), ... +and multiply all these products together. We wish to +prove that (with a suitable hypothesis) the result is similar to the +product of all the \(\alpha\)'s and all the \(\beta\)'s taken together +as one class; <i>i.e.</i> if we call \(\lambda\) the class of products +\(\alpha _{1} \times \beta _{1}\), \(\alpha _{2} \times \beta _{2}\), +\(\alpha _{3} \times \beta _{3}\), ..., and \(\mu\) the class whose +members are \(\alpha _{1}\), \(\alpha _{2}\), \(\alpha _{3}\), ..., +\(\beta _{1}\), \(\beta _{2}\), \(\beta _{3}\), ..., we wish to prove +\[ +\Pi \text{Nc}ʻ\lambda = \Pi \text{Nc}ʻ\mu . +\] +In order to express this proposition in symbols, let \(S\) be the +correlator of the \(\alpha\)'s and \(\beta\)'s, so that \(\beta_{\nu} =Sʻ\alpha_{\nu}\). +(The suffix \(\nu\) will not be used further, since it implies that the +number of \(\alpha\)'s and of \(\beta\)'s is finite or denumerable.) +Then our class of products of the form \(\alpha \times \beta\) is +\[ +\hat{\mu} \{(\exists \alpha ).\alpha \in \gamma .\mu = \alpha \times Sʻ\alpha\}, +\]<span class="pagenum" id="Page_130">[Pg 130]</span> +where \(\gamma\) is the class of all the \(\alpha\)'s ; and the product +of this class of products is +\[ +{\in}_{\Delta}ʻ\hat{\mu} \{(\exists \alpha ).\alpha \in \gamma .\mu = \alpha \times Sʻ\alpha\}. +\] +On the other hand, the class of all the \(\alpha\)'s and \(\beta\)'s is +\(\gamma \cup Sʻʻ\gamma\), and the product of this class is +\[ +{\in}_{\Delta}ʻ(\gamma \cup Sʻʻ\gamma ). +\] +Thus what we have to prove (with a suitable hypothesis) is +\[ +{\in}_{\Delta}ʻ\hat{\mu} \{(\exists \alpha ).\alpha \in \gamma .\mu = \alpha \times Sʻ\alpha\} \text{ sm } {\in}_{\Delta}ʻ(\gamma \cup Sʻʻ\gamma ). +\] +The hypothesis required is +\[ +S\upharpoonright \gamma \in 1\rightarrow 1.\gamma \subset \text{ᗡ}ʻS.\gamma \cap Sʻʻ\gamma = \Lambda . +\]</p> + +<p>A smaller hypothesis suffices, however, for a proposition which, in +virtue of <a href="#*114·301">*114·301</a>, is closely allied to the above, namely +\[ +{\in}_{\Delta}ʻ\gamma \times {\in}_{\Delta}ʻSʻʻ\gamma \text{ sm } {\in}_{\Delta}ʻ\hat{\mu} \{(\exists \alpha ).\alpha \in \gamma .\mu = \alpha \times Sʻ\alpha\}. +\] +For this, a sufficient hypothesis is +\[ +S\upharpoonright \gamma \in 1\rightarrow 1.\gamma \subset \text{ᗡ}ʻS. +\] +Thus <i>e.g.</i> we may write \(I\) for \(S\), and we find +\[ +\vdash .{\in}_{\Delta}ʻ\gamma \times {\in}_{\Delta}ʻ\gamma \text{ sm } {\in}_{\Delta}ʻ\hat{\mu} \{(\exists \alpha ).\alpha \in \gamma .\mu = \alpha \times \alpha\}. +\]</p> + +<p>We shall now prove the above propositions. What follows, down to +<a href="#*114·621">*114·621</a>, consists of lemmas.</p> + +<p>For convenience, we write \(S_{\times }ʻ\alpha\) for \(\alpha \times Sʻ\alpha\) +in the course of these lemmas; this notation is introduced in the +hypotheses of the lemmas.</p> + +<p class="nind"><b><a id="*114·601">*114·601</a>.</b> \[\begin{aligned}\vdash \colon\ldotp S\upharpoonright \gamma &\in 1\rightarrow 1.\gamma \subset \text{ᗡ}ʻS.\Lambda {\sim}\in \gamma .S_{\times } + = \hat{\mu} \hat{\alpha} (\alpha \in \gamma .\mu = \alpha \times Sʻ\alpha ).\supset :\\ +&S_{\times }\in 1\rightarrow 1.\text{ᗡ}ʻS_{\times } = \gamma .\text{D}ʻS_{\times } = \hat{\mu} \{(\exists \alpha ).\alpha \in \gamma .\mu = \alpha \times Sʻ\alpha\}:\\ +&\alpha \in \gamma .\supset _{\alpha }.S_{\times }ʻ\alpha = \alpha \times Sʻ\alpha\end{aligned}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*33·11. &\supset \vdash :\text{Hp}.\supset .\text{D}ʻS_{\times } = \hat{\mu} \{(\exists \alpha ).\alpha \in \gamma .\mu = \alpha \times Sʻ\alpha\} &\qquad \text{(1)}\\ +\vdash .*21·33 \supset \vdash \colon\ldotp \text{Hp}.\alpha \in \gamma .&\supset :\mu (S_{\times })\alpha . \equiv _{\mu } .\mu = \alpha \times Sʻ\alpha :\\ +[*30·3] &\supset :S_{\times }ʻ\alpha = \alpha \times Sʻ\alpha &\qquad \text{(2)}\\ +\vdash .(2).*14·204. \supset \vdash \colon\ldotp \text{Hp}.\supset :\alpha \in \gamma .&\supset _{\alpha }.\text{E}!S_{\times }ʻ\alpha . &\qquad \text{(3)}\\ +[*33·43] &\supset _{\alpha }.\alpha \in \text{ᗡ}ʻS_{\times } &\qquad \text{(4)}\\ +\vdash .*21·33.*33·131.&\supset \vdash \colon\ldotp \text{Hp}.\supset :\alpha \in \text{ᗡ}ʻS_{\times }.\supset _{\alpha }.\alpha \in \gamma &\qquad \text{(5)}\\ +\vdash .(4).(5). \supset \vdash :\text{Hp}.&\supset .\text{ᗡ}ʻS_{\times } = \gamma . &\qquad \text{(6)}\\ +[(3).*71·16] &\supset .S_{\times }\in 1\rightarrow \text{Cls} &\qquad \text{(7)}\\ +\vdash .*113·181. \supset \vdash \colon\ldotp \text{Hp}.&\supset :\alpha ,\alpha'\in \gamma .\alpha \times Sʻ\alpha = \alpha'\times Sʻ\alpha'.\supset .Sʻ\alpha = Sʻ\alpha'.\\ +[*71·59] &\supset .\alpha = \alpha' &\qquad \text{(8)}\\ +\vdash .(8).*71·55.(2).(6).(7).&\supset \vdash :\text{Hp}.\supset .S_{\times }\in 1\rightarrow 1 &\qquad \text{(9)}\\ +\vdash .(1).(2).(6).(9).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_131">[Pg 131]</span></p> + +<p class="nind"><b>*114·602.</b> \(\vdash :\text{Hp}*114·601.A = \hat{R} \hat{\alpha} \{\alpha \in \gamma .R = (Sʻ\alpha )\downarrow \alpha\}.\supset .A\in 1\rightarrow 1.\text{ᗡ}ʻA = \gamma\)</p> + +<p><i>Dem.</i></p> + +<p>As in *114·601, we prove</p> + +<p>\[ +\begin{array}{l} +\vdash :\text{Hp}.&\supset .A\in 1\rightarrow \text{Cls}.\text{ᗡ}ʻA = \gamma &\qquad \text{(1)}\\ +\vdash .*21·33.*13·171.\supset \vdash \colon\ldotp \text{Hp}.\supset :RA\alpha .RA\beta .&\supset .(Sʻ\alpha )\downarrow \alpha = (Sʻ\beta )\downarrow \beta .\\ +[*55·202] &\supset .\alpha = \beta &\qquad \text{(2)}\\ +\vdash .(1).(2).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*114·603.</b> \(\vdash :\text{Hp}*114·602.X\in {\in}_{\Delta}ʻ\gamma .Y\in {\in}_{\Delta}ʻSʻʻ\gamma .P = (Y\parallel \breve{X} )\mid A\mid \breve{S} _{\times }.\supset + .P\in {\in}_{\Delta}ʻ\text{D}ʻS_{\times }\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*43·122.*71·166.*144·601·602.&\supset \vdash :\text{Hp}.\supset .P\in 1\rightarrow \text{Cls} &\qquad \text{(1)}\\ +\vdash .*43·122.*37·32·322.*33·431.\supset \vdash :\text{Hp}.\supset .\text{ᗡ}ʻP &= S_{\times }ʻʻ\text{ᗡ}ʻA\\ +[*114·601·602] &= \text{D}ʻS_{\times } &\qquad \text{(2)}\\ +\vdash .*34·1.\supset \colon\ldotp \text{Hp}.\supset :\\ +MP\mu . &\equiv .(\exists R,\alpha ).M = Y\mid R\mid \breve{X} .R = (Sʻ\alpha )\downarrow \alpha .\alpha \in \gamma .\mu = S_{\times }ʻ\alpha .\\ +[*113·123.*80·14] &\equiv .(\exists \alpha ).M = (YʻSʻ\alpha )\downarrow (Xʻ\alpha ).\mu = S_{\times }ʻ\alpha .\alpha \in \gamma .\\ +[*13·195.*114·601] &\equiv .(\exists \alpha ,\beta ).\beta = Sʻ\alpha .\alpha \in \gamma .M = (Yʻ\beta )\downarrow (Xʻ\alpha ).\mu = \alpha \times \beta .\\ +[*83·2] &\supset .(\exists \alpha ,\beta ,u,v).\beta = Sʻ\alpha .\alpha \in \gamma .u\in \alpha .v\in \beta .\\ +&M = (v\downarrow u).\mu = \alpha \times \beta .\\ +[*113·101] &\supset .M\in \mu &\qquad \text{(3)}\\ +\vdash .(1).(2).(3).*80·14.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*114·604.</b> \[\begin{aligned}\vdash :\text{Hp}*114·602.T = &\hat{P} \hat{Q}\{(\exists X,Y).X\in {\in}_{\Delta}ʻ\gamma .Y\in {\in}_{\Delta}ʻSʻʻ\gamma\}.\\ +&\{Q = Y\downarrow X.P = (Y\parallel \breve{X} )\mid A\mid \breve{S} _{\times }\}.\\ +&\supset .T\in 1\rightarrow \text{Cls}.\text{ᗡ}ʻT = {\in}_{\Delta}ʻ\gamma \times {\in}_{\Delta}ʻSʻʻ\gamma + .\text{D}ʻT\subset {\in}_{\Delta}ʻ\text{D}ʻS_{\times }\end{aligned}\]</p> + +<p>The relation \(T\) here defined is the correlator required for proving +\[ +{\in}_{\Delta}ʻ\hat{\mu}\{(\exists \alpha ).\alpha \in \gamma .\mu = \alpha \times Sʻ\alpha\}\text{ sm }{\in}_{\Delta}ʻ\gamma \times {\in}_{\Delta}ʻSʻʻ\gamma . +\]</p> + +<p>Besides what is proved in the present proposition, we shall have to prove +\[ +T\in \text{Cls}\rightarrow 1.{\in}_{\Delta}ʻ\text{D}ʻS_{\times }\subset \text{D}ʻT. +\]</p> + +<p>The proof of the present proposition is as follows.</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*21·33.*13·171.\supset \vdash \colon\ldotp \text{Hp}.\supset :\\ +PTQ.P'TQ.&\supset .(\exists X,Y,X',Y').Y\downarrow X = Y'\downarrow X'.P = (Y\parallel \breve{X} )\mid A\mid \breve{S} _{\times }.\\ +&P' = (Y'\parallel \breve{X}')\mid A\mid \breve{S} _{\times }.\\ +[*55·202] &\supset .P = P' &\qquad \text{(1)}\\ +\vdash .*21·33.*114·603.&\supset \vdash \colon\ldotp \text{Hp}.\supset :PTQ.\supset .P\in {\in}_{\Delta}ʻ\text{D}ʻS_{\times } &\qquad \text{(2)}\\ +\vdash .(1).(2).*113·101.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_132">[Pg 132]</span></p> + +<p class="nind"><b>*114·605.</b> \(\vdash :\text{Hp}*114·604.\supset .T\in \text{Cls}\rightarrow 1\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*114·601.&\supset \vdash :\text{Hp}.\supset .S_{\times }\in 1\rightarrow 1 &\qquad \text{(1)}\\ +\vdash .(1).*74·71.*114·601·602.\supset \\ +\vdash \colon\ldotp \text{Hp}.X,X'\in {\in}_{\Delta}ʻ\gamma .&Y,Y'\in {\in}_{\Delta}ʻSʻʻ\gamma .(Y\parallel \breve{X})\mid A\mid \breve{S} _{\times } + = (Y'\parallel \breve{X}')\mid A\mid \breve{S} _{\times }.\supset :\\ +&(Y\parallel \breve{X} )\mid A = (Y'\parallel \breve{X}')\mid A:\\ +[*74·7] &\supset :(Y\parallel \breve{X} )\upharpoonright \text{D}ʻA = (Y'\parallel \breve{X}')\upharpoonright \text{D}ʻA:\\ +[*114·602] \supset :\alpha \in \gamma .&\supset _{\alpha }.(Y\parallel \breve{X})ʻ(Sʻ\alpha )\downarrow \alpha = (Y'\parallel \breve{X}')ʻ(Sʻ\alpha )\downarrow \alpha .\\ +[*113·123] &\supset _{\alpha }.(YʻSʻ\alpha )\downarrow (Xʻ\alpha ) = (Y'ʻSʻ\alpha )\downarrow (X'ʻ\alpha ).\\ +[*55·202] &\supset _{\alpha }.Xʻ\alpha = X'ʻ\alpha .YʻSʻ\alpha = Y'ʻSʻ\alpha :\\ +[*80·14.*33·45] &\supset :X = X'.Y = Y':\\ +[*55·202] &\supset :Y\downarrow X = Y'\downarrow X' &\qquad \text{(2)}\\ +\vdash . (2) . *13·22. *21·33. \supset \vdash \colon\ldotp \text{Hp}.&\supset :PTQ.PTQ'.\supset .Q = Q'\colon\ldotp \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p>The following propositions are required for proving that, with the same +hypothesis, \({\in}_{\Delta}ʻ\text{D}ʻS_{\times }\subset \text{D}ʻT\).</p> + +<p class="nind"><b>*114·61.</b> \[\begin{aligned}\vdash :\text{Hp}*114·602.P\in {\in}_{\Delta}ʻ\text{D}ʻS_{\times }.&X = \breve{\iota} \mid \text{ᗡ}\mid P\mid S_{\times }.Y + = \breve{\iota} \mid \text{D}\mid P\mid S_{\times }\mid \breve{S} .\supset .\\ +&X\in {\in}_{\Delta}ʻ\gamma .Y\in {\in}_{\Delta}ʻSʻʻ\gamma\end{aligned}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*72·181·13·131.*80·14.*114·601.\supset \vdash :\text{Hp}.\supset .X,Y\in 1\rightarrow \text{Cls} &\qquad \text{(1)}\\ +\vdash .*72·2·181·13·131.*80·14.*114·601.\supset\\ +\vdash \colon\ldotp \text{Hp}.\supset :xX\alpha . \equiv .x = \breve{\iota} ʻ\text{ᗡ}ʻPʻS_{\times }ʻ\alpha . &\qquad \text{(2)}\\ +[*51·53] \supset .x\in \text{ᗡ}ʻPʻS_{\times }ʻ\alpha .\\ +[*83·2. *114·601] \supset .(\exists R).R\in \alpha \times Sʻ\alpha .x\in \text{ᗡ}ʻR.\\ +[*113·142] \supset .x\in \alpha &\qquad \text{(3)}\\ +\vdash . *114·601. \supset \vdash \colon\ldotp \text{Hp}.\supset :\alpha \in \gamma . \equiv .S_{\times }ʻ\alpha \in \text{D}ʻS_{\times }.\\ +[*83·2] \equiv .\text{E}!PʻS_{\times }ʻ\alpha &\qquad \text{(4)}\\ +\vdash . *83·2. \supset \vdash \colon\ldotp \text{Hp}.\supset :\text{E}!PʻS_{\times }ʻ\alpha . \equiv .PʻS_{\times }ʻ\alpha \in S_{\times }ʻ\alpha .\\ +[*113·142] \supset .\text{ᗡ}ʻPʻS_{\times }ʻ\alpha \in 1.\\ +[*52·15] \supset .\text{E}!\breve{\iota} ʻ\text{ᗡ}ʻPʻS_{\times }ʻ\alpha &\qquad \text{(5)}\\ +\vdash .(2).(4).(5). \supset \vdash :\text{Hp}.\supset .\gamma \subset \text{ᗡ}ʻX &\qquad \text{(6)}\\ +\vdash . *34·36. *114·601. \supset \vdash :\text{Hp}.\supset .\text{ᗡ}ʻX\subset \gamma &\qquad \text{(7)}\\ +\vdash .(1).(3).(6).(7). \supset \vdash :\text{Hp}.\supset .X\in {\in}_{\Delta}ʻ\gamma &\qquad \text{(8)}\\ +\text{Similarly}\qquad \vdash :\text{Hp}.\supset .Y\in {\in}_{\Delta}ʻSʻʻ\gamma &\qquad \text{(9)}\\ +\vdash .(8).(9).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*114·611.</b> \(\vdash \colon\ldotp \text{Hp}*114·61.\supset :\alpha \in \gamma .\supset .(YʻSʻ\alpha )\downarrow (Xʻ\alpha ) = PʻS_{\times }ʻ\alpha\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*72·2. \supset \vdash :\text{Hp}.\alpha \in \gamma .&\supset .Xʻ\alpha = \breve{\iota} ʻ\text{ᗡ}ʻPʻS_{\times }ʻ\alpha .YʻSʻ\alpha = \breve{\iota} ʻ\text{D}ʻPʻS_{\times }ʻ\alpha + .\\ +[*55·16. *51·51] &\supset .(YʻSʻ\alpha )\downarrow (Xʻ\alpha ) = PʻS_{\times }ʻ\alpha :\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_133">[Pg 133]</span></p> + +<p class="nind"><b>*114·612.</b> \(\vdash :\text{Hp} *114·61.\supset .(Y\parallel \breve{X} )\mid A\mid \breve{S} _{\times } = P\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*83·15.\supset \vdash :\text{Hp}.\dot{\exists} !P.&\supset .\exists !\text{D}ʻS_{\times } .\\ +[*114·601] &\supset .\exists !\gamma &\qquad \text{(1)}\\ +\vdash .*34·1.\supset \vdash \colon\ldotp \text{Hp}.\supset :&M\{(Y\parallel \breve{X} )\mid A\mid \breve{S} _{\times }\}\mu . \equiv .\\ +&(\exists Q,\alpha ).M = (Y\parallel \breve{X} )ʻQ.QA\alpha .\mu = S_{\times }ʻ\alpha .\\ +[*114·601·602] &\equiv .(\exists \alpha ).M = (Y\parallel \breve{X} )ʻ(Sʻ\alpha )\downarrow \alpha .\mu = S_{\times }ʻ\alpha .\alpha \in \gamma .\\ +[*113·123] &\equiv .(\exists \alpha ).M = (YʻSʻ\alpha )\downarrow (Xʻ\alpha ).\mu = S_{\times }ʻ\alpha .\alpha \in \gamma .\\ +[*114·611] &\equiv .(\exists \alpha ).M = PʻS_{\times }ʻ\alpha .\mu = S_{\times }ʻ\alpha .\alpha \in \gamma .\\ +[*13·193.*114·601.*71·16] &\equiv .M = Pʻ\mu .\exists !\gamma .\\ +[*71·36.*80·14.(1)] &\equiv .MP\mu \colon\ldotp \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*114·613.</b> \[\begin{aligned}\vdash :\text{Hp}*114·61.&\text{Hp}*114·604.\supset .\\ +&P = Tʻ(Y\downarrow X).(Y\downarrow X)\in {\in}_{\Delta}ʻ\gamma \times {\in}_{\Delta}ʻSʻʻ\gamma \end{aligned}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*21·33.*114·604.\supset \vdash \colon\ldotp \text{Hp}*114·604.\supset :\\ +X\in {\in}_{\Delta}ʻ\gamma .Y\in {\in}_{\Delta}ʻSʻʻ\gamma .\supset .Tʻ(Y\downarrow X) = (Y\parallel \breve{X} )\mid A\mid \breve{S} _{\times } &\qquad \text{(1)}\\ +\vdash .(1).*114·61·612.*113·106.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*114·614.</b> \(\vdash :\text{Hp}*114·604.\supset .{\in}_{\Delta}ʻ\text{D}ʻS_{\times }\subset \text{D}ʻT\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*114·613.\supset \vdash \colon\ldotp \text{Hp}.&\supset :P\in {\in}_{\Delta}ʻ\text{D}ʻS_{\times }.\supset .(\exists Q).P = TʻQ.\\ +[*33·43] &\supset .P\in \text{D}ʻT\colon\ldotp \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*114·62.</b> \[\begin{aligned}&\vdash :\text{Hp}*114·604.\supset .T\in 1\rightarrow 1.\text{D}ʻT = {\in}_{\Delta}ʻ\text{D}ʻS_{\times }.\text{ᗡ}ʻT = {\in}_{\Delta}ʻ\gamma + \times {\in}_{\Delta}ʻSʻʻ\gamma \\ +&[*114·604·605·614]\end{aligned}\]</p> + + +<p class="nind"><b><a id="*114·621">*114·621</a>.</b> \[\begin{aligned}\vdash :S\upharpoonright \gamma \in 1\rightarrow 1.&\gamma \subset \text{ᗡ}ʻS.\Lambda {\sim}\in \gamma .\supset .\\ +&{\in}_{\Delta}ʻ\hat{\mu}\{(\exists \alpha ).\alpha \in \gamma .\mu = \alpha \times Sʻ\alpha\}\text{ sm }{\in}_{\Delta}ʻ\gamma \times {\in}_{\Delta}ʻSʻʻ\gamma \\ +&[*114·62·601]\end{aligned}\]</p> + +<p>The hypothesis \(\Lambda {\sim}\in \gamma\) is not necessary, since, +when \(\Lambda \in \gamma\), +\[ +{\in}_{\Delta}ʻ\hat{\mu}\{(\exists \alpha ).\alpha \in \gamma .\mu = \alpha \times Sʻ\alpha\}\,\,\text{and}\,\,{\in}_{\Delta}ʻ\gamma \times {\in}_{\Delta}ʻSʻʻ\gamma +\]</p> + +<p class="nind">are both \(\Lambda\). This is proved in <a href="#*114·63">*114·63</a>.</p> + +<p class="nind"><b><a id="*114·63">*114·63</a>.</b> \[\begin{aligned}\vdash :S\upharpoonright \gamma \in 1\rightarrow& 1.\gamma \subset \text{ᗡ}ʻS.\supset .\\ +&{\in}_{\Delta}ʻ\hat{\mu}\{(\exists \alpha ).\alpha \in \gamma .\mu = \alpha \times Sʻ\alpha\}\text{ sm }{\in}_{\Delta}ʻ\gamma + \times {\in}_{\Delta}ʻSʻʻ\gamma\end{aligned}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*10·24.*83·11.\supset \\ +\vdash :\text{Hp}.\Lambda \in \gamma .&\supset .\Lambda \times Sʻ\Lambda \in \hat{\mu}\{(\exists \alpha ).\alpha \in \gamma .\mu = \alpha \times Sʻ\alpha\}.{\in}_{\Delta}ʻ\gamma + = \Lambda .\\ +[*113·114] &\supset .\Lambda \in \hat{\mu}\{(\exists \alpha ).\alpha \in \gamma .\mu = \alpha \times Sʻ\alpha\}.{\in}_{\Delta}ʻ\gamma \times {\in}_{\Delta}ʻSʻʻ\gamma + = \Lambda .\\ +[*83·11] &\supset .{\in}_{\Delta}ʻ\hat{\mu} \{(\exists \alpha ).\alpha \in \gamma .\mu = \alpha \times Sʻ\alpha\} = \Lambda .{\in}_{\Delta}ʻ\gamma \times {\in}_{\Delta}ʻSʻʻ\gamma + = \Lambda &\qquad \text{(1)}\\ +\vdash .(1).*73·47.*114·621.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_134">[Pg 134]</span></p> + +<p>The above is one of the two variants of the associative law for +\({\in}_{\Delta}\) and \(\times\).</p> + +<p class="nind"><b>*114·631.</b> \(\vdash .{\in}_{\Delta}ʻ\hat{\mu}\{(\exists \alpha ).\alpha \in \gamma .\mu = \alpha \times \alpha \}\text{ sm }{\in}_{\Delta}ʻ\alpha \times {\in}_{\Delta}ʻ\alpha + \quad\left[*114·63 \frac{I}{S}\right]\)</p> + +<p class="nind"><b>*114·632.</b> \[\begin{aligned}\vdash :S\upharpoonright &\gamma \in 1\rightarrow 1.\gamma \subset \text{ᗡ}ʻS.\gamma \cap Sʻʻ\gamma = \Lambda .\supset .\\ +&{\in}_{\Delta}ʻ\hat{\mu}\{(\exists \alpha ).\alpha \in \gamma .\mu = \alpha \times Sʻ\alpha\}\text{ sm }{\in}_{\Delta}ʻ(\gamma \cup Sʻʻ\gamma ) \quad[*114·63·301]\end{aligned}\]</p> + +<p>This is the second variant of the associative law for +\({\in}_{\Delta}\) and \(\times\).</p> + +<p class="nind"><b><a id="*114·64">*114·64</a>.</b> \[\begin{align}\vdash :(Rʻʻ\gamma )&\upharpoonleft R,S\upharpoonright \gamma \in 1\rightarrow 1.\gamma \subset \text{ᗡ}ʻR.\gamma \subset \text{ᗡ}ʻS.\supset .\\ +&{\in}_{\Delta}ʻRʻʻ\gamma \times {\in}_{\Delta}ʻSʻʻ\gamma \text{ sm }{\in}_{\Delta}ʻ\hat{\mu} \{(\exists z).z\in \gamma .\mu = Rʻz \times Sʻz\}\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *114·63 \frac{S\mid\, \breve{R},\,Rʻʻ\gamma}{S,\,\gamma}. \supset \\ +\vdash :S\mid \breve{R} \upharpoonright &Rʻʻ\gamma \in 1\rightarrow 1.Rʻʻ\gamma \subset \text{ᗡ}ʻ(S\mid \breve{R} ).\supset .\\ +&{\in}_{\Delta}ʻRʻʻ\gamma \times {\in}_{\Delta}ʻSʻʻRʻʻRʻʻ\gamma \text{ sm }{\in}_{\Delta}ʻ\hat{\mu} + \{(\exists \alpha ).\alpha \in Rʻʻ\gamma .\mu = \alpha \times (S\mid \breve{R} )ʻ\alpha\} &\qquad \text{(1)}\\ +\vdash .*74·14 .*35·354. \supset \vdash :\text{Hp}.&\supset .S\mid \breve{R} \upharpoonright Rʻʻ\gamma = S\upharpoonright \gamma \mid \gamma \upharpoonleft \breve{R} .\breve{R} \upharpoonright Rʻʻ\gamma = \gamma \upharpoonleft \breve{R} .\\ +[*71·252] &\supset .S\mid \breve{R} \upharpoonright Rʻʻ\gamma \in 1\rightarrow 1 &\qquad \text{(2)}\\ +\vdash .*37·2.\supset \vdash :\text{Hp}.&\supset .Rʻʻ\gamma \subset Rʻʻ\text{ᗡ}ʻS.\\ +[*37·32] &\supset .Rʻʻ\gamma \subset \text{ᗡ}ʻ(S\mid \breve{R} ) &\qquad \text{(3)}\\ +\vdash .*74·171. &\supset \vdash :\text{Hp}.\supset .\breve{R} ʻʻRʻʻ\gamma = \gamma &\qquad \text{(4)}\\ +\vdash .(4).*74·14. \supset \vdash :\text{Hp}.&\supset .(Rʻʻ\gamma )\upharpoonleft R = R\upharpoonright \gamma .\\ +[*35·7.*71·4] &\supset .\hat{\mu} \{(\exists \alpha ).\alpha \in Rʻʻ\gamma .\mu = \alpha \times (S\mid \breve{R} )ʻ\alpha \}\\ +&= \hat{\mu} \{(\exists z).z\in \gamma .\mu = Rʻz \times Sʻ\breve{R} ʻRʻz\}\\ +[*74·53] &= \hat{\mu} \{(\exists z).z\in \gamma .\mu = Rʻz \times Sʻz\} &\qquad \text{(5)}\\ +\vdash .(1).(2).(3).(4).(5).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p>In the above proposition, the hypothesis has to be such as to yield +\(\breve{R} ʻʻRʻʻ\gamma = \gamma\). Various other forms of hypothesis +will secure this result, and will give other forms of the above +proposition. This subject is treated in *74, above.</p> + +<p class="nind"><b><a id="*114·65">*114·65</a>.</b> + \[\begin{align}\vdash \colon\ldotp &(Rʻʻ\gamma )\upharpoonleft R,S\upharpoonright \gamma \in 1\rightarrow 1.\gamma \subset \text{ᗡ}ʻR.\gamma \subset \text{ᗡ}ʻS.Rʻʻ\gamma \cap Sʻʻ\gamma = \Lambda .\supset .\\ +&{\in}_{\Delta}ʻ(Rʻʻ\gamma \cup Sʻʻ\gamma )\text{ sm }{\in}_{\Delta}ʻ\hat{\mu} \{(\exists z).z\in \gamma .\mu = Rʻz \times Sʻz\}\\ +&[*114·64·301]\end{align}\]</p> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_135">[Pg 135]</span></p> +<h2 class="nobreak" id="115_MULTIPLICATIVE_CLASSES_AND_ARITHMETICAL_CLASSES"> +*115. MULTIPLICATIVE CLASSES AND ARITHMETICAL CLASSES.</h2> +</div> + + +<p class="nind"><i>Summary of</i> *115.</p> + +<p>Whenever \(\kappa\) is a class of mutually exclusive +classes, \({\in}_{\Delta}ʻ\kappa\) is similar to +\(\text{D}ʻʻ{\in}_{\Delta}ʻ\kappa\); hence +\[ +\Pi \text{Nc}ʻ\kappa = \text{Nc}ʻ\text{D}ʻʻ{\in}_{\Delta}ʻ\kappa . +\]</p> + +<p>Now \(\text{D}ʻʻ{\in}_{\Delta}ʻ\kappa\) is of the same type as +\(\kappa\); and when \(\kappa\) is a class of mutually exclusive +classes, \(\text{D}ʻʻ{\in}_{\Delta}ʻ\kappa\) consists of all classes +formed by selecting one representative from each member of \(\kappa\). +It often happens that \(\text{D}ʻʻ{\in}_{\Delta}ʻ\kappa\) is easier +to deal with than \({\in}_{\Delta}ʻ\kappa\); hence when possible +(<i>i.e.</i> when \(\kappa \in \text{Cls}^{2} \text{excl})\), it +is convenient to use \(\text{D}ʻʻ{\in}_{\Delta}ʻ\kappa\), rather +than \({\in}_{\Delta}ʻ\kappa\), as the standard member of +\(\Pi\text{Nc}ʻ\kappa\). We therefore put +\[ +\text{Prod}ʻ\kappa = \text{D}ʻʻ{\in}_{\Delta}ʻ\kappa \quad\text{Df}. +\]</p> + +<p>We shall call \(\text{Prod}ʻ\kappa\) the "multiplicative class" of +\(\kappa\).</p> + +<p>The associative law, +\[ +\text{Prod}ʻsʻ\kappa \text{ sm } \text{Prod}ʻ\text{Prod}ʻʻ\kappa , +\] +requires not merely \(\kappa \in \text{Cls}^{2} \text{excl}\), but also +\(sʻ\kappa \in \text{Cls}^{2} \text{excl}\). The combination of these +two hypotheses gives a completely disjointed class of classes of +classes, <i>i.e.</i> a class of classes of classes \(\kappa\) which can +be obtained by dividing a given class \((sʻsʻ\kappa)\) into mutually +exclusive portions, and then dividing each of those portions into +mutually exclusive portions. For example, take a square (a class of +points) and divide it by horizontal lines, and then divide each of the +resulting rectangles by vertical lines; then the resulting rows of +little rectangles form such a class, each row of rectangles being one +member of the class. Such a class we call an "arithmetical" class, and +denote by "\(\text{Cls}^{3}\,\text{arithm}\)."</p> + +<p>The present number is concerned with the properties of multiplicative +classes and arithmetical classes. Some of these properties will be +useful in dealing with exponentiation.</p> + +<p>The present number begins with various propositions concerning +\(\text{Prod}ʻ\kappa\) which are merely repetitions of previous +propositions of *83, *84, *85 or <a href="#*113">*113</a>. Thus we have</p> + +<p class="nind"><b>*115·141.</b> \(\vdash :\exists !\text{Prod}ʻ\kappa .\supset .sʻ\text{Prod}ʻ\kappa = sʻ\kappa \quad\text{by *83·66}\),</p> + +<p class="nind"><b>*115·142.</b> \(\vdash .\text{Prod}ʻ\iota ʻ\alpha = \iotaʻʻ\alpha \quad\text{by *83·7}\),</p> + +<p class="nind"><b>*115·143.</b> \(\vdash .\text{Prod}ʻ\iotaʻʻ\alpha = \iota ʻ\alpha \quad\text{by *83·71}\),</p> + +<p class="nind"><b>*115·16.</b> \(\vdash .\kappa \in \text{Cls}^{2} \text{excl}.\supset .\text{Prod}ʻ\kappa \subset \text{Nc}ʻ\kappa \quad\text{by *100·64}\),</p> + +<p class="nind">and various other properties.</p> + +<p><span class="pagenum" id="Page_136">[Pg 136]</span></p> + +<p>We then proceed to consider \(\text{Cls}^{3}\,\text{arithm}\). We prove</p> + +<p class="nind"><b>*115·22.</b> \(\vdash \colon\ldotp \kappa \in \text{Cls}^{3}\,\text{arithm}.\supset :sʻʻ\kappa \in \text{Cls}^{2} + \text{excl}:\alpha ,\beta \in \kappa .\exists !sʻ\alpha \cap sʻ\beta .\supset _{\alpha ,\beta }.\alpha =\beta\)</p> + +<p class="nind"> +and <a href="#*115·23">*115·23</a> gives a similar proposition substituting "\(\text{Prod}\)" +for \(s\).</p> + +<p>After a few more propositions on \(\text{Cls}^{3}\,\text{arithm}\), +we proceed to the associative law for \(\text{Prod}\) (<a href="#*115·34">*115·34</a>), +<i>i.e.</i> +\[ +\vdash :\kappa \in \text{Cls}^{3}\,\text{arithm}.\supset .\text{Prod}ʻ\text{Prod}ʻʻ\kappa \text{ sm }\text{Prod}ʻsʻ\kappa . +\]</p> + +<p>(This proposition, <a href="#*115·34">*115·34</a>, also states that, with the same hypothesis, +\(\text{Prod}ʻsʻ\kappa \text{ sm }{\in}_{\Delta}ʻsʻ\kappa)\). Hence we +have</p> + +<p class="nind"><b>*115·35.</b> \[\begin{align}\vdash :\kappa \in \text{Cls}^{3}\,\text{arithm}.\supset + .\text{Nc}ʻ\text{Prod}ʻ\text{Prod}ʻʻ\kappa &=\text{Nc}ʻ\text{Prod}ʻsʻ\kappa =\Pi \text{Nc}ʻ\text{Prod}ʻʻ\kappa \\ +&=\Pi \text{Nc}ʻ{\in}_{\Delta}ʻʻ\kappa =\Pi \text{Nc}ʻsʻ\kappa\end{align}\]</p> + +<p>We have also</p> + +<p class="nind"><b>*115·42.</b> \(\vdash :\kappa \in \text{Cls}^{3}\,\text{arithm}.\supset .\text{Prod}ʻ\text{Prod}ʻʻ\kappa =\text{D}ʻʻʻ\text{Prod}ʻ{\in}_{\Delta}ʻʻ\kappa + =\text{D}ʻʻʻ\text{D}ʻʻ{\in}_{\Delta}ʻ{\in}_{\Delta}ʻʻ\kappa\)</p> + +<p class="nind"><b>*115·44.</b> \(\vdash :\kappa \in \text{Cls}^{3}\,\text{arithm}.\supset .\text{Prod}ʻsʻ\kappa =sʻʻ\text{Prod}ʻ\text{Prod}ʻʻ\kappa\)</p> + +<p>We have next to prove that if two classes of classes have double +similarity, so have their multiplicative classes. The proof is simple, +since the double correlator is the same as for the original classes, +<i>i.e.</i></p> + +<p class="nind"><b>*115·502.</b> + \(\vdash :T\upharpoonright sʻ\lambda \in \kappa \,\overline{\text{ sm }}\, \,\overline{\text{ sm }}\, \lambda .\supset .T\upharpoonright sʻ\text{Prod}ʻ\lambda \in (\text{Prod}ʻ\kappa )\,\overline{\text{ sm }}\, \,\overline{\text{ sm }}\, (\text{Prod}ʻ\lambda )\)</p> + +<p class="nind">whence</p> + +<p class="nind"><b>*115·51.</b> \(\vdash :\kappa \text{ sm }\text{ sm }\lambda .\supset .\text{Prod}ʻ\kappa \text{ sm }\text{ sm }\text{Prod}ʻ\lambda\)</p> + +<p>The number ends with some propositions which result from <a href="#*114·64">*114·64</a><a href="#*114·65">·65</a> and +are analogous to them. One of these is used in the following number, in +proving \(\mu ^{\varpi }\times _{c}\nu ^{\varpi }=(\mu \times _{c}\nu)^{\varpi }\), +namely,</p> + +<p class="nind"><b>*115·6.</b> + \[\begin{align}\vdash :(Rʻʻ\gamma )&\upharpoonleft R,S\upharpoonright \gamma \in 1\rightarrow 1.\gamma \subset \text{ᗡ}ʻR.\gamma \subset \text{ᗡ}ʻS.Rʻʻ\gamma ,Sʻʻ\gamma \in \text{Cls}^{2} + \text{excl}.\supset .\\ +&\text{Prod}ʻRʻʻ\gamma \times \text{Prod}ʻSʻʻ\gamma \text{ sm }{\in}_{\Delta}ʻ\hat{\mu} \{(\exists z).z\in \gamma .\mu =Rʻz\times Sʻz\}\end{align}\]</p> + +<p>The subject of this number will be useful in dealing with +exponentiation, since we shall define \(\mu ^\nu\) by means +of \(\text{Prod}ʻ\alpha\downarrow_{,,}ʻʻ\beta\), where +\(\mu=\text{N}_{0}\text{c}ʻ\alpha\) and \(\nu =\text{N}_{0}\text{c}ʻ\beta\).</p> + +<hr class="tb"> + +<p class="nind"><b>*115·01.</b> \(\text{Prod}ʻ\kappa =\text{D}ʻʻ{\in}_{\Delta}ʻ\kappa \quad\text{Df}\)</p> + +<p class="nind"><b>*115·02.</b> \(\text{Cls}^{3}\,\text{arithm}=\hat{\kappa} (\kappa ,sʻ\kappa \in \text{Cls}^{2} \text{excl}) \quad\text{Df}\)</p> + +<p class="nind"><b>*115·1.</b> \(\vdash .\text{Prod}ʻ\kappa =\text{D}ʻʻ{\in}_{\Delta}ʻ\kappa \quad[(*115·01)]\)</p> + +<p class="nind"><b>*115·101.</b> \(\vdash \colon\ldotp \alpha \in \kappa .\supset _{\alpha }.\varpi \cap \alpha \in 1:\varpi \subset sʻ\kappa :\supset .\varpi \in \text{Prod}ʻ\kappa \quad[*84·411]\)</p> + +<p class="nind"><b>*115·11.</b> \[\begin{align}&\vdash \colon\colon \kappa \in \text{Cls}^{2} + \text{excl}.\supset \colon\ldotp \varpi \in \text{Prod}ʻ\kappa .\equiv :\alpha \in \kappa .\supset _{\alpha }.\varpi \cap \alpha \in 1:\varpi \subset sʻ\kappa\\ +&[*84·412]\end{align}\]</p> + +<p><span class="pagenum" id="Page_137">[Pg 137]</span></p> + +<p>Owing to this proposition, \(\text{Prod}ʻ\kappa\) can be treated +without any reference to \({\in}_{\Delta}ʻ\kappa\) whenever +\(\kappa\in \text{Cls}^{2} \text{excl}\).</p> + +<p class="nind"><b><a id="*115·12">*115·12</a>.</b> \(\vdash : \kappa \in \text{Cls}^{2} \text{excl} . \supset . \text{Prod}ʻ\kappa \in \Pi \text{Nc}ʻ\kappa . \text{Prod}ʻ\kappa \text{ sm } {\in}_{\Delta}ʻ\kappa + \quad[*84·41]\)</p> + +<p>It is this proposition that makes the notation \(\text{Prod}ʻ\kappa\) +appropriate for the multiplicative class.</p> + +<p class="nind"><b>*115·13.</b> \(\vdash : \alpha \cap \beta = \Lambda . \supset . \text{Prod}ʻ(\iota ʻ\alpha \cup \iota ʻ\beta ) \text{ sm } (\alpha \times \beta ) \quad[*113·152]\)</p> + +<p class="nind"><b>*115·131.</b> \(\vdash : \alpha \neq \beta . \supset . \text{Prod}ʻ(\iota ʻ\alpha \cup \iota ʻ\beta ) = Cʻʻ(\alpha \times \beta ) \quad[*113·151]\)</p> + +<p class="nind"><b>*115·14.</b> \[\begin{align}\vdash \colon\ldotp &\kappa \cap \lambda = \Lambda . \lor . sʻ\kappa \cap sʻ\lambda = \Lambda : \supset :\\ +&\varpi \in \text{Prod}ʻ(\kappa \cup \lambda ) . \equiv . (\exists \rho , \sigma ) . \rho \in \text{Prod}ʻ\kappa . \sigma \in \text{Prod}ʻ\lambda . \varpi = \rho \cup \sigma\\ +&[*83·64·641]\end{align}\]</p> + +<p class="nind"><b>*115·141.</b> \(\vdash : \exists ! \text{Prod}ʻ\kappa . \supset . sʻ\text{Prod}ʻ\kappa = sʻ\kappa \quad[*83·66]\)</p> + +<p class="nind"><b>*115·142.</b> \(\vdash . \text{Prod}ʻ\iota ʻ\alpha = \iotaʻʻ\alpha \quad[*83·7]\)</p> + +<p class="nind"><b>*115·143.</b> \(\vdash . \text{Prod}ʻ\iotaʻʻ\alpha = \iota ʻ\alpha \quad[*83·71]\)</p> + +<p class="nind"><b>*115·144.</b> \(\vdash : \kappa \subset 1 . \supset . \text{Prod}ʻ\kappa = \iota ʻsʻ\kappa \quad[*83·72]\)</p> + +<p class="nind"><b>*115·145.</b> \[\begin{align}&\vdash \colon\ldotp \kappa \in \text{Cls}^{2} + \text{excl} . \alpha \in \kappa . \mu \cap \alpha \in 1 . \supset : \mu - \alpha \in \text{Prod}ʻ(\kappa - \iota ʻ\alpha ) . \equiv . \mu \in \text{Prod}ʻ\kappa\\ +&[*84·422]\end{align}\]</p> + +<p class="nind"><b>*115·15.</b> \[\begin{align}&\vdash \colon\ldotp \kappa , \lambda \in \text{Cls}^{2} + \text{excl} . sʻ\kappa = sʻ\lambda . \supset : \kappa \subset \text{Prod}ʻ\lambda . \equiv . \lambda \subset \text{Prod}ʻ\kappa \\ +&[*84·43]\end{align}\]</p> + +<p class="nind"><b>*115·151.</b> \(\vdash : \kappa \in \text{Cls}^{2} \text{excl} . \supset . {\in}_{\Delta}ʻsʻ\kappa = \dot{s}ʻʻ\text{Prod}ʻ{\in}_{\Delta}ʻʻ\kappa + \quad[*85·28]\)</p> + +<p class="nind"><b>*115·152.</b> \(\vdash . P_{\Delta }ʻ\alpha \text{ sm } \text{Prod}ʻP \unicode{x21A7}ʻʻ\alpha \quad[*85·55]\)</p> + +<p class="nind"><b>*115·153.</b> \(\vdash . {\in}_{\Delta}ʻ\kappa \text{ sm } \text{Prod}ʻ\in \unicode{x21A7}ʻʻ\kappa \quad[*115·152]\)</p> + +<p class="nind"><b>*115·154.</b> \(\vdash . \text{Prod}ʻ\in \unicode{x21A7}ʻʻ\kappa \in \Pi \text{Nc}ʻ\kappa \quad[*115·153]\)</p> + +<p class="nind"><b>*115·16.</b> \(\vdash : \kappa \in \text{Cls}^{2} \text{excl} . \supset . \text{Prod}ʻ\kappa \subset \text{Nc}ʻ\kappa \quad[*100·64]\)</p> + +<p>The following proposition is used in the theory of well-ordered series +(*250·5).</p> + +<p class="nind"><b>*115·17.</b> \(\vdash : \exists ! {\in}_{\Delta}ʻ\text{Cl ex}ʻ\alpha . \supset . \text{Prod}ʻ\text{Cl ex}ʻ\alpha = \iota ʻ\alpha\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *80·14 . *115·1 . *37·45 . &\supset \vdash : \text{Hp} . \supset . \exists ! \text{Prod}ʻ\text{Cl ex}ʻ\alpha &\qquad \text{(1)}\\ +\vdash . *60·61 . \text{Fact} . \supset \\ +\vdash \colon\ldotp R \in 1 \rightarrow \text{Cls}. R \,\unicode{x2abd}\, \in . \text{ᗡ}ʻR = \text{Cl ex}ʻ\alpha . &\supset : R \in 1 \rightarrow \text{Cls} . R \,\unicode{x2abd}\, \in . \iotaʻʻ\alpha \subset \text{ᗡ}ʻR :\\ +[*51·15] &\supset : x \in \alpha . \supset _{x} . xR(\iota ʻx) :\\ +[*33·14] &\supset : \alpha \subset \text{D}ʻR &\qquad \text{(2)}\\ +\vdash . *83·21 . \supset \vdash : \text{Hp} (2) . &\supset . \text{D}ʻR \subset sʻ\text{Cl ex}ʻ\alpha .\\ +[*60·501] &\supset . \text{D}ʻR \subset \alpha &\qquad \text{(3)}\\ +\vdash . (2) . (3) . &\supset \vdash : R \in 1 \rightarrow \text{Cls}. R \,\unicode{x2abd}\, \in . \text{ᗡ}ʻR = \text{Cl ex}ʻ\alpha . \supset . \text{D}ʻR = \alpha &\qquad \text{(4)}\\ +\vdash . (4) . *115·1 . *80·14 . &\supset \vdash . \text{Prod}ʻ\text{Cl ex}ʻ\alpha \subset \iota ʻ\alpha &\qquad \text{(5)}\\ +\vdash . (1) . (5) . *51·4 . \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_138">[Pg 138]</span></p> + +<p class="nind"><b>*115·18.</b> \(\vdash .tʻ\text{Prod}ʻ\kappa = tʻ\kappa \quad[*83·81]\)</p> + +<p class="nind"><b>*115·2.</b> \(\vdash :\kappa \in \text{Cls}^{3}\,\text{arithm}. \equiv .\kappa ,sʻ\kappa \in \text{Cls}^{2} \text{excl} \quad[(*115·02)]\)</p> + +<p class="nind"><b>*115·21.</b> \[\begin{align}\vdash \colon\ldotp \kappa \in \text{Cls}^{3}\,\text{arithm}. \equiv :&\alpha ,\beta \in \kappa .\exists !\alpha \cap \beta .\supset _{\alpha ,\beta }.\alpha + = \beta :\\ +&\alpha ,\beta \in \kappa .\rho \in \alpha .\sigma \in \beta .\exists !\rho \cap \sigma .\supset _{\alpha ,\beta ,\rho ,\sigma }.\rho = \sigma \\ +[*115·2.*84·11]\end{align}\]</p> + +<p class="nind"><b>*115·211.</b> \(\vdash :\kappa \in \text{Cls}^{3}\,\text{arithm}.\alpha ,\beta \in \kappa .\rho \in \alpha .\sigma \in \beta .\exists !\rho \cap \sigma .\supset .\alpha = \beta\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*115·21 .\supset \vdash :\text{Hp}.&\supset .\rho = \sigma .\rho \in \alpha .\sigma \in \beta .\\ +[*13·13] &\supset .\rho \in \alpha \cap \beta .\\ +[*115·21] &\supset .\alpha = \beta :\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*115·22.</b> \(\vdash \colon\ldotp \kappa \in \text{Cls}^{3}\,\text{arithm}.\supset :sʻʻ\kappa \in \text{Cls}^{2} + \text{excl}:\alpha ,\beta \in \kappa .\exists !sʻ\alpha \cap sʻ\beta .\supset _{\alpha ,\beta }.\alpha = \beta\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*40·11. \supset \vdash :\exists !sʻ\alpha \cap sʻ\beta . &\equiv .(\exists x,\rho ,\sigma ).\rho \in \alpha .\sigma \in \beta .x\in \rho .x\in \sigma .\\ +[*10·35] &\equiv .(\exists \rho ,\sigma ).\rho \in \alpha .\sigma \in \beta .\exists !\rho \cap \sigma &\qquad \text{(1)}\\ +\vdash .(1).*115·211.\supset \\ +\vdash \colon\ldotp \text{Hp}.&\supset :\alpha ,\beta \in \kappa .\exists !sʻ\alpha \cap sʻ\beta .\supset .\alpha = \beta . &\qquad \text{(2)}\\ +[*30·37] &\supset .sʻ\alpha = sʻ\beta &\qquad \text{(3)}\\ +\vdash .(2).(3).*84·11.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p>Observe that, although "\(sʻʻ\kappa \in \text{Cls}^{2} \text{excl}\)" +follows from +\[ +\unicode{x201c}\alpha ,\beta \in \kappa .\exists !sʻ\alpha \cap sʻ\beta .\supset _{\alpha ,\beta }.\alpha = \beta ,\unicode{x201d} +\] +the converse implication does not hold. If there were two different +classes \(\alpha\) and \(\beta\) having the same sum, we might have +\(\exists !sʻ\alpha \cap sʻ\beta\), <i>i.e.</i> \(\exists !sʻ\alpha\), +without having \(\alpha = \beta\), in spite of "\(sʻʻ\kappa \in\text{Cls}^{2} \text{excl}\)." +In proofs, less use can be made of "\(sʻʻ\kappa \in \text{Cls}^{2} \text{excl}\)" +than of "\(\alpha ,\beta \in \kappa .\exists !sʻ\alpha \cap sʻ\beta.\supset_{\alpha,\beta }.\alpha = \beta\)." +If \(\Lambda {\sim} \in\kappa\) or \(\iota ʻ\Lambda {\sim} \in \kappa\), +the latter implies \(s\upharpoonright \kappa \in 1\rightarrow 1\).</p> + +<p class="nind"><b><a id="*115·23">*115·23</a>.</b> \[\begin{align}\vdash \colon\ldotp \kappa \in &\text{Cls}^{3}\,\text{arithm}.\supset :\\ +&\text{Prod}ʻʻ\kappa \in \text{Cls}^{2} \text{excl}:\alpha ,\beta \in \kappa .\exists !\text{Prod}ʻ\alpha \cap \text{Prod}ʻ\beta .\supset _{\alpha ,\beta }.\alpha + = \beta\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*83·62. &\supset \vdash :\varpi \in \text{Prod}ʻ\alpha \cap \text{Prod}ʻ\beta .\supset .\varpi \subset sʻ\alpha \cap sʻ\beta &\qquad \text{(1)}\\ +\vdash .(1).*24·58. &\supset \vdash :\varpi \in \text{Prod}ʻ\alpha \cap \text{Prod}ʻ\beta .\exists !\varpi .\supset .\exists !sʻ\alpha \cap sʻ\beta &\qquad \text{(2)}\\ +\vdash .(2).*115·22. &\supset \vdash :\text{Hp}.\alpha ,\beta \in \kappa .\varpi \in \text{Prod}ʻ\alpha \cap \text{Prod}ʻ\beta .\exists !\varpi .\supset .\alpha = \beta &\qquad \text{(3)}\\ +\vdash .*83·16. \text{Transp}.&\supset \vdash :\Lambda \in \text{Prod}ʻ\alpha \cap \text{Prod}ʻ\beta .\supset .\alpha = \Lambda .\beta = \Lambda &\qquad \text{(4)}\\ +\vdash .(3).(4). \supset \vdash \colon\ldotp \text{Hp}.&\supset :\alpha ,\beta \in \kappa .\exists !\text{Prod}ʻ\alpha \cap \text{Prod}ʻ\beta .\supset .\alpha = \beta . &\qquad \text{(5)}\\ +[*30·37] &\supset .\text{Prod}ʻ\alpha = \text{Prod}ʻ\beta &\qquad \text{(6)}\\ +\vdash .(5).(6).*84·11.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*115·24.</b> \(\vdash :\kappa \in \text{Cls}^{3}\,\text{arithm}. \equiv .\in \upharpoonright \kappa ,\in \upharpoonright sʻ\kappa \in \text{Cls}\rightarrow 1 \quad[*115·2.*84·14]\)</p> + +<p><span class="pagenum" id="Page_139">[Pg 139]</span></p> + +<p class="nind"><b>*115·25.</b> \(\vdash :\kappa \in \text{Cls}^{3}\,\text{arithm}.\supset .{\in}_{\Delta}ʻ\kappa \subset 1\rightarrow 1.{\in}_{\Delta}ʻsʻ\kappa + \subset 1\rightarrow 1 \quad[*84·3.*115·2]\)</p> + +<p class="nind"><b>*115·26.</b> \[\begin{align}\vdash :\kappa \in \text{Cls}^{3}\,\text{arithm}.&\supset .\\ +&{\in}_{\Delta}ʻsʻʻ\kappa \subset 1\rightarrow 1.{\in}_{\Delta}ʻ{\in}_{\Delta}ʻʻ\kappa \subset 1\rightarrow 1.{\in}_{\Delta}ʻ\text{Prod}ʻʻ\kappa + \subset 1\rightarrow 1\\ +&[*84·3.*115·22.*84·55.*115·23]\end{align}\]</p> + +<p>In the above proposition, \({\in}_{\Delta}ʻ{\in}_{\Delta}ʻʻ\kappa\subset 1\rightarrow 1\) +does not require the hypothesis \(\kappa\in \text{Cls}^{3}\,\text{arithm}\), +being true always. It is merely included here for convenience of +reference.</p> + +<p class="nind"><b>*115·27.</b> \(\vdash :\kappa \in \text{Cls}^{3}\,\text{arithm}.\supset .\kappa \subset \text{Cls}^{2} \text{excl} \quad[*115·2.*84·25.*40·13]\)</p> + +<p>We have now to prove the associative law for "\(\text{Prod}\)," +<i>i.e.</i> +\[ +\kappa \in \text{Cls}^{3}\,\text{arithm}.\supset .\text{Prod}ʻsʻ\kappa \text{ sm } \text{Prod}ʻ\text{Prod}ʻʻ\kappa . +\]</p> + +<p>In virtue of <a href="#*115·12">*115·12</a>, we have only to prove (under the hypothesis) +\[ +{\in}_{\Delta}ʻsʻ\kappa \text{ sm }{\in}_{\Delta}ʻ\text{Prod}ʻʻ\kappa +\] +which, by *85·44, will follow from +\[ +{\in}_{\Delta}ʻ{\in}_{\Delta}ʻʻ\kappa \text{ sm }{\in}_{\Delta}ʻ\text{Prod}ʻʻ\kappa +\] +which, by <a href="#*114·52">*114·52</a>, will follow from +\[ +{\in}_{\Delta}ʻʻ\kappa \text{ sm }\text{ sm }\text{Prod}ʻʻ\kappa . +\]</p> + +<p>Now +\[ +\text{Prod}ʻʻ\kappa = \text{D}_{\in}ʻʻ{\in}_{\Delta}ʻʻ\kappa . +\]</p> + +<p>Thus the correlator which will give our proposition will be +\(\text{D}\upharpoonright sʻ{\in}_{\Delta}ʻʻ\kappa\). We have only to +prove that this is a \(1\rightarrow 1\), and the rest follows.</p> + +<p class="nind"><b>*115·3.</b> \(\vdash :\kappa \in \text{Cls}^{3}\,\text{arithm}.R,S\in sʻ{\in}_{\Delta}ʻʻ\kappa .\text{D}ʻR = \text{D}ʻS.\supset .R = S\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *115·23. \supset \vdash :\kappa \in \text{Cls}^{3}\,\text{arithm}.\alpha ,\beta \in \kappa .R\in {\in}_{\Delta}ʻ\alpha .S\in {\in}_{\Delta}ʻ\beta + .\text{D}ʻR = \text{D}ʻS.\supset .\alpha = \beta &\qquad \text{(1)}\\ +\vdash .*115·27.*84·4. \supset \vdash :\kappa \in \text{Cls}^{3}\,\text{arithm}.\alpha \in \kappa .R,S\in {\in}_{\Delta}ʻ\alpha + .\text{D}ʻR = \text{D}ʻS.\supset .R = S &\qquad \text{(2)}\\ +\vdash . (1) .(2).\supset \vdash :\kappa \in \text{Cls}^{3}\,\text{arithm}.\alpha ,\beta \in \kappa .R\in {\in}_{\Delta}ʻ\alpha .S\in {\in}_{\Delta}ʻ\beta + .\text{D}ʻR = \text{D}ʻS.\supset .R = S &\qquad \text{(3)}\\ +\vdash .(3).*10·11·23·35.*40·11.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*115·31.</b> \(\vdash :\kappa \in \text{Cls}^{3}\,\text{arithm}.\supset .\text{Prod}ʻʻ\kappa \text{ sm }\text{ sm }{\in}_{\Delta}ʻʻ\kappa\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*115·3.*71·55.*72·13.&\supset \vdash :\text{Hp}.\supset .\text{D}\upharpoonright sʻ{\in}_{\Delta}ʻʻ\kappa \in 1\rightarrow 1 &\qquad \text{(1)}\\ +\vdash . *33·431. &\supset \vdash .sʻ{\in}_{\Delta}ʻʻ\kappa \subset \text{ᗡ}ʻ\text{D} &\qquad \text{(2)}\\ +\vdash .*37·11.*115·1. &\supset \vdash .\text{Prod}ʻʻ\kappa = \text{D}_{\in}ʻʻ{\in}_{\Delta}ʻʻ\kappa &\qquad \text{(3)}\\ +\vdash .(1).(2).(3).*111·402. &\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*115·32.</b> \(\vdash :\kappa \in \text{Cls}^{3}\,\text{arithm}.\supset .{\in}_{\Delta}ʻ\text{Prod}ʻʻ\kappa \text{ sm }{\in}_{\Delta}ʻ{\in}_{\Delta}ʻʻ\kappa + \quad[*115·31.*114·52]\)</p> + +<p class="nind"><b>*115·33.</b> \(\vdash :\kappa \in \text{Cls}^{3}\,\text{arithm}.\supset .{\in}_{\Delta}ʻ\text{Prod}ʻʻ\kappa \text{ sm }{\in}_{\Delta}ʻsʻ\kappa + \quad[*115·32. *85·44]\)</p> + +<p class="nind"><b><a id="*115·34">*115·34</a>.</b> \[\begin{align}&\vdash :\kappa \in \text{Cls}^{3}\,\text{arithm}.\supset + .\text{Prod}ʻ\text{Prod}ʻʻ\kappa \text{ sm }\text{Prod}ʻsʻ\kappa .\text{Prod}ʻsʻ\kappa \text{ sm }{\in}_{\Delta}ʻsʻ\kappa\\ +&[*115·33·12·23]\end{align}\]</p> + +<p>This proposition gives the associative law for "\(\text{Prod}\)."</p> + +<p>The following proposition embodies the last three propositions.</p> + +<p><span class="pagenum" id="Page_140">[Pg 140]</span></p> + +<p class="nind"><b>*115·35.</b> \[\begin{align}&\vdash : \kappa \in \text{Cls}^{3}\,\text{arithm} . \supset .\\ +&\text{Nc}ʻ\text{Prod}ʻ\text{Prod}ʻʻ\kappa = \text{Nc}ʻ\text{Prod}ʻsʻ\kappa = \Pi \text{Nc}ʻ\text{Prod}ʻʻ\kappa = \Pi \text{Nc}ʻ{\in}_{\Delta}ʻʻ\kappa = \Pi \text{Nc}ʻsʻk\\ +&[*115·34·33·32]\end{align}\]</p> + +<p>In connection with \(\text{Prod}ʻsʻ\kappa\) and +\(\text{Prod}ʻ\text{Prod}ʻʻ\kappa\), there remain two propositions of +sufficient interest to deserve proof, namely +\[ +\kappa \in \text{Cls}^{3}\,\text{arithm} . \supset . \text{Prod}ʻsʻ\kappa = sʻʻ\text{Prod}ʻ\text{Prod}ʻʻ\kappa +\] +and +\[ +\kappa \in \text{Cls}^{3}\,\text{arithm} . \supset . \text{Prod}ʻ\text{Prod}ʻʻ\kappa = \text{D}ʻʻʻ\text{D}ʻʻ{\in}_{\Delta}ʻ{\in}_{\Delta}ʻʻ\kappa +\]</p> + +<p>Of these, the first is deduced from the second, while the second is +proved by means of <a href="#*114·51">*114·51</a>, putting \(\text{D}\) for the \(T\) which +appears in that proposition, and \({\in}_{\Delta}ʻʻ\kappa\) for the +\(\lambda\) of that proposition.</p> + +<p class="nind"><b>*115·4.</b> \(\vdash : T \upharpoonright sʻ\lambda \in 1 \rightarrow 1 . sʻ\lambda \subset \text{ᗡ}ʻT . \supset . \text{Prod}ʻTʻʻʻ\lambda = Tʻʻʻ\text{Prod}ʻ\lambda\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *111·14 . *37·103 . \supset \vdash : \text{Hp} . \kappa = Tʻʻʻ\lambda . &\supset . T \upharpoonright sʻ\lambda \in \kappa \,\overline{\text{ sm }}\, \,\overline{\text{ sm }}\, \lambda .\\ +[*114·51 . *73·142] &\supset . {\in}_{\Delta}ʻ\kappa = (T \parallel \breve{T} _{\in })ʻʻ{\in}_{\Delta}ʻ\lambda &\qquad \text{(1)}\\ +\vdash . (1) . *115·1 . &\supset \vdash : \text{Hp} . \supset . \text{Prod}ʻTʻʻʻ\lambda = \text{D}ʻʻ(T \parallel \breve{T} _{\in })ʻʻ{\in}_{\Delta}ʻ\lambda + &\qquad \text{(2)}\\ +\vdash . *37·321·231 . \supset \vdash . \text{D}ʻ(T \mid R \mid \breve{T} _{\in }) &= \text{D}ʻ(T \mid R)\\ +[*37·32] &= Tʻʻ\text{D}ʻR &\qquad \text{(3)}\\ +\vdash . (3) . *43·112 . \supset \vdash . \text{D}ʻʻ(T \parallel \breve{T} _{\in })ʻʻ{\in}_{\Delta}ʻ\lambda &= Tʻʻʻ\text{D}ʻʻ{\in}_{\Delta}ʻ\lambda \\ +[*115·1] &= Tʻʻʻ\text{Prod}ʻ\lambda &\qquad \text{(4)}\\ +\vdash . (2) . (4) . \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*115·41.</b> \[\begin{align}&\vdash \colon\ldotp R, S \in sʻ\lambda . \text{D}ʻR = \text{D}ʻS . \supset _{R, S} + . R = S : \supset . \text{Prod}ʻ\text{D}ʻʻʻ\lambda = \text{D}ʻʻʻ\text{Prod}ʻ\lambda \\ +&\left[*115·4 \frac{\text{D}}{T}. *71·55 . *72·13\right]\end{align}\]</p> + +<p class="nind"><b>*115·42.</b> \[\begin{align}\vdash : \kappa \in \text{Cls}^{3}\,\text{arithm} + . \supset . \text{Prod}ʻ\text{Prod}ʻʻ\kappa &= \text{D}ʻʻʻ\text{Prod}ʻ{\in}_{\Delta}ʻʻ\kappa\\ +&= \text{D}ʻʻʻ\text{D}ʻʻ{\in}_{\Delta}ʻ{\in}_{\Delta}ʻʻ\kappa\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *115·1 . &\supset \vdash . \text{Prod}ʻ\text{Prod}ʻʻ\kappa = \text{Prod}ʻ\text{D}ʻʻʻ{\in}_{\Delta}ʻʻ\kappa &\qquad \text{(1)}\\ +\vdash . *115·3·41 . \supset \vdash : \text{Hp} . \supset . \text{Prod}ʻ\text{D}ʻʻʻ{\in}_{\Delta}ʻʻ\kappa & = \text{D}ʻʻʻ\text{Prod}ʻ{\in}_{\Delta}ʻʻ\kappa + &\qquad \text{(2)}\\ +[*115·1] &= \text{D}ʻʻʻ\text{D}ʻʻ{\in}_{\Delta}ʻ{\in}_{\Delta}ʻʻ\kappa &\qquad \text{(3)}\\ +\vdash . (1) . (2) . (3) . \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*115·43.</b> \(\vdash : \kappa \in \text{Cls}^{2} \text{excl} . \supset . \text{Prod}ʻsʻ\kappa = sʻʻ\text{D}ʻʻʻ\text{D}ʻʻ{\in}_{\Delta}ʻ{\in}_{\Delta}ʻʻ\kappa\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *115·1 . *85·28 . \supset \\ +\vdash : \text{Hp} . \supset . \text{Prod}ʻsʻ\kappa &= \text{D}ʻʻ\dot{s} ʻʻ\text{D}ʻʻ{\in}_{\Delta}ʻ{\in}_{\Delta}ʻʻ\kappa \\ +[*41·43] &= sʻʻ\text{D}ʻʻʻ\text{D}ʻʻ{\in}_{\Delta}ʻ{\in}_{\Delta}ʻʻ\kappa : \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*115·44.</b> \(\vdash : \kappa \in \text{Cls}^{3}\,\text{arithm} . \supset . \text{Prod}ʻsʻ\kappa = sʻʻ\text{Prod}ʻ\text{Prod}ʻʻ\kappa \quad[*115·43·42]\)</p> + +<p>The following proposition is a lemma for <a href="#*115·46">*115·46</a>.</p> + +<p><span class="pagenum" id="Page_141">[Pg 141]</span></p> + +<p class="nind"><b>*115·45.</b> \[\begin{align}\vdash \colon\ldotp \alpha ,\beta \in \kappa .\exists !&sʻ\alpha \cap sʻ\beta .\supset _{\alpha ,\beta }.\alpha =\beta :\supset .\\ +&(s\mid \text{D})\upharpoonright {\in}_{\Delta}ʻ\kappa \in 1\rightarrow 1.s\upharpoonright \text{Prod}ʻ\kappa \in 1\rightarrow 1\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*83·2.*40·13.&\supset \vdash :R\in {\in}_{\Delta}ʻ\kappa .\alpha \in \kappa .\supset .Rʻ\alpha \subset sʻ\alpha &\qquad \text{(1)}\\ +\vdash .*83·2.*33·43.&\supset \vdash :R\in {\in}_{\Delta}ʻ\kappa .\alpha \in \kappa .\supset .Rʻ\alpha \subset sʻ\text{D}ʻR &\qquad \text{(2)}\\ +\vdash .*83·23.\supset \\ +\vdash :R\in {\in}_{\Delta}ʻ\kappa .\alpha \in \kappa .x\in (sʻ\text{D}ʻR\cap sʻ\alpha).&\supset .(\exists \beta ).\beta \in \kappa .x\in Rʻ\beta .x\in sʻ\alpha .\\ +[(1)] &\supset .(\exists \beta ).\beta \in \kappa .x\in Rʻ\beta .x\in sʻ\beta . x\in sʻ\alpha &\qquad \text{(3)}\\ +\vdash .(3).\supset \vdash \colon\ldotp \text{Hp}.R\in {\in}_{\Delta}ʻ\kappa + .\alpha \in \kappa .\supset :x\in (sʻ\text{D}ʻR\cap sʻ\alpha ).&\supset .(\exists \beta ).x\in Rʻ\beta .\beta =\alpha . +[*13·195] \supset .x\in Rʻ\alpha &\qquad \text{(4)}\\ +\vdash .(1).(2).(4).&\supset \vdash \colon\ldotp \text{Hp}.\supset :R\in {\in}_{\Delta}ʻ\kappa .\alpha \in \kappa .\supset .Rʻ\alpha =sʻ\text{D}ʻR\cap sʻ\alpha &\qquad \text{(5)}\\ +\vdash .(5).\supset \vdash \colon\colon \text{Hp}.&\supset \colon\ldotp R,S\in {\in}_{\Delta}ʻ\kappa .sʻ\text{D}ʻR=sʻ\text{D}ʻS.\supset :\alpha \in \kappa .\supset _{\alpha }.Rʻ\alpha + =Sʻ\alpha :\\ +[*33·45.*80·14] &\supset :R=S\colon\ldotp &\qquad \text{(6)}\\ +[*71·55.*72·13·161]&\supset \colon\ldotp (s\mid \text{D})\upharpoonright {\in}_{\Delta}ʻ\kappa \in 1\rightarrow 1 &\qquad \text{(7)}\\ +\vdash .(6).*37·63.*115·1.*30·37.\supset \vdash \colon\ldotp \text{Hp}.&\supset :\mu ,\nu \in \text{Prod}ʻ\kappa .sʻ\mu = sʻ\nu .\supset .\mu =\nu :\\ +[*71·55.*72·161] &\supset :s\upharpoonright \text{Prod}ʻ\kappa \in 1\rightarrow 1 &\qquad \text{(8)}\\ +\vdash .(7).(8).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*115·46">*115·46</a>.</b> \(\vdash :\kappa \in \text{Cls}^{3}\,\text{arithm}.\supset .s\upharpoonright \text{Prod}ʻ\text{Prod}ʻʻ\kappa \in 1\rightarrow 1\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*115·141.\supset \\ +\vdash :\alpha ,\beta \in \kappa .\exists !sʻ\text{Prod}ʻ\alpha \cap sʻ\text{Prod}ʻ\beta .\supset .\exists !sʻ\alpha \cap sʻ\beta &&\qquad \text{(1)}\\ +\vdash .(1).*115·22.\supset \\ +\vdash \colon\ldotp \kappa \in \text{Cls}^{3}\,\text{arithm}.\supset :\alpha ,\beta \in \kappa .\exists !sʻ\text{Prod}ʻ\alpha \cap sʻ\text{Prod}ʻ\beta .&\supset .\alpha =\beta .\\ +[*30·37] &\supset .\text{Prod}ʻ\alpha =\text{Prod}ʻ\beta :\\ +[*37·63] \supset :\mu ,\nu \in \text{Prod}ʻʻ\kappa .\exists !sʻ\mu \cap sʻ\nu .\supset . \mu =\nu :\\ +[*115·45] \supset :s\upharpoonright \text{Prod}ʻ\text{Prod}ʻʻ\kappa \in 1\rightarrow 1\colon\ldotp \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p>The above proposition is used in dealing with products in +relation-arithmetic (<a href="#*174·42">*174·42</a>).</p> + +<p class="nind"><b>*115·5.</b> \(\vdash :T\upharpoonright sʻ\lambda \in \kappa \,\overline{\text{ sm }}\, \,\overline{\text{ sm }}\, \lambda .\supset .\text{Prod}ʻ\kappa =T_{\in }ʻʻ\text{Prod}ʻ\lambda + \quad[*115·4.*111·14]\)</p> + +<p class="nind"><b>*115·501.</b> + \[\begin{align}&\vdash :T\upharpoonright sʻ\lambda \in \kappa \,\overline{\text{ sm }}\, \,\overline{\text{ sm }}\, \lambda .\exists !\text{Prod}ʻ\lambda .\supset .T\upharpoonright sʻ\lambda \in (\text{Prod}ʻ\kappa )\,\overline{\text{ sm }}\, \,\overline{\text{ sm }}\, (\text{Prod}ʻ\lambda )\\ +&[*115·5·141.*111·14]\end{align}\]</p> + +<p class="nind"><b><a id="*115·502">*115·502</a>.</b> + \(\vdash :T\upharpoonright sʻ\lambda \in \kappa \,\overline{\text{ sm }}\, \,\overline{\text{ sm }}\, \lambda .\supset .T\upharpoonright sʻ\text{Prod}ʻ\lambda \in (\text{Prod}ʻ\kappa )\,\overline{\text{ sm }}\, \,\overline{\text{ sm }}\, (\text{Prod}ʻ\lambda )\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*35·75.\supset \vdash :{\sim}\exists !\text{Prod}ʻ\lambda .\supset .T\upharpoonright sʻ\text{Prod}ʻ\lambda =\dot{\Lambda} &\qquad \text{(1)}\\ +\vdash .*115·5.*37·29.\supset \vdash :\text{Hp}.{\sim}\exists !\text{Prod}ʻ\lambda .\supset .\text{Prod}ʻ\kappa =\Lambda .\\ +[*37·29.*40·21] \supset .sʻ\text{Prod}ʻ\kappa =(T\upharpoonright sʻ\text{Prod}ʻ\lambda )ʻʻsʻ\text{Prod}ʻ\lambda &\qquad \text{(2)}\\ +\vdash .(1).*72·1.(2).*115·5.*111·1.\supset \\ +\qquad\qquad\vdash :\text{Hp}.{\sim}\exists !\text{Prod}ʻ\lambda .\supset .T\upharpoonright sʻ\text{Prod}ʻ\lambda \in (\text{Prod}ʻ\kappa )\,\overline{\text{ sm }}\, \,\overline{\text{ sm }}\, (\text{Prod}ʻ\lambda ) &\qquad \text{(3)}\\ +\vdash .(3).*115·501·141.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_142">[Pg 142]</span></p> + +<p class="nind"><b><a id="*115·51">*115·51</a>.</b> \(\vdash : \kappa \text{ sm } \text{ sm } \lambda . \supset . \text{Prod}ʻ\kappa \text{ sm } \text{ sm } \text{Prod}ʻ\lambda \quad[*115·502]\)</p> + +<p>The above propositions show how, in certain respects, +\(\text{Prod}ʻ\kappa\) is more convenient than +\({\in}_{\Delta}ʻ\kappa\). We cannot have \({\in}_{\Delta}ʻ\kappa\text{ sm } \text{ sm } {\in}_{\Delta}ʻ\lambda\), +because \({\in}_{\Delta}ʻ\kappa\) is a class of <i>relations</i>, not a +class of classes; and the correlator of \({\in}_{\Delta}ʻ\kappa\) and +\({\in}_{\Delta}ʻ\lambda\) is by no means so simple a function of the +correlator of \(\kappa\) and \(\lambda\) as \(T_{\in } \upharpoonright\text{Prod}ʻ\lambda\), +which correlates \(\text{Prod}ʻ\kappa\) and +\(\text{Prod}ʻ\lambda\), in virtue of <a href="#*115·502">*115·502</a>.</p> + +<p>The following propositions are a continuation of those given in +<a href="#*114·601">*114·601</a> ff.</p> + +<p class="nind"><b><a id="*115·6">*115·6</a>.</b> + \[\begin{align}\vdash : &(Rʻʻ\gamma ) \upharpoonleft R, S \upharpoonright \gamma \in 1 \rightarrow 1 . \gamma \subset \text{ᗡ}ʻR . \gamma \subset \text{ᗡ}ʻS . Rʻʻ\gamma , Sʻʻ\gamma \in \text{Cls}^{2} \text{excl} . \supset .\\ +&\text{Prod}ʻRʻʻ\gamma \times \text{Prod}ʻSʻʻ\gamma \text{ sm } {\in}_{\Delta}ʻ\hat{\mu} \{(\exists z) . z \in \gamma . \mu = Rʻz \times Sʻz\}\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *115·12 . *113·13 . \supset \\ +\vdash : \text{Hp} . \supset . \text{Prod}ʻRʻʻ\gamma \times \text{Prod}ʻSʻʻ\gamma \text{ sm } {\in}_{\Delta}ʻRʻʻ\gamma \times {\in}_{\Delta}ʻSʻʻ\gamma &\qquad \text{(1)}\\ +\vdash . (1) . *114·64 . \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*115·601.</b> + \[\begin{align}\vdash : (Rʻʻ\gamma ) &\upharpoonleft R, S \upharpoonright \gamma \in 1 \rightarrow 1 . \gamma \subset \text{ᗡ}ʻR . \gamma \subset \text{ᗡ}ʻS . Rʻʻ\gamma \in \text{Cls}^{2} \text{excl} . \supset .\\ +&\hat{\mu} \{(\exists z) . z \in \gamma . \mu = Rʻz \times Sʻz\} \in \text{Cls}^{2} \text{excl}\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *113·19 . \supset \vdash \colon\ldotp \text{Hp} . \supset :\\ +z, w \in \gamma . \exists ! (Rʻz \times Sʻz) \cap (Rʻw \times Sʻw) . &\supset . \exists ! Rʻz \cap Rʻw .\\ +[*84·11] &\supset . Rʻz = Rʻw .\\ +[*74·53 . *30·37] &\supset . z = w .\\ +[*30·37] &\supset . Rʻz \times Sʻz = Rʻw \times Sʻw &\qquad \text{(1)}\\ +\vdash . (1) . *84·11 . \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*115·602.</b> + \[\begin{align}\vdash : &(Rʻʻ\gamma ) \upharpoonleft R, S \upharpoonright \gamma \in 1 \rightarrow 1 . \gamma \subset \text{ᗡ}ʻR . \gamma \subset \text{ᗡ}ʻS . Sʻʻ\gamma \in \text{Cls}^{2} \text{excl} . \supset .\\ +&\hat{\mu} \{(\exists z) . z \in \gamma . \mu = Rʻz \times Sʻz\} \in \text{Cls}^{2} \text{excl}\\ +&[\text{Proof as in *115·601}]\end{align}\]</p> + +<p class="nind"><b>*115·61.</b> + \[\begin{align}\vdash \colon\ldotp &(Rʻʻ\gamma ) \upharpoonleft R, S \upharpoonright \gamma \in 1 \rightarrow 1 . \gamma \subset \text{ᗡ}ʻR . \gamma \subset \text{ᗡ}ʻS . Rʻʻ\gamma \cap Sʻʻ\gamma = \Lambda :\\ +&Rʻʻ\gamma \in \text{Cls}^{2} \text{excl} . \lor . Sʻʻ\gamma \in \text{Cls}^{2} \text{excl} : \supset .\\ +&{\in}_{\Delta}ʻ(Rʻʻ\gamma \cup Sʻʻ\gamma ) \text{ sm } \text{Prod}ʻ\hat{\mu} \{(\exists z) . z \in \gamma . \mu = Rʻz \times Sʻz\}\\ +&[*115·601·602·12 . *114·65]\end{align}\]</p> + +<p class="nind"><b>*115·62.</b> + \[\begin{align}\vdash : &(Rʻʻ\gamma ) \upharpoonleft R, S \upharpoonright \gamma \in 1 \rightarrow 1 . \gamma \subset \text{ᗡ}ʻR . \gamma \subset \text{ᗡ}ʻS . Rʻʻ\gamma \cap Sʻʻ\gamma = \Lambda .\\ +&(Rʻʻ\gamma \cup Sʻʻ\gamma ) \in \text{Cls}^{2} \text{excl} . \supset .\\ +&\text{Prod}ʻ(Rʻʻ\gamma \cup Sʻʻ\gamma ) \text{ sm } \text{Prod}ʻ\hat{\mu} \{(\exists z) . z \in \gamma . \mu = Rʻz \times Sʻz\}\\ +&[*115·61·12 . *84·25]\end{align}\]</p> + +<p class="nind"><b>*115·63.</b> + \[\begin{align}&\vdash : (Rʻʻ\gamma ) \upharpoonleft R, S \upharpoonright \gamma \in 1 \rightarrow 1 . \gamma \subset \text{ᗡ}ʻR . \gamma \subset \text{ᗡ}ʻS . Rʻʻ\gamma , Sʻʻ\gamma \in \text{Cls}^{2} \text{excl} . \supset .\\ +&\text{Prod}ʻRʻʻ\gamma \times \text{Prod}ʻSʻʻ\gamma \text{ sm } \text{Prod}ʻ\hat{\mu}\{(\exists z) . z \in \gamma . \mu = Rʻz \times Sʻz\}\\ +&[*115·6·601·12]\end{align} +\]</p> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_143">[Pg 143]</span></p> +<h2 class="nobreak" id="*116">*116. EXPONENTIATION.</h2> +</div> + + +<p class="nind"><i>Summary of</i> *116.</p> + +<p>In this number, we define "\(\alpha\, \,\text{exp}\,\, \beta\)," meaning +"\(\alpha\) to the exponent \(\beta\)," where \(\alpha\) and \(\beta\) +are classes, as +\[ +\text{Prod}ʻ\alpha \downarrow_{,,}ʻʻ\beta . +\] +Now \(\text{Prod}ʻ\alpha \downarrow_{,,}ʻʻ\beta\) consists of +all ways of selecting one each from the members of +\(\alpha\downarrow_{,,}ʻʻ\beta\), <i>i.e.</i> from the classes +\(\downarrow yʻʻ\alpha\), where \(y\in \beta\). Thus to get a member of +\(\text{Prod}ʻ\alpha \downarrow_{,,}ʻʻ\beta\), take a set of couples +\(x\downarrow y\), where \(x\) is always an \(\alpha\), and there +is only one \(x\) for a given \(y\), and \(y\) is each member of +\(\beta\) in succession. Thus for each member of \(\beta\), we have +\(\text{Nc}ʻ\alpha\) possible referents; hence it is plain that the +number of possible sets of couples consists of \(\text{Nc}ʻ\beta\) +factors each equal to \(\text{Nc}ʻ\alpha\), and is therefore fit to be +taken as defining \((\text{Nc}ʻ\alpha )^{\text{Nc}ʻ\beta}\).</p> + +<p>The definitions of \(\mu ^\nu\) and \((\text{Nc}ʻ\alpha)^{\text{Nc}ʻ\beta}\) +are derived from the definition of \(\alpha \,\,\text{exp}\,\, \beta\) +exactly as the definitions of \(\mu +_{c} \nu\) +and \(\text{Nc}ʻ\alpha +_{c} \text{Nc}ʻ\beta\), or of \(\mu \times_{c} \nu\) +and \(\text{Nc}ʻ\alpha \times _{c} \text{Nc}ʻ\beta\), were +derived respectively from \(\alpha + \beta\) and \(\alpha \times\beta\).</p> + +<p>The chief difficulty in this number lies in the proof of the three +formal laws of exponentiation, namely +\[ +\begin{aligned} +\mu ^\nu \times _{c} \mu ^{\varpi } &= \mu ^{\nu +_{c} \varpi },\\ +\mu ^{\varpi } \times _{c} \nu ^{\varpi } &= (\mu \times _{c} \nu )^{\varpi },\\ +\text{and}\qquad (\mu ^\nu )^{\varpi } &= \mu ^{\nu \times _{c} \varpi }. +\end{aligned} +\] +The proofs of the second and third of these, in particular, require +various lemmas; but there is no difficulty involved except the +complexity of the classes and relations concerned.</p> + +<p>The definition of \(\mu ^\nu\) is so framed as to minimize the +necessity for the multiplicative axiom (see the note on *<a href="#*113·31">*113·31</a> in the +introduction to *113). We have</p> + +<p class="nind"><b>*116·36.</b> \(\vdash \colon\ldotp \text{Mult ax}.\supset :\mu ,\nu \in \text{NC} - \iota ʻ\Lambda .\kappa \in \nu \cap \text{Cl}ʻ\mu .\supset .\Pi \text{Nc}ʻ\kappa = \mu ^\nu\)</p> + +<p class="nind"> +that is, assuming the multiplicative axiom, the product of \(\nu\) +factors each equal to \(\mu\) is \(\mu ^\nu\) (assuming \(\mu\) and +\(\nu\) to be cardinals which are not null).</p> + +<p>If we had <i>defined</i> \(\mu ^\nu\) as the product of \(\nu\) factors +each equal to \(\mu\), we should<span class="pagenum" id="Page_144">[Pg 144]</span> have required the multiplicative +axiom for almost all propositions on \(\mu ^\nu\) but by taking the +particular class \(\alpha \downarrow_{,,}ʻʻ\beta\), we avoid the +multiplicative axiom except in a few propositions. Among these few is +the above proposition connecting exponentiation with multiplication.</p> + +<p>Cantor has defined \(\mu ^\nu\) by means of the class of "Belegungen," +<i>i.e.</i> the class +\[ +\hat{R} (R\in 1\rightarrow \text{Cls}.\text{D}ʻR\subset \alpha .\text{ᗡ}ʻR = \beta ) +\] +which (<a href="#*116·12">*116·12</a>) = \((\alpha \uparrow \beta )_{\Delta }ʻ\beta\). +By *85·53 and <a href="#*113·103">*113·103</a>, this class is equal to \(\dot{s}ʻʻ(\alpha\,\text{exp}\,\,\beta)\) +(as is proved in <a href="#*116·13">*116·13</a>), whence, since +\(\dot{s} \upharpoonright \alpha \,\text{exp}\,\, \beta \in 1\rightarrow1\), +it follows (<a href="#*116·15">*116·15</a>) that the class of "Belegungen" is similar to +\(\alpha\, \,\text{exp}\,\, \beta\). Hence our definition gives the same +value of \(\mu^\nu\) as Cantor's.</p> + +<p>The propositions of the present number begin with various simple +properties of \(\alpha\, \,\text{exp}\,\, \beta\). Its existence follows +from</p> + +<p class="nind"><b>*116·152.</b> \(\vdash :x\in \alpha .\supset .x\downarrow ʻʻ\beta \in \alpha\, \,\text{exp}\,\, \beta\)</p> + +<p class="nind"> +whence (<a href="#*116·16">*116·16</a>) \(\vdash .\text{Cnv}ʻʻʻ\beta \downarrow_{,,}ʻʻ\alpha \subset \alpha\, \,\text{exp}\,\, \beta\), and</p> + +<p class="nind"><b>*116·18.</b> \(\vdash \colon\ldotp \exists !\alpha .\lor.\beta = \Lambda : \equiv .\exists !\alpha\, \,\text{exp}\,\, \beta\)</p> + +<p>We have</p> + +<p class="nind"><b>*116·19.</b> \(\vdash :\alpha \text{ sm }\gamma .\beta \text{ sm }\delta .\supset .(\alpha\, \,\text{exp}\,\, \beta)\text{ sm } \text{ sm }(\gamma \, \,\text{exp}\,\, \delta \)</p> + +<p class="nind"> +in virtue of <a href="#*113·13">*113·13</a> and <a href="#*115·51">*115·51</a>. <a href="#*116·192">*116·192</a> shows that, if +\(R\upharpoonright \gamma\) correlates \(\alpha\) with \(\gamma\), and +\(S\upharpoonright \delta\) correlates \(\beta\) with \(\delta\), then +\((R\parallel \breve{S})\upharpoonright (\delta \times \gamma)\) is a +double correlator of \(\alpha\, \,\text{exp}\,\, \beta\) with \((\gamma\,\,\text{exp}\,\, \delta)\).</p> + +<p>We then proceed to a set of propositions on \(\mu ^\nu\) which are +analogous to <a href="#*113·2">*113·2</a> ff. on \(\mu \times _{c }\nu\). We have</p> + +<p class="nind"><b>*116·203.</b> \(\vdash :\exists !\mu ^\nu .\supset .\mu ,\nu \in \text{NC} - \iota ʻ\Lambda .\mu ,\nu \in \text{N}_{0}\text{C}\)</p> + +<p class="nind"><b>*116·25.</b> \(\vdash .(\text{Nc}ʻ\gamma )^{\text{Nc}ʻ\delta } = \text{Nc}ʻ(\gamma \, \,\text{exp}\,\, \delta)\)</p> + +<p class="nind">and various other less useful propositions.</p> + +<p>We then have various propositions on 0 and 1 and 2. We prove</p> + +<p class="nind"><b>*116·301.</b> \(\vdash :\mu \in \text{NC} - \iota ʻ\Lambda .\supset .\mu ^{0} = 1\)</p> + +<p class="nind"><b>*116·311.</b> \(\vdash :\nu \in \text{NC} - \iota ʻ\Lambda - \iota ʻ0.\supset .0^\nu = 0\)</p> + +<p class="nind"><b>*116·321.</b> \(\vdash :\mu \in \text{NC} - \iota ʻ\Lambda .\supset .\mu ^{1} = \text{ sm }ʻʻ\mu\)</p> + +<p>(Observe that \(\text{ sm }ʻʻ\mu\) is the same cardinal as \(\mu\), but +rendered typically ambiguous.)</p> + +<p class="nind"><b>*116·331.</b> \(\vdash :\mu \in \text{NC} - \iota ʻ\Lambda .\supset .1^\mu = 1\)</p> + +<p class="nind"><b>*116·34.</b> \(\vdash .\mu ^{2} = \mu \times _{c} \mu\)</p> + +<p>(This proposition does not require that \(\mu\) should be a cardinal.)</p> + +<p><span class="pagenum" id="Page_145">[Pg 145]</span></p> + +<p>After the proposition (<a href="#*116·36">*116·36</a>) already quoted, on the connection of +exponentiation and multiplication, we proceed to a set of propositions +on the case where a number of classes are all given as similar (by +assignable correlations) to a given class. In <a href="#*116·411">*116·411</a>, we prove that +if \(\kappa\) is a class of mutually exclusive classes, each of which +is similar to a given class \(\gamma\), and if, when \(\alpha \in\kappa\), +\(Mʻ\alpha\) is a correlator of \(\alpha\) and \(\gamma\), +and \(T\) is the sum of \(Mʻʻ\kappa\), then +\[ +\text{Nc}ʻ{\in}_{\Delta}ʻ\overrightarrow{T}ʻʻ\gamma = \text{Nc}ʻT_{\Delta }ʻ\gamma = \text{Nc}ʻ(\kappa \,\text{exp}\,\, \gamma ) = (\text{Nc}ʻ\kappa )^{\text{Nc}ʻ\gamma }. +\] +This is a further connection of multiplication and exponentiation. (On +the purport of this and following propositions, see the explanation +preceding <a href="#*116·4">*116·4</a>.) In <a href="#*116·43">*116·43</a>, the hypothesis is somewhat modified. +We still have a set \(\kappa\) of classes which are all similar to +\(\gamma\), but the correlator for a given class \(\alpha\) is not +given as \(Mʻ\alpha\), but is given as \(Mʻw\), where \(w\) is a member +of a class \(\delta\) which is similar to \(\kappa\). Then +\(\kappa =\text{D}ʻʻMʻʻ\delta\). We assume that \(M\upharpoonright \delta\) is a +one-one, and that if \(Mʻw\) and \(Mʻv\) have domains which overlap, +then \(w = v\). Thus \(\kappa\) is a class of mutually exclusive +classes, each of which has \(\text{Nc}ʻ\gamma\) terms, while \(\kappa\) +has \(\text{Nc}ʻ\delta\) terms. Then it is proved in *116·43 that +\[ +\text{Prod}ʻ\text{D}ʻʻMʻʻ\delta \text{ sm } \text{ sm } (\gamma \,\text{exp}\,\, \delta ).\Pi \text{Nc}ʻ\text{D}ʻʻMʻʻ\delta = (\text{Nc}ʻ\gamma )^{\text{Nc}ʻ\delta }. +\] +This proposition and another (<a href="#*116·45">*116·45</a>) which follows from it are useful +in proving the formal laws of exponentiation. The proof of these +occupies the following propositions from <a href="#*116·5">*116·5</a> to <a href="#*116·68">*116·68</a>. We have</p> + +<p class="nind"><b>*116·52.</b> \(\vdash .\mu ^\nu \times _{c} \mu ^{\varpi } = \mu ^{\nu +_{c} \varpi }\)</p> + +<p class="nind"><b>*116·55.</b> \(\vdash .\mu ^{\varpi } x_{c} \nu ^{\varpi } = (\mu \times _{c} \nu )^{\varpi }\)</p> + +<p class="nind"><b>*116·63.</b> \(\vdash .\mu ^{\nu \times _{c} \varpi } = (\mu ^\nu )^{\varpi }\)</p> + +<p>An extension of the first of these is</p> + +<p class="nind"><b>*116·661.</b> \(\vdash .\Pi \text{Nc}ʻ(\alpha \,\text{exp}\,)ʻʻ\kappa = (\text{Nc}ʻ\alpha )^{\Sigma \text{Nc}ʻ\kappa }\)</p> + +<p>Here the number of members of \(\kappa\) need not be finite. The +purport of the proposition is as follows: Let \(\beta\), \(\gamma\), +\(\delta\), ... be the members of \(\kappa\); form \(\alpha\,\,\text{exp}\,\,\beta\), +\(\alpha \, \,\text{exp}\,\, \gamma\), \(\alpha \,\,\text{exp}\,\, \delta\), ..., +and take the product of the numbers of all these; then the resulting +number is the same as if we first took the sum of the numbers of all +the members of \(\kappa\), thus obtaining (say) a number \(\mu\), and +raised \(\text{Nc}ʻ\alpha\) to the \(\mu\)th power.</p> + +<p>An extension of <a href="#*116·55">*116·55</a> is given by <a href="#*116·68">*116·68</a>, where we prove +\[ +\vdash :\kappa \in \text{Cls}^{2} \text{excl}.\supset .\Pi \text{Nc}ʻ\,\text{exp}\,\gamma ʻʻ\kappa = (\Pi \text{Nc}ʻ\kappa )^{\text{Nc}ʻ\gamma }. +\]</p> + +<p>There is no analogous extension of <a href="#*116·63">*116·63</a>.</p> + +<p>We prove next Cantor's proposition (which is very useful)</p> + +<p class="nind"><b>*116·72.</b> \(\vdash .\text{Nc}ʻ\text{Cl}ʻ\alpha = 2^{\text{Nc}ʻ\alpha }\)</p> + +<p><i>I.e.</i> the number of combinations of \(\mu\) things any number +at a time is \(2^\mu\). (Observe that \(\mu\) need not be finite.) +The remainder of the number is concerned with consequences of this +proposition.</p> + +<p><span class="pagenum" id="Page_146">[Pg 146]</span></p> + +<hr class="tb"> + +<p class="nind"><b>*116·01.</b> \(\alpha \,\text{exp}\,\,\beta = \text{Prod}ʻ\alpha \downarrow_{,,}ʻʻ\beta \quad\text{Df}\)</p> + +<p class="nind"><b>*116·02.</b> \(\mu ^\nu =\hat{\gamma}\{(\exists \alpha ,\beta ).\mu = \text{N}_{0}\text{c}ʻ\alpha .\nu = \text{N}_{0}\text{c}ʻ\beta + .\gamma \text{ sm }(\alpha \,\text{exp}\,\,\beta )\} \quad\text{Df}\)</p> + +<p class="nind"><b><a id="*116·03">*116·03</a>.</b> \((\text{Nc}ʻ\alpha )^\nu = (\text{N}_{0}\text{c}ʻ\alpha )^\nu \quad\text{Df}\)</p> + +<p class="nind"><b><a id="*116·04">*116·04</a>.</b> \(\mu ^{\text{Nc}ʻ\beta } = \mu ^{\text{N}_{0}\text{c}ʻ\beta } \quad\text{Df}\)</p> + +<p class="nind"><b>*116·1.</b> \[\begin{align}&\vdash :\xi \in (\alpha \,\text{exp}\,\, \beta ). \equiv .(\exists R).R\in {\in}_{\Delta}ʻ\alpha \downarrow_{,,}ʻʻ\beta .\xi = \text{D}ʻR\\ +&[*115·1.(*116·01)]\end{align}\]</p> + +<p class="nind"><b>*116·11.</b> \(\vdash \colon\ldotp \xi \in (\alpha \,\text{exp}\,\, \beta ). \equiv :y\in \beta .\supset _{y}.\alpha + \cap \hat{x} (x\downarrow y\in \xi )\in 1:\xi \subset \beta \times \alpha \)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*113·111.*115·11.\supset \\ +\vdash \colon\ldotp \xi \in (\alpha \,\text{exp}\,\, \beta ). &\equiv :\rho \in \alpha \downarrow_{,,}ʻʻ\beta .\supset _{\rho }.\rho + \cap \xi \in 1:\xi \subset sʻ\alpha \downarrow_{,,}ʻʻ\beta :\\ +[*38·2.*113·1] &\equiv :y\in \beta .\supset _{y}.\downarrow yʻʻ\alpha \cap \xi \in 1:\xi \subset \beta \times \alpha &\qquad \text{(1)}\\ +\vdash .*37·6.\supset \\ +\vdash :\downarrow yʻʻ\alpha \cap \xi \in 1.& \equiv .\hat{R}\{(\exists x).x\in \alpha .R = x\downarrow y.R\in \xi\}\in 1.\\ +[*13·193] &\equiv .\hat{R}\{(\exists x).x\in \alpha .x\downarrow y\in \xi .R = x\downarrow y\}\in 1.\\ +[*37·6] &\equiv .\downarrow yʻʻ\hat{x} (x\in \alpha .x\downarrow y\in \xi )\in 1.\\ +[*73·611·44] &\equiv .\hat{x} (x\in \alpha .x\downarrow y\in \xi )\in 1 &\qquad \text{(2)}\\ +\vdash .(1).(2).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*116·12">*116·12</a>.</b> \(\vdash .(\alpha \uparrow \beta )_{\Delta }ʻ\beta = \hat{R} \{R\in 1\rightarrow \text{Cls}.\text{D}ʻR\subset \alpha .\text{ᗡ}ʻR = \beta\}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*80·14.\supset \vdash :R\in (\alpha \uparrow \beta )_{\Delta }ʻ\beta . &\equiv .R\in 1\rightarrow \text{Cls}.R\,\unicode{x2abd}\, \alpha \uparrow \beta .\text{ᗡ}ʻR = \beta .\\ +[*35·83] &\equiv .R\in 1\rightarrow \text{Cls}.\text{D}ʻR\subset \alpha .\text{ᗡ}ʻR\subset \beta .\text{ᗡ}ʻR = \beta .\\ +[*22·42] &\equiv .R\in 1\rightarrow \text{Cls}.\text{D}ʻR\subset \alpha .\text{ᗡ}ʻR = \beta :\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*116·13">*116·13</a>.</b> \(\vdash .sʻʻ(\alpha \,\text{exp}\,\, \beta ) = (\alpha\uparrow \beta )_{\Delta }ʻ\beta\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*85·53.\supset \vdash .(\alpha \uparrow \beta )_{\Delta }ʻ\beta &= \dot{s}ʻʻ\text{D}ʻʻ{\in}_{\Delta}ʻ(\alpha \uparrow \beta )\unicode{x21A7}ʻʻ\beta \\ +[*113·103] &= \dot{s}ʻʻ\text{D}ʻʻ{\in}_{\Delta}ʻ\alpha \downarrow_{,,}ʻʻ\beta \\ +[*115·1.(*116·01)] &= \dot{s}ʻʻ(\alpha \,\text{exp}\,\, \beta ).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p>\((\alpha \uparrow \beta )_{\Delta }ʻ\beta\) is the class of +one-many relations whose converse domain is \(\beta\) and whose +domain is contained in \(\alpha\). This is what Cantor calls +the "Belegungsmenge," and is used by him as the definition of +exponentiation. In virtue of <a href="#*116·15">*116·15</a>, his definition gives the same +results as ours.</p> + +<p class="nind"><b><a id="*116·131">*116·131</a>.</b> \(\vdash .\dot{s} \upharpoonright (\alpha \,\,\text{exp}\,\, \beta )\in \{(\alpha \uparrow \beta )_{\Delta }ʻ\beta\} + \,\overline{\text{ sm }}\, (\alpha \,\,\text{exp}\,\, \beta )\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*84·241.*113·103.&\supset \vdash .\iotaʻʻ\beta \in \text{Cls}^{2} \text{excl} .\alpha \downarrow_{,,}ʻʻ\beta = (\alpha \uparrow \beta )_{\Delta }ʻʻ\iotaʻʻ\beta + &\qquad \text{(1)}\\ +\vdash .(1).*85·42.&\supset \vdash :M,N\in {\in}_{\Delta}ʻ\alpha \downarrow_{,,}ʻʻ\beta .\dot{s} ʻ\text{D}ʻM = \dot{s} ʻ\text{D}ʻN.\supset .M = N.\\ +[*30·37] &\supset .\text{D}ʻM = \text{D}ʻN &\qquad \text{(2)}\\ +\vdash .(2).*37·63.*115·1.(*116·01).&\supset \vdash :\mu ,\nu \in (\alpha \,\,\text{exp}\,\, \beta ).\dot{s} ʻ\mu = \dot{s} ʻ\nu .\supset .\mu = \nu :\\ +[*71·55.*72·163] &\supset \vdash .\dot{s} \upharpoonright (\alpha \,\,\text{exp}\,\, \beta )\in 1\rightarrow 1 &\qquad \text{(3)}\\ +\vdash .(3).*116·13.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_147">[Pg 147]</span></p> + +<p class="nind"><b>*116·14.</b> \(\vdash .(\alpha \,\,\text{exp}\,\, \beta )\text{ sm }{\in}_{\Delta} ʻ\alpha \downarrow_{,,}ʻʻ\beta \quad[*115·12.*113·111]\)</p> + +<p class="nind"><b><a id="*116·15">*116·15</a>.</b> \(\vdash .(\alpha \,\,\text{exp}\,\, \beta )\text{ sm }(\alpha \uparrow \beta )_{\Delta }ʻ\beta \quad[*116·131]\)</p> + +<p>*116·151 is a lemma for *116·152.</p> + +<p class="nind"><b>*116·151.</b> \( \vdash :x \in \alpha .\supset .x\downarrow \mid \text{Cnv}ʻ(\alpha \downarrow_{,,}\upharpoonright \beta ) \in {\in}_{\Delta}ʻ\alpha + \downarrow_{,,}ʻʻ\beta\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*113·105.*72·184.\supset \vdash :\text{Hp}.&\supset .x\downarrow \mid \text{Cnv}ʻ(\alpha \downarrow_{,,}\upharpoonright + \beta ) \in 1\rightarrow \text{Cls} &\qquad \text{(1)}\\ +\vdash .*34·1.*38·1. &\supset \vdash \colon\ldotp \text{Hp}.\supset :R{x\downarrow \mid \text{Cnv}ʻ(\alpha \downarrow_{,,}\upharpoonright \beta )} \lambda .\equiv .\\ +&(\exists y).R=x\downarrow y.y \in \beta . \lambda =\alpha \downarrow_{,,}y.x \in \alpha .\\ +[*38·21] &\supset .R \in \lambda &\qquad \text{(2)}\\ +\vdash .*37·322·401. &\supset \vdash .\text{ᗡ}ʻ{x\downarrow \mid \text{Cnv}ʻ(\alpha \downarrow_{,,}\upharpoonright \beta )}=\alpha \downarrow_{,,}ʻʻ\beta &\qquad \text{(3)}\\ +\vdash .(1).(2).(3).*80·14.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*116·152.</b> \(\vdash :x \in \alpha .\supset .x\downarrowʻʻ\beta \in (\alpha \,\,\text{exp}\,\,\beta )\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*37·32.*35·65.&\supset \vdash .\text{D}ʻ{x\downarrow \mid \text{Cnv}ʻ(\alpha \downarrow_{,,}\upharpoonright \beta )}=x\downarrowʻʻ\beta &\qquad \text{(1)}\\ +\vdash .(1).*116·151·1.&\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*116·16">*116·16</a>.</b> \(\vdash .\text{Cnv}ʻʻʻ\beta \downarrow_{,,}ʻʻ\alpha \subset \alpha \,\,\text{exp}\,\,\beta\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*116·152.*55·14.\supset \vdash :x \in \alpha .&\supset .\text{Cnv}ʻʻ\downarrow xʻʻ\beta \in (\alpha \,\,\text{exp}\,\,\beta ).\\ +[*38·2] &\supset .\text{Cnv}ʻʻ\beta \downarrow_{,,}x \in (\alpha \,\,\text{exp}\,\,\beta ):\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p>The above propositions are useful in establishing existence-theorems, +as appears in the following propositions.</p> + +<p class="nind"><b>*116·17.</b> \(\vdash :\exists !\beta \downarrow_{,,}ʻʻ\alpha .\supset .\exists !\alpha \,\,\text{exp}\,\,\beta \quad[*116·16.*37·47]\)</p> + +<p class="nind"><b>*116·171.</b> \(\vdash \colon\ldotp \exists !\alpha .\lor.\beta =\Lambda :\supset .\exists !\alpha \,\,\text{exp}\,\,\beta\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*113·113.*83·15.*51·161.&\supset \vdash :\beta =\Lambda .\supset .\exists !\alpha \,\,\text{exp}\,\,\beta &\qquad \text{(1)}\\ +\vdash .*116·152. &\supset \vdash :\exists !\alpha .\supset .\exists !\alpha \,\,\text{exp}\,\,\beta &\qquad \text{(2)}\\ +\vdash .(1).(2).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*116·172.</b> \(\vdash \colon\ldotp \exists !\alpha \,\,\text{exp}\,\,\beta .\supset :\exists !\alpha .\lor.\beta =\Lambda\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*83·11.\supset \vdash \colon\ldotp \text{Hp}.&\supset :\Lambda {\sim}\in \alpha \downarrow_{,,}ʻʻ\beta :\\ +[*113·112] &\supset :{\sim}(\alpha =\Lambda .\exists !\beta ):\\ +[*24·51] &\supset :\exists !\alpha .\lor.\beta =\Lambda \colon\ldotp \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*116·18.</b> \(\vdash \colon\ldotp \exists !\alpha .\lor.\beta =\Lambda :\equiv .\exists !\alpha \,\,\text{exp}\,\,\beta \quad[*116·171·172]\)</p> + +<p class="nind"><b>*116·181.</b> \(\vdash .\alpha \,\,\text{exp}\,\,\Lambda =\iota ʻ\Lambda\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*113·113.\supset \vdash .\alpha \,\,\text{exp}\,\,\Lambda &=\text{Prod}ʻ\Lambda \\ +[*83·15.*33·241] &=\iota ʻ\Lambda .\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_148">[Pg 148]</span></p> + +<p class="nind"><b>*116·182.</b> \(\vdash : \exists ! \beta . \supset . \Lambda \,\,\text{exp}\,\, \beta = \Lambda \quad[*113·112 . *83·11]\)</p> + +<p class="nind"><b>*116·183.</b> \(\vdash . sʻ(\alpha \,\,\text{exp}\,\, \beta ) = \beta \times \alpha\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *115·141 . *116·18 . \supset \vdash \colon\ldotp \exists ! \alpha . \lor . \beta &= \Lambda : \supset . sʻ(\alpha \,\,\text{exp}\,\, \beta ) = sʻ\alpha \downarrow_{,,}ʻʻ\beta + \\ +[*113·1] &= \beta \times \alpha &\qquad \text{(1)}\\ +\vdash . *116·182 . \supset \vdash \colon\ldotp \alpha &= \Lambda . \exists ! \beta . \supset . sʻ(\alpha \,\,\text{exp}\,\, \beta ) = \Lambda \\ +[*113·114] &= \beta \times \alpha &\qquad \text{(2)}\\ +\vdash . (1) . (2) . \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*116·19.</b> \(\vdash : \alpha \text{ sm } \gamma . \beta \text{ sm } \delta . \supset . (\alpha \,\,\text{exp}\,\, \beta ) \text{ sm } \text{ sm } (\gamma \,\,\text{exp}\,\, \delta )\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *113·13 . \supset \vdash : \text{Hp} . &\supset . \alpha \downarrow_{,,}ʻʻ\beta \text{ sm } \text{ sm } \gamma \downarrow_{,,}ʻʻ\delta .\\ +[*115·51] &\supset . (\alpha \,\,\text{exp}\,\, \beta ) \text{ sm } \text{ sm } (\gamma \,\,\text{exp}\,\, \delta ) : \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*116·191.</b> + \[\begin{align}\vdash : R \in \alpha \overline{\text{ sm }} \gamma . S \in \beta \,\overline{\text{ sm }}\, \delta . \supset . &(R \parallel \breve{S} ) \upharpoonright (\delta \times \gamma ) \in (\alpha \,\,\text{exp}\,\, \beta ) \,\overline{\text{ sm }}\, \,\overline{\text{ sm }}\, (\gamma \,\,\text{exp}\,\, \delta ) .\\ +&(R \parallel \breve{S} )_{\in}ʻʻ(\gamma \,\,\text{exp}\,\, \delta ) = \alpha \,\,\text{exp}\,\, \beta \\ +[*113·127 . *115·502 . *116·183]\end{align}\]</p> + +<p class="nind"><b><a id="*116·192">*116·192</a>.</b> + \[\begin{align}\vdash : R \upharpoonright \gamma \in \alpha \,\overline{\text{ sm }}\, \gamma . S \upharpoonright \delta \in \beta &\,\overline{\text{ sm }}\, \delta . \supset .\\ +&(R \parallel \breve{S} ) \upharpoonright (\delta \times \gamma ) \in (\alpha \,\,\text{exp}\,\, \beta ) \,\overline{\text{ sm }}\, \,\overline{\text{ sm }}\, (\gamma \,\,\text{exp}\,\, \delta ) .\\ +&(R \parallel \breve{S} )_{\in } \upharpoonright (\gamma \,\,\text{exp}\,\, \delta ) \in (\alpha \,\,\text{exp}\,\, \beta ) \,\overline{\text{ sm }}\, (\gamma \,\,\text{exp}\,\, \delta )\\ +&[*113·127 . *115·502 . *116·183 . *111·15]\end{align}\]</p> + +<p class="nind"><b>*116·194.</b> + \[\begin{align}\vdash : R \upharpoonright \gamma \in \alpha \,\overline{\text{ sm }}\, \gamma . &S \upharpoonright \delta \in \beta \,\overline{\text{ sm }}\, \delta . \supset .\\ +&(R \parallel \breve{S} ) \upharpoonright {(\gamma \uparrow \delta )_{\Delta }ʻ\delta } \in {(\alpha \uparrow \beta )_{\Delta }ʻ\beta + } \,\overline{\text{ sm }}\, {(\gamma \uparrow \delta )_{\Delta }ʻ\delta }\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *116·12 . \supset \vdash : \text{Hp}. &\supset .sʻ\text{D}ʻʻ(\gamma \uparrow \delta )_{\Delta }ʻ\delta + \subset \gamma . sʻ\text{ᗡ}ʻʻ(\gamma \uparrow \delta )_{\Delta }ʻ\delta \subset \delta .\\ +[*74·773 . *73·142] \supset . (R \parallel \breve{S} ) &\upharpoonright {(\gamma \uparrow \delta )_{\Delta }ʻ\delta } \in \\ +&\{(R \parallel \breve{S} )ʻʻ(\gamma \uparrow \delta )_{\Delta }ʻ\delta\} \,\overline{\text{ sm }}\, \{(\gamma \uparrow \delta )_{\Delta }ʻ\delta\} &\qquad \text{(1)}\\ +\vdash . *116·192 . *111·14 . \supset \vdash : \text{Hp} . &\supset . \alpha \,\,\text{exp}\,\, \beta = (R \parallel \breve{S} )_{\in}ʻʻ(\gamma \,\,\text{exp}\,\, \delta ) .\\ +[*116·13] \supset . (\alpha \uparrow \beta )_{\Delta }ʻ\beta &= \dot{s}ʻʻ(R \parallel \breve{S} )_{\in}ʻʻ(\gamma \,\,\text{exp}\,\, \delta )\\ +[*43·43] &= (R \parallel \breve{S} )ʻʻ\dot{s}ʻʻ(\gamma \,\,\text{exp}\,\, \delta )\\ +[*116·13] &= (R \parallel \breve{S} )ʻʻ(\gamma \uparrow \delta )_{\Delta }ʻ\delta &\qquad \text{(2)}\\ +\vdash . (1) . (2) . \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p>The following propositions (down to <a href="#*116·27">*116·27</a> exclusive) are the +analogues of propositions with the same decimal part in <a href="#*113">*113</a>.</p> + +<p class="nind"><b>*116·2.</b> \(\vdash : \xi \in \mu ^\nu . \equiv . (\exists \alpha , \beta ) . \mu = \text{N}_{0}\text{c}ʻ\alpha . \nu = \text{N}_{0}\text{c}ʻ\beta + . \xi \text{ sm } (\alpha \,\,\text{exp}\,\, \beta ) \quad[(*116·02)]\)</p> + +<p class="nind"><b>*116·201.</b> \[\begin{align}&\vdash \colon\ldotp \xi \in \mu ^\nu + . \equiv : \mu , \nu \in \text{NC} : (\exists \alpha , \beta ) . \alpha \in \mu . \beta \in \nu . \xi \text{ sm } (\alpha \,\,\text{exp}\,\, \beta )\\ +&[*116·2 . *103·27]\end{align}\]</p> + +<p><span class="pagenum" id="Page_149">[Pg 149]</span></p> + +<p class="nind"><b>*116·202.</b> \[\begin{align}&\vdash \colon\ldotp \xi \in \mu ^\nu + . \equiv :\exists !\mu .\exists !\nu :(\exists \alpha ,\beta ).\mu = \text{Nc}ʻ\alpha .\nu = \text{Nc}ʻ\beta .\xi \text{ sm } (\alpha \,\,\text{exp}\,\, \beta )\\ +&[\text{Proof as in}\, *113·202]\end{align}\]</p> + +<p class="nind"><b>*116·203.</b> \(\vdash :\exists !\mu ^\nu .\supset .\mu ,\nu \in \text{NC}-\iota ʻ\Lambda .\mu ,\nu \in \text{N}_0\text{C} \quad[*116·201·202·2]\)</p> + +<p class="nind"><b><a id="*116·204">*116·204</a>.</b> \(\vdash \colon\ldotp \mu = \Lambda .\lor.\nu = \Lambda .\lor.{\sim}(\mu ,\nu \in \text{NC}):\supset .\mu ^\nu = \Lambda \quad[*116·203]\)</p> + +<p class="nind"><b>*116·205.</b> \(\vdash :{\sim}(\mu ,\nu \in \text{N}_0\text{C}).\supset .\mu ^\nu = \Lambda \quad [*116·203]\)</p> + +<p class="nind"><b>*116*21.</b> \(\vdash \colon\ldotp \mu ,\nu \in \text{NC}.\supset :\xi \in \mu ^\nu + . \equiv .(\exists \alpha ,\beta ).\alpha \in \mu .\beta \in \nu .\xi \text{ sm } (\alpha \,\,\text{exp}\,\, \beta ) \quad[*116·201]\)</p> + +<p class="nind"><b>*116·22.</b> \[\begin{align}&\vdash :\xi \in {\text{Nc}(\eta )ʻ\gamma }^{\text{Nc}(\zeta )ʻ^\delta + }. \equiv .\exists !\text{Nc}(\eta )ʻ\gamma .\exists !\text{Nc}(\zeta )ʻ\delta .\xi \text{ sm } (\gamma \,\,\text{exp}\,\, \delta )\\ +&[\text{Proof as in *113·22, using *116·19 in place of *113·13}]\end{align}\]</p> + +<p class="nind"><b>*116·221.</b> \[\begin{align}&\vdash :\exists !\text{Nc}(\eta )ʻ\gamma .\exists !\text{Nc}(\zeta )ʻ\delta .\supset .{\text{Nc}(\eta )ʻ\gamma }^{\text{Nc}(\zeta + )ʻ\delta } = \text{Nc}ʻ(\gamma \,\,\text{exp}\,\, \delta )\\ +&[*116·22]\end{align}\]</p> + +<p class="nind"><b>*116·222.</b> \(\vdash .(\text{N}_0\text{c}ʻ\gamma )^{\text{N}_0\text{c}ʻ\delta } = \text{Nc}ʻ(\gamma \,\,\text{exp}\,\, \delta ) \quad[\text{Proof as in *113·222}]\)</p> + +<p class="nind"><b><a id="*116·23">*116·23</a>.</b> \(\vdash .\mu ^\nu \in \text{NC} \quad[\text{Proof as in *113·23}]\)</p> + +<p class="nind"><b>*116·24.</b> \(\vdash .(\text{Nc}ʻ\gamma )^{\text{Nc}ʻ\delta } = (\text{N}_0\text{c}ʻ\gamma )^{\text{N}_0\text{c}ʻ\delta } \quad[(*116·03·04)]\)</p> + +<p class="nind"><b>*116·25.</b> \(\vdash .(\text{Nc}ʻ\gamma )^{\text{Nc}ʻ\delta } = \text{Nc}ʻ(\gamma \,\,\text{exp}\,\, \delta ) \quad[*116·24·222]\)</p> + +<p class="nind"><b>*116·251.</b> \(\vdash .(\gamma \,\,\text{exp}\,\, \delta )\in (\text{Nc}ʻ\gamma )^{\text{Nc}ʻ\delta } \quad[*116·25.*100·3]\)</p> + +<p class="nind"><b><a id="*116·26">*116·26</a>.</b> \[\begin{align}&\vdash :\mu ,\nu \in \text{NC}.\exists !\text{ sm }_\eta ʻʻ\mu .\exists !\text{ sm }_\zetaʻʻ\nu .\supset .\mu ^\nu = (\text{ sm }_\eta ʻʻ\mu )^{\text{ sm }_\zetaʻʻ\nu + }\\ +&[\text{Proof as in *113·26}]\end{align}\]</p> + +<p>This proposition shows that we may raise or lower the types of \(\mu\) +and \(\nu\) as we please, without affecting the value of \(\mu ^\nu\) +provided \(\mu\) and \(\nu\), or rather \(\text{ sm }ʻʻ\mu\) and +\(\text{ sm }ʻʻ\nu\), exist in the new types.</p> + +<p class="nind"><b>*116·261.</b> \(\vdash :\mu ,\nu \in \text{NC}.\supset .\mu ^\nu = \{\mu ^{(1)}\}^{\nu ^{(1)}} = \{\mu _{(00)}\}^{\nu _{(00)}} + = etc. \quad[\text{Proof as in *113·261}]\)</p> + +<p>Here "etc." covers any derivative of \(\mu\) or \(\nu\) whose existence +follows from that of \(\mu\) or \(\nu\).</p> + +<p class="nind"><b><a id="*116·27">*116·27</a>.</b> \[\begin{align}&\vdash .\mu ^\nu + = \hat{\xi} \{(\exists \alpha ,\beta ).\mu = \text{N}_0\text{c}ʻ\alpha .\nu = \text{N}_0\text{c}ʻ\beta .\xi \text{ sm } (\alpha \uparrow \beta )_{\Delta }ʻ\beta\}\\ +&[*116·15.*73·37.(*116·02)]\end{align}\]</p> + +<p class="nind"><b>*116·271.</b> \(\vdash :\mu ,\nu \in \text{NC}.\alpha \in \mu .\beta \in \nu .\supset .(\alpha \,\,\text{exp}\,\, \beta )\in \mu ^\nu \quad[*116·21]\)</p> + +<p class="nind"><b>*116·3.</b> \(\vdash .(\text{Nc}ʻ\alpha )^{0} = 1\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*101·1.*116·25.\supset \vdash .(\text{Nc}ʻ\alpha )^{0} &= \text{Nc}ʻ(\alpha \,\,\text{exp}\,\, \Lambda )\\ +[*116·181] &= \text{Nc}ʻ\iota ʻ\Lambda \\ +[*101·2] & = 1.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*116·301.</b> \(\vdash :\mu \in \text{NC}-\iota ʻ\Lambda .\supset .\mu ^{0} = 1 \quad[\text{Proof as in *113·601}]\)</p> + +<p class="nind"><b>*116·31.</b> \(\vdash :\beta \neq \Lambda .\supset .0^{\text{Nc}ʻ\beta } = 0\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*101·1.*116·25.&\supset \vdash .0^{\text{Nc}ʻ\beta } = \text{Nc}ʻ(\Lambda \,\,\text{exp}\,\, \beta )\\ +[*116·182] \supset \vdash :\text{Hp}.\supset .0^{\text{Nc}ʻ\beta } &= \text{Nc}ʻ\Lambda\\ +[*101·1] &= 0:\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_150">[Pg 150]</span></p> + +<p class="nind"><b>*116·311.</b> \(\vdash :\nu \in \text{NC}-\iota ʻ\Lambda -\iota ʻ0.\supset .0^{\nu }=0\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*103·34.*101·1.\supset \vdash :\text{Hp}.&\supset .(\exists \beta ).\beta \neq \Lambda .\nu =\text{N}_{0}\text{c}ʻ\beta .\\ +[*13·12·15] \supset .(\exists \beta ).\beta \neq \Lambda .0^{\nu }&=0^{\text{N}_{0}\text{c}ʻ\beta }\\ +[*116·31.(*116·04)] &=0:\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*116·32.</b> \(\vdash .(\text{Nc}ʻ\alpha )^{1}=\text{Nc}ʻ\alpha\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*116·25.*101·2.\supset \vdash .(\text{Nc}ʻ\alpha )^{1}&=\text{Nc}ʻ{\alpha \,\,\text{exp}\,\, (\iota ʻx)}\\ +[(*116·01)] &=\text{Nc}ʻ\text{Prod}ʻ\alpha \downarrow_{,,}ʻʻ\iota ʻx\\ +[*115·142.*53·31] &=\text{Nc}ʻ\iota ʻʻ\alpha \downarrow_{,,}x\\ +[*113·11.*100·6] &=\text{Nc}ʻ\alpha .\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*116·321">*116·321</a>.</b> \(\vdash .\mu \in \text{NC}-\iota ʻ\Lambda .\supset +.\mu ^{1}=\text{ sm }ʻʻ\mu \quad[*116·32]\)</p> + +<p>It would not be an error to write "\(\mu ^{1}=\mu\)" instead of +"\(\mu ^{1}=\text{ sm }ʻʻ\mu\)" in the above proposition. For if the +"\(\text{ sm }\)" is typically determined so that \(\text{sm}ʻʻ\mu \inʻ\mu\), +then \(\text{ sm }ʻʻ\mu =\mu\). Thus in virtue of <a href="#*116·321">*116·321</a>, +\(\mu ^{1}=\mu\) is true whenever it is significant. But the above form +gives more information, since it preserves the typical ambiguity of +\(\mu ^{1}\) and \(\text{ sm }ʻʻ\mu\).</p> + +<p class="nind"><b>*116·33.</b> \(\vdash .1^{\text{Nc}ʻ\beta }=1\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*113·11. \supset \vdash :\alpha \in 1.&\supset .\alpha \downarrow_{,,}ʻʻ\beta \subset 1.\\ +[*115·144.*101·2] &\supset .\text{Nc}ʻ\text{Prod}ʻ\alpha \downarrow_{,,}ʻʻ\beta =1 &\qquad \text{(1)}\\ +\vdash .(1).*101·2. &\supset \vdash .\text{Nc}ʻ{(\iota ʻx) \,\,\text{exp}\,\, \beta }=1 &\qquad \text{(2)}\\ +\vdash .*101·2.*116·25.&\supset \vdash .1^{\text{Nc}ʻ\beta }=\text{Nc}ʻ\{(\iota ʻx) \,\,\text{exp}\,\, \beta\} &\qquad \text{(3)}\\ +\vdash .(2).(3).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*116·311.</b> \(\vdash :\mu \in \text{NC}-\iota ʻ\Lambda .\supset .1^{\mu }=1\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*103·34.\supset \vdash :\text{Hp}.&\supset .(\exists \beta ).\mu =\text{N}_{0}\text{c}ʻ\beta .\\ +[*13·12·15] &\supset .(\exists \beta ).1^{\mu }=1^{\text{N}_{0}\text{c}ʻ\beta }.\\ +[(*116·04)] &\supset .(\exists \beta ).1^{\mu }=1^{\text{Nc}ʻ\beta }.\\ +[*116·33] &\supset .1^{\mu }=1:\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*116·34.</b> \(\vdash .\mu ^{2}=\mu \times _{c}\mu\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*24·1.*101·3.\supset \vdash .\iota ʻ\Lambda \cup \iota ʻ\text{V}\in 2.\\ +[*116·222] \supset \vdash :\mu &=\text{N}_{0}\text{c}ʻ\alpha .\supset .\mu ^{2}=\text{Nc}ʻ\text{Prod}ʻ\alpha \downarrow_{,,}ʻʻ(\iota ʻ\Lambda \cup \iota ʻ\text{V})\\ +[*53·32] &=\text{Nc}ʻ\text{Prod}ʻ(\iota ʻ\alpha \downarrow_{,,}\Lambda \cup \iota ʻ\alpha \downarrow_{,,}\text{V})\\ +[*115·13.*55·233.*38·2] &=\text{Nc}ʻ(\alpha \downarrow_{,,}\Lambda \times \alpha \downarrow_{,,}\text{V})\\ +[*113·11·25·13] &=\text{Nc}ʻ\alpha \times _{c}\text{Nc}ʻ\alpha \\ +[*113·24] &=\mu \times _{c}\mu &\qquad \text{(1)}\\ +\vdash .(1).*103·2.&\supset \vdash :\mu \in \text{N}_{0}\text{C}.\supset .\mu ^{2}=\mu \times _{c}\mu &\qquad \text{(2)}\\ +\vdash .*116·205. \supset \vdash :\mu {\sim}\in \text{N}_{0}\text{C}.\supset .\mu ^{2}&=\Lambda \\ +[*113·205] &=\mu \times _{c}\mu &\qquad \text{(3)}\\ +\vdash .(2).(3).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_151">[Pg 151]</span></p> + +<p class="nind"><b>*116·35.</b> \(\vdash :\mu ^{\nu }=0.\equiv .\mu =0.\nu \in \text{NC}-\iota ʻ0-\iota ʻ\Lambda\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*116·311.&\supset \vdash :\mu =0.\nu \in \text{NC}-\iota ʻ0-\iota ʻ\Lambda .\supset .\mu ^{\nu }=0 &\qquad \text{(1)}\\ +\vdash .*101·12. \supset \vdash :\mu ^{\nu }=0.&\supset .\exists !\mu ^{\nu }.\\ +[*116·203] &\supset .\mu ,\nu \in \text{NC}-\iota ʻ\Lambda &\qquad \text{(2)}\\ +\vdash .(2)*116·21.*54·102.&\supset \\ +\vdash \colon\ldotp \mu ^{\nu }=0.&\supset :\xi =\Lambda .\equiv .(\exists \alpha ,\beta ).\alpha \in \mu .\beta \in \nu .\xi \text{ sm } (\alpha \,\,\text{exp}\,\, \beta ):\\ +[*73·47] &\supset :(\exists \alpha ,\beta ).\alpha \in \mu .\beta \in \nu .\alpha \,\,\text{exp}\,\, \beta =\Lambda :\\ +[*116·18] &\supset :(\exists \alpha ,\beta ).\alpha \in \mu .\beta \in \nu .\alpha =\Lambda .\beta \neq \Lambda :\\ +[*13·195] &\supset :\Lambda \in \mu .\nu \neq \iota ʻ\Lambda .\exists !\nu :\\ +[*101·1.*100·45.(2)] &\supset :\mu =0.\nu \in \text{NC}-\iota ʻ\Lambda -\iota ʻ0 &\qquad \text{(3)}\\ +\vdash .(1).(3).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*116·351.</b> \[\begin{align}&\vdash :\mu \in \text{NC}-\iota ʻ\Lambda .\kappa =\Lambda .\nu =0.\supset .\mu ^\nu =\Pi \text{Nc}ʻ\kappa =1\\ +&[*116·301.*114·2]\end{align}\]</p> + +<p class="nind"><b>*116·352.</b> \[\begin{align}&\vdash :\mu =0.\nu \in \text{NC}-\iota ʻ\Lambda .\kappa \in \nu .\Lambda \in \kappa .\supset .\mu ^\nu =\Pi \text{Nc}ʻ\kappa =0\\ +&[*116·311.*114·23]\end{align}\]</p> + +<p class="nind"><b>*116·353.</b> \(\vdash :\mu =0.\nu \in \text{NC}-\iota ʻ\Lambda .\kappa \in \nu \cap \text{Cl}ʻ\mu .\supset .\mu ^\nu =\Pi \text{Nc}ʻ\kappa\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*60·362.*54·1. &\supset \vdash \colon\ldotp \text{Hp}.\supset :\kappa =\Lambda .\lor.\kappa =\iota ʻ\Lambda &\qquad \text{(1)}\\ +\vdash .*100·45.*101·1.\supset \vdash :\text{Hp}.\kappa =\Lambda .&\supset .\nu =0.\\ +[*116·351] &\supset .\mu ^\nu =\Pi \text{Nc}ʻ\kappa &\qquad \text{(2)}\\ +\vdash .*51·16. &\supset \vdash :\text{Hp}.\kappa =\iota ʻ\Lambda .\supset .\Lambda \in \kappa .\\ +[*116·352] &\supset .\mu ^\nu =\Pi \text{Nc}ʻ\kappa &\qquad \text{(3)}\\ +\vdash .(1).(2).(3).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*116·36">*116·36</a>.</b> \(\vdash \colon\ldotp \text{Mult ax}.\supset :\mu ,\nu \in \text{NC}-\iota ʻ\Lambda .\kappa \in \nu \cap \text{Cl}ʻ\mu .\supset .\Pi \text{Nc}ʻ\kappa =\mu ^\nu\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*113·12.*100·45.\supset \vdash :\mu ,\nu \in \text{NC}.\alpha \in \mu .\beta \in \nu .\exists !\alpha .\supset .\alpha \downarrow_{,,}ʻʻ\beta + \in \nu \cap \text{Cl}ʻ\mu &\qquad \text{(1)}\\ +\vdash .(1).*114·571. \supset \vdash \colon\ldotp \text{Mult ax}.\supset :\\ +\mu ,\nu \in \text{NC}.\alpha \in \mu .\beta \in \nu .\exists !\alpha .\kappa \in \nu \cap \text{Cl}ʻ\mu .\supset .\Pi \text{Nc}ʻ\kappa =\Pi \text{Nc}ʻ\alpha \downarrow_{,,}ʻʻ\beta \\ +[*116·14.*114·1] =\text{Nc}ʻ(\alpha \,\,\text{exp}\,\, \beta )\\ +[*116·271] =\mu ^\nu &\qquad \text{(2)}\\ +\vdash .(2).\supset \vdash \colon\ldotp \text{Mult ax}.\supset :\mu ,\nu \in \text{NC}-\iota ʻ\Lambda .\exists !\mu -\iota ʻ\Lambda .\kappa \in \nu \cap \text{Cl}ʻ\mu .\supset .\Pi \text{Nc}ʻ\kappa =\mu ^\nu &\qquad \text{(3)}\\ +\vdash .*51·4.*54·1.\supset \vdash :\mu \in \text{NC}-\iota ʻ\Lambda .{\sim}\exists !\mu -\iota ʻ\Lambda .\supset .\mu =0 &\qquad \text{(4)}\\ +\vdash .(4).*116·353.\supset \\ +\vdash :\mu ,\nu \in \text{Nc}-\iota ʻ\Lambda .{\sim}\exists !\mu -\iota ʻ\Lambda .\kappa \in \nu \cap \text{Cl}ʻ\mu .\supset .\Pi \text{Nc}ʻ\kappa =\mu ^\nu &\qquad \text{(5)}\\ +\vdash .(3).(5).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_152">[Pg 152]</span></p> + +<p>In the above proposition, "\(\nu \in \text{NC}\)" is sufficient +hypothesis as to \(\nu\), since "\(\nu \neq \Lambda\)" is implied by +\(\kappa \in \nu \cap \text{Cl}ʻ\mu\). But \(\mu \neq \Lambda\) is +essential, since if \(\mu = \Lambda\), \(\mu ^\nu = \Lambda\) and \(\kappa= \Lambda\) +(provided \(\nu = 0)\), whence \(\Pi \text{Nc}ʻ\kappa = 1\).</p> + +<p>The above proposition connects exponentiation with multiplication.</p> + +<p class="nind"><b>*116·361.</b> + \(\vdash \colon\ldotp \text{Mult ax} . \supset : \mu , \nu \in \text{NC} - \iota ʻ\Lambda . \kappa \in \nu \cap \text{Cl excl}ʻ\mu . \supset . \text{Prod}ʻ\kappa \in \mu ^\nu\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *115·12 . \supset \vdash : \kappa \in \nu \cap \text{Cl excl}ʻ\mu . \supset . \text{Prod}ʻ\kappa \in \Pi \text{Nc}ʻ\kappa &\qquad \text{(1)}\\ +\vdash . (1) . *116·36 . \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<figure class="figcenter width500" id="i_152" style="width: 1600px;"> +<img src="images/i_152.jpg" width="1600" height="1062" alt="Diagram +showing a cylindrical structure with elements labeled z, w, δ, γ, and +various relations R, D', D''. The top ellipse γ=Cl'R connects to lower +levels via R=M'w arrows. Left shows w-ordered set δ. Right side lists +equations: R=M'w, D'R=D'M'z, κ=D''λ=D''M''w."> +</figure> + +<p>The following propositions, which illustrate certain generalizations +of the relations of rows and columns, may be made clearer by the +accompanying figure, in which, for the sake of simplicity, all the +classes concerned are taken to be finite.</p> + +<p>Let \(\kappa\) be a set of classes, constituted by four rows of +five dots in the figure, which are each given as similar to a given +class \(\gamma\), represented by the top row of five dots in the +figure, namely the row enclosed in an oval. We assume that an actual +correlating relation is given correlating each member of \(\kappa\) +with \(\gamma\). Let \(\lambda\) be the class of these relations, and +assume that \(\lambda\) consists of one correlator for each member +of \(\kappa\), and that \(\kappa \in \text{Cls}^{2} \text{excl}\). +Thus \(\text{D}ʻʻ\lambda = \kappa\), and \(R \in \lambda . \supset. \text{ᗡ}ʻR = \gamma\). +Put \(T = \dot{s} ʻ\lambda\). Then, if \(z\in \gamma\), \(T\) +relates to \(z\) every member of the column below \(z\), <i>i.e.</i> +\(\overrightarrow{T}ʻz\) consists of the four dots which are +vertically below \(z\); assuming, what in the circumstances is +possible, that each dot is placed below its correlate in \(\gamma\). +Thus \(\overrightarrow{T}ʻʻ\gamma\) represents the columns, while +\(\text{D}ʻʻ\lambda\) represents the rows.</p> + +<p>We prove, in <a href="#*116·41">*116·41</a>, that \(\overrightarrow{T}ʻʻ\gamma\), +the class of rows, has double similarity with \(\lambda\downarrow_{,,}ʻʻ\gamma\), +or, what comes to the same thing, with \(\kappa\downarrow_{,,}ʻʻ\gamma\). +Hence it follows that \(Tʻʻ\gamma\), +which is the whole class of dots, is similar to \(\gamma\times \lambda\) +or \(\gamma \times \kappa\), and that +\(\text{Nc}ʻ{\in}_{\Delta}ʻ\overrightarrow{T}ʻʻ\gamma\), which is the product of +<span class="pagenum" id="Page_153">[Pg 153]</span>the numbers of the columns, is equal to \((\text{Nc}ʻ\lambda)^{\text{Nc}ʻ\gamma}\) +or \((\text{Nc}ʻ\kappa )^{\text{Nc}ʻ\gamma}\). +The correlator which is used for proving these propositions is \(W\), +where, if \(R\) is a member of \(\lambda\) and \(z\) is a member of +\(\gamma\), \(W\) correlates \(Rʻz\) with \(R\downarrow z\).</p> + +<p>Similarly, by correlating \(Rʻz\) with \(z\downarrow R\), calling the +correlator \(U\), we have \(Uʻʻ\downarrow Rʻʻ\gamma =Rʻʻ\gamma\), +<i>i.e.</i> \(U_{\in }ʻ\gamma \downarrow_{,,}R=\text{D}ʻR\), whence +\(U_{\in }ʻʻ\gamma \downarrow_{,,}ʻʻ\lambda =\text{D}ʻʻ\lambda\). +Hence \(\text{D}ʻʻ\lambda\), <i>i.e.</i> the class of rows, has +double similarity with \(\gamma \downarrow_{,,}ʻʻ\lambda\) or +\(\gamma \downarrow_{,,}ʻʻ\kappa\), whence the product of the numbers +of the rows is \((\text{Nc}ʻ\gamma)^{\text{Nc}ʻ\lambda}\) or +\((\text{Nc}ʻ\gamma )^{\text{Nc}ʻ\kappa}\).</p> + +<p>Finally, we take a class \(\delta\) similar to \(\kappa\) or +\(\lambda\) (illustrated in the figure by the column of dots enclosed +in an oval), and calling \(M\) a correlator of \(\lambda\) and +\(\delta\), we replace \(\lambda\) by \(Mʻʻ\delta\) and \(\kappa\) +by \(\text{D}ʻʻMʻʻ\delta\). We thus find that, if \(M\upharpoonright\delta\) +correlates with \(\delta\) a class of relations whose domains +are mutually exclusive, and which each correlate their domains with +a given class \(\gamma\), then \(\text{D}ʻʻMʻʻ\delta\) has double +similarity with \(\gamma \downarrow_{,,}ʻʻ\delta\), whence the same +results as before with \(\delta\) in place of \(\kappa\) or \(\lambda\).</p> + +<p>The following propositions are useful in connecting multiplication with +exponentiation, and in proving the formal laws of exponentiation.</p> + +<p><a href="#*116·4">*116·4</a><a href="#*116·401">·401</a> are lemmas for <a href="#*116·41">*116·41</a>.</p> + +<p class="nind"><b><a id="*116·4">*116·4</a>.</b> \[\begin{align}\vdash \colon\ldotp \lambda \subset 1&\rightarrow 1:R,S\in \lambda .\exists !\text{D}ʻR\cap \text{D}ʻS.\supset _{R,S}.R=S:\\ +&\text{ᗡ}ʻʻ\lambda \subset \iota ʻ\gamma .W=\hat{x} \hat{P} \{(\exists R,z).R\in \lambda .x=Rʻz.P=R\downarrow z\}:\\ +&\supset .W\in 1\rightarrow 1.\text{ᗡ}ʻW=\gamma \times \lambda .\text{D}ʻW=\text{D}ʻ\dot{s} ʻ\lambda\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*21·33.\supset \vdash \colon\ldotp \text{Hp}.&\supset :xWP.xWQ.\equiv .\\ +&(\exists R,S,z,w).R,S\in \lambda .x=Rʻz=Sʻw.P=R\downarrow z.Q=S\downarrow w.\\ +[*33·43] &\equiv .(\exists R,S,z,w).R,S\in \lambda .x=Rʻz=Sʻw.x\in \text{D}ʻR\cap \text{D}ʻS.\\ +&P=R\downarrow z.Q=S\downarrow w.\\ +[\text{Hp}.*13·195] &\supset .(\exists R,z,w).R\in \lambda .x=Rʻz=Rʻw.P=R\downarrow z.Q=R\downarrow w.\\ +[*71·532.*13·195] &\supset .(\exists R,z).R\in \lambda .x=Rʻz.P=R\downarrow z.Q=R\downarrow z.\\ +[*13·172] &\supset .P=Q &\qquad \text{(1)}\\ +\vdash .*21·33.\supset \vdash \colon\ldotp \text{Hp}.\supset :xWP.yWP.&\equiv .\\ +&(\exists R,S,z,w).R,S\in \lambda .x=Rʻz.y=Sʻw.P=R\downarrow z=Q\downarrow w.\\ +[*55·202] &\supset .(\exists R,S,z,w).R,S\in \lambda .x=Rʻz.y=Sʻw.R=S.z=w.\\ +[*13·22·172] &\supset .x=y &\qquad \text{(2)}\\ +\vdash .*33·131.\supset \vdash \colon\ldotp \text{Hp}.\supset :P\in \text{ᗡ}ʻW.&\equiv .(\exists x,R,z).R\in \lambda .x=Rʻz.P=R\downarrow z.\\ +[*71·411] &\equiv .(\exists R,z).R\in \lambda .z\in \text{ᗡ}ʻR.P=R\downarrow z.\\ +[\text{Hp}] &\equiv .(\exists R,z).R\in \lambda .z\in \gamma .P=R\downarrow z.\\ +[*113·101] &\equiv .P\in \gamma \times \lambda &\qquad \text{(3)}\\ +\vdash .*33·13. \supset \vdash \colon\ldotp \text{Hp}.\supset :x\in \text{D}ʻW.&\equiv .(\exists P,R,z).R\in \lambda .x=Rʻz.P=R\downarrow z.\\ +[*55·12.*71·36] &\equiv .(\exists R,z).R\in \lambda .xRz.\\ +[*41·11.*33·13] &\equiv .x\in \text{D}ʻ\dot{s} ʻ\lambda &\qquad \text{(4)}\\ +\vdash .(1).(2).(3).(4).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_154">[Pg 154]</span></p> + +<p class="nind"><b><a id="*116·401">*116·401</a>.</b> \(\vdash : \text{Hp} *116·4 . T = \dot{s} ʻ\lambda . \supset . \overrightarrow{T}ʻʻ\gamma = W_{\in }ʻʻ\lambda \downarrow_{,,} ʻʻ\gamma\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *37·11·1 . *38·2 . &\supset \vdash \colon\ldotp \text{Hp} . z \in \gamma . \supset :\\ +x \in W_{\in }ʻ\lambda \downarrow_{,,} z . &\equiv . (\exists R) . R \in \lambda . x W (R \downarrow z) .\\ +[*21·33] &\equiv . (\exists R, S, w) . R, S \in \lambda . x = Sʻw . R \downarrow z = S \downarrow w .\\ +[*55·202 . *13·22] &\equiv . (\exists R) . R \in \lambda . x = Rʻz .\\ +[*71·36] &\equiv . (\exists R) . R \in \lambda . xRz .\\ +[*41·11] &\equiv . x (\dot{s} ʻ\lambda ) z .\\ +[\text{Hp} . *32·18] &\equiv . x \in \overrightarrow{T}ʻz &\qquad \text{(1)}\\ +\vdash . (1) . *37·68 . \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*116·41">*116·41</a>.</b> + \[\begin{align}&\vdash \colon\ldotp \lambda \subset 1 \rightarrow 1 . \text{ᗡ}ʻʻ\lambda \subset \iota ʻ\gamma : R, S \in \lambda . \exists ! \text{D}ʻR \cap \text{D}ʻS . \supset _{R, S} . R = S : T = \dot{s} ʻ\lambda :\\ +&\supset . \overrightarrow{T}ʻʻ\gamma \text{ sm } \text{ sm } \lambda \downarrow_{,,} + ʻʻ\gamma . Tʻʻ\gamma \text{ sm } \gamma \times \lambda . T \in \text{Cls} \rightarrow 1 . \overrightarrow{T}ʻʻ\gamma \in \text{Cls}^{2} \text{excl} .\\ +&\text{Nc}ʻ{\in}_{\Delta}ʻ\overrightarrow{T}ʻʻ\gamma = \text{Nc}ʻT_{\Delta }ʻ\gamma + = \text{Nc}ʻ\text{Prod}ʻ\overrightarrow{T}ʻʻ\gamma = \text{Nc}ʻ(\lambda \,\,\text{exp}\,\, \gamma ) = (\text{Nc}ʻ\lambda )^{\text{Nc}ʻ\gamma }\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *116·4·401 . *111·4 . *113·1 . \supset \vdash : \text{Hp} . &\supset . \overrightarrow{T}ʻʻ\gamma \text{ sm } \text{ sm } \lambda \downarrow_{,,} + ʻʻ\gamma . &\qquad \text{(1)}\\ +[*111·44 . *40·5] &\supset . Tʻʻ\gamma \text{ sm } \gamma \times \lambda &\qquad \text{(2)}\\ +\vdash . *72·321 . *85·14 . &\supset \vdash : \text{Hp} . \supset . T \in \text{Cls} \rightarrow 1 . \text{Nc}ʻ{\in}_{\Delta}ʻ\overrightarrow{T}ʻʻ\gamma = \text{Nc}ʻT_{\Delta }ʻ\gamma + &\qquad \text{(3)}\\ +\vdash . (3) . *84·51 . \supset \vdash : \text{Hp} . &\supset .\overrightarrow{T}ʻʻ\gamma \in \text{Cls}^{2} \text{excl} . &\qquad \text{(4)}\\ +[*115·12] &\supset . \text{Nc}ʻ{\in}_{\Delta}ʻ\overrightarrow{T}ʻʻ\gamma = \text{Nc}ʻ\text{Prod}ʻ\overrightarrow{T}ʻʻ\gamma &\qquad \text{(5)}\\ +\vdash . (1) . *114·52 . \supset \vdash : \text{Hp} . \supset . \text{Nc}ʻ{\in}_{\Delta}ʻ\overrightarrow{T}ʻʻ\gamma &= \text{Nc}ʻ{\in}_{\Delta}ʻ\lambda \downarrow_{,,} + ʻʻ\gamma \\ +[*116·14] &= \text{Nc}ʻ(\lambda \,\,\text{exp}\,\, \gamma ) &\qquad \text{(6)}\\ +[*116·25] &= (\text{Nc}ʻ\lambda )^{\text{Nc}ʻ\gamma } &\qquad \text{(7)}\\ +\vdash . (1) . (2) . (3) . (4) . (5) . (6) . (7) . \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p>The following proposition is merely another form of <a href="#*116·41">*116·41</a>.</p> + +<p class="nind"><b><a id="*116·411">*116·411</a>.</b> \[\begin{align}&\vdash \colon\ldotp \kappa \in \text{Cls}^{2} \text{excl} : \alpha \in \kappa . \supset _{\alpha } + . Mʻ\alpha \in \alpha \,\overline{\text{ sm }}\, \gamma : T = \dot{s} ʻMʻʻ\kappa ʻʻ: \supset .\\ +&\overrightarrow{T}ʻʻ\gamma \text{ sm } \text{ sm } \kappa \downarrow_{,,} + ʻʻ\gamma . Tʻʻ\gamma \text{ sm } \gamma \times \kappa . T \in \text{Cls} \rightarrow 1 . \overrightarrow{T}ʻʻ\gamma \in \text{Cls}^{2} \text{excl} .\\ +&\text{Nc}ʻ{\in}_{\Delta}ʻ\overrightarrow{T}ʻʻ\gamma = \text{Nc}ʻT_{\Delta }ʻ\gamma + = \text{Nc}ʻ\text{Prod}ʻ\overrightarrow{T}ʻʻ\gamma = \text{Nc}ʻ(\kappa \,\,\text{exp}\,\, \gamma ) = (\text{Nc}ʻ\kappa )^{\text{Nc}ʻ\gamma }\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *73·03 . &\supset \vdash : \text{Hp} . \supset . Mʻʻ\kappa \subset 1 \rightarrow 1 . \text{ᗡ}ʻʻMʻʻ\kappa \subset \iota ʻ\gamma &\qquad \text{(1)}\\ +\vdash . *111·16 . &\supset \vdash \colon\ldotp \text{Hp} . \supset : \alpha , \beta \in \kappa . Mʻ\alpha = Mʻ\beta . \supset . \alpha = \beta &\qquad \text{(2)}\\ +\vdash . *14·21 . &\supset \vdash \colon\ldotp \text{Hp} . \supset : \alpha \in \kappa . \supset . \text{E}! Mʻ\alpha &\qquad \text{(3)}\\ +\vdash . (2) . (3) . *73·24 . &\supset \vdash : \text{Hp} . \supset . Mʻʻ\kappa \text{ sm } \kappa &\qquad \text{(4)}\\ +\vdash . *73·03 . &\supset \vdash \colon\ldotp \text{Hp} . \supset : \alpha \in \kappa . \supset . \text{D}ʻMʻ\alpha = \alpha :\\ +[*13·12] &\supset : \alpha , \beta \in \kappa . \exists ! \text{D} ʻMʻ\alpha \cap \text{D}ʻMʻ\beta . \supset . \exists ! \alpha \cap \beta .\\ +[*84·11] &\supset . \alpha = \beta .\\ +[*30·37 . (3)] &\supset . Mʻ\alpha = Mʻ\beta :\\ +[*37·63] &\supset : R, S \in Mʻʻ\kappa . \exists ! \text{D}ʻR \cap \text{D}ʻS . \supset . R = S &\qquad \text{(5)}\\ +\vdash . (1) . (4) . (5) . *116·41 \frac{Mʻʻ\kappa}{\lambda}. *113·13 . *116·19 . \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_155">[Pg 155]</span></p> + +<p>*116·412·413 are lemmas for *116·414.</p> + +<p class="nind"><b>*116·412.</b> \[\begin{align}&\vdash \colon\ldotp \lambda \subset 1 \rightarrow 1 : R, S \in \lambda . \exists ! \text{D}ʻR \cap \text{D}ʻS . \supset _{R, S} + . R = S : \text{ᗡ}ʻʻ\lambda \subset \iota ʻ\gamma :\\ +&U = \hat{x} \hat{P}\{(\exists R, z) . R \in \lambda . x = Rʻz . P = z \downarrow R\} :\\ +&\supset . U \in (sʻ\text{D}ʻʻ\lambda ) \,\overline{\text{ sm }}\, (\lambda \times \gamma ) \quad[\text{Proof as in *116·4}]\end{align}\]</p> + +<p class="nind"><b>*116·413.</b> \(\vdash : \text{Hp} *116·412 . \supset . \text{D}ʻʻ\lambda = U_{\in}ʻʻ\gamma \downarrow_{,,}ʻʻ\lambda \quad[\text{Proof as in *116·401}]\)</p> + +<p class="nind"><b>*116·414.</b> \[\begin{align}&\vdash : \text{Hp} *116·412 . \supset . U \in (\text{D}ʻʻ\lambda ) \,\overline{\text{ sm }}\, \,\overline{\text{ sm }}\, (\gamma \downarrow_{,,}ʻʻ\lambda + ) . (\text{D}ʻʻ\lambda ) \text{ sm } \text{ sm } (\gamma \downarrow_{,,}ʻʻ\lambda )\\ +&[*116·412·413]\end{align}\]</p> + +<p class="nind"><b>*116·42.</b> \[\begin{align}&\vdash \colon\ldotp \lambda \subset 1 \rightarrow 1 : R, S \in \lambda . \exists ! \text{D}ʻR \cap \text{D}ʻS . \supset _{R,S} + . R = S : \text{ᗡ}ʻʻ\lambda \subset \iota ʻ\gamma :\\ +&\supset . \text{D}ʻʻ\lambda \text{ sm } \text{ sm } (\gamma \downarrow_{,,}ʻʻ\lambda ) . (\text{D}ʻ\dot{s} ʻ\lambda ) \text{ sm } (\lambda \times \gamma ) . ({\in}_{\Delta}ʻ\text{D}ʻʻ\lambda + ) \text{ sm } (\gamma \,\,\text{exp}\,\, \lambda ) .\\ +&\text{Nc}ʻ\text{Prod}ʻ\text{D}ʻʻ\lambda = \Pi \text{Nc}ʻ\text{D}ʻʻ\lambda = (\text{Nc}ʻ\gamma )^{\text{Nc}ʻ\lambda }\\ +&[*116·414·25 . *115·51 . *111·44 . *41·43]\end{align}\]</p> + +<p class="nind"><b>*116·422.</b> \[\begin{align}&\vdash \colon\ldotp M \upharpoonright \delta \in 1 \rightarrow 1 : w, v \in \delta . \exists ! \text{D}ʻMʻw \cap \text{D}ʻMʻv . \supset _{w,v} + . w = v :\\ +&w \in \delta . \supset _{w} + . Mʻw \in 1 \rightarrow 1 . \text{ᗡ}ʻMʻw = \gamma : \supset . \text{D}ʻʻMʻʻ\delta \text{ sm } \text{ sm } \gamma \downarrow_{,,}ʻʻ\delta\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *116·42 \frac{Mʻʻ\delta}{\lambda}. \supset \\ +\vdash \colon\ldotp Mʻʻ\delta \subset 1 \rightarrow 1 : R, S \in Mʻʻ\delta . \exists ! \text{D}ʻR \cap \text{D}ʻS . &\supset _{R, S} + . R = S : \text{ᗡ}ʻʻMʻʻ\delta \subset \iota ʻ\gamma :\\ +&\supset . \text{D}ʻʻMʻʻ\delta \text{ sm } \text{ sm } \gamma \downarrow_{,,}ʻʻMʻʻ\delta &\qquad \text{(1)}\\ +\vdash . *14·21 . \supset \vdash \colon\ldotp \text{Hp} . &\supset : w \in \delta . \supset . \text{E}! Mʻw : &\qquad \text{(2)}\\ +[*33·43] &\supset : \delta \subset \text{ᗡ}ʻM :\\ +[*73·15] &\supset : (Mʻʻ\delta ) \text{ sm } \delta &\qquad \text{(3)}\\ +\vdash . *51·15 . \supset \vdash \colon\ldotp \text{Hp} . &\supset : w \in \delta . \supset . \text{ᗡ}ʻMʻw \in \iota ʻ\gamma :\\ +[*37·61] &\supset : \text{ᗡ}ʻʻMʻʻ\delta \subset \iota ʻ\gamma &\qquad \text{(4)}\\ +\vdash . (2) . *30·37 . \supset \vdash \colon\ldotp \text{Hp} . &\supset : w,v \in \delta . \exists ! \text{D}ʻMʻw \cap \text{D}ʻMʻv . \supset _{w,v} . Mʻw = Mʻv :\\ +[*37·63] &\supset : R, S \in Mʻʻ\delta . \exists ! \text{D}ʻR \cap \text{D}ʻS . \supset _{R, S} . R = S &\qquad \text{(5)}\\ +\vdash . (1) . (4) . (5) . \supset \vdash : \text{Hp} . &\supset . \text{D}ʻʻMʻʻ\delta \text{ sm } \text{ sm } \gamma \downarrow_{,,}ʻʻMʻʻ\delta .\\ +[(3) . *113·13] &\supset . \text{D}ʻʻMʻʻ\delta \text{ sm } \text{ sm } \gamma \downarrow_{,,}ʻʻ\delta : \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_156">[Pg 156]</span></p> + +<p class="nind"><b><a id="*116·43">*116·43</a>.</b> \[\begin{align}&\vdash \colon\ldotp M\upharpoonright \delta \in 1\rightarrow 1:w,v\in \delta .\exists !\text{D}ʻMʻw\cap \text{D}ʻMʻv.\supset _{w,v}.w = v:\\ +&w\in \delta .\supset _{w}.Mʻw\in 1\rightarrow 1.\text{ᗡ}ʻMʻw = \gamma :\\ +&\supset .\text{Prod}ʻ\text{D}ʻʻMʻʻ\delta \text{ sm }\text{ sm }(\gamma \,\,\text{exp}\,\,\delta ).\Pi \text{Nc}ʻ\text{D}ʻʻMʻʻ\delta = (\text{Nc}ʻ\gamma )^{\text{Nc}ʻ\delta }\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*115·51.*116·422.&\supset \vdash :\text{Hp}.\supset .\text{Prod}ʻ\text{D}ʻʻMʻʻ\delta \text{ sm }\text{ sm }(\gamma \,\,\text{exp}\,\,\delta ) &\qquad \text{(1)}\\ +\vdash .*116·422.*114·52.&\supset \vdash :\text{Hp}.\supset .\Pi \text{Nc}ʻ\text{D}ʻʻMʻʻ\delta = \Pi \text{Nc}ʻ\gamma \downarrow_{,,}ʻʻ\delta &\qquad \text{(2)}\\ +\vdash .*116·14·25.&\supset \vdash .\Pi \text{Nc}ʻ\gamma \downarrow_{,,}ʻʻ\delta = (\text{Nc}ʻ\gamma )^{\text{Nc}ʻ\delta } &\qquad \text{(3)}\\ +\vdash .(1).(2).(3).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p>The above proposition is used in <a href="#*116·534">*116·534</a><a href="#*116·61">·61</a>.</p> + +<p class="nind"><b>*116·44.</b> \[\begin{align}&\vdash \colon\ldotp \exists !\gamma :(z).Mʻz\in 1\rightarrow 1.\text{ᗡ}ʻMʻz = \text{V}:\\ +&w,v\in \delta .\exists !(Mʻw)ʻʻ\gamma \cap (Mʻv)ʻʻ\gamma .\supset _{w,v}.w = v:\\ +&\supset .\text{D}ʻʻ\upharpoonright \gammaʻʻMʻʻ\delta \text{ sm }\text{ sm }\gamma \downarrow_{,,}ʻʻ\delta + .\text{Prod}ʻ\text{D}ʻʻ\upharpoonright \gamma ʻʻMʻʻ\delta \text{ sm }\text{ sm }(\gamma \,\,\text{exp}\,\,\delta )\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*71·29.*35·65.\supset\\ +\vdash \colon\ldotp \text{Hp}:(z).Nʻz = (Mʻz)\upharpoonright \gamma :\supset .(z).Nʻz\in 1\rightarrow 1.\text{ᗡ}ʻNʻz = \gamma &\qquad \text{(1)}\\ +\vdash .*37·401.\supset \vdash \colon\ldotp \text{Hp}.\text{Hp}(1).\supset :w,v\in \delta .\exists !\text{D}ʻNʻw\cap \text{D}ʻNʻv.\supset _{w,v}.w = v &\qquad \text{(2)}\\ +\vdash .*35·7.\supset \vdash \colon\ldotp \text{Hp}.\text{Hp}(1).\supset :x\in \gamma .w,v\in \delta .Nʻw = Nʻv.\supset .(Nʻw)ʻx = (Nʻv)ʻx.\\ +[(2)] \supset .w = v &\qquad \text{(3)}\\ +\vdash .(3).*10·11·23·35.\supset \vdash \colon\ldotp \text{Hp}.\text{Hp}(1).\supset :w,v\in \delta .Nʻw = Nʻv.\supset .w = v:\\ +[*71·55·166] \supset :N\upharpoonright \delta \in 1\rightarrow 1 &\qquad \text{(4)}\\ +\vdash .(1).(2).(4).*116·422.*115·51.\supset\\ +\vdash :\text{Hp}.\text{Hp}(1).\supset .\text{D}ʻʻNʻʻ\delta \text{ sm }\text{ sm }\gamma \downarrow_{,,}ʻʻ\delta + .\text{Prod}ʻ\text{D}ʻʻNʻʻ\delta \text{ sm }\text{ sm }(\gamma \,\,\text{exp}\,\,\delta ) &\qquad \text{(5)}\\ +\vdash .*38·11.\supset \vdash :\text{Hp}.\text{Hp}(1).\supset .\text{D}ʻNʻz = \text{D}ʻ\upharpoonright \gamma ʻMʻz.\\ +[*37·353] \supset .\text{D}ʻʻNʻʻ\delta = \text{D}ʻʻ\upharpoonright \gamma ʻʻMʻʻ\delta &\qquad \text{(6)}\\ +\vdash .(5).(6).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*116·45">*116·45</a>.</b> \[\begin{align}&\vdash \colon\ldotp (z).Mʻz\in 1\rightarrow 1.\text{ᗡ}ʻMʻz = \text{V}:\\ +&w,v\in \delta .\exists !(Mʻw)ʻʻ\gamma \cap (Mʻv)ʻʻ\gamma .\supset _{w,v}.w\\ +& = v:\supset .\text{Prod}ʻ\text{D}ʻʻ\upharpoonright \gamma ʻʻMʻʻ\delta \text{ sm }(\gamma \,\,\text{exp}\,\,\delta )\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*116·182.*115·142.*37·29.\supset\\ +\vdash :\text{Hp}.\gamma &= \Lambda .\exists !\delta .\supset .\text{Prod}ʻ\text{D}ʻʻ\upharpoonright \gamma ʻʻMʻʻ\delta = \Lambda .\gamma \,\,\text{exp}\,\,\delta = \Lambda &\qquad \text{(1)}\\ +\vdash .*115·1.*83·15.*116·181.\supset\\ +\vdash :\text{Hp}.\delta & = \Lambda .\supset .\text{Prod}ʻ\text{D}ʻʻ\upharpoonright \gamma ʻʻMʻʻ\delta = \iota ʻ\Lambda .\gamma \,\,\text{exp}\,\,\delta = \iota ʻ\Lambda &\qquad \text{(2)}\\ +\vdash .(1).(2).*116·44.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p>The above proposition is used in <a href="#*116·676">*116·676</a>.</p> + +<p>We have now to prove the three formal laws of exponentiation, namely +\[ +\begin{aligned} +\mu ^\nu \times _{c} \mu ^{\varpi } &= \mu ^{\nu +_{c} \varpi },\\ +\mu ^{\varpi } \times _{c} \nu ^{\varpi } &= (\mu \times _{c} \nu )^{\varpi },\\ +\text{and}\qquad (\mu ^\nu )^{\varpi } &= \mu ^{\nu \times _{c} \varpi }. +\end{aligned} +\]<span class="pagenum" id="Page_157">[Pg 157]</span> +Of these the first is an immediate consequence of the distributive law, +while the second and third result from forms of the associative law of +multiplication.</p> + +<p class="nind"><b><a id="*116·5">*116·5</a>.</b> + \(\vdash : \beta \cap \gamma = \Lambda . \supset . (\alpha \,\,\text{exp}\,\, \beta ) \times (\alpha \,\,\text{exp}\,\, \gamma ) \text{ sm } \alpha \,\,\text{exp}\,\, (\beta \cup \gamma )\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *113·191 . \supset \\ +\vdash : \text{Hp} . \exists ! \alpha . &\supset . \alpha \downarrow_{,,}ʻʻ\beta \cap \alpha \downarrow_{,,}ʻʻ\gamma = \Lambda .\\ +[*114·301] &\supset . {\in}_{\Delta}ʻ\alpha \downarrow_{,,}ʻʻ\beta \times {\in}_{\Delta}ʻ\alpha \downarrow_{,,}ʻʻ\gamma \text{ sm } {\in}_{\Delta}ʻ(\alpha + \downarrow_{,,}ʻʻ\beta \cup \alpha \downarrow_{,,}ʻʻ\gamma .\\ +[*116·14 . *113·13] &\supset . (\alpha \,\,\text{exp}\,\, \beta ) \times (\alpha \,\,\text{exp}\,\, \gamma ) \text{ sm } {\in}_{\Delta}ʻ(\alpha \downarrow_{,,}ʻʻ\beta + \cup \alpha \downarrow_{,,}ʻʻ\gamma ) .\\ +[*37·22] &\supset . (\alpha \,\,\text{exp}\,\, \beta ) \times (\alpha \,\,\text{exp}\,\, \gamma ) \text{ sm } {\in}_{\Delta}ʻ\alpha \downarrow_{,,}ʻʻ(\beta \cup \gamma ) .\\ +[*116·14] &\supset . (\alpha \,\,\text{exp}\,\, \beta ) \times (\alpha \,\,\text{exp}\,\, \gamma ) \text{ sm } \alpha \,\,\text{exp}\,\, (\beta \cup \gamma ) &\qquad \text{(1)}\\ +\vdash . *116·182 . &\supset \vdash : \alpha = \Lambda . \exists ! \beta . \supset . \alpha \,\,\text{exp}\,\, \beta = \Lambda .\\ +[*113·114] &\supset . (\alpha \,\,\text{exp}\,\, \beta ) \times (\alpha \,\,\text{exp}\,\, \gamma ) = \Lambda &\qquad \text{(2)}\\ +\vdash . *116·182 . *24·56 . &\supset \vdash : \alpha = \Lambda . \exists ! \beta . \supset . \alpha \,\,\text{exp}\,\, (\beta \cup \gamma ) = \Lambda &\qquad \text{(3)}\\ +\vdash . (2) . (3) . &\supset \vdash : \alpha = \Lambda . \exists ! \beta . \supset . (\alpha \,\,\text{exp}\,\, \beta ) \times (\alpha \,\,\text{exp}\,\, \gamma ) \text{ sm } \alpha \,\,\text{exp}\,\, (\beta \cup \gamma ) &\qquad \text{(4)}\\ +\text{Similarly}\quad &\vdash : \alpha = \Lambda . \exists ! \gamma . \supset . (\alpha \,\,\text{exp}\,\, \beta ) \times (\alpha \,\,\text{exp}\,\, \gamma ) \text{ sm } \alpha \,\,\text{exp}\,\,(\beta \cup \gamma ) &\qquad \text{(5)}\\ +\vdash . *116·181 . &\supset \vdash : \alpha = \Lambda . \beta = \Lambda . \gamma = \Lambda . \supset . (\alpha \,\,\text{exp}\,\, \beta ) \times (\alpha \,\,\text{exp}\,\, \gamma ) = \iota ʻ\Lambda \times \iota ʻ\Lambda &\qquad \text{(6)}\\ +\vdash . *116·181 . &\supset \vdash : \alpha = \Lambda . \beta = \Lambda . \gamma = \Lambda . \supset . \alpha \,\,\text{exp}\,\, (\beta \cup \gamma ) = \iota ʻ\Lambda &\qquad \text{(7)}\\ +\vdash . (6) . (7) . *113·611 . *73·43 . \supset \\ +&\vdash : \alpha = \Lambda . \beta = \Lambda . \gamma = \Lambda . \supset . (\alpha \,\,\text{exp}\,\, \beta ) \times (\alpha \,\,\text{exp}\,\, \gamma ) \text{ sm } \alpha \,\,\text{exp}\,\, (\beta \cup \gamma ) &\qquad \text{(8)}\\ +\vdash . (1) . (4) . (5) . (8) . \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p>In the last line of the above proof, *73·43 is required because the two +\(\Lambda\)'s involved have not been proved to be of the same type. +They are in fact of the same type, but it is unnecessary to prove this.</p> + +<p class="nind"><b>*116·51.</b> \(\vdash . (\alpha \,\,\text{exp}\,\, \beta ) \times (\alpha \,\,\text{exp}\,\, \gamma ) \text{ sm } \alpha \,\,\text{exp}\,\, (\beta + \gamma )\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *116·19 . *110·12 . \supset \vdash . &(\alpha \,\,\text{exp}\,\, \beta ) \text{ sm } (\alpha \,\,\text{exp}\,\, \downarrow \Lambda _{\gamma } ʻʻ\iota ʻʻ\beta ) .\\ +&(\alpha \,\,\text{exp}\,\, \gamma ) \text{ sm } (\alpha \,\,\text{exp}\,\, \Lambda _{\beta } \downarrow ʻʻ\iotaʻʻ\gamma ) .\\ +[*113·13] \supset \vdash . &(\alpha \,\,\text{exp}\,\, \beta ) \times (\alpha \,\,\text{exp}\,\, \gamma ) \text{ sm }\\ +&(\alpha \,\,\text{exp}\,\, \downarrow \Lambda _{\gamma }ʻʻ\iotaʻʻ\beta ) \times (\alpha \,\,\text{exp}\,\, \Lambda _{\beta } \downarrow ʻʻ\iotaʻʻ\gamma ) .\\ +[*110·11 . *116·5] \supset \vdash . &(\alpha \,\,\text{exp}\,\, \beta ) \times (\alpha \,\,\text{exp}\,\, \gamma ) \text{ sm }\\ +&\alpha \,\,\text{exp}\,\, (\downarrow \Lambda _{\gamma } ʻʻ\iotaʻʻ\beta \cup \Lambda _{\beta } \downarrow ʻʻ\iotaʻʻ\gamma ) &\qquad \text{(1)}\\ +\vdash . (1) . (*110·01) . \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_158">[Pg 158]</span></p> + +<p class="nind"><b><a id="*116·52">*116·52</a>.</b> \(\vdash .\mu ^\nu \times _{c} \mu ^{\varpi } = \mu ^{v +_{c}\varpi }\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*116·51 .*110·22.\supset \\ +\vdash .(\text{N}_{0}\text{c}ʻ\alpha )^{\text{N}_{0}\text{c}ʻ\beta } \times _{c} (\text{N}_{0}\text{c}ʻ\alpha )^{\text{N}_{0}\text{c}ʻ\gamma + } = (\text{N}_{0}\text{c}ʻ\alpha )^{\text{N}_{0}\text{c}ʻ\beta +_{c} \text{N}_{0}\text{c}ʻ\gamma } &\qquad \text{(1)}\\ +\vdash .(1).*103·2.\supset \vdash :\mu ,\nu ,\varpi \in \text{N}_{0}\text{C}.\supset .\mu ^\nu \times _{c} \mu ^{\varpi } = \mu ^{\nu +_{c}\varpi } + &\qquad \text{(2)}\\ +\vdash .*116·205.*113·204.\supset \\ +\vdash : \mu {\sim} \in \text{N}_{0}\text{C}.\supset .\mu ^\nu \times _{c} \mu ^{\varpi } = \Lambda = \mu ^{\nu +_{c}\varpi } + &\qquad \text{(3)}\\ +\vdash .*116·205.*113·204.\supset\\ +\vdash : {\sim} (\nu ,\varpi \in \text{N}_{0}\text{C}).\supset .\mu ^\nu \times _{c} \mu ^{\varpi } = \Lambda &\qquad \text{(4)}\\ +\vdash .*110·4.*116·204.\supset \vdash :{\sim}(\nu ,\varpi \in \text{N}_{0}\text{C}).\supset .\mu ^{\nu +_{c}\varpi } = \Lambda &\qquad \text{(5)}\\ +\vdash . (4).(5).\supset \vdash :{\sim}(\nu ,\varpi \in \text{N}_{0}\text{C}).\supset .\mu ^\nu \times _{c} \mu ^{\varpi } = \mu ^{\nu +_{c}\varpi } + &\qquad \text{(6)}\\ +\vdash .(2).(3).(6).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p>The following propositions are lemmas for +\[ +\mu ^{\varpi } \times _{c} \nu ^{\varpi } = (\mu \times _{c} \nu )^{\varpi }. +\] +The principal previous propositions used in the proof are <a href="#*115·6">*115·6</a> and +<a href="#*116·43">*116·43</a>. The proof proceeds as follows. +\[ +(\alpha \,\,\text{exp}\,\,\gamma ) \times (\beta \,\,\text{exp}\,\,\gamma )\,\text{is}\,\text{Prod}ʻ\alpha \downarrow_{,,}ʻʻ\gamma \times \text{Prod}ʻ\beta \downarrow_{,,}ʻʻ\gamma. +\] +This, using *115·6, and putting \(\alpha \downarrow_{,,}\), \(\beta +\downarrow_{,,}\) in place of \(R\) and \(S\) of that proposition, is +similar to +\[ +{\in}_{\Delta}ʻ\hat{\mu} \{\exists z).z\in \gamma .\mu = \alpha \downarrow_{,,}z \times \beta \downarrow_{,,}z\},\, \textit{i.e.}\, \text{to}\, {\in}_{\Delta}ʻ\hat{\mu} + \{(\exists z).z\in \gamma .\mu = \downarrow zʻʻ\alpha \times \downarrow zʻʻ\beta\}. +\] +Now by <a href="#*113·65">*113·65</a>, putting \(R\dagger = R\parallel \breve{R}\quad\text{Dft}\), +\(\downarrow zʻʻ\alpha \times \downarrow zʻʻ\beta =(\downarrow z)\dagger ʻʻ(\alpha \times \beta)\). +We now apply <a href="#*116·43">*116·43</a>, taking \((\downarrow z)\dagger\) as the \(Mʻz\) +of that proposition, or rather, taking \((\downarrow z)\dagger\upharpoonright (\alpha \times \beta)\). +Thus we find +\[ +{\in}_{\Delta}ʻ\hat{\mu} \{(\exists z).z\in \gamma .\mu = \downarrow zʻʻ\alpha \times \downarrow zʻʻ\beta \}\text{ sm }(\alpha \times \beta )\,\,\text{exp}\,\,\gamma . +\] +Hence our proposition follows.</p> + +<p class="nind"><b>*116·529.</b> \(R\dagger = R\parallel \breve{R} \quad\text{Dft}[*116]\)</p> + +<p>In <a href="#*150">*150</a>, this notation will be introduced as a permanent definition. +For the present, we only introduce it to avoid \((\downarrow z\parallel\text{Cnv}ʻ\downarrow z)\), +which is awkward.</p> + +<p class="nind"><b><a id="*116·53">*116·53</a>.</b> \[\begin{align}\vdash :\exists !&\alpha .\exists !\beta .\supset .\\ +&(\alpha \,\,\text{exp}\,\,\gamma ) \times (\beta \,\,\text{exp}\,\,\gamma )\text{ sm }{\in}_{\Delta}ʻ\hat{\mu} + \{(\exists z).z\in \gamma .\mu = \downarrow zʻʻ\alpha \times \downarrow zʻʻ\beta\}\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*113·104·111.&\supset \vdash .\gamma \subset \text{ᗡ}ʻ\alpha \downarrow_{,,}.\gamma \subset \text{ᗡ}ʻ\beta \downarrow_{,,}.\alpha \downarrow_{,,}ʻʻ\gamma + ,\beta \downarrow_{,,}ʻʻ\gamma \in \text{Cls}^{2} \text{excl} &\qquad \text{(1)}\\ +\vdash .*113·105. &\supset \vdash :\text{Hp}.\supset .(\alpha \downarrow_{,,}ʻʻ\gamma )\upharpoonleft \alpha \downarrow_{,,}, \beta \downarrow_{,,}\upharpoonright + \gamma \in 1\rightarrow 1 &\qquad \text{(2)}\\ +\vdash .(1).(2).*115·6 \frac{\alpha \downarrow_{,,},\,\beta \downarrow_{,,}}{R,\,S}.\supset \\ +\vdash : \text{Hp}.&\supset .(\alpha \,\,\text{exp}\,\,\gamma ) \times (\beta \,\,\text{exp}\,\,\gamma )\text{ sm }{\in}_{\Delta}ʻ\hat{\mu}\{(\exists z).z\in \gamma .\mu = \alpha \downarrow_{,,}z + \times \beta \downarrow_{,,}z\} &\qquad \text{(3)}\\ +\vdash .(3).*38·2.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_159">[Pg 159]</span></p> + +<p>The hypothesis \(\exists !\alpha .\exists !\beta\) is not necessary in +the above proposition; but the proof is simpler with the hypothesis, +and we do not need the proposition without the hypothesis.</p> + +<p class="nind"><b>*116·531.</b> \[\begin{align}\vdash \colon\ldotp &M = \hat{R} \hat{z} \{z\in \gamma .R = (\downarrow z)\dagger \upharpoonright (\alpha \times \beta )\}.\supset :\\ +&z\in \gamma .\supset _{z}.Mʻz = (\downarrow z)\dagger \upharpoonright (\alpha \times \beta ).Mʻz\in 1\rightarrow 1.\text{ᗡ}ʻMʻz = \alpha \times \beta \end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*74·772.*55·12.*72·184 .&\supset \vdash .(\downarrow z)\dagger \in 1\rightarrow 1 &\qquad \text{(1)}\\ +\vdash .*21·33.&\supset \vdash \colon\ldotp \text{Hp}.z\in \gamma .\supset :RMz. \equiv .R = (\downarrow z)\dagger \upharpoonright (\alpha \times \beta ):\\ +[*30·3] &\supset :Mʻz = (\downarrow z)\dagger \upharpoonright (\alpha \times \beta ): &\qquad \text{(2)}\\ +[(1).*43·122] &\supset :Mʻz\in 1\rightarrow 1.\text{ᗡ}ʻMʻz = \alpha \times \beta &\qquad \text{(3)}\\ +\vdash .(2).(3).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*116·532.</b> \(\vdash :\text{Hp}*116·531.\exists !\alpha .\exists !\beta .\supset .M\in 1\rightarrow 1.\text{ᗡ}ʻM = \gamma\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*116·531.*14·21.*71·16.&\supset \vdash :\text{Hp}.\supset .M\in 1\rightarrow \text{Cls} &\qquad \text{(1)}\\ +\vdash . *116·531.\supset \vdash \colon\ldotp \text{Hp}.z,w\in \gamma .Mʻz = Mʻw.\supset :\\ +&(\downarrow z)\dagger \upharpoonright (\alpha \times \beta ) = (\downarrow w)\dagger \upharpoonright (\alpha \times \beta ):\\ +[*71·35] &\supset :R\in (\alpha \times \beta ).\supset .(\downarrow z)\dagger ʻR = (\downarrow w)\dagger ʻR:\\ +[*113·101] \supset :x\in \alpha .y\in \beta .&\supset .(\downarrow z)\dagger ʻ(y\downarrow x) = (\downarrow w)\dagger ʻ(y\downarrow x).\\ +[*113·123] &\supset .(y\downarrow z)\downarrow (x\downarrow z) = (y\downarrow w)\downarrow (x\downarrow w).\\ +[*55·202] &\supset .z = w &\qquad \text{(2)}\\ +\vdash .(2). &\supset \vdash \colon\ldotp \text{Hp}.\supset :z,w\in \gamma .Mʻz = Mʻw.\supset .z = w &\qquad \text{(3)}\\ +\vdash .*116·531.*14·21.*33·43.&\supset \vdash :\text{Hp}.z\in \gamma .\supset .z\in \text{ᗡ}ʻR &\qquad \text{(4)}\\ +\vdash .*21·33.&\supset \vdash \colon\ldotp \text{Hp}.\supset :RMz.\supset _{R,z}.z\in \gamma :\\ +[*33·351] &\supset :\text{ᗡ}ʻR\subset \gamma &\qquad \text{(5)}\\ +\vdash .(1).(3).(4).(5).*71·55.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*116·533.</b> + \[\begin{align}\vdash \colon\ldotp \text{Hp}*116·531.\supset :\text{D}ʻʻMʻʻ\gamma = \hat{\mu} \{(\exists z).z\in \gamma .\mu = \downarrow zʻʻ\alpha \times \downarrow zʻʻ\beta\}:\\ +z,w\in \gamma .\exists !\text{D}ʻMʻz\cap \text{D}ʻMʻw.\supset _{z,w}.z = w\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*116·531.\supset \vdash :\text{Hp}.z\in \gamma .\supset .\text{D}ʻMʻz &= \text{D}ʻ\{(\downarrow z)\dagger \upharpoonright (\alpha \times \beta )\}\\ +[*37·401] &= (\downarrow z)\dagger ʻʻ(\alpha \times \beta )\\ +[*113·65] &= \downarrow zʻʻ\alpha \times \downarrow zʻʻ\beta &\qquad \text{(1)}\\ +\vdash .(1).*37·6.&\supset \vdash :\text{Hp}.\supset .\text{D}ʻʻMʻʻ\gamma = \hat{\mu} \{(\exists z).z\in \gamma .\mu = \downarrow zʻʻ\alpha \times \downarrow zʻʻ\beta\} &\qquad \text{(2)}\\ +\vdash .*113·19.\supset \vdash :&\exists !(\downarrow zʻʻ\alpha \times \downarrow zʻʻ\beta )\cap (\downarrow wʻʻ\alpha \times \downarrow wʻʻ\beta ).\supset .\\ +&\exists !\downarrow zʻʻ\alpha \cap \downarrow wʻʻ\alpha .\\ +[*55·232] &\supset .z = w &\qquad \text{(3)}\\ +\vdash .(1).(2).(3).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_160">[Pg 160]</span></p> + +<p class="nind"><b><a id="*116·534">*116·534</a>.</b> \(\vdash :\text{Hp}*116·532.\supset .{\in}_{\Delta}ʻ\text{D}ʻʻMʻʻ\gamma \text{ sm }(\alpha \times \beta )\,\,\text{exp}\,\,\gamma\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*116·531·532·533.\supset \\ +\vdash \colon\ldotp \text{Hp}.\supset :M\in 1\rightarrow 1:z,w\in \gamma .&\exists !\text{D}ʻMʻz\cap \text{D}ʻMʻw.\supset _{z,\omega }.z = w:\\ +&z\in \gamma .\supset _{z}.Mʻz\in 1\rightarrow 1.\text{ᗡ}ʻMʻz = \alpha \times \beta :\\ +[*116·43] &\supset :\text{Prod}ʻ\text{D}ʻʻMʻʻ\gamma \text{ sm }(\alpha \times \beta )\,\,\text{exp}\,\,\gamma :\\ +[*115·12.*30·37.*84·11]&\supset :{\in}_{\Delta}ʻ\text{D}ʻʻMʻʻ\gamma \text{ sm }(\alpha \times \beta )\,\,\text{exp}\,\, \gamma \colon\ldotp \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*116·535.</b> + \[\begin{align}&\vdash :\exists !\alpha .\exists !\beta .\supset .(\alpha \,\,\text{exp}\,\,\gamma ) \times (\beta \,\,\text{exp}\,\,\gamma )\text{ sm }(\alpha \times \beta )\,\,\text{exp}\,\,\gamma\\ +&[*116·53·533·534]\end{align}\]</p> + +<p>The hypothesis \(\exists !\alpha .\exists !\beta\) is not necessary, as +we shall now prove.</p> + +<p class="nind"><b><a id="*116·54">*116·54</a>.</b> \(\vdash .(\alpha \,\,\text{exp}\,\,\gamma ) \times (\beta \,\,\text{exp}\,\,\gamma )\text{ sm }(\alpha \times \beta )\,\,\text{exp}\,\,\gamma \)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*116·182.\supset \vdash :\alpha = \Lambda .\exists !\gamma .&\supset .\alpha \,\,\text{exp}\,\,\gamma = \Lambda .\\ +[*113·114] &\supset .(\alpha \,\,\text{exp}\,\,\gamma ) \times (\beta \,\,\text{exp}\,\,\gamma ) = \Lambda &\qquad \text{(1)}\\ +\vdash .*113·114.*116·182.&\supset \vdash :\alpha = \Lambda .\exists !\gamma .\supset .(\alpha \times \beta )\,\,\text{exp}\,\,\gamma = \Lambda &\qquad \text{(2)}\\ +\vdash .(1).(2). &\supset \vdash :\alpha = \Lambda .\exists !\gamma .\supset .(\alpha \,\,\text{exp}\,\,\gamma ) \times (\beta \,\,\text{exp}\,\,\gamma )\text{ sm }(\alpha \times \beta )\,\,\text{exp}\,\,\gamma &\qquad \text{(3)}\\ +\text{Similarly}\quad &\vdash :\beta = \Lambda .\exists !\gamma .\supset .(\alpha \,\,\text{exp}\,\,\gamma ) \times (\beta \,\,\text{exp}\,\,\gamma )\text{ sm }(\alpha \times \beta )\,\,\text{exp}\,\,\gamma &\qquad \text{(4)}\\ +\vdash .*116·181. \supset \vdash :\gamma = \Lambda .&\supset .(\alpha \,\,\text{exp}\,\,\gamma ) \times (\beta \,\,\text{exp}\,\,\gamma ) = \iota ʻ\Lambda \times \iota ʻ\Lambda .\\ +[*113·611.*73·43] &\supset .(\alpha \,\,\text{exp}\,\,\gamma ) \times (\beta \,\,\text{exp}\,\,\gamma )\text{ sm }\iota ʻ\Lambda &\qquad \text{(5)}\\ +\vdash .*116·181. \supset \vdash :\gamma = \Lambda .&\supset .(\alpha \times \beta )\,\,\text{exp}\,\,\gamma = \iota ʻ\Lambda .\\ +[(5)] &\supset .(\alpha \,\,\text{exp}\,\,\gamma ) \times (\beta \,\,\text{exp}\,\,\gamma )\text{ sm }(\alpha \times \beta )\,\,\text{exp}\,\,\gamma &\qquad \text{(6)}\\ +\vdash .(3).(4).(6).*116·535.&\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p>In obtaining (5), we use *73·43 as well as <a href="#*113·611">*113·611</a>, because +\(\Lambda\)'s of different types are involved.</p> + +<p class="nind"><b><a id="*116·55">*116·55</a>.</b> \(\vdash .\mu ^{\varpi } \times _{c} \nu ^{\varpi } = (\mu \times _{c} \nu )^{\varpi }\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*116·54·222.*113·222.&\supset \vdash .(\text{N}_{0}\text{c}ʻ\alpha )^{\text{N}_{0}\text{c}ʻ\gamma } \times _{c} (\text{N}_{0}\text{c}ʻ\beta + )^{\text{N}_{0}\text{c}ʻ\gamma }\\ +&=(\text{N}_{0}\text{c}ʻ\alpha \times _{c} \text{N}_{0}\text{c}ʻ\beta )^{\text{N}_{0}\text{c}ʻ\gamma } &\qquad \text{(1)}\\ +\vdash .(1).*103·2. &\supset \vdash :\mu ,\nu ,\varpi \in \text{N}_{0}\text{C}.\supset .\mu ^{\varpi } \times _{c} \nu ^{\varpi } = (\mu \times _{c} + \nu )^{\varpi } &\qquad \text{(2)}\\ +\vdash .*116·205.*113·204.&\supset \vdash :\varpi {\sim}\in \text{N}_{0}\text{C}.\supset .\mu ^{\varpi } \times _{c} \nu ^{\varpi } + = \Lambda = (\mu \times _{c} \nu )^{\varpi } &\qquad \text{(3)}\\ +\vdash .*116·205.*113·204.&\supset \vdash :{\sim}(\mu ,\nu \in \text{N}_{0}\text{C}).\supset .\mu ^{\varpi } \times _{c} \nu ^{\varpi } + = \Lambda &\qquad \text{(4)}\\ +\vdash .*113·204.*116·204.&\supset \vdash :{\sim}(\mu ,\nu \in \text{N}_{0}\text{C}).\supset .(\mu \times _{c} \nu )^{\varpi } = \Lambda &\qquad \text{(5)}\\ +\vdash .(4).(5) &\supset \vdash :{\sim}(\mu ,\nu \in \text{N}_{0}\text{C}).\supset .\mu ^{\varpi } \times _{c} \nu ^{\varpi } = (\mu \times _{c} + \nu )^{\varpi } &\qquad \text{(6)}\\ +\vdash .(2).(3).(6). \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p>This completes the proof of the second of the formal laws of +exponentiation. The following propositions are lemmas for the third of +these laws, namely +\[ +(\mu ^{\nu })^{\varpi } = \mu ^{\nu \times _{c} \varpi }. +\]</p> + +<p><span class="pagenum" id="Page_161">[Pg 161]</span></p> + +<p class="nind"><b>*116·6.</b> \[\begin{align}\vdash :\exists !\alpha .\supset .\alpha \,\,\text{exp}\,\,(\beta \times \gamma )\text{ sm } \text{Prod}ʻ\text{Prod}ʻʻ&\alpha \downarrow_{,,}ʻʻʻ\beta + \downarrow_{,,}ʻʻ\gamma .\\ +&\alpha \downarrow_{,,}ʻʻʻ\beta \downarrow_{,,}ʻʻ\gamma \in \text{Cls}^{3}\,\text{arithm}\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*113·105.*84·53 \frac{\alpha \downarrow_{,,}}{R}.*113·111.&\supset \vdash :\text{Hp}.\supset .\alpha \downarrow_{,,}ʻʻʻ\beta \downarrow_{,,}ʻʻ\gamma + \in \text{Cls}^{2} \text{excl} &\qquad \text{(1)}\\ +\vdash .*40·38. &\supset \vdash .sʻ\alpha \downarrow_{,,}ʻʻʻ\beta \downarrow_{,,}ʻʻ\gamma = \alpha \downarrow_{,,}ʻʻsʻ\beta \downarrow_{,,}ʻʻ\gamma + &\qquad \text{(2)}\\ +[*113·111] &\supset \vdash .sʻ\alpha \downarrow_{,,}ʻʻʻ\beta \downarrow_{,,}ʻʻ\gamma \in \text{Cls}^{2} \text{excl} &\qquad \text{(3)}\\ +\vdash .(1).(3).*115·2. &\supset \vdash :\text{Hp}.\supset .\alpha \downarrow_{,,}ʻʻʻ\beta \downarrow_{,,}ʻʻ\gamma \in \text{Cls}^{3}\,\text{arithm} + &\qquad \text{(4)}\\ +\vdash .*113·141.*116·19. \supset \vdash .&\text{Nc}ʻ\{\alpha \,\,\text{exp}\,\,(\beta \times \gamma )\} = \text{Nc}ʻ\{\alpha \,\,\text{exp}\,\,(\gamma \times \beta )\}\\ +[(*116·01.*113·02)] &= \text{Nc}ʻ\text{Prod}ʻ\alpha \downarrow_{,,}ʻʻsʻ\beta \downarrow_{,,}ʻʻ\gamma \\ +[(2)] &= \text{Nc}ʻ\text{Prod}ʻsʻ\alpha \downarrow_{,,}ʻʻʻ\beta \downarrow_{,,}ʻʻ\gamma \\ +[*115·35.(4)] &= \text{Nc}ʻ\text{Prod}ʻ\text{Prod}ʻʻ\alpha \downarrow_{,,}ʻʻʻ\beta \downarrow_{,,}ʻʻ\gamma &\qquad \text{(5)}\\ +\vdash .(4).(5).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*116·601.</b> \(\vdash .\mid (\text{Cnv}ʻ\downarrow z)\in 1\rightarrow 1 \quad[*74·774.*72·184]\)</p> + +<p class="nind"><b>*116·602.</b> \[\begin{align}\vdash \colon\ldotp M = \hat{R} \hat{z} &[z\in \gamma .R = \{\mid (\text{Cnv}ʻ\downarrow z)\}_{\in }\upharpoonright + (\alpha \,\,\text{exp}\,\,\beta )].\supset :\\ +&z\in \gamma .\supset .Mʻz = \{\mid (\text{Cnv}ʻ\downarrow z)\}_{\in }\upharpoonright (\alpha \,\,\text{exp}\,\,\beta ):\text{ᗡ}ʻM = \gamma \end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*21·33.\supset \\ +\vdash \colon\colon \text{Hp}.\supset \colon\ldotp z\in \gamma .\supset :RMz. \equiv .R = {\mid (\text{Cnv}ʻ\downarrow z)}_{\in }\upharpoonright + (\alpha \,\,\text{exp}\,\,\beta ) &\qquad \text{(1)}\\ +\vdash .(1).*30·3.\supset \vdash \colon\ldotp \text{Hp}.\supset :z\in \gamma .\supset .Mʻz = {\mid (\text{Cnv}ʻ\downarrow z)}_{\in }\upharpoonright + (\alpha \,\,\text{exp}\,\,\beta ) &\qquad \text{(2)}\\ +\vdash .*21·33.*33·131.\supset \vdash :\text{Hp}.\supset .\text{ᗡ}ʻM = \gamma &\qquad \text{(3)}\\ +\vdash .(2).(3).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*116·603.</b> \[\begin{align}&\vdash \colon\ldotp \text{Hp}*116·602.\supset :z\in \gamma .\supset .\text{ᗡ}ʻMʻz = \alpha \,\,\text{exp}\,\,\beta \\ +&[*116·602.*37·231.*35·65]\end{align}\]</p> + +<p class="nind"><b>*116·604.</b> \(\vdash \colon\ldotp \text{Hp}*116·602.\supset :z\in \gamma .\supset .\text{D}ʻMʻz = \text{Prod}ʻ\alpha \downarrow_{,,}ʻʻ\beta \downarrow_{,,}z\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*37·401.*116·602.\supset \\ +\vdash :\text{Hp}.z\in \gamma .\supset .\text{D}ʻMʻz &= \{\mid (\text{Cnv}ʻ\downarrow z)\}_{\in}ʻʻ(\alpha \,\,\text{exp}\,\,\beta )\\ +[*115·4.*116·601.*43·301] &= \text{Prod}ʻ\{\mid (\text{Cnv}ʻ\downarrow z)\}_{\in}ʻʻ\alpha \downarrow_{,,}ʻʻ\beta \\ +[*113·125 \frac{I}{R}.*50·75·16] &= \text{Prod}ʻ\alpha \downarrow_{,,}ʻʻ\downarrow zʻʻ\beta \\ +[*38·2] &= \text{Prod}ʻ\alpha \downarrow_{,,}ʻʻ\beta \downarrow_{,,}z:\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*116·605.</b> \(\vdash \colon\ldotp \text{Hp}*116·602.\supset :z\in \gamma .\supset .Mʻz\in 1\rightarrow 1\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*116·601.*72·451.\supset \\ +\vdash .{\mid (\text{Cnv}ʻ\downarrow z)}_{\in }\upharpoonright \text{Cl}ʻ\text{ᗡ}ʻ\mid (\text{Cnv}ʻ\downarrow z)\in 1\rightarrow 1.\\ +[*43·301]\supset \vdash .{\mid (\text{Cnv}ʻ\downarrow z)}_{\in }\upharpoonright (\alpha \,\,\text{exp}\,\,\beta )\in 1\rightarrow 1 &\qquad \text{(1)}\\ +\vdash .(1).*116·602.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_162">[Pg 162]</span></p> + +<p class="nind"><b>*116·606.</b> \[\begin{align}\vdash \colon\ldotp \text{Hp}*116·602.&\exists !\alpha .\exists !\beta .\supset :\\ +&M\in 1\rightarrow 1:z,w\in \gamma .\text{D}ʻMʻz = \text{D}ʻMʻw.\supset _{z,w}.z = w\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*116·602.*14·21.&\supset \vdash \colon\ldotp \text{Hp}.\supset :z\in \text{ᗡ}ʻM.\supset _{z}.\text{E}!Mʻz:\\ +[*71·16] &\supset :M\in 1\rightarrow \text{Cls} &\qquad \text{(1)}\\ +\vdash .*30·37.&\supset \vdash \colon\ldotp \text{Hp}.\supset :z,w\in \gamma .Mʻz = Mʻw.\supset .\text{D}ʻMʻz = \text{D}ʻMʻw &\qquad \text{(2)}\\ +\vdash .*116·604.&\supset \vdash \colon\ldotp \text{Hp}.\supset :z,w\in \gamma .\text{D}ʻMʻz = \text{D}ʻMʻw.\supset .\\ +&\text{Prod}ʻ\alpha \downarrow_{,,}ʻʻ\beta \downarrow_{,,}z = \text{Prod}ʻ\alpha \downarrow_{,,}ʻʻ\beta \downarrow_{,,}w.\\ +[*30·37] &\supset .sʻ\text{Prod}ʻ\alpha \downarrow_{,,}ʻʻ\beta \downarrow_{,,}z = sʻ\text{Prod}ʻ\alpha \downarrow_{,,}ʻʻ\beta \downarrow_{,,}w.\\ +[*116·171.*115·141.(*116·01)] &\supset .sʻ\alpha \downarrow_{,,}ʻʻ\beta \downarrow_{,,}z = sʻ\alpha \downarrow_{,,}ʻʻ\beta \downarrow_{,,}w.\\ +[*113·1] &\supset .\beta \downarrow_{,,}z \times \alpha = \beta \downarrow_{,,}w \times \alpha .\\ +[*113·182] &\supset .\beta \downarrow_{,,}z = \beta \downarrow_{,,}w.\\ +[*113·105.\text{Hp}] &\supset .z = w &\qquad \text{(3)}\\ +\vdash .(1).(2).(3).*71·55.*116·602.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*116·607.</b> \[\begin{align}&\vdash \colon\ldotp \text{Hp}*116·602.\exists !\alpha .\exists !\beta .\supset :\\ +&M\in 1\rightarrow 1.\text{D}ʻʻMʻʻ\gamma = \text{Prod}ʻʻ\alpha \downarrow_{,,}ʻʻʻ\beta \downarrow_{,,}ʻʻ\gamma :\\ +&z,w\in \gamma .\text{D}ʻMʻz = \text{D}ʻMʻw.\supset _{z,w}.z = w:\\ +&z\in \gamma .\supset _{z}.Mʻz\in 1\rightarrow 1.\text{ᗡ}ʻMʻz = \alpha \,\,\text{exp}\,\,\beta \quad[*116·606·604·605·603]\end{align}\]</p> + +<p class="nind"><b><a id="*116·61">*116·61</a>.</b> \[\begin{align}&\vdash :\exists !\alpha .\exists !\beta .\supset .\text{Prod}ʻ\text{Prod}ʻʻ\alpha \downarrow_{,,}ʻʻʻ\beta \downarrow_{,,}ʻʻ\gamma + \text{ sm }(\alpha \,\,\text{exp}\,\,\beta )\,\,\text{exp}\,\,\gamma \\ +&[*116·607·43]\end{align}\]</p> + +<p class="nind"><b>*116·611.</b> \(\vdash :\exists !\alpha .\exists !\beta .\supset .\alpha \,\,\text{exp}\,\,(\beta \times \gamma )\text{ sm }(\alpha \,\,\text{exp}\,\,\beta )\,\,\text{exp}\,\,\gamma \quad[116·6·61]\)</p> + +<p class="nind"><b>*116·62.</b> \(\vdash .\alpha \,\,\text{exp}\,\,(\beta \times \gamma )\text{ sm }(\alpha \,\,\text{exp}\,\,\beta )\,\,\text{exp}\,\,\gamma\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*116·181.*113·114.&\supset \vdash :\beta = \Lambda .\supset .\alpha \,\,\text{exp}\,\,(\beta \times \gamma ) = \iota ʻ\Lambda &\qquad \text{(1)}\\ +\vdash .*116·181.&\supset \vdash :\beta = \Lambda .\supset .(\alpha \,\,\text{exp}\,\,\beta )\,\,\text{exp}\,\,\gamma = (\iota ʻ\Lambda )\,\,\text{exp}\,\,\gamma &\qquad \text{(2)}\\ +\vdash .*116·33·25.&\supset \vdash .\text{Nc}ʻ\{(\iota ʻ\Lambda )\,\,\text{exp}\,\,\gamma\}= 1 &\qquad \text{(3)}\\ +\vdash . (1).(2).(3).*55·22.*100·31.&\supset \vdash :\beta = \Lambda .\supset .\alpha \,\,\text{exp}\,\,(\beta \times \gamma )\text{ sm }(\alpha \,\,\text{exp}\,\,\beta )\,\,\text{exp}\,\,\gamma &\qquad \text{(4)}\\ +\vdash . *113·107.*116·182.&\supset \vdash :\alpha = \Lambda .\exists !\beta .\exists !\gamma .\supset .\alpha \,\,\text{exp}\,\,(\beta \times \gamma ) = \Lambda &\qquad \text{(5)}\\ +\vdash .*116·182.&\supset \vdash :\alpha = \Lambda .\exists !\beta .\exists !\gamma .\supset .(\alpha \,\,\text{exp}\,\,\beta )\,\,\text{exp}\,\,\gamma = \Lambda &\qquad \text{(6)}\\ +\vdash .(5).(6).&\supset \vdash :\alpha = \Lambda .\exists !\beta .\exists !\gamma .\supset .\alpha \,\,\text{exp}\,\,(\beta \times \gamma )\text{ sm }(\alpha \,\,\text{exp}\,\,\beta )\,\,\text{exp}\,\,\gamma &\qquad \text{(7)}\\ +\vdash .*113·114.*116·181.&\supset \vdash :\gamma = \Lambda .\supset .\alpha \,\,\text{exp}\,\,(\beta \times \gamma ) = \iota ʻ\Lambda .(\alpha \,\,\text{exp}\,\,\beta )\,\,\text{exp}\,\,\gamma = \iota ʻ\Lambda .\\ +[*73·43] &\supset .\alpha \,\,\text{exp}\,\,(\beta \times \gamma )\text{ sm }(\alpha \,\,\text{exp}\,\,\beta )\,\,\text{exp}\,\,\gamma &\qquad \text{(8)}\\ +\vdash .(4).(7).(8).&\supset \vdash \colon\ldotp \alpha = \Lambda .\lor.\beta = \Lambda .\lor.\gamma = \Lambda :\supset .\\ +&\alpha \,\,\text{exp}\,\,(\beta \times \gamma )\text{ sm }(\alpha \,\,\text{exp}\,\,\beta )\,\,\text{exp}\,\,\gamma &\qquad \text{(9)}\\ +\vdash .(9).*116·611.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_163">[Pg 163]</span></p> + +<p class="nind"><b><a id="*116·63">*116·63</a>.</b> \(\vdash . \mu ^{\nu \times _{c} \varpi } = (\mu ^\nu )^{\varpi }\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *113·222 . \supset \vdash . (\text{N}_{0}\text{c}ʻ\alpha )^{\text{N}_{0}\text{c}ʻ\beta \times _{c} \text{N}_{0}\text{c}ʻ\gamma } &= (\text{N}_{0}\text{c}ʻ\alpha + )^{\text{Nc}ʻ(\beta \times \gamma )}\\ +[*116·222 . (*116·04)] &= \text{Nc}ʻ\{\alpha \,\,\text{exp}\,\, (\beta \times \gamma )\}\\ +[*116·62] &= \text{Nc}ʻ\{(\alpha \,\,\text{exp}\,\, \beta ) \,\,\text{exp}\,\, \gamma\}\\ +[*116·222] &= \{\text{N}_{0}\text{c}ʻ(\alpha \,\,\text{exp}\,\, \beta )\}^{\text{N}_{0}\text{c}ʻ\gamma }\\ +[*116·222 . (*116· 03)] &= \{(\text{N}_{0}\text{c}ʻ\alpha )^{\text{N}_{0}\text{c}ʻ\beta }\}^{\text{N}_{0}\text{c}ʻ\gamma } &\qquad \text{(1)}\\ +\vdash . (1) . *103·2 . &\supset \vdash : \mu , \nu , \varpi \in \text{N}_{0}\text{C} . \supset . \mu ^{\nu \times _{c} \varpi } = (\mu ^\nu )^{\varpi } + &\qquad \text{(2)}\\ +\vdash . *116·204·205 . &\supset \vdash : {\sim} (\mu , \nu \in \text{N}_{0}\text{C}) . \supset . (\mu ^\nu )^{\varpi } = \Lambda &\qquad \text{(3)}\\ +\vdash . *113·205 . *116·204·205 . &\supset \vdash : {\sim}(\mu , \nu \in \text{N}_{0}\text{C}) . \supset . \mu ^{\nu \times _{c} \varpi } + = \Lambda &\qquad \text{(4)}\\ +\vdash . *116·205 . &\supset \vdash : \varpi {\sim} \in \text{N}_{0}\text{C} . \supset . (\mu ^\nu )^{\varpi } = \Lambda &\qquad \text{(5)}\\ +\vdash . *113·205 . *116·204 . &\supset \vdash : \varpi {\sim} \in \text{N}_{0}\text{C} . \supset . \mu ^{\nu \times _{c} \varpi } + = \Lambda &\qquad \text{(6)}\\ +\vdash . (3) . (4) . (5) . (6) . &\supset \vdash : {\sim} (\mu , \nu , \varpi \in \text{N}_{0}\text{C}) . \supset . \mu ^{\nu \times _{c} \varpi } = (\mu ^\nu + )^{\varpi } &\qquad \text{(7)}\\ +\vdash . (2) . (7) . \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p>This completes the proof of the third of the formal laws of +exponentiation.</p> + +<p class="nind"><b>*116·64.</b> \(\vdash . (\mu ^\nu )^{\varpi } = (\mu ^{\varpi })^\nu \quad[*116·63 . *113·27]\)</p> + +<p class="nind"><b>*116·651.</b> \(\vdash : Q \in \text{Cls} \rightarrow 1 . \kappa \in \text{Cls}^{2} \text{excl} . \supset . {\in}_{\Delta}ʻP_{\Delta }ʻʻQʻʻʻ\kappa + \text{ sm } P_{\Delta }ʻQʻʻsʻ\kappa \)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *84·53 . \supset \vdash : \text{Hp} . &\supset . Qʻʻʻ\kappa \in \text{Cls}^{2} \text{excl}.\\ +[*85·43] &\supset . {\in}_{\Delta}ʻP_{\Delta }ʻʻQʻʻʻ\kappa \text{ sm } P_{\Delta }ʻsʻQʻʻʻ\kappa .\\ +[*40·38] &\supset . {\in}_{\Delta}ʻP_{\Delta }ʻʻQʻʻʻ\kappa \text{ sm } P_{\Delta }ʻQʻʻsʻ\kappa : \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*116·652.</b> \[\begin{align}&\vdash : Q \in \text{Cls} \rightarrow 1 . \kappa \in \text{Cls}^{2} \text{excl} . \supset . {\in}_{\Delta}ʻ{\in}_{\Delta}ʻʻQʻʻʻ\kappa + \text{ sm } {\in}_{\Delta}ʻQʻʻsʻ\kappa \\ +&\left[*116·651 \frac{\in}{P}\right]\end{align}\]</p> + +<p>The following propositions are lemmas for <a href="#*116·661">*116·661</a>, which is an +extension of <a href="#*116·52">*116·52</a>.</p> + +<p class="nind"><b>*116·653.</b> \(\vdash : \kappa \in \text{Cls}^{2} \text{excl} . \supset . \alpha \downarrow_{,,}ʻʻʻ\kappa \in \text{Cls}^{2} \text{arithm}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *113·105 . *84·53 . &\supset \vdash : \text{Hp} . \exists ! \alpha . \supset . \alpha \downarrow_{,,}ʻʻʻ\kappa \in \text{Cls}^{2} + \text{excl} &\qquad \text{(1)}\\ +\vdash . *113·111 . &\supset \vdash . \alpha \downarrow_{,,}ʻʻsʻ\kappa \in \text{Cls}^{2} \text{excl} .\\ +[*40·38] &\supset \vdash . sʻ\alpha \downarrow_{,,}ʻʻʻ\kappa \in \text{Cls}^{2} \text{excl} . &\qquad \text{(2)}\\ +\vdash . *113·112·113 . \supset \vdash \colon\ldotp \alpha = \Lambda . \supset : &\beta \in \kappa . \exists ! \beta . \supset . \alpha \downarrow_{,,}ʻʻ\beta + = \iota ʻ\Lambda :\\ +&\beta \in \kappa . \beta = \Lambda . \supset . \alpha \downarrow_{,,}ʻʻ\beta = \Lambda &\qquad \text{(3)}\\ +\vdash . (3) . \supset \vdash \colon\ldotp \alpha = \Lambda . &\supset : \alpha \downarrow_{,,}ʻʻʻ\kappa \subset \iota ʻ\iota ʻ\Lambda \cup \iota ʻ\Lambda :\\ +[*24·43·561] &\supset : \rho , \sigma \in \alpha \downarrow_{,,}ʻʻʻ\kappa . \exists ! \rho \cap \sigma . \supset . \rho , \sigma \in \iota ʻ\iota ʻ\Lambda .\\ +[*51·15] &\supset . \rho = \sigma :\\ +[*84·11] &\supset : \alpha \downarrow_{,,}ʻʻʻ\kappa \in \text{Cls}^{2} \text{excl} &\qquad \text{(4)}\\ +\vdash . (1) . (4) . &\supset \vdash : \text{Hp} . \supset . \alpha \downarrow_{,,}ʻʻʻ\kappa \in \text{Cls}^{2} \text{excl} &\qquad \text{(5)}\\ +\vdash . (2) . (5) . \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_164">[Pg 164]</span></p> + +<p class="nind"><b>*116·654.</b> \(\vdash :\kappa \in \text{Cls}^{2} \text{excl}.\supset .\{\text{Prod}ʻ(\alpha \,\,\text{exp}\,\,)ʻʻ\kappa \}\text{ sm }\{\alpha \,\,\text{exp}\,\,(sʻ\kappa )\}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*38·13.(*116·01).\supset \vdash .\text{Prod}ʻ(\alpha \,\,\text{exp}\,\,)ʻʻ\kappa = \text{Prod}ʻ\text{Prod}ʻʻ\alpha \downarrow_{,,}ʻʻʻ\kappa &\qquad \text{(1)}\\ +\vdash .(1).*116·653.*115·34.\supset \\ +\vdash .\{\text{Prod}ʻ(\alpha \,\,\text{exp}\,\,)ʻʻ\kappa \}\text{ sm }\{\text{Prod}ʻsʻ\alpha \downarrow_{,,}ʻʻʻ\kappa\} &\qquad \text{(2)}\\ +\vdash .(2).*40·38.(*116·01).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*116·655.</b> \(\vdash :\kappa \in \text{Cls}^{2} \text{excl}.\supset .\Pi \text{Nc}ʻ(\alpha \,\,\text{exp}\,\,)ʻʻ\kappa = (\text{Nc}ʻ\alpha )^{\Sigma \text{Nc}ʻ\kappa + } \quad[*116·654]\)</p> + +<p>This proposition is an extension of <a href="#*116·5">*116·5</a>.</p> + +<p>The hypothesis \(\kappa \in \text{Cls}^{2} \text{excl}\) is unnecessary +in the above proposition, as we shall now prove.</p> + +<p class="nind"><b>*116·656.</b> \(\vdash :\exists !\alpha \,\,\text{exp}\,\,\beta \cap \alpha \,\,\text{exp}\,\,\gamma .\supset .\beta = \gamma \)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*116·11.*52·16.\supset \\ +\vdash \colon\ldotp \mu \in (\alpha \,\,\text{exp}\,\, \beta )\cap (\alpha \,\,\text{exp}\,\, \gamma ).&\supset :y\in \beta .\supset .(\exists x).x\in \alpha .x\downarrow y\in \mu :\mu \subset \gamma \times \alpha :\\ +[*113·101] &\supset :y\in \beta .\supset .(\exists x).x\in \alpha .x\downarrow y\in \mu :x\downarrow y\in \mu .\supset .y\in \gamma :\\ +[\text{Syll}] &\supset :y\in \beta .\supset .(\exists x).x\in \alpha .x\downarrow y\in \mu .y\in \gamma .\\ +[*10·35] &\supset .y\in \gamma &\qquad \text{(1)}\\ +\text{Similarly}\quad &\vdash \colon\ldotp \mu \in \alpha \,\,\text{exp}\,\, \beta \cap \alpha \,\,\text{exp}\,\, \gamma .\supset :y\in \gamma .\supset .y\in \beta &\qquad \text{(2)}\\ +\vdash .(1).(2).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*116·657.</b> \(\vdash .(\alpha \,\,\text{exp}\,\,)ʻʻ\kappa \in \text{Cls}^{2} \text{excl} \quad[*116·656]\)</p> + +<p class="nind"><b>*116·658.</b> \(\vdash .\alpha \,\,\text{exp}\,\, (\in \unicode{x21A7}\beta ) = {\mid (\text{Cnv}ʻ\downarrow \beta )}_{\in}ʻʻ(\alpha \,\,\text{exp}\,\, \beta )\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*116·602·604.*37·401.\supset \vdash .\{\mid (\text{Cnv}ʻ\downarrow \beta )\}_{\in}ʻʻ(\alpha \,\,\text{exp}\,\,\beta ) &= \alpha \,\,\text{exp}\,\,(\beta \downarrow_{,,}\beta )\\ +[*85.601] &= \alpha \,\,\text{exp}\,\,(\in \unicode{x21A7}\beta ).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*116·659.</b> \[\begin{align}T = \hat{\nu} \hat{\mu}\{(\exists \beta ).\beta \in \kappa .\mu \in \alpha \,\,\text{exp}\,\,\beta .&\nu = \mid (\text{Cnv}ʻ\downarrow \beta )ʻʻ\mu\}.\supset .\\ +&T\in (\alpha \,\,\text{exp}\,\,)ʻʻ\in \unicode{x21A7}ʻʻ\kappa \,\overline{\text{ sm }}\, \,\overline{\text{ sm }}\, (\alpha \,\,\text{exp}\,\,)ʻʻ\kappa\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*40·4. \supset \vdash :\text{Hp}.&\supset .\text{ᗡ}ʻT = sʻ(\alpha \,\,\text{exp}\,\,)ʻʻ\kappa &\qquad \text{(1)}\\ +\vdash .*21·33.\supset \vdash \colon\ldotp \text{Hp}.&\supset :\nu T\mu .\varpi T\mu .\supset .\\ +&(\exists \beta ,\gamma ).\beta ,\gamma \in \kappa .\mu \in \alpha \,\,\text{exp}\,\,\beta .\mu \in \alpha \,\,\text{exp}\,\,\gamma .\\ +&\nu = \mid (\text{Cnv}ʻ\downarrow \beta )ʻʻ\mu .\varpi = \mid (\text{Cnv}ʻ\downarrow \gamma )ʻʻ\mu .\\ +[*116·656] &\supset .(\exists \beta ,\gamma ).\beta = \gamma .\nu = \mid (\text{Cnv}ʻ\downarrow \beta )ʻʻ\mu .\varpi = \mid (\text{Cnv}ʻ\downarrow \gamma )ʻʻ\mu .\\ +[*13·195] &\supset .\nu = \varpi &\qquad \text{(2)}\\ +\vdash .*21·33.&\supset \vdash \colon\ldotp \text{Hp}.\supset :\varpi T\mu .\varpi T\nu .\supset .\\ +&(\exists \beta ,\gamma ).\beta ,\gamma \in \kappa .\mu \in \alpha \,\,\text{exp}\,\,\beta .\nu \in \alpha \,\,\text{exp}\,\,\gamma .\\ +&\varpi = \mid (\text{Cnv}ʻ\downarrow \beta )ʻʻ\mu = \mid (\text{Cnv}ʻ\downarrow \gamma )ʻʻ\nu .\\ +[*116·658] &\supset .(\exists \beta ,\gamma ).\beta ,\gamma \in \kappa .\mu \in \alpha \,\,\text{exp}\,\, \beta .\nu \in \alpha \,\,\text{exp}\,\, \gamma .\\ +&\varpi = \mid (\text{Cnv}ʻ\downarrow \beta )ʻʻ\mu = \mid (\text{Cnv}ʻ\downarrow \gamma )ʻʻ\nu .\\ +&\varpi \in \alpha \,\,\text{exp}\,\, (\in \unicode{x21A7}\beta )\cap \alpha \,\,\text{exp}\,\, (\in \unicode{x21A7}\gamma ).\\ +[*116·656] & \supset .(\exists \beta ,\gamma ).\beta ,\gamma \in \kappa .\mid (\text{Cnv}ʻ\downarrow \beta )ʻʻ\mu = \mid (\text{Cnv}ʻ\downarrow \gamma )ʻʻ\nu .\\ +&\in \unicode{x21A7}\beta = \in \unicode{x21A7}\gamma .\\ +[*85·601] &\supset .(\exists \beta ).\beta \in \kappa .\mid (\text{Cnv}ʻ\downarrow \beta )ʻʻ\mu = \mid (\text{Cnv}ʻ\downarrow \beta )ʻʻ\nu .\\ +[*116·601.*72·441] &\supset .\mu = \nu &\qquad \text{(3)}\\ +\vdash .(2).(3). &\supset \vdash :\text{Hp}.\supset .T\in 1\rightarrow 1 &\qquad \text{(4)}\\ +\vdash .*116·658.&\supset \vdash \colon\ldotp \text{Hp}.\supset :\beta \in \kappa .\supset .Tʻʻ(\alpha \,\,\text{exp}\,\, \beta ) = \alpha \,\,\text{exp}\,\, (\in \unicode{x21A7}\beta ):\\ +[*37·69] &\supset :\tau _{\in }ʻʻ(\alpha \,\,\text{exp}\,\,)ʻʻ\kappa = (\alpha \,\,\text{exp}\,\,)ʻʻ\in \unicode{x21A7}ʻʻ\kappa &\qquad \text{(5)}\\ +\vdash .(1).(4).(5).*111·1 .\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_165">[Pg 165]</span></p> + +<p class="nind"><b>*116·66.</b> \(\vdash .\text{Prod}ʻ(\alpha \,\,\text{exp}\,)ʻʻ\kappa \text{ sm } \{\alpha \,\,\text{exp}\,\, (\Sigma ʻ\kappa )\}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*116·659.*115·51. &\supset \vdash .\text{Prod}ʻ(\alpha \,\,\text{exp}\,\,)ʻʻ\kappa \text{ sm } \text{Prod}ʻ(\alpha \,\,\text{exp}\,\,)ʻʻ\in \unicode{x21A7}ʻʻ\kappa &\qquad \text{(1)}\\ +\vdash .*85·61.*116·654. &\supset \vdash .\text{Prod}ʻ(\alpha \,\,\text{exp}\,\,)ʻʻ\in \unicode{x21A7}ʻʻ\kappa \text{ sm } \{\alpha \,\,\text{exp}\,\, (sʻ\in \unicode{x21A7}ʻʻ\kappa )\} &\qquad \text{(2)}\\ +\vdash .(1).(2).*112·1. \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*116·661">*116·661</a>.</b> \(\vdash .\Pi \text{Nc}ʻ(\alpha \,\,\text{exp}\,\,)ʻʻ\kappa = (\text{Nc}ʻ\alpha )^{\Sigma \text{Nc}ʻ\kappa } \quad[*116·66·657.*115·12.*112·101]\)</p> + +<p>This proposition is an extension of <a href="#*116·52">*116·52</a>.</p> + +<p>The following propositions are concerned in proving <a href="#*116·68">*116·68</a>, which is +an extension of <a href="#*116·54">*116·54</a>, where the \(\alpha\) and \(\beta\) of that +proposition are replaced by the members of a class \(\kappa\).</p> + +<p class="nind"><b>*116·67.</b> \(\vdash \colon\ldotp \rho = \hat{\lambda} \{(\exists \alpha ).\alpha \in \kappa .\lambda = \alpha \downarrow ʻʻ\gamma\}.\supset :\kappa \in \text{Cls}^{2} + \text{excl}.\supset .\rho \in \text{Cls}^{3}\,\text{arithm}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*20·3 .\supset \vdash :&\text{Hp}.\lambda ,\mu \in \rho .\exists !\lambda \cap \mu .\supset .\\ +&(\exists \alpha ,\beta ).\alpha ,\beta \in \kappa .\lambda = \alpha \downarrow_{,,}ʻʻ\gamma .\mu = \beta \downarrow_{,,}ʻʻ\gamma .\exists !\lambda \cap \mu .\\ +[*37·6] &\supset .(\exists \alpha ,\beta ,z,w).\alpha ,\beta \in \kappa .z,w\in \gamma .\alpha \downarrow_{,,}z = \beta \downarrow_{,,}w.\lambda = \alpha \downarrow_{,,}ʻʻ\gamma + .\\ +&\mu = \beta \downarrow_{,,}ʻʻ\gamma .\\ +[*55·262.*38·2]&\supset .(\exists \alpha ,z,w).\alpha \in \kappa .z,w\in \gamma .\lambda = \alpha \downarrow_{,,}ʻʻ\gamma .\mu = \alpha \downarrow_{,,}ʻʻ\gamma .\\ +[*13·172] &\supset .\lambda =\mu &\qquad \text{(1)}\\ +\vdash .*37·6.*40·11.\supset \\ +\vdash :\text{Hp}.\xi ,\eta \in sʻ\rho .\exists !\xi \cap \eta .\supset .\\ +&(\exists \alpha ,\beta ,z,w).\alpha ,\beta \in \kappa .z,w\in \gamma .\xi = \alpha \downarrow_{,,}z.\eta = \beta \downarrow_{,,}w.\exists !\xi \cap \eta .\\ +[*55·232.*38·2]&\supset .(\exists \alpha ,\beta ,z).\alpha ,\beta \in \kappa .z\in \gamma .\xi = \alpha \downarrow_{,,}z.\eta = \beta \downarrow_{,,}z.\exists + !\alpha \cap \beta &\qquad \text{(2)}\\ +\vdash .(2).*84·11.\supset \\ +\vdash :\text{Hp}.\kappa \in \text{Cls}^{2} \text{excl}.\xi ,\eta \in sʻ\rho .\exists !\xi \cap \eta .&\supset .(\exists \alpha ,\beta ,z).\xi = \alpha \downarrow_{,,}z.\eta + = \beta \downarrow_{,,}z.\alpha = \beta .\\ +[*13·195·172] &\supset :\xi = \eta &\qquad \text{(3)}\\ +\vdash .(1).(3).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_166">[Pg 166]</span></p> + +<p class="nind"><b>*116·671.</b> \(\vdash \colon\ldotp \sigma = \hat{\mu} \{(\exists z) . z \in \gamma . \mu = \downarrow zʻʻʻ\kappa\} . \supset : \text{Hp} *116·67 . \supset . sʻ\rho = sʻ\sigma\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *40·11 . \supset \vdash \colon\ldotp \text{Hp} *116·67 . \supset : \xi \in sʻ\rho . &\equiv . (\exists \alpha ) . \alpha \in \kappa . \xi \in \alpha \downarrow_{,,} + ʻʻ\gamma .\\ +[*38·3] &\equiv . (\exists \alpha , z) . \alpha \in \kappa . z \in \gamma . \xi = \downarrow zʻʻ\alpha .\\ +[*37·103] &\equiv . (\exists z) . z \in \gamma . \xi \in \downarrow zʻʻʻ\kappa .\\ +[*40·11] &\equiv . \xi \in sʻ\hat{\mu}\{(\exists z) . z \in \gamma . \mu = \downarrow zʻʻʻ\kappa\} \colon\ldotp \\ +\supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*116·672.</b> \(\vdash : \text{Hp} *116·671 . \kappa \in \text{Cls}^{2} \text{excl} . \Lambda {\sim} \in \kappa . \supset . \sigma \in \text{Cls}^{2} \text{excl}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *37·103 . \supset \vdash : &\text{Hp} . \mu , \nu \in \sigma . \exists ! \mu \cap \nu . \supset .\\ +&(\exists z, w, \alpha , \beta ) . z, w \in \gamma . \alpha , \beta \in \kappa . \downarrow zʻʻ\alpha = \downarrow wʻʻ\beta .\\ +\mu = \downarrow zʻʻʻ\alpha . \nu = \downarrow wʻʻʻ\beta .\\ +[*55·262] &\supset . (\exists z, w, \alpha ) . z, w \in \gamma . \alpha \in \kappa . \downarrow zʻʻ\alpha = \downarrow wʻʻ\alpha .\\ +\mu = \downarrow zʻʻʻ\alpha . \nu = \downarrow wʻʻʻ\alpha .\\ +[*113·105 . *38·2 . \text{Hp}] &\supset . (\exists z, \alpha ) . \mu = \downarrow zʻʻʻ\alpha . \nu = \downarrow zʻʻʻ\alpha .\\ +[*13·172] &\supset . \mu = \nu : \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*116·673.</b> \(\vdash : \text{Hp} *116·672 . \supset . {\in}_{\Delta}ʻ(\,\,\text{exp}\,\, \gamma )ʻʻ\kappa \text{ sm } {\in}_{\Delta}ʻ{\in}_{\Delta}ʻʻ\sigma\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *38·131 . (*116·01) . \supset \vdash . {\in}_{\Delta}ʻ(\,\,\text{exp}\,\, \gamma )ʻʻ\kappa &= {\in}_{\Delta}ʻ\hat{\xi} + \{(\exists \alpha ) . \alpha \in \kappa . \xi = \text{Prod}ʻ\alpha \downarrow_{,,} ʻʻ\gamma\}\\ +[*37·6] &= {\in}_{\Delta}ʻ\text{Prod}ʻʻ\hat{\lambda} \{(\exists \alpha ) . \alpha \in \kappa . \lambda = \alpha \downarrow_{,,} ʻʻ\gamma\} &\qquad \text{(1)}\\ +\vdash . (1) . *115·33 . *116·67 . \supset \\ +\vdash : \text{Hp} . &\supset . {\in}_{\Delta}ʻ(\,\,\text{exp}\,\, \gamma )ʻʻ\kappa \text{ sm } {\in}_{\Delta}ʻsʻ\hat{\lambda} + \{(\exists \alpha ) . \alpha \in \kappa . \lambda = \alpha \downarrow_{,,} ʻʻ\gamma\} .\\ +[*116·671] &\supset . {\in}_{\Delta}ʻ(\,\,\text{exp}\,\, \gamma )ʻʻ\kappa \text{ sm } {\in}_{\Delta}ʻsʻ\sigma .\\ +[*85·44 . *116·672] &\supset . {\in}_{\Delta}ʻ(\,\,\text{exp}\,\, \gamma )ʻʻ\kappa \text{ sm } {\in}_{\Delta}ʻ{\in}_{\Delta}ʻʻ\sigma : \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*116·674.</b> \[\begin{align}\vdash \colon\ldotp &M = \hat{R} \hat{z} \{R = (\downarrow z) \parallel \text{Cnv}ʻ(\downarrow z)_{\in }\} . \supset :\\ +&(z) . Mʻz \in 1 \rightarrow 1 . \text{D}ʻ(Mʻz) \upharpoonright {\in}_{\Delta}ʻ\kappa = {\in}_{\Delta}ʻ \downarrow zʻʻʻ\kappa\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *30·3 . &\supset \vdash : \text{Hp} . \supset . Mʻz = (\downarrow z) \parallel \text{Cnv}ʻ(\downarrow z)_{\in } &\qquad \text{(1)}\\ +\vdash . *72·184 . *111·14 . &\supset \vdash . \downarrow z \upharpoonright \kappa \in (\downarrow zʻʻʻ\kappa ) \,\overline{\text{ sm }}\, \,\overline{\text{ sm }}\, \kappa .\\ +[*114·51] &\supset \vdash . {\downarrow z \parallel \text{Cnv}ʻ(\downarrow z)_{\in }} \upharpoonright {\in}_{\Delta}ʻ\kappa \in ({\in}_{\Delta}ʻ + \downarrow zʻʻʻ\kappa ) \,\overline{\text{ sm }}\, ({\in}_{\Delta}ʻ\kappa ) &\qquad \text{(2)}\\ +\vdash . (1) . (2) . *73·03 . &\supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*116·675 .</b> \(\vdash \colon\ldotp \text{Hp} *116·674 . \exists ! sʻ\kappa . \supset : \exists ! (Mʻw)ʻʻ{\in}_{\Delta}ʻ\kappa \cap (Mʻ\nu )ʻʻ{\in}_{\Delta}ʻ\kappa + . \supset . w = v\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *116·674 . \supset \vdash \colon\ldotp \text{Hp} . &\supset : \exists ! (Mʻw)ʻʻ{\in}_{\Delta}ʻ\kappa \cap (Mʻv)ʻʻ{\in}_{\Delta}ʻ\kappa . \supset .\\ +&\exists ! {\in}_{\Delta}ʻ \downarrow wʻʻʻ\kappa \cap {\in}_{\Delta}ʻ \downarrow vʻʻʻ\kappa .\\ +[*80·32] &\supset . \downarrow wʻʻʻ\kappa = \downarrow vʻʻʻ\kappa .\\ +[*40·38] &\supset . \downarrow wʻʻsʻ\kappa = \downarrow vʻʻsʻ\kappa .\\ +[*113·105 . *38·2] &\supset . w = v \colon\ldotp \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_167">[Pg 167]</span></p> + +<p class="nind"><b><a id="*116·676">*116·676</a>.</b> \[\begin{align}\vdash : \text{Hp} *116·672·675 . \supset . \text{Prod}ʻ&\text{D}ʻʻ\upharpoonright ({\in}_{\Delta}ʻ\kappa )ʻʻMʻʻ\gamma \text{ sm } ({\in}_{\Delta}ʻ\kappa + ) \,\,\text{exp}\,\, \gamma .\\ +&\text{D}ʻʻ \upharpoonright ({\in}_{\Delta}ʻ\kappa )ʻʻMʻʻ\gamma = {\in}_{\Delta}ʻʻ\sigma\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *116·674·675 . *116·45 \frac{{\in}_{\Delta}ʻ\kappa,\,\gamma} {\gamma,\,\delta}. \supset \\ +\vdash : \text{Hp} . &\supset . \text{Prod}ʻ\text{D}ʻʻ \upharpoonright ({\in}_{\Delta}ʻ\kappa )ʻʻMʻʻ\gamma \text{ sm } ({\in}_{\Delta}ʻ\kappa + ) \,\,\text{exp}\,\, \gamma &\qquad \text{(1)}\\ +\vdash . *116·674 . &\supset \vdash : \text{Hp} . \supset . \text{D}ʻʻ \upharpoonright ({\in}_{\Delta}ʻ\kappa + )ʻʻMʻʻ\gamma = \hat{\mu} \{(\exists z) . z \in \gamma . \mu = {\in}_{\Delta}ʻ \downarrow zʻʻʻ\kappa\}\\ +[*37·6 . \text{Hp}] &= {\in}_{\Delta}ʻʻ\sigma &\qquad \text{(2)}\\ +\vdash . (1) . (2) . \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*116·68">*116·68</a>.</b> \[\begin{align}\vdash : \kappa \in \text{Cls}^{2} \text{excl} . \supset . {\in}_{\Delta}ʻ(\text{exp}\,\, \gamma )ʻʻ\kappa \text{ sm } &({\in}_{\Delta}ʻ\kappa + ) \,\,\text{exp}\,\, \gamma .\\ +&\Pi \text{Nc}ʻ(\text{exp}\,\, \gamma )ʻʻ\kappa = (\Pi \text{Nc}ʻ\kappa )^{\text{Nc}ʻ\gamma }\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *115·12 . *84·55 . &\supset \vdash . \text{Prod}ʻ{\in}_{\Delta}ʻʻ\sigma \text{ sm } {\in}_{\Delta}ʻ{\in}_{\Delta}ʻʻ\sigma &\qquad \text{(1)}\\ +\vdash . (1) . *116·673·676 . \supset \\ +\vdash : \text{Hp} . \Lambda {\sim} \in \kappa . \exists ! sʻ\kappa . &\supset . {\in}_{\Delta}ʻ(\text{exp}\,\, \gamma)ʻʻk \text{ sm } ({\in}_{\Delta}ʻ\kappa + ) \,\,\text{exp}\,\, \gamma &\qquad \text{(2)}\\ +\vdash . *53·24 . \supset \vdash : \Lambda {\sim} \in \kappa . {\sim} \exists ! sʻ\kappa . &\supset . \kappa = \Lambda .\\ +[*83·15 . *116·33] &\supset . {\in}_{\Delta}ʻ(\text{exp}\,\, \gamma )ʻʻ\kappa = \iota ʻ\dot{\Lambda} . ({\in}_{\Delta}ʻ\kappa ) \,\,\text{exp}\,\, \gamma \in 1 .\\ +[*73·45] &\supset . {\in}_{\Delta}ʻ(\text{exp}\,\, \gamma )ʻʻ\kappa \text{ sm } ({\in}_{\Delta}ʻ\kappa ) \,\,\text{exp}\,\, \gamma &\qquad \text{(3)}\\ +\vdash . *83·11 . *116·182 . \supset \vdash : \Lambda \in \kappa . \exists ! \gamma . &\supset . {\in}_{\Delta}ʻ\kappa + = \Lambda . \Lambda \in (\text{exp}\,\, \gamma )ʻʻ\kappa .\\ +[*116·182 . *83·11] &\supset . ({\in}_{\Delta}ʻ\kappa ) \,\,\text{exp}\,\, \gamma = \Lambda . {\in}_{\Delta}ʻ(\text{exp}\,\, \gamma )ʻʻ\kappa = \Lambda &\qquad \text{(4)}\\ +\vdash . *116·181 . &\supset \vdash : \Lambda \in \kappa . \gamma = \Lambda . \supset . ({\in}_{\Delta}ʻ\kappa ) \,\,\text{exp}\,\, \gamma = \iota ʻ\Lambda &\qquad \text{(5)}\\ +\vdash . *116·181 . &\supset \vdash : \Lambda \in \kappa . \gamma = \Lambda . \supset . (\text{exp}\,\, \gamma )ʻʻ\kappa = \iota ʻ\iota ʻ\Lambda .\\ +[*83·41] &\supset . {\in}_{\Delta}ʻ(\text{exp}\,\, \gamma )ʻʻ\kappa \text{ sm } \iota ʻ\Lambda &\qquad \text{(6)}\\ +\vdash . (4) . (5) . (6) . &\supset \vdash : \Lambda \in \kappa . \supset . {\in}_{\Delta}ʻ(\text{exp}\,\, \gamma )ʻʻ\kappa \text{ sm } ({\in}_{\Delta}ʻ\kappa + ) \,\,\text{exp}\,\, \gamma &\qquad \text{(7)}\\ +\vdash . (2) . (3) . (7) . *114·1 . *116·25 . \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p>The above proposition is an extension of <a href="#*116·54">*116·54</a><a href="#*116·55">·55</a>.</p> + +<p>The following propositions are lemmas for +\[ +\text{Nc}ʻ\text{Cl}ʻ\alpha = 2^{\text{Nc}ʻ\alpha } . +\]</p> + +<p>The proposition and its proof are due to Cantor.</p> + +<p class="nind"><b>*116·7.</b> \(\vdash . \text{Nc}ʻ\{(\iota ʻ\Lambda \cup \iota ʻ\text{V}) \uparrow \alpha \}_{\Delta }ʻ\alpha = 2^{\text{Nc}ʻ\alpha }\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *24·1 .*101·3 . \supset \vdash . \text{Nc}ʻ(\iota ʻ\Lambda \cup \iota ʻ\text{V}) &= 2 &\qquad \text{(1)}\\ +\vdash . *116·15 . \supset \vdash . \text{Nc}ʻ{(\iota ʻ\Lambda \cup \iota ʻ\text{V}) \uparrow \alpha }_{\Delta }ʻ\alpha + &= \text{Nc}ʻ\{(\iota ʻ\Lambda \cup \iota ʻ\text{V}) \,\,\text{exp}\,\, \alpha\}\\ +[*116·25] &= {\text{Nc}ʻ(\iota ʻ\Lambda \cup \iota ʻ\text{V})}^{\text{Nc}ʻ\alpha }\\ +[(1)] &= 2^{\text{Nc}ʻ\alpha } . \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_168">[Pg 168]</span></p> + +<p>In this and following propositions, the class \(\iota ʻ\Lambda \cup\iota ʻ\text{V}\) +is introduced solely as a known class consisting of two terms. Any +other class of two terms will serve equally well.</p> + +<p class="nind"><b>*116·71.</b> \(\vdash : R \in \{(\iota ʻ\Lambda \cup \iota ʻ\text{V}) \uparrow \alpha\}_{\Delta }ʻ\alpha + . \supset . \overleftarrow{R}ʻ\text{V} = \alpha - \overleftarrow{R}ʻ\Lambda \)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *116·12 . \supset \vdash : \text{Hp} . &\supset . R \in 1 \rightarrow \text{Cls} . \text{D}ʻR \subset \iota ʻ\Lambda \cup \iota ʻ\text{V} . \text{ᗡ}ʻR = \alpha &\qquad \text{(1)}\\ +[*37·271] \supset . \alpha &= \breve{R} ʻʻ(\iota ʻ\Lambda \cup \iota ʻ\text{V})\\ +[*53·302] &= \overleftarrow{R}ʻ\Lambda \cup \overleftarrow{R}ʻ\text{V} &\qquad \text{(2)}\\ +\vdash . (1) . *71·18 . &\supset \vdash : \text{Hp} . \supset . \overleftarrow{R}ʻ\Lambda \cap \overleftarrow{R}ʻ\text{V} = \Lambda &\qquad \text{(3)}\\ +\vdash . (2) . (3) . *24·47 . \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*116·711.</b> \(\vdash : R, S \in \{(\iota ʻ\Lambda \cup \iota ʻ\text{V}) \uparrow \alpha\}_{\Delta }ʻ\alpha + . \overleftarrow{R}ʻ\Lambda = \overleftarrow{S}ʻ\Lambda . \supset . R = S\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *116·71 . &\supset \vdash : \text{Hp} . \supset . \overleftarrow{R}ʻ\text{V} = \overleftarrow{S}ʻ\text{V} &\qquad \text{(1)}\\ +\vdash . (1) . *116·12 . &\supset \vdash \colon\ldotp \text{Hp} . \supset : \gamma \in \text{D}ʻR \cup \text{D}ʻS . \supset _{\gamma } + . \overleftarrow{R}ʻ\gamma = \overleftarrow{S}ʻ\gamma :\\ +[*33·48] &\supset : R = S \colon\ldotp \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*116·712.</b> \[\begin{align}&\vdash \colon\ldotp T = \hat{\mu} \hat{R} [R \in \{(\iota ʻ\Lambda \cup \iota ʻ\text{V}) \uparrow \alpha \}_{\Delta }ʻ\alpha + . \mu = \overleftarrow{R}ʻ\Lambda ] . \supset :\\ +&R \in \{(\iota ʻ\Lambda \cup \iota ʻ\text{V}) \uparrow \alpha\}_{\Delta }ʻ\alpha + . \supset . TʻR = \overleftarrow{R}ʻ\Lambda :\text{ᗡ}ʻT = \{(\iota ʻ\Lambda \cup \iota ʻ\text{V}) \uparrow \alpha\}_{\Delta }ʻ\alpha\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *21·33 .& \supset \vdash \colon\colon \text{Hp} . \supset \colon\ldotp R \in {(\iota ʻ\Lambda \cup \iota ʻ\text{V}) \uparrow \alpha }_{\Delta }ʻ\alpha + . \supset : \mu TR . \equiv _{\mu } . \mu = \overleftarrow{R}ʻ\Lambda :\\ +[*30.3] &\supset : TʻR = \overleftarrow{R}ʻ\Lambda &\qquad \text{(1)}\\ +\vdash . (1) . *14·21 . &\supset \vdash \colon\ldotp \text{Hp} . \supset : R \in {(\iota ʻ\Lambda \cup \iota ʻ\text{V}) \uparrow \alpha }_{\Delta }ʻ\alpha + . \supset . \text{E}! TʻR .\\ +[*33·43] &\supset . R \in \text{ᗡ}ʻT &\qquad \text{(2)}\\ +\vdash . *21·33 . *33·131 . &\supset \vdash : \text{Hp} . \supset · \text{ᗡ}ʻT \subset {(\iota ʻ\Lambda \cup \iota ʻ\text{V}) \uparrow \alpha }_{\Delta }ʻ\alpha + &\qquad \text{(3)}\\ +\vdash (1) . (2) . (3) . \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*116·713.</b> \(\vdash : \text{Hp} *116·712 . \supset . T \in 1 \rightarrow 1\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *116·712 . *14·21 . \supset \vdash \colon\ldotp \text{Hp} . &\supset : R \in \text{ᗡ}ʻT . \supset . \text{E}! TʻR :\\ +[*71·16] &\supset : R \in 1 \rightarrow \text{Cls} &\qquad \text{(1)}\\ +\vdash . *116·712.711 . &\supset \vdash \colon\ldotp \text{Hp} . \supset : R, S \in \text{ᗡ}ʻT . TʻR = TʻS . \supset . R = S &\qquad \text{(2)}\\ +\vdash . (1) . (2) . *71·55 . \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*116.714.</b> + \[\begin{align}\vdash : \text{Hp} *116·712 . \mu \in \text{Cl}ʻ\alpha . &R = \hat{\gamma} \hat{x} \{\gamma = \Lambda . x \in \mu . \lor . \gamma = \text{V} . x \in \alpha - \mu\} . \supset .\\ +&R \in \{(\iota ʻ\Lambda \cup \iota ʻ\text{V}) \uparrow \alpha\}_{\Delta }ʻ\alpha . \mu = TʻR\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *21·33 . *33·13 . &\supset \vdash \colon\ldotp \text{Hp} . \supset : \gamma \in \text{D}ʻR . \supset . \gamma \in \iota ʻ\Lambda \cup \iota ʻ\text{V} &\qquad \text{(1)}\\ +\vdash . *21·33 . *33·131 . &\supset \vdash \colon\colon \text{Hp} . \supset \colon\ldotp x \in \text{ᗡ}ʻR . \equiv :\\ +&(\exists \gamma ) : \gamma = \Lambda . x \in \mu . \lor . \gamma = \text{V} . x \in \alpha - \mu :\\ +[*10·42 . *13·19] &\equiv : x \in \mu . \lor . x \in \alpha - \mu : &\qquad \text{(2)}\\ +[*24·411 . \text{Hp}] &\equiv : x \in \alpha &\qquad \text{(3)}\\ +\vdash .*21·33.*30·3.&\supset \vdash \colon\ldotp \text{Hp}.\supset :x\in \mu .\supset _{x}.Rʻx=\Lambda :x\in \alpha -\mu .\supset _{x}.Rʻx =\text{V}:\\ +[(2).*14·21] &\supset :x\in \text{ᗡ}ʻR.\supset _{x}.\text{E}!Rʻx:\\ +[*71·16] &\supset :R\in 1\rightarrow \text{Cls} &\qquad \text{(4)}\\ +\vdash .*21·33. &\supset \vdash \colon\colon \text{Hp}.\supset \colon\ldotp \gamma =\Lambda .\supset :\gamma Rx.\equiv _{x}.x\in \mu :\\ +[*32·181] &\supset :\overleftarrow{R}ʻ\gamma =\mu &\qquad \text{(5)}\\ +\vdash .(1).(3).(4).*116·12.&\supset \vdash :\text{Hp}.\supset .R\in \{(\iota ʻ\Lambda \cup \iota ʻ\text{V})\uparrow \alpha\}_{\Delta }ʻ\alpha &\qquad \text{(6)}\\ +\vdash .(5).(6).*116·712. & \supset \vdash :\text{Hp}.\supset .\mu =TʻR &\qquad \text{(7)}\\ +\vdash .(6).(7).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_169">[Pg 169]</span></p> + +<p class="nind"><b>*116·715.</b> \(\vdash :\text{Hp}*116·712.\supset .\text{D}ʻT=\text{Cl}ʻ\alpha \)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*116·714.*33·43.&\supset \vdash :\text{Hp}.\supset .\text{Cl}ʻ\alpha \subset \text{D}ʻT &\qquad \text{(1)}\\ +\vdash .*21·33. *33·13.\supset \\ +\vdash \colon\ldotp \text{Hp}.&\supset :\mu \in \text{D}ʻT.\supset .(\exists R).R\in {(\iota ʻ\Lambda \cup \iota ʻ\text{V})\uparrow \alpha }_{\Delta }ʻ\alpha + .\mu =\overleftarrow{R}ʻ\Lambda .\\ +[*33·151] &\supset .(\exists R).R\in {(\iota ʻ\Lambda \cup \iota ʻ\text{V})\uparrow \alpha }_{\Delta }ʻ\alpha .\mu \subset \text{ᗡ}ʻR.\\ +[*80·14] &\supset .\mu \subset \alpha &\qquad \text{(2)}\\ +\vdash .(1).(2).\supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*116·72">*116·72</a>.</b> \(\vdash .\text{Nc}ʻ\text{Cl}ʻ\alpha =2^{\text{Nc}ʻ\alpha}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*116·712·713·715.\supset \vdash .\text{Cl}ʻ\alpha \text{ sm }\{(\iota ʻ\Lambda \cup \iota ʻ\text{V})\uparrow \alpha\}_{\Delta }ʻ\alpha &\qquad \text{(1)}\\ +\vdash .(1).*116·7.\supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*116·8.</b> \(\vdash .\text{Rl}ʻ(\rho \uparrow \sigma )=\dot{s} ʻʻ\text{Cl}ʻ(\sigma \times \rho )\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*60·2.&\supset \vdash \colon\ldotp R\in \dot{s} ʻʻ\text{Cl}ʻ(\sigma x\rho ).\equiv :(\exists \lambda ).\lambda \subset \sigma \times \rho .R=\dot{s} ʻ\lambda :\\ +[*113·101]&\equiv :(\exists \lambda ):P\in \lambda .\supset _{P}.(\exists x,y).x\in \rho .y\in \sigma .P=x\downarrow y:R=\dot{s} ʻ\lambda :\\ +[*41·11]&\equiv :(\exists \lambda ):P\in \lambda .\supset _{P}.(\exists x,y).x\in \rho .y\in \sigma .P=x\downarrow y:\\ +uRv.\equiv _{u,v}.(\exists P).P\in \lambda .uPv:\\ +[*10·56] &\supset :uRv.\supset _{u,v}.(\exists x,y).x\in \rho .y\in \sigma .u(x\downarrow y)v:\\ +[*55·13] &\supset :uRv.\supset _{u,v}.u\in \rho .v\in \sigma :\\ +[*35·103]. &\supset :R\,\unicode{x2abd}\, \rho \uparrow \sigma &\qquad \text{(1)}\\ +\vdash .*35·103.*113·101.\supset \\ +\vdash :R\,\unicode{x2abd}\, \rho \uparrow \sigma .\lambda& =\hat{P} \{(\exists x,y).xRy.P=x\downarrow y\}.\supset .\lambda \subset \sigma \times \rho &\qquad \text{(2)}\\ +\vdash .*41·11.*13·195.\supset \\ +\vdash \colon\ldotp \text{Hp}(2).\supset :u(\dot{s} ʻ\lambda )v.&\equiv .(\exists x,y).xRy.u(x\downarrow y)v.\\ +[*55·13] &\equiv .uRv &\qquad \text{(3)}\\ +\vdash .(2).(3).&\supset \vdash :R\,\unicode{x2abd}\, \rho \uparrow \sigma .\supset .(\exists \lambda ).\lambda \subset \sigma \times \rho .R=\dot{s} ʻ\lambda &\qquad \text{(4)}\\ +\vdash .(1).(4).&\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_170">[Pg 170]</span></p> + +<p class="nind"><b>*116·81.</b> \(\vdash .\dot{s} \upharpoonright \text{Cl}ʻ(\sigma \times \rho )\in 1\rightarrow 1\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*41·13.&\supset \vdash \colon\ldotp \alpha ,\beta \in \text{Cl}ʻ(\sigma \times \rho ).\dot{s} ʻ\alpha =\dot{s} ʻ\beta .x\downarrow y\in \alpha .\supset :x\downarrow y\,\unicode{x2abd}\, \dot{s} ʻ\beta :\\ +[*41·11] &\supset :(\exists P).P\in \beta .x\downarrow y\,\unicode{x2abd}\, P:\\ +[*113·101.\text{Hp}] &\supset :(\exists P,u,v).P\in \beta .P=u\downarrow v.x\downarrow y\,\unicode{x2abd}\, u\downarrow v:\\ +[*55·134·34] &\supset :(\exists P,u,v).P\in \beta .P=u\downarrow v.x\downarrow y=u\downarrow v:\\ +[*13·172·13] &\supset :x\downarrow y\in \beta &\qquad \text{(1)}\\ +\vdash .(1)\frac{\beta,\,\alpha}{\alpha,\,\beta}.&\supset \vdash :\alpha ,\beta \in \text{Cl}ʻ(\sigma \times \rho ).\dot{s} ʻ\alpha =\dot{s} ʻ\beta .x\downarrow y\in \beta .\supset .x\downarrow y\in \alpha &\qquad \text{(2)}\\ +\vdash .(1).(2).*113·101.&\supset \vdash :\alpha ,\beta \in \text{Cl}ʻ(\sigma \times \rho ).\dot{s} ʻ\alpha =\dot{s} ʻ\beta .\supset .\alpha =\beta &\qquad \text{(3)}\\ +\vdash .(3).*71·55.*72·163.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*116·82.</b> \(\vdash .\text{Rl}ʻ(\rho \uparrow \sigma )\text{ sm } \text{Cl}ʻ(\sigma \times \rho ) \quad[*116·8·81]\)</p> + +<p class="nind"><b>*116·83.</b> \(\vdash .\text{Nc}ʻ\text{Rl}ʻ(\rho \uparrow \sigma )=2^{\text{Nc}ʻ\rho \times _{c}\text{Nc}ʻ\sigma } \quad[*116·82·72 .*113·25]\)</p> + +<p class="nind"><b>*116·9.</b> \(\vdash :\text{Nc}ʻtʻx=\mu .\supset .\text{Nc}ʻt^{2}ʻx=2^\mu \quad[*116·72.*63·66]\)</p> + +<p class="nind"><b>*116·901.</b> \(\vdash :\text{Nc}ʻt_{0}ʻ\alpha =\mu .\supset .\text{Nc}ʻtʻ\alpha =2^\mu \quad[*116·72.*63·65]\)</p> + +<p class="nind"><b>*116·91.</b> \(\vdash :\text{Nc}ʻt_{0}ʻ\alpha =\mu .\supset .\text{Nc}ʻt_{00}ʻ\alpha =2^{\mu ^{2}} \quad[*116·83.*64·5·11]\)</p> + +<p class="nind"><b>*116·92.</b> \[\begin{align}&\vdash :\text{Nc}ʻt_{0}ʻ\alpha =\mu .\supset .\text{Nc}ʻt_{0}^{1}ʻ\alpha =2^{\mu \times _{c}2^\mu + }.\text{Nc}ʻt^{11}ʻ\alpha =2^{2^\mu \times _{c}2^\mu }.\text{etc}.\\ +&[*116·83.*64·16.*116·901]\end{align}\]</p> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_171">[Pg 171]</span></p> +<h2 class="nobreak" id="*117"> +*117. GREATER AND LESS.</h2> +</div> + + +<p class="nind"><i>Summary of</i> *117.</p> + +<p>A cardinal \(\mu\) is said to be greater than another cardinal \(\nu\) +when there is a class \(\alpha\) which has \(\mu\) terms and has +a part which has \(\nu\) terms, while there is no class \(\beta\) +which has \(\nu\) terms and has a part which has \(\mu\) terms. The +relation "greater than" is transitive and asymmetrical; and by the +Schröder-Bernstein theorem, if \(\mu\) is greater than or equal to +\(\nu\), and \(\nu\) is greater than or equal to \(\mu\), then \(\mu = +\nu\). But we cannot prove that of any two cardinals one must be the +greater, unless we assume the multiplicative axiom. The proof then +follows from Zermelo's theorem that on that assumption every class can +be well-ordered. This subject will be dealt with at a later stage.</p> + +<p>The form of the definitions is so arranged as to allow of the +inequality of two cardinals in different types. The relevant +considerations are the same as for the definitions of addition, +multiplication and exponentiation.</p> + +<p>Our definition of "\(\mu \gt \nu\)" is</p> + +<p class="nind"><b>*117·01.</b> \[\begin{align}\mu \gt \nu .=.(\exists \alpha ,\beta ).\mu =\text{N}_{0}\text{c}ʻ\alpha .\nu =&\text{N}_{0}\text{c}ʻ\beta .\\ +&\exists !\text{Cl}ʻ\alpha \cap \text{Nc}ʻ\beta .{\sim}\exists !\text{Cl}ʻ\beta \cap \text{Nc}ʻ\alpha \quad\text{Df}\end{align}\]</p> + +<p>We also define "\(\mu \gt \text{Nc}ʻ\alpha\)" as meaning "\(\mu \gt \text{N}_{0}\text{c}ʻ\alpha\)," +and "\(\text{Nc}ʻ\alpha \gt \nu\)" as meaning +"\(\text{N}_{0}\text{c}ʻ\alpha \gt \nu\)," for the reasons explained in +<a href="#*110">*110</a>. It then easily follows that if \(\mu \gt \nu\), \(\mu\) and \(\nu\) +must be homogeneous cardinals (this is part of <a href="#*117·15">*117·15</a>); that if +\(\mu\) and \(\nu\) are homogeneous cardinals, and \(\mu \gt \nu\), the +same holds if we substitute \(\text{ sm }ʻʻ\mu\) and \(\text{ sm }ʻʻ\nu\) +for one or both of \(\mu\) and \(\nu\) (<a href="#*117·16">*117·16</a>); that</p> + +<p class="nind"><b>*117·13.</b> \(\vdash :\text{Nc}ʻ\alpha \gt \text{Nc}ʻ\beta .\equiv .\exists !\text{Cl}ʻ\alpha \cap \text{Nc}ʻ\beta .{\sim}\exists !\text{Cl}ʻ\beta \cap \text{Nc}ʻ\alpha\)</p> + +<p class="nind">and that</p> + +<p class="nind"><b>*117·14.</b> \(\vdash :\mu \gt \nu .\equiv .(\exists \alpha ,\beta ).\mu = \text{N}_{0}\text{c}ʻ\alpha .\nu =\text{N}_{0}\text{c}ʻ\beta + .\text{Nc}ʻ\alpha \gt \text{Nc}ʻ\beta\)</p> + +<p>We cannot define "\(\mu \geq \nu\)" as "\(\mu \gt \nu .\lor.\mu =\nu\)," +because "\(\mu =\nu\)" restricts \(\mu\) and \(\nu\) too much by +requiring that they should be of the same type, and restricts them too +little by not requiring that they should both be existent cardinals. To +avoid both these inconveniences, we put</p> + +<p class="nind"><b>*117·05.</b> \(\mu \geq \nu .=:\mu \gt \nu .\lor.\mu ,\nu \in \text{N}_{0}\text{C}.\mu =\text{ sm }ʻʻ\nu \quad\text{Df}\)</p> + +<p>The use of this definition is chiefly through the propositions</p> + +<p class="nind"><b>*117·108.</b> \(\vdash \colon\ldotp \text{Nc}ʻ\alpha \geq \text{Nc}ʻ\beta .\equiv :\text{Nc}ʻ\alpha \gt \text{Nc}ʻ\beta .\lor.\text{Nc}ʻ\alpha =\text{Nc}ʻ\beta\)</p> + +<p><span class="pagenum" id="Page_172">[Pg 172]</span></p> + +<p class="nind"><b>*117·24.</b> \(\vdash :\mu \geq \nu .\equiv .(\exists \alpha ,\beta ).\mu =\text{N}_{0}\text{c}ʻ\alpha .\nu =\text{N}_{0}\text{c}ʻ\beta + .\text{Nc}ʻ\alpha \geq \text{Nc}ʻ\beta\)</p> + +<p>In <a href="#*117·2">*117·2</a>, we repeat the Schröder-Bernstein theorem (*73·88), which is +required in most of the remaining propositions of this number. It leads +at once to the propositions</p> + +<p class="nind"><b>*117·22.</b> \(\vdash :\exists !\text{Cl}ʻ\alpha \cap \text{Nc}ʻ\beta .\equiv .\text{Nc}ʻ\alpha \geq \text{Nc}ʻ\beta\)</p> + +<p class="nind">(which practically supersedes the definition of "\(\geq\)")</p> + +<p class="nind"><b>*117·221.</b> \(\vdash :\text{Nc}ʻ\alpha \geq \text{Nc}ʻ\beta .\equiv .(\exists \rho ).\rho \subset \alpha .\rho \text{ sm }\beta\)</p> + +<p class="nind"><b>*117·222.</b> \(\vdash :\beta \subset \alpha .\supset .\text{Nc}ʻ\alpha \geq \text{Nc}ʻ\beta\)</p> + +<p class="nind"><b>*117·23.</b> \(\vdash :\text{Nc}ʻ\alpha \geq \text{Nc}ʻ\beta .\text{Nc}ʻ\beta \geq \text{Nc}ʻ\alpha .\equiv .\text{Nc}ʻ\alpha =\text{Nc}ʻ\beta\)</p> + +<p>This last proposition may be called the Schröder-Bernstein theorem with +as much propriety as *73·88; the two are scarcely different.</p> + +<p>If we now revert to the definition of \(\mu \gt \nu\), or to <a href="#*117·13">*117·13</a>, +and apply <a href="#*117·22">*117·22</a>, we see (<a href="#*117·26">*117·26</a>) that "\(\text{Nc}ʻ\alpha>\text{Nc}ʻ\beta\)" +may be conveniently regarded as asserting \(\text{Nc}ʻ\alpha \geq\text{Nc}ʻ\beta .{\sim}(\text{Nc}ʻ\beta \geq \text{Nc}ʻ\alpha)\); +in fact, the best ideas to work with are \(\geq\) and its converse +\(\leq\), which for practical purposes we regard as defined by +*117·22, and from which we derive \(\gt\) and \(\lt\) . The relation \(\gt\) will be the +product of \(\geq\) into the negation of its converse; this holds for +\(\mu\) and \(\nu\) (<a href="#*117·281">*117·281</a>) as well as for \(\text{Nc}ʻ\alpha\) and +\(\text{Nc}ʻ\beta\).</p> + +<p><a href="#*117·3">*117·3</a><a href="#*117·31">·31</a> constitute an important use of <a href="#*110·72">*110·72</a>, namely to prove that +one existent cardinal is greater than another or equal to it when the +first can be obtained by adding to the second (where what is added must +be a cardinal). That is to say, we have</p> + +<p class="nind"><b>*117·3.</b> \(\vdash :\text{Nc}ʻ\alpha \geq \text{Nc}ʻ\beta .\equiv .(\exists \varpi ).\varpi \in \text{NC}.\text{Nc}ʻ\alpha =\text{Nc}ʻ\beta +_{c}\varpi\)</p> + +<p class="nind"><b>*117·31.</b> \(\vdash \colon\ldotp \mu \geq \nu .\equiv :\mu ,\nu \in \text{N}_{0}\text{C}:(\exists \varpi ).\varpi \in \text{NC}.\mu =\nu +_{c}\varpi\)</p> + +<p><a href="#*117·4">*117·4</a>—<a href="#*117·471">·471</a> are concerned in proving that \(\gt\) and \(\geq\) are +transitive, that \(\gt\) is asymmetrical (<a href="#*117·42">*117·42</a>), and allied propositions.</p> + +<p>Our next set of propositions is concerned with 0 and 1 and 2. We prove +that a homogeneous cardinal is whatever is greater than or equal to +0 (<a href="#*117·501">*117·501</a>); that a homogeneous cardinal other than 0 is whatever +is greater than 0 (<a href="#*117·511">*117·511</a>); that a homogeneous cardinal other than +0 is whatever is greater than or equal to 1 (<a href="#*117·531">*117·531</a>); and that a +homogeneous cardinal other than 0 and 1 is whatever is greater than 1 +(<a href="#*117·55">*117·55</a>), and is whatever is greater than or equal to 2 (<a href="#*117·551">*117·551</a>).</p> + +<p>We next prove a set of propositions concerning \(\geq\) which have no +analogues for \(\gt\), except when the cardinals concerned are finite. Thus +<i>e.g.</i> we prove</p> + +<p class="nind"><b>*117·561.</b> \(\vdash :\mu \geq \nu .\varpi \in \text{N}_{0}\text{C}.\supset .\mu +_{c}\varpi \geq \nu +_{c}\varpi\)</p> + +<p>If we substitute \(\gt\) for \(\geq\), this no longer holds. Thus <i>e.g.</i> +put \(\mu =2\), \(\nu =1\), \(\varpi =\aleph _{0}\) (cf. <a href="#*123">*123</a>); then +\(\mu >\nu\), but \(\mu +_{c}\varpi =\nu +_{c}\varpi =\varpi\). Similar +remarks apply to the analogous propositions (<a href="#*117·571">*117·571</a><a href="#*117·581">·581</a><a href="#*117·591">·591</a>) on +multiplication and exponentiation.</p> + +<p><span class="pagenum" id="Page_173">[Pg 173]</span></p> + +<p>We prove next that a sum is greater than or equal to either of its +summands (<a href="#*117·6">*117·6</a>); that a product neither of whose factors vanishes +is greater than or equal to either of its factors (<a href="#*117·62">*117·62</a>); that, +assuming \(\mu\) and \(\nu\) are existent cardinals, then if they are +neither 0 nor 1, their product is greater than or equal to their sum +(<a href="#*117·631">*117·631</a>), and if \(\mu\) is neither 0 nor 1, then +\(\mu ^\nu \geq \mu\times _{c} \nu\) (<a href="#*117·652">*117·652</a>).</p> + +<p>The last important proposition in this number is Cantor's theorem</p> + +<p class="nind"><b>*117·661.</b> \(\vdash : \mu \in \text{N}_{0}\text{C} . \supset . 2^\mu > \mu\)</p> + +<p class="nind">which follows immediately from <a href="#*102·72">*102·72</a> and <a href="#*116·72">*116·72</a>.</p> + +<p>The propositions of this number are much used in the following section, +on finite and infinite.</p> + +<hr class="tb"> + +<p class="nind"><b><a id="*117·01">*117·01</a>.</b> \[\begin{align}\mu \gt \nu . = . (\exists \alpha , \beta ) . &\mu = \text{N}_{0}\text{c}ʻ\alpha . \nu = \text{N}_{0}\text{c}ʻ\beta .\\ +&\exists ! \text{Cl}ʻ\alpha \cap \text{Nc}ʻ\beta . {\sim} \exists ! \text{Cl}ʻ\beta \cap \text{Nc}ʻ\alpha \quad\text{Df}\end{align}\]</p> + +<p class="nind"><b><a id="*117·02">*117·02</a>.</b> \(\mu \gt \text{Nc}ʻ\alpha . = . \mu \gt \text{N}_{0}\text{c}ʻ\alpha \quad\text{Df}\)</p> + +<p class="nind"><b><a id="*117·03">*117·03</a>.</b> \(\text{Nc}ʻ\alpha \gt \nu . = . \text{N}_{0}\text{c}ʻ\alpha \gt \nu \quad\text{Df}\)</p> + +<p class="nind"><b><a id="*117·04">*117·04</a>.</b> \(\mu \lt \nu . = . \nu \gt \mu \quad\text{Df}\)</p> + +<p class="nind"><b><a id="*117·05">*117·05</a>.</b> \(\mu \geq \nu . = : \mu \gt \nu . \lor . \mu , \nu \in \text{N}_{0}\text{C} .\mu = \text{ sm }ʻʻ\nu \quad\text{Df}\)</p> + +<p class="nind"><b><a id="*117·06">*117·06</a>.</b> \(\mu \leq \nu . = . \nu \geq \mu \quad\text{Df}\)</p> + +<p>The analogues of <a href="#*117·02">*117·02</a><a href="#*117·03">·03</a> are to be applied also to <a href="#*117·04">*117·04</a><a href="#*117·05">·05</a><a href="#*117·06">·06</a>.</p> + +<p class="nind"><b>*117·1.</b> \[\begin{align}\vdash : \mu \gt \nu . &\equiv . (\exists \alpha , \beta ) . \mu = \text{N}_{0}\text{c}ʻ\alpha . \nu = \text{N}_{0}\text{c}ʻ\beta .\\ +&\exists ! \text{Cl}ʻ\alpha \cap \text{Nc}ʻ\beta . {\sim} \exists ! \text{Cl}ʻ\beta \cap \text{Nc}ʻ\alpha \quad[(*117·01)]\end{align}\]</p> + +<p class="nind"><b>*117·101.</b> \(\vdash : \mu \gt \text{Nc}ʻ\beta . \equiv . \mu \gt \text{N}_{0}\text{c}ʻ\beta \quad[(*117·02)]\)</p> + +<p class="nind"><b>*117·102.</b> \(\vdash : \text{Nc}ʻ\alpha > \nu . \equiv . \text{N}_{0}\text{c}ʻ\alpha > \nu \quad[(*117·03)]\)</p> + +<p class="nind"><b>*117·103.</b> \(\vdash : \mu < \nu . \equiv . \nu > \mu \quad[(*117·04)]\)</p> + +<p class="nind"><b><a id="*117·104">*117·104</a>.</b> \(\vdash \colon\ldotp \mu \geq \nu . \equiv : \mu > \nu . \lor . \mu , \nu \in \text{N}_{0}\text{C} . \mu = \text{ sm }ʻʻ\nu \quad[(*117·05)]\)</p> + +<p class="nind"><b>*117·105.</b> \(\vdash : \mu \leq \nu . \equiv . \nu \geq \mu \quad[(*117·06)]\)</p> + +<p class="nind"><b>*117·106.</b> \(\vdash : \text{Nc}ʻ\alpha > \text{Nc}ʻ\beta . \equiv . \text{N}_{0}\text{c}ʻ\alpha > \text{N}_{0}\text{c}ʻ\beta \quad [*117·101·102]\)</p> + +<p class="nind"><b>*117·107.</b> \(\vdash : \text{Nc}ʻ\alpha \geq \text{Nc}ʻ\beta . \equiv . \text{N}_{0}\text{c}ʻ\alpha \geq \text{N}_{0}\text{c}ʻ\beta\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *117·104·106 . \supset \\ +\vdash \colon\ldotp \text{Nc}ʻ\alpha \geq \text{Nc}ʻ\beta . &\equiv : \text{N}_{0}\text{c}ʻ\alpha > \text{N}_{0}\text{c}ʻ\beta + . \lor . \text{Nc}ʻ\alpha , \text{Nc}ʻ\beta \in \text{N}_{0}\text{C} . \text{Nc}ʻ\alpha = \text{ sm }ʻʻ\text{Nc}ʻ\beta :\\ +[*100·511 . *103·22] &\equiv : \text{N}_{0}\text{c}ʻ\alpha > \text{N}_{0}\text{c}ʻ\beta . \lor . \text{Nc}ʻ\alpha , \text{Nc}ʻ\beta \in \text{N}_{0}\text{C} + . \text{Nc}ʻ\alpha = \text{Nc}ʻ\beta :\\ +[*103·16] &\equiv : \text{N}_{0}\text{c}ʻ\alpha \gt \text{N}_{0}\text{c}ʻ\beta . \lor . \text{Nc}ʻ\alpha , \text{Nc}ʻ\beta \in \text{N}_{0}\text{C} + . \text{Nc}ʻ\alpha = \text{N}_{0}\text{c}ʻ\beta :\\ +[*103·21] &\equiv : \text{N}_{0}\text{c}ʻ\alpha \gt \text{N}_{0}\text{c}ʻ\beta . \lor . \text{Nc}ʻ\beta \in \text{N}_{0}\text{C} . \text{Nc}ʻ\alpha = \text{N}_{0}\text{c}ʻ\beta + :\\ +[*103·16] &\equiv : \text{N}_{0}\text{c}ʻ\alpha \gt \text{N}_{0}\text{c}ʻ\beta . \lor . \text{Nc}ʻ\beta \in \text{N}_{0}\text{C} . \text{N}_{0}\text{c}ʻ\alpha + = \text{Nc}ʻ\beta :\\ +[*103·2] &\equiv : \text{N}_{0}\text{c}ʻ\alpha \gt \text{N}_{0}\text{c}ʻ\beta . \lor . \text{N}_{0}\text{c}ʻ\alpha = \text{Nc}ʻ\beta :\\ +[*103·4] &\equiv : \text{N}_{0}\text{c}ʻ\alpha \gt \text{N}_{0}\text{c}ʻ\beta . \lor . \text{N}_{0}\text{c}ʻ\alpha = \text{ sm }ʻʻ\text{N}_{0}\text{c}ʻ\beta + :\\ +[*103·21 . *117·104] &\equiv : \text{N}_{0}\text{c}ʻ\alpha \geq \text{N}_{0}\text{c}ʻ\beta \colon\ldotp \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_174">[Pg 174]</span></p> + +<p class="nind"><b><a id="*117·108">*117·108</a>.</b> \[\begin{align}&\vdash \colon\ldotp \text{Nc}ʻ\alpha \geq \text{Nc}ʻ\beta .\equiv :\text{Nc}ʻ\alpha \gt \text{Nc}ʻ\beta .\lor.\text{Nc}ʻ\alpha =\text{Nc}ʻ\beta \\ +&[*117·107·106·104. *103·16·4]\end{align}\]</p> + +<p class="nind"><b>*117·11.</b> + \(\vdash \colon\ldotp \alpha \text{ sm } \alpha'.\beta \text{ sm } \beta'. \supset : \exists !\text{Cl}ʻ\alpha \cap \text{Nc}ʻ\beta .\equiv .\exists !\text{Cl}ʻ\alpha'\cap \text{Nc}ʻ\beta'\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*100·321.\supset \vdash \colon\ldotp \text{Hp}.&\supset :\exists !\text{Cl}ʻ\alpha'\cap \text{Nc}ʻ\beta .\equiv .\exists !\text{Cl}ʻ\alpha'\cap \text{Nc}ʻ\beta' &\qquad \text{(1)}\\ +\vdash .*73·21.\supset \\ +\vdash : R\in 1\rightarrow &1.\text{D}ʻR=\alpha .\text{ᗡ}ʻR=\alpha'.\gamma \subset \alpha .\gamma \in \text{Nc}ʻ\beta .\supset .\\ +&\breve{R} ʻʻ\gamma \subset \alpha'.\breve{R} ʻʻ\gamma \in \text{Nc}ʻ\beta .\\ +[*60·2]&\supset .\exists !\text{Cl}ʻ\alpha'\cap \text{Nc}ʻ\beta &\qquad \text{(2)}\\ +\vdash .(2).*10·11·23·35.*73·1.\supset \\ +\vdash :\alpha \text{ sm } \alpha'.\exists !\text{Cl}ʻ\alpha \cap \text{Nc}ʻ\beta .&\supset .\exists !\text{Cl}ʻ\alpha'\cap \text{Nc}ʻ\beta &\qquad \text{(3)}\\ +\vdash .(3)\frac{\alpha',\,\alpha}{\alpha,\,\alpha'}.&\supset \vdash :\alpha \text{ sm } \alpha'.\exists !\text{Cl}ʻ\alpha'\cap \text{Nc}ʻ\beta .\supset .\exists !\text{Cl}ʻ\alpha \cap \text{Nc}ʻ\beta &\qquad \text{(4)}\\ +\vdash .(3).(4).&\supset \vdash \colon\ldotp \alpha \text{ sm } \alpha '.\supset :\exists !\text{Cl}ʻ\alpha \cap \text{Nc}ʻ\beta .\equiv .\exists !\text{Cl}ʻ\alpha ʻ\cap \text{Nc}ʻ\beta &\qquad \text{(5)}\\ +\vdash .(1).(5).&\supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*117·12">*117·12</a>.</b> \[\begin{align}\vdash \colon\ldotp \mu >&\nu .\equiv :\mu ,\nu \in \text{N}_{0}\text{C}:\\ +&\gamma \in \mu .\delta \in \nu .\supset _{\gamma ,\delta }.\exists !\text{Cl}ʻ\gamma \cap \text{Nc}ʻ\delta .{\sim}\exists !\text{Cl}ʻ\delta \cap \text{Nc}ʻ\gamma\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*117·1·11.\supset \\ +\vdash \colon\ldotp \mu >\nu .\equiv :(\exists \alpha ,\beta ):&\mu =\text{N}_{0}\text{c}ʻ\alpha .\nu =\text{N}_{0}\text{c}ʻ\beta + .\exists !\text{Cl}ʻ\alpha \cap \text{Nc}ʻ\beta .{\sim}\exists !\text{Cl}ʻ\beta \cap \text{Nc}ʻ\alpha :\\ +&\gamma \in \mu .\delta \in \nu .\supset _{\gamma ,\delta }.\exists !\text{Cl}ʻ\gamma \cap \text{Nc}ʻ\delta .{\sim}\exists !\text{Cl}ʻ\delta \cap \text{Nc}ʻ\gamma :\\ +[*103·12]&\equiv :(\exists \alpha ,\beta ):\mu =\text{N}_{0}\text{c}ʻ\alpha .\nu =\text{N}_{0}\text{c}ʻ\beta + .\alpha \in \mu .\beta \in \nu .\exists !\text{Cl}ʻ\alpha \cap \text{Nc}ʻ\beta .\\ +&{\sim}\exists !\text{Cl}ʻ\beta \cap \text{Nc}ʻ\alpha :\\ +&\gamma \in \mu .\delta \in \nu .\supset _{\gamma ,\delta }.\exists !\text{Cl}ʻ\gamma \cap \text{Nc}ʻ\delta .{\sim}\exists !\text{Cl}ʻ\delta \cap \text{Nc}ʻ\gamma :\\ +[*10·55] &\equiv :(\exists \alpha ,\beta ):\mu =\text{N}_{0}\text{c}ʻ\alpha .\nu =\text{N}_{0}\text{c}ʻ\beta .\alpha \in \mu .\beta \in \nu :\\ +&\gamma \in \mu .\delta \in \nu .\supset _{\gamma ,\delta }.\exists !\text{Cl}ʻ\gamma \cap \text{Nc}ʻ\delta .{\sim}\exists !\text{Cl}ʻ\delta \cap \text{Nc}ʻ\gamma :\\ +[*103·12·2]&\equiv :\mu ,\nu \in \text{N}_{0}\text{C}:\gamma \in \mu .\delta \in \nu .\supset _{\gamma ,\delta }.\exists + !\text{Cl}ʻ\gamma \cap \text{Nc}ʻ\delta .{\sim}\exists !\text{Cl}ʻ\delta \cap \text{Nc}ʻ\gamma \colon\ldotp \\ +&\supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*117·121.</b> \[\begin{align}\vdash \colon\ldotp \mu >&\nu .\equiv :\mu ,\nu \in \text{N}_{0}\text{C}:\\ +&\alpha \in \mu .\supset _{\alpha }.(\exists \beta ).\beta \in \nu .\exists !\text{Cl}ʻ\alpha \cap \text{Nc}ʻ\beta .{\sim}\exists !\text{Cl}ʻ\beta \cap \text{Nc}ʻ\alpha \end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*117·1·11.\supset \\ +\vdash \colon\ldotp \mu >\nu .\equiv :&(\exists \alpha ,\beta ):\mu =\text{N}_{0}\text{c}ʻ\alpha .\nu =\text{N}_{0}\text{c}ʻ\beta + .\exists !\text{Cl}ʻ\alpha \cap \text{Nc}ʻ\beta .\\ +&{\sim}\exists !\text{Cl}ʻ\beta \cap \text{Nc}ʻ\alpha :\\ +&\gamma \in \mu .\supset _{\gamma }.(\exists \delta ).\delta \in \nu .\exists !\text{Cl}ʻ\gamma \cap \text{Nc}ʻ\delta .{\sim}\exists !\text{Cl}ʻ\delta \cap \text{Nc}ʻ\gamma \\ +[*103·12.*10·55]&\equiv :(\exists \alpha ,\beta ):\mu =\text{N}_{0}\text{c}ʻ\alpha .\nu =\text{N}_{0}\text{c}ʻ\beta .\alpha \in \mu :\\ +&\gamma \in \mu .\supset _{\gamma }.(\exists \delta ).\delta \in \nu .\exists !\text{Cl}ʻ\gamma \cap \text{Nc}ʻ\delta .{\sim}\exists !\text{Cl}ʻ\delta \cap \text{Nc}^\gamma \\ +[*103·12·2] &\equiv :\mu ,\nu \in \text{N}_{0}\text{C}:\gamma \in \mu .\supset _{\gamma }.(\exists \delta ).\delta \in \nu .\exists !\text{Cl}ʻ\gamma \cap \text{Nc}ʻ\delta .\\ +&{\sim}\exists !\text{Cl}ʻ\delta \cap \text{Nc}ʻ\gamma \colon\ldotp \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_175">[Pg 175]</span></p> + +<p>The above proof is given shortly because it proceeds on the same lines +as <a href="#*117·12">*117·12</a>. In applying *10·55, the \(\phi x\) of that proposition is +replaced by \(\alpha \in \mu\), and the \(\psi x\) is replaced by +\[ +(\exists \beta ).\beta \in \nu .\exists !\text{Cl}ʻ\alpha \cap \text{Nc}ʻ\beta .{\sim}\exists !\text{Cl}ʻ\beta \cap \text{Nc}ʻ\alpha . +\]</p> + +<p class="nind"><b><a id="*117·13">*117·13</a>.</b> \(\vdash :\text{Nc}ʻ\alpha >\text{Nc}ʻ\beta .\equiv .\exists !\text{Cl}ʻ\alpha \cap \text{Nc}ʻ\beta .{\sim}\exists !\text{Cl}ʻ\beta \cap \text{Nc}ʻ\alpha\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*117·106.\supset \\ +\vdash \colon\ldotp \text{Nc}ʻ\alpha >\text{Nc}ʻ\beta .&\equiv :\text{N}_{0}\text{c}ʻ\alpha >\text{N}_{0}\text{c}ʻ\beta :\\ +[*103·2.*117·12] &\equiv :\gamma \in \text{N}_{0}\text{c}ʻ\alpha .\delta \in \text{N}_{0}\text{c}ʻ\beta .\supset _{\gamma ,\delta }.\\ +&\exists !\text{Cl}ʻ\gamma \cap \text{Nc}ʻ\delta .{\sim}\exists !\text{Cl}ʻ\delta \cap \text{Nc}ʻ\gamma :\\ +[*100·31.*117·11] &\equiv :\gamma \in \text{N}_{0}\text{c}ʻ\alpha .\delta \in \text{N}_{0}\text{c}ʻ\beta .\supset _{\gamma ,\delta }.\\ +&\exists !\text{Cl}ʻ\alpha \cap \text{Nc}ʻ\beta .{\sim}\exists !\text{Cl}ʻ\beta \cap \text{Nc}ʻ\alpha :\\ +[*10·23] &\equiv :\exists !\text{N}_{0}\text{c}ʻ\alpha .\exists !\text{N}_{0}\text{c}ʻ\beta .\supset .\\ +&\exists !\text{Cl}ʻ\alpha \cap \text{Nc}ʻ\beta .{\sim}\exists !\text{Cl}ʻ\beta \cap \text{Nc}ʻ\alpha :\\ +[*103·13] &\equiv :\exists !\text{Cl}ʻ\alpha \cap \text{Nc}ʻ\beta .{\sim}\exists !\text{Cl}ʻ\beta \cap \text{Nc}ʻ\alpha \colon\ldotp \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*117·14">*117·14</a>.</b> \[\begin{align}&\vdash :\mu \gt \nu .\equiv .(\exists \alpha ,\beta ).\mu =\text{N}_{0}\text{c}ʻ\alpha .\nu =\text{N}_{0}\text{c}ʻ\beta + .\text{Nc}ʻ\alpha \gt \text{Nc}ʻ\beta \\ +&[*117·1·13]\end{align}\]</p> + +<p class="nind"><b><a id="*117·15">*117·15</a>.</b> \(\vdash :\mu \gt \nu .\equiv .\mu ,\nu \in \text{N}_{0}\text{C}.\exists !sʻ\text{Cl}ʻʻ\mu \cap \text{ sm }ʻʻ\nu .{\sim}\exists !sʻ\text{Cl}ʻʻ\nu \cap \text{ sm }ʻʻ\mu\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*103·4.*117·1.\supset \\ +\vdash \colon\ldotp \mu >\nu .&\equiv :(\exists \alpha ,\beta ).\mu =\text{N}_{0}\text{c}ʻ\alpha .\nu =\text{N}_{0}\text{c}ʻ\beta + .\exists !\text{Cl}ʻ\alpha \cap \text{ sm }ʻʻ\nu .\\ +&{\sim}\exists !\text{Cl}ʻ\beta \cap \text{ sm }ʻʻ\mu :\\ +[*103·2·26] &\equiv :\mu ,\nu \in \text{N}_{0}\text{C}:(\exists \alpha ,\beta ).\alpha \in \mu .\beta \in \nu .\exists !\text{Cl}ʻ\alpha \cap \text{ sm }ʻʻ\nu .\\ +&{\sim}\exists !\text{Cl}ʻ\beta \cap \text{ sm }ʻʻ\mu :\\ +[*117·11] &\equiv :\mu ,\nu \in \text{N}_{0}\text{C}:(\exists \alpha ,\beta ).\alpha \in \mu .\beta \in \nu .\exists !\text{Cl}ʻ\alpha \cap \text{ sm }ʻʻ\nu :\\ +&\delta \in \nu .\supset _{\delta }.{\sim}\exists !\text{Cl}ʻ\delta \cap \text{ sm }ʻʻ\mu :\\ +[*103·13.*10·51] &\equiv :\mu ,\nu \in \text{N}_{0}\text{C}:(\exists \alpha ).\alpha \in \mu .\exists !\text{Cl}ʻ\alpha \cap \text{ sm }ʻʻ\nu :\\ +&{\sim}(\exists \delta ).\delta \in \nu .\exists !\text{Cl}ʻ\delta \cap \text{ sm }ʻʻ\mu :\\ +[*40·4.*60·2] &\equiv :\mu ,\nu \in \text{N}_{0}\text{C}.\exists !sʻ\text{Cl}ʻʻ\mu \cap \text{ sm }ʻʻ\nu .\\ +&{\sim}\exists !sʻ\text{Cl}ʻʻ\nu \cap \text{ sm }ʻʻ\mu \colon\ldotp \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p>The advantage of this proposition is that it expresses "\(\mu \gt \nu\)" +in terms of \(\mu\) and \(\nu\) alone, without the auxiliary \(\alpha\) +and \(\beta\) of the definition.</p> + +<p class="nind"><b><a id="*117·16">*117·16</a>.</b> \[\begin{align}&\vdash \colon\ldotp \mu ,\nu \in \text{N}_{0}\text{C}.\supset + :\mu \gt \nu .\equiv .\text{ sm }ʻʻ\mu \gt \nu .\equiv .\mu >\text{ sm }ʻʻ\nu .\equiv .\text{ sm }ʻʻ\mu >\text{ sm }ʻʻ\nu \\ +&[*117·14.*103·4]\end{align}\]</p> + +<p class="nind"><b><a id="*117·2">*117·2</a>.</b> \(\vdash :\alpha \text{ sm } \alpha ʻ.\beta \text{ sm } \beta ʻ. \beta ʻ\subset \alpha .\alpha ʻ\subset \beta .\supset .\alpha \text{ sm } \beta \quad[*73·88]\)</p> + +<p><span class="pagenum" id="Page_176">[Pg 176]</span></p> + +<p>This proposition (which is the Schröder-Bernstein theorem) is +fundamental in the theory of greater and less.</p> + +<p class="nind"><b><a id="*117·21">*117·21</a>.</b> \[\begin{align}&\vdash :\exists !\text{Cl}ʻ\alpha \cap \text{Nc}ʻ\beta .\exists !\text{Cl}ʻ\beta \cap \text{Nc}ʻ\alpha .\supset .\text{Nc}ʻ\alpha =\text{Nc}ʻ\beta \\ +&[*117·2.*100·321]\end{align}\]</p> + +<p class="nind"><b><a id="*117·211">*117·211</a>.</b> \(\vdash :\exists !\text{Cl}ʻ\alpha \cap \text{Nc}ʻ\beta .\exists !\text{Cl}ʻ\beta \cap \text{Nc}ʻ\alpha .\equiv .\text{Nc}ʻ\alpha =\text{Nc}ʻ\beta\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*100·3.*60·34.\supset \vdash :\text{Nc}ʻ\alpha =\text{Nc}ʻ\beta .\supset .\alpha \in \text{Cl}ʻ\alpha \cap \text{Nc}ʻ\beta .\beta \in \text{Cl}ʻ\beta\\ \cap \text{Nc}ʻ\alpha &\quad \text{(1)}\\ +\vdash .(1).*117·21.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*117·22">*117·22</a>.</b> \(\vdash :\exists !\text{Cl}ʻ\alpha \cap \text{Nc}ʻ\beta .\equiv .\text{Nc}ʻ\alpha \geq \text{Nc}ʻ\beta\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*117·13. &\supset \vdash :\text{Hp}.{\sim}\exists !\text{Cl}ʻ\beta \cap \text{Nc}ʻ\alpha .\equiv .\text{Nc}ʻ\alpha >\text{Nc}ʻ\beta &\qquad \text{(1)}\\ +\vdash .*117·211.&\supset \vdash :\text{Hp}.\exists !\text{Cl}ʻ\beta \cap \text{Nc}ʻ\alpha .\equiv .\text{Nc}ʻ\alpha =\text{Nc}ʻ\beta &\qquad \text{(2)}\\ +\vdash .(1).(2).*117·108.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*117·221.</b> \(\vdash :\text{Nc}ʻ\alpha \geq \text{Nc}ʻ\beta .\equiv .(\exists \rho ).\rho \subset \alpha .\rho \text{ sm }\beta \quad[*117·22.*60·2.*100·1]\)</p> + +<p class="nind"><b>*117·222.</b> \(\vdash :\beta \subset \alpha .\supset .\text{Nc}ʻ\alpha \geq \text{Nc}ʻ\beta \quad[*117·221]\)</p> + +<p class="nind"><b><a id="*117·23">*117·23</a>.</b> \(\vdash :\text{Nc}ʻ\alpha \geq \text{Nc}ʻ\beta .\text{Nc}ʻ\beta \geq \text{Nc}ʻ\alpha .\equiv .\text{Nc}ʻ\alpha =\text{Nc}ʻ\beta \quad[*117·211·22]\)</p> + +<p class="nind"><b><a id="*117·24">*117·24</a>.</b> \(\vdash :\mu \geq \nu .\equiv .(\exists \alpha ,\beta ).\mu =\text{N}_{0}\text{c}ʻ\alpha .\nu =\text{N}_{0}\text{c}ʻ\beta + .\text{Nc}ʻ\alpha \geq \text{Nc}ʻ\beta\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*117·104·14.\supset \vdash \colon\ldotp \mu \geq \nu .&\equiv :(\exists \alpha ,\beta ).\mu =\text{N}_{0}\text{c}ʻ\alpha .\nu +=\text{N}_{0}\text{c}ʻ\beta .\text{Nc}ʻ\alpha >\text{Nc}ʻ\beta .\lor.\\ +&(\exists \alpha ,\beta ).\mu =\text{N}_{0}\text{c}ʻ\alpha .\nu =\text{N}_{0}\text{c}ʻ\beta .\mu =\text{ sm }ʻʻ\nu \\ +[*103·4.*13·193] &\equiv :(\exists \alpha ,\beta ).\mu =\text{N}_{0}\text{c}ʻ\alpha .\nu =\text{N}_{0}\text{c}ʻ\beta .\text{Nc}ʻ\alpha >\text{Nc}ʻ\beta .\lor.\\ +&(\exists \alpha ,\beta ).\mu =\text{N}_{0}\text{c}ʻ\alpha .\nu =\text{N}_{0}\text{c}ʻ\beta .\text{N}_{0}\text{c}ʻ\alpha =\text{Nc}ʻ\beta :\\ +[*103·16] &\equiv :(\exists \alpha ,\beta ).\mu =\text{N}_{0}\text{c}ʻ\alpha .\nu =\text{N}_{0}\text{c}ʻ\beta .\text{Nc}ʻ\alpha >\text{Nc}ʻ\beta .\lor.\\ +&(\exists \alpha ,\beta ).\mu =\text{N}_{0}\text{c}ʻ\alpha .\nu =\text{N}_{0}\text{c}ʻ\beta .\text{Nc}ʻ\alpha =\text{Nc}ʻ\beta :\\ +[*11·41.*117·108] &\equiv :(\exists \alpha ,\beta ).\mu =\text{N}_{0}\text{c}ʻ\alpha .\nu =\text{N}_{0}\text{c}ʻ\beta .\text{Nc}ʻ\alpha \geq \text{Nc}ʻ\beta \colon\ldotp \\ +&\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*117·241">*117·241</a>.</b> \[\begin{align}&\vdash :\mu \geq \nu .\equiv .(\exists \alpha ,\beta ).\mu =\text{N}_{0}\text{c}ʻ\alpha .\nu =\text{N}_{0}\text{c}ʻ\beta + .\exists !\text{Cl}ʻ\alpha \cap \text{Nc}ʻ\beta \\ +&[*117·24·22]\end{align}\]</p> + +<p class="nind"><b>*117·242.</b> + \[\begin{align}&\vdash \colon\ldotp \mu ,\nu \in \text{NC}.\supset :\mu \geq \nu .\equiv .(\exists \alpha ,\beta ).\alpha \in \mu .\beta \in \nu .\exists !\text{Cl}ʻ\alpha \cap \text{Nc}ʻ\beta \\ +&[*117·241.*103·26]\end{align}\]</p> + +<p class="nind"><b><a id="*117·243">*117·243</a>.</b> \[\begin{align}&\vdash \colon\ldotp \mu \geq \nu .\equiv :(\exists \alpha ,\beta ):\mu =\text{N}_{0}\text{c}ʻ\alpha .\nu =\text{N}_{0}\text{c}ʻ\beta + :(\exists \rho ).\rho \subset \alpha .\rho \text{ sm }\beta \\ +&[*117·24·221]\end{align}\]</p> + +<p class="nind"><b>*117·244.</b> \[\begin{align}\vdash \colon\ldotp \mu ,\nu \in \text{N}_{0}\text{C}.\supset + :\mu \geq \nu .\equiv .&\text{ sm }ʻʻ\mu \geq \nu .\equiv .\mu \geq \text{ sm }ʻʻ\nu .\equiv .\\ +&\text{ sm }ʻʻ\mu \geq \text{ sm }ʻʻ\nu \quad[*117·24.*103·4]\end{align}\]</p> + +<p><span class="pagenum" id="Page_177">[Pg 177]</span></p> + +<p class="nind"><b><a id="*117·25">*117·25</a>.</b> \(\vdash :\mu \geq \nu .\nu \geq \mu .\equiv .\mu ,\nu \in \text{N}_{0}\text{C}.\text{ sm }ʻʻ\mu =\text{ sm }ʻʻ\nu\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*117·24.\supset \\ +\vdash :\mu \geq \nu .\nu \geq \mu .&\equiv .(\exists \alpha ,\beta ,\gamma ,\delta ).\mu =\text{N}_{0}\text{c}ʻ\alpha =\text{N}_{0}\text{c}ʻ\gamma .\nu =\text{N}_{0}\text{c}ʻ\beta + =\text{N}_{0}\text{c}ʻ\delta .\\ +&\text{Nc}ʻ\alpha \geq \text{Nc}ʻ\beta .\text{Nc}ʻ\delta \geq \text{Nc}ʻ\gamma .\\ +[*117·107] &\equiv .(\exists \alpha ,\beta ,\gamma,\delta ).\mu =\text{N}_{0}\text{c}ʻ\alpha =\text{N}_{0}\text{c}ʻ\gamma .\nu =\text{N}_{0}\text{c}ʻ\beta =\text{N}_{0}\text{c}ʻ\delta + .\\ +&\text{N}_{0}\text{c}ʻ\alpha \geq \text{N}_{0}\text{c}ʻ\beta .\text{N}_{0}\text{c}ʻ\delta \geq \text{N}_{0}\text{c}ʻ\gamma .\\ +[*13·193] &\equiv .(\exists \alpha ,\beta ,\gamma ,\delta ).\mu =\text{N}_{0}\text{c}ʻ\alpha =\text{N}_{0}\text{c}ʻ\gamma .\nu =\text{N}_{0}\text{c}ʻ\beta =\text{N}_{0}\text{c}ʻ\delta + .\\ +&\text{N}_{0}\text{c}ʻ\alpha \geq \text{N}_{0}\text{c}ʻ\beta .\text{N}_{0}\text{c}ʻ\beta \geq \text{N}_{0}\text{c}ʻ\alpha .\\ +[*117·107·23] &\equiv .(\exists \alpha ,\beta ,\gamma ,\delta ).\mu =\text{N}_{0}\text{c}ʻ\alpha =\text{N}_{0}\text{c}ʻ\gamma .\nu =\text{N}_{0}\text{c}ʻ\beta + =\text{N}_{0}\text{c}ʻ\delta .\\ +&\text{Nc}ʻ\alpha =\text{Nc}ʻ\beta .\\ +[*11·45.*103·2] &\equiv .(\exists \alpha ,\beta ).\mu =\text{N}_{0}\text{c}ʻ\alpha .\nu =\text{N}_{0}\text{c}ʻ\beta .\mu ,\nu \in \text{N}_{0}\text{C}.\text{Nc}ʻ\alpha + =\text{Nc}ʻ\beta .\\ +[*103·4] &\equiv .(\exists \alpha ,\beta ).\mu =\text{N}_{0}\text{c}ʻ\alpha .\nu =\text{N}_{0}\text{c}ʻ\beta .\mu ,\nu \in \text{N}_{0}\text{C}.\text{ sm }ʻʻ\mu + =\text{ sm }ʻʻ\nu .\\ +[*11·45.*103·2] &\equiv .\mu ,\nu \in \text{N}_{0}\text{C}.\text{ sm }ʻʻ\mu =\text{ sm }ʻʻ\nu :\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*117·26">*117·26</a>.</b> \(\vdash :\text{Nc}ʻ\alpha >\text{Nc}ʻ\beta .\equiv .\text{Nc}ʻ\alpha \geq \text{Nc}ʻ\beta .\text{Nc}ʻ\alpha \neq \text{Nc}ʻ\beta\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*117·13.*13·12.\text{Transp}.&\supset \vdash :\text{Nc}ʻ\alpha >\text{Nc}ʻ\beta .\supset .\text{Nc}ʻ\alpha \neq \text{Nc}ʻ\beta :\\ +[*117·108] &\supset \vdash :\text{Nc}ʻ\alpha >\text{Nc}ʻ\beta .\supset .\text{Nc}ʻ\alpha \geq \text{Nc}ʻ\beta .\text{Nc}ʻ\alpha \neq \text{Nc}ʻ\beta &\qquad \text{(1)}\\ +\vdash .*117·108.*5·6.&\supset \vdash :\text{Nc}ʻ\alpha \geq \text{Nc}ʻ\beta .\text{Nc}ʻ\alpha \neq \text{Nc}ʻ\beta .\supset .\text{Nc}ʻ\alpha >\text{Nc}ʻ\beta &\qquad \text{(2)}\\ +\vdash .(1).(2).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*117·27">*117·27</a>.</b> \[\begin{align}&\vdash :\text{Nc}ʻ\alpha <\text{Nc}ʻ\beta .\equiv .\text{Nc}ʻ\alpha \leq \text{Nc}ʻ\beta .\text{Nc}ʻ\alpha \neq \text{Nc}ʻ\beta \\ +&[*117·26·103·105]\end{align}\]</p> + +<p class="nind"><b>*117·28.</b> \[\begin{align}&\vdash :\text{Nc}ʻ\alpha >\text{Nc}ʻ\beta .\equiv .\text{Nc}ʻ\alpha \geq \text{Nc}ʻ\beta .{\sim}(\text{Nc}ʻ\beta \geq \text{Nc}ʻ\alpha )\\ +&[*117·22·13]\end{align}\]</p> + +<p class="nind"><b><a id="*117·281">*117·2</a>.</b> \(\vdash :\mu >\nu .\equiv .\mu \geq \nu .{\sim}(\nu \geq \mu ) \quad[*117·14·28·24]\)</p> + +<p class="nind"><b>*117·29.</b> \(\vdash :\text{Nc}ʻ\alpha <\text{Nc}ʻ\beta .\equiv .\text{Nc}ʻ\alpha \leq \text{Nc}ʻ\beta .{\sim}(\text{Nc}ʻ\beta \leq \text{Nc}ʻ\alpha ) \quad[*117·28]\)</p> + +<p class="nind"><b>*117·291.</b> \(\vdash :\mu <\nu .\equiv .\mu \leq \nu .{\sim}(\nu \leq \mu ) \quad[*117·281]\)</p> + +<p class="nind"><b><a id="*117·3">*117·3</a>.</b> \(\vdash :\text{Nc}ʻ\alpha \geq \text{Nc}ʻ\beta .\equiv .(\exists \varpi ).\varpi \in \text{NC}.\text{Nc}ʻ\alpha =\text{Nc}ʻ\beta +_{c}\varpi\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*117·221.\supset \vdash :\text{Nc}ʻ\alpha \geq \text{Nc}ʻ\beta &.\equiv .(\exists \delta ).\delta \text{ sm }\beta .\delta \subset \alpha .\\ +[*110·72] &\equiv .(\exists \varpi ).\varpi \in \text{NC}.\text{Nc}ʻ\alpha =\text{Nc}ʻ\beta +_{c}\varpi :\\ +&\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_178">[Pg 178]</span></p> + +<p class="nind"><b><a id="*117·31">*117·31</a>.</b> \(\vdash \colon\ldotp \mu \geq \nu .\equiv :\mu ,\nu \in \text{N}_{0}\text{C}:(\exists \varpi ).\varpi \in \text{NC}.\mu =\nu +_{c}\varpi\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*117·24·3.\supset \\ +\vdash \colon\ldotp \mu \geq \nu .&\equiv :(\exists \alpha ,\beta ,\varpi ).\mu =\text{N}_{0}\text{c}ʻ\alpha .\nu =\text{N}_{0}\text{c}ʻ\beta + .\text{Nc}ʻ\alpha =\text{Nc}ʻ\beta +_{c}\varpi :\\ +[(*110·03)] &\equiv :(\exists \alpha ,\beta ,\varpi ).\mu =\text{N}_{0}\text{c}ʻ\alpha .\nu =\text{N}_{0}\text{c}ʻ\beta .\text{Nc}ʻ\alpha =\nu +_{c}\varpi :\\ +[*103·16.*110·42] &\equiv :(\exists \alpha ,\beta ,\varpi ).\mu =\text{N}_{0}\text{c}ʻ\alpha .\nu =\text{N}_{0}\text{c}ʻ\beta .\mu =\nu +_{c}\varpi :\\ +[*103·2] &\equiv :\mu ,\nu \in \text{N}_{0}\text{C}:(\exists \varpi ).\mu =\nu +_{c}\varpi \colon\ldotp \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*117·32">*117·32</a>.</b> \(\vdash :\mu \geq \nu .\exists !\text{ sm }ʻʻ\mu \cap tʻ\alpha .\supset .\exists !\text{ sm }ʻʻ\nu \cap tʻ\alpha\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*117·241.*103·4.\supset \\ +\vdash :\text{Hp}.\supset .(\exists \beta ,\gamma ).\mu =\text{N}_{0}\text{c}ʻ\beta .\nu =\text{N}_{0}\text{c}ʻ\gamma + .\exists !\text{Cl}ʻ\beta \cap \text{Nc}ʻ\gamma .\text{ sm }ʻʻ\mu =\text{Nc}ʻ\beta .\\ +\text{ sm }ʻʻ\nu =\text{Nc}ʻ\gamma &\qquad \text{(1)}\\ +\vdash .*63·105·371.*73·12.\supset \\ +\vdash :R\in \rho \,\overline{\text{ sm }}\, \beta .\rho \in tʻ\alpha .\sigma \subset \beta .\sigma \text{ sm }\gamma .\supset .Rʻʻ\sigma \in tʻ\alpha .Rʻʻ\sigma \text{ sm }\gamma &\qquad \text{(2)}\\ +\vdash .(2).*73·04.\supset \vdash :\rho \in \text{Nc}ʻ\beta \cap tʻ\alpha .\sigma \in \text{Cl}ʻ\beta \cap \text{Nc}ʻ\gamma .\supset .\exists !\text{Nc}ʻ\gamma \cap tʻ\alpha &\qquad \text{(3)}\\ +\vdash .(1).(3).\supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p>The above proposition shows that if a cardinal \(\mu\) exists in a +given type, so do all smaller cardinals.</p> + +<p class="nind"><b><a id="*117·4">*117·4</a>.</b> \(\vdash :\mu \geq \nu .\nu \geq \varpi .\supset .\mu \geq \varpi\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*117·243.\supset \vdash \colon\ldotp \text{Hp}.&\supset :(\exists \alpha ,\beta ,\gamma ):\mu =\text{N}_{0}\text{c}ʻ\alpha .\nu =\text{N}_{0}\text{c}ʻ\beta + .\varpi =\text{N}_{0}\text{c}ʻ\gamma :\\ +&(\exists \rho ).\rho \subset \alpha .\rho \text{ sm }\beta :(\exists \sigma ).\sigma \subset \beta .\sigma \text{ sm }\gamma :\\ +[*117·11.*60·2.*100·1] &\supset :(\exists \alpha ,\beta ,\gamma ):\mu =\text{N}_{0}\text{c}ʻ\alpha .\nu =\text{N}_{0}\text{c}ʻ\beta .\varpi =\text{N}_{0}\text{c}ʻ\gamma + :\\ +&(\exists \rho ,\tau ).\rho \subset \alpha .\rho \text{ sm }\beta .\tau \subset \rho .\tau \text{ sm }\gamma :\\ +[*22·44] &\supset :(\exists \alpha ,\gamma ):\mu =\text{N}_{0}\text{c}ʻ\alpha .\varpi =\text{N}_{0}\text{c}ʻ\gamma :(\exists \tau ).\tau \subset \alpha .\tau \text{ sm }\gamma :\\ +[*117·243] &\supset :\mu \geq \varpi \colon\ldotp \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*117·41.</b> \(\vdash :\mu \leq \nu .\nu \leq \varpi .\supset .\mu \leq \varpi \quad[*117·4]\)</p> + +<p class="nind"><b><a id="*117·42">*117·42</a>.</b> \(\vdash :{\sim}(\mu >\mu ).{\sim}(\mu <\mu )\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*117·15.*13·12.\text{Transp}.\supset \vdash :\mu >\nu .\supset .\mu \neq \nu :\supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*117·43.</b> \(\vdash :\mu \geq \nu .{\sim}(\mu \geq \varpi ).\supset .{\sim}(\nu \geq \varpi ) \quad[*117·4.\text{Transp}]\)</p> + +<p class="nind"><b>*117·44.</b> \(\vdash :\nu \geq \varpi .{\sim}(\mu \geq \varpi ).\supset .{\sim}(\mu \geq \nu ) \quad[*117·4.\text{Transp}]\)</p> + +<p class="nind"><b>*117·45.</b> \(\vdash :\mu \geq \nu .\nu >\varpi .\supset .\mu >\varpi\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*117·281.\supset \vdash :\text{Hp}.&\supset .\mu \geq \nu .\nu \geq \varpi .{\sim}(\varpi \geq \nu ).\\ +[*117·4·44] &\supset .\mu \geq \varpi .{\sim}(\varpi \geq \mu ).\\ +[*117·281] &\supset .\mu >\varpi :\supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*117·46.</b> \(\vdash :\mu >\nu .\nu \geq \varpi .\supset .\mu >\varpi \quad[\text{Proof as in *117·45}]\)</p> + +<p class="nind"><b>*117·47.</b> \(\vdash :\mu >\nu .\nu >\varpi .\supset .\mu >\varpi \quad[*117·45·104]\)</p> + +<p><span class="pagenum" id="Page_179">[Pg 179]</span></p> + +<p class="nind"><b><a id="*117·471">*117·471</a>.</b> \(\vdash :\mu <\nu .\nu <\varpi .\supset .\mu <\varpi \quad[*117·47·103]\)</p> + +<p class="nind"><b>*117·5.</b> \(\vdash :\mu \in \text{N}_{0}\text{C}.\supset .\mu \geq 0\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*60·3.*100·3.&\supset \vdash .\exists !\text{Cl}ʻ\alpha \cap \text{Nc}ʻ\Lambda .\\ +[*117·22] &\supset \vdash .\text{Nc}ʻ\alpha \geq \text{Nc}ʻ\Lambda .\\ +[*117·107.*101·1]&\supset \vdash .\text{N}_{0}\text{c}ʻ\alpha \geq 0 &\qquad \text{(1)}\\ +\vdash .(1).*103·2. &\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*117·501">*117·501</a>.</b> \(\vdash :\mu \in \text{N}_{0}\text{C}.\equiv .\mu \geq 0 \quad[*117·5·104]\)</p> + +<p class="nind"><b>*117·51.</b> \(\vdash :\mu \in \text{N}_{0}\text{C}-\iota ʻ0.\supset .\mu >0\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*101·15.\supset \vdash :\text{Hp}.\supset .\mu \neq \text{ sm }ʻʻ0 &\qquad \text{(1)}\\ +\vdash .(1).*117·5·104.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*117·511">*117·511</a>.</b> \(\vdash :\mu \in \text{N}_{0}\text{C}-\iota ʻ0.\equiv .\mu >0 \quad[*117·51·15·42]\)</p> + +<p class="nind"><b>*117·52.</b> \(\vdash :\exists !\xi .\supset .\text{Nc}ʻ\xi \geq 1\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*51·2.\supset \vdash :\text{Hp}.&\supset .(\exists x).\iota ʻx\subset \xi .\\ +[*117·222] &\supset .(\exists x).\text{Nc}ʻ\xi \geq \text{Nc}ʻ\iota ʻx.\\ +[*101·2] &\supset .\text{Nc}ʻ\xi \geq 1:\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*117·53.</b> \(\vdash :\mu \in \text{N}_{0}\text{C}-\iota ʻ0.\supset .\mu \geq 1\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*101·16.*103·2.\supset \vdash :\text{Hp}.&\supset .(\exists \alpha ).\text{N}_{0}\text{c}ʻ\alpha =\mu .\exists !\alpha .\\ +[*117·52] &\supset .(\exists \alpha ).\text{N}_{0}\text{c}ʻ\alpha =\mu .\text{Nc}ʻ\alpha \geq 1.\\ +[*117·107] &\supset .\mu \geq 1:\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*117·531">*117·531</a>.</b> \(\vdash :\mu \in \text{N}_{0}\text{C}-\iota ʻ0.\equiv .\mu \geq 1\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*117·104. &\supset \vdash :\mu \geq 1.\supset .\mu \in \text{N}_{0}\text{C} &\qquad \text{(1)}\\ +\vdash .*117·51.*101·22.&\supset \vdash .1>0.\\ +[*117·45] &\supset \vdash :\mu \geq 1.\supset .\mu >0.\\ +[*117·42] &\supset .\mu \neq 0 &\qquad \text{(2)}\\ +\vdash .(1).(2).*117·53.&\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*117·54">*117·54</a>.</b> \(\vdash \colon\ldotp 1\geq \mu .\equiv :\mu =0.\lor.\mu =1\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*117·241.*101·2.*52·22.\supset \\ +\vdash \colon\ldotp 1\geq \mu .&\equiv :(\exists \alpha ,x).\mu =\text{N}_{0}\text{c}ʻ\alpha .\exists !\text{Nc}ʻ\alpha \cap \text{Cl}ʻ\iota ʻx:\\ +[*60·362] &\equiv :(\exists \alpha ,x):\mu =\text{N}_{0}\text{c}ʻ\alpha :\exists !\text{Nc}ʻ\alpha \cap \iota ʻ\Lambda .\lor.\exists !\text{Nc}ʻ\alpha \cap \iota ʻ\iota ʻx:\\ +[*51·31] &\equiv :(\exists \alpha ,x):\mu =\text{N}_{0}\text{c}ʻ\alpha :\Lambda \in \text{Nc}ʻ\alpha .\lor.\iota ʻx\in \text{Nc}ʻ\alpha :\\ +[*101·17·29] &\equiv :(\exists \alpha ,x):\mu =\text{N}_{0}\text{c}ʻ\alpha :\text{Nc}ʻ\alpha =\text{Nc}ʻ\Lambda .\lor.\text{Nc}ʻ\alpha =\text{Nc}ʻ\iota ʻx:\\ +[*103·16] &\equiv :(\exists \alpha ,x).\mu =\text{N}_{0}\text{c}ʻ\alpha :\mu =\text{Nc}ʻ\Lambda .\lor.\mu =\text{Nc}ʻ\iota ʻx:\\ +[*101·1·2] &\equiv :(\exists \alpha ).\mu =\text{N}_{0}\text{c}ʻ\alpha :\mu =0.\lor.\mu =1:\\ +[*103·2·5·51] &\equiv :\mu =0.\lor.\mu =1\colon\ldotp \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_180">[Pg 180]</span></p> + +<p class="nind"><b><a id="*117·55">*117·55</a>.</b> \(\vdash : \mu > 1 . \equiv . \mu \in \text{N}_{0}\text{C} - \iota ʻ0 - \iota ʻ1\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *117·281 . \supset \vdash : \mu > 1 . &\equiv . \mu \geq 1 . {\sim} (1 \geq \mu ) .\\ +[*117·531·54] &\equiv . \mu \in \text{N}_{0}\text{C} - \iota ʻ0 . \mu \neq 0 . \mu \neq 1 .\\ +[*51·15] &\equiv . \mu \in \text{N}_{0}\text{C} - \iota ʻ0 - \iota ʻ1 : \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*117·551">*117·551</a>.</b> \[\begin{align}\vdash \colon\ldotp \mu \in \text{N}_{0}\text{C} - \iota ʻ0 - &\iota ʻ1 . \equiv :\\ +&(\exists \alpha ) : \mu = \text{N}_{0}\text{c}ʻ\alpha : (\exists x, y) . x, y \in \alpha . x \neq y : \equiv . \mu \geq 2\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *103·2 . \supset \vdash \colon\ldotp &\mu \in \text{N}_{0}\text{C} - \iota ʻ0 - \iota ʻ1 . \equiv :\\ +&(\exists \alpha ) . \mu = \text{N}_{0}\text{c}ʻ\alpha . \text{N}_{0}\text{c}ʻ\alpha \neq 0 . \text{N}_{0}\text{c}ʻ\alpha \neq 1 :\\ +[*101·14] &\equiv : (\exists \alpha ) . \mu = \text{N}_{0}\text{c}ʻ\alpha . \exists ! \alpha . \text{N}_{0}\text{c}ʻ\alpha \neq 1 :\\ +[*103·26] &\equiv : (\exists \alpha ) . \mu = \text{N}_{0}\text{c}ʻ\alpha . \exists ! \alpha . \alpha {\sim} \in 1 :\\ +[*52·41] &\equiv : (\exists \alpha ) : \mu = \text{N}_{0}\text{c}ʻ\alpha : (\exists x, y) . x, y \in \alpha . x \neq y : &\qquad \text{(1)}\\ +[*54·26 . *51·2] &\equiv : (\exists \alpha ) : \mu = \text{N}_{0}\text{c}ʻ\alpha : (\exists x, y) . \iota ʻx \cup \iota ʻy \subset \alpha . \iota ʻx \cup \iota ʻy \in 2 :\\ +[*13·195] &\equiv : (\exists \alpha ) : \mu = \text{N}_{0}\text{c}ʻ\alpha : (\exists x, y, \beta ) . \beta = \iota ʻx \cup \iota ʻy . \beta \subset \alpha . \beta \in 2 :\\ +[*54·101] &\equiv : (\exists \alpha ) : \mu = \text{N}_{0}\text{c}ʻ\alpha : (\exists \beta ) . \beta \subset \alpha . \beta \in 2 :\\ +[*117·241] &\equiv : \mu \geq 2 &\qquad \text{(2)}\\ +\vdash . (1) . (2) . \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*117·56">*117·56</a>.</b> \(\vdash : \text{Nc}ʻ\alpha \geq \text{Nc}ʻ\beta . \supset . \text{Nc}ʻ\alpha +_{c} \text{Nc}ʻ\gamma \geq \text{Nc}ʻ\beta +_{c} \text{Nc}ʻ\gamma\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *110·12 . *117·221 . \supset \\ +\vdash : \text{Hp} . &\supset . (\exists \delta ) . \delta \subset \downarrow \Lambda _{\gamma }ʻʻ\iotaʻʻ\alpha . \delta \text{ sm } \downarrow \Lambda _{\gamma }ʻʻ\iotaʻʻ\beta + .\\ +[*110·11 . *73·71 . (*110·01)] &\supset . (\exists \delta ) . \delta \cup \Lambda _{\alpha } \downarrow ʻʻ\iotaʻʻ\gamma \subset \alpha + \gamma .\\ +&\delta \cup \Lambda _{\alpha } \downarrow ʻʻ\iotaʻʻ\gamma \text{ sm } (\beta + \gamma ) .\\ +[*117·221] &\supset . \text{Nc}ʻ(\alpha + \gamma ) \geq \text{Nc}ʻ(\beta + \gamma ) .\\ +[*110·3] &\supset . \text{Nc}ʻ\alpha +_{c} \text{Nc}ʻ\gamma \geq \text{Nc}ʻ\beta +_{c} \text{Nc}ʻ\gamma : \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*117·561">*117·561</a>.</b> \(\vdash : \mu \geq \nu . \varpi \in \text{N}_{0}\text{C} . \supset . \mu +_{c} \varpi \geq \nu +_{c} \varpi \quad[*117·56]\)</p> + +<p>The proof of <a href="#*117·561">*117·561</a> follows from <a href="#*117·56">*117·56</a> in the same way as the proof +of <a href="#*117·31">*117·31</a> follows from <a href="#*117·3">*117·3</a>. In the remainder of this number we +shall omit proofs of this kind.</p> + +<p class="nind"><b>*117·57.</b> \(\vdash : \text{Nc}ʻ\alpha \geq \text{Nc}ʻ\beta . \supset . \text{Nc}ʻ\alpha \times _{c} \text{Nc}ʻ\gamma \geq \text{Nc}ʻ\beta \times _{c} + \text{Nc}ʻ\gamma\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *37·2 . &\supset \vdash : \rho \subset \alpha . \supset . \gamma \downarrow_{,,}ʻʻ\rho \subset \gamma \downarrow_{,,}ʻʻ\alpha .\\ +[*40·161 . *113·1] &\supset . \rho \times \gamma \subset \alpha \times \gamma &\qquad \text{(1)}\\ +\vdash . *113·13 . &\supset \vdash \rho \text{ sm } \beta . \supset . \rho \times \gamma \text{ sm } \beta \times \gamma &\qquad \text{(2)}\\ +\vdash . (1) . (2) . &\supset \vdash : \rho \subset \alpha . \rho \text{ sm } \beta . \supset . \rho \times \gamma \subset \alpha \times \gamma . \rho \times \gamma \text{ sm } \beta \times \gamma .\\ +[*117·221] &\supset . \text{Nc}ʻ(\alpha \times \gamma ) \geq \text{Nc}ʻ(\beta \times \gamma ) &\qquad \text{(3)}\\ +\vdash . (3) . *117·221 . \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_181">[Pg 181]</span></p> + +<p class="nind"><b><a id="*117·571">*117·571</a>.</b> \(\vdash .\mu \geq \nu .\varpi \in \text{N}_{0}\text{C}.\supset .\mu \times _{c}\varpi \geq \nu \times _{c}\varpi \quad[*117·57]\)</p> + +<p class="nind"><b><a id="*117·58">*117·58</a>.</b> \(\vdash :\text{Nc}ʻ\alpha \geq \text{Nc}ʻ\beta .\supset .(\text{Nc}ʻ\alpha )^{\text{Nc}ʻ\gamma }\geq (\text{Nc}ʻ\beta )^{\text{Nc}ʻ\gamma }\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*35·432·82.\supset \vdash : \rho \subset \alpha .&\supset .\rho \uparrow \gamma \,\unicode{x2abd}\, \alpha \uparrow \gamma .\\ +[*80·15] &\supset .(\rho \uparrow \gamma )_{\Delta }ʻ\gamma \subset (\alpha \uparrow \gamma )_{\Delta }ʻ\gamma &\qquad \text{(1)}\\ +\vdash .*116·15·19.&\supset \vdash :\rho \text{ sm }\beta .\supset .(\rho \uparrow \gamma )_{\Delta }ʻ\gamma \text{ sm }(\beta \uparrow \gamma )_{\Delta }ʻ\gamma + &\qquad \text{(2)}\\ +\vdash .(1).(2).*117·221.\supset \\ +\vdash :\rho \subset \alpha .\rho \text{ sm }\beta .&\supset .\text{Nc}ʻ(\alpha \uparrow \gamma )_{\Delta }ʻ\gamma \geq \text{Nc}ʻ(\beta \uparrow \gamma )_{\Delta }ʻ\gamma .\\ +[*116·15·25] &\supset .(\text{Nc}ʻ\alpha )^{\text{Nc}ʻ\gamma }\geq (\text{Nc}ʻ\beta )^{\text{Nc}ʻ\gamma } &\qquad \text{(3)}\\ +\vdash .(3).*117·221.&\supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*117·581">*117·581</a>.</b> \(\vdash :\mu \geq \nu .\varpi \in \text{N}_{0}\text{C}.\supset .\mu ^{\varpi }\geq \nu ^{\varpi } \quad[*117·58]\)</p> + +<p>The two following propositions are lemmas for <a href="#*117·59">*117·59</a>.</p> + +<p class="nind"><b>*117·582.</b> + \[\begin{align}\vdash :\exists !\gamma .\beta \subset \alpha .\sigma \in \gamma \,\,\text{exp}\,\,(\alpha -\beta ).\supset .&(\cup \sigma )\upharpoonright (\gamma \,\,\text{exp}\,\,\beta )\in 1\rightarrow 1.\\ +&(\cup \sigma )ʻʻ(\gamma \,\,\text{exp}\,\,\beta )\subset \gamma \,\,\text{exp}\,\,\alpha \end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*116·183.&\supset \vdash :\rho \in (\gamma \,\,\text{exp}\,\,\beta ).\sigma \in \gamma \,\,\text{exp}\,\,(\alpha -\beta ).\supset .\rho \subset \beta \times \gamma .\sigma \subset (\alpha -\beta )\times \gamma .\\ +[*113·19.*24·21] &\supset .\rho \cap \sigma =\Lambda &\qquad \text{(1)}\\ +\vdash .(1).*24·481.&\supset \vdash \colon\colon \text{Hp}.\supset \colon\ldotp \rho ,\rho ʻ\in (\gamma \,\,\text{exp}\,\,\beta ).\supset :\rho \cup \sigma =\rho ʻ\cup \sigma .\equiv .\rho =\rho ʻ\colon\ldotp \\ +[*71·58] &\supset \colon\ldotp (\cup \sigma )\upharpoonright (\gamma \,\,\text{exp}\,\,\beta )\in 1\rightarrow 1 &\qquad \text{(2)}\\ +\vdash .*113·191.&\supset \vdash \colon\ldotp \text{Hp}.\supset :\gamma \downarrow_{,,}ʻʻ\beta \cap \gamma \downarrow_{,,}ʻʻ(\alpha -\beta )=\Lambda :\\ +[*115·14.(*116·01)] &\supset :\rho \in (\gamma \,\,\text{exp}\,\,\beta ).\supset .\rho \cup \sigma \in \text{Prod}ʻ\{\gamma \downarrow_{,,}ʻʻ\beta \cup \gamma \downarrow_{,,}ʻʻ(\alpha + -\beta )\}.\\ +[*37·22.*24·411] &\supset .\rho \cup \sigma \in (\gamma \,\,\text{exp}\,\,\alpha ):\\ +[*37·61] & \supset :(\cup \sigma )ʻʻ(\gamma \,\,\text{exp}\,\,\beta )\subset \gamma \,\,\text{exp}\,\,\alpha &\qquad \text{(3)}\\ +\vdash .(2).(3).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*117·583.</b> \(\vdash :\beta \subset \alpha .\exists !\gamma .\supset .(\exists \tau ).\tau \subset \gamma \,\,\text{exp}\,\,\alpha .\tau \text{ sm }(\gamma \,\,\text{exp}\,\,\beta )\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*116·171.\supset \vdash :\text{Hp}.\supset .\exists !\gamma \,\,\text{exp}\,\,(\alpha -\beta ) &\qquad \text{(1)}\\ +\vdash .(1).*117·582.*73·15.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*117·59">*117·59</a>.</b> \(\vdash :\text{Nc}ʻ\alpha \geq \text{Nc}ʻ\beta .\exists !\gamma .\supset .(\text{Nc}ʻ\gamma )^{\text{Nc}ʻ\alpha }\geq (\text{Nc}ʻ\gamma )^{\text{Nc}ʻ\beta }\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*117·221.\supset \vdash \colon\ldotp \text{Hp}.&\supset :(\exists \rho ).\rho \subset \alpha .\rho \text{ sm }\beta :\exists !\gamma :\\ +[*117·583] &\supset :(\exists \rho ,\tau ).\rho \subset \alpha .\rho \text{ sm }\beta .\tau \subset \gamma \,\,\text{exp}\,\,\alpha .\tau \text{ sm }(\gamma \,\,\text{exp}\,\,\rho ):\\ +[*116·19] &\supset :(\exists \tau ).\tau \subset \gamma \,\,\text{exp}\,\,\alpha .\tau \text{ sm }(\gamma \,\,\text{exp}\,\,\beta ):\\ +[*117·221] &\supset :\text{Nc}ʻ(\gamma \,\,\text{exp}\,\,\alpha )\geq \text{Nc}ʻ(\gamma \,\,\text{exp}\,\,\beta ) &\qquad \text{(1)}\\ +\vdash .(1).*116·25.\supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_182">[Pg 182]</span></p> + +<p>The hypothesis is essential in the above proposition, for \(0^{0} = 1\) +while \(0^{1} = 0\), so that \(0^{0} > 0^{1}\).</p> + +<p class="nind"><b><a id="*117·591">*117·591</a>.</b> \(\vdash :\mu \geq \nu .\varpi \in \text{N}_{0}\text{C}-\iota ʻ0.\supset .\varpi ^{\mu } \geq \varpi ^{\nu } \quad[*117·59]\)</p> + +<p class="nind"><b><a id="*117·592">*117·592</a>.</b> \(\vdash :\alpha ^{\delta } = 1.\alpha \neq 0.\alpha \neq 1.\supset .\delta = 0\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*116·203.\supset \vdash \colon\ldotp \text{Hp}.&\supset :\alpha ,\delta \in \text{N}_{0}\text{C}:\\ +[*117·551·53] &\supset :\alpha \geq 2:\delta \neq 0.\supset .\delta \geq 1:\\ +[*117·581·591] &\supset :\delta \neq O.\supset .\alpha ^{\delta } \geq 2^{1}.\\ +[*116·321.*117·244] &\supset .\alpha ^{\delta } \geq 2.\\ +[*117·551] &\supset .\alpha ^{\delta } \neq 1 &\qquad \text{(1)}\\ +\vdash .(1).\text{Transp}.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p>The above proposition is used in <a href="#*120·53">*120·53</a>.</p> + +<p class="nind"><b><a id="*117·6">*117·6</a>.</b> \(\vdash :\mu ,\nu \in \text{N}_{0}\text{C}.\supset .\mu +_{c}\nu \geq \mu .\mu +_{c}\nu \geq \nu\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*117·561·5.\supset \vdash :\text{Hp}.\supset .\mu +_{c}\nu \geq \mu +_{c}0.\mu +_{c}\nu \geq 0+_{c}\nu &\qquad \text{(1)}\\ +\vdash .(1).*110·6.*117·244.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*117·61.</b> \(\vdash :\nu > \mu .\supset .\mu +_{c} \nu > \mu \quad[*117·6·45]\)</p> + +<p class="nind"><b><a id="*117·62">*117·62</a>.</b> \(\vdash :\mu ,\nu \in \text{N}_{0}\text{C}-\iota ʻ0.\supset .\mu \times _{c}\nu \geq \mu .\mu \times _{c}\nu \geq \nu\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*117·571·53.\supset \vdash :\text{Hp}.\supset .\mu \times _{c}\nu \geq \mu \times _{c}1.\mu \times _{c}\nu \geq 1\times _{c}\nu &\qquad \text{(1)}\\ +\vdash .(1).*113·621.*117·244.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*117·63.</b> \(\vdash :\alpha ,\beta {\sim}\in 0\cup 1.\supset .\text{Nc}ʻ\alpha \times _{c}\text{Nc}ʻ\beta \geq \text{Nc}ʻ\alpha +_{c}\text{Nc}ʻ\beta\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*52·4.\text{Transp}.&\supset \vdash :\text{Hp}.\supset .(\exists x,x',y,y').x,x'\in \alpha .y,y'\in \beta .x \neq x'.y \neq y' &\qquad \text{(1)}\\ +\vdash .*113·101. &\supset \vdash :\text{Hp}.x,x'\in \alpha .y,y'\in \beta .x \neq x'.y \neq y'.\rho = \downarrow yʻʻ\alpha .\\ +&\sigma = x\downarrow ʻʻ(\beta -\iota ʻy)\cup \iota ʻx'\downarrow y'.\supset .\rho \cup \sigma \subset \beta \times \alpha &\qquad \text{(2)}\\ +\vdash .*55·15.&\supset \vdash \colon\ldotp \text{Hp}(2).\supset :R\in \rho .\supset _{R}.\text{ᗡ}ʻR = \iota ʻy:\\ +&S\in x\downarrow ʻʻ(\beta -\iota ʻy).\supset _{S}.\text{ᗡ}ʻS\in \beta -\iota ʻy:\text{ᗡ}ʻx'\downarrow y' = \iota ʻy':\\ +[*51·23] &\supset :R\in \rho .S\in \sigma .\supset _{R,S}.\text{ᗡ}ʻR \neq \text{ᗡ}ʻS:\\ +[*24·37.*30·37] &\supset :\rho \cap \sigma = \Lambda &\qquad \text{(3)}\\ +\vdash .*73·61·611. &\supset \vdash :\text{Hp}(2).\supset .\rho \text{ sm } \alpha .x\downarrow ʻʻ(\beta -\iota ʻy) \text{ sm } (\beta -\iota ʻy) &\qquad \text{(4)}\\ +\vdash .*55·202. &\supset \vdash :\text{Hp}(2).\supset .x'\downarrow y'{\sim}\in x\downarrow ʻʻ(\beta -\iota ʻy) &\qquad \text{(5)}\\ +\vdash .(4).(5).*73·71. &\supset \vdash :\text{Hp}(2).\supset .\rho \text{ sm } \alpha .\sigma \text{ sm } \beta &\qquad \text{(6)}\\ +\vdash .(3).(6).*110·13. &\supset \vdash :\text{Hp}(2).\supset .\rho \cup \sigma \in \text{Nc}ʻ(\alpha +\beta ) &\qquad \text{(7)}\\ +\vdash .(2).(7).*117·221. &\supset \vdash :\text{Hp}(2).\supset .\text{Nc}ʻ(\beta \times \alpha ) \geq \text{Nc}ʻ(\alpha +\beta ) &\qquad \text{(8)}\\ +\vdash .(1).(8).*113·141·25.*110·3.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_183">[Pg 183]</span></p> + +<p class="nind"><b><a id="*117·631">*117·631</a>.</b> \(\vdash :\mu ,\nu \in \text{N}_{0}\text{C}-\iota ʻ0-\iota ʻ1.\supset .\mu \times _{c}\nu \geq \mu +_{c}\nu \quad[*117·63]\)</p> + +<p>The two following propositions are lemmas for <a href="#*117·64">*117·64</a>.</p> + +<p class="nind"><b>*117·632.</b> \[\begin{align}&\vdash :\kappa \in \text{Cls}^{2} \text{excl}.\kappa {\sim}\in 0\cup 1.\rho ,\sigma \in \text{Prod}ʻ\kappa .\rho \cap \sigma =\Lambda .\\ +&T=\hat{\mu} \hat{x} \{(\exists \alpha ,\beta ).\alpha ,\beta \in \kappa .\alpha \neq \beta .x\in \beta .\mu =(\rho -\alpha -\beta )\cup (\sigma \cap \alpha )\cup \iota ʻx\}.\\ +&\supset .T\in 1\rightarrow 1.\text{D}ʻT\subset \text{Prod}ʻ\kappa .\text{ᗡ}ʻT=sʻ\kappa\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*115·11·145. &\supset \vdash \colon\ldotp \text{Hp}.\alpha ,\beta \in \kappa .\alpha \neq \beta .\supset :\rho -\alpha -\beta \in \text{Prod}ʻ(\kappa -\iota ʻ\alpha -\iota ʻ\beta ):\\ +[*115·11·145] &\supset :(\rho -\alpha -\beta )\cup (\sigma \cap \alpha )\in \text{Prod}ʻ(\kappa -\iota ʻ\beta ):\\ +[*115·145] &\supset :x\in \beta .\supset .(\rho -\alpha -\beta )\cup (\sigma \cap \alpha )\cup \iota ʻx\in \text{Prod}ʻ\kappa &\qquad \text{(1)}\\ +\vdash .(1).*21·33. &\supset \vdash :\text{Hp}.\mu Tx.\supset .\mu \in \text{Prod}ʻ\kappa &\qquad \text{(2)}\\ +\vdash .*52·4.\text{Transp}. &\supset \vdash \colon\ldotp \text{Hp}.\supset :\beta \in \kappa .x\in \beta .\supset .(\exists \alpha ).\alpha \in \kappa .\alpha \neq \beta .\\ +[*21·33.*33·131] &\supset .x\in \text{ᗡ}ʻT &\qquad \text{(3)}\\ +\vdash .*21·33.*33·131. &\supset \vdash \colon\ldotp \text{Hp}.\supset :x \in\text{ᗡ}ʻT.\supset .(\exists \beta ).\beta \in \kappa .x\in \beta &\qquad \text{(4)}\\ +\vdash .(3).(4). &\supset \vdash :\text{Hp}.\supset .\text{ᗡ}ʻT=sʻ\kappa &\qquad \text{(5)}\\ +\vdash :*21·33.*13·172. &\supset \vdash \colon\ldotp \text{Hp}.\supset :\mu Tx.\nu Tx.\supset .\mu =\nu &\qquad \text{(6)}\\ +\vdash .*21·33.*13·171. &\supset \vdash \colon\ldotp \text{Hp}.\supset :\mu Tx.\mu Tx'.\supset .\\ +&(\exists \alpha ,\alpha',\beta ,\beta').\alpha ,\alpha'\in \kappa .\beta ,\beta'\in \kappa .\alpha \neq \beta .\alpha'\neq \beta'.\\ +&(\rho -\alpha -\beta )\cup (\sigma \cap \alpha )\cup \iota ʻx=(\rho -\alpha'-\beta')\cap (\sigma \cap \alpha')\cup \iota ʻx'.\\ +[*24·48.\text{Hp}] &\supset .\iota ʻx=\iota ʻx' &\qquad \text{(7)}\\ +\vdash .(2).(5).(6).(7).\supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*117·633.</b> \[\begin{align}&\vdash \colon\ldotp \kappa \in \text{Cls}^{2} + \text{excl}.\kappa {\sim}\in 0\cup 1:(\exists \rho ,\sigma ).\rho ,\sigma \in \text{Prod}ʻ\kappa .\rho \cap \sigma =\Lambda :\supset .\\ +&\Pi \text{Nc}ʻ\kappa \geq \Sigma \text{Nc}ʻ\kappa\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*117·632.\supset \vdash :\text{Hp}.&\supset .(\exists \gamma ).\gamma \subset \text{Prod}ʻ\kappa .\gamma \text{ sm } sʻ\kappa .\\ +[*117·221] &\supset .\text{Nc}ʻ\text{Prod}ʻ\kappa \geq \text{Nc}ʻsʻ\kappa &\qquad \text{(1)}\\ +\vdash .(1).*115·12.*112·15.\supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*117·64">*117·64</a>.</b> \[\begin{align}\vdash \colon\ldotp \kappa \in \text{Cls}^{2} + \text{excl}:(\exists \rho ,\sigma ).\rho ,\sigma \in \text{Prod}ʻ\kappa .\rho \cap \sigma =&\Lambda :\supset .\\ +&\Pi \text{Nc}ʻ\kappa \geq \Sigma \text{Nc}ʻ\kappa\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*112·321.*114·21. \supset \vdash :\kappa \in 1.&\supset .\Pi \text{Nc}ʻ\kappa =\Sigma \text{Nc}ʻ\kappa &\qquad \text{(1)}\\ +\vdash .*114·2.*112·3. \supset \vdash :\kappa \in 0.&\supset .\Pi \text{Nc}ʻ\kappa =1.\Sigma \text{Nc}ʻ\kappa =0.\\ +[*117·51] &\supset .\Pi \text{Nc}ʻ\kappa >\Sigma \text{Nc}ʻ\kappa &\qquad \text{(2)}\\ +\vdash .(1).(2).*117·633.\supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*117·651.</b> \(\vdash :\alpha {\sim}\in 0\cup 1.\supset .(\text{Nc}ʻ\alpha )^{\text{Nc}ʻ\beta }\geq \text{Nc}ʻ\alpha x_{c}\text{Nc}ʻ\beta\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*52·4.\text{Transp}. &\supset \vdash :\text{Hp}.\supset .(\exists x,y).x,y\in \alpha .x\neq y &\qquad \text{(1)}\\ +\vdash .*116·152.*55·23·202. \supset \vdash :x,y\in \alpha .x\neq y.\supset .&x\downarrowʻʻ\beta ,y\downarrowʻʻ\beta \in (\alpha \,\,\text{exp}\,\,\beta ).\\ +&x\downarrowʻʻ\beta \cap y\downarrowʻʻ\beta =\Lambda &\qquad \text{(2)}\\ +\vdash .*113·111. &\supset \vdash .\alpha \downarrow_{,,}ʻʻ\beta \in \text{Cls}^{2} \text{excl} &\qquad \text{(3)}\\ +\vdash .(1).(2).(3).*117·64.*113·1·141·25.*116·25.(*116·01).\supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_184">[Pg 184]</span></p> + +<p class="nind"><b><a id="*117·652">*117·652</a>.</b> \(\vdash :\mu \in \text{N}_{0}\text{C}-\iota ʻ0-\iota ʻ1.\nu \in \text{N}_{0}\text{C}.\supset .\mu ^\nu \geq \mu \times _{c} + \nu \quad[*117·651]\)</p> + +<p class="nind"><b>*117·66.</b> \(\vdash .\text{Nc}ʻ\text{Cl}ʻ\alpha >\text{Nc}ʻ\alpha\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*102·72. &\supset \vdash .{\sim}(\exists \beta ).\beta \subset \alpha .\beta \text{ sm }\text{Cl}ʻ\alpha &\qquad \text{(1)}\\ +\vdash .*100·6.*60·61. &\supset \vdash .\iotaʻʻ\alpha \subset \text{Cl}ʻ\alpha .\iotaʻʻ\alpha \text{ sm }\alpha &\qquad \text{(2)}\\ +\vdash .(1).(2).*117·13. &\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*117·661.</b> \(\vdash :\mu \in \text{N}_{0}\text{C}.\supset .2^\mu >\mu \quad[*117·66.*116·72]\)</p> + +<p>The above proposition is important.</p> + +<p class="nind"><b>*117·67.</b> \(\vdash :\kappa \in \text{Cls}^{2} \text{excl}.\exists !\text{Prod}ʻ\kappa .\supset .\text{Nc}ʻsʻ\kappa \geq \text{Nc}ʻ\kappa\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*115·16·11.&\supset \vdash :\kappa \in \text{Cls}^{2} \text{excl}.\mu \in \text{Prod}ʻ\kappa .\supset .\mu \text{ sm }\kappa .\mu \subset sʻ\kappa \\ . +[*117·22] &\supset .\text{Nc}ʻsʻ\kappa \geq \text{Nc}ʻ\kappa :\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*117·68.</b> \[\begin{align}\vdash :R,S\in {\in}_{\Delta}ʻ\kappa + .R\dot{\cap} S=\dot{\Lambda} .T=\hat{P} &\hat{\rho} \{\rho \in \kappa .P=R\upharpoonright -\iota ʻ\rho \unicode{x228d} S\upharpoonright \iota ʻ\rho\}.\\ +&\supset .T\in 1\rightarrow 1.\text{D}ʻT\subset {\in}_{\Delta}ʻ\kappa .\text{ᗡ}ʻT=\kappa\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*21·33.*13·172. &\supset \vdash \colon\ldotp \text{Hp}.\supset :PT\rho .QT\rho .\supset .P=Q &\qquad \text{(1)}\\ +\vdash .*23·631. &\supset \vdash :\text{Hp}.\rho \in \kappa .\supset .(Tʻ\rho )\dot{\cap} S=S\upharpoonright \iota ʻ\rho :\\ +[*13·17] &\supset \vdash :\text{Hp}.\rho ,\sigma \in \kappa .Tʻ\rho =Tʻ\sigma .\supset .S\upharpoonright \iota ʻ\rho =S\upharpoonright \iota ʻ\sigma .\\ +[*35·65] &\supset .\iota ʻ\rho =\iota ʻ\sigma .\\ +[*51·23] &\supset .\rho =\sigma &\qquad \text{(2)}\\ +\vdash .(1).(2). &\supset \vdash :\text{Hp}.\supset .T\in 1\rightarrow 1 &\qquad \text{(3)}\\ +\vdash .*21·33.*33·131. &\supset \vdash :\text{Hp}.\supset .\text{ᗡ}ʻT=\kappa &\qquad \text{(4)}\\ +\vdash .*80·36. &\supset \vdash :\text{Hp}.\supset .\text{D}ʻT\subset {\in}_{\Delta}ʻ\kappa &\qquad \text{(5)}\\ +\vdash .(3).(4).(5). \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*117·681.</b> \(\vdash :(\exists R,S).R,S\in {\in}_{\Delta}ʻ\kappa .R\dot{\cap} S=\dot{\Lambda} .\supset .\text{Nc}ʻ{\in}_{\Delta}ʻ\kappa + \geq \text{Nc}ʻ\kappa \quad [*117·68·22]\)</p> + +<p class="nind"><b>*117·682.</b> \(\vdash :\kappa \subset \lambda .\exists !{\in}_{\Delta}ʻ(\lambda -\kappa ).\supset .\text{Nc}ʻ{\in}_{\Delta}ʻ\lambda + \geq \text{Nc}ʻ{\in}_{\Delta}ʻ\kappa\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*80·65. &\supset \vdash \colon\ldotp \text{Hp}.\supset :R \in {\in}_{\Delta}ʻ\kappa .S\in {\in}_{\Delta}ʻ(\lambda + -\kappa ).\supset .R\unicode{x228d} S\in {\in}_{\Delta}ʻ\lambda &\qquad \text{(1)}\\ +\vdash .*80·14. &\supset \vdash :R\in {\in}_{\Delta}ʻ\lambda .S\in {\in}_{\Delta}ʻ(\lambda -\kappa ).\supset .\text{ᗡ}ʻR\cap \text{ᗡ}ʻS=\Lambda .\\ +[*33·33] &\supset .R\dot{\cap} S=\dot{\Lambda} .\\ +[*25·4] &\supset .(R\unicode{x228d} S)\dot{-} S=R &\qquad \text{(2)}\\ +\vdash .(2).*13·171. &\supset \vdash :Q,R\in {\in}_{\Delta}ʻ\lambda .S\in {\in}_{\Delta}ʻ(\lambda + -\kappa ).Q\unicode{x228d} S=R\unicode{x228d} S.\supset .Q=R &\qquad \text{(3)}\\ +\vdash .(1).(3). &\supset \vdash :\text{Hp}.S\in {\in}_{\Delta}ʻ(\lambda -\kappa ).\supset .(\unicode{x228d} S)\upharpoonright {\in}_{\Delta}ʻ\kappa + \in 1\rightarrow 1.(\unicode{x228d} S)ʻʻ{\in}_{\Delta}ʻ\kappa \subset {\in}_{\Delta}ʻ\lambda .\\ +[*117·22] &\supset .\text{Nc}ʻ{\in}_{\Delta}ʻ\lambda \geq \text{Nc}ʻ{\in}_{\Delta}ʻ\kappa :\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*117·683.</b> \[\begin{align}\vdash \colon\ldotp \kappa \subset \lambda .\exists !{\in}_{\Delta}ʻ(\lambda -\kappa ):(\exists R,S).&R,S\in {\in}_{\Delta}ʻ\kappa + .R\dot{\cap} S=\dot{\Lambda} :\supset .\\ +&\text{Nc}ʻ{\in}_{\Delta}ʻ\lambda \geq \text{Nc}ʻ\kappa \quad[*117·681·682]\end{align}\]</p> + +<p class="nind"><b>*117·684.</b> \[\begin{align}\vdash :\kappa \subset \lambda .\exists !{\in}_{\Delta}ʻ\lambda :(\exists R,S).&R,S\in {\in}_{\Delta}ʻ\kappa + .R\dot{\cap} S=\dot{\Lambda} :\supset .\\ +&\text{Nc}ʻ{\in}_{\Delta}ʻ\lambda \geq \text{Nc}ʻ\kappa \quad[*117·683.*88·22]\end{align}\]</p> + +<p>The above proposition is used in <a href="#*120·765">*120·765</a>.</p> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_185">[Pg 185]</span></p> +<h2 class="nobreak" id="GENERAL_NOTE_ON_CARDINAL_CORRELATORS"> +GENERAL NOTE ON CARDINAL CORRELATORS.</h2> +</div> + +<p>The correlators established at various stages throughout Section B +present certain analogies to each other, and they or others closely +resembling them will be found to be the correlators required in +relation-arithmetic (Part IV). We shall therefore here collect together +the most important propositions hitherto proved on correlators.</p> + +<p>When we have to deal with correlators of two different functions +of a single class, as <i>e.g.</i> \({\in}_{\Delta}ʻ\kappa\) and +\(\text{Prod}ʻ\kappa\), the correlator is usually \(\text{D}\) or +\(\dot{s}\) or \(\dot{s} \mid \text{D}\), with a suitable limitation on +the converse domain. Sometimes it is \(\breve{\iota} \mid \text{D}\) +or \(\breve{\in}\mid \text{D}\). Thus for example the class +\({\in}\unicode{x21A7}ʻʻ\kappa\), by means of which \(\Sigma ʻ\kappa\) is +defined (<a href="#*112">*112</a>), has double similarity with \(\kappa\) if +\(\kappa\in \text{Cls}^{2} \text{excl}\) (<a href="#*112·14">*112·14</a>); in this case, the +double correlator is \(\breve{\iota} \mid \text{D}\) with its converse +domain limited, <i>i.e.</i> +\[ +\vdash :\kappa \in \text{Cls}^{2} + \text{excl}.\supset .\breve{\iota} \mid \text{D}\upharpoonright \Sigma ʻ\kappa \in \kappa \,\overline{\text{ sm }}\, \,\overline{\text{ sm }}\, (\in \unicode{x21A7}ʻʻ\kappa ) . +\] +In the case of \(\text{Prod}ʻ\kappa\) and \({\in}_{\Delta}ʻ\kappa\), +the correlator is \(\text{D}\), <i>i.e.</i> +\[ +\vdash :\kappa \in \text{Cls}^{2} \text{excl}.\supset .\text{D}\upharpoonright {\in}_{\Delta}ʻ\kappa \in (\text{Prod}ʻ\kappa )\,\overline{\text{ sm }}\, ({\in}_{\Delta}ʻ\kappa ) . +\] +In the case of \({\in}_{\Delta}ʻsʻ\kappa\) and +\({\in}_{\Delta}ʻ{\in}_{\Delta}ʻʻ\kappa\), the correlator is +\(\dot{s}\mid \text{D}\), <i>i.e.</i> +\[ +\vdash :\kappa \in \text{Cls}^{2} \text{excl}.\supset .\dot{s} \mid \text{D}\upharpoonright {\in}_{\Delta}ʻ{\in}_{\Delta}ʻʻ\kappa \in ({\in}_{\Delta}ʻsʻ\kappa + )\,\overline{\text{ sm }}\, ({\in}_{\Delta}ʻ{\in}_{\Delta}ʻʻ\kappa ). +\] +\(\dot{s} \mid \text{D}\) also correlates \({\in}_{\Delta}ʻ\kappa\) +with \({\in}_{\Delta}ʻ\in \unicode{x21A7}ʻʻ\kappa\) (*85·61) and +\(P_{\Delta }ʻ\alpha\) with \({\in}_{\Delta}ʻP\unicode{x21A7}ʻʻ\alpha\) +(*85·53), and \(P_{\Delta }ʻsʻ\kappa\) with \({\in}_{\Delta}ʻP_{\Delta}ʻʻ\kappa\) +(*85·27·42) if \(\kappa \in \text{Cls}^{2} \text{excl}\).</p> + +<p>The correlator of \((\alpha \uparrow \beta )_{\Delta }ʻ\beta\) with +\((\alpha \,\,\text{exp}\,\,\beta)\) is \(\dot{s}\) (<a href="#*116·131">*116·131</a>).</p> + +<p>Another kind of correlators arises where we are given a correlator +of \(\kappa\) and \(\lambda\), and we wish to construct a correlator +for some associated classes \(Wʻ\kappa\) and \(Wʻ\lambda\) where we +are given correlators of \(\alpha\) with \(\gamma\) and of \(\beta\) +with \(\delta\), and we wish to construct a correlator of +\(\alpha\unicode{x2640}\beta\) with \(\gamma \unicode{x2640}\delta\), where +\(\unicode{x2640}\) is some double descriptive function in the sense +of *38. In this case, the correlator will usually be of the form +\(R\parallel \breve{S}\) (with a limited converse domain). Sometimes +\(R\) and \(S\) will be identical; sometimes \(S\) will be \(R_{\in}\). +Such correlators always depend upon</p> + +<p class="nind"><b>*55·61.</b> \(\vdash :\text{E}!Rʻx.\text{E}!Sʻy.\supset .(R\parallel \breve{S} )ʻ(x\downarrow y)=(Rʻx)\downarrow (Sʻy)\)</p> + +<p class="nind">together with the propositions *74·77 <i>seq.</i> giving cases +in which \((R\parallel \breve{S})\upharpoonright \lambda\) is a +one-one relation. It follows from *55·61 that if \(R\) and \(S\) +are correlators whose converse domains include the domain and +converse domain respectively<span class="pagenum" id="Page_186">[Pg 186]</span> of a relation \(P\), then \((R\parallel\breve{S})ʻP\) +will be a relation holding between \(Rʻx\) and \(Sʻy\) +whenever \(P\) holds between \(x\) and \(y\). Examples of such +correlators as \(R\parallel \breve{S}\) are</p> + +<p class="nind"><b>*112·153.</b> \(\vdash :T\in \kappa \,\overline{\text{ sm }}\, \,\overline{\text{ sm }}\, \lambda .\supset .(T\parallel \breve{T} _{\in })\upharpoonright + sʻ\in \unicode{x21A7}ʻʻ\lambda \in (\in \unicode{x21A7}ʻʻ\kappa )\,\overline{\text{ sm }}\, \,\overline{\text{ sm }}\, (\in \unicode{x21A7}ʻʻ\lambda )\)</p> + +<p class="nind"><b>*113·127.</b> \[\begin{align}\vdash :R\upharpoonright \gamma \in \alpha \,\overline{\text{ sm }}\, \gamma .S\upharpoonright \delta \in &\beta \,\overline{\text{ sm }}\, \delta .\supset .\\ +&(R\parallel \breve{S} )\upharpoonright (\delta \times \gamma )\in (\alpha \downarrow_{,,}ʻʻ\beta )\,\overline{\text{ sm }}\, \,\overline{\text{ sm }}\, (\gamma \downarrow_{,,}ʻʻ\delta + )\end{align}\]</p> + +<p class="nind"><b>*113·65.</b> \(\vdash .\downarrow zʻʻ\alpha \times \downarrow zʻʻ\beta =(\downarrow z\parallel \text{Cnv}ʻ\downarrow z)ʻʻ(\alpha \times \beta )\)</p> + +<p class="nind"><b>*114·51.</b> \(\vdash :T\upharpoonright sʻ\lambda \in \kappa \,\overline{\text{ sm }}\, \,\overline{\text{ sm }}\, \lambda .\supset .(T\parallel \breve{T} _{\in })\upharpoonright {\in}_{\Delta}ʻ\lambda + \in ({\in}_{\Delta}ʻ\kappa )\,\overline{\text{ sm }}\, ({\in}_{\Delta}ʻ\lambda )\)</p> + +<p class="nind"><b>*116·192.</b> \[\begin{align}\vdash :R\upharpoonright \gamma \in \alpha \,\overline{\text{ sm }}\, \gamma .S\upharpoonright \delta \in &\beta \,\overline{\text{ sm }}\, \delta .\supset .\\ +&(R\parallel \breve{S} )\upharpoonright (\delta \times \gamma )\in (\alpha \,\,\text{exp}\,\,\beta )\,\overline{\text{ sm }}\, \,\overline{\text{ sm }}\, (\gamma \,\,\text{exp}\,\,\delta ).\\ +&(R\parallel \breve{S} )_{\in }\upharpoonright (\gamma \,\,\text{exp}\,\,\delta )\in (\alpha \,\,\text{exp}\,\,\beta )\,\overline{\text{ sm }}\, (\gamma \,\,\text{exp}\,\,\delta )\end{align}\]</p> + +<p>An exceptionally simple correlator is given by</p> + +<p class="nind"><b>*115·502.</b> + \(\vdash :T\upharpoonright sʻ\lambda \in \kappa \,\overline{\text{ sm }}\, \,\overline{\text{ sm }}\, \lambda .\supset .T\upharpoonright sʻ\text{Prod}ʻ\lambda \in (\text{Prod}ʻ\kappa )\,\overline{\text{ sm }}\, \,\overline{\text{ sm }}\, (\text{Prod}ʻ\lambda )\)</p> + +<p>Another exceptionally simple case is</p> + +<p class="nind"><b>*73·63.</b> + \[\begin{align}\vdash :S\in \alpha \,\overline{\text{ sm }}\, \beta .T\upharpoonright \alpha ,T\upharpoonright \beta \in 1\rightarrow 1.\alpha \cup &\beta \subset \text{ᗡ}ʻT.\supset .\\ +&T\mid S\mid \breve{T} \in (Tʻʻ\alpha )\,\overline{\text{ sm }}\, (Tʻʻ\beta )\end{align}\]</p> + +<p>By means of the above correlators, most correlators that are required +can be calculated. Thus it will be seen that <a href="#*116·192">*116·192</a> in the above list +is an immediate consequence of <a href="#*113·127">*113·127</a> and <a href="#*115·502">*115·502</a>, since +\[ +\alpha \,\,\text{exp}\,\,\beta =\text{Prod}ʻ\alpha \downarrow_{,,}ʻʻ\beta\,\text{and}\,sʻ\text{Prod}ʻ\gamma \downarrow_{,,}ʻʻ\delta =\delta \times \gamma . +\]</p> + +<p>In order to develop the subject, it is almost always necessary, +not merely to prove that two classes are similar, but actually to +construct a correlator of the two classes. This applies equally to +relation-arithmetic, in which analogous correlators are used to prove +ordinal similarity.</p> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_187">[Pg 187]</span></p> +<h2 class="nobreak" id="SECTION_C">SECTION C.<br> +<span class="allsmcap">FINITE AND INFINITE.</span></h2> +</div> + + +<p><i>Summary of Section C.</i></p> + +<p>The distinction of finite and infinite is not required, as appears from +Section B, for the definition of the arithmetical operations or for the +proof of their formal laws. There are, however, many important respects +in which finite cardinals and classes differ respectively from infinite +cardinals and classes, and these differences must now be investigated.</p> + +<p>There are two different ways in which we may define the finite and the +infinite, and these two ways cannot (so far as is known at present) +be shown to be equivalent except by assuming the multiplicative +axiom. As there seems no good reason for regarding one of these ways +as giving more exactly than the other what is usually meant by the +words "finite" and "infinite," we shall, to avoid confusion, give +other names than these to each of the two ways of dividing classes and +cardinals. The division effected by the first method of definition we +shall call the division into <i>inductive</i> and <i>non-inductive</i>; +that effected by the second method we shall call the division into +<i>non-reflexive</i> and <i>reflexive</i>.</p> + +<p>The division into inductive and non-inductive, which is treated in +*120, is defined as follows. An inductive cardinal is one which can +be reached from 0 by successive additions of 1; that is, an inductive +cardinal is one which has to 0 the relation \((+_{c}1)_{*}\), where (by +*38·02) \(+_{c}1\) is the relation of \(\alpha +_{c}1\) to \(\alpha\), and the +subscript asterisk has the meaning defined in *90. Hence we put +\[ +\text{NC induct}=\hat{\alpha}\{\alpha (+_{c}1)_{*}0\} \quad\text{Df}. +\] +By applying the definition of *90, this gives +\[ +\vdash \colon\colon \alpha _{\in }\text{NC induct}.\equiv \colon\ldotp \xi \in \mu .\supset _{\xi }.\xi +_{c}1\in \mu :0\in \mu :\supset _{\mu }.\alpha \in \mu . +\] +This proposition may be regarded as stating that an inductive cardinal +is one which obeys mathematical induction starting from 0, <i>i.e.</i> +it is one which possesses every property possessed by 0 and by the +numbers obtained by adding 1 to numbers possessing the property. +In elementary mathematics, it is customary to regard mathematical +induction, as applied to the series of natural numbers, as a principle +rather than a definition, but according to<span class="pagenum" id="Page_188">[Pg 188]</span> the above procedure it +becomes a definition rather than a principle. This procedure is +unavoidable as soon as it is perceived that there are cardinals which +do not obey mathematical induction starting from 0. (This only holds +on the assumption that the total number of objects in any one type is +not one of the inductive cardinals. This assumption, in a slightly +different form, is introduced below as the "axiom of infinity.") +Thus for example \(0\neq 1\), and \(\xi \neq \xi +_{c}1.\supset .\xi+_{c}1\neq \xi +_{c}2\). +Hence if \(\alpha\) is any inductive cardinal, +\(\alpha \neq \alpha +_{c}1\). But we know that \(\aleph _{0}\), +the first of Cantor's transfinite cardinals<a id="FNanchor_6" href="#Footnote_6" class="fnanchor">[6]</a>, satisfies +\(\aleph_{0}=\aleph _{0}+_{c}1\). Thus mathematical induction starting from 0 +cannot be validly applied to prove properties of \(\aleph _{0}\). It +follows that the inductive cardinals as above defined are only some +among cardinals; nor does it appear that there is any way of defining +them except as those that obey mathematical induction starting from 0. +It follows that mathematical induction is not a principle, to be either +proved or assumed as an axiom, but is merely a characteristic defining +a certain class of cardinals, namely the class of inductive cardinals.</p> + +<p>By a syllogism in Barbara, it is evident that 0 is an inductive +cardinal; hence by the definition 1 is an inductive cardinal, and +hence 2, 3, ... are inductive cardinals. Thus any given cardinal +in the series of natural numbers can be shown to be an inductive +cardinal. The usual elementary properties of inductive cardinals, such +as the uniqueness of subtraction and division, are easily proved by +mathematical induction.</p> + +<p>We define an inductive class as a class the number of whose terms is +an inductive cardinal. More simply, we put +\[ +\text{Cls induct}=sʻ\text{NC induct} \quad\text{Df}. +\] +It is then easily shown that an inductive class is one which can be +reached from \(\Lambda\) by successive additions of single members. +That is, if we put +\[ +\begin{aligned} +M=\hat{\eta} \hat{\zeta} \{(\exists y).\zeta =\eta \cup \iota ʻy\},\\ +\text{then}\quad \text{Cls induct}=\overleftarrow{M}_{*}ʻ\Lambda . +\end{aligned} +\]</p> + +<p>Thus we have +\[ +\vdash \colon\colon \rho \in \text{Cls induct}.\equiv \colon\ldotp \eta \in \mu .\supset _{\eta ,y}.\eta \cup \iota ʻy\in \mu :\Lambda \in \mu :\supset _{\mu }.\rho \in \mu . +\] +We might equally well have begun by defining inductive classes, and +proceeded to define inductive cardinals as the cardinals of inductive +classes; in that case, we should have used the above relation \(M\) to +define inductive classes.</p> + +<p>Some of the properties which we expect inductive cardinals to possess, +such for example as \(\alpha \neq \alpha +_{c}1\), can only be proved +by assuming that no inductive cardinal is null, <i>i.e.</i> that +\[ +\alpha \in \text{NC induct}.\supset _{\alpha }.\exists !\alpha . +\] +This amounts to the assumption that, in any fixed type, a class +can be found<span class="pagenum" id="Page_189">[Pg 189]</span> having any assigned inductive number of terms. If +this were false, there would have to be some definite member of the +series of natural numbers which gave the total number of objects of +the type in question. Thus suppose there were exactly n individuals +in the universe, and no more, where n is an inductive cardinal. We +should then have \(2^{n}\) classes, \(2^{2^{n}}\) classes of classes, +and so on. In that case, in the type of individuals we should have +\(n+_{c}1=\Lambda\), \(n+_{c}2=\Lambda\), etc. Hence we should have +\[ +n+_{c}1=(n+_{c}1)+_{c}1,\,\text{etc}. +\] +In the type of classes, we should get similar results for \(2^{n}\), and +so on. It is plain (though not demonstrable except in each particular +case) that if the assumption \(\alpha \in \text{NC induct}.\supset_{\alpha }.\exists !\alpha\) +fails in any one type, it fails in any other type in the same +hierarchy, and if it holds in any one, it holds in any other; for if +n be the total number of individuals, then if \(n\) is an inductive +cardinal, the total number of any other type is an inductive cardinal, +while if \(n\) is not an inductive cardinal, no more is the total +number of any other type. Hence the assumption \(\alpha \in \text{NC induct}.\supset _{\alpha }.\exists !\alpha\) +is either true in any type or false in any type in one hierarchy. We +shall call it the "axiom of infinity," putting +\[ +\text{Infin ax}.=:\alpha \in \text{NC induct}.\supset _{\alpha }.\exists !\alpha \quad\text{Df}. +\] +This assumption, like the multiplicative axiom, will be adduced as +a hypothesis whenever it is relevant. It seems plain that there is +nothing in logic to necessitate its truth or falsehood, and that it can +only be legitimately believed or disbelieved on empirical grounds. When +we wish to use a typically definite form of the axiom, we shall employ +the definition +\[ +\text{Infin ax}(x).=:\alpha \in \text{NC induct}.\supset _{\alpha }.\exists !\alpha (x) \quad\text{Df} +\] +which asserts that, if \(\alpha\) is any inductive cardinal, there are +at least \(\alpha\) terms of the same type as \(x\).</p> + +<p>It is important to observe that, although the axiom of infinity cannot +(so far as appears) be proved a priori, we can prove that any given +inductive cardinal exists in a sufficiently high type. For if the total +number of individuals be \(n\), the numbers of objects in succeeding +types are \(2^{n}\), \(2^{2^{n}}\), etc., and these numbers grow beyond +any assigned inductive cardinal. Owing, however, to the fact that we +cannot add together an infinite number of classes whose types increase +without limit, we cannot hence show that there is a type in which +every inductive cardinal exists, though we can show of every inductive +cardinal that there is a type in which it exists. <i>i.e.</i> if +\(\alpha\) is any inductive cardinal, there must be a type for \(x\) +such that \(\exists !\alpha (x)\) is true; but there need not be a type +for \(x\) such that if \(\alpha\) is any inductive cardinal, \(\exists!\alpha (x)\) +is true.</p> + +<p>The axiom of infinity suffices to prove the existence, in appropriate +types, of \(\aleph _{0}\), \(2^{\aleph _{0}}\), \(2^{2^{{}\aleph_{0}}}\), +... \(\aleph _{1}\), \(\aleph _{2}\), ...<a id="FNanchor_7" href="#Footnote_7" class="fnanchor">[7]</a>. It does not +<span class="pagenum" id="Page_190">[Pg 190]</span>suffice, so far as we know, to prove the existence of \(\aleph_{\omega}\) +or any Aleph with a greater suffix than \(\omega\), because +the existences of \(\aleph _{1}\), \(\aleph _{2}\), ... are proved in +successively rising types, and no meaning can be found for a type whose +order is infinite.</p> + +<p>The other definition of finite and infinite is of less importance +in practice than the definition by induction. It is dealt with in +<a href="#*124">*124</a>. According to this definition, we call a class reflexive when it +contains a proper part similar to itself, <i>i.e.</i> we put +\[ +\text{Cls refl}=\hat{\alpha} \{(\exists R).R\in 1\rightarrow 1.\text{D}ʻR=\alpha .\text{ᗡ}ʻR\subset \alpha .\text{ᗡ}ʻR\neq \alpha\} \quad\text{Df}, +\] +or, what comes to the same thing, +\[ +\text{Cls refl}=\hat{\alpha}\{(\exists R).R\in 1\rightarrow 1.\text{ᗡ}ʻR\subset \text{D}ʻR.\exists !\overrightarrow{B}ʻR.\alpha =\text{D}ʻR\} \quad\text{Df}. +\] +We call a cardinal reflexive when it is the homogeneous cardinal of a +reflexive class, <i>i.e.</i> we put +\[ +\text{NC refl} = N_{0}ʻʻ\text{Cls refl} \quad\text{Df}. +\] +It is easy to show that +\[ +\text{NC refl} =\hat{\alpha} \{\exists !\alpha .\alpha =\alpha +_{c}1\}. +\] +We find that inductive classes and cardinals are non-reflexive, and +reflexive classes and cardinals are non-inductive. We find also that +reflexive cardinals are those that are equal to or greater than +\(\aleph _{0}\), while inductive cardinals are those that are less than +\(\aleph _{0}\). By assuming the multiplicative axiom, we can show +that every cardinal is equal to, greater than, or less than \(\aleph_{0}\), +whence it follows that every cardinal is either reflexive or +inductive, thus identifying the two definitions of finite and infinite. +But so long as we refrain from assuming either the multiplicative axiom +or some special axiom <i>ad hoc</i>, it remains possible (so far as is +known at present) that there may be cardinals neither greater than, +nor equal to, nor less than \(\aleph _{0}\). Such cardinals, if they +exist, are neither inductive nor reflexive: they are infinite if we +define infinity by the negation of induction, but finite if we define +infinity by reflexiveness. It is possible that further investigation +may either prove or disprove the existence of such cardinals; for the +present, their existence must remain an open question, except for those +who regard the multiplicative axiom as a self-evident truth.</p> + +<p>In <a href="#*121">*121</a> we shall consider <i>intervals</i> in a discrete series; +<i>i.e.</i> in a series generated by a one-one relation between +consecutive terms. If \(P\) be the generating relation of such a +series, and \(x\) and \(y\) be two members of the series, of which +\(y\) is the later, the terms which lie between \(x\) and \(y\) are the +terms \(z\) for which we have +\[ +xP_{\text{po}}z.zP_{\text{po}}y, +\] +where \(P_{\text{po}}\) has the meaning defined in *91. Hence we put +\[ +P(x-y)=\overleftarrow{P}_{\text{po}}ʻx\cap \overrightarrow{P}_{\text{po}}ʻy \quad\text{Df}, +\] +where "\(P(x-y)\)" means "the \(P\)-interval between \(x\) and \(y\)." +We want<span class="pagenum" id="Page_191">[Pg 191]</span> also symbols for the interval together with one or both of its +end-points. For these we put +\[ +\begin{aligned} +P(x\unicode{x22A3} y)&=\overleftarrow{P}_{\text{po}}ʻx\cap \overrightarrow{P}_{*}ʻy &\quad\text{Df},\\ +P(x\unicode{x27dd}y)&=\overleftarrow{P}_{*}ʻx\cap \overrightarrow{P}_{\text{po}}ʻy &\quad\text{Df},\\ +P(x\vdash\dashv y)&=\overleftarrow{P}_{*}ʻx\cap \overrightarrow{P}_{*}ʻy &\quad\text{Df}. +\end{aligned} +\] +<a id="FNanchor_8" href="#Footnote_8" class="fnanchor">[8]</a> +Thus, for example, if \(x\) and \(y\) be inductive cardinals, and +\(P\) be the relation of \(n\) to \(n+_{c}1\), and \(x < y\), +\(P(x-y)\) will be the numbers greater than \(x\) and less than \(y\), +while \(P(x\unicode{x27dd} y)\) will be these numbers together with \(y\), +\(P(x\unicode{x22A2}y)\) will be these numbers together with \(x\), and +\(P(x\vdash\dashv y)\) will be these numbers together with both \(x\) and +\(y\). By means of intervals, we define a class of relations +\(P_{\nu}\) (where \(\nu\) is any inductive cardinal), where "\(xP_{\nu }z\)" +means that we can pass from \(x\) to \(z\) in \(\nu\) steps. In order +to fit the case in which \(x\) and \(z\) are identical, and to insure +that no relation such as \(P_{\nu}\) shall hold between terms which do +not both belong to the field of \(P\), we put +\[ +P_{\nu }=\hat{x} \hat{y} \{\text{Nc}ʻP(x\vdash\dashv y)=\nu +_{c}1\} \quad\text{Df}. +\] +Then, provided \(P_{\text{po}}\,\unicode{x2abd}\, J\), \(P_{0}=I\upharpoonright CʻP\), +and if further \(P\in 1\rightarrow 1\), then \(P_{1}= P\), +\(P_{2}=P^{2}\), etc. If \(P\) is a transitive serial relation, \(P_{1}\) +is the relation "immediately preceding," which has great importance +in well-ordered series. In this case, \(P_{1}=P\dot{-} P^{2}\). If +\(P\) is a transitive serial relation generating a finite series or +a progression or a series of the type of the negative and positive +integers in order of magnitude, we have +\[ +P=(P_{1})_{\text{po}}. +\]</p> + +<p>In <a href="#*121">*121</a> we shall only consider \(P_{\nu}\) in the case where +\[ +P\in (1\rightarrow \text{Cls})\cup (\text{Cls}\rightarrow 1), +\] +and generally we shall have the further hypothesis +\(P_{\text{po}}\,\unicode{x2abd}\, J\). We can then prove that the interval +between \(x\) and \(y\) is always an inductive class (it will be null +unless \(xP_{*}y\)); this proposition is useful in its application to +the number-series and to progressions generally.</p> + +<p>When \(P\in (1\rightarrow \text{Cls})\cup (\text{Cls}\rightarrow 1).P_{\text{po}}\,\unicode{x2abd}\, J\), +<span class="pagenum" id="Page_192">[Pg 192]</span>the class of such relations as \(P_{\nu }\) (where \(\nu\) is an +inductive cardinal) is identical with \(\text{Potid}ʻP\), the class +of powers of \(P\) (cf. *91 <i>seq.</i>). This identification (which +does not hold in general without the above hypothesis) leads to many +useful propositions. In *91 <i>seq.</i>, we treated powers of a +relation without the use of numbers, <i>i.e.</i> without defining the +\(\nu\)th power of \(P\). When the powers of \(P\) are the class of +such relations as \(P_{\nu }\), we can of course take \(P_{\nu }\) as +the \(\nu\)th power of \(P\). The general definition of the \(\nu\)th +power of \(P\) (where \(\nu\) is an inductive cardinal) will be given +later, in *301; we shall denote it by \(P^\nu\), thereby including the +notation \(P^{2}\) already defined.</p> + +<p>In <a href="#*122">*122</a> we shall deal with progressions, <i>i.e.</i> with series of the +type of the series of natural numbers. In this number, we shall deal +with such series as generated by one-one relations; they will be dealt +with at a later stage (*263) as generated by transitive relations. +We define a progression as a one-one relation whose domain is the +posterity of its first term, <i>i.e.</i> +\[ +\text{Prog} = (1 \rightarrow 1) \cap \hat{R} (\text{D}ʻR = \overleftarrow{R}_{*}ʻBʻR) \quad\text{Df}. +\] +According to this definition, there must be a first term \(BʻR\); +\(\text{ᗡ}ʻR\) will be \(\breve{R} ʻʻ\overleftarrow{R}_{*}ʻBʻR\), +<i>i.e.</i> \(\overleftarrow{R}_{\text{po}}ʻBʻR\), which is contained +in \(\overleftarrow{R}_{*}ʻBʻR\), <i>i.e.</i> in \(\text{D}ʻR\); +since \(\text{ᗡ}ʻR \subset \text{D}ʻR\), every term of the field of +\(R\) has a successor, so that there is no end to the series; since +\(CʻR = \text{D}ʻR = \overleftarrow{R}_{*}ʻBʻR\), every term of the +series can be reached from the beginning by successive steps. These +characteristics suffice to define progressions.</p> + +<p>In <a href="#*123">*123</a> we proceed to the definition and discussion of \(\aleph _{0}\), +the smallest of reflexive cardinals. This is the cardinal number of any +class whose terms can be arranged in a progression; hence it is the +class of domains of progressions, <i>i.e.</i> we may put +\[ +\aleph _{0} = \text{D}ʻʻ\text{Prog} \quad\text{Df}. +\] +With this definition, remembering that \(\Lambda\) is a cardinal, +we can prove that \(\aleph _{0}\) is a cardinal; but to prove that +\(\aleph _{0}\) is an <i>existent</i> cardinal, we need the axiom of +infinity. The existence-theorem for \(\aleph _{0}\) is then derived +from the inductive cardinals, which, if no one of them is null, form a +progression when arranged in order of magnitude. It should be observed +that this existence-theorem is for a higher type than that for which +the axiom of infinity is assumed. In order to get an existence-theorem +for the same type, we need the multiplicative axiom as well.</p> + +<p>After a number on reflexive classes and cardinals (<a href="#*124">*124</a>) and a number +on the axiom of infinity (<a href="#*125">*125</a>), the Section ends with a number (<a href="#*126">*126</a>) +on "typically indefinite inductive cardinals." The constant inductive +cardinals are the typically ambiguous symbols 0, 1, 2, ...; thus we +want to define the class of inductive cardinals in such a way that a +variable member of the class shall be typically ambiguous. This is not +possible without a sacrifice of rigour, but in *126 it is shown how +to minimize the sacrifice of rigour, and how to obviate the resulting +logical dangers. A variable whose values are typically ambiguous is +said to be "typically indefinite."</p> + +<p>A proof that all inductive cardinals exist has often been derived from +<a href="#*120·57">*120·57</a> (below). But according to the doctrine of types, this proof is +invalid, since "\(\mu +_{0} 1\)" in *120·57 is necessarily of higher +type than "\(\mu\)."</p> + + +<div class="footnotes"><h3>FOOTNOTES:</h3> + +<div class="footnote"> + +<p class="nind"> +<a id="Footnote_6" href="#FNanchor_6" class="label">[6]</a> +For the definition of \(\aleph _{0}\), cf. <a href="#*123·01">*123·01</a> and <a href="#Page_192">p.192</a> +of this summary.</p> + +</div> + +<div class="footnote"> + +<p class="nind"> +<a id="Footnote_7" href="#FNanchor_7" class="label">[7]</a> +For the definitions of \(\aleph _{1}\), \(\aleph _{2}\), +etc., see *265.</p> + +</div> + +<div class="footnote"> + +<p class="nind"> +<a id="Footnote_8" href="#FNanchor_8" class="label">[8]</a> +These symbols are suggested by those given in Peano's +<i>Formulaire</i>, Vol. <span class="allsmcap">IV</span>. p. 116. (<i>Algèbre</i>, § 46.)</p> + +</div> +</div> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_193">[Pg 193]</span></p> +<h2 class="nobreak" id="*118"> +*118. ARITHMETICAL SUBSTITUTION AND UNIFORM +FORMAL NUMBERS.</h2> +</div> + + +<p class="nind"><i>Summary of</i> *118.</p> + +<p>A difficulty arises respecting substitution in arithmetic. For +if \(\mu\) is a formal number and its occurrence in \(f\mu\) is +arithmetical, then by \(\text{IIT}\) \(\mu\) is always to be taken +in an existential type. Hence we can only substitute a real variable +\(\xi\) for \(\mu\) under the hypothesis \(\exists !\xi\), and we can +only substitute another formal number \(\sigma\) for \(\mu\) provided +that the equation \(\mu = \sigma\), which justifies the substitution, +is arithmetical, <i>i.e.</i> provided that in this equation the type of +\(\mu\) is such that \(\exists !\mu\).</p> + +<p>The result is that the application of *20·18 is apt to lead to +fallacies owing to the different meanings which a formal number may +possess in different occurrences. Hitherto we have considered each case +in detail, <i>e.g.</i> note on <a href="#*110·61">*110·61</a>, and proof of <a href="#*110·56">*110·56</a>.</p> + +<p>The condition for the safe application of *20·18 is given in *118·01, +namely</p> + +<p class="nind"><b>*118·01.</b> \(\vdash \colon\ldotp \exists !\mu .\mu =\sigma .\supset :f\mu .\equiv .f\sigma \quad[*20·18]\)</p> + +<p>This question is more fully discussed in the prefatory statement of +this volume. The first reference to *118·01 is in <a href="#*120·222">*120·222</a>. Another +way of evading the difficulty is to work with formal numbers which, +together with all their components, are of the same type. This leads to +the consideration of Uniform Formal Numbers, which with the exception +of *118·01 occupies the rest of the number.</p> + +<p>The <i>dominant type</i> of a formal number as used in any context +is the type of the formal number itself in that context, and the +<i>subordinate types</i> of the formal number are the dominant types of +its component formal numbers.</p> + +<p>When the dominant types of some of the formal numbers are not expressly +indicated by an explicit notation (cf. *65), the rules according to +which the dominant types thus left ambiguous are to be related, so far +as they are related, including the rules governing the relation of +subordinate types, if left ambiguous, to dominant types, are given by +conventions \(\text{IT}\), \(\text{IIT}\), and \(\text{AT}\) of the +prefatory statement in this volume.</p> + +<p>We have now to consider an important special case which arises when +types are explicitly indicated by the use of *65·01·03. A formal +number,<span class="pagenum" id="Page_194">[Pg 194]</span> whose subordinate types are the same as its dominant type, is +called <i>uniform</i>; and if some of its subordinate types are the +same as its dominant type, it is called <i>partially uniform</i>. A +formal number can only be partially uniform, or at least so designated +as to be necessarily partially uniform, when the dominant type and +those subordinate types identical with it are expressly indicated by +*65·01·03. For otherwise the conventions \(\text{IT}\), \(\text{IIT}\), +and perhaps also \(\text{AT}\), apply; and these do not secure +uniformity, and may perhaps in some contexts be inconsistent with it.</p> + +<p>Common sense in its consideration of arithmetic habitually disregards +the possibility of a formal number representing \(\Lambda\). In other +words, it always applies conventions \(\text{IIT}\) and \(\text{AT}\). +But also, owing to its disregard of types, it assumes that the formal +numbers are all uniform. The assumption which is really essential to +this common sense reasoning, so far as the form of its arithmetical +conclusions are concerned, is the assumption that none of the numerical +symbols represent \(\Lambda\). This assumption is secured here, when +no types are expressly indicated, by \(\text{IIT}\) and \(\text{AT}\). +We have now to consider the effect on arithmetical operations of the +other assumption, that the formal numbers are uniform, or partially +uniform. There is no difficulty arising from any change of convention +for symbolism, since, as stated above, partial or complete uniformity +is secured by express indication of type. Accordingly conventions +\(\text{IT}\), \(\text{IIT}\) continue, as always, to apply when the +types of formal numbers are left ambiguous.</p> + +<p>Convention \(\text{AT}\) will not be applied either in <a href="#*118">*118</a> or <a href="#*119">*119</a> or +<a href="#*120">*120</a>: in *118 the fact is entirely unimportant since the dominant types +of equational occurrences are always indicated, so that no case arises +when it could apply.</p> + +<p>Apart from its intrinsic interest and its bearing on substitution, +the arithmetic of uniform formal numbers is necessary for *120, where +the fundamental arithmetical properties of inductive numbers are +investigated.</p> + +<p>The propositions of this number are proved by the use of the results of +<a href="#*117">*117</a>. The basis of the reasoning is</p> + +<p class="nind"><b>*118·13.</b> \(\vdash \colon\ldotp \mu \leq \nu .\supset :\exists !\text{ sm }_{\xi }ʻʻ\nu .\supset .\exists !\text{ sm }_{\xi }ʻʻ\mu\)</p> + +<p>In *118·2·3·4 the meaning of the symbolism for dominant types is +stated, namely</p> + +<p class="nind"><b>*118·2.</b> \(\vdash .(\mu +_{c}\nu )_{\xi }=\hat{\eta} \{(\exists \alpha ,\beta ).\mu =\text{N}_{0}\text{c}ʻ\alpha .\nu =\text{N}_{0}\text{c}ʻ\beta + .\eta \text{ sm }_{\xi }(\alpha +\beta )\}\)</p> + +<p class="nind"><b>*118·3.</b> \(\vdash .(\mu \times _{c}\nu )_{\xi }=\hat{\eta} \{(\exists \alpha ,\beta ).\mu =\text{N}_{0}\text{c}ʻ\alpha .\nu =\text{N}_{0}\text{c}ʻ\beta + .\eta \text{ sm }_{\xi }(\alpha \times \beta )\}\)</p> + +<p class="nind"><b>*118·4.</b> \(\vdash .(\mu ^\nu )_{\xi }=\hat{\eta} \{(\exists \alpha ,\beta ).\mu =\text{N}_{0}\text{c}ʻ\alpha .\nu =\text{N}_{0}\text{c}ʻ\beta + .\eta \text{ sm }_{\xi }(\alpha \,\,\text{exp}\,\,\beta )\}\)</p> + +<p>The important propositions which are finally reached for addition are</p> + +<p class="nind"><b>*118·23.</b> \(\vdash :\mu ,\nu \in \text{NC}.\supset .(\mu +_{c}\nu )_{\xi }=(\text{ sm }_{\xi }ʻʻ\mu +_{c}\text{ sm }_{\xi }ʻʻ\nu)_{\xi }\)</p> + +<p><span class="pagenum" id="Page_195">[Pg 195]</span></p> + +<p class="nind"><b>*118·24.</b> \(\vdash :\nu \in \text{NC}.\supset .(\mu +_{c}\nu )_{\xi }=(\mu +_{c}\text{ sm }_{\xi }ʻʻ\nu )_{\xi }\)</p> + +<p class="nind"><b>*118·241.</b> \(\vdash :\mu \in \text{NC}.\supset .(\mu +_{c}\nu )_{\xi }=(\text{ sm }_{\xi }ʻʻ\mu +_{c}\nu )_{\xi }\)</p> + +<p class="nind"><b>*118·25.</b> \(\vdash .(\mu +_{c}\nu +_{c}\varpi )_{\xi }=\{(\mu +_{c}\nu )_{\xi }+_{c}\varpi\}_{\xi }=\{\mu +_{c}(\nu +_{c}\varpi + )_{\xi }\}_{\xi }\)</p> + +<p>The important propositions for multiplication are</p> + +<p class="nind"><b>*118·33.</b> \(\vdash :\mu ,\nu \in \text{NC}-\iota ʻ0.\supset .(\mu \times _{c}\nu )_{\xi }=(\text{ sm }_{\xi }ʻʻ\mu \times _{c}\text{ sm }_{\xi }ʻʻ\nu + )_{\xi }\)</p> + +<p class="nind"><b>*118·34.</b> \(\vdash :\nu \in \text{NC}.\mu \neq 0.\supset .(\mu \times _{c}\nu )_{\xi }=(\mu \times _{c}\text{ sm }_{\xi }ʻʻ\nu )_{\xi }\)</p> + +<p class="nind"><b>*118·341.</b> \(\vdash :\mu \in \text{NC}.\nu \neq 0.\supset .(\mu \times _{c}\nu )_{\xi }=(\text{ sm }_{\xi }ʻʻ\mu \times _{c}\nu )_{\xi }\)</p> + +<p class="nind"><b>*118·35.</b> \(\vdash :\varpi \neq 0.\supset .(\mu \times _{c}\nu \times _{c}\varpi )_{\xi }=\{(\mu \times _{c}\nu )_{\xi }\times + _{c}\varpi\}_{\xi }\)</p> + +<p class="nind"><b>*118·351.</b> \(\vdash :\mu \neq 0.\supset .(\mu \times _{c}\nu \times _{c}\varpi )_{\xi }=\{\mu \times _{c}(\nu \times _{c}\varpi + )_{\xi }\}_{\xi}\)</p> + +<p>The important propositions for exponentiation are</p> + +<p class="nind"><b>*118·43.</b> \(\vdash :\mu ,\nu \in \text{NC}-\iota ʻ0.\mu \neq 1.\supset .(\mu ^\nu )_{\xi }=\{(\text{ sm }_{\xi }ʻʻ\mu )^{\text{ sm }_{\xi }ʻʻ\nu + }\}_{\xi}\)</p> + +<p class="nind"><b>*118·44.</b> \(\vdash :\nu \in \text{NC}.\mu \neq 0.\mu \neq 1.\supset .(\mu ^\nu )_{\xi }=(\mu ^{\text{ sm }_{\xi }ʻʻ\nu })_{\xi }\)</p> + +<p class="nind"><b>*118·441.</b> \(\vdash :\mu \in \text{NC}.\nu \neq 0.\supset .(\mu ^\nu )_{\xi }=\{(\text{ sm }_{\xi }ʻʻ\mu )^\nu \}_{\xi}\)</p> + +<p class="nind"><b>*118·45.</b> \(\vdash :\mu \neq 0.\mu \neq 1.\supset .(\mu ^{\nu \times _{c}\varpi })_{\xi }=\{\mu ^{(\nu \times _{c}\varpi + )_{\xi }}\}_{\xi}\)</p> + +<p class="nind"><b>*118·451.</b> \(\vdash :\varpi \neq 0.\supset .(\mu ^{\nu \times _{c}\varpi })_{\xi }=[\{(\mu ^\nu )_{\xi }\}^{\varpi}]_{\xi }\)</p> + +<p class="nind"><b>*118·46.</b> \(\vdash :\mu \neq 0.\mu \neq 1.\supset .(\mu ^{\nu +_{c}\varpi })_{\xi }=\{\mu ^{(\nu +_{c}\varpi )_{\xi }}\}_{\xi }\)</p> + +<p class="nind"><b>*118·461.</b> \(\vdash .(\mu ^{\nu +_{c}\varpi })_{\xi }=\{(\mu ^\nu )_{\xi }\times _{c}(\mu + ^{\varpi })_{\xi }\}_{\xi }\)</p> + +<p>with two analogous propositions *118·462·463,</p> + +<p class="nind"><b>*118·47.</b> \(\vdash :\varpi \neq 0.\supset .{(\mu \times _{c}\nu )^{\varpi }}_{\xi }=[\{(\mu \times _{c}\nu + )_{\xi }\}^{\varpi }]_{\xi }\)</p> + +<p class="nind"><b>*118·471.</b> \[\begin{align}\vdash \colon\ldotp \mu \neq 0.\nu \neq 0.\lor.\varpi =0.\lor.{\sim}&(\mu ,\nu ,\varpi \in \text{N}_{0}\text{C}):\supset .\\ +&\{(\mu \times _{c}\nu )^{\varpi }\}_{\xi }=\{(\mu ^{\varpi })_{\xi }\times _{c}(\nu + ^{\varpi })_{\xi }\}_{\xi }\end{align}\]</p> + +<p class="nind">with two analogous propositions <a href="#*118·472">*118·472</a><a href="#*118·473">·473</a>.</p> + +<p>It is thus seen that, apart from some exceptional cases connected with +0 and 1, in all arithmetical operations uniform, or partially uniform, +formal numbers can replace those constructed in obedience to convention +\(\text{IIT}\).</p> + +<hr class="tb"> + +<p class="nind"><b><a id="*118·01">*118·01</a>.</b> \(\vdash \colon\ldotp \exists !\mu .\mu =\sigma .\supset :f\mu .\equiv .f\sigma \quad[*20·18]\)</p> + +<p><span class="pagenum" id="Page_196">[Pg 196]</span></p> + +<p>As far as the symbolism is concerned, this proposition with the +omission of \(\exists !\mu\) from the hypothesis is a transcript +of *20·18. But if \(\mu\) or \(\sigma\) (not excluding both) is a +formal number, \(\exists !\mu\) is required in case the occurrence of +\(\mu\) in \(f\mu\) is arithmetical. In fact this proposition embodies +the three fundamental propositions of the Principle of Arithmetical +Substitution arrived at in the Prefatory Explanations on Types. Its +necessity arises from the convention \(\text{IIT}\) which is explained +there.</p> + +<p class="nind"><b>*118·11.</b> \(\vdash :\exists !\text{Nc}(\xi )ʻ\beta .\alpha \subset \beta .\supset .\exists !\text{Nc}(\xi )ʻ\alpha\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*100·31.&\supset \vdash \colon\ldotp \text{Hp}.\supset :\\ +\gamma \in \text{Nc}(\xi )ʻ\beta .&\supset .\gamma \text{ sm }_{\xi }\beta .\\ +[*73·1] &\supset .(\exists R).R\in 1(\xi )\rightarrow 1.\gamma =\text{D}ʻR.\beta =\text{ᗡ}ʻR.\\ +[*22·55] &\supset .(\exists R).R\in 1(\xi )\rightarrow 1.\alpha \subset \text{ᗡ}ʻRʻʻ\alpha =Rʻʻ\alpha .\\ +[*73·12] &\supset .(\exists R).Rʻʻ\alpha \text{ sm }_{\xi }\alpha .\\ +[*100·31] &\supset .\exists !\text{Nc}(\xi )ʻ\alpha :\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*118·12.</b> \[\begin{align}&\vdash \colon\ldotp \text{Nc}ʻ\alpha \leq \text{Nc}ʻ\beta .\supset :\exists !\text{Nc}(\xi )ʻ\beta .\supset .\exists !\text{Nc}(\xi )ʻ\alpha \\ +&[*117·32·107.*100·511]\end{align}\]</p> + +<p class="nind"><b>*118·13.</b> \(\vdash \colon\ldotp \mu \leq \nu .\supset :\exists !\text{ sm }_{\xi }ʻʻ\nu .\supset .\exists !\text{ sm }_{\xi }ʻʻ\mu \quad[*117·32]\)</p> + +<p class="nind"><b>*118·2.</b> \[\begin{align}&\vdash .(\mu +_{c}\nu )_{\xi }=\hat{\eta} \{(\exists \alpha ,\beta ).\mu =\text{N}_{0}\text{c}ʻ\alpha .\nu =\text{N}_{0}\text{c}ʻ\beta + .\eta \text{ sm }_{\xi }(\alpha +\beta )\}\\ +&[(*65·01·03).*110·2]\end{align}\]</p> + +<p class="nind"><b>*118·201.</b> \[\begin{align}&\vdash :\exists !(\mu +_{c}\nu ).\supset .\text{ sm }_{\xi }ʻʻ(\mu +_{c}\nu )=(\mu +_{c}\nu )_{\xi }\\ +&[\text{*110·44. Note Erratum in enunciation}]\end{align}\]</p> + +<p class="nind"><b>*118·21.</b> \(\vdash :\exists !(\mu +_{c}\nu )_{\xi }.\supset .\exists !\text{ sm }_{\xi }ʻʻ\mu .\exists !\text{ sm }_{\xi }ʻʻ\nu\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*110·4.*118·2.\supset \vdash :\text{Hp}.&\supset .\mu ,\nu \in \text{N}_{0}\text{C}.\\ +[*117·6] &\supset .\mu +_{c}\nu \geq \mu .\mu +_{c}\nu \geq \nu .\\ +[*118·13·201.(\text{IIT})] &\supset .\exists !\text{ sm }_{\xi }ʻʻ\mu .\exists !\text{ sm }_{\xi }ʻʻ\nu :\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p>Here the reference \(\text{IIT}\) is to the convention \(\text{IIT}\) +explained in the prefatory statement.</p> + +<p class="nind"><b>*118·22.</b> \[\begin{align}\vdash \colon\ldotp \mu ,\nu \in \text{NC}.\supset :&\exists !(\mu +_{c}\nu )_{\xi }.\equiv .\exists !(\text{ sm }_{\xi }ʻʻ\mu + +_{c}\text{ sm }_{\xi }ʻʻ\nu )_{\xi }.\equiv .\\ +&\exists !(\mu +_{c}\text{ sm }_{\xi }ʻʻ\nu )_{\xi }.\equiv .\exists !(\text{ sm }_{\xi }ʻʻ\mu +_{c}\nu )_{\xi }\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*118·21.\supset \vdash \colon\ldotp \text{Hp}.\supset :\exists !(\mu +_{c}\nu )_{\xi }.&\equiv .\exists !(\mu +_{c}\nu )_{\xi }.\exists !\text{ sm }_{\xi }ʻʻ\mu + .\exists !\text{ sm }_{\xi }ʻʻ\nu .\\ +[*110·25·4] &\equiv .\exists !(\text{ sm }_{\xi }ʻʻ\mu +_{c}\text{ sm }_{\xi }ʻʻ\nu )_{\xi } &\qquad \text{(1)}\\ +\vdash .*118·21.*103·43.*110·4.\supset \\ +\vdash \colon\ldotp \text{Hp}.\supset :\exists !(\mu +_{c}\nu )_{\xi }.&\equiv .\exists !(\mu +_{c}\nu )_{\xi }.\exists !\text{ sm }ʻʻ\mu \cap t_{0}ʻ\mu + .\exists !\text{ sm }_{\xi }ʻʻ\nu .\\ +[*103·43.*110·25·4] &\equiv .\exists !(\mu +_{c}\text{ sm }_{\xi }ʻʻ\nu )_{\xi } &\qquad \text{(2)}\\ +\text{Similarly}\qquad \vdash \colon\ldotp \text{Hp}.&\supset :\exists !(\mu +_{c}\nu )_{\xi }.\equiv .\exists !(\text{ sm }_{\xi }ʻʻ\mu +_{c}\nu )_{\xi } + &\qquad \text{(3)}\\ +\vdash .(1).(2).(3).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*118·23.</b> \(\vdash :\mu ,\nu \in \text{NC}.\supset .(\mu +_{c}\nu )_{\xi }=(\text{ sm }_{\xi }ʻʻ\mu +_{c}\text{ sm }_{\xi }ʻʻ\nu )_{\xi }\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*118·21.*110·4·25.&\supset \vdash :\exists !(\mu +_{c}\nu )_{\xi }.\supset .(\mu +_{c}\nu )_{\xi }=(\text{ sm }_{\xi }ʻʻ\mu + +_{c}\text{ sm }_{\xi }ʻʻ\nu )_{\xi } &\qquad \text{(1)}\\ +\vdash .*118·22.&\supset \vdash :\text{Hp}.{\sim}\exists !(\mu +_{c}\nu )_{\xi }.\supset .(\mu +_{c}\nu )_{\xi }=(\text{ sm }_{\xi }ʻʻ\mu + +_{c}\text{ sm }_{\xi }ʻʻ\nu )_{\xi } &\qquad \text{(2)}\\ +\vdash .(1).(2).&\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_197">[Pg 197]</span></p> + +<p class="nind"><b>*118·24.</b> \(\vdash : \nu \in \text{NC} . \supset . (\mu +_{c} \nu )_{\xi } = (\mu +_{c} \text{ sm }_{\xi }ʻʻ\nu )_{\xi }\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *118·21 . *110·4·25 . *103·43 . \supset \\ +&\vdash : \exists ! (\mu +_{c} \nu )_{\xi } . \supset . (\mu +_{c} \nu )_{\xi } = (\mu +_{c} \text{ sm }_{\xi }ʻʻ\nu )_{\xi } + &\qquad \text{(1)}\\ +\vdash . *110·4 . &\supset \vdash : \mu {\sim} \in \text{NC} . \supset . (\mu +_{c} \nu )_{\xi } = (\mu +_{c} \text{ sm }_{\xi }ʻʻ\nu )_{\xi } + &\qquad \text{(2)}\\ +\vdash . *118·22 . &\supset \vdash : \text{Hp} . \mu \in \text{NC} . {\sim} \exists ! (\mu +_{c} \nu )_{\xi } . \supset . (\mu +_{c} \nu )_{\xi } + = (\mu +_{c} \text{ sm }_{\xi }ʻʻ\nu )_{\xi } &\qquad \text{(3)}\\ +\vdash . (1) . (2) . (3) . \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*118·241">*118·241</a>.</b> \(\vdash : \mu \in \text{NC} . \supset . (\mu +_{c} \nu )_{\xi } = (\text{ sm }_{\xi }ʻʻ\mu +_{c} \nu )_{\xi } + \quad[*118·24 . *110·51]\)</p> + +<p class="nind"><b>*118·25.</b> \(\vdash . (\mu +_{c} \nu +_{c} \varpi )_{\xi } = \{(\mu +_{c} \nu )_{\xi } +_{c} \varpi\}_{\xi } = \{\mu +_{c} + (\nu +_{c} \varpi )_{\xi }\}_{\xi}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *110·42 . *118·241·201 . (\text{IIT}) . \supset \\ +&\vdash : \mu , \nu \in \text{N}_{0}\text{C} . \supset . (\mu +_{c} \nu +_{c} \varpi )_{\xi } = \{(\mu +_{c} \nu )_{\xi } +_{c}\varpi\}_{\xi } &\qquad \text{(1)}\\ +\vdash . *110·4 . &\supset \vdash : {\sim} (\mu , \nu \in \text{N}_{0}\text{C}) . \supset . \mu +_{c} \nu = \Lambda . (\mu +_{c} \nu )_{\xi } + = \Lambda .\\ +[*110·4] &\supset . (\mu +_{c} \nu +_{c} \varpi )_{\xi } = \{(\mu +_{c} \nu )_{\xi } +_{c} \varpi\}_{\xi } &\qquad \text{(2)}\\ +\vdash . (1) . (2) . &\supset \vdash . (\mu +_{c} \nu +_{c} \varpi )_{\xi } = \{(\mu +_{c} \nu )_{\xi } +_{c} \varpi\}_{\xi } + &\qquad \text{(3)}\\ +\text{Similarly}\qquad &\vdash . (\mu +_{c} \nu +_{c} \varpi )_{\xi } = \{\mu +_{c} (\nu +_{c} \varpi)_{\xi }\}_{\xi } + &\qquad \text{(4)}\\ +\vdash . (3) . (4) . &\supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*118·3.</b> \[\begin{align}&\vdash . (\mu \times _{c} \nu )_{\xi } = \hat{\eta} \{(\exists \alpha , \beta ) . \mu = \text{N}_{0}\text{c}ʻ\alpha + . \nu = \text{N}_{0}\text{c}ʻ\beta . \eta \text{ sm }_{\xi } (\alpha \times \beta )\}\\ +&[(*65·01·03) . *113·2]\end{align}\]</p> + +<p class="nind"><b>*118·301.</b> \(\vdash : \exists ! (\mu \times _{c} \nu ) . \supset . \text{ sm }_{\xi }ʻʻ(\mu \times _{c} \nu ) = (\mu \times _{c} \nu )_{\xi } + \quad[\text{Proof as in *118·201}]\)</p> + +<p class="nind"><b>*118·31.</b> \(\vdash : \exists ! (\mu \times _{c} \nu )_{\xi } . \nu \neq 0 . \supset . \exists ! \text{ sm }_{\xi }ʻʻ\mu \)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *101·15·12. &\supset \vdash : \mu = 0 . \supset . \exists ! \text{ sm }_{\xi }ʻʻ\mu &\qquad \text{(1)}\\ +\vdash . *113·203 . *118·3 . &\supset \vdash : \text{Hp} . \mu \neq 0 . \supset . \mu , \nu \in \text{N}_{0}\text{C} - \iota ʻ0 .\\ +[*117·62] &\supset . \mu \times _{c} \nu \geq \mu .\\ +[*118·13·301 . (\text{IIT})] &\supset . \exists ! \text{ sm }_{\xi }ʻʻ\mu &\qquad \text{(2)}\\ +\vdash . (1) . (2) . \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*118·311.</b> \(\vdash : \exists ! (\mu \times _{c} \nu )_{\xi } . \mu \neq 0 . \supset . \exists ! \text{ sm }_{\xi }ʻʻ\nu \quad[*118·31 . *113·27]\)</p> + + +<p class="nind"><b>*118·32.</b> \(\vdash \colon\ldotp \nu \in \text{NC} . \mu \neq 0 . \supset : \exists ! (\mu \times _{c} \nu )_{\xi } . \equiv . \exists ! (\mu \times _{c} + \text{ sm }_{\xi }ʻʻ\nu )_{\xi }\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *113·203 . &\supset \vdash : \exists ! (\mu \times _{c} \nu )_{\xi } . \supset . \mu \in \text{NC} &\qquad \text{(1)}\\ +\vdash . *113·203 . &\supset \vdash : \exists ! (\mu \times _{c} \text{ sm }_{\xi }ʻʻ\nu )_{\xi } . \supset . \mu \in \text{NC} &\qquad \text{(2)}\\ +\vdash . *113·203 . *118·311 . \supset \\ +\vdash \colon\ldotp \text{Hp} . &\supset : \exists ! (\mu \times _{c} \nu )_{\xi } . \supset . \exists ! \mu . \exists ! \text{ sm }_{\xi }ʻʻ\nu .\\ +[*103·43] &\supset . \exists ! \text{ sm }ʻʻ\mu \cap t_{0}ʻ\mu . \exists ! \text{ sm }_{\xi }ʻʻ\nu .\\ +[(1) . *113·26 . *103·43] &\supset . \exists ! (\mu \times _{c} \text{ sm }_{\xi }ʻʻ\nu )_{\xi } &\qquad \text{(3)}\\ +\vdash . *113·203 . *103·43 . \supset \\ +\vdash \colon\ldotp \text{Hp} . &\supset : \exists ! (\mu \times _{c} \text{ sm }_{\xi }ʻʻ\nu )_{\xi } . \supset . \exists ! \text{ sm }ʻʻ\mu \cap t_{0}ʻ\mu + . \exists ! \text{ sm }_{\xi }ʻʻ\nu .\\ +[(2) . *113·26 . *103·43] &\supset . \exists ! (\mu \times _{c} \nu )_{\xi } &\qquad \text{(4)}\\ +\vdash . (3) . (4) . \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_198">[Pg 198]</span></p> + +<p class="nind"><b>*118·33.</b> \[\begin{align}&\vdash :\mu ,\nu \in \text{NC}-\iota ʻ0.\supset .(\mu \times _{c}\nu )_{\xi }=(\text{ sm }_{\xi }ʻʻ\mu + \times _{c}\text{ sm }_{\xi }ʻʻ\nu )_{\xi }\\ +&[\text{Proof as in *118·23, using *118·31·311. *113·203·26}]\end{align}\]</p> + +<p class="nind"><b>*118·34.</b> \(\vdash :\nu \in \text{NC}.\mu \neq 0.\supset .(\mu \times _{c}\nu )_{\xi }=(\mu \times _{c}\text{ sm }_{\xi }ʻʻ\nu )_{\xi }\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*118·311.*113·203·26.*103·43.\supset \\ +&\vdash :\exists !(\mu \times _{c}\nu )_{\xi }.\mu \neq 0.\supset .(\mu \times _{c}\nu )_{\xi }=(\mu \times _{c}\text{ sm }_{\xi }ʻʻ\nu )_{\xi }&\qquad + \text{(1)}\\ +\vdash .*118·32.&\supset \vdash :\text{Hp}.{\sim}\exists !(\mu \times _{c}\nu )_{\xi }.\supset .{\sim}\exists !(\mu \times _{c}\text{ sm }_{\xi }ʻʻ\nu + )_{\xi }.\\ +[*24·51] &\supset .(\mu \times _{c}\nu )_{\xi }=(\mu \times _{c}\text{ sm }_{\xi }ʻʻ\nu )_{\xi } &\qquad \text{(2)}\\ +\vdash .(1).(2).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*118·341.</b> \(\vdash :\mu \in \text{NC}.\nu \neq 0.\supset .(\mu \times _{c}\nu )_{\xi }=(\text{sm }_{\xi }ʻʻ\mu \times _{c}\nu )_{\xi } + \quad[*118·34.*113·27]\)</p> + +<p class="nind"><b>*118·35.</b> \[\begin{align}&\vdash :\varpi \neq 0.\supset .(\mu \times _{c}\nu \times _{c}\varpi )_{\xi }=\{(\mu \times _{c}\nu )_{\xi }\times _{c}\varpi + \}_{\xi }\\ +&[\text{Proof similar to *118·25, using *118·341·301.*113·203·23}]\end{align}\]</p> + +<p class="nind"><b>*118·351.</b> \(\vdash :\mu \neq 0.\supset .(\mu \times _{c}\nu \times _{c}\varpi )_{\xi }=\{\mu \times _{c}(\nu \times _{c}\varpi + )_{\xi }\}_{\xi } \quad[*118·35.*113·27]\)</p> + +<p class="nind"><b>*118·352.</b> \[\begin{align}&\vdash :\mu \neq 0.\varpi \neq 0.\supset .\{\mu \times _{c}(\nu \times _{c}\varpi )_{\xi }\}_{\xi }=\{(\mu \times _{c}\nu + )_{\xi }\times _{c}\varpi \}_{\xi }\\ +&[*118·35·351]\end{align}\]</p> + +<p class="nind"><b>*118·4.</b> \[\begin{align}&\vdash .(\mu ^\nu )_{\xi }=\hat{\eta} \{(\exists \alpha ,\beta ).\mu =\text{N}_{0}\text{c}ʻ\alpha .\nu =\text{N}_{0}\text{c}ʻ\beta + .\eta \text{ sm }_{\xi }(\alpha \,\,\text{exp}\,\,\beta )\}\\ +&[(*65·01·03).*116·2]\end{align}\]</p> + +<p class="nind"><b>*118·401.</b> \(\vdash :\exists !\mu ^\nu .\supset .\text{ sm }_{\xi }ʻʻ\mu ^\nu =(\mu ^\nu )_{\xi } \quad[\text{Proof as in *118·201}]\)</p> + +<p class="nind"><b>*118·402.</b> \(\vdash \colon\ldotp \mu ,\nu \in \text{N}_{0}\text{C}.\mu \neq 0.\mu \neq 1.\supset :\exists !(\mu ^\nu )_{\xi }.\supset .\exists !(\mu \times _{c}\nu + )_{\xi }\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*103·2. \supset \vdash \colon\ldotp \text{Hp}.&\supset :(\exists \alpha ,\beta ).\mu =\text{N}_{0}\text{c}ʻ\alpha .\nu =\text{N}_{0}\text{c}ʻ\beta + .\alpha {\sim}\in 0\cup 1:\\ +[*117·651] &\supset :(\exists \alpha ,\beta ).\mu =\text{N}_{0}\text{c}ʻ\alpha .\nu =\text{N}_{0}\text{c}ʻ\beta .\\ +&(\text{N}_{0}\text{c}ʻ\alpha )^{\text{N}_{0}\text{c}ʻ\beta }\geq \text{N}_{0}\text{c}ʻ\alpha \times _{c}\text{N}_{0}\text{c}ʻ\beta :\\ +[*118·13·301·401.(\text{IIT})]&\supset :\exists !(\mu ^\nu )_{\xi }.\supset .\exists !(\mu \times _{c}\nu )_{\xi }\colon\ldotp \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*118·41.</b> \(\vdash :\exists !(\mu ^\nu )_{\xi }.\nu \neq 0.\supset .\exists !\text{ sm }_{\xi }ʻʻ\mu\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*118·402·31. \supset \vdash :\text{Hp}.\mu \neq 1.\mu \neq 0.\supset .\exists !\text{ sm }_{\xi }ʻʻ\mu &\qquad \text{(1)}\\ +\vdash .*101·12·15·241·28.\supset \vdash \colon\ldotp \mu =0.\lor.\mu =1:\supset .\exists !\text{ sm }_{\xi }ʻʻ\mu &\qquad \text{(2)}\\ +\vdash .(1).(2).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*118·411.</b> \(\vdash :\exists !(\mu ^\nu )_{\xi }.\mu \neq 0.\mu \neq 1.\supset .\exists !\text{ sm }_{\xi }ʻʻ\nu \quad[*118·402·311]\)</p> + +<p class="nind"><b>*118·42.</b> \[\begin{align}&\vdash \colon\ldotp \nu \in \text{NC}.\mu \neq 0.\mu \neq 1.\supset :\exists !(\mu ^\nu )_{\xi }.\equiv + .\exists !(\mu ^{\text{ sm }_{\xi }ʻʻ\nu })_{\xi }\\ +&[\text{Proof as in *118·32, using *116·203·26.*118·411}]\end{align}\]</p> + +<p class="nind"><b>*118·421.</b> \[\begin{align}&\vdash \colon\ldotp \mu \in \text{NC}.\nu \neq 0.\supset :\exists !(\mu ^\nu )_{\xi }.\equiv .\exists !\{(\text{ sm }_{\xi }ʻʻ\mu + )^\nu\}_{\xi }\\ +&[\text{Proof as in *118·32, using *116·203·26.*118·41}]\end{align}\]</p> + +<p class="nind"><b>*118·43.</b> \[\begin{align}&\vdash :\mu ,\nu \in \text{NC}-\iota ʻ0.\mu \neq 1.\supset .(\mu ^\nu )_{\xi }=\{(\text{ sm }_{\xi }ʻʻ\mu )^{\text{ sm }_{\xi }ʻʻ\nu + }\}_{\xi }\\ +&[\text{Proof as in *118·23, using *118·41·411.*116·203·26}]\end{align}\]</p> + +<p><span class="pagenum" id="Page_199">[Pg 199]</span></p> + +<p class="nind"><b>*118·44.</b> \[\begin{align}&\vdash :\nu \in \text{NC}.\mu \neq 0.\mu \neq 1.\supset .(\mu ^\nu )_{\xi }=(\mu ^{\text{ sm }_{\xi }ʻʻ\nu })_{\xi }\\ +&[\text{Proof as in *118·34, using *116·203·26.*118·411·42}]\end{align}\]</p> + +<p class="nind"><b>*118·441.</b> \[\begin{align}&\vdash :\mu \in \text{NC}.\nu \neq 0.\supset .(\mu ^\nu )_{\xi }=\{(\text{ sm }_{\xi }ʻʻ\mu )^\nu\}_{\xi }\\ +&[\text{Proof as in *118·34, using *116·203·26.*118·41·421}]\end{align}\]</p> + +<p class="nind"><b>*118·45.</b> \(\vdash :\mu \neq 0.\mu \neq 1.\supset .(\mu ^{\nu \times _{c}\varpi })_{\xi }=\{\mu ^{(\nu \times _{c}\varpi + )_{\xi }}\}_{\xi }\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*113·23.*118·44·301.(\text{IIT}).\supset\\ +\vdash :\text{Hp}.\nu ,\varpi \in \text{N}_{0}\text{C}.\supset .(\mu ^{\nu \times _{c}\varpi })_{\xi }=\{\mu ^{(\nu \times _{c}\varpi + )_{\xi }}\}_{\xi } &&\qquad \text{(1)}\\ +\vdash .*113·203.\supset \vdash :{\sim}(\nu ,\varpi \in \text{N}_{0}\text{C}).&\supset .\nu \times _{c}\varpi =\Lambda .\\ +[*116·203] &\supset . (\mu ^{\nu \times _{c}\varpi })_{\xi }=\{\mu ^{(\nu \times _{c}\varpi )_{\xi }}\}_{\xi } &\qquad \text{(2)}\\ +\vdash .(1).(2).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*118·451.</b> \(\vdash :\varpi \neq 0.\supset .(\mu ^{\nu \times _{c}\varpi })_{\xi } =[\{(\mu ^\nu )_{\xi }\}^\varpi ]_{\xi }\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*116·63. \supset \vdash :\text{Hp}.\mu \in \text{NC}.\supset .(\mu ^{\nu \times _{c}\varpi })_{\xi } &=\{(\mu ^\nu )^\varpi\}_{\xi }\\ +[*116·23.*118·441·401.(\text{IIT})] &=[\{(\mu ^\nu )_{\xi }\}^\varpi ]_{\xi } &\qquad \text{(1)}\\ +\vdash .*116·204.\supset \vdash :\mu {\sim}\in \text{NC}.\supset .(\mu ^{\nu \times _{c}\varpi })_{\xi } &=[\{(\mu ^\nu )_{\xi }\}^\varpi + ]_{\xi } &\qquad \text{(2)}\\ +\vdash .(1).(2).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*118·46.</b> \[\begin{align}&\vdash :\mu \neq 0.\mu \neq 1.\supset .(\mu ^{\nu +_{c}\varpi })_{\xi }=\{\mu ^{(\nu +_{c}\varpi )}_{\xi}\}_{\xi }\\ +&[\text{Proof as in *118·45, using *118·44·201.*116·203.*110·4·42}]\end{align}\]</p> + +<p class="nind"><b>*118·461.</b> \(\vdash .(\mu ^{\nu +_{c}\varpi })_{\xi }=\{(\mu ^\nu )_{\xi }\times _{c}(\mu ^\varpi + )_{\xi }\}_{\xi }\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*116·52.\supset \vdash :\mu \neq 0.\supset .(\mu ^{\nu +_{c}\varpi })_{\xi }=(\mu ^\nu \times _{c}\mu ^\varpi )_{\xi }\\ +[*116·35·23.*118·33·401.(\text{IIT})] =\{(\mu ^\nu )_{\xi }\times _{c}(\mu ^\varpi )_{\xi }\}_{\xi } &\qquad \text{(1)}\\ +\vdash .*110·4.*113·203.*116·203.\supset \\ +\vdash :{\sim}(\nu ,\varpi \in \text{N}_{0}\text{C}).\supset .(\mu ^{\nu +_{c}\varpi })_{\xi }=\{(\mu ^\nu )_{\xi }\times _{c}(\mu ^\varpi + )_{\xi }\}_{\xi } &\qquad \text{(2)}\\ +\vdash .*116·311.*113·601.*110·62.\supset \\ +\vdash :\nu ,\varpi \in \text{N}_{0}\text{C}-\iota ʻ0.\mu =0.\supset .(\mu ^{\nu +_{c}\varpi })_{\xi }=\{(\mu ^\nu )_{\xi }\times _{c}(\mu + ^\varpi )_{\xi }\}_{\xi } &\qquad \text{(3)}\\ +\vdash .*116·311·301.*110·6.*113·601.\supset \\ +\vdash :\nu \in \text{N}_{0}\text{C}-\iota ʻ0.\varpi =0.\mu =0.\supset .(\mu ^{\nu +_{c}\varpi })_{\xi }=\{(\mu ^\nu )_{\xi }\times _{c}(\mu + ^\varpi )_{\xi }\}_{\xi } &\qquad \text{(4)}\\ +\text{Similarly}\qquad \vdash :\varpi \in \text{N}_{0}\text{C}-\iota ʻ0.\nu =0.\mu =0.\supset .(\mu ^{\nu \times _{c}\varpi })_{\xi }=\{(\mu ^\nu )_{\xi }\times + _{c}(\mu ^\varpi )_{\xi }\}_{\xi } &\qquad \text{(5)}\\ +\vdash .*116·301.*113·621.\supset \\ +\vdash :\nu =0.\varpi =0.\mu =0.\supset .(\mu ^{\nu +_{c}\varpi })_{\xi }=\{(\mu ^\nu )_{\xi }\times _{c}(\mu ^\varpi + )_{\xi }\}_{\xi } &\qquad \text{(6)}\\ +\vdash .(1).(2).(3).(4).(5).(6).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*118·462.</b> \(\vdash .(\mu ^{\nu +_{c}\varpi })_{\xi }=\{\mu ^\nu \times _{c}(\mu ^\varpi )_{\xi }\}_{\xi } + \quad[\text{Proof as in *118·461, using *118·34}]\)</p> + +<p><span class="pagenum" id="Page_200">[Pg 200]</span></p> + +<p class="nind"><b>*118·463.</b> \(\vdash .(\mu ^{\nu +_{c}\varpi })_{\xi }=\{(\mu ^\nu )_{\xi }\times _{c}\mu ^\varpi\}_{\xi } + \quad[\text{Proof as in *118·461, using *118·341}]\)</p> + +<p class="nind"><b>*118·47.</b> \[\begin{align}&\vdash :\varpi \neq 0.\supset .\{(\mu _{c}\nu )^\varpi\}_{\xi }=[\{(\mu \times _{c}\nu )_{\xi }\}^\varpi + ]_{\xi }\\ +&[\text{Proof as in *118·45, using *118·441}]\end{align}\]</p> + +<p class="nind"><b>*118·471.</b> \[\begin{align}\vdash \colon\ldotp \mu \neq 0.\nu \neq 0.\lor.\varpi =0.\lor.{\sim}&(\mu ,\nu ,\varpi \in \text{N}_{0}\text{C}):\supset .\\ +&\{(\mu \times _{c}\nu )^\varpi\}_{\xi }=\{(\mu ^\varpi )_{\xi }\times _{c}(\nu ^\varpi + )_{\xi }\}_{\xi }\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*116·55.\supset \vdash :\mu \neq 0.\nu \neq 0.\supset .\{(\mu \times _{c}\nu )^\varpi\}_{\xi }=\{\mu ^\varpi \times _{c}\nu + ^\varpi\}_{\xi }\\ +[*116·35·23.*118·33·401.(\text{IIT})] =\{(\mu ^\varpi )_{\xi }\times _{c}(\nu ^\varpi )_{\xi }\}_{\xi } &\qquad \text{(1)}\\ +\vdash .*110·4.*113·203.*116·203.\supset\\ +\vdash :{\sim}(\mu ,\nu ,\varpi \in \text{N}_{0}\text{C}).\supset .\{(\mu \times _{c}\nu )^\varpi\}_{\xi }=\{(\mu ^\varpi )_{\xi }\times _{c}(\nu + ^\varpi )_{\xi }\}_{\xi } &\qquad \text{(2)}\\ +\vdash .*116·301.*113·621.\supset \\ +\vdash :\mu ,\nu \in \text{N}_{0}\text{C}.\varpi =0.\supset .\{(\mu \times _{c}\nu )^\varpi\}_{\xi }=\{(\mu ^\varpi )_{\xi }\times _{c}(\nu + ^\varpi )_{\xi }\}_{\xi } &\qquad \text{(3)}\\ +\vdash .(1).(2).(3).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*118·472">*118·472</a>.</b> \[\begin{align}&\vdash \colon\ldotp \mu \neq 0.\lor.\varpi =0.\lor.{\sim}(\mu ,\nu ,\varpi \in \text{N}_{0}\text{C}):\supset .\{(\mu \times _{c}\nu + )^\varpi\}_{\xi }=\{\mu ^\varpi \times _{c}\nu ^\varpi )_{\xi }\}_{\xi }\\ +&[\text{Proof as in *118·471, using *118·34}]\end{align}\]</p> + +<p class="nind"><b><a id="*118·473">*118·473</a>.</b> \[\begin{align}&\vdash \colon\ldotp \nu \neq 0.\lor.\varpi =0.\lor.{\sim}(\mu ,\nu ,\varpi \in \text{N}_{0}\text{C}):\supset .\{(\mu \times _{c}\nu + )^\varpi\}_{\xi }=\{(\mu ^\varpi )_{\xi }\times _{c}\nu ^\varpi\}_{\xi }\\ +&[\text{Proof as in *118·471, using *118·341}]\end{align}\]</p> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_201">[Pg 201]</span></p> +<h2 class="nobreak" id="*119">*119. SUBTRACTION.</h2> +</div> + + +<p class="nind"><i>Summary of</i> *119.</p> + +<p>The treatment of subtraction follows the same general lines as that of +addition, and is simplified by the results in <a href="#*110">*110</a>. A difficulty arises +from the fact that subtraction (in any ordinary sense of the term) +is not always possible; and also from the fact that the result, when +possible, is not always a cardinal number.</p> + +<p>We put</p> + +<p class="nind"><b>*119·01.</b> \(\gamma -_{c}\nu =\hat{\xi} \{\text{Nc}ʻ\xi +_{c}\nu =\gamma .\exists !\text{Nc}ʻ\xi +_{c}\nu\} \quad\text{Df}\)</p> + +<p>Thus when subtraction (in the ordinary sense of the term) is not +possible, +\[ +\gamma -_{c}\nu =\Lambda . +\]</p> + +<p>The question of existential adjustment of types is dealt with by +\(\text{IIT}\) of the prefatory statement combined with the following +definitions:</p> + +<p class="nind"><b>*119·02.</b> \(\text{Nc}ʻ\alpha -_{c}\nu =\text{N}_{0}\text{c}ʻ\alpha -_{c}\nu \quad\text{Df}\)</p> + +<p class="nind"><b>*119·03.</b> \(\gamma -_{c}\text{Nc}ʻ\beta =\gamma -_{c}\text{N}_{0}\text{c}ʻ\beta \quad\text{Df}\)</p> + +<p>We then proceed to deduce the elementary properties derivable from +these definitions.</p> + +<p class="nind"><b>*119·11.</b> \(\vdash :\exists !\gamma -_{c}\nu .\supset .\gamma ,\nu \in \text{N}_{0}\text{C}\)</p> + +<p class="nind"><b>*119·12</b> \(\vdash :\xi \in \text{Nc}ʻ\alpha -_{c}\text{Nc}ʻ\beta .\equiv .\alpha \text{ sm }\xi +\beta\)</p> + +<p class="nind"><b>*119·14.</b> \(\vdash :\xi \in \gamma -_{c}\nu .\supset .\text{N}_{0}\text{c}ʻ\xi \subset \gamma -_{c}\nu\)</p> + +<p class="nind"><b>*119·25.</b> \(\vdash :\gamma \geq \nu .\supset .\exists !(\gamma -_{c}\nu )\cap t_{0}ʻ\gamma\)</p> + +<p class="nind"><b>*119·26.</b> \(\vdash :\exists !\gamma -_{c}\nu .\supset .\gamma \geq \nu\)</p> + +<p>The next group of propositions is concerned with some simple results of +subtraction.</p> + +<p class="nind"><b>*119·32.</b> \(\vdash :(\gamma +_{c}\nu )-_{c}\nu \in \text{N}_{0}\text{C}.\supset .\text{ sm }ʻʻ\gamma =(\gamma +_{c}\nu )-_{c}\nu\)</p> + +<p class="nind"><b>*119·34.</b> \(\vdash :\gamma -_{c}\nu \in \text{N}_{0}\text{C}.\supset .(\gamma -_{c}\nu )+_{c}\nu =\text{ sm }ʻʻ\gamma\)</p> + +<p class="nind"><b>*119·35.</b> \(\vdash :\gamma -_{c}\nu \in \text{N}_{0}\text{C}.\supset .\alpha +_{c}\gamma =(\alpha +_{c}\nu )+_{c}(\gamma -_{c}\nu )\)</p> + +<p>Associative laws are then considered.</p> + +<p class="nind"><b>*119·44.</b> \(\vdash :\mu +_{c}(\nu -_{c}\varpi )\subset (\mu +_{c}\nu )-_{c}\varpi\)</p> + +<p class="nind"><b>*119·45.</b> \(\vdash :(\mu +_{c}\nu )-_{c}\varpi \in \text{NC}.\exists !\{\mu +_{c}(\nu -_{c}\varpi )\}.\supset .\mu +_{c}(\nu -_{c}\varpi + )=(\mu +_{c}\nu )-_{c}\varpi\)</p> + +<p>The question of types is then dealt with:</p> + +<p><span class="pagenum" id="Page_202">[Pg 202]</span></p> + +<p class="nind"><b>*119·52.</b> \(\vdash :\text{ sm }_{\delta ,\gamma }ʻʻ(\mu -_{c}\nu )_{\gamma }=(\mu -_{c}\nu )_{\delta }\cap + \text{D}ʻ\text{ sm }_{\delta ,\gamma }\)</p> + +<p>A difficulty arises from the fact that if \(\tau _{1}\) and \(\tau_{2}\) +are two complete types whose members are classes, we cannot +prove that, either \(\tau _{1} = \text{ sm }ʻʻ\tau _{2}\) or +\(\tau _{2}= \text{ sm }ʻʻ\tau _{1}\). We put</p> + +<p class="nind"><b>*119·54.</b> \(\text{SM}(\delta ,\gamma ).=:tʻ\delta =\text{D}ʻ\text{ sm }_{\delta ,\gamma }.\lor.tʻ\gamma =\text{D}ʻ\text{ sm }_{\gamma ,\delta } \quad\text{Df}\)</p> + +<p>Then we obtain</p> + +<p class="nind"><b>*119·541.</b> \[\begin{align}\vdash :\text{SM}(\delta ,\gamma ).(\mu -_{c}\nu )_{\gamma }&\in \text{N}_{0}\text{C}.(\mu -_{c}\nu )_{\delta }\in + \text{NC}.\supset .\\ +&\text{ sm }_{\delta ,\gamma }ʻʻ(\mu -_{c}\nu )_{\gamma }=(\mu -_{c}\nu )_{\delta }\end{align}\]</p> + +<p>Finally we show that any existential adjustment of types will suffice +for the components:</p> + +<p class="nind"><b>*119·61.</b> \(\vdash :\mu \in \text{N}_{0}\text{C}.\exists !\text{ sm }_{\xi }ʻʻ\mu .\supset .\mu -_{c}\nu =\text{ sm }_{\xi }ʻʻ\mu -_{c}\nu\)</p> + +<p class="nind"><b>*119·62.</b> \(\vdash :\nu \in \text{N}_{0}\text{C}.\exists !\text{ sm }_{\xi }ʻʻ\nu .\supset .\mu -_{c}\nu =\mu -_{c}\text{ sm }_{\xi }ʻʻ\nu\)</p> + +<p>Also <a href="#*119·25">*119·25</a><a href="#*119·26">·26</a> are now extended to</p> + +<p class="nind"><b>*119·64.</b> \(\vdash \colon\ldotp \exists !\text{ sm }_{\xi }ʻʻ\mu .\supset :\mu \geq \nu .\equiv .\exists !(\mu -_{c}\nu )_{\xi }\)</p> + +<p>The only applications of the propositions of this number are in +connection with Inductive Cardinals (cf. <a href="#*120">*120</a>).</p> + +<hr class="tb"> + +<p class="nind"><b>*119·01.</b> \(\gamma -_{c}\nu =\hat{\xi} \{\text{Nc}ʻ\xi +_{c}\nu =\gamma .\exists !\text{Nc}ʻ\xi +_{c}\nu\} \quad\text{Df}\)</p> + +<p>Here the suffix to the sign of subtraction is introduced to show that +we are concerned with cardinal subtraction. It will be found that +\(\gamma -_{c}\nu\) is not an \(\text{NC}\) except under hypotheses for +\(\gamma\) and \(\nu\).</p> + +<p class="nind"><b><a id="*119·02">*119·02</a>.</b> \(\text{Nc}ʻ\alpha -_{c}\nu =\text{N}_{0}\text{c}ʻ\alpha -_{c}\nu \quad\text{Df}\)</p> + +<p class="nind"><b><a id="*119·03">*119·03</a>.</b> \(\gamma -_{c}\text{Nc}ʻ\beta =\gamma -_{c}\text{N}_{0}\text{c}ʻ\beta \quad\text{Df}\)</p> + +<p class="nind"><b>*119·04.</b> \(\vdash .\text{Nc}ʻ\alpha -_{c}\text{Nc}ʻ\beta =\text{N}_{0}\text{c}ʻ\alpha -_{c}\text{N}_{0}\text{c}ʻ\beta \quad[*119·02·03]\)</p> + +<p>Note that the occurrence of a formal number in the place of \(\gamma\) +or \(\nu\) in \(\gamma -_{c}\nu\) is an arithmetic occurrence, and +accordingly \(\text{IIT}\) applies to it.</p> + +<p class="nind"><b>*119·1.</b> \(\vdash :\xi \in \gamma -_{c}\nu .\equiv .\text{Nc}ʻ\xi +_{c}\nu =\gamma .\exists !\text{Nc}ʻ\xi +_{c}\nu \quad [(*119·01)]\)</p> + +<p class="nind"><b>*119·101.</b> \(\vdash :\xi \in \text{Nc}ʻ\alpha -_{c}\nu .\equiv .\text{Nc}ʻ\xi +_{c}\nu =\text{N}_{0}\text{c}ʻ\alpha \quad[(*119·02).*103·13]\)</p> + +<p class="nind"><b>*119·102.</b> \[\begin{align}&\vdash :\xi \in \gamma -_{c}\text{Nc}ʻ\beta .\equiv .\text{Nc}ʻ\xi +_{c}\text{Nc}ʻ\beta + =\gamma .\exists !\text{Nc}ʻ\xi +_{c}\text{Nc}ʻ\beta\\ +&[(*119·03).*110·3]\end{align}\]</p> + +<p class="nind"><b>*119·103.</b> \[\begin{align}&\vdash :\xi \in \text{Nc}ʻ\alpha -_{c}\text{Nc}ʻ\beta .\equiv .\text{Nc}ʻ\xi +_{c}\text{Nc}ʻ\beta =\text{N}_{0}\text{c}ʻ\alpha\\ +&[*119·04.*110·3.*103·13]\end{align}\]</p> + +<p class="nind"><b>*119·11.</b> \(\vdash :\exists !\gamma -_{c}\nu .\supset .\gamma ,\nu \in \text{N}_{0}\text{C} \quad[*110·4·42.*103·34]\)</p> + +<p class="nind"><b><a id="*119·12">*119·12</a>.</b> \(\vdash :\xi \in \text{Nc}ʻ\alpha -_{c}\text{Nc}ʻ\beta .\equiv .\alpha \text{ sm }\xi +\beta\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*119·103.\supset \vdash :\xi \in \text{Nc}ʻ\alpha -_{c}\text{Nc}ʻ\beta .&\equiv .\text{Nc}ʻ\xi +_{c}\text{Nc}ʻ\beta =\text{N}_{0}\text{c}ʻ\alpha .\\ +[*110·3] &\equiv .\text{Nc}ʻ(\xi +\beta )=\text{N}_{0}\text{c}ʻ\alpha .\\ +[*100·35.*103·13] &\equiv .\alpha \text{ sm }\xi +\beta :\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_203">[Pg 203]</span></p> + +<p>Thus \(\text{Nc}ʻ\alpha -_{c}\text{Nc}ʻ\beta\) is an \(\text{NC}\) when +\(\hat{\xi} (\alpha \text{ sm }\xi +\beta )\) is an \(\text{NC}\).</p> + +<p class="nind"><b>*119·13.</b> \(\vdash :\text{N}_{0}\text{c}ʻ\gamma \subset \text{Nc}ʻa-_{c}\text{Nc}ʻ\beta .\equiv .\alpha \text{ sm }(\gamma +\beta )\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*22·1.\supset \vdash \colon\ldotp \text{N}_{0}\text{c}ʻ\gamma \subset \text{Nc}ʻa-_{c}\text{Nc}ʻ\beta .&\equiv :\xi \in \text{N}_{0}\text{c}ʻ\gamma + .\supset _{\xi }.\xi \in \text{Nc}ʻ\alpha -_{c}\text{Nc}ʻ\beta :\\ +[*103·12.*119·12] &\supset :\alpha \text{ sm }(\gamma +\beta ) &\qquad \text{(1)}\\ +\vdash .*110·15.*100·31.&\supset \vdash \colon\ldotp \alpha \text{ sm }(\gamma +\beta ).\supset :\xi \in \text{N}_{0}\text{c}ʻ\gamma .\supset .(\xi +\beta )\text{ sm }(\gamma +\beta ).\\ +[*73·32] &\supset .\alpha \text{ sm }(\xi +\beta ).\\ +[*119·12] &\supset .\xi \in \text{Nc}ʻ\alpha -_{c}\text{Nc}ʻ\beta &\qquad \text{(2)}\\ +\vdash .(1).(2).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*119·14.</b> \(\vdash :\xi \in \gamma -_{c}\nu .\supset .\text{N}_{0}\text{c}ʻ\xi \subset \gamma -_{c}\nu \quad[*119·1.*100·31·321]\)</p> + +<p class="nind"><b>*119·21.</b> \(\vdash :\beta \subset \alpha .\supset .\exists !(\text{Nc}ʻ\alpha -_{c}\text{Nc}ʻ\beta )_{\alpha }\)</p> + +<p>The notation is defined in *65·01.</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*24·411·21.\supset \vdash :\text{Hp}.&\supset .\alpha =\beta \cup (\alpha -\beta ).\beta \cap (\alpha -\beta )=\Lambda .\\ +[*110·32] &\supset .\text{Nc}ʻ\alpha =\text{Nc}ʻ\beta +_{c}\text{Nc}ʻ(\alpha -\beta ).\\ +[*10·24] &\supset .(\exists \xi ).\xi \in tʻ\alpha .\text{N}_{0}\text{c}ʻ\alpha =\text{Nc}ʻ\beta +_{c}\text{Nc}ʻ\xi .\\ +[*119·103] &\supset .\exists !(\text{Nc}ʻ\alpha -_{c}\text{Nc}ʻ\beta )_{\alpha }:\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*119·22.</b> \(\vdash :\text{Nc}ʻ\alpha \geq \text{Nc}ʻ\beta .\supset .\exists !(\text{Nc}ʻ\alpha -_{c}\text{Nc}ʻ\beta )_{\alpha }\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*117·221.&\supset \vdash :\text{Hp}.\supset .(\exists \rho ).\rho \subset \alpha .\rho \text{ sm }\beta .\\ +[*119·21] &\supset .(\exists \rho ).\exists !(\text{Nc}ʻ\alpha -_{c}\text{Nc}ʻ\rho )_{\alpha }.\rho \text{ sm }\beta .\\ +[*100·35.*119·04] &\supset .\exists !(\text{Nc}ʻ\alpha -_{c}\text{Nc}ʻ\beta )_{\alpha }:\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*119·23.</b> \(\vdash :\exists !(\text{Nc}ʻ\alpha -_{c}\text{Nc}ʻ\beta ).\supset .(\exists \delta ).\delta \text{ sm }\beta .\delta \subset \alpha\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*119·103.\supset \vdash :\text{Hp}.&\supset .(\exists \xi ).\text{N}_{0}\text{c}ʻ\alpha =\text{Nc}ʻ\beta +_{c}\text{Nc}ʻ\xi .\\ +[*110·71] &\supset .(\exists \delta ).\delta \text{ sm }\beta .\delta \subset \alpha :\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*119·24.</b> \(\vdash :\exists !(\text{Nc}ʻ\alpha -_{c}\text{Nc}ʻ\beta ).\supset .\text{Nc}ʻ\alpha \geq \text{Nc}ʻ\beta \quad[*119·23.*117·221]\)</p> + +<p class="nind"><b><a id="*119·25">*119·25</a>.</b> \(\vdash :\gamma \geq \nu .\supset .\exists !(\gamma -_{c}\nu )\cap t_{0}ʻ\gamma\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*117·24.\supset \vdash :\text{Hp}.&\supset .(\exists \alpha ,\beta ).\gamma =\text{N}_{0}\text{c}ʻ\alpha .\nu =\text{N}_{0}\text{c}ʻ\beta .\text{N}_{0}\text{c}ʻ\alpha + \geq \text{N}_{0}\text{c}ʻ\beta .\\ +[*117·107] &\supset .(\exists \alpha ,\beta ).\gamma =\text{N}_{0}\text{c}ʻ\alpha .\nu =\text{N}_{0}\text{c}ʻ\beta .\text{Nc}ʻ\alpha \geq \text{Nc}ʻ\beta .\\ +[*119·22·04] &\supset .(\exists \alpha ,\beta ).\gamma =\text{N}_{0}\text{c}ʻ\alpha .\nu =\text{N}_{0}\text{c}ʻ\beta .\exists !(\text{N}_{0}\text{c}ʻ\alpha + -_{c}\text{N}_{0}\text{c}ʻ\beta )_{\alpha }.\\ +[(*63·02).*13·193] &\supset ·\exists !(\gamma -_{c}\nu )\cap t_{0}ʻ\gamma :\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*119·26">*119·26</a>.</b> \(\vdash :\exists !\gamma -_{c}\nu .\supset .\gamma \geq \nu\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*119·11.\supset \vdash :\text{Hp}.&\supset .(\exists \alpha ,\beta ).\gamma =\text{N}_{0}\text{c}ʻ\alpha .\nu =\text{N}_{0}\text{c}ʻ\beta .\exists !(\text{N}_{0}\text{c}ʻ\alpha + -_{c}\text{N}_{0}\text{c}ʻ\beta ).\\ +[*119·04·24] &\supset .(\exists \alpha ,\beta ).\gamma =\text{N}_{0}\text{c}ʻ\alpha .\nu =\text{N}_{0}\text{c}ʻ\beta .\text{Nc}ʻ\alpha \geq \text{Nc}ʻ\beta .\\ +[*117·107.*13·193] &\supset .\gamma \geq \nu :\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*119·27.</b> \(\vdash :\gamma \geq \nu .\equiv .\exists !(\gamma -_{c}\nu )\cap t_{0}ʻ\gamma \quad[*119·25·26]\)</p> + +<p>For the extension of this theorem cf. <a href="#*119·64">*119·64</a>.</p> + +<p><span class="pagenum" id="Page_204">[Pg 204]</span></p> + +<p class="nind"><b>*119·31.</b> \(\vdash :\gamma ,\nu \in \text{N}_{0}\text{C}.\supset .\text{ sm }ʻʻ\gamma \subset (\gamma +_{c}\nu )-_{c}\nu\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*119·1.(\text{IIT}).\supset \vdash :\xi \in (\gamma +_{c}\nu )-_{c}\nu .&\equiv .\text{Nc}ʻ\xi +_{c}\nu =\gamma +_{c}\nu .\exists !\gamma +_{c}\nu + &\qquad \text{(1)}\\ +\vdash .*100·51·521. &\supset \vdash \colon\ldotp \text{Hp}.\supset :\xi \in \text{ sm }ʻʻ\gamma .\supset .\text{Nc}ʻ\xi =\gamma .\\ +[*103·22.*118·01] &\supset .\text{Nc}ʻ\xi +_{c}\nu =\gamma +_{c}\nu .\\ +[*110·22·03.*103·13] &\supset .\text{Nc}ʻ\xi +_{c}\nu =\gamma +_{c}\nu .\exists !\gamma +_{c}\nu .\\ +[(1)] &\supset .\xi \in (\gamma +_{c}\nu )-_{c}\nu :\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p>The penultimate step in the proof employs the principle, explained in +the prefatory statement, that, since in the previous line the equation +\[ +\text{Nc}ʻ\xi +_{c}\nu =\gamma +_{c}\nu +\] +has its sides undetermined in type by the conventions \(\text{IT}\) and +\(\text{IIT}\), any convenient type can be chosen for them. The type +chosen in this line is such that \(\exists !\gamma +_{c}\nu\), and the +references indicate the existence of at least one such type.</p> + +<p class="nind"><b>*119·32.</b> \[\begin{align}&\vdash :(\gamma +_{c}\nu )-_{c}\nu \in \text{N}_{0}\text{C}.\supset .\text{ sm }ʻʻ\gamma =(\gamma +_{c}\nu )-_{c}\nu\\ +&[*119·11·31.*103·22.*100·52·42]\end{align}\]</p> + +<p class="nind"><b>*119·33.</b> \(\vdash :\text{Nc}ʻ\alpha -_{c}\text{Nc}ʻ\beta \in \text{N}_{0}\text{C}.\supset .(\text{Nc}ʻ\alpha -_{c}\text{Nc}ʻ\beta )+_{c}\text{Nc}ʻ\beta + =\text{Nc}ʻ\alpha\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*119·13.&\supset \vdash :\text{N}_{0}\text{c}ʻ\gamma =\text{Nc}ʻ\alpha -_{c}\text{Nc}ʻ\beta .\supset .\alpha \text{ sm }(\gamma +\beta ) &\qquad \text{(1)}\\ +\vdash .*20·18.*118·01.\supset \vdash \colon\ldotp \text{Hp}(1).\supset : +(\text{Nc}ʻ\alpha -_{c}\text{Nc}ʻ\beta )+_{c}\text{Nc}ʻ\beta =\text{N}_{0}\text{c}ʻ\xi .&\equiv _{\xi }.\text{Nc}ʻ\gamma +_{c}\text{Nc}ʻ\beta + =\text{N}_{0}\text{c}ʻ\xi .\\ +[*110·3.*100·35] &\equiv _{\xi }.\xi \text{ sm }(\gamma +\beta ).\\ +[(1).*103·42] &\equiv _{\xi }.\text{N}_{0}\text{c}ʻ\xi =\text{Nc}ʻ\alpha &\qquad \text{(2)}\\ +\vdash .*103·2·34.&\supset \vdash \colon\ldotp \text{Hp}.\supset :\exists !\text{Nc}ʻ\alpha .\supset .(\exists \xi ).\text{N}_{0}\text{c}ʻ\xi =\text{Nc}ʻ\alpha .\\ +[(2).*10·1] &\supset .(\text{Nc}ʻ\alpha -_{c}\text{Nc}ʻ\beta )+_{c}\text{Nc}ʻ\beta =\text{Nc}ʻ\alpha &\qquad \text{(3)}\\ +\vdash .*110·42.*103·34·2.&\supset \vdash \colon\ldotp \text{Hp}.\supset :\\ +&\exists !\{(\text{Nc}ʻ\alpha -_{c}\text{Nc}ʻ\beta )+_{c}\text{Nc}ʻ\beta \}.\supset .(\exists \xi ).\text{N}_{0}\text{c}ʻ\xi =(\text{Nc}ʻ\alpha -_{c}\text{Nc}ʻ\beta + )+_{c}\text{Nc}ʻ\beta .\\ +[(2).*10·1] &\supset .(\text{Nc}ʻ\alpha -_{c}\text{Nc}ʻ\beta )+_{c}\text{Nc}ʻ\beta =\text{Nc}ʻ\alpha &\qquad \text{(4)}\\ +\vdash .(3).(4).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*119·34">*119·34</a>.</b> \[\begin{align}&\vdash :\gamma -_{c}\nu \in \text{N}_{0}\text{C}.\supset .(\gamma -_{c}\nu )+_{c}\nu =\text{ sm }ʻʻ\gamma\\ +&[*119·11·33.*103·2.*100·51.*118·01]\end{align}\]</p> + +<p class="nind"><b>*119·35.</b> \(\vdash :\gamma -_{c}\nu \in \text{N}_{0}\text{C}.\supset .\alpha +_{c}\gamma =(\alpha +_{c}\nu )+_{c}(\gamma -_{c}\nu )\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*110·51·56.\supset \vdash :\text{Hp}.\supset .(\alpha +_{c}\nu )+_{c}(\gamma -_{c}\nu )&=\alpha +_{c}\{(\gamma -_{c}\nu )+_{c}\nu\}\\ +[*119·34] &=\alpha +_{c}\text{ sm }ʻʻ\gamma \\ +[*118·24.*119·11] &=\alpha +_{c}\gamma :\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*119·41.</b> \[\begin{align}\vdash \colon\ldotp \delta \in \text{Nc}ʻ\beta& -_{c}\text{Nc}ʻ\gamma .\supset :\\ +&\xi \in (\text{Nc}ʻ\alpha +_{c}\text{Nc}ʻ\beta )-_{c}\text{Nc}ʻ\gamma .\equiv .{(\alpha +\delta )+\gamma }\text{ sm }(\xi +\gamma )\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*119·12.*110·3. &\supset \vdash :\xi \in (\text{Nc}ʻ\alpha +_{c}\text{Nc}ʻ\beta )-_{c}\text{Nc}ʻ\gamma + .\equiv .(\alpha +\beta )\text{ sm }(\xi +\gamma ) &\qquad \text{(1)}\\ +\vdash .*119·12. &\supset \vdash :\text{Hp}.\equiv .\beta \text{ sm }(\delta +\gamma ) &\qquad \text{(2)}\\ +\vdash .(1).(2).*110·15·53.\supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_205">[Pg 205]</span></p> + +<p class="nind"><b>*119·42.</b> \[\begin{align}\vdash \colon\ldotp \text{Nc}ʻ\beta -_{c}\text{Nc}ʻ\gamma &\in \text{N}_{0}\text{C}.\eta \in \text{Nc}ʻ\alpha +_{c}(\text{Nc}ʻ\beta + -_{c}\text{Nc}ʻ\gamma ).\supset :\\ +&\xi \in (\text{Nc}ʻ\alpha +_{c}\text{Nc}ʻ\beta )-_{c}\text{Nc}ʻ\gamma .\equiv .(\eta +\gamma )\text{ sm }(\xi +\gamma )\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*118·01.*110·3.*103·2.*100·31.&\supset \vdash \colon\ldotp \text{N}_{0}\text{c}ʻ\delta =\text{Nc}ʻ\beta -_{c}\text{Nc}ʻ\gamma .\supset:\\ +&\eta \in \text{Nc}ʻ\alpha +_{c}(\text{Nc}ʻ\beta -_{c}\text{Nc}ʻ\gamma ).\equiv .\eta \text{ sm }(a+\delta ) &\qquad \text{(1)}\\ +\vdash .*119·41.(1).*103·12.*110·15.&\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p>Note that if \(\gamma\) be an infinite class, it does not follow from +\((\eta +\gamma)\text{ sm }(\xi +\gamma)\) that \(\eta \text{ sm }\xi\). +This will be proved, however, when \(\gamma\) is an inductive class (cf. +<a href="#*120·41">*120·41</a>).</p> + +<p class="nind"><b>*119·43.</b> \[\begin{align}\vdash :\text{Nc}ʻ\beta -_{c}\text{Nc}ʻ\gamma &\in \text{N}_{0}\text{C}.\supset .\\ +&\text{Nc}ʻ\alpha +_{c}(\text{Nc}ʻ\beta -_{c}\text{Nc}ʻ\gamma )\subset (\text{Nc}ʻ\alpha +_{c}\text{Nc}ʻ\beta )-_{c}\text{Nc}ʻ\gamma \end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*119·42.\supset \vdash \colon\ldotp \text{Hp}.&\eta \in \text{Nc}ʻ\alpha +_{c}(\text{Nc}ʻ\beta -_{c}\text{Nc}ʻ\gamma ).\supset :\\ +&\eta \in (\text{Nc}ʻ\alpha +_{c}\text{Nc}ʻ\beta )-_{c}\text{Nc}ʻ\gamma .\equiv .(\eta +\gamma )\text{ sm }(\eta +\gamma ):\\ +[*73·3] \supset :&\eta \in (\text{Nc}ʻ\alpha +_{c}\text{Nc}ʻ\beta )-_{c} \text{Nc}ʻ\gamma &\qquad \text{(1)}\\ +\vdash .(1).*22·1.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*119·44.</b> \(\vdash :\mu +_{c}(\nu -_{c}\varpi )\subset (\mu +_{c}\nu )-_{c}\varpi\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*119·11·43.*103·2.&\supset \\ +\vdash : \nu -_{c} \varpi \in \text{N}_{0}\text{C}.\mu \in \text{N}_{0}\text{C}.&\supset .\mu +_{c}(\nu -_{c}\varpi )\subset (\mu +_{c}\nu )-_{c}\varpi + &\qquad \text{(1)}\\ +\vdash . *110·4·42.*119·11.&\supset \\ +\vdash :{\sim}\{\nu -_{c}\varpi \in \text{N}_{0}\text{C}.\mu \in \text{N}_{0}\text{C}\}.&\supset .\mu +_{c}(\nu -_{c}\varpi )=\Lambda .\\ +[*24·12] &\supset .\mu +_{c}(\nu -_{c}\varpi )\subset (\mu +_{c}\nu )-_{c}\varpi &\qquad \text{(2)}\\ +\vdash .(1).(2).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*119·45.</b> \[\begin{align}&\vdash :(\mu +_{c} \nu )-_{c}\varpi \in \text{NC}.\exists !\{\mu +_{c}(\nu -_{c}\varpi )\}.\supset .\mu +_{c}(\nu + -_{c}\varpi )=(\mu +_{c}\nu )-_{c}\varpi \\ +&[*119·44.*100·33·321.*110·42]\end{align}\]</p> + +<p class="nind"><b>*119·51.</b> \(\vdash :\text{ sm }_{\delta ,\gamma }ʻʻ(\text{Nc}ʻ\alpha -_{c}\text{Nc}ʻ\beta )_{\gamma }=(\text{Nc}ʻ\alpha -_{c}\text{Nc}ʻ\beta )_{\delta }\cap + \text{D}ʻ\text{ sm }_{\delta ,\gamma }\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*119·12. \supset \vdash :\eta \in (\text{Nc}ʻ\alpha -_{c}\text{Nc}ʻ\beta )_{\gamma }.\zeta \text{ sm }_{\delta ,\gamma }\eta + .&\equiv .\alpha \text{ sm }\eta +\beta .\zeta \text{ sm }_{\delta ,\gamma }\eta .\\ +[*110·15] &\equiv .\alpha \text{ sm }\zeta +\beta .\zeta \text{ sm }_{\delta ,\gamma }\eta .\\ +[*119·12] &\equiv .\zeta \in (\text{Nc}ʻ\alpha -_{c}\text{Nc}ʻ\beta )_{\delta }.\zeta \text{ sm }_{\delta ,\gamma }\eta :\\ +[*37·1.*33·13]\supset \vdash .\text{ sm }_{\delta ,\gamma }ʻʻ(\text{Nc}ʻ\alpha -_{c}\text{Nc}ʻ\beta )_{\gamma }&=(\text{Nc}ʻ\alpha -_{c}\text{Nc}ʻ\beta )_{\delta }\cap + \text{D}ʻ\text{ sm }_{\delta ,\gamma }:\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*119·52.</b> \(\vdash :\text{ sm }_{\delta ,\gamma }ʻʻ(\mu -_{c}\nu )_{\gamma }=(\mu -_{c}\nu )_{\delta }\cap \text{D}ʻ\text{ sm }_{\delta ,\gamma } + \quad[*119·51·11]\)</p> + +<p>The difficulty in respect to types, which arises from the fact that +\(\text{ sm }_{\delta ,\gamma }ʻʻ(\mu -_{c}\nu )_{\gamma }\) and \((\mu-_{c}\nu )_{\delta }\) +have not been proved to be identical, does not +exist when \(\nu\) is an "inductive number"; cf. <a href="#*120·413">*120·413</a>.</p> + +<p class="nind"><b>*119·53.</b> \(\vdash \colon\ldotp tʻ\delta =\text{D}ʻ\text{ sm }_{\delta ,\gamma }.\supset :\text{ sm }_{\delta ,\gamma }ʻʻ(\mu -_{c}\nu )_{\gamma }=(\mu -_{c}\nu + )_{\delta } \quad[*119·52.(*65·01)]\)</p> + +<p class="nind"><b>*119·531.</b> \(\vdash :tʻ\delta =\text{D}ʻ\text{ sm }_{\delta ,\gamma }.(\mu -_{c}\nu )_{\delta }\in \text{N}_{0}\text{C}.\supset .\text{ sm }_{\gamma ,\delta }ʻʻ(\mu + -_{c}\nu )_{\delta }\in \text{N}_{0}\text{C}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*65·13. \supset \vdash :\text{Hp}.&\supset .(\mu -_{c}\nu )_{\delta }\subset \text{D}ʻ\text{ sm }_{\delta ,\gamma }.\\ +[*37·43.*103·22.(*65·1)]&\supset .\exists !\text{ sm }_{\gamma ,\delta }ʻʻ(\mu -_{c}\nu )_{\delta }.\\ +[*100·52.*103·34] &\supset .\text{ sm }_{\gamma ,\delta }ʻʻ(\mu -_{c}\nu )_{\delta }\in \text{N}_{0}\text{C}:\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_206">[Pg 206]</span></p> + +<p class="nind"><b>*119·532.</b> \[\begin{align}\vdash : tʻ\delta = \text{D}ʻ\text{ sm }_{\delta ,\gamma }.(\mu -_{c}\nu )_{\delta } \in \text{N}_{0}\text{C}.&(\mu -_{c}\nu + )_{\gamma } \in \text{NC}.\supset .\\ +&\text{ sm }_{\gamma ,\delta }ʻʻ(\mu -_{c}\nu )_{\delta } = (\mu -_{c}\nu )_{\gamma }\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *119·52·531 .\supset \vdash : \text{Hp} . &\supset . \exists ! (\mu -_{c}\nu )_{\gamma }.\\ +[*119·52·531·*100·34 &\supset . \text{ sm }_{\gamma ,\delta }ʻʻ(\mu -_{c}\nu )_{\delta } = (\mu -_{c}\nu )_{\gamma } : \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*119·54.</b> \(\text{SM} (\delta ,\gamma ). = : tʻ\delta = \text{D}ʻ\text{ sm }_{\delta ,\gamma } . \lor . tʻ\gamma = \text{D}ʻ\text{ sm }_{\gamma ,\delta } \quad\text{Df}\)</p> + +<p class="nind"><b>*119·541.</b> \[\begin{align}\vdash : SM(\delta ,\gamma ).(\mu -_{c}\nu )_{\gamma } \in &\text{N}_{0}\text{C} . (\mu -_{c}\nu )_{\delta } + \in \text{NC}.\supset .\\ +&\text{ sm }_{\delta ,\gamma }ʻʻ(\mu -_{c}\nu )_{\gamma } = (\mu -_{c}\nu )_{\delta } \quad[*119·53·532]\end{align}\]</p> + +<p class="nind"><b><a id="*119·61">*119·61</a>.</b> \(\vdash : \mu \in \text{N}_{0}\text{C} . \exists ! \text{ sm }_{\xi }ʻʻ\mu . \supset . \mu -_{c}\nu = \text{ sm }_{\xi }ʻʻ\mu + -_{c}\nu\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *119·1 .\supset \vdash \colon\ldotp \text{Hp}.\supset : \eta \in \mu -_{c}\nu .&\equiv .\text{Nc}ʻ\eta +_{c}\nu = \mu . \exists ! \mu .\\ +[*103·16.*118·201.*37·29] &\equiv .(\text{Nc}ʻ\eta +_{c}\nu )_{\xi } = \text{ sm }_{\xi }ʻʻ\mu .\\ +[*119·1] &\equiv . \eta \in \text{ sm }_{\xi }ʻʻ\mu -_{c}\nu \colon\ldotp \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*119·62">*119·62</a>.</b> \(\vdash : \nu \in \text{N}_{0}\text{C} . \exists ! \text{ sm }_{\xi }ʻʻ\nu . \supset .\mu -_{c}\nu = \mu -_{c} + \text{ sm }_{\xi }ʻʻ\nu\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *119·1 .\supset \vdash \colon\ldotp \text{Hp}.\supset : \eta \in \mu -_{c}\nu .&\equiv .\text{Nc}ʻ\eta +_{c}\nu = \mu . \exists ! \mu .\\ +[*110·25] &\equiv .\text{Nc}ʻ\eta +_{c} \text{ sm }_{\xi }ʻʻ\nu = \mu .\exists !\mu .\\ +[*119·1] &\equiv . \eta \in \mu -_{c}\text{ sm }_{\xi }ʻʻ\nu \colon\ldotp \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*119·63.</b> \(\vdash : \mu , \nu \in \text{N}_{0}\text{C} . \exists ! \text{ sm }_{\xi }ʻʻ\mu .\supset .\mu -_{c}\nu = \text{ sm }_{\xi }ʻʻ\mu -_{c} + \text{ sm }_{\xi }ʻʻ\nu\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *119·26 . &\supset \vdash : \text{Hp} . \exists !\mu -_{c}\nu . \supset . \mu \geq \nu .\\ +[*118·13] & \supset . \exists !\text{ sm }_{\xi }ʻʻ\nu .\\ +[*119·61·62] &\supset . \mu -_{c}\nu = \text{ sm }_{\xi }ʻʻ\mu -_{c}\text{ sm }_{\xi }ʻʻ\nu &\qquad \text{(1)}\\ +\vdash . *119·11.*103·13. \supset \\ +\vdash : \text{Hp}.\exists ! \text{ sm }_{\xi }ʻʻ\mu -_{c}\text{ sm }_{\xi }ʻʻ\nu . &\supset . \exists ! \text{ sm }_{\xi }ʻʻ\nu .\\ +[*119·61·62] &\supset . \mu -_{c}\nu = \text{ sm }_{\xi }ʻʻ\mu -_{c} \text{ sm }_{\xi }ʻʻ\nu &\qquad \text{(2)}\\ +\vdash . (1).(2).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*119·64">*119·64</a>.</b> \(\vdash \colon\ldotp \exists ! \text{ sm }_{\xi }ʻʻ\mu .\supset :\mu \geq \nu .\equiv . \exists !(\mu -_{c}\nu )_{\xi }\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *117·24. \supset \vdash \colon\ldotp \text{Hp}.\supset :\mu \geq \nu .&\supset .\mu ,\nu \in \text{N}_{0}\text{C} . \exists ! \text{ sm }_{\xi }ʻʻ\mu .\\ +[*119·61] & \supset .(\mu -_{c}\nu )_{\xi } = (\text{ sm }_{\xi }ʻʻ\mu -_{c}\nu )_{\xi } &\qquad \text{(1)}\\ +\vdash . *117·24·244 .\supset \vdash \colon\ldotp \text{Hp}.\supset :\mu \geq \nu .&\supset .\text{ sm }_{\xi }ʻʻ\mu \geq \nu .\\ +[*119·27] &\supset . \exists ! (\text{ sm }_{\xi }ʻʻ\mu -_{c}\nu )_{\xi }.\\ +[(1)] &\supset . \exists !(\mu -_{c}\nu )_{\xi } &\qquad \text{(2)}\\ +\vdash .(2).*119·26.\supset \vdash .\text{Prop} +\end{array} +\]</p> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_207">[Pg 207]</span></p> +<h2 class="nobreak" id="*120"> +*120. INDUCTIVE CARDINALS.</h2> +</div> + + +<p class="nind"><i>Summary of</i> *120.</p> + +<p>Inductive Cardinals are those that obey mathematical induction starting +from 0, <i>i.e.</i> in the language of Part II, Section E, they are +the posterity of 0 with respect to the relation of \(\nu\) to \(\nu+_{c}1\), +or, in more popular language, they are those that can be +reached from 0 by successive additions of 1. In former days, these +were supposed to be all the cardinals, and mathematical induction was +treated as a kind of self-evident axiom. We now know that only certain +cardinals obey mathematical induction starting from 0. It is these +cardinals which are to be considered in this number. They embrace 0, 1, +2, ... and generally all those cardinals which would be commonly called +finite, all those which can be expressed in the usual Arabic system of +numeration, and no others. The propositions to be proved concerning +them in this number are elementary and familiar; the interest lies +entirely in the definition and method of proof, not in the propositions +themselves.</p> + +<p>Put \(\text{NC induct} = \hat{\alpha} \{\alpha (+_{c}1)_{*}0\}\quad\text{Df}.\)</p> + +<p>Since \((+_{c}1)_{*}\) has necessarily its domain and converse domain +of the same type, it is important to be careful in noting the relations +of type. Accordingly we also put +\[ +\text{N}_{\xi }\text{C induct} = \hat{\alpha} \{\alpha (+_{c}1)_{*}0_{\xi }\} \quad\text{Df}. +\]</p> + +<p>We begin by applying the propositions of *90. Thus we have</p> + +<p class="nind"><b>*120·11.</b> \(\vdash \colon\ldotp \alpha \in \text{N}_{\eta }\text{C induct}:\phi \xi .\supset _{\xi }.\phi (\xi +_{c}1):\phi 0_{\eta }:\supset + .\phi \alpha\)</p> + +<p class="nind"><b>*120·12.</b> \(\vdash .0\in \text{NC induct}\)</p> + +<p class="nind"><b>*120·121.</b> \(\vdash :\alpha \in \text{N}_{\xi }\text{C induct}.\supset .(\alpha +_{c}1)_{\xi }\in \text{N}_{\xi }\text{C induct}\)</p> + +<p class="nind"><b>*120·13.</b> \(\vdash \colon\ldotp \alpha \in \text{N}_{\eta }\text{C induct}:\xi \in \text{N}_{\eta }\text{C induct}.\phi \xi .\supset _{\xi }.\phi (\xi +_{c}1):\phi + 0_{\eta }:\supset .\phi \alpha \)</p> + +<p class="nind"><b>*120·15.</b> \(\vdash :\alpha \in \text{NC induct}.\exists !\alpha .\supset .\text{ sm }ʻʻ\alpha \in \text{NC induct}\)</p> + +<p class="nind"><b>*120·151.</b> \(\vdash :\alpha \in \text{NC induct}.\exists !\alpha .\supset .a+_{c}1\in \text{NC induct}\)</p> + +<p class="nind"><b>*120·152.</b> \(\vdash :\alpha \in \text{NC}.\text{ sm }ʻʻ\alpha \in \text{NC induct}-\iota ʻ\Lambda .\supset .\alpha \in \text{NC induct}-\iota ʻ\Lambda\)</p> + +<p>We then proceed to deduce the elementary properties of inductive +<i>classes</i>, putting +\[ +\text{Cls induct} = sʻ\text{NC induct}. +\]</p> + +<p>We have</p> + +<p><span class="pagenum" id="Page_208">[Pg 208]</span></p> + +<p class="nind"><b>*120·21.</b> \(\vdash :\rho \in \text{Cls induct}.\equiv .\text{N}_{0}\text{c}ʻ\rho \in \text{NC induct}\)</p> + +<p class="nind"><b>*120·211.</b> \(\vdash :\text{Nc}ʻ\rho \in \text{NC induct}-\iota ʻ\Lambda .\supset .\rho \in \text{Cls induct}\)</p> + +<p>(We do not have an equivalence here, because, for aught we know, it +might be possible to determine the ambiguity of \(\text{Nc}ʻ\rho\) +so that \(\text{Nc}ʻ\rho = \Lambda\), even when \(\rho \in \text{Cls induct}\). +his will not be possible, however, if the axiom of infinity +is assumed.)</p> + +<p class="nind"><b>*120·212·213.</b> \(\vdash .\Lambda ,\iota ʻx\in \text{Cls induct}\)</p> + +<p class="nind"><b>*120·214.</b> \(\vdash \colon\ldotp \rho \text{ sm }\sigma .\supset :\rho \in \text{Cls induct}.\equiv .\sigma \in \text{Cls induct}\)</p> + +<p>We have a set of propositions applying induction to classes directly, +and not through the intermediary of cardinals. Thus we have</p> + +<p class="nind"><b>*120·251.</b> \(\vdash :\eta \in \text{Cls induct}.\supset .\eta \cup \iota ʻy\in \text{Cls induct}\)</p> + +<p class="nind"><b>*120·26.</b> \(\vdash \colon\ldotp \rho \in \text{Cls induct}:\phi\eta .\supset _{\eta ,x}.\phi(\eta \cup \iota ʻx):\phi\Lambda :\supset .\phi\rho\)</p> + +<p>We then state the axiom of infinity, and prove (<a href="#*120·33">*120·33</a>) that it +is equivalent to the assumption that if \(\alpha\) is an inductive +cardinal, \(\alpha \neq \alpha +_{c}1\). To prove this, we first +prove various propositions about \(\alpha +_{c}1\), among others the +following:</p> + +<p class="nind"><b>*120·311.</b> \(\vdash :\exists !\alpha +_{c}1.a+_{c}1=\beta +_{c}1.\supset .\alpha =\text{ sm }ʻʻ\beta .\exists !\alpha\)</p> + +<p class="nind"><b>*120·322.</b> \(\vdash \colon\ldotp \alpha \in \text{NC induct}.\supset :\exists !\alpha .\equiv .\alpha \neq \alpha +_{c}1\)</p> + +<p>We then proceed to consider subtraction (<a href="#*120·41">*120·41</a>—<a href="#*120·418">·418</a>), which only gives +a cardinal number when the subtrahend is an inductive cardinal. We have</p> + +<p class="nind"><b>*120·41.</b> \(\vdash \colon\ldotp \nu \in \text{NC induct}.\exists !\alpha +_{c}\nu .\supset :\alpha +_{c}\nu =\beta +_{c}\nu .\supset .\alpha =\text{ sm }ʻʻ\beta\)</p> + +<p>We might validly put \(\alpha = \beta\) instead of \(\alpha =\text{ sm }ʻʻ\beta\), +since \(\alpha = \beta\) will be true whenever it +is significant.</p> + +<p>We have</p> + +<p class="nind"><b>*120·411.</b> \[\begin{align}\vdash \colon\ldotp \nu \in &\text{NC induct}.\supset :\\ +&\exists !\gamma -_{c}\nu .\supset .\gamma -_{c}\nu \in \text{N}_{0}\text{C}:\gamma \geq \nu .\equiv .(\gamma -_{c}\nu )\cap t_{0}ʻ\gamma + \in \text{N}_{0}\text{C}\end{align}\]</p> + +<p class="nind"><b>*120·4111.</b> \(\vdash \colon\ldotp \nu \in \text{NC induct}.\exists !\text{ sm }_{\xi }ʻʻ\gamma .\supset :\gamma \geq \nu .\equiv .(\gamma -_{c}\nu )_{\xi }\in + \text{N}_{0}\text{C}\)</p> + +<p>Hence we arrive at the conditions requisite for the usual point of view +of subtraction; namely,</p> + +<p class="nind"><b>*120·412.</b> \(\vdash :\nu \in \text{NC induct}.\gamma \geq \nu .\exists !\text{ sm }_{\xi }ʻʻ\gamma .\supset .(\gamma -_{c}\nu )_\xi =\{(\iota \alpha )(\alpha +_{c}\nu + =\gamma )\}_{\xi }\)</p> + +<p>Also from <a href="#*120·4111">*120·4111</a> we deduce</p> + +<p class="nind"><b>*120·414.</b> \(\vdash :\mu \in \text{N}_{0}\text{C}-\iota ʻ0.\exists !\text{ sm }_{\xi }ʻʻ\mu .\supset .(\mu -_{c}1)_{\xi }\in \text{N}_{0}\text{C}\)</p> + +<p>And from <a href="#*120·411">*120·411</a>.<a href="#*119·34">*119·34</a>, we find</p> + +<p class="nind"><b>*120·416.</b> \(\vdash :\nu \in \text{NC induct}.\exists !\gamma -_{c}\nu .\supset .(\gamma -_{c}\nu )+_{c}\nu =\text{ sm }ʻʻ\gamma\)</p> + +<p>We prove next that no proper part of an inductive class is similar +to the whole (<a href="#*120·426">*120·426</a>), <i>i.e.</i> that inductive classes are +non-reflexive, and various connected propositions, <i>e.g.</i></p> + +<p><span class="pagenum" id="Page_209">[Pg 209]</span></p> + +<p class="nind"><b>*120·423.</b> \(\vdash :\alpha \in \text{N}_{\eta }\text{C induct} -\iota ʻ0.\equiv .(\exists \beta ).\beta \in \text{N}_{\eta }\text{C + induct}.\alpha =(\beta +_{c}1)_{\eta }\)</p> + +<p class="nind"><b>*120·4232.</b> \(\vdash :\alpha \in \text{N}_{\eta }\text{C induct} -\iota ʻ0.\equiv .(\exists \beta ).\beta \in \text{N}_{\eta }\text{C + induct} -\iota ʻ\Lambda .\alpha =(\beta +_{c}1)_{\eta }\)</p> + +<p class="nind"><b>*120·428.</b> \(\vdash :\nu \in \text{NC induct}.\exists !\alpha +_{c}\nu .\alpha \neq 0.\supset .\alpha +_{c}\nu >\nu\)</p> + +<p class="nind"><b>*120·429.</b> \(\vdash \colon\ldotp \nu \in \text{NC induct}.\supset :\mu >\nu .\equiv .\mu \geq \nu +_{c}1\)</p> + +<p>The last two of the above propositions do not hold in general when +\(\nu\) is a cardinal which is not inductive.</p> + +<p>We prove next that if \(\alpha\) is an existent inductive cardinal, +then any existent cardinal is greater than, equal to, or less than +\(\alpha\) (<a href="#*120·441">*120·441</a>); that if \(\alpha\), \(\beta\) are inductive +cardinals, so is \(\alpha +_{c}\beta\) (<a href="#*120·45">*120·45</a><a href="#*120·4501">·4501</a>), and if \(\alpha+_{c}\beta\) +is an inductive cardinal other than \(\Lambda\), so are +\(\alpha\) and \(\beta\) (<a href="#*120·452">*120·452</a>). We then have some propositions +dealing with mathematical induction starting from 1 or 2, <i>e.g.</i></p> + +<p class="nind"><b>*120·4622.</b> \[\begin{align}\vdash \colon\ldotp \alpha \in \text{NC}.\beta \in &\text{NC}(\eta ).\exists !\text{ sm }_{\xi }ʻʻ\beta .\supset :\\ +&\beta (+_{c} 1)_{*}\text{ sm }_{\eta }ʻʻ\alpha .\equiv .\text{ sm }_{\xi }ʻʻ\beta (+_{c}1)_{*}\text{ sm }_{\xi }ʻʻ\alpha\end{align}\]</p> + +<p class="nind"><b>*120·47.</b> \(\vdash \colon\colon \beta \in \text{N}_{\eta }\text{C induct} -\iota ʻ0.\equiv \colon\ldotp \xi \in \mu .\supset _{\xi }.(\xi +_{c}1)_{\eta }\in + \mu :1_{\eta }\in \mu :\supset _{\mu }.\beta \in \mu\)</p> + +<p>From <a href="#*120·452">*120·452</a> we deduce</p> + +<p class="nind"><b>*120·48.</b> \(\vdash :\beta \in \text{NC induct}.\beta \geq \alpha .\supset .\alpha \in \text{NC induct}-\iota ʻ\Lambda\)</p> + +<p class="nind">so that any number less than an inductive number is inductive. Hence</p> + +<p class="nind"><b>*120·481.</b> \(\vdash :\eta \in \text{Cls induct}.\xi \subset \eta .\supset .\xi \in \text{Cls induct}\)</p> + +<p class="nind">which is a proposition constantly used, and</p> + +<p class="nind"><b>*120·491.</b> \(\vdash \colon\ldotp \xi {\sim}\in \text{Cls induct}.\equiv :\beta \in \text{NC induct}.\supset _{\beta }.\exists !\beta \cap \text{Cl}ʻ\xi\)</p> + +<p>We then prove that if \(\alpha\), \(\beta\) are inductive cardinals, +\(\alpha \times _{c}\beta\) and \(\alpha ^\beta\) are either inductive +cardinals or \(\Lambda\) (<a href="#*120·5">*120·5</a><a href="#*120·52">*120·52</a>), while conversely if \(\alpha\times _{c} \beta\) +or \(\alpha ^\beta\) is an existent inductive cardinal, \(\alpha\) and +\(\beta\) are so also, with exceptions for 0 and 1 (<a href="#*120·512">*120·512</a><a href="#*120·56">·56</a><a href="#*120·561">·561</a>). +Hence we infer the uniqueness of division and the taking of roots +(<a href="#*120·51">*120·51</a><a href="#*120·53">·53</a><a href="#*120·55">·55</a>) so long as inductive numbers are concerned.</p> + +<p>We have next a set of propositions on the axiom of infinity and the +multiplicative axiom. We prove (<a href="#*120·61">*120·61</a>) that if there is any existent +cardinal which is not inductive, the axiom of infinity is true. From +*83·9·904, we infer by induction that if \(\kappa\) is an inductive +class of which \(\Lambda\) is not a number, \({\in}_{\Delta}ʻ\kappa\) +exists (<a href="#*120·62">*120·62</a>), whence it follows that either the multiplicative +axiom or the axiom of infinity must be true (<a href="#*120·64">*120·64</a>).</p> + +<p>Finally, we have a set of propositions on inductive classes. We prove</p> + +<p class="nind"><b>*120·71.</b> \(\vdash :\rho ,\sigma \in \text{Cls induct}.\equiv .\rho \cup \sigma \in \text{Cls induct}.\equiv .\rho +\sigma \in \text{Cls induct}\)</p> + +<p class="nind"><b>*120·74.</b> \(\vdash :\rho \in \text{Cls induct}.\equiv .\text{Cl}ʻ\rho \in \text{Cls induct}\)</p> + +<p class="nind"><b>*120·75.</b> \(\vdash :sʻ\kappa \in \text{Cls induct}.\equiv .\kappa \in \text{Cls induct}.\kappa \subset \text{Cls induct}\)</p> + +<p>with analogous propositions (involving however a hypothesis as to +<span class="pagenum" id="Page_210">[Pg 210]</span>\(\kappa\)) on the subject of \({\in}_{\Delta}ʻ\kappa\).</p> + +<p>The propositions of the present number are essential to the ordinary +arithmetic of finite numbers. In the present work, however, they are +not much used after the present section until we reach Part V, Section +E, where we deal with the ordinal theory of finite and infinite.</p> + +<hr class="tb"> + +<p class="nind"><b>*120·01.</b> \(\text{NC induct} = \hat{\alpha}\{\alpha (+_{c}1)_{*}0\} \quad\text{Df}\)</p> + +<p>Note that in virtue of our general conventions for descriptive +functions of two arguments (*38), +\[ ++_{c}1=\hat{\alpha} \hat{\beta} (a=\beta +_{c}1). +\] +That is, \(+_{c}1\) is the relation of a cardinal to its immediate +predecessor. It is the number written in the usual mathematical +notation as +1 in the series of positive and negative integers, just as +its converse is the number -1. (It should be observed that if \(\nu\) +is any cardinal, +\(\nu\) is not identical with \(\nu\), since +\(\nu\) +is a relation, while \(\nu\) is a class of classes.)</p> + +<p class="nind"><b>*120·011.</b> \(\text{N}_{\xi }\text{C induct} =\hat{\alpha} \{\alpha (+_{c}1)_{*}0_{\xi }\} \quad\text{Df}\)</p> + +<p>All members of \(\text{N}_{\xi }\text{C induct}\) belong to the +same type as \(0_{\xi }\), so that, if \(\alpha\) is any member of +\(\text{N}_{\xi }\text{C induct}\), "\(\xi \in \alpha\)" is significant.</p> + +<p class="nind"><b>*120·02.</b> \(\text{Cls induct} = sʻ\text{NC induct} \quad\text{Df}\)</p> + +<p class="nind"><b>*120·021.</b> \(\text{Cls}_{\xi }\text{induct} = sʻ\text{N}_{\xi }\text{C induct} \quad\text{Df}\)</p> + +<p>In virtue of these definitions an inductive class is one whose cardinal +is an inductive cardinal.</p> + +<p class="nind"><b><a id="*120·03">*120·03</a>.</b> \(\text{Infin ax}.=:\alpha \in \text{NC induct}.\supset _{a}.\exists !\alpha \quad\text{Df}\)</p> + +<p>"\(\text{Infin ax}\)," like "\(\text{Mult ax}\)," is an arithmetical +hypothesis which some will consider self-evident, but which we prefer +to keep as a hypothesis, and to adduce in that form whenever it is +relevant. Like "\(\text{Mult ax}\)," it states an existence-theorem. +In the above form, it states that, if \(\alpha\) is any inductive +cardinal, there is at least one class (of the type in question) +which has \(\alpha\) terms. An equivalent assumption would be that, +if \(\rho\) is any inductive class, there are objects which are not +members of \(\rho\). For in that case, if \(x\) be such an object, +\(\text{Nc}ʻ(\rho \cup \iota ʻx)=\text{Nc}ʻ\rho +_{c}1\). Hence by +induction, every inductive cardinal must exist. Another equivalent +assumption would be that \(\text{V}\) (the class of all objects of +the type in question) is not an inductive class. The assumption that +\(\aleph _{0}\) exists in the type in question is, as we shall see, a +stronger assumption than the above, unless we assume the multiplicative +axiom.</p> + +<p>If the axiom of infinity is true, the inductive cardinals are all +different one from another, <i>i.e.</i> \(\alpha +_{c}\beta\), where +\(\alpha\) and \(\beta\) are inductive cardinals, is not equal to +\(\alpha\) unless \(\beta = 0\). But if the axiom of infinity is false, +then, in any assigned type, all the cardinals after a certain one are +\(\Lambda\). (Except in the lowest type, the last existent cardinal +must be a power of 2.) That is, if (say) 8 were the largest existent +cardinal in the type in question, we should<span class="pagenum" id="Page_211">[Pg 211]</span> have, in that type, +\(9 = \Lambda\), and the same would hold of 10, 11, .... This +possibility has to be taken account of in what follows.</p> + +<p>In order to give typical definiteness to the axiom of infinity, we write</p> + +<p class="nind"><b><a id="*120·04">*120·04</a>.</b> \(\text{Infin ax}(x).=:\alpha \in \text{NC induct}.\supset _{a}.\exists !\alpha (x) \quad\text{Df}\)</p> + +<p>Then "\(\text{Infin ax} (x)\)" states that, if \(\alpha\) is any +inductive cardinal, there are at least \(\alpha\) objects of the same +type as \(x\).</p> + +<p class="nind"><b><a id="*120·1">*120·1</a>.</b> \(\vdash :\alpha \in \text{NC induct}.\equiv .\alpha (+_{c}1)_{*}0 \quad[(*120·01)]\)</p> + +<p class="nind"><b><a id="*120·101">*120·101</a>.</b> \[\begin{align}&\vdash \colon\colon \alpha \in \text{NC induct}. \equiv \colon\ldotp \xi \in \mu .\supset _{\xi }.\xi +_{c}1\in + \mu :0\in \mu :\supset _{\mu }.\alpha \in \mu \\ +&[*120·1.*90·131.*38·12]\end{align}\]</p> + +<p>The right-hand side of the above equivalence gives the usual formula +for mathematical induction. Observe that the conditions of significance +require that \(\xi +_{c}1\) should be taken in the same type as +\(\xi\). This fact is specially relevant in the proof of <a href="#*120·15">*120·15</a>.</p> + +<p>The symbol "\(\text{NC induct}\)" is of ambiguous type not necessarily +the same in different occurrences; also, according to the convention +explained in the prefatory statement as holding for \(\text{NC}\) and +\(\text{NC induct}\), "\(\alpha ,\beta \in \text{NC induct}\)" will not +imply that \(\alpha\) and \(\beta\) are of the same type. Accordingly +to avoid error in connection with <a href="#*120·1">*120·1</a><a href="#*120·101">·101</a> typical definiteness is +required as in the three following propositions.</p> + +<p class="nind"><b>*120·102.</b> \(\vdash :\alpha \in \text{N}_{\eta }\text{C induct}. \equiv .\alpha (+_{c} 1)_{*}0_{\eta } \quad[(*120·011)]\)</p> + +<p class="nind"><b>*120·103.</b> \[\begin{align}&\vdash \colon\colon \alpha \in \text{N}_{\eta }\text{C induct}. \equiv \colon\ldotp \xi \in \mu .\supset _{\xi }.(\xi + +_{c}1)_{\eta }\in \mu :0_{\eta }\in \mu :\supset _{\mu }.\alpha \in \mu \\ +&[*120·101]\end{align}\]</p> + +<p class="nind"><b><a id="*120·11">*120·11</a>.</b> \[\begin{align}&\vdash \colon\ldotp \alpha \in \text{N}_{\eta }\text{C induct}:\phi\xi .\supset _{\xi }.\phi (\xi +_{c}1):\phi 0_{\eta }:\supset + .\phi \alpha \\ +&[*120·102.*90·112]\end{align}\]</p> + +<p class="nind"><b><a id="*120·12">*120·12</a>.</b> \(\vdash .0\in \text{NC induct} \quad\left[*120·101 \frac{0}{\alpha}\right]\)</p> + +<p class="nind"><b><a id="*120·121">*120·121</a>.</b> \(\vdash :\alpha \in \text{N}_{\xi }\text{C induct}.\supset .(\alpha +_{c}1)_{\xi }\in \text{N}_{\xi }\text{C induct} \quad[*90·172.*120·102]\)</p> + +<p>By means of this proposition and <a href="#*120·12">*120·12</a>, any assigned cardinal in the +series of natural numbers can be shown to be an inductive cardinal; +thus <i>e.g.</i> to show that 27 is an inductive cardinal, we shall +only have to use <a href="#*120·121">*120·121</a> twenty-seven times in succession.</p> + +<p class="nind"><b>*120·122.</b> \(\vdash .1\in \text{NC induct} \quad[*120·12·121.*110·641]\)</p> + +<p class="nind"><b><a id="*120·123">*120·123</a>.</b> \(\vdash .2\in \text{NC induct}. \text{etc}. \quad[*120·122·121.*110·643]\)</p> + +<p class="nind"><b>*120·124.</b> \(\vdash .\alpha +_{c}1\neq 0\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*110·4. \text{Transp}.&\supset \vdash :\alpha {\sim}\in \text{NC}.\supset .\alpha +_{c}1=\Lambda .\\ +[*101·12] &\supset .\alpha +_{c}1\neq 0 &\qquad \text{(1)}\\ +\vdash .*110·632. &\supset \vdash \colon\ldotp \alpha \in \text{NC}.\supset :\xi \in \alpha +_{c}1.\supset .\exists !\xi :\\ +[*24·63] &\supset :\Lambda {\sim}\in \alpha +_{c}1:\\ +[*54·102] &\supset :\alpha +_{c}1\neq 0 &\qquad \text{(2)}\\ +\vdash .(1).(2).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_212">[Pg 212]</span></p> + +<p class="nind"><b>*120·13.</b> \(\vdash \colon\ldotp \alpha \in \text{N}_{\eta }\text{C induct}:\xi \in \text{N}_{\eta }\text{C induct}.\phi \xi .\supset _{\xi }.\phi (\xi +_{c}1):\phi + 0_{\eta }:\supset .\phi \alpha\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*120·121. &\supset \vdash \colon\ldotp \xi \in \text{N}_{\eta }\text{C induct}.\phi \xi .\supset _{\xi }.\phi (\xi +_{c}1):\supset :\\ +&\xi \in \text{N}_{\eta }\text{C induct}.\phi \xi .\supset _{\xi }.(\xi +_{c}1)_{\eta }\in \text{N}_{\eta }\text{C induct}.\phi (\xi +_{c}1) + &\qquad \text{(1)}\\ +\vdash .*120·12. &\supset \vdash :\phi 0_{\eta }.\supset .0_{\eta }\in \text{N}_{\eta }\text{C induct}.\phi 0_{\eta } &\qquad \text{(2)}\\ +\vdash .(1).(2). &\supset \vdash \colon\ldotp \text{Hp}.\supset :\\ +\xi \in \text{N}_{\eta }\text{C induct}.\phi \xi .&\supset _{\xi }.(\xi +_{c}1)_{\eta }\in \text{N}_{\eta }\text{C induct}.\phi (\xi +_{c}1):0_{\eta }\in + \text{N}_{\eta }\text{C induct}.\phi 0_{\eta }:\\ +\left[*120·11 \frac{\xi \in \text{N}_{\eta }\text{C induct}.\phi \xi}{\phi \xi }\right]&\supset :\alpha \in \text{N}_{\eta }\text{C + induct}.\phi \alpha \colon\ldotp \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p>The above proposition is often convenient for inductive proofs.</p> + +<p class="nind"><b>*120·14.</b> \(\vdash .\text{NC induct} \subset \text{NC}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *110·42. \text{Simp}.\supset \vdash :\alpha \in \text{NC}.\supset .\alpha +_{c}1\in \text{NC} &\qquad \text{(1)}\\ +\vdash .(1).*101·11. *120·11 \frac{\alpha \in \text{NC}}{\phi \alpha} .\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p>This proposition does not show that every inductive cardinal is an +<i>existent</i> cardinal; to obtain this, we require the axiom of +infinity.</p> + +<p class="nind"><b><a id="*120·15">*120·15</a>.</b> \(\vdash :\alpha \in \text{NC induct}.\exists !\alpha .\supset .\text{ sm }ʻʻ\alpha \in \text{NC induct}\)</p> + +<p><i>i.e.</i> a cardinal which is not null and is inductive in any one +type is also inductive in any other type.</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*101·15.*120·12.&\supset \vdash .\text{ sm }_{\eta }ʻʻ0_{\xi }\in \text{N}_{\eta }\text{C induct} &\qquad \text{(1)}\\ +\vdash .*110·4. &\supset \vdash .\alpha =\Lambda _{\xi }.\supset .(\alpha +_{c}1)_{\xi } = \Lambda _{\xi } &\qquad \text{(2)}\\ +\vdash .*118·201. &\supset \vdash :\exists !(\alpha +_{c}1)_{\xi }.\supset .\text{ sm }_{\eta }ʻʻ(\alpha +_{c}1)_{\xi }=(\alpha + +_{c}1)_{\eta }\\ +[*118·241.*110·4] &= (\text{ sm }_{\eta }ʻʻ\alpha +_{c} 1)_{\eta } &\qquad \text{(3)}\\ +\vdash .*120·121. &\supset \vdash :\exists !(\alpha +_{c}1)_{\xi }.\text{ sm }_{\eta }ʻʻ\alpha \in \text{N}_{\eta }\text{C induct}.\supset .(\text{ sm }_{\eta }ʻʻ\alpha + +_{c}1)_{\eta }\in \text{N}_{\eta }\text{C induct}.\\ +[(3)] &\supset .\text{ sm }_{\eta }ʻʻ(\alpha +_{c}1)_{\xi }\in 1\,\text{N}_{C}\text{induct} &\qquad \text{(4)}\\ +\vdash .(4).*2·2. & \supset \vdash \colon\ldotp \text{ sm }_{\eta }ʻʻ\alpha \in \text{N}_{\eta }\text{C induct}.\supset :\\\\ +&(\alpha +_{c}1)_{\xi }=\Lambda _{\xi }.\lor.\text{ sm }_{\eta }ʻʻ(\alpha +_{c}1)_{\xi }\in \text{N}_{\eta }\text{C + induct} &\qquad \text{(5)}\\ +\vdash .(2).(5).*3·48.&\supset \vdash \colon\ldotp \alpha =\Lambda _{\xi }.\lor.\text{ sm }_{\eta }ʻʻ\alpha \in \text{N}_{\eta }\text{C induct}:\supset :\\ +&(\alpha +_{c}1)_{\xi }=\Lambda _{\xi }.\lor.\text{ sm }_{\eta }ʻʻ(\alpha +_{c}1)_{\xi }\in \text{N}_{\eta }\text{C + induct} &\qquad \text{(6)}\\ +\vdash .(1).(6).*120·11.*4·6.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*120·151.</b> \(\vdash :\alpha \in \text{NC induct}.\exists !\alpha .\supset .\alpha +_{c}1\in \text{NC induct}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*120·15. \supset \vdash :\alpha \in \text{N}_{\xi }\text{C induct}.\exists !\alpha .&\supset .\text{ sm }_\eta ʻʻ\alpha \in \text{N}_{\eta }\text{C induct}.\\ +[*120·121] &\supset .(\text{ sm }_{\eta }ʻʻ\alpha +_{c}1)_{\eta }\in \text{N}_{\eta }\text{C induct}.\\ +[*118·241.*120·14] & \supset .(\alpha +_{c}1)_{\eta }\in \text{N}_{\eta }\text{C induct}:\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*120·152.</b> \(\vdash :\alpha \in \text{NC}.\text{ sm }ʻʻ\alpha \in \text{NC induct}-\iota ʻ\Lambda .\supset .\alpha \in \text{NC induct}-\iota ʻ\Lambda\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*100·521. \supset \vdash :\text{Hp}.&\supset .\text{ sm }ʻʻ\text{ sm }ʻʻ\alpha =\alpha .\\ +[*120·15] &\supset .\alpha \in \text{NC induct} &\qquad \text{(1)}\\ +\vdash .*37·29. &\supset \vdash :\text{Hp}.\supset .\exists !\alpha &\qquad \text{(2)}\\ +\vdash .(1).(2).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_213">[Pg 213]</span></p> + +<p>The following propositions, giving alternative forms for the definition +of inductive <i>classes</i>, are inserted in order to show that the +theory of inductive classes might be treated in a less arithmetical +manner than we have adopted.</p> + +<p class="nind"><b>*120·2.</b> \(\vdash :\rho \in \text{Cls induct}.\equiv .(\exists \alpha ).\alpha \in \text{NC induct}.\rho \in \alpha \quad[(*120·02)]\)</p> + +<p class="nind"><b>*120·201.</b> \(\vdash \colon\ldotp \rho \text{ sm }\sigma .\supset :\text{N}_{0}\text{c}ʻ\rho \in \text{NC induct}.\equiv .\text{N}_{0}\text{c}ʻ\sigma \in \text{NC induct}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*100·35.*103·13.*100·511.\supset \\ +\vdash :\text{Hp}.\supset .\text{N}_{0}\text{c}ʻ\rho =\text{ sm }ʻʻ\text{N}_{0}\text{c}ʻ\sigma .\text{N}_{0}\text{c}ʻ\sigma =\text{ sm }ʻʻ\text{N}_{0}\text{c}ʻ\rho :\\ +[*120·152.*103·13] \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*120·21.</b> \(\vdash :\rho \in \text{Cls induct}.\equiv .\text{N}_{0}\text{c}ʻ\rho \in \text{NC induct}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*120·14·2. \supset \vdash :\rho \in \text{Cls induct}.&\equiv .(\exists \alpha ).\alpha \in \text{NC induct}.\alpha \in \text{NC}.\rho \in \alpha .\\ +[*103·27] &\equiv .(\exists \alpha ).\alpha \in \text{NC induct}. \text{N}_{0}\text{c}ʻ\rho =\alpha .\\ +[*13·195] &\equiv .\text{N}_{0}\text{c}ʻ\rho \in \text{NC induct}:\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p>Note that "\(\rho \in \text{Cls induct}.\equiv .\text{Nc}ʻ\rho \in\text{NC induct}\)" +is not proved above. The proof encounters the +difficulty that we may have \(\text{Nc}ʻ\rho = \Lambda\); in order +to establish our proposition in this case, we have to show that if +\(\Lambda \in \text{NC induct}\), then <i>every</i> class is an +inductive class. We can however prove the following implication.</p> + +<p class="nind"><b>*120·211.</b> \(\vdash :\text{Nc}ʻ\rho \in \text{NC induct} - \iota ʻ\Lambda .\supset .\rho \in \text{Cls induct}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*100·511.&\supset \vdash :\text{Hp}.\supset.\text{ sm }ʻʻ\text{Nc}ʻp=\text{N}_{0}\text{c}ʻ\rho .\\ +[*120·15] &\supset . \text{N}_{0}\text{c}ʻ\rho \in \text{NC induct}.\\ +[*120·21] &\supset .\rho \in \text{Cls induct}:\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*120·212.</b> \(\vdash .\Lambda \in \text{Cls induct} \quad[*120·211·12]\)</p> + +<p class="nind"><b>*120·213.</b> \(\vdash .\iota ʻ\alpha \in \text{Cls induct} \quad [*120·211·122]\)</p> + +<p class="nind"><b>*120·214.</b> \(\vdash \colon\ldotp \rho \text{ sm }\sigma .\supset :\rho \in \text{Cls induct}.\equiv .\sigma \in \text{Cls induct} \quad[*120·201·21]\)</p> + +<p>The following propositions are lemmas for <a href="#*120·24">*120·24</a>.</p> + +<p class="nind"><b>*120·22.</b> \(\vdash \colon\colon \eta \in \mu .\supset _{\eta ,y}.\eta \cup \iota ʻy\in \mu :\Lambda \in \mu :\supset _{\mu }.\rho + \in \mu \colon\ldotp \supset .\rho \in \text{Cls induct}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*120·212. &\supset \vdash .\Lambda \in \text{Cls induct} &\qquad \text{(1)}\\ +\vdash .*51·2. &\supset \vdash \colon\ldotp y\in \eta .\supset :\eta \cup \iota ʻy=\eta :\\ +[*13·12] &\supset :\eta \in \text{Cls induct}.\supset .\eta \cup \iota ʻy\in \text{Cls induct} &\qquad \text{(2)}\\ +\vdash .*110·63. &\supset \vdash \colon\ldotp y{\sim}\in \eta .\supset :\text{Nc}ʻ(\eta \cup \iota ʻy)=\text{Nc}ʻ\eta +_{c}\\ +[(*110·03)] &= \text{N}_{0}\text{c}ʻ\eta +_{c}1:\\ +[*120·121] & \supset :\text{N}_{0}\text{c}ʻ\eta \in \text{NC induct}.\supset .\text{N}_{0}\text{c}ʻ(\eta \cup \iota ʻy)\in \text{NC induct}:\\ +[*120·21·211] & \supset :\eta \in \text{Cls induct}.\supset .\eta \cup \iota ʻy\in \text{Cls induct} &\qquad \text{(3)}\\ +\vdash .(2).(3). &\supset \vdash :\eta \in \text{Cls induct}. \supset . \eta \cup \iota ʻy\in \text{Cls induct} &\qquad \text{(4)}\\ +\vdash .*10·1.(1).(4).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_214">[Pg 214]</span></p> + +<p class="nind"><b>*120·221.</b> \(\vdash \colon\ldotp \eta \in \mu .\supset _{\eta ,y}.\eta \cup \iota ʻy\in \mu :\text{Nc}ʻ\rho \subset \mu :\supset .\text{Nc}ʻ\rho +_{c}1\subset \mu \)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*110·63.*100·31.\supset \\ +\vdash :\zeta \in \text{Nc}ʻ\rho +_{c}1.&\equiv .(\exists \eta ,y).\eta \in \text{Nc}ʻ\rho .y{\sim}\in \eta .\zeta =\eta \cup \iota ʻy &\qquad \text{(1)}\\ +\vdash .*22·1. \supset \vdash \colon\ldotp \text{Hp}.&\supset :\eta \in \text{Nc}ʻ\rho .\supset .\eta \in \mu .\\ +[*10·1] & \supset .\eta \cup \iota ʻy\in \mu :\\ +[*3·41] & \supset :\eta \in \text{Nc}ʻ\rho .y{\sim}\in \eta .\supset .\eta \cup \iota ʻy\in \mu :\\ +[*13·12] &\supset :\eta \in \text{Nc}ʻ\rho .y{\sim}\in \eta .\zeta =\eta \cup \iota ʻy.\supset .\zeta \in \mu &\qquad \text{(2)}\\ +\vdash .(1).(2).&\supset \vdash \colon\ldotp \text{Hp}.\supset :\zeta \in \text{Nc}ʻ\rho +_{c}1.\supset .\zeta \in \mu \colon\ldotp \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*120·222">*120·222</a>.</b> \(\vdash \colon\ldotp \eta \in \mu .\supset _{\eta ,y}.\eta \cup \iota ʻy\in \mu :\xi \in \text{NC}.\xi \subset \mu :\supset .\xi +_{c}1\subset \mu\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*100·4.\supset \vdash :\text{Hp}.\exists !\xi .&\supset .(\exists \alpha ).\xi =\text{Nc}(\zeta )ʻ\alpha .\text{Nc}(\zeta )ʻ\alpha \subset \mu .\\ +[*120·221] &\supset .(\exists \alpha ).\xi =\text{Nc}(\zeta )ʻ\alpha .\text{Nc}ʻ\alpha +_{c}1\subset \mu .\\ +[*118·01] &\supset .\xi +_{c}1\subset \mu &\qquad \text{(1)}\\ +\vdash .*110·4.&\supset \vdash :{\sim}\exists !\xi .\supset .\xi +_{c}1\subset \mu &\qquad \text{(2)}\\ +\vdash .(1).(2).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p>The proof of this proposition might also proceed by the use of uniform +formal numbers, employing <a href="#*118·241">*118·241</a>.</p> + +<p class="nind"><b>*120·23.</b> \(\vdash \colon\ldotp \eta \in \mu .\supset _{\eta ,y}.\eta \cup \iota ʻy\in \mu :\Lambda \in \mu :\supset .\text{Cls induct}\subset \mu\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*51·2.*54·1. &\supset \vdash :\text{Hp}.\supset .0\subset \mu &\qquad \text{(1)}\\ +\vdash .*120·222·14. & \supset \vdash \colon\ldotp \text{Hp}.\supset :\xi \in \text{NC induct}.\xi \subset \mu .\supset _{\xi }.\xi +_{c}1\subset \mu &\qquad \text{(2)}\\ +\vdash .(1).(2).*120·13. &\supset \vdash \colon\ldotp \text{Hp}.\supset :\xi \in \text{NC induct}.\supset .\xi \subset \mu :\\ +[*40·151.(*120·02)] &\supset :\text{Cls induct}\subset \mu \colon\ldotp \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*120·24">*120·24</a>.</b> \(\vdash \colon\colon \rho \in \text{Cls induct}.\equiv \colon\ldotp \eta \in \mu .\supset _{\eta ,y}.\eta \cup \iota ʻy\in \mu :\Lambda \in \mu :\supset _{\mu }.\rho + \in \mu\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*120·23.\supset \vdash \colon\colon \rho \in \text{Cls induct}.\supset \colon\ldotp \eta \in \mu .\supset _{\eta ,y}.\mu + \cup \iota ʻy\in \mu :\Lambda \in \mu :\supset .\rho \in \mu &\qquad \text{(1)}\\ +\vdash .(1).*120·22.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p>This proposition might be used to define inductive classes. It gives +a form of mathematical induction applicable to classes instead of +to numbers. Virtually it states that an inductive class is one +which can be formed by adding members one at a time, starting from +\(\Lambda\). This is made more explicit in <a href="#*120·25">*120·25</a>. Instead of +\(\eta\in \mu .\supset _{\eta ,y}.\mu \cup \iota ʻy\in \mu\), +in the above propositions, as well as in those that follow, we may +plainly substitute +\[ +\eta \in \mu .y{\sim}\in \eta .\supset _{\eta ,y}.\eta \cup \iota ʻy\in \mu . +\]</p> + +<p class="nind"><b><a id="*120·25">*120·25</a>.</b> \[\begin{align}&\vdash :M=\hat{\eta} \hat{\zeta}\{(\exists y).\zeta =\eta \cup \iota ʻy\}.\supset .\text{Cls induct}=\overleftarrow{M}_{*}ʻ\Lambda\\ +&[*120·24.*90·131]\end{align}\]</p> + +<p class="nind"><b>*120·251.</b> \(\vdash :\eta \in \text{Cls induct}.\supset .\eta \cup \iota ʻy\in \text{Cls induct} \quad[*90·172.*120·25]\)</p> + +<p class="nind"><b>*120*26.</b> \[\begin{align}&\vdash \colon\ldotp \rho \in \text{Cls induct}:\phi\eta .\supset _{\eta ,x}.\phi(\eta \cup \iota ʻx):\phi\Lambda :\supset \phi\rho \\ +&[*120·25.*90·112]\end{align}\]</p> + +<p><span class="pagenum" id="Page_215">[Pg 215]</span></p> + +<p class="nind"><b>*120·261.</b> \[\begin{align}&\vdash \colon\ldotp \rho \in \text{Cls induct}:\eta \in \text{Cls induct}.\phi\eta .\supset _{\eta ,x}.\phi(\eta + \cup \iota ʻx):\phi\Lambda :\supset .\phi\rho \\ +&[*120·26·251·212]\end{align}\]</p> + +<p class="nind"><b>*120·27.</b> \(\vdash :\rho \in \text{Cls induct}.\supset .\text{Nc}ʻ\rho \cap tʻ\gamma \in \text{NC induct}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*120·12. &\supset \vdash .\text{Nc}ʻ\Lambda \cap tʻ\gamma \in \text{NC induct} &\qquad \text{(1)}\\ +\vdash .*13·12. &\supset \vdash :\text{Nc}ʻ\eta \cap tʻ\gamma \in \text{NC induct}.y\in \eta .\supset .\\ +&\text{Nc}ʻ(\eta \cup \iota ʻy)\cap tʻ\gamma \in \text{NC induct} &\qquad \text{(2)}\\ +\vdash .*110·63.*120·121.\supset\\ +&\vdash :\text{Nc}ʻ\eta \cap tʻ\gamma \in \text{NC induct}.y{\sim}\in \eta .\supset .\text{Nc}ʻ(\eta \cup \iota ʻy)\cap tʻ\gamma \in \text{NC induct} &\qquad \text{(3)}\\ +\vdash .(1).(2).(3).*120·26.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p>This proposition also follows immediately from *12·21·15.</p> + +<p class="nind"><b>*120·3.</b> \(\vdash \colon\ldotp \text{Infin ax}.\equiv :\alpha \in \text{NC induct}.\supset _{\alpha }.\exists !\alpha \quad[(*120·03)]\)</p> + +<p class="nind"><b>*120·301.</b> \(\vdash \colon\ldotp \text{Infin ax}(x).\equiv :\alpha \in \text{NC induct}.\supset _{\alpha }.\exists !\alpha (x) \quad[(*120·04)]\)</p> + +<p class="nind"><b><a id="*120·31">*120·31</a>.</b> \(\vdash :\exists !\text{Nc}ʻ\alpha +_{c}1.\text{Nc}ʻ\alpha +_{c}1=\text{Nc}ʻ\beta +_{c}1.\supset + .\text{Nc}ʻ\alpha =\text{Nc}ʻ\beta .\alpha \text{ sm }\beta \)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*110·63. &\supset \vdash \colon\ldotp \text{Nc}ʻ\alpha +_{c}1=\text{Nc}ʻ\beta +_{c}1.\equiv :\\ +&(\exists \gamma ,y).\gamma \text{ sm }\alpha .y{\sim}\in \gamma .\xi =\gamma \cup \iota ʻy.\equiv _{\xi }.(\exists + \delta ,z).\delta \text{ sm }\beta .z{\sim}\in \delta .\xi =\delta \cup \iota ʻz:\\ +[*10·1] &\supset :\gamma \text{ sm }\alpha .y{\sim}\in \gamma .\supset .(\exists \delta ,z).\delta \text{ sm }\beta .z{\sim}\in \delta .\gamma \cup \iota ʻy=\delta \cup \iota ʻz.\\ +[*73·72·3] &\supset .(\exists \delta ).\delta \text{ sm }\beta .\gamma \text{ sm }\delta .\\ +[*73·32] &\supset .\gamma \text{ sm }\beta .\\ +[*73·32] &\supset .\alpha \text{ sm }\beta &\qquad \text{(1)}\\ +\vdash .*110·63. &\supset \vdash :\text{Hp}.\supset .(\exists \gamma ,y).\gamma \text{ sm }\alpha .y{\sim}\in \gamma &\qquad \text{(2)}\\ +\vdash .(1).(2).*100·321.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*120·311">*120·311</a>.</b> \[\begin{align}&\vdash :\exists !\alpha +_{c}1.\alpha +_{c}1=\beta +_{c}1.\supset .\alpha =\text{ sm }ʻʻ\beta .\exists !\alpha\\ +&[*120·31.*110·4.*103·16·4·2]\end{align}\]</p> + +<p class="nind"><b><a id="*120·32">*120·32</a>.</b> \(\vdash :\alpha \in \text{NC induct}.\exists !\alpha .\supset .\alpha \neq \alpha +_{c}1\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*101·22.*110·641.&\supset \vdash .0_{\xi }\neq 0_{\xi }+_{c}1 &\qquad \text{(1)}\\ +\vdash .*120·311.*110·44.&\supset \vdash :\alpha \in \text{NC}.\exists !\alpha +_{c}1.\alpha +_{c}1=\alpha +_{c}1+_{c}1.\supset .\alpha =\alpha +_{c}1:\\ +[\text{Transp}] &\supset \vdash :\alpha \in \text{NC}.\exists !\alpha +_{c}1.\alpha \neq \alpha +_{c}1.\supset .\alpha +_{c}1\neq \alpha +_{c}1+_{c}1:\\ +[*118·2·25] &\supset \vdash :\alpha \in \text{NC}(\xi ).\exists !(\alpha +_{c}1)_{\xi }.\alpha \neq (\alpha +_{c}1)_{\xi }.\supset + .(\alpha +_{c}1)_{\xi }\neq \{(\alpha +_{c}1)_{\xi }+_{c}1\}_{\xi} &\qquad \text{(2)}\\ +\vdash .(2).&\supset \vdash \colon\ldotp \alpha \in \text{NC}(\xi ).\alpha \neq (\alpha +_{c}1)_{\xi }.\supset :(\alpha +_{c}1)_{\xi }=\Lambda + .\lor.(\alpha +_{c}1)_{\xi }\neq \{(\alpha +_{c}1)_{\xi }+_{c}1\}_{\xi} &\qquad \text{(3)}\\ +\vdash .*110·4. \text{Transp}.&\supset \vdash \colon\ldotp \alpha {\sim}\in \text{NC}(\xi ).\lor.\alpha =\Lambda _{\xi }:\supset .(\alpha +_{c}1)_{\xi }=\Lambda _{\xi } + &\qquad \text{(4)}\\ +\vdash .(3).(4).&\supset \vdash \colon\ldotp \alpha =\Lambda _{\xi }.\lor.\alpha \neq (\alpha +_{c}1)_{\xi }:\supset :\\ +&(\alpha +_{c}1)_{\xi }=\Lambda _{\xi }.\lor.(\alpha+_{c}1)_{\xi }\neq {(\alpha +_{c}1)_{\xi }+_{c}1}_{\xi } + &\qquad \text{(5)}\\ +\vdash .(1).(5).*120·11.&\supset \vdash \colon\ldotp \alpha \in \text{N}_{\xi }\text{C induct}.\supset :\alpha =\Lambda _{\xi }.\lor.\alpha \neq (\alpha +_{c}1)_{\xi }\colon\ldotp + \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_216">[Pg 216]</span></p> + +<p class="nind"><b>*120·321.</b> \(\vdash :\alpha \neq \alpha +_{c}1.\supset .\exists !\alpha\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*110·4. \text{Transp}.&\supset \vdash :\alpha =\Lambda .\supset .\alpha +_{c}1=\Lambda &\qquad \text{(1)}\\ +\vdash .(1).\text{Transp}. &\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*120·322">*120·322</a>.</b> \(\vdash \colon\ldotp \alpha \in \text{NC induct}.\supset :\exists !\alpha .\equiv .\alpha \neq \alpha +_{c}1 \quad[*120·32·321]\)</p> + +<p class="nind"><b><a id="*120·33">*120·33</a>.</b> \(\vdash \colon\ldotp \text{Infin ax}.\equiv :\alpha \in \text{NC induct}.\supset _{\alpha }.\alpha \neq \alpha +_{c}1 \quad[*120·3·322]\)</p> + +<p class="nind"><b><a id="*120·41">*120·41</a>.</b> \(\vdash \colon\ldotp \nu \in \text{NC induct}.\exists !\alpha +_{c}\nu .\supset :\alpha +_{c}\nu =\beta +_{c}\nu .\supset .\alpha =\text{ sm }ʻʻ\beta\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*110·4. \text{Transp}.*118·25.&\supset \vdash :(\alpha +_{c}\nu )_{\xi }=\Lambda .\supset .\{\alpha +_{c}(\nu +_{c}1)_{\xi }\}_{\xi }=\Lambda + &\qquad \text{(1)}\\ +\vdash .*118·25 .&\supset \vdash \colon\colon \exists !{\alpha +_{c}(\nu +_{c}1)_{\xi }}_{\xi }.\supset \colon\ldotp \exists !\{(\alpha +_{c}\nu + )_{\xi }+_{c}1\}_{\xi}\colon\ldotp \\ +[*120·311.*110·4.*118·201] +&\supset \colon\ldotp \{(\alpha +_{c}\nu )_{\xi }+_{c}1\}_{\xi}=\{(\beta +_{c}\nu )_{\xi }+_{c}1\}_{\xi }.\supset .(\alpha +_{c}\nu + )_{\xi}=(\beta +_{c}\nu )_{\xi }\colon\ldotp \\ +[\text{Syll}.*118·25] &\supset \colon\ldotp (\alpha +_{c}\nu )_{\xi }=(\beta +_{c}\nu )_{\xi }.\supset .\alpha =\text{ sm }ʻʻ\beta :\supset :\\ +&\{\alpha +_{c}(\nu +_{c}1)_{\xi }\}_{\xi}=\{\beta +_{c}(\nu +_{c}1)_{\xi }\}_{\xi}.\supset + .\alpha =\text{ sm }ʻʻ\beta &\qquad \text{(2)}\\ +\vdash .(2).\text{Comm}.&\supset \vdash \colon\colon (\alpha +_{c}\nu )_{\xi }=(\beta +_{c}\nu )_{\xi }.\supset .\alpha =\text{ sm }ʻʻ\beta :\supset \colon\ldotp \\ +&\{\alpha +_{c}(\nu +_{c}1)_{\xi }\}_{\xi} = \Lambda :\lor:\{\alpha +_{c}(\nu +_{c}1)_{\xi }\}_{\xi}=\{\beta +_{c}(\nu + +_{c}1)_{\xi }\}_{\xi }.\supset .\alpha =\text{ sm }ʻʻ\beta &\qquad \text{(3)}\\ +\vdash .(1).(3).&\supset \vdash \colon\colon (\alpha +_{c}\nu )_{\xi }=\Lambda :\lor:(\alpha +_{c}\nu )_{\xi }=(\beta +_{c}\nu )_{\xi }.\supset + .\alpha =\text{ sm }ʻʻ\beta \colon\ldotp \supset \colon\ldotp \\ +&\{\alpha +_{c}(\nu +_{c}1)_{\xi }\}_{\xi}=\Lambda :\lor:\{\alpha +_{c}(\nu +_{c}1)_{\xi }\}_{\xi }=\{\beta +_{c}(\nu + +_{c}1)_{\xi }\}_{\xi }.\supset .\alpha =\text{ sm }ʻʻ\beta &\qquad \text{(4)}\\ +\vdash .*110·4.*118·21.&\supset \vdash \colon\ldotp \exists !(\beta +_{c}0)_{\xi }.\supset :\beta \in \text{NC}.\exists !\text{ sm }_{\xi }ʻʻ\beta :\\ +[*102·87.*100·51] &\supset :\text{ sm }_{\xi }ʻʻ\alpha =\text{ sm }_{\xi }ʻʻ\beta .\supset .\alpha =\text{ sm }ʻʻ\beta &\qquad \text{(5)}\\ +\vdash .*110·6·4.&\supset \vdash :\exists !(\alpha +_{c}0)_{\xi }.(\alpha +_{c}0)_{\xi }=(\beta +_{c}0)_{\xi }.\supset .\text{ sm }_{\xi }ʻʻ\alpha + =\text{ sm }_{\xi }ʻʻ\beta .\\ +[(5)] &\supset .\alpha =\text{ sm }ʻʻ\beta &\qquad \text{(6)}\\ +\vdash .(6).\text{Exp}.*4·6.&\supset \vdash \colon\ldotp (\alpha +_{c}0)_{\xi }=\Lambda :\lor:(\alpha +_{c}0)_{\xi }=(\beta +_{c}0)_{\xi }.\supset + .\alpha =\text{ sm }ʻʻ\beta &\qquad \text{(7)}\\ +\vdash .(4).(7).*120·11.\supset \\ +\vdash \colon\colon \nu \in \text{N}_{\xi }\text{C induct}.&\supset \colon\ldotp (\alpha +_{c}\nu )_{\xi }=\Lambda :\lor:(\alpha +_{c}\nu )_{\xi }=(\beta + +_{c}\nu )_{\xi }.\supset .\alpha =\text{ sm }ʻʻ\beta &\qquad \text{(8)}\\) +\vdash .*110·4. &\supset \vdash :\nu =\Lambda_\eta.\supset .(\alpha +_c{\nu} )_\xi=\Lambda &\qquad \text{(9)}\\ +\vdash .*120·15.&\supset \vdash \colon\colon \nu \in \text{N}_\eta \text{C induct}-\iota ʻ\Lambda .\supset \colon\ldotp \text{ sm }_\xiʻʻ\nu =\text{N}_\xi\text{C + induct}\colon\ldotp \\ +[(8)] &\supset \colon\ldotp (\alpha +_c\text{ sm }_\xiʻʻ\nu )_\xi=\Lambda :\lor:(\alpha +_c\text{ sm }_\xiʻʻ\nu )_\xi=(\beta + +_c\text{ sm }_\xiʻʻ\nu )_\xi.\supset .\alpha =\text{ sm }ʻʻ\beta \colon\ldotp \\ +[*118·24] &\supset \colon\ldotp (\alpha +_c\nu )_\xi=\Lambda :\lor:(\alpha +_c\nu )_\xi=(\beta +_c\nu )_\xi.\supset + .\alpha =\text{ sm }ʻʻ\beta &\qquad \text{(10)}\\ +\vdash .(9).(10).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_217">[Pg 217]</span></p> + +<p>The above proposition establishes (with the natural limitations) the +uniqueness (within each type) of subtraction (conceived as in <a href="#*120·412">*120·412</a>) +when the subtrahend is an inductive cardinal. (When the subtrahend is +a non-inductive cardinal, subtraction ceases to give a unique result.) +Hence we are led to the following extensions of <a href="#*118">*118</a> for the case of +inductive cardinals:</p> + +<p class="nind"><b><a id="*120·411">*120·411</a>.</b> \[\begin{align}\vdash \colon\ldotp \nu \in &\text{NC induct}.\supset :\\ +&\exists !\gamma -_{c}\nu .\supset .\gamma -_{c}\nu \in \text{N}_{0}\text{C}:\gamma \geq \nu .\equiv .(\gamma -_{c}\nu )\cap t_{0}ʻ\gamma + \in \text{N}_{0}\text{C}\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*119·1. \supset \vdash \colon\ldotp &\nu \in \text{NC induct}.\supset :\\ +&\xi ,\eta \in \gamma -_{c}\nu .\supset .\text{Nc}ʻ\xi +_{c}\nu =\gamma .\text{Nc}ʻ\eta +_{c}\nu =\gamma .\exists !\text{Nc}ʻ\xi +_{c}\nu .\\ +[*20·22] &\supset .\text{Nc}ʻ\xi +_{c}\nu =\text{Nc}ʻ\eta +_{c}\nu .\exists !\text{Nc}ʻ\xi +_{c}\nu .\\ +[*120·41.*100·511.(*110·03)] &\supset .\text{Nc}ʻ\xi =\text{Nc}ʻ\eta &\qquad \text{(1)}\\ +\vdash .(1).*119·14. &\supset \vdash \colon\ldotp \text{Hp}.\supset :\exists !\gamma -_{c}\nu .\supset .\gamma -_{c}\nu \in \text{N}_{0}\text{C} + &\qquad \text{(2)}\\ +\vdash .*119·27.(2). &\supset \vdash \colon\ldotp \text{Hp}.\supset :\gamma \geq \nu .\supset .(\gamma -_{c}\nu )\cap t_{0}ʻ\gamma \in \text{N}_{0}\text{C} + &\qquad \text{(3)}\\ +\vdash .*103·22.*119·27. &\supset \vdash \colon\ldotp \text{Hp}.\supset :(\gamma -_{c}\nu )\cap t_{0}ʻ\gamma \in \text{N}_{0}\text{C}.\supset + .\gamma \geq \nu &\qquad \text{(4)}\\ +\vdash .(2).(3).(4).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*120·4111">*120·4111</a>.</b> \(\vdash \colon\ldotp \nu \in \text{NC induct}.\exists !\text{ sm }_{\xi }ʻʻ\gamma .\supset :\gamma \geq \nu .\equiv .(\gamma -_{c}\nu )_{\xi }\in + \text{N}_{0}\text{C}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*119·64. \supset \vdash \colon\ldotp \text{Hp}.\supset :\gamma \geq \nu .&\supset .\exists !(\gamma -_{c}\nu )_{\xi }.\\ +[*120·411] &\supset .(\gamma -_{c}\nu )_{\xi }\in \text{N}_{0}\text{C} &\qquad \text{(1)}\\ +\vdash .(1).*119·26.*103·13.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*120·412">*120·412</a>.</b> \(\vdash :\nu \in \text{NC induct}.\gamma \geq \nu .\exists !\text{ sm }_{\xi }ʻʻ\gamma .\supset .(\gamma -_{c}\nu )_{\xi }=\{({℩}\alpha )(\alpha +_{c}\nu + =\gamma )\}_{\xi }\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*120·4111.\supset \vdash \colon\ldotp \text{Hp}.&\supset .(\gamma -_{c}\nu )_{\xi }\in \text{N}_{0}\text{C}.\\ +[*119·34] &\supset .(\gamma -_{c}\nu )_{\xi }+_{c}\nu = \gamma &\qquad \text{(1)}\\ +\vdash .*120·41.*103·43.*37·29.&\supset \vdash \colon\ldotp \text{Hp}.\supset :\alpha +_{c}\nu =\gamma .\beta +_{c}\nu =\gamma .\supset _{\alpha ,\beta }.\alpha + =\beta &\qquad \text{(2)}\\ +\vdash .(1).(2).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*120·413">*120·413</a>.</b> \(\vdash :\mu \in \text{N}_{0}\text{C}.\supset .\mu -_{c}0=\text{ sm }ʻʻ\mu\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*119·1 .\supset \vdash \colon\ldotp \text{Hp}.\supset :\xi \in \mu -_{c}0.&\equiv .\text{N}_{0}\text{c}ʻ\xi +_{c}0=\mu .\exists !\mu .\\ +[*110·61.*103·13] &\equiv .\text{Nc}ʻ\xi =\mu .\\ +[*103·44·4] &\equiv .\text{N}_{0}\text{c}ʻ\xi =\text{ sm }ʻʻ\mu .\\ +[*103·26] &\equiv .\xi \in \text{ sm }ʻʻ\mu \colon\ldotp \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*120·414.</b> \[\begin{align}&\vdash :\mu \in \text{N}_{0}\text{C}-\iota ʻ0.\exists !\text{ sm }_{\xi }ʻʻ\mu .\supset .(\mu -_{c}1)_{\xi }\in + \text{N}_{0}\text{C}\\ +&[*120·4111.*117·53]\end{align}\]</p> + +<p class="nind"><b>*120·415.</b> \[\begin{align}&\vdash :\mu \in \text{N}_{0}\text{C}-\iota ʻ0-\iota ʻ1.\exists !\text{ sm }_{\xi }ʻʻ\mu .\supset .(\mu -_{c}2)_{\xi }\in + \text{N}_{0}\text{C}\\ +&[*120·4111.*117·551]\end{align}\]</p> + +<p class="nind"><b>*120·416.</b> \[\begin{align}&\vdash :\nu \in \text{NC induct}.\exists !\gamma -_{c}\nu .\supset .(\gamma -_{c}\nu )+_{c}\nu =\text{ sm }ʻʻ\gamma \\ +&[*120·411.*119·34]\end{align}\]</p> + +<p class="nind"><b>*120·417.</b> \[\begin{align}&\vdash :\mu \in \text{N}_{0}\text{C}-\iota ʻ0 .\exists !\text{ sm }_{\xi }ʻʻ\gamma .\supset .\alpha +_{c}\gamma + =(\alpha +_{c}1)+_{c}(\gamma -_{c}1)_{\xi }\\ +&[*120·414.*119·35]\end{align}\]</p> + +<p class="nind"><b><a id="*120·418">*120·418</a>.</b> \[\begin{align}&\vdash :\nu \in \text{NC induct}.\exists !\text{ sm }_{\xi }ʻʻ\gamma .\gamma \geq \nu .\supset .\alpha +_{c}\gamma =(\alpha +_{c}\nu + )+_{c}(\gamma -_{c}\nu )_{\xi }\\ +&[*120·4111.*119·35]\end{align}\]</p> + +<p><span class="pagenum" id="Page_218">[Pg 218]</span></p> + +<p class="nind"><b>*120·42.</b> \(\vdash :\nu \in \text{NC induct}.\exists !\nu .\alpha \neq 0.\supset .\nu \neq \alpha +_{c}\nu \)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*110·61.*120·14.\supset \vdash :\nu\in \text{NC induct}.\supset .\nu =0+_{c}\nu &\qquad \text{(1)}\\ +\vdash .*120·41.\supset \vdash :\nu \in \text{NC induct}.\exists !0+_{c}\nu .0+_{c}\nu =\alpha +_{c}\nu .\supset .0=\alpha &\qquad \text{(2)}\\ +\vdash .(1).(2).\supset \vdash :\nu \in \text{NC induct}.\exists !\nu .\nu =\alpha +_{c}\nu .\supset .\alpha =0:\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*120·422.</b> \(\vdash :\alpha +_{c}1\in \text{NC induct}-\iota ʻ\Lambda .\supset .\alpha \in \text{NC induct}-\iota ʻ\Lambda\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*120·1·124.*91·542.&\supset \vdash :\alpha +_{c}1\in \text{NC induct}.\supset .(\alpha +_{c}1)(+_{c}1)_{\text{po}}0.\\ +[*91·52] &\supset .(\exists \beta ).(\alpha +_{c}1)(+_{c}1)\beta .\beta (+_{c}1)_{*}0.\\ +[*120·1] &\supset .(\exists \beta ).\alpha +_{c}1=\beta +_{c}1.\beta \in \text{NC induct} &\qquad \text{(1)}\\ +\vdash .*120·311. & \supset \vdash \colon\ldotp \text{Hp}.\supset :\alpha +_{c}1=\beta +_{c}1.\supset .\alpha =\text{ sm }ʻʻ\beta .\exists !\alpha &\qquad \text{(2)}\\ +\vdash .(1).(2).*120·15.&\supset \vdash :\text{Hp}.\supset .\alpha \in \text{NC induct} &\qquad \text{(3)}\\ +\vdash .(3).*110·4.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*120·423.</b> \(\vdash :\alpha \in \text{N}_{\eta }\text{C induct}-\iota ʻ0.\equiv .(\exists \beta ).\beta \in \text{N}_{\eta }\text{C induct}.\alpha =(\beta +_{c}1)_{\eta }\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*120·121·124. &\supset \vdash :\beta \in \text{N}_{\eta }\text{C induct}.\alpha =(\beta +_{c}1)_{\eta }.\supset .\alpha \in \text{N}_{\eta }\text{C induct}-\iota ʻ0 &\qquad \text{(1)}\\ +\vdash .*120·102.*91·542.&\supset \vdash :\alpha \in \text{N}_{\eta }\text{C induct}-\iota ʻ0.\supset .\alpha (+_{c}1)_{\text{po}}0_{\eta }.\\ +[*91·52] &\supset .(\exists \beta ).\alpha (+_{c}1)\beta .\beta (+_{c}1)_{*}0_{\eta }.\\ +[*120·102] &\supset .(\exists \beta ).\beta \in \text{N}_{\eta }\text{C induct}.\alpha =(\beta +_{c}1)_{\eta } &\qquad \text{(2)}\\ +\vdash .(1).(2).\supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*120·4231.</b> \(\vdash :\alpha \in \text{N}_{\eta }\text{C induct}.\supset .(\exists \beta ).\beta \in \text{N}_{\eta }\text{C induct}-\iota ʻ\Lambda .(\alpha +_{c}1)_{\eta }=(\beta +_{c} 1)_{\eta }\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +&\vdash .*10·24.*101·12.*120·12.\supset \\ +&\vdash .(\exists \beta ).\beta \in \text{N}_{\eta }\text{C induct}-\iota ʻ\Lambda .(0+_{c}1)_{\eta }=(\beta +_{c} 1)_{\eta } &\qquad \text{(1)}\\ +\vdash .*120·121.&\supset \vdash \colon\ldotp \exists !\xi .\supset :\\ +&\beta \in \text{N}_{\eta }\text{C induct}-\iota ʻ\Lambda .\xi =(\beta +_{c}1)_{\eta }.\supset .\xi \in \text{N}_{\eta }\text{C induct}-\iota ʻ\Lambda .(\xi +_{c}1)_{\eta }=(\xi +_{c}1)_{\eta }:\\ +[*10·23·24] &\supset :(\exists \beta ).\beta \in \text{N}_{\eta }\text{C induct}-\iota ʻ\Lambda .\xi =(\beta +_{c}1)_{\eta }.\supset .\\ +&(\exists \gamma ).\gamma \in \text{N}_{\eta }\text{C induct}-\iota ʻ\Lambda .(\xi +_{c}1)_{\eta }=(\gamma +_{c}1)_{\eta } &\qquad \text{(2)}\\ +\vdash .*110·4.*13·17.\supset \\ +\vdash \colon\ldotp {\sim}\exists !\xi .&\supset :\beta \in \text{N}_{\eta }\text{C induct}-\iota ʻ\Lambda .\xi =(\beta +_{c}1)_{\eta }.\supset + .(\xi +_{c}1)_{\eta }=(\beta +_{c}1)_{\eta }:\\ +[*10·28] &\supset :(\exists \beta ).\beta \in \text{N}_{\eta }\text{C induct}-\iota ʻ\Lambda .\xi =(\beta +_{c}1)_{\eta }.\supset .\\ +&(\exists \beta ).\beta \in \text{N}_{\eta }\text{C induct}-\iota ʻ\Lambda .(\xi +_{c}1)_{\eta }=(\beta +_{c}1)_{\eta } &\qquad \text{(3)}\\ +\vdash .(2).(3).&\supset \vdash :(\exists \beta ).\beta \in \text{N}_{\eta }\text{C induct}-\iota ʻ\Lambda .\xi =(\beta +_{c}1)_{\eta }.\supset .\\ +&(\exists \beta ).\beta \in \text{N}_{\eta }\text{C induct}-\iota ʻ\Lambda .(\xi +_{c}1)_{\eta }=(\beta +_{c}1)_{\eta } &\qquad \text{(4)}\\ +\vdash .(1).(4) \frac{(\xi +_{c}1)_{\eta }}{\xi}.*120·11.\supset \\ +&\vdash :\alpha \in \text{N}_{\eta }\text{C induct}.\supset .(\exists \beta ).\beta \in \text{N}_{\eta }\text{C induct}-\iota ʻ\Lambda .(\alpha +_{c}1)_{\eta }=(\beta + +_{c}1)_{\eta }:\\ +\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*120·4232">*120·4232</a>.</b> \[\begin{align}&\vdash :\alpha \in \text{N}_{\eta }\text{C induct}-\iota ʻ0.\equiv .(\exists \beta ).\beta \in \text{N}_{\eta }\text{C induct}-\iota ʻ\Lambda .\alpha =(\beta +_{c}1)_{\eta }\\ +&[*120·423·4231]\end{align}\]</p> + +<p><span class="pagenum" id="Page_219">[Pg 219]</span></p> + +<p class="nind"><b><a id="*120·424">*120·424</a>.</b> \(\vdash :\beta \neq 0.\exists !(\alpha +_{c}\beta )_{\xi }.\supset .(\alpha +_{c}\beta )_{\xi }-_{c}1=\alpha +_{c}(\beta + -_{c}1)_{\xi }\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*110·42·62.\supset \vdash :\text{Hp}.\supset .(\alpha +_{c}\beta )_{\xi }\in \text{NC}-\iota ʻ\Lambda -\iota ʻ0.\\ +[*120·414.*103·13] \supset .\exists !(\alpha +_{c}\beta )_{\xi }-_{c}1 &\qquad \text{(1)}\\ +\vdash .*110·4.*118·21.*120·414.*103·13.\supset \vdash :\text{Hp}.\supset .\exists !(\beta -_{c}1)_{\xi } &\qquad \text{(2)}\\ +\vdash .(1).(2).*120·416.\supset \\ +\vdash :\text{Hp}. \supset .\{(\alpha +_{c}\beta )_{\xi }-_{c}1\}+_{c}1=\alpha +_{c}\beta .(\beta -_{c}1)_{\xi }+_{c}1=\beta + . &\qquad \text{(3)}\\ +[*110·56]\supset .\{(\alpha +_{c}\beta )_{\xi }-_{c}1\}+_{c}1=\{\alpha +_{c}(\beta -_{c}1)_{\xi }\}+_{c}1 &\qquad \text{(4)}\\ +\vdash .(3).\supset \vdash :\text{Hp}.\supset .\exists ![\{(\alpha +_{c}\beta )_{\xi\}-_{c}1}+_{c}1]_{\xi } &\qquad \text{(5)}\\ +\vdash .(4).(5).*120·311.*110·44.\supset \\ +\vdash :\text{Hp}.\supset .(\alpha +_{c}\beta )_{\xi }-_{c}1=\alpha +_{c}(\beta -_{c}1)_{\xi }:\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*120·425.</b> \[\begin{align}\vdash \colon\ldotp &(\alpha +_{c}\beta )_{\xi }\in \text{N}_{0}\text{C}-\iota ʻ0.\supset :\\ +&(\alpha +_{c}\beta )_{\xi }-_{c}1=\alpha +_{c}(\beta -_{c}1)_{\xi }.\lor.(\alpha +_{c}\beta )_{\xi }-_{c}1=(\alpha + -_{c}1)_{\xi }+_{c}\beta \end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*110·62.*103·22.\supset \vdash \colon\ldotp \text{Hp}.\supset :\alpha \neq 0.\lor.\beta \neq 0:\exists !(\alpha +_{c}\beta )_{\xi } &\qquad \text{(1)}\\ +\vdash .(1).*120·424.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*120·426">*120·426</a>.</b> \(\vdash :\rho \in \text{Cls induct}.\rho \subset \sigma .\exists !\sigma -\rho .\supset .{\sim}(\rho \text{ sm }\sigma ).\text{Nc}ʻ\rho <\text{Nc}ʻ\sigma\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*110·32.&\supset \vdash :\text{Hp}.\supset .\text{Nc}ʻ\sigma =\text{Nc}ʻ\rho +_{c}\text{Nc}ʻ(\sigma -\rho ) &\qquad \text{(1)}\\ +\vdash .*101·14.&\supset \vdash :\text{Hp}.\supset .\text{Nc}ʻ(\sigma -\rho )\neq 0 &\qquad \text{(2)}\\ +\vdash .(1).(2).*120·42.*117·222·26.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*120·427.</b> \[\begin{align}&\vdash :R\in 1\rightarrow 1.\text{ᗡ}ʻR\subset \text{D}ʻR.\exists !\text{D}ʻR-\text{ᗡ}ʻR.\supset .\text{D}ʻR{\sim}\in \text{Cls induct}\\ +&[*120·426.\text{Transp}]\end{align}\]</p> + +<p>The above proposition shows that no reflexive class is inductive.</p> + +<p class="nind"><b>*120·428.</b> \(\vdash :\nu \in \text{NC induct}.\exists !\alpha +_{c}\nu .\alpha \neq 0.\supset .\alpha +_{c}\nu >\nu \)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*117·511.*110·4.\supset \vdash :\text{Hp}.&\supset .\alpha >0.\nu \in \text{N}_{0}\text{C} .\\ +[*117·561.*110·6] &\supset .\alpha +_{c}\nu \geq \nu &\qquad \text{(1)}\\ +\vdash .*120·42.*110·4. &\supset \vdash :\text{Hp}.\supset .\alpha +_{c}\nu \neq \nu &\qquad \text{(2)}\\ +\vdash .(1).(2).*117·26.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*120·429.</b> \(\vdash \colon\ldotp \nu \in \text{NC induct}.\supset :\mu >\nu .\equiv .\mu \geq \nu +_{c}1\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*120·428. \supset \vdash \colon\ldotp \text{Hp}.&\supset :\mu \in \text{N}_{0}\text{C}.\mu =\nu +_{c}1.\supset .\mu >\nu : &\qquad \text{(1)}\\ +[*117·47·12] &\supset :\mu >\nu +_{c}1.\supset .\mu >\nu &\qquad \text{(2)}\\ +\vdash .*117·31.&\supset \vdash :\mu >\nu .\supset .(\exists \varpi ).\varpi \in \text{N}_{0}\text{C}.\mu =\nu +_{c}\varpi &\qquad \text{(3)}\\ +\vdash .*117·26·12.&\supset \vdash :\mu >\nu .\supset .\mu \neq \nu +_{c}0 &\qquad \text{(4)}\\ +\vdash .(3).(4). &\supset \vdash :\mu >\nu .\supset .(\exists \varpi ).\varpi \in \text{N}_{0}\text{C}-\iota ʻ0.\mu =\nu +_{c}\varpi .\\ +[*117·531] &\supset .(\exists \varpi ).\varpi \geq 1.\mu =\nu +_{c}\varpi .\\ +[*117·31] &\supset .(\exists \varpi ,\rho ).\rho \in \text{N}_{0}\text{C}.\varpi =\rho +_{c}1.\mu =\nu +_{c}\varpi .\\ +[*13·195] &\supset .(\exists \rho ).\rho \in \text{N}_{0}\text{C}.\mu =\nu +_{c}\rho +_{c}1.\\ +[*117·31] &\supset .\mu \geq \nu +_{c}1 &\qquad \text{(5)}\\ +\vdash .(1).(2).(5).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_220">[Pg 220]</span></p> + +<p>The following definition, in which "\(\text{spec}\)" stands for +"\(species\)," defines the "species" of a cardinal \(\beta\) as all +cardinals which are less than, equal to, or greater than \(\beta\). +We cannot prove, unless by assuming the multiplicative axiom, that +all cardinals belong to the species of \(\beta\), except in the case +where \(\beta\) is an inductive cardinal. In all other cases there may, +so far as is known at present, be other cardinals which are neither +greater nor less than \(\beta\).</p> + +<p class="nind"><b>*120·43.</b> \(\text{spec}ʻ\beta =\hat{\alpha} \{\alpha <\beta .\lor.\alpha \geq \beta\} \quad\text{Df}\)</p> + +<p class="nind"><b>*120·431.</b> \(\vdash \colon\ldotp \alpha \in \text{spec}ʻ\beta .\equiv :\alpha <\beta .\lor.\alpha \geq \beta \quad[(*120·43)]\)</p> + +<p class="nind"><b>*120·432.</b> \(\vdash \colon\ldotp \alpha \in \text{spec}ʻ\beta .\equiv :\alpha \leq \beta .\lor.\alpha \geq \beta \quad[*117·281.*120·431]\)</p> + +<p class="nind"><b>*120·433.</b> + \[\begin{align}&\vdash \colon\ldotp \text{Nc}ʻ\rho \in \text{spec}ʻ\text{Nc}ʻ\sigma .\equiv :\exists !\text{Cl}ʻ\rho \cap \text{Nc}ʻ\sigma .\lor.\exists !\text{Cl}ʻ\sigma \cap \text{Nc}ʻ\rho \\ +&[*117·22.*120·432]\end{align}\]</p> + +<p class="nind"><b>*120·434.</b> \(\vdash .\text{spec}ʻ\beta \subset \text{N}_{0}\text{C} \quad[*117·105·104·12.*120·432]\)</p> + +<p class="nind"><b><a id="*120·435">*120·435</a>.</b> \(\vdash :\beta \in \text{N}_{0}\text{C}.\equiv .\beta \in \text{spec}ʻ\beta .\equiv .\exists !\text{spec}ʻ\beta \quad[*117·104.*120·434]\)</p> + +<p class="nind"><b>*120·436.</b> \[\begin{align}&\vdash \colon\ldotp \alpha \in \text{spec}ʻ\beta .\equiv :\alpha ,\beta \in \text{N}_{0}\text{C}:(\exists \gamma ):\alpha +_{c}\gamma + =\beta .\lor.\beta +_{c}\gamma =\alpha \\ +&[*120·432 . *117·31]\end{align}\]</p> + +<p class="nind"><b>*120·437.</b> \(\vdash :\beta \in \text{N}_{0}\text{C}.\supset .0\in \text{spec}ʻ\beta \quad[*117·5.*120·432]\)</p> + +<p class="nind"><b>*120·438.</b> \(\vdash :\alpha \in \text{spec}ʻ\beta .\exists !\alpha +_{c}1.\supset .\alpha +_{c}1\in \text{spec}ʻ\beta\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*120·436.*110·4.&\supset \vdash \colon\ldotp \text{Hp}.\equiv :\alpha ,\beta \in \text{N}_{0}\text{C}.\exists !\alpha +_{c}1:\\ +&(\exists \gamma ):\gamma \in \text{N}_{0}\text{C}:\alpha +_{c}\gamma =\beta .\lor.\beta +_{c}\gamma =\alpha &\qquad \text{(1)}\\ +\vdash .*110·61. &\supset \vdash :\alpha ,\beta \in \text{N}_{0}\text{C}.\alpha +_{c}0=\beta .\supset .\alpha =\beta .\\ +[*13·12·15] &\supset .\alpha +_{c}1=\beta +_{c}1 .\\ +[*120·436] &\supset .\alpha +_{c}1\in \text{spec}ʻ\beta &\qquad \text{(2)}\\ +\vdash .*120·417.&\supset \vdash .\alpha ,\beta ,\gamma \in \text{N}_{0}\text{C}.\gamma \neq 0.\alpha +_{c}\gamma =\beta .\supset .\alpha +_{c}1+_{c}(\gamma + -_{c}1)=\beta .\\ +[*120·436] &\supset .\alpha +_{c}1\in \text{spec}ʻ\beta &\qquad \text{(3)}\\ +\vdash .*13·12·15. &\supset \vdash :\alpha ,\beta ,\gamma \in \text{N}_{0}\text{C}.\beta +_{c}\gamma =\alpha .\exists !\alpha +_{c}1.\supset .\beta +_{c}\gamma + +_{c}1=\alpha +_{c}1.\\ +[*120·436] &\supset .\alpha +_{c}1\in \text{spec}ʻ\beta &\qquad \text{(4)}\\ +\vdash .(1).(2).(3).(4).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*120·44.</b> \(\vdash :\beta \in \text{N}_{0}\text{C}.\supset .\text{NC induct}-\iota ʻ\Lambda \subset \text{spec}ʻ\beta\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*120·437. &\supset \vdash :\text{Hp}.\supset .0\in \text{spec}ʻ\beta &\qquad \text{(1)}\\ +\vdash .*120·438.*110·4.&\supset \vdash \colon\colon \text{Hp}.\supset \colon\ldotp \alpha =\Lambda .\lor.\alpha \in \text{spec}ʻ\beta :\supset :\\ +&\alpha +_{c}1=\Lambda .\lor.\alpha +_{c}1\in \text{spec}ʻ\beta &\qquad \text{(2)}\\ +\vdash .(1).(2).*120.11.&\supset \vdash \colon\colon \text{Hp}.\supset \colon\ldotp \alpha \in \text{NC induct}.\supset :\\ +&\alpha =\Lambda .\lor.\alpha \in \text{spec}ʻ\beta \colon\colon \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*120·441">*120·441</a>.</b> + \[\begin{align}&\vdash \colon\ldotp \alpha \in \text{NC induct}-\iota ʻ\Lambda .\beta \in \text{NC}-\iota ʻ\Lambda .\supset :\alpha <\beta .\lor.\alpha =\text{ sm }ʻʻ\beta .\lor.\alpha >\beta \\ +&[*120·44.*103·34]\end{align}\]</p> + +<p><span class="pagenum" id="Page_221">[Pg 221]</span></p> + +<p class="nind"><b>*120·442.</b> \[\begin{align}\vdash \colon\ldotp \alpha \in \text{NC induct}&-\iota ʻ\Lambda .\beta \in \text{NC}-\iota ʻ\Lambda .\supset :\\ +&\alpha < \beta .\equiv .{\sim}(\alpha \geq \beta ):\alpha >\beta .\equiv .{\sim}(\alpha \leq \beta )\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*117·104.*120·441.&\supset \vdash \colon\ldotp \text{Hp}.\supset :\alpha <\beta .\lor.\alpha \geq \beta &\qquad \text{(1)}\\ +\vdash .*117·291. &\supset \vdash :\alpha <\beta .\supset .{\sim}(\alpha \geq \beta ) &\qquad \text{(2)}\\ +\vdash .(1).(2).*5·17.&\supset \vdash \colon\ldotp \text{Hp}.\supset :\alpha <\beta .\equiv .{\sim}(\alpha \geq \beta ) &\qquad \text{(3)}\\ +\text{Similarly}\quad &\vdash \colon\ldotp \text{Hp}.\supset :\alpha >\beta .\equiv .{\sim}(\alpha \leq \beta ) &\qquad \text{(4)}\\ +\vdash .(3).(4).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*120·45">*120·45</a>.</b> \(\vdash :\alpha ,\beta \in \text{N}_{\xi }\text{C induct}.\supset .(\alpha +_{c}\beta )_{\xi }\in \text{N}_{\xi }\text{C induct}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*110·6.\supset \vdash :\alpha \in \text{N}_{\xi }\text{C induct}.\supset .(\alpha +_{c}0_{\xi })_{\xi }\in \text{N}_{\xi }\text{C induct} &\qquad \text{(1)}\\ +\vdash .*120·121.*118·25.\supset\\ +\vdash :(\alpha +_{c}\beta )_{\xi }\in \text{N}_{\xi }\text{C induct}.\supset .\{\alpha +_{c}(\beta +_{c}1)_{\xi }\}_{\xi }\in + \text{N}_{\xi }\text{C induct} &\qquad \text{(2)}\\ +\vdash .(1).(2).*120·11.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*120·4501">*120·4501</a>.</b> \(\vdash :\alpha ,\beta \in \text{NC induct}-\iota ʻ\Lambda .\supset .\alpha +_{c}\beta \in \text{NC induct}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*120·15.\supset \vdash :\text{Hp}.&\supset .\text{ sm }_{\xi }ʻʻ\alpha ,\text{ sm }_{\xi }ʻʻ\beta \in \text{N}_{\xi }\text{C induct}.\\ +[*120·45] &\supset .(\text{ sm }_{\xi }ʻʻ\alpha +_{c}\text{ sm }_{\xi }ʻʻ\beta )_{\xi }\in \text{N}_{\xi }\text{C induct}.\\ +[*118·23] &\supset .(\alpha +_{c}\beta )_{\xi }\in \text{N}_{\xi }\text{C induct}:\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p>The following proposition is a lemma in the proof of <a href="#*120·452">*120·452</a>.</p> + +<p class="nind"><b><a id="*120·451">*120·451</a>.</b> \[\begin{align}&\vdash \colon\ldotp \gamma =(\alpha +_{c}\beta )_{\xi }.\supset _{\alpha ,\beta }.\alpha ,\beta \in \text{NC induct}-\iota ʻ\Lambda :\\ +&\exists !(\gamma +_{c}1)_{\xi }.(\gamma +_{c}1)_{\xi }=(\alpha ʻ+_{c}\beta ʻ)_{\xi }:\supset + .\alpha ʻ,\beta ʻ\in \text{NC induct}-\iota ʻ\Lambda\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*120·414·124.*110·42.&\supset \vdash :\exists !(\gamma +_{c}1)_{\xi }.\supset .\{(\gamma +_{c}1)_{\xi }-_{c}1\}_{\xi}\in + \text{N}_{0}\text{C}.\\ +[*119·32] &\supset .\gamma =\{(\gamma +_{c}1)_{\xi}-_{c}1\}_{\xi } &\qquad \text{(1)}\\ +\vdash .(1).*120·124.&\supset \vdash \colon\ldotp \text{Hp}.\supset :\gamma =\{(\alpha'+ \beta')_{\xi}-_{c}1\}_{\xi }.(\alpha'+ \beta')_{\xi }\neq + 0.\exists !(\alpha'+ \beta')_{\xi }:\\ +[*120·425]&\supset :\gamma =\{\alpha'+_{c}(\beta'-_{c}1)_{\xi }\}_{\xi }.\lor.\gamma =\{(\alpha'-_{c}1)_{\xi }+_{c}\beta'\}_{\xi }:\\ +[\text{Hp}] &\supset :\alpha',(\beta'-_{c}1)_{\xi }\in \text{NC induct}-\iota ʻ\Lambda .\lor.(\alpha'-_{c}1)_{\xi },\beta'\in \text{NC induct}-\iota ʻ\Lambda :\\ +[*119·11] &\supset :\alpha',\beta'\in \text{NC induct}-\iota ʻ\Lambda \colon\ldotp \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p>This proposition could be extended to greater generality as regards +types; but its sole use is as a lemma.</p> + +<p class="nind"><b><a id="*120·452">*120·452</a>.</b> \(\vdash :\alpha +_{c}\beta \in \text{NC induct}-\iota ʻ\Lambda .\supset .\alpha ,\beta \in \text{NC induct}-\iota ʻ\Lambda \)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*110·4.\text{Transp}.&\supset \vdash :\gamma =\Lambda .\supset .(\gamma +_{c}1)_{\eta }=\Lambda &\qquad \text{(1)}\\ +\vdash .*120·451.&\supset \vdash \colon\colon \gamma =(\alpha +_{c}\beta )_{\eta }.\supset _{\alpha ,\beta }.\alpha + ,\beta \in \text{NC induct}-\iota ʻ\Lambda :\supset \colon\ldotp \\ +&(\gamma +_{c}1)_{\eta }=\Lambda :\lor:(\gamma +_{c}1)_{\eta }=(\alpha ʻ+_{c}\beta ʻ)_{\eta }.\supset _{\alpha ʻ,\beta ʻ}.\alpha + ʻ,\beta ʻ\in \text{NC induct}-\iota ʻ\Lambda &\qquad \text{(2)}\\ +\vdash .(1).(2). &\supset \vdash \colon\colon \gamma =\Lambda :\lor:\gamma =(\alpha +_{c}\beta )_{\eta }.\supset _{\alpha ,\beta }.\alpha + ,\beta \in \text{NC induct}-\iota ʻ\Lambda \colon\ldotp \supset \colon\ldotp\\ +&(\gamma +_{c}1)_{\eta }=\Lambda :\lor:(\gamma +_{c}1)_{\eta }=(\alpha ʻ+_{c}\beta ʻ)_\eta .\supset _{\alpha ʻ,\beta ʻ}.\alpha + ʻ,\beta ʻ\in \text{NC induct}-\iota ʻ\Lambda &\qquad \text{(3)}\\ +\vdash .*110·62.*120·12.&\supset \vdash :0=(\alpha +_{c}\beta )_{\eta }.\supset _{\alpha ,\beta }.\alpha + ,\beta \in \text{NC induct}-\iota ʻ\Lambda &\qquad \text{(4)}\\ +\vdash .(3).(4).*120·11.&\supset \vdash \colon\colon \gamma \in \text{N}_{\eta }\text{C induct}.\supset \colon\ldotp \\ +&\gamma =\Lambda :\lor:\gamma =(\alpha +_{c}\beta )_{\eta }.\supset _{\alpha ,\beta }.\alpha ,\beta \in \text{NC induct}-\iota ʻ\Lambda \colon\colon \\ +[*13·15] &\supset \vdash \colon\ldotp (\alpha +_{c}\beta )_{\eta }\in \text{N}_{\eta }\text{C induct}.\supset :\\ +&(\alpha +_{c}\beta )_{\eta }=\Lambda .\lor.\alpha ,\beta \in \text{NC induct}-\iota ʻ\Lambda \colon\ldotp \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_222">[Pg 222]</span></p> + +<p>In the last line but one of the above proof, we substitute for the +\(\phi \xi\) of <a href="#*120·11">*120·11</a> the function +\[ +\xi =\Lambda :\lor:\xi =(\alpha +_{c}\beta )_{\eta }.\supset _{\alpha ,\beta }.\alpha ,\beta \in \text{NC induct}-\iota ʻ\Lambda . +\]</p> + +<p>The following propositions are chiefly required as leading to +<a href="#*120·4621">*120·4621</a><a href="#*120·4622">·4622</a><a href="#*120·47">·47</a>, which are useful in proving propositions concerning +all inductive cardinals other than zero.</p> + +<p class="nind"><b>*120·46.</b> \(\vdash :\alpha \in \text{NC}.\gamma \in \text{N}_{\eta }\text{C induct}.\supset .(\alpha +_{c}\gamma + )_{\eta }(+_{c}1)_{*}\text{ sm }_{\eta }ʻʻ\alpha \)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*110·6.*118·241.&\supset \vdash :\alpha \in \text{NC}.\supset .(\alpha +_{c}0)_{\eta }(+_{c}1)_{*}\text{ sm }_{\eta }ʻʻ\alpha + &\qquad \text{(1)}\\ +\vdash .*90·172.*118·25.&\supset \vdash :(\alpha +_{c}\gamma )_{\eta }(+_{c}1)_{*}\text{ sm }_{\eta }ʻʻ\alpha .\supset .\\ +&\{\alpha +_{c}(\gamma +_{c}1)_{\eta }\}_{\eta }(+_{c}1)_{*}\text{ sm }_{\eta }ʻʻ\alpha &\qquad \text{(2)}\\ +\vdash .(1).(2).*120·11.&\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*120·461.</b> \(\vdash :\alpha \in \text{NC}.\beta (+_{c}1)_{*}\text{ sm }_{\eta }ʻʻ\alpha .\supset .(\exists \gamma ).\gamma \in \text{N}_{\eta }\text{C induct}.\beta =(\alpha +_{c}\gamma )_{\eta }\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*110·6.*118·23. &\supset \vdash :\alpha \in \text{NC}.\beta =\text{ sm }_{\eta }ʻʻ\alpha .\supset .\beta =(a+_{c}0)_{\eta } &\qquad \text{(1)}\\ +\vdash .*120·121.*118·25.&\supset \vdash :\beta =(\alpha +_{c}\gamma )_{\eta }.\gamma \in \text{N}_{\eta }\text{C induct}.\supset .\\ +&(\beta +_{c}1)_{\eta }=\{\alpha +_{c}(\gamma +_{c}1)_{\eta }\}_{\eta }.(\gamma +_{c}1)_{\eta }\in \text{N}_{\eta }\text{C induct} &\qquad \text{(2)}\\ +\vdash .(1).(2).*90·112.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*120·462.</b> \[\begin{align}&\vdash \colon\ldotp \alpha \in \text{NC}.\supset :(\exists \gamma ).\gamma \in \text{N}_{\eta }\text{C induct}.\beta =(\alpha +_{c}\gamma + )_{\eta }.\equiv .\beta (+_{c}1)_{*}\text{ sm }_{\eta }ʻʻ\alpha \\ +&[*120·46·461]\end{align}\]</p> + +<p class="nind"><b><a id="*120·4621">*120·4621</a>.</b> \(\vdash \colon\ldotp \alpha \in \text{NC}.\exists !\beta .\supset :\beta (+_{c}1)_{*}\text{ sm }_{\eta }ʻʻ\alpha .\supset .\text{ sm }_{\xi }ʻʻ\beta + (+_{c}1)_{*}\text{ sm }_{\xi }ʻʻ\alpha \)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*120·461.\supset \vdash \colon\ldotp \text{Hp}.\supset :\\ +\beta (+_{c}1)_{*}\text{ sm }_{\eta }ʻʻ\alpha .&\supset .(\exists \gamma ).\gamma \in \text{N}_{\eta }\text{C induct}.\beta =(\alpha +_{c}\gamma + )_{\eta }.\\ +[*110·4] &\supset .(\exists \gamma ).\gamma \in \text{N}_{\eta }\text{C induct}-\iota ʻ\Lambda .\beta =(\alpha +_{c}\gamma )_{\eta }.\\ +[*120·15.*118·201] &\supset .(\exists \gamma ).\text{ sm }_{\xi }ʻʻ\gamma \in \text{N}_{\xi }\text{C induct}.\text{ sm }_{\xi }ʻʻ\beta =(\alpha +_{c}\gamma + )_{\xi }.\\ +[*118·24.*120·14] &\supset .(\exists \gamma').\gamma'\in \text{N}_{\xi }\text{C induct}.\text{ sm }_{\xi }ʻʻ\beta =(\alpha +_{c}\gamma')_{\xi }.\\ +[*120·462]& \supset .\text{ sm }_{\xi }ʻʻ\beta (+_{c}1)_{*}\text{ sm }_{\xi }ʻʻ\alpha \colon\ldotp \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*120·4622">*120·4622</a>.</b> \[\begin{align}&\vdash \colon\ldotp \alpha \in \text{NC}.\beta \in \text{NC}(\eta ).\exists !\text{ sm }_{\xi }ʻʻ\beta .\supset :\\ +&\beta (+_{c}1)_{*}\text{ sm }_{\eta }ʻʻ\alpha .\equiv .\text{ sm }_{\xi }ʻʻ\beta (+_{c}1)_{*}\text{ sm }_{\xi }ʻʻ\alpha \end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*110·4.*37·29.*120·461.\supset \\ +\vdash \colon\ldotp \text{Hp}.&\supset :\text{ sm }_{\xi }ʻʻ\beta (+_{c}1)_{*}\text{ sm }_{\xi }ʻʻ\alpha .\supset .\exists !\text{ sm }_{\xi }ʻʻ\alpha + .\exists !\alpha . &\qquad \text{(1)}\\ +[*100·52] &\supset .\text{ sm }_{\xi }ʻʻ\alpha \in \text{NC} &\qquad \text{(2)}\\ +\vdash .*120·4621.(2).&\supset \vdash \colon\ldotp \text{Hp}.\supset :\\ +&\text{ sm }_{\xi }ʻʻ\beta (+_{c}1)_{*}\text{ sm }_{\xi }ʻʻ\alpha .\supset .\text{ sm }_{\eta }ʻʻ\text{ sm }_{\xi }ʻʻ\beta + (+_{c}1)_{*}\text{ sm }_{\eta }ʻʻ\text{ sm }_{\xi }ʻʻ\alpha .\\ +[*102·87.\text{Hp}.(1)] &\supset .\text{ sm }_{\eta }ʻʻ\beta (+_{c}1)_{*}\text{ sm }_{\eta }ʻʻ\alpha .\\ +[*103·34] &\supset .\beta (+_{c}1)_{*}\text{ sm }_{\eta }ʻʻ\alpha &\qquad \text{(3)}\\ +\vdash .*37·29.*120·4621.&\supset \vdash \colon\ldotp \text{Hp}.\supset :\beta (+_{c}1)_{*}\text{ sm }_{\eta }ʻʻ\alpha .\supset .\text{ sm }_{\xi }ʻʻ\beta + (+_{c}1)_{*}\text{ sm }_{\xi }ʻʻ\alpha &\qquad \text{(4)}\\ +\vdash .(3).(4).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_223">[Pg 223]</span></p> + +<p>It is on this proposition that the irrelevance of types in the +consideration of inductive cardinals depends.</p> + +<p class="nind"><b>*120·463.</b> \[\begin{align}\vdash \colon\colon .\alpha \in \text{NC}.\supset \colon\colon &(\exists \gamma ).\gamma \in \text{N}_{\eta }\text{C induct}.\beta =(\alpha +_{c}\gamma + )_{\eta }.\equiv \colon\ldotp \\ +&\xi \in \mu .\supset _{\xi }.(\xi +_{c}1)_{\eta }\in \mu :\text{ sm }_{\eta }ʻʻ\alpha \in \mu :\supset _{\mu }.\beta \in \mu \\ +[*120·462.*90·11]\end{align}\]</p> + +<p class="nind"><b><a id="*120·47">*120·47</a>.</b> \[\begin{align}&\vdash \colon\colon \beta \in \text{N}_{\eta }\text{C induct}-\iota ʻ0.\equiv \colon\ldotp \xi \in \mu .\supset _{\xi }.(\xi + +_{c}1)_{\eta }\in \mu :1_{\eta }\in \mu :\supset _{\mu }.\beta \in \mu \\ +&[*120·423·463]\end{align}\]</p> + +<p>Thus mathematical induction starting from 1 will apply to all inductive +cardinals except 0. Similar propositions can be similarly proved for 2, +3, ....</p> + +<p class="nind"><b>*120·471.</b> \(\vdash :(\exists \alpha ).\alpha \in \text{NC induct}-\iota ʻ0.f\alpha .\equiv .(\exists \beta ).\beta \in \text{NC induct}.f(\beta +_{c}1)\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*120·423.\supset \\ +\vdash :(\exists \alpha ).\alpha \in \text{NC induct}-\iota ʻ0.f\alpha .&\equiv .(\exists \beta ).\beta \in \text{NC induct}.\alpha =\beta +_{c}1.f\alpha .\\ +[*13·195] &\equiv .(\exists \beta ).\beta \in \text{NC induct}.f(\beta +_{c}1):\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*120·472">*120·472</a>.</b> \[\begin{align}\vdash :&(\exists \alpha ).\alpha \in \text{NC induct}-\iota ʻ0-\iota ʻ1.f\alpha .\equiv .\\ +&(\exists \beta ).\beta \in \text{NC induct}-\iota ʻ0.f(\beta +_{c}1).\equiv .(\exists \gamma ).\gamma \in \text{NC induct}.f(\gamma +_{c}2)\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*120·471.\supset \\ +\vdash :(\exists \alpha ).\alpha \in &\text{NC induct}-\iota ʻ0-\iota ʻ1.f\alpha .\equiv .\\ +&(\exists \beta ).\beta \in \text{NC induct}.\beta +_{c}1\neq 1.f(\beta +_{c}1).\\ +[*120·42.*110·641] &\equiv .(\exists \beta ).\beta \in \text{NC induct}-\iota ʻ0.f(\beta +_{c}1). &\qquad \text{(1)}\\ +[*120·471] &\equiv .(\exists \gamma ).\gamma \in \text{NC induct}.f(\gamma +_{c}1+_{c}1).\\ +[*110·643] &\equiv .(\exists \gamma ).\gamma \in \text{NC induct}.f(\gamma +_{c}2) &\qquad \text{(2)}\\ +\vdash .(1).(2).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*120·473">*120·473</a>.</b> \[\begin{align}\vdash \colon\ldotp \phi 1:\xi \in \text{N}_{\eta }\text{C induct}-\iota ʻ0.\phi \xi .&\supset _{\xi }.\phi (\xi +_{c}1):\supset :\\ +&\xi \in \text{N}_{\eta }\text{C induct}-\iota ʻ0.\supset .\phi \xi\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*120·122.*101·22.&\supset \vdash :\phi 1.\supset .1\in \text{N}_{\eta }\text{C induct}-\iota ʻ0.\phi 1 &\qquad \text{(1)}\\ +\vdash .*120·121·124.&\supset \vdash :\xi \in \text{N}_{\eta }\text{C induct}-\iota ʻ0.\supset .\xi +_{c}1\in \text{N}_{\eta }\text{C + induct}-\iota ʻ0 &\qquad \text{(2)}\\ +\vdash .(1).(2).\supset \\ +\vdash \colon\ldotp \text{Hp}.&\supset :1\in \text{N}_{\eta }\text{C induct}-\iota ʻ0.\phi 1:\xi \in \text{N}_{\eta }\text{C induct}-\iota ʻ0.\phi \xi .\supset _{\xi }.\\ +&\xi +_{c}1\in \text{N}_{\eta }\text{C induct}-\iota ʻ0.\phi (\xi +_{c}1) &\qquad \text{(3)}\\ +\vdash .(3).*120·47 \frac{\hat{\xi} (\xi \in \text{N}_{\eta }\text{C induct}-\iota ʻ0.\phi \xi )}{\mu}.&\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_224">[Pg 224]</span></p> + +<p class="nind"><b>*120·48.</b> \[\begin{align}&\vdash :\beta \in \text{NC induct}.\beta \geq \alpha .\supset .\alpha \in \text{NC induct}-\iota ʻ\Lambda \\ +&[*120·452.*117·31]\end{align}\]</p> + +<p>Thus every cardinal which is not greater than every inductive cardinal +is an inductive cardinal.</p> + +<p class="nind"><b>*120·481.</b> \(\vdash :\eta \in \text{Cls induct}.\xi \subset \eta .\supset .\xi \in \text{Cls induct} \quad[*117·222.*120·21·48]\)</p> + +<p>Thus if any inductive class can be found which contains a given class, +the given class is also inductive.</p> + +<p class="nind"><b>*120·49.</b> \(\vdash :\alpha \in \text{NC}-\text{NC induct}-\iota ʻ\Lambda .\beta \in \text{NC induct}-\iota ʻ\Lambda .\supset .\alpha >\beta\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*120·48.\text{Transp}.&\supset \vdash :\text{Hp}.\supset .{\sim}(\beta \geq \alpha ) &\qquad \text{(1)}\\ +\vdash .*120·441. &\supset \vdash \colon\ldotp \text{Hp}.\supset :\alpha >\beta .\lor.\beta \geq \alpha &\qquad \text{(2)}\\ +\vdash .(1).(2).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p>Thus every non-inductive cardinal (except \(\Lambda\)) is greater than +every inductive cardinal (except \(\Lambda\)).</p> + +<p class="nind"><b><a id="*120·491">*120·491</a>.</b> \(\vdash \colon\ldotp \xi {\sim}\in \text{Cls induct}.\equiv :\beta \in \text{NC induct}.\supset _{\beta }.\exists !\beta \cap \text{Cl}ʻ\xi\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*120·49.&\supset \vdash :\xi {\sim}\in \text{Cls induct}.\beta \in \text{NC induct}-\iota ʻ\Lambda .\supset .\text{N}_{0}\text{c}ʻ\xi >\beta .\\ +[*120·429.*117·12]&\supset .\text{N}_{0}\text{c}ʻ\xi \geq \beta +_{c}1.\exists !\beta \cap \text{Cl}ʻ\xi &\qquad \text{(1)}\\ +\vdash .(1).*117·104·12.*103·13.&\supset +\vdash :\xi {\sim}\in \text{Cls induct}.\beta \in \text{NC induct}-\iota ʻ\Lambda .\supset .\beta +_{c}1\neq \Lambda &\qquad \text{(2)}\\ +\vdash .(2).*101·12.*102·13.\supset \\ +\vdash \colon\ldotp \xi {\sim}\in \text{Cls induct}.&\supset :\beta \in \text{NC induct}.\supset .\beta \neq \Lambda &\qquad \text{(3)}\\ +\vdash .(1).(3).&\supset \vdash :\xi {\sim}\in \text{Cls induct}.\beta \in \text{NC induct}.\supset .\exists !\beta \cap \text{Cl}ʻ\xi &\qquad \text{(4)}\\ +\vdash .*120·121.\supset \\ +\vdash \colon\ldotp \beta \in \text{NC induct}.&\supset _{\beta }.\exists !\beta \cap \text{Cl}ʻ\xi :\supset :\beta \in \text{NC induct}.\supset _{\beta }.\exists !(\beta +_{c}1)\cap + \text{Cl}ʻ\xi .\\ +[*117·242.*120·429] &\supset _{\beta }.\text{Nc}ʻ\xi > \beta .\\ +[*117·42.(*117·03)] &\supset _{\beta }.\text{N}_{0}\text{c}ʻ\xi \neq \beta :\\ +[*13·196] &\supset :\text{N}_{0}\text{c}ʻ\xi {\sim}\in \text{NC induct}:\\ +[*120·21] &\supset :\xi {\sim}\in \text{Cls induct} &\qquad \text{(5)}\\ +\vdash .(4).(5).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*120·492.</b> \[\begin{align}&\vdash :\alpha \in \text{NC}-\text{NC induct}.\beta \geq \alpha .\supset .\beta \in \text{NC}-\text{NC induct}\\ +&[*120·48.\text{Transp}]\end{align}\]</p> + +<p>In virtue of <a href="#*120·491">*120·491</a>, a class \(\xi\) which is not inductive contains +sub-classes having 0, 1, 2, 3, ... terms. If we take the successive +classes of sub-classes +\[ +0\cap \text{Cl}ʻ\xi , 1\cap \text{Cl}ʻ\xi , 2\cap \text{Cl}ʻ\xi , ..., +\] +these are mutually exclusive, and all exist provided \(\Lambda\) +is not an inductive<span class="pagenum" id="Page_225">[Pg 225]</span> cardinal, <i>i.e.</i> provided the axiom of +infinity holds. Thus if the axiom of infinity holds, we get \(\aleph_{0}\) +classes of sub-classes contained in any non-inductive class. +It follows, as we shall see later, that if \(\xi\) is a non-inductive +class, \(\text{Cl}ʻ\text{Cl}ʻ\xi\) is a reflexive class. This seems to +be the nearest approach possible to identifying the two definitions of +finite and infinite when the multiplicative axiom is not assumed. When +the multiplicative axiom is assumed as well as the axiom of infinity, +we pick out one class from \(1\cap \text{Cl}ʻ\xi\), one from +\(2\cap\text{Cl}ʻ\xi\), and so on; then, forming the logical sum of +all these classes, we get \(\aleph _{0}\) terms which are members of +\(\xi\). Hence it follows that \(\xi\) is a reflexive class; for, as we +shall see later, a reflexive class is one which contains sub-classes of +\(\aleph _{0}\) terms. Thus with the help of the multiplicative axiom, +the two definitions of finite and infinite can be identified.</p> + +<p class="nind"><b>*120·493.</b> \[\begin{align}&\vdash \colon\ldotp \sigma \in \text{Cls induct}.\supset :\\ +&\text{Nc}ʻ\xi <\text{Nc}ʻ\sigma .\equiv .(\exists \rho ).\rho \text{ sm }\xi .\rho \subset \sigma .\exists !\sigma -\rho .\equiv .\exists !\text{Nc}ʻ\xi \cap \text{Cl}ʻ\sigma -\iota ʻ\sigma\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*117·26·221.&\supset \vdash \colon\ldotp \text{Nc}ʻ\xi <\text{Nc}ʻ\sigma .\supset :{\sim}(\xi \text{ sm }\sigma ):(\exists \rho ).\rho \text{ sm }\xi .\rho \subset \sigma :\\ +[*73·3·37] &\supset :(\exists \rho ).\rho \text{ sm }\xi .\rho \subset \sigma .\rho \neq \sigma &\qquad \text{(1)}\\ +\vdash .*120·481.&\supset \vdash \colon\ldotp \text{Hp}.\supset :\rho \subset \sigma .\exists !\sigma -\rho .\supset .\rho \in \text{Cls} \text{induct}.\rho \subset \sigma .\exists !\sigma -\rho .\\ +[*120·426] &\supset .\text{Nc}ʻ\rho <\text{Nc}ʻ\sigma :\\ +[*100*321] &\supset :\rho \text{ sm }\xi .\rho \subset \sigma .\exists !\sigma -\rho .\supset .\text{Nc}ʻ\xi <\text{Nc}ʻ\sigma &\qquad \text{(2)}\\ +\vdash .(1).(2).*24·6.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*120·5">*120·5</a>.</b> \(\vdash :\alpha ,\beta \in \text{NC induct}.\exists !\alpha \times _{c}\beta .\supset .\alpha \times _{c}\beta \in \text{NC induct}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*113·203.&\supset \vdash :\alpha \in \text{NC induct}.\exists !\alpha \times _{c}0.\supset .\alpha \in \text{NC}-\iota ʻ\Lambda .\\ +[*113·601] &\supset .\alpha \times _{c}0=0.\\ +[*120·12] &\supset .\alpha \times _{c}0\in \text{NC induct} &\qquad \text{(1)}\\ +\vdash .*113·671. &\supset \vdash .\alpha \times _{c}(\beta +_{c}1)=(\alpha \times _{c}\beta )+_{c}\alpha .\\ +[*120·4501.*113·203] &\supset \vdash :\alpha \in \text{NC induct}.\alpha \times _{c}\beta \in \text{NC induct}-\iota ʻ\Lambda .\supset .\\ +&\alpha \times _{c}(\beta +_{c}1)\in \text{NC induct} &\qquad \text{(2)}\\ +\vdash .(1).(2).*120·13.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p>The restriction involved in \(\exists !\alpha \times _{c}\beta\) in +the hypothesis of the above proposition is not necessary if we assume +that the axiom of infinity must fail in any one type if it fails in any +other, <i>i.e.</i> +\[ +\Lambda \cap tʻ\alpha \in \text{NC induct}.\supset .\Lambda \cap tʻ\beta \in \text{NC induct}, +\] +<span class="pagenum" id="Page_226">[Pg 226]</span>where \(\alpha\) and \(\beta\) are any two objects of any two +types. To prove this proposition would require assumptions, as to +the interrelation of various types, which have not been made in our +previous proofs.</p> + +<p class="nind"><b><a id="*120·51">*120·51</a>.</b> \(\vdash :\alpha ,\beta ,\gamma ,\in \text{NC induct}.\alpha \neq 0.\exists !\alpha \times _{c}\beta .\alpha\times _{c}\beta =\alpha \times _{c}\gamma + .\supset .\beta =\text{ sm }ʻʻ\gamma\)</p> + +<p>This proposition establishes the uniqueness of division among inductive +cardinals.</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *120·44·436. &\supset \vdash \colon\ldotp \text{Hp}.\supset :(\exists \delta ):\beta =\gamma +_{c}\delta .\lor.\gamma =\beta +_{c}\delta &\qquad \text{(1)}\\ +\vdash . *113·43. &\supset \vdash :\text{Hp}.\beta =\gamma +_{c}\delta .\supset .\alpha \times _{c}\gamma =(\alpha \times _{c}\gamma )+_{c}(\alpha \times _{c}\delta + ).\\ +[*120·42.\text{Transp}] &\supset .\alpha \times _{c}\delta =0.\\ +[*113·602] &\supset .\delta =0.\\ +[*110·6] &\supset .\beta = \text{ sm }ʻʻ\gamma &\qquad \text{(2)}\\ +\text{Similarly}\quad &\vdash :\text{Hp}.\gamma =\beta +_{c}\delta .\supset .\gamma =\text{ sm }ʻʻ\beta .\\ +[*100·53.*113·203] &\supset .\beta =\text{ sm }ʻʻ\gamma &\qquad \text{(3)}\\ +\vdash .(1).(2).(3).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p>If \(\beta\), \(\gamma\) in the above are typically ambiguous symbols, +such as +\[ +0, 1, 2, ... \text{Nc}ʻ\rho , \text{Nc}ʻ\sigma , ..., +\] +we have \(\beta = \gamma\); for in this case, \(\beta =\text{ sm }ʻʻ\beta.\gamma =\text{ sm }ʻʻ\gamma\). +Also if \(\beta\) and \(\gamma\) are of the same type, we have \(\beta=\gamma\), +in virtue of <a href="#*103·43">*103·43</a>. Hence "\(\beta =\gamma\)" may, with +truth, be substituted for "\(\beta =\text{ sm }ʻʻ\gamma\)" in the above +proposition, since the result is true whenever significant. But in this +form the proposition gives less information, since it tells us nothing +as to what happens when \(\beta\) and \(\gamma\) are not of the same +type.</p> + +<p class="nind"><b>*120·511.</b> \(\vdash :\alpha ,\beta \in \text{NC induct}.\alpha \neq 0.\exists !\alpha .\alpha \times _{c}\beta =\alpha .\supset .\beta =1\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*113·621. \supset \vdash :\text{Hp}.\supset .\alpha \times _{c}\beta =\alpha \times _{c} 1 &\qquad \text{(1)}\\ +\vdash .(1).*120·51.*101·28.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*120·512">*120·512</a>.</b> \(\vdash :\alpha \times _{c}\beta \in \text{NC induct}-\iota ʻ0-\iota ʻ\Lambda .\supset .\alpha ,\beta \in \text{NC induct}-\iota ʻ0-\iota ʻ\Lambda \)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *113·602·203.\supset \vdash :\text{Hp}.&\supset .\alpha ,\beta \in \text{NC}-\iota ʻ0-\iota ʻ\Lambda &\qquad \text{(1)}\\ +\vdash .(1).*117·62.\supset \vdash :\text{Hp}.&\supset .\alpha \times _{c}\beta \geq \alpha .\alpha \times _{c}\beta \geq \beta .\\ +[*120·48] &\supset .\alpha ,\beta \in \text{NC induct} &\qquad \text{(2)}\\ +\vdash .(1).(2).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*120·513.</b> \(\vdash :\alpha \in \text{NC induct}-\iota ʻ0-\iota ʻ\Lambda .\alpha \times _{c}\beta =\alpha .\supset .\beta =1 \quad[*120·511·512]\)</p> + +<p>This proposition does not hold when \(\alpha\) is a non-inductive +cardinal.</p> + +<p class="nind"><b><a id="*120·52">*120·52</a>.</b> \(\vdash :\alpha ,\beta \in \text{NC induct}.\exists !\alpha ^\beta .\supset .\alpha ^\beta \in \text{NC induct}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*116·203·301.&\supset \vdash :\alpha \in \text{NC induct}.\exists !\alpha ^{0}.\supset .\alpha ^{0}=1.\\ +[*120·122] &\supset .\alpha ^{0}\in \text{NC induct} &\qquad \text{(1)}\\ +\vdash . *116·321·52. &\supset \vdash :\exists !\alpha ^{\beta +_{c}1}.\supset .\alpha ^{\beta +_{c}1}=\alpha ^\beta \times _{c}\alpha .\\ +[*120·5] &\supset \vdash :\alpha \in \text{NC induct}.\alpha ^\beta \in \text{NC induct}.\exists !\alpha ^{\beta +_{c}1}.\supset .\\ +&\alpha ^{\beta +_{c}1}\in \text{NC induct} &\qquad \text{(2)}\\ +\vdash .*116·52.*113·204.&\supset \vdash :\alpha ^\beta =\Lambda .\supset .\alpha ^{\beta +_{c}1}=\Lambda &\qquad \text{(3)}\\ +\vdash .(1).(2).(3).*120·11.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_227">[Pg 227]</span></p> + +<p class="nind"><b><a id="*120·53">*120·53</a>.</b> \(\vdash :\alpha ,\beta ,\gamma \in \text{NC induct}.\alpha \neq 0.a\neq 1.\exists !\alpha ^\beta .\alpha ^\beta =\alpha ^\gamma + .\supset .\beta =\text{ sm }ʻʻ\gamma \)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*116·203. &\supset \vdash :\exists !\alpha ^\beta .\supset .\exists !\beta &\qquad \text{(1)}\\ +\vdash .*120·44·436. &\supset \vdash \colon\ldotp \text{Hp}.\supset :(\exists \delta ):\beta =\gamma +_{c}\delta .\lor.\gamma =\beta +_{c}\delta &\qquad \text{(2)}\\ +\vdash .*118·01.*116·52. &\supset \vdash :\beta =\gamma +_{c}\delta .\exists !\beta .\supset .\alpha ^\beta =\alpha ^\gamma \times _{c}\alpha ^\delta + :\\ +[*13·171.*118·01.(1)] & \supset \vdash :\alpha ^\beta =\alpha ^\gamma .\beta =\gamma +_{c}\delta .\exists !\alpha ^\beta .\supset .\alpha ^\gamma + =\alpha ^\gamma \times _{c}\alpha ^\delta &\qquad \text{(3)}\\ +\vdash .*120·52.*116·35.(1).&\supset :\text{Hp}.\beta =\gamma +_{c}\delta .\supset .a^\gamma \in \text{NC induct}-\iota ʻ\Lambda -\iota ʻ0.\exists !\beta .\\ +[(3).*120·513] & \supset .\alpha ^\delta =1.\\ +[*117·592] &\supset .\delta =0.\\ +[*110·6] &\supset .\beta =\text{ sm }ʻʻ\gamma &\qquad \text{(4)}\\ +\text{Similarly}\quad &\vdash :\text{Hp}.\gamma =\beta +_{c}\delta .\supset .\gamma =\text{ sm }ʻʻ\beta .\\ +[*100·53.(1)] &\supset .\beta =\text{ sm }ʻʻ\gamma &\qquad \text{(5)}\\ +\vdash .(2).(4).(5).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p>If \(\alpha\), \(\beta\), \(\gamma\) are typically ambiguous symbols, +we have \(\beta =\gamma\) in the conclusion of the above proposition, +instead of \(\beta =\text{ sm }ʻʻ\gamma\). Also if \(\beta\) and +\(\gamma\) are of the same type, \(\beta =\gamma\); thus \(\beta=\gamma\) +whenever "\(\beta =\gamma\)" is significant.</p> + +<p class="nind"><b>*120·54.</b> \(\vdash :\xi ,\rho \in \text{Cls induct}.\exists !\xi .\rho \subset \sigma .\exists !\sigma - \rho .\supset .(\text{Nc}ʻ\rho )^{\text{Nc}ʻ\xi + }<(\text{Nc}ʻ\sigma )^{\text{Nc}ʻ\xi }\)</p> + +<p>For the proof, which is here given shortly, compare <a href="#*117·58">*117·58</a>.</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*35·432·82.*80·15.*116·12. &\supset \vdash :\text{Hp}.\supset .(\rho \uparrow \xi )_{\Delta }ʻ\xi \subset (\sigma \uparrow \xi )_{\Delta }ʻ\xi .\\ +&\exists !(\sigma \uparrow \xi )_{\Delta }ʻ\xi -(\rho \uparrow \xi )_{\Delta }ʻ\xi &\qquad \text{(1)}\\ +\vdash .*120·52.*116·15·251.*120·2.&\supset \vdash :\text{Hp}.\supset .(\rho \uparrow \xi )_{\Delta }ʻ\xi \in \text{Cls induct} &\qquad \text{(2)}\\ +\vdash .(1).(2).*120·426.&\supset \vdash :\text{Hp}.\supset .\text{Nc}ʻ(\rho \uparrow \xi )_{\Delta }ʻ\xi < \text{Nc}ʻ(\sigma \uparrow \xi )_{\Delta }ʻ\xi + :\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*120·541.</b> \(\vdash :\alpha ,\beta \in \text{NC induct}-\iota ʻ\Lambda .\alpha \neq 0.\beta <\gamma .\supset .\beta ^\alpha <\gamma ^\alpha + \quad[*120·54·493]\)</p> + +<p class="nind"><b>*120·542.</b> \(\vdash :\alpha ,\gamma \in \text{NC induct}-\iota ʻ\Lambda .\alpha \neq 0.\beta >\gamma .\supset .\beta ^\alpha >\gamma ^\alpha \quad[*120·541]\)</p> + +<p class="nind"><b><a id="*120·55">*120·55</a>.</b> \(\vdash :\alpha ,\beta ,\gamma \in \text{NC induct}.\alpha \neq 0.\exists !\beta ^\alpha .\beta ^\alpha =\gamma ^\alpha + .\supset .\beta =\text{ sm }ʻʻ\gamma\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*120·541·542.\supset \vdash :\text{Hp}.&\supset .{\sim}(\beta <\gamma ).{\sim}(\beta >\gamma ).\\ +[*120·441] &\supset .\beta =\text{ sm }ʻʻ\gamma :\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*120·56">*120·56</a>.</b> \(\vdash :\alpha \geq 2.\alpha ^\beta \in \text{NC induct}-\iota ʻ\Lambda .\supset .\beta \in \text{NC induct}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*117·581.\supset \vdash :\text{Hp}.&\supset .\alpha ^\beta \geq 2^\beta .\\ +[*117·661] & \supset .\alpha ^\beta >\beta &\qquad \text{(1)}\\ +\vdash .(1).*120·48.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*120·561">*120·561</a>.</b> \(\vdash :\beta \geq 1.\alpha ^\beta \in \text{NC induct}-\iota ʻ\Lambda .\supset .\alpha \in \text{NC induct}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*117·591.*116·321.\supset \vdash :\text{Hp}.\supset .\alpha ^\beta \geq \alpha &\qquad \text{(1)}\\ +\vdash .(1).*120·48.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_228">[Pg 228]</span></p> + +<p class="nind"><b><a id="*120·57">*120·57</a>.</b> \(\vdash :\mu \in \text{NC induct}-\iota ʻ\Lambda .\supset .\text{Nc}ʻ\hat{\nu} (\nu \leq \mu )=\mu +_{c}1\)</p> + +<p>Here "\(\mu +_{c}1\)" is necessarily in a higher type than "\(\mu\)," +because it applies to a class of which \(\mu\) is a member.</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*117·511.&\supset \vdash .\text{Nc}ʻ\hat{\nu} (\nu \leq 0)\in 1 &\qquad \text{(1)}\\ +\vdash . *110·4. &\supset \vdash :\mu =\Lambda .\supset .\mu +_{c}1=\Lambda &\qquad \text{(2)}\\ +\vdash .*120·429·442.\supset \\ +&\vdash :\mu \in \text{NC induct}.\exists !\mu +_{c}1.\supset .\hat{\nu} (\nu \leq \mu )=\hat{\nu} (\nu <\mu +_{c}1).\\ +[*117·104·105] &\supset .\hat{\nu} (\nu \leq \mu +_{c}1)=\hat{\nu} (\nu \leq \mu )\cup \iota ʻ(\mu +_{c}1) &\qquad \text{(3)}\\ +\vdash .*120·428.&\supset \vdash :\text{Hp}(3).\supset .\mu +_{c}1{\sim}\in \hat{\nu} (\nu \leq \mu ) &\qquad \text{(4)}\\ +\vdash .(3).(4).*110·631.\supset \\ +&\vdash :\text{Hp}(3).\text{Nc}ʻ\hat{\nu} (\nu \leq \mu )=\mu +_{c}1.\supset .\text{Nc}ʻ\hat{\nu} (\nu \leq \mu +_{c}1)=\mu +_{c}2 &\qquad \text{(5)}\\ +\vdash .(2).(5).&\supset \vdash \colon\ldotp \mu \in \text{NC induct}:\mu =\Lambda .\lor.\text{Nc}ʻ\hat{\nu} (\nu \leq \mu )=\mu +_{c}1:\supset :\\ +&\mu +_{c}1=\Lambda .\lor.\text{Nc}ʻ\hat{\nu} (\nu \leq \mu +_{c}1)=\mu +_{c}2 &\qquad \text{(6)}\\ +\vdash .(1).(6).*120·13.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*120·6.</b> \(\vdash :(\exists \gamma ).\gamma >\alpha .\gamma \subset tʻ\eta .\supset .\exists !(\alpha +_{c}1)\cap tʻ\eta \)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*117·1.\supset \\ +\vdash \colon\ldotp \text{Hp}.&\supset :(\exists \gamma ,\rho ,\sigma ).\text{N}_{0}\text{c}ʻ\rho =\alpha .\text{N}_{0}\text{c}ʻ\sigma + =\gamma .\exists !\text{Nc}ʻ\rho \cap \text{Cl}ʻ\sigma .{\sim}\exists !\text{Nc}ʻ\sigma \cap \text{Cl}ʻ\rho :\\ +[*100·1]&\supset :(\exists \gamma ,\rho ,\sigma ,\xi ).\text{N}_{0}\text{c}ʻp=\alpha .\text{N}_{0}\text{c}ʻ\sigma + =\gamma .\xi \text{ sm }\rho .\xi \subset \sigma .\xi \neq \sigma :\\ +[*24·6] &\supset :(\exists \gamma ,\rho ,\sigma ,\xi ,x).\text{N}_{0}\text{c}ʻp=\alpha .\text{N}_{0}\text{c}ʻ\sigma =\gamma .\xi \text{ sm }\rho .x\in \sigma -\xi :\\ +[*110·631]&\supset :(\exists \xi ,x).\xi \cup \iota ʻx\in \alpha +_{c}1\cap tʻ\eta \colon\ldotp \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*120·61">*120·61</a>.</b> \(\vdash :\exists !\text{N}_{0}\text{C}\cap t^{3}ʻx-\text{NC induct}.\supset .\text{Infin ax}(x)\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*120·49.&\supset \vdash \colon\ldotp \gamma \in \text{N}_{0}\text{C}\cap t^{3}ʻx-\text{NC induct}.\supset :\\ +&\alpha \in \text{NC induct}.\exists !\alpha . \supset _{\alpha }.\gamma >\alpha .\gamma \subset t^{2}ʻx.\\ +[*120·6] &\supset _{\alpha }.\exists !\alpha +_{c}1\cap t^{2}ʻx &\qquad \text{(1)}\\ +\vdash .(1).*101·12.*120·13.\supset \\ +\vdash \colon\ldotp \gamma \in \text{N}_{0}\text{C}-\text{NC induct}.&\supset :\alpha \in \text{NC induct}. \supset _{\alpha }.\exists + !\alpha (x)\colon\ldotp \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*120·611">*120·611</a>.</b> \(\vdash :\beta \in \text{Cls induct}.\beta \subset \text{ᗡ}ʻP.\supset .\exists !P_{\Delta }ʻ\beta\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*80·26.&\supset \vdash .\exists !P_{\Delta }ʻ\Lambda .\\ +[\text{Simp}] &\supset \vdash :\Lambda \subset \text{ᗡ}ʻP.\supset .\exists !P_{\Delta }ʻ\Lambda &\qquad \text{(1)}\\ +\vdash .*80·94.&\supset \vdash :\exists !P_{\Delta }ʻ\beta .z\in \text{ᗡ}ʻP.\supset .\exists !P_{\Delta }ʻ(\beta \cup \iota ʻz):\\ +[\text{Syll}] &\supset \vdash \colon\ldotp \beta \subset \text{ᗡ}ʻP.\supset .\exists !P_{\Delta }ʻ\beta :\supset :\beta \subset \text{ᗡ}ʻP.z\in \text{ᗡ}ʻP.\supset .\exists !P_{\Delta }ʻ(\beta + \cup \iota ʻz):\\ +[*51·238] &\supset :\beta \cup \iota ʻz\subset \text{ᗡ}ʻP.\supset .\exists !P_{\Delta }ʻ(\beta \cup \iota ʻz) &\qquad \text{(2)}\\ +\vdash .(1).(2).*120·26.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_229">[Pg 229]</span></p> + +<p class="nind"><b><a id="*120·62">*120·62</a>.</b> \(\vdash :\kappa \in \text{Cls induct}.\Lambda {\sim}\in \kappa .\supset .\exists !{\in}_{\Delta}ʻ\kappa \)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*83·9. &\supset \vdash .\exists !{\in}_{\Delta}ʻ\Lambda &\qquad \text{(1)}\\ +\vdash . *83·904. &\supset \vdash :\exists !{\in}_{\Delta}ʻ\kappa .\exists !\alpha .\supset .\exists !{\in}_{\Delta}ʻ(\kappa \cup \iota ʻ\alpha ):\\ +[\text{Syll}] &\supset \vdash \colon\ldotp \Lambda {\sim}\in \kappa .\supset .\exists !{\in}_{\Delta}ʻ\kappa + :\supset :\Lambda {\sim}\in \kappa .\exists !\alpha .\supset .\exists !{\in}_{\Delta}ʻ(\kappa \cup \iota ʻ\alpha ):\\ +[*24·54] &\supset :\Lambda {\sim}\in (\kappa \cup \iota ʻ\alpha ).\supset .\exists !{\in}_{\Delta}ʻ(\kappa \cup \iota ʻ\alpha ) &\qquad \text{(2)}\\ +\vdash .(1).(2).*120·26.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p>The above proposition may also be deduced from <a href="#*120·611">*120·611</a>, by *62·231.</p> + +<p class="nind"><b>*120·63.</b> \(\vdash .\text{Cls induct} - \overleftarrow{{\in}}ʻ\Lambda \subset \text{Cls}^{2}\, \text{mult} \quad[*120·62.*88·2]\)</p> + +<p>In virtue of this proposition the multiplicative axiom is not required +in dealing with a finite number of factors, even when some or all of +the factors are themselves infinite.</p> + +<p class="nind"><b><a id="*120·64">*120·64</a>.</b> \(\vdash : \text{Infin ax}.\lor.\text{Mult ax}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*120·61.\text{Transp}.&\supset \vdash \colon\ldotp {\sim}\text{Infin ax}.\supset :\text{N}_{0}\text{C}\subset \text{NC induct}:\\ +[*120·21] &\supset :(\kappa ).\kappa \in \text{Cls induct}:\\ +[*120·62] &\supset :(\kappa ):\Lambda {\sim}\in \kappa .\supset .\exists !{\in}_{\Delta}ʻ\kappa :\\ +[*88·37] &\supset :\text{Mult ax}\colon\ldotp \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p>Thus of our two arithmetical axioms, the multiplicative axiom and the +axiom of infinity, at least one must be true.</p> + +<p class="nind"><b>*120·7.</b> \(\vdash :\alpha \in \text{Cls induct}.\alpha \subset \beta .\alpha \neq \beta .\supset .\text{Nc}ʻ\alpha < \text{Nc}ʻ\beta \quad[*120·426.*24·6]\)</p> + +<p class="nind"><b>*120·71.</b> \(\vdash :\rho ,\sigma \in \text{Cls induct}.\equiv .\rho \cup \sigma \in \text{Cls induct}.\equiv .\rho + \sigma \in \text{Cls induct}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*120·481.&\supset \vdash :\rho \cup \sigma \in \text{Cls induct}.\supset .\rho ,\sigma \in \text{Cls induct} &\qquad \text{(1)}\\ +\vdash .*120·481.&\supset \vdash :\rho ,\sigma \in \text{Cls induct}.\supset .\rho ,\sigma - \rho \in \text{Cls induct}.\\ +[*120·21] & \supset .\text{N}_{0}\text{c}ʻ\rho ,\text{N}_{0}\text{c}ʻ(\sigma - \rho )\in \text{NC induct}.\\ +[*120.45] &\supset .\text{N}_{0}\text{c}ʻ\rho +_{c} \text{N}_{0}\text{c}ʻ(\sigma - \rho )\in \text{NC induct}.\\ +[*110·32] & \supset .\text{Nc}ʻ(\rho \cup \sigma )\in \text{NC induct}.\\ +[*120·211] & \supset .\rho \cup \sigma \in \text{Cls induct} &\qquad \text{(2)}\\ +\vdash .(1).(2). & \supset \vdash :\rho ,\sigma \in \text{Cls induct}.\equiv .\rho \cup \sigma \in \text{Cls induct} &\qquad \text{(3)}\\ +\vdash .*110·12.*120·214.&\supset \vdash :\rho ,\sigma \in \text{Cls induct}.\equiv .\\ +&\downarrow (\Lambda \cap \sigma )ʻʻ\iota ʻʻ\rho ,(\Lambda \cap \rho )\downarrow ʻʻ\iota ʻʻ\sigma \in \text{Cls induct}.\\ +[(3).(*110·01)] & \equiv .\rho + \sigma \in \text{Cls induct} &\qquad \text{(4)}\\ +\vdash .(3).(4).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p>The above proposition is frequently used.</p> + +<p><span class="pagenum" id="Page_230">[Pg 230]</span></p> + +<p class="nind"><b>*120·72.</b> \(\vdash :\rho ,\sigma \in \text{Cls induct}.\supset .\rho \times \sigma \in \text{Cls induct}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*120·21.\supset \vdash :\text{Hp}.&\supset .\text{N}_{0}\text{c}ʻ\rho ,\text{N}_{0}\text{c}ʻ\sigma \in \text{NC induct}.\\ +[*120·5] &\supset .\text{Nc}ʻ(\rho \times \sigma )\in \text{NC induct}.\\ +[*120·211] & \supset .\rho \times \sigma \in \text{Cls induct}:\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*120·721.</b> \(\vdash \colon\ldotp \exists !\rho .\exists !\sigma .\supset :\rho ,\sigma \in \text{Cls induct}.\equiv .\rho \times \sigma \in \text{Cls induct}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*120·512.*113·107.\supset \\ +\vdash \colon\ldotp \text{Hp}.&\supset :\rho \times \sigma \in \text{Cls induct}.\supset .\text{Nc}ʻ\rho ,\text{Nc}ʻ\sigma \in \text{NC induct}.\\ +[*120·211] &\supset .\rho ,\sigma \in \text{Cls induct} &\qquad \text{(1)}\\ +\vdash .(1).*120·72.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*120·73.</b> \(\vdash :\rho ,\sigma \in \text{Cls induct}.\supset .(\rho \,\,\text{exp}\,\,\sigma )\in \text{Cls induct} \quad[*120·52.*116·251]\)</p> + +<p class="nind"><b>*120·731.</b> + \[\begin{align}&\vdash \colon\ldotp \exists !\rho .\exists !\sigma .\rho {\sim}\in 1.\supset :\rho ,\sigma \in \text{Cls induct}.\equiv .(\rho \,\,\text{exp}\,\,\sigma )\in \text{Cls induct}\\ +&[*120·56·561·73]\end{align}\]</p> + +<p class="nind"><b><a id="*120·74">*120·74</a>.</b> \(\vdash :\rho \in \text{Cls induct}.\equiv .\text{Cl}ʻ\rho \in \text{Cls induct}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*116·72.*120·21.\supset \vdash :\text{Cl}ʻ\rho \in \text{Cls induct}.&\equiv .2^{\text{Nc}ʻ\rho }\cap tʻ\text{Cl}ʻ\rho \in \text{NC induct}.\\ +[*120·123·52·56.*116·72.(*116·04)] &\equiv .\text{N}_{0}\text{c}ʻ\rho \in \text{NC induct}.\\ +[*120·21] &\equiv .\rho \in \text{Cls induct}.:\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*120·741.</b> \(\vdash :sʻ\kappa \in \text{Cls induct}.\supset .\kappa \in \text{Cls induct}.\kappa \subset \text{Cls induct}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*120·74. &\supset \vdash :\text{Hp}.\supset .\text{Cl}ʻsʻ\kappa \in \text{Cls induct}.\\ +[*60·57.*120·481] &\supset .\kappa \in \text{Cls induct} &\qquad \text{(1)}\\ +\vdash .*40·13.*120·481.&\supset \vdash \colon\ldotp \text{Hp}.\supset :\rho \in \kappa .\supset .\rho \in \text{Cls induct} &\qquad \text{(2)}\\ +\vdash .(1).(2).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*120·75">*120·75</a>.</b> \(\vdash :sʻ\kappa \in \text{Cls induct}.\equiv .\kappa \in \text{Cls induct}.\kappa \subset \text{Cls induct}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*22·58. &\supset \vdash :\exists !\kappa -\text{Cls induct}.\supset .\exists !(\kappa \cup \iota ʻ\alpha )-\text{Cls induct} &\qquad \text{(1)}\\ +\vdash .*120·71.*53·15.&\supset \vdash :sʻ\kappa \in \text{Cls induct}.\alpha \in \text{Cls induct}.\supset .\\ +&sʻ(\kappa \cup \iota ʻ\alpha )\in \text{Cls induct}:\\ +[*5·6] &\supset \vdash \colon\ldotp sʻ\kappa \in \text{Cls induct}.\supset :\\ +&\alpha {\sim}\in \text{Cls induct}.\lor.sʻ(\kappa \cup \iota ʻ\alpha )\in \text{Cls induct}:\\ +[*51·16]&\supset :\exists !(\kappa \cup \iota ʻ\alpha )-\text{Cls induct}.\lor.sʻ(\kappa \cup \iota ʻ\alpha )\in \text{Cls induct} &\qquad \text{(2)}\\ +\vdash .(1).(2).&\supset \vdash \colon\ldotp \exists !\kappa -\text{Cls induct}.\lor.sʻ\kappa \in \text{Cls induct}:\supset :\\ +&\exists !(\kappa \cup \iota ʻ\alpha )-\text{Cls induct}.\lor.sʻ(\kappa \cup \iota ʻ\alpha )\in \text{Cls induct} &\qquad \text{(3)}\\ +\vdash .*40·21.*120·212.&\supset \vdash .sʻ\Lambda \in \text{Cls induct} &\qquad \text{(4)}\\ +\vdash .(3).(4).*120·26.&\supset \vdash \colon\ldotp \kappa \in \text{Cls induct}.\supset :\\ +&\exists !\kappa -\text{Cls induct}.\lor.sʻ\kappa \in \text{Cls induct}\colon\ldotp \\ +[*5·6]&\supset \vdash :\kappa \in \text{Cls induct}.\kappa \subset \text{Cls induct}.\supset .sʻ\kappa \in \text{Cls induct} &\qquad \text{(5)}\\ +\vdash .(5).*120·741.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_231">[Pg 231]</span></p> + +<p class="nind"><b><a id="*120·76">*120·76</a>.</b> \(\vdash :\kappa \in \text{Cls induct}.\kappa \subset \text{Cls induct}.\supset .{\in}_{\Delta}ʻ\kappa \in \text{Cls induct}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*51·2. &\supset \vdash \colon\ldotp \alpha \in \kappa .\supset :\kappa =\kappa \cup \iota ʻ\alpha :\\ +[*13·12] &\supset :{\in}_{\Delta}ʻ\kappa \in \text{Cls induct}.\supset .{\in}_{\Delta}ʻ(\kappa \cup \iota ʻ\alpha )\in \text{Cls induct} &\qquad \text{(1)}\\ +\vdash .*83·41.*114·301. &\supset \vdash :\alpha {\sim}\in \kappa .\supset .{\in}_{\Delta}ʻ(\kappa \cup \iota ʻ\alpha )\text{ sm }{\in}_{\Delta}ʻ\kappa + \times \alpha &\qquad \text{(2)}\\ +\vdash .(2).*120·214. &\supset \vdash \colon\ldotp \alpha {\sim}\in \kappa .\supset :\\ +&{\in}_{\Delta}ʻ(\kappa \cup \iota ʻ\alpha )\in \text{Cls induct}.\equiv .{\in}_{\Delta}ʻ\kappa \times \alpha \in \text{Cls induct} &\qquad \text{(3)}\\ +\vdash .(3).*120·72. &\supset \vdash \colon\ldotp \alpha {\sim}\in \kappa .\supset :\\ +&{\in}_{\Delta}ʻ\kappa ,\alpha \in \text{Cls induct}.\supset .{\in}_{\Delta}ʻ(\kappa \cup \iota ʻ\alpha )\in \text{Cls induct} &\qquad \text{(4)}\\ +\vdash .(1).(4). &\supset \vdash :{\in}_{\Delta}ʻ\kappa ,\alpha \in \text{Cls induct}.\supset .{\in}_{\Delta}ʻ(\kappa + \cup \iota ʻ\alpha )\in \text{Cls induct} &\qquad \text{(5)}\\ +\vdash .(5).*51·2.\text{Syll}. &\supset \vdash \colon\ldotp \kappa \subset \text{Cls induct}.\supset .{\in}_{\Delta}ʻ\kappa \in \text{Cls induct}:\supset :\\ +&\kappa \cup \iota ʻ\alpha \subset \text{Cls induct}.\supset .{\in}_{\Delta}ʻ(\kappa \cup \iota ʻ\alpha )\in \text{Cls induct} &\qquad \text{(6)}\\ +\vdash .*83·15.*120·213. &\supset \vdash .{\in}_{\Delta}ʻ\Lambda \in \text{Cls induct}.\\ +[\text{Simp}] &\supset \vdash :\Lambda \subset \text{Cls induct}.\supset .{\in}_{\Delta}ʻ\Lambda \in \text{Cls induct} &\qquad \text{(7)}\\ +\vdash .(6).(7).*120·26. &\supset \vdash \colon\ldotp \kappa \in \text{Cls induct}.\supset :\\ +&\kappa \subset \text{Cls induct}.\supset .{\in}_{\Delta}ʻ\kappa \in \text{Cls induct}\colon\ldotp \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p>The following propositions are concerned in establishing the converse +of <a href="#*120·76">*120·76</a> subject to a suitable hypothesis. The final outcome is given +in <a href="#*120·77">*120·77</a>.</p> + +<p class="nind"><b>*120·761.</b> \(\vdash :\exists !{\in}_{\Delta}ʻ\kappa .{\in}_{\Delta}ʻ\kappa \in \text{Cls induct}.\supset .\kappa \subset \text{Cls induct}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*83·41.*114·301. &\supset \vdash \colon\ldotp \alpha \in \kappa .\supset :{\in}_{\Delta}ʻ\kappa \text{ sm }\alpha \times {\in}_{\Delta}ʻ(\kappa + -\iota ʻ\alpha ): &\qquad \text{(1)}\\ +[*120·214] &\supset :{\in}_{\Delta}ʻ\kappa \in \text{Cls induct}.\equiv .\alpha \times {\in}_{\Delta}ʻ(\kappa -\iota ʻ\alpha )\in \text{Cls induct} &\qquad \text{(2)}\\ +\vdash .(1).*113·114. &\supset \vdash :\exists !{\in}_{\Delta}ʻ\kappa .\alpha \in \kappa .\supset .\exists !\alpha .\exists !{\in}_{\Delta}ʻ(\kappa + -\iota ʻ\alpha ) &\qquad \text{(3)}\\ +\vdash .(2).(3).*120·721. & \supset \vdash \colon\ldotp \exists !{\in}_{\Delta}ʻ\kappa .\alpha \in \kappa .\supset :\\ +&{\in}_{\Delta}ʻ\kappa \in \text{Cls induct}.\supset .\alpha \in \text{Cls induct} &\qquad \text{(4)}\\ +\vdash .(4).\text{Comm}.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*120·762.</b> \(\vdash :\kappa \in \text{Cls induct}.\Lambda {\sim}\in \kappa .{\sim}\exists !1\cap \kappa .\supset .(\exists R,S).R,S\in {\in}_{\Delta}ʻ\kappa + .R\dot{\cap} S=\dot{\Lambda} \)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*51·2. & \supset \vdash :R,S\in {\in}_{\Delta}ʻ\kappa .R\dot{\cap} S=\dot{\Lambda} .\alpha \in \kappa .\supset .R,S\in {\in}_{\Delta}ʻ(\kappa + \cup \iota ʻ\alpha ).R\dot{\cap} S=\dot{\Lambda} &\qquad \text{(1)}\\ +\vdash .*83·5.*55·201.\supset \\ +&\vdash :R,S\in {\in}_{\Delta}ʻ\kappa .R\dot{\cap} S=\dot{\Lambda} .x,y\in \alpha .x\neq y.\alpha {\sim}\in \kappa .\supset .\\ +&R\unicode{x228d} x\downarrow \alpha ,S\unicode{x228d} y\downarrow \alpha \in {\in}_{\Delta}ʻ(\kappa + \cup \iota ʻ\alpha ).(R\unicode{x228d} x\downarrow \alpha )\dot{\cap} (S\unicode{x228d} y\downarrow \alpha )=\dot{\Lambda} &\qquad \text{(2)}\\ +\vdash .(1).(2).*52·41. &\supset \vdash :R,S\in {\in}_{\Delta}ʻ\kappa .R\dot{\cap} S=\dot{\Lambda} .\alpha \neq \Lambda .\alpha {\sim}\in 1.\supset .\\ +&(\exists P,Q).P,Q\in {\in}_{\Delta}ʻ(\kappa \cup \iota ʻ\alpha ).P\dot{\cap} Q=\dot{\Lambda} &\qquad \text{(3)}\\ +\vdash .*51·16.&\supset \vdash \colon\ldotp \alpha =\Lambda .\lor.\alpha \in 1:\supset :\Lambda \in (\kappa \cup \iota ʻ\alpha ).\lor.\exists !1\cap (\kappa \cup \iota ʻ\alpha ) &\qquad \text{(4)}\\ +\vdash .*22·58.&\supset \vdash \colon\ldotp \Lambda \in \kappa .\lor.\exists !1\cap \kappa :\supset :\Lambda \in (\kappa \cup \iota ʻ\alpha ).\lor.\exists !1\cap (\kappa \cup \iota ʻ\alpha ) &\qquad \text{(5)}\\ +\vdash .(3).(4).(5).&\supset \vdash \colon\ldotp \Lambda \in \kappa .\lor.\exists !1\cap \kappa .\lor.(\exists R,S).R,S\in {\in}_{\Delta}ʻ\kappa .R\dot{\cap} S=\dot{\Lambda} :\supset :\\ +&\Lambda \in (\kappa \cup \iota ʻ\alpha ).\lor.\exists !1\cap (\kappa \cup \iota ʻ\alpha ).\lor.(\exists R,S).R,S\in {\in}_{\Delta}ʻ(\kappa + \cup \iota ʻ\alpha ).R\dot{\cap} S=\dot{\Lambda} &\qquad \text{(6)}\\ +\vdash .*83·15.&\supset \vdash .(\exists R,S).R,S\in {\in}_{\Delta}ʻ\Lambda .R\dot{\cap} S=\dot{\Lambda} &\qquad \text{(7)}\\ +\vdash .(6).(7).*120·26.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_232">[Pg 232]</span></p> + +<p class="nind"><b>*120·764.</b> \[\begin{align}&\vdash :\kappa \in \text{Cls induct}.\Lambda {\sim}\in \kappa .{\sim}\exists !(1\cap \kappa ).\supset .\text{Nc}ʻ{\in}_{\Delta}ʻ\kappa + \geq \text{Nc}ʻ\kappa \\ +&[*120·762.*117·681]\end{align}\]</p> + +<p class="nind"><b><a id="*120·765">*120·765</a>.</b> \[\begin{align}\vdash :\kappa \in \text{Cls induct}.\Lambda {\sim}\in \kappa .{\sim}&\exists !(1\cap \kappa ).\kappa \subset \lambda .\exists !{\in}_{\Delta}ʻ\lambda + .\supset .\\ +&\text{Nc}ʻ{\in}_{\Delta}ʻ\lambda \geq \text{Nc}ʻ\kappa \quad[*120·762.*117·684]\end{align}\]</p> + +<p class="nind"><b>*120·766.</b> \[\begin{align}\vdash :\lambda {\sim}\in \text{Cls induct}.\Lambda {\sim}\in \lambda.{\sim}\exists !(1\cap \lambda ).&\exists !{\in}_{\Delta}ʻ\lambda .\supset .\\ +&\text{Nc}ʻ{\in}_{\Delta}ʻ\lambda {\sim}\in \text{NC induct}\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*120·491.\supset \vdash \colon\ldotp \text{Hp}.&\supset :\nu \in \text{NC induct}.\supset .\\ +&(\exists \kappa ).\kappa \subset \lambda .\text{Nc}ʻ\kappa =\nu .\Lambda {\sim}\in \kappa .{\sim}\exists !(1\cap \kappa ).\\ +[*120·765] &\supset .\text{Nc}ʻ{\in}_{\Delta}ʻ\lambda \geq \nu :\\ +[*120·121] &\supset :\nu \in \text{NC induct}.\supset .\text{Nc}ʻ{\in}_{\Delta}ʻ\lambda \geq \nu +_{c}1.\\ +[*120·429] &\supset .\text{Nc}ʻ{\in}_{\Delta}ʻ\lambda >\nu :\\ +[*117·42] &\supset :\text{Nc}ʻ{\in}_{\Delta}ʻ\lambda {\sim}\in \text{NC induct}\colon\ldotp \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*120·767.</b> \[\begin{align}&\vdash :{\in}_{\Delta}ʻ\lambda \in \text{Cls induct}.\Lambda {\sim}\in \lambda .{\sim}\exists !(1\cap \lambda ).\exists !{\in}_{\Delta}ʻ\lambda + .\supset .\lambda \in \text{Cls induct}\\ +&[*120·766.\text{Transp}]\end{align}\]</p> + +<p class="nind"><b><a id="*120·77">*120·77</a>.</b> \[\begin{align}\vdash \colon\ldotp \Lambda {\sim}\in \kappa .{\sim}&\exists !(1\cap \kappa ).\exists !{\in}_{\Delta}ʻ\kappa .\supset :\\ +&{\in}_{\Delta}ʻ\kappa \in \text{Cls induct}.\equiv .\kappa \in \text{Cls induct}.\kappa \subset \text{Cls induct}\\ +&[*120·76·761·767]\end{align}\]</p> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_233">[Pg 233]</span></p> +<h2 class="nobreak" id="*121">*121. INTERVALS.</h2> +</div> + + +<p class="nind"><i>Summary of</i> *121.</p> + +<p>The present number is concerned with the class of terms between +\(x\) and \(y\) with respect to some relation \(P\), <i>i.e.</i> +those terms which lie on a road from \(x\) to \(y\) on which any two +consecutive terms have the relation \(P\). Such a road may be called +a \(P\)-road, and if \(zPw\), the step from \(z\) to \(w\) may be +called a \(P\)-step. In order that a \(P\)-road from \(x\) to \(y\) +should exist, it is necessary and sufficient that we should have +\(xP_{\text{po}}y\). When this condition is fulfilled, there will +in general be many \(P\)-roads from \(x\) to \(y\). But if \(P\in +\text{Cls}\rightarrow 1.{\sim}(yP_{\text{po}}y)\), or if \(P\in +1\rightarrow \text{Cls}.{\sim}(xP_{\text{po}}x)\),then at most one +road leads from \(x\) to \(y\). This follows from the propositions of +*96. In virtue of those propositions, if \(P\in \text{Cls}\rightarrow +1.{\sim}(yP_{\text{po}}y).xP_{\text{po}}y\), \(P\) is \(1\rightarrow +1\) throughout the road from \(x\) to \(y\), and this road forms an +open series. The two other possibilities with a \(\text{Cls}\rightarrow 1\) +are (assuming \(xP_{\text{po}}y\)) +\[ +\begin{aligned} +&(1)\quad xP_{\text{po}}x,\\ +&(2) \quad yP_{\text{po}}y.{\sim}(xP_{\text{po}}x). +\end{aligned} +\]</p> + +<p>In the first case, there is a cyclic road from \(x\) to \(x\), and +there are two roads from \(x\) to \(y\), one consisting of that part +of the cycle which is required to reach \(y\), the other consisting of +this part together with the whole cycle required to travel from \(y\) +back to \(y\). Thus the class of terms which can be reached in some +journey from \(x\) to \(y\) is the whole class of descendants of \(x\), +<i>i.e.</i> the class \(\overleftarrow{R}_{*}ʻx\), which is the cycle +composing the road from \(x\) to \(x\).</p> + +<p>In the second case, the descendants of \(x\) form a \(Q\), and \(y\) is +in the circular part of the \(Q\). Here, as before, there are two roads +from \(x\) to \(y\), of which the first stops as soon as it reaches +\(y\), while the second proceeds to travel round the circle until it +comes to \(y\) again. Thus here again, all the descendants of \(x\) lie +on some road between \(x\) and \(y\).</p> + +<p>The interval between \(x\) and \(y\) is defined as the class of terms +lying on some road from \(x\) to \(y\). There will be four kinds of +interval, according as we do or do not include the end-points as such. +We denote the kind including both end-points by +\[ +P(x\vdash\dashv y), +\] +that excluding both by +\[ +P(x\unicode{x2013} y) +\] +and the other two respectively by +\[ +P(x\dashv y), P(x \unicode{x27dd} y). +\]</p> + +<p><span class="pagenum" id="Page_234">[Pg 234]</span></p> + +<p>The definitions are +\[ +\overleftarrow{P}_{*}ʻx\cap \overrightarrow{P}_{*}ʻy, \quad\overleftarrow{P}_{\text{po}}ʻx\cap \overrightarrow{P}_{\text{po}}ʻy, \quad\overleftarrow{P}_{\text{po}}ʻx\cap + \overrightarrow{P}_{*}ʻy, \quad\overleftarrow{P}_{*}ʻx\cap \overrightarrow{P}_{\text{po}}ʻy. +\]</p> + +<p>If \(P\) is either one-many or many-one, it will be one-one +throughout the interval \(P(x\vdash\dashv y)\), except at most at one +exceptional point, namely the junction of the tail and circle of the +\(Q\). If \(xP_{\text{po}}x\) or \({\sim}(yP_{\text{po}}y)\), the +interval between \(x\) and \(y\) cannot be \(Q\)-shaped, but must +be either open or cyclic; in either case, \(P\) is \(1\rightarrow 1\) +throughout \(P(x\vdash\dashv y)\), with no exceptions; for +if \(P\in \text{Cls}\rightarrow 1\), \(P\) is \(1\rightarrow 1\) +throughout the interval because the interval is contained in +\(\overleftarrow{P}_{*}ʻx\), and if \(P\in 1\rightarrow \text{Cls}\), +because the interval is contained in \(\overrightarrow{P}_{*}ʻy\). Thus +throughout this number we shall constantly have the hypothesis +\(P\in(\text{Cls}\rightarrow 1)\cup (1\rightarrow \text{Cls})\); if +\(P\in\text{Cls}\rightarrow 1\), the interval is to be supposed traversed +from \(x\) to \(y\), while if \(P\in 1\rightarrow \text{Cls}\), it +is to be supposed traversed from \(y\) to \(x\). In either case the +interval between \(x\) and \(y\) must be an inductive class. This is +proved in <a href="#*121·47">*121·47</a>. If, however, \(P\) is serial (cf. <a href="#*204">*204</a>), and thus +neither many-one nor one-many, the interval between \(x\) and \(y\) +is the stretch of the series between \(x\) and \(y\), with or without +end-points according to the definition chosen, and need not be an +inductive class.</p> + +<p>If the interval between \(x\) and \(y\) (both included) has +\(\nu +_{c}1\) members, we say that \(xP_{\nu }y\). Thus if +there is only one road from \(x\) to \(y\), "\(xP_{\nu }y\)" +means that it requires \(\nu\) steps to get from \(x\) to \(y\). +Assuming \(P\in \text{Cls}\rightarrow 1\), if we also have +\(P_{\text{po}}\,\unicode{x2abd}\, J\) (<i>i.e.</i> if none of the families +of \(P\) are cyclic), then if \(xP_{\nu }y\) and \(yPz\), we shall have +\(xP_{\nu +_{c}1}z\). On this basis an inductive theory of \(P_{\nu}\) +is built up, and it is shown that the class of such relations +as \(P_{\nu }\) for different inductive values of \(\nu\) is the +same as \(\text{Potid}ʻP\), the class of powers of \(P\) including +\(I\upharpoonright CʻP\) (<a href="#*121·5">*121·5</a>). The definition of \(P_{\nu }\) is +\[ +P_{\nu } = \hat{x} \hat{y} \{\text{N}_{0}\text{c}ʻP(x\vdash\dashv y)=v+_{c}1\} \quad\text{Df}. +\]</p> + +<p>The whole class of such relations as \(P_{\nu}\) for different +inductive values of \(\nu\) is called \(\text{finid}ʻP\), <i>i.e.</i> +we put +\[ +\text{finid}ʻP=\hat{R} \{(\exists \nu ).\nu \in \text{NC induct}-\iota ʻ\Lambda .R=P_{\nu }\} \quad\text{Df}. +\]</p> + +<p>If \(BʻP\) exists, and if \(P\in \text{Cls}\rightarrow 1\), then the +descendants of \(BʻP\), so long as we do not reach a term \(y\) for which +\(yP_{\text{po}}y\), may be unambiguously described as the 2nd, 3rd, +... \(\nu\)th, ... terms of the posterity of \(BʻP\), \(BʻP\) itself +being the 1st term. The correlation thus effected with the inductive +cardinals is the logical essence of the process of counting; the last +cardinal used in the correlation is the cardinal number of terms +counted. We will call these terms \(1_{P}\), \(2_{P}\), ... \(\nu_{P}\), +..., defining \(\nu _{P}\) as follows: +\[ +\nu _{P} = \breve{P} _{\nu -_{c}1}ʻBʻP \quad\text{Df}, +\]</p> + +<p>This notation does not conflict with \(\nu _{\xi }\) as defined in +*65.01. There \(\xi\) must be a class if \(\nu\) is a cardinal, here +\(\nu\) must be a cardinal and \(P\) a relation.</p> + +<p><span class="pagenum" id="Page_235">[Pg 235]</span></p> + +<p>Hence whenever \(\nu _{P}\) exists, the number of terms from the +beginning to \(\nu _{P}\) (both included) is \(\nu\). This is +the fact upon which counting relies. If \(P\) is a many-one and +\(P_{\text{po}}\) is contained in diversity, and \(\nu\) is any +inductive cardinal other than 0, then \(\nu _{P}\) exists when and +only when \(\overleftarrow{P}_{*}ʻBʻP\) has at least \(\nu\) members; +<i>i.e.</i> roughly speaking, \(\nu _{P}\) exists whenever it could +possibly be expected to exist. In this case the whole posterity of +\(BʻP\) is contained in the series \(1_{P}\), \(2_{P}\), ... \(\nu_{P}\), +... (<a href="#*121·62">*121·62</a>). If the posterity is an inductive class, this +series stops; if not, it forms a <i>progression</i> (cf. <a href="#*122">*122</a>).</p> + +<p>The propositions of the present number are very useful, not only in +this section, but in the ordinal theory of finite and infinite and in +parts of the book subsequent to that theory.</p> + +<p>After some propositions which merely repeat definitions and give +immediate consequences, we proceed (<a href="#*121·3">*121·3</a> ff.) to the theory of +\(P_{\nu }\). We have</p> + +<p class="nind"><b>*121·302.</b> \(\vdash :P_{\text{po}}\,\unicode{x2abd}\, J.\supset .P_{0}=I\upharpoonright CʻP\)</p> + +<p class="nind"><b>*121·305.</b> \(\vdash :P_{\text{po}}\,\unicode{x2abd}\, J.\supset .P_{1}\,\unicode{x2abd}\, P\)</p> + +<p class="nind"><b>*121·31.</b> \(\vdash :P\in (l\rightarrow \text{Cls})\cup (\text{Cls}\rightarrow 1).P_{\text{po}}\,\unicode{x2abd}\, J.\supset .P_{1}=P\)</p> + +<p>When \(P\) is a transitive serial relation, we shall have +\(P_{1}=P\dot{-} P^{2}\).</p> + +<p class="nind"><b>*121·321.</b> \(\vdash :\nu >0.\supset .P_{\nu }\,\unicode{x2abd}\, P_{\text{po}}\)</p> + +<p class="nind"><b>*121·333.</b> \(\vdash :P\in \text{Cls}\rightarrow 1.P_{\text{po}}\,\unicode{x2abd}\, J.\supset .P_{\nu +_{c}1}=P\mid P_{\nu }\)</p> + +<p class="nind"><b>*121·35·351·352.</b> \[\begin{align}\vdash :P\in (1\rightarrow \text{Cls})\cup (\text{Cls}\rightarrow 1).&P_{\text{po}}\,\unicode{x2abd}\, J.\mu ,\nu \in \text{NC induct}.\supset .\\ +&P_{\mu }\mid P_{\nu }=P_{\nu }\mid P_{\mu }=P_{\mu +_{c}\nu }\end{align}\]</p> + +<p>A similar result holds for \((P_{\mu })_{\nu }\), which = \(P_{\mu x_{c}\nu }\) +n the same circumstances.</p> + +<p>We next proceed to the proof that an interval (under a similar +hypothesis) is always an inductive class. This occupies <a href="#*121·4">*121·4</a>—<a href="#*121·47">·47</a>, +being summed up in the proposition</p> + +<p class="nind"><b>*121·47.</b> \(\vdash :R\in (\text{Cls}\rightarrow 1)\cup (1\rightarrow \text{Cls}).\supset .R(x \vdash\dashv z)\in \text{Cls induct}\)</p> + +<p>This is an important proposition. It leads to</p> + +<p class="nind"><b>*121·481.</b> \[\begin{align}\vdash \colon\ldotp R\in \text{Cls}\rightarrow 1.\supset :\text{Nc}ʻR(x\vdash\dashv y)\leq \text{Nc}ʻ&R(x \vdash\dashv z).\equiv .\\ +&R(x \vdash\dashv y)\subset R(x \vdash\dashv z)\end{align}\]</p> + +<p>with a similar proposition if \(R\in 1\rightarrow \text{Cls}\).</p> + +<p>The next set of propositions (<a href="#*121·5">*121·5</a>—<a href="#*121·52">·52</a>) is concerned with +\(\text{finid}ʻP\). Assuming \(P\in (\text{Cls}\rightarrow 1)\cup +(1\rightarrow \text{Cls}).P_{\text{po}}\,\unicode{x2abd}\, J\), we prove +that \(\text{finid}ʻP=\text{Potid}ʻP\) and \(\text{finid}ʻP-\iotaʻP_{0}\subset \text{Pot}ʻP\) +(<a href="#*121·5">*121·5</a>); that if \(P\) is not null, +\(\text{finid}ʻP-\iota ʻP_{0}=\text{Pot}ʻP\) (<a href="#*121·501">*121·501</a>); that \(\dot{s}ʻ\text{finid}ʻP=P_{*}\) +(<a href="#*121·52">*121·52</a>) and \(\dot{s} ʻ(\text{finid}ʻP-\iotaʻP_{0})=P_{\text{po}}\) +(<a href="#*121·502">*121·502</a>); and that \(P_{2}=P^{2}.P_{3}=P^{3}\) +etc. (<a href="#*121·51">*121·51</a>).</p> + +<p><span class="pagenum" id="Page_236">[Pg 236]</span></p> + +<p>Our next set of propositions is concerned with \(\nu _{P}\) +(<a href="#*121·6">*121·6</a>—<a href="#*121·638">·638</a>). We have</p> + +<p class="nind"><b>*121·601.</b> \(\vdash : \text{E}! BʻP . \supset . BʻP = 1_{P} . {\sim} \{(BʻP) P_{\text{po}} (BʻP)\}\)</p> + +<p class="nind"><b>*121·602.</b> \(\vdash : \text{E}! BʻP . P \in 1 \rightarrow 1 . \supset . \breve{P} ʻBʻP = 2_{P}\)</p> + +<p class="nind"><b>*121·634.</b> \[\begin{align}\vdash \colon\ldotp P \in \text{Cls} \rightarrow 1 . P_{\text{po}} \,\unicode{x2abd}\, J . \nu \in &\text{NC induct} - \iota ʻ0 . \supset :\\ +&\nu _{P} \in \text{D} ʻP . \equiv . \text{E}! (\nu +_{c} 1)_{P}\end{align}\]</p> + +<p>Finally we have three propositions (<a href="#*121·7">*121·7</a>—<a href="#*121·72">*121·72</a>) on +\(\overrightarrow{R}_{*}ʻx\), of which the most useful is</p> + +<p class="nind"><b>*121·7.</b> \(\vdash : R \in 1 \rightarrow 1 . aBR . aR_{*}x . \supset . \overrightarrow{R}_{*}ʻx = R(a \vdash\dashv x) . \overrightarrow{R}_{*}ʻx + \in \text{Cls induct}\)</p> + +<hr class="tb"> + +<p class="nind"><b>*121·01.</b> \(P(x \unicode{x2013} y) = \overleftarrow{P}_{\text{po}}ʻx \cap \overrightarrow{P}_{\text{po}}ʻy \quad\text{Df}\)</p> + +<p class="nind"><b>*121·011.</b> \(P(x \dashv y) = \overleftarrow{P}_{\text{po}} ʻx \cap \overrightarrow{P}_{*} ʻy \quad\text{Df}\)</p> + +<p class="nind"><b>*121·012.</b> \(P(x \unicode{x27dd} y) = \overleftarrow{P}_{*} ʻx \cap \overrightarrow{P}_{\text{po}} ʻy \quad\text{Df}\)</p> + +<p class="nind"><b>*121·013.</b> \(P(x \vdash\dashv y) = \overleftarrow{P}_{*} ʻx \cap \overrightarrow{P}_{*} ʻy \quad\text{Df}\)</p> + +<p class="nind"><b><a id="*121·02">*121·02</a>.</b> \(P_\nu = \hat{x} \hat{y} \{\text{N}_0\text{c} ʻP (x \vdash\dashv y) = \nu +_{c} 1\} \quad\text{Df}\)</p> + +<p class="nind"><b>*121·03.</b> \(\text{finid} ʻP = \hat{R} \{(\exists \nu ) . \nu \in \text{NC induct} - \iota ʻ\Lambda . R = P_{\nu }\} \quad\text{Df}\)</p> + +<p class="nind"><b>*121·031.</b> \(\text{fin} ʻP = \hat{R} \{(\exists \nu ). \nu \in \text{NC induct} - \iota ʻ\Lambda - \iota ʻ0 . R = P_{\nu }\} \quad\text{Df}\)</p> + +<p class="nind"><b>*121·04.</b> \(\nu _{P} = \breve{P} _{\nu -_{c} 1} ʻBʻP \quad\text{Df}\)</p> + +<p class="nind"><b>*121·1.</b> \(\vdash : z \in P (x \unicode{x2013} y) . \equiv . xP_{\text{po}}z . zP_{\text{po}}y \quad[(*121·01)]\)</p> + +<p class="nind"><b>*121·101.</b> \(\vdash : z \in P (x \dashv y) . \equiv . xP_{\text{po}}z . zP_{*}y\)</p> + +<p class="nind"><b>*121·102.</b> \(\vdash : z \in P (x \unicode{x27dd} y) . \equiv . xP_{*}z . zP_{\text{po}}y\)</p> + +<p class="nind"><b>*121·103.</b> \(\vdash : z \in P (x \vdash\dashv y) . \equiv . xP_{*}z . zP_{*}y\)</p> + +<p class="nind"><b>*121·11.</b> \(\vdash : xP_{\nu }y . \equiv . \text{N}_0\text{c} ʻP (x \vdash\dashv y) = \nu +_{c} 1 \quad[(*121·02)]\)</p> + +<p class="nind"><b>*121·12.</b> \(\vdash : R \in \text{finid} ʻP . \equiv . (\exists \nu ) . \nu \in \text{NC induct} - \iota ʻ\Lambda . R = P_\nu \quad[(*121·03)]\)</p> + +<p class="nind"><b>*121·121.</b> \(\vdash : R \in \text{fin} ʻP . \equiv . (\exists \nu ) . \nu \in \text{NC induct} - \iota ʻ\Lambda - \iota ʻ0 . R = P_\nu \quad[(*121·031)]\)</p> + +<p class="nind"><b>*121·13.</b> \(\vdash : f (\nu _{P}) . \equiv . f(\breve{P} _{\nu -_{c} 1} ʻBʻP) \quad[(*121·04)]\)</p> + +<p class="nind"><b>*121*131.</b> \(\vdash : \text{E}! \breve{P} _{\nu -_{c} 1} ʻBʻP . \supset . \nu _{P} = \breve{P} _{\nu -_{c} 1}ʻBʻP \quad[*121·13.*14·28]\)</p> + +<p class="nind"><b>*121·14.</b> \(\vdash . P (x \unicode{x2013} y) = \breve{P} (y \unicode{x2013} x) \quad[*121·1 . *91·53]\)</p> + +<p class="nind"><b>*121·141.</b> \(\vdash . P (x \dashv y) = \breve{P} (y \unicode{x27dd} x)\)</p> + +<p class="nind"><b>*121·142.</b> \(\vdash . P (x \unicode{x27dd} y) = \breve{P} (y \dashv x)\)</p> + +<p class="nind"><b><a id="*121·143">*121·143</a>.</b> \(\vdash . P (x \vdash\dashv y) = \breve{P} (y \vdash\dashv x)\)</p> + +<p class="nind"><b>*121·2.</b> \(\vdash : {\sim} (xP_{\text{po}}x) . \supset . x {\sim} \in P (x \unicode{x2013} y) \quad[*121·1]\)</p> + +<p><span class="pagenum" id="Page_237">[Pg 237]</span></p> + +<p class="nind"><b>*121·201.</b> \(\vdash : {\sim} (yP_{\text{po}}y) . \supset . y {\sim} \in P (x \unicode{x2013} y)\)</p> + +<p class="nind"><b>*121·202.</b> \(\vdash :P_{\text{po}}\,\unicode{x2abd}\, J.\supset .x,y{\sim}\in P(x\unicode{x2013}y) \quad[*121·2·201]\)</p> + +<p class="nind"><b>*121*21.</b> \(\vdash :xP_{\text{po}}y.\equiv .y\in P(x\dashv y).\equiv .\exists !P(x\dashv y)\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*90·12.*91·54.\supset \vdash :xP_{\text{po}}y.&\equiv .xP_{\text{po}}y.yP_{*}y.\\ +[*121·101] &\equiv .y\in P(x\dashv y) &\qquad \text{(1)}\\ +\vdash .*121·101.\supset \vdash :\exists !P(x\dashv y).&\equiv .xP_{\text{po}}\mid P_{*}y.\\ +[*91·574] &\equiv .xP_{\text{po}}y &\qquad \text{(2)}\\ +\vdash .(1).(2).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*121·22.</b> \(\vdash :xP_{\text{po}}y.\equiv .x\in P(x\unicode{x27dd} y).\equiv .\exists !P(x\unicode{x27dd} y)\)</p> + +<p class="nind"><b>*121·23.</b> \(\vdash :xP_{*}y.\equiv .x,y\in P(x\vdash\dashv y).\equiv .\exists !P(x\vdash\dashv y)\)</p> + +<p class="nind"><b>*121·231.</b> \(\vdash :x\in CʻP.\equiv .x\in P(x\vdash\dashv x).\equiv .\exists !P(x\vdash\dashv x) \quad[*121·23.*90·12]\)</p> + +<p class="nind"><b>*121·24.</b> \(\vdash :xP_{\text{po}}y.\supset .P(x\dashv y)=P(x\unicode{x2013}y)\cup \iota ʻy\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*91·54.*121·101.\supset \\ +\vdash \colon\ldotp z\in P(x\dashv y). &\equiv :xP_{\text{po}}z:zP_{\text{po}}y.\lor.z=y.y\in CʻP:\\ +[*13·193.*91·504]&\equiv:xP_{\text{po}}z.zP_{\text{po}}y.\lor.xP_{\text{po}}y.z=y &\qquad \text{(1)}\\ +\vdash .(1).*4·73.\supset \vdash \colon\colon \text{Hp}.\supset \colon\ldotp z\in P(x\dashv y).&\equiv :xP_{\text{po}}z.zP_{\text{po}}y.\lor.z=y\colon\colon \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*121·241.</b> \(\vdash :xP_{\text{po}}y.\supset .P(x \unicode{x27dd} y)=P(x\unicode{x2013}y)\cup \iota ʻx\)</p> + +<p class="nind"><b>*121·242.</b> \[\begin{align}\vdash :xP_{*}y.\supset .P(x\vdash\dashv y)&=P(x\dashv y)\cup \iota ʻx=P(x \unicode{x27dd} y)\cup \iota ʻy\\ +&=P(x-y)\cup \iota ʻx\cup \iota ʻy\end{align}\]</p> + +<p class="nind"><b>*121*25.</b> \(\vdash .P_{\text{po}}(x\unicode{x2013}y)=P(x\unicode{x2013}y) \quad[*91·601.*121·1]\)</p> + +<p class="nind"><b>*121·251.</b> \(\vdash .P_{\text{po}}(x\dashv y)=P(x\dashv y)\)</p> + +<p class="nind"><b>*121·252.</b> \(\vdash .P_{\text{po}}(x\unicode{x27dd} y)=P(x \unicode{x27dd} y)\)</p> + +<p class="nind"><b>*121·253.</b> \(\vdash .P_{\text{po}}(x\vdash\dashv y)=P(x\vdash\dashv y)\)</p> + +<p class="nind"><b><a id="*121·254">*121·254</a>.</b> \(\vdash .P_{\nu }=(P_{\text{po}})_{\nu } \quad[*121·253·11]\)</p> + +<p>*121·254 is frequently used in the theory of series.</p> + +<p class="nind"><b>*121·26.</b> \(\vdash .\breve{P} _{\nu }=(\breve{P} )_{\nu }\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*121·11·143.\supset \vdash :x\breve{P} _{\nu }y.&\equiv .\text{N}_{0}\text{c}ʻ\breve{P} (x\vdash\dashv y)=\nu +_{c}1.\\ +[*90·132.*121·11] &\equiv .x(\breve{P} )_{\nu }y:\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*121·27.</b> \(\vdash :xP_{\nu }y.\supset .\nu ,\nu +_{c}1\in \text{NC}-\iota ʻ\Lambda\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*121·11.*103·12.\supset \vdash :\text{Hp}.\supset .P(x\vdash\dashv y)\in \nu +_{c}1 &\qquad \text{(1)}\\ +\vdash .(1).*110·4·42.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_238">[Pg 238]</span></p> + +<p class="nind"><b>*121·271.</b> \(\vdash :{\sim}(\nu ,\nu +_{c}1\in \text{NC}-\iota ʻ\Lambda ).\supset .P_{\nu }=\dot{\Lambda} \quad[*121·27.\text{Transp}]\)</p> + +<p class="nind"><b>*121·272.</b> \(\vdash :\dot{\exists} !P_{\nu }.\supset .\nu \geq 0.\nu +_{c}1>0.\nu +_{c}1\geq 1\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*117·5.*121·27.\supset \vdash :\text{Hp}.&\supset .\nu \geq 0. &\qquad \text{(1)}\\ +[*117·561.*110·641] &\supset .\nu +_{c}1\geq 1. &\qquad \text{(2)}\\ +[*117·511·531] &\supset .\nu +_{c}1>0 &\qquad \text{(3)}\\ +\vdash .(1).(2).(3).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*121·273.</b> \(\vdash :\dot{\exists} !P_{\nu +_{c}1}.\supset .\nu +_{c}1>0\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*121·27.*110·4.\supset \vdash :\text{Hp}.&\supset .\nu \in \text{NC}-{℩}ʻ\Lambda .\\ +[*117·6] &\supset .\nu +_{c}1\geq 1.\\ +[*117·511·531] &\supset .\nu +_{c}1>0:\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*121·3">*121·3</a>.</b> \(\vdash .P_{0}\,\unicode{x2abd}\, 1\upharpoonright CʻP\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*121·11.\supset \vdash :xP_{0}y.&\equiv .P(x\vdash\dashv y)\in 1.\\ +[*121·23] &\supset .xP_{*}y.x=y.\\ +[*90·12] &\supset .x(I\upharpoonright CʻP)y:\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*121·301.</b> \(\vdash :{\sim}(xP_{\text{po}}x).\supset :xP_{0}y.\equiv .x\in CʻP.x=y\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*91·542·56. &\supset \vdash :xP_{*}z.zP_{*}x.x\neq z.\supset .xP_{\text{po}}x &\qquad \text{(1)}\\ +\vdash .(1).\text{Transp}. \supset \vdash \colon\ldotp \text{Hp}.&\supset :xP_{*}z.zP_{*}x.\supset _{x,z}.x=z:\\ +[*121·231] &\supset :x\in CʻP.\supset .P(x\vdash\dashv x)={℩}ʻx:\\ +[*13·12.*52·22] &\supset :x\in CʻP.x=y.\supset .P(x\vdash\dashv y)\in 1.\\ +[*121·11] &\supset .xP_{0}y &\qquad \text{(2)}\\ +\vdash .(2).*121·3.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*121·302.</b> \(\vdash :P_{\text{po}}\,\unicode{x2abd}\, J.\supset .P_{0}=I\upharpoonright CʻP \quad[*121·301]\)</p> + +<p class="nind"><b>*121·303.</b> \(\vdash :\text{Nc}ʻP(x\vdash\dashv y)>1.\supset .xP_{\text{po}}y\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*121·23.*52·22.*117·42.\supset \vdash \colon\ldotp \text{Hp}.&\supset :x\in P(x\vdash\dashv y).P(x\vdash\dashv y)\neq {℩}ʻx:\\ +[*51·4.\text{Transp}] &\supset :(\exists z).z\neq x.z\in P(x\vdash\dashv y):\\ +[*121·103.*91·542] &\supset :(\exists z).xP_{\text{po}}z.zP_{*}y:\\ +[*91·574] &\supset :xP_{\text{po}}y\colon\ldotp \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*121·304">*121·304</a>.</b> \(\vdash \colon\ldotp P_{\text{po}}\,\unicode{x2abd}\, J.\supset :xP_{1}y.\equiv .P(x\vdash\dashv y)={℩}ʻx\cup {℩}ʻy.x\neq y\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*121·303·11.\supset \vdash :\text{Hp}.xP_{1}y.&\supset .xP_{\text{po}}y.\\ +[\text{Hp}] &\supset .x\neq y &\qquad \text{(1)}\\ +\vdash .(1).*54·53·101.*121·23·11.&\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_239">[Pg 239]</span></p> + +<p class="nind"><b><a id="*121·305">*121·305</a>.</b> \(\vdash :P_{\text{po}}\,\unicode{x2abd}\, J.\supset .P_{1}\,\unicode{x2abd}\, P\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*121·303.\supset \vdash :\text{Hp}.xP_{1}y.&\supset .xP_{\text{po}}y.\\ +[*91·52] &\supset .(\exists z).xPz.zP_{*}y &\qquad \text{(1)}\\ +\vdash .*121·304.*91·542.\supset \\ +\vdash \colon\ldotp \text{Hp}.xP_{1}y.&\supset :xP_{\text{po}}z.zP_{*}y.\supset .z=y:\\ +[*91*502] &\supset :xPz.zP_{*}y.\supset .z=y &\qquad \text{(2)}\\ +\vdash .(1).(2).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*121·306.</b> \(\vdash :P\in 1\rightarrow \text{Cls}.{\sim}(xP_{\text{po}}x).xPy.\supset .P(x\vdash\dashv y)=\iota ʻx\cup \iota ʻy.x\neq y\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*91·542. &\supset \vdash :xP_{*}z.zP_{*}y.z\neq x.z\neq y.xPy.\supset :xP_{\text{po}}z.zP_{\text{po}}y.xPy:\\ +[*34·1] &\supset :xP_{\text{po}}z.zP_{\text{po}}\mid \breve{P} x:\\ +[*92·11] &\supset :P\in 1\rightarrow \text{Cls}.\supset .xP_{\text{po}}z.zP_{*}x:\\ +[*91·574] &\supset :P\in 1\rightarrow \text{Cls}.\supset .xP_{\text{po}}x &\qquad \text{(1)}\\ +\vdash .(1).\text{Transp}.&\supset \vdash \colon\colon \text{Hp}.\supset \colon\ldotp xP_{*}z.zP_{*}y.\supset _{z}:z=x.\lor.z=y &\qquad \text{(2)}\\ +\vdash .*121·23. &\supset \vdash :\text{Hp}.\supset .x,y\in P(x\vdash\dashv y) &\qquad \text{(3)}\\ +\vdash .*91·502.& \supset \vdash :\text{Hp}.\supset .x\neq y &\qquad \text{(4)}\\ +\vdash .(2).(3).(4).*121·103.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*121·307.</b> \[\begin{align}&\vdash :P\in \text{Cls}\rightarrow 1.{\sim}(yP_{\text{po}}y).xPy.\supset .P(x\vdash\dashv y)=\iota ʻx\cup \iota ʻy.x\neq y\\ +&[*121·306·143]\end{align}\]</p> + +<p class="nind"><b>*121·308.</b> \[\begin{align}&\vdash :P\in (1\rightarrow \text{Cls})\cup (\text{Cls}\rightarrow 1).P_{\text{po}}\,\unicode{x2abd}\, J.\supset .P\,\unicode{x2abd}\, P_{1}\\ +&[*121·306·307·11.*54·101]\end{align}\]</p> + +<p class="nind"><b>*121·31.</b> \(\vdash :P\in (1\rightarrow \text{Cls})\cup (\text{Cls}\rightarrow 1).P_{\text{po}}\,\unicode{x2abd}\, J.\supset .P_{1}=P \quad[*121·305·308]\)</p> + +<p class="nind"><b>*121·32.</b> \(\vdash .P_{\nu }\,\unicode{x2abd}\, P_{*}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*121·11.*120·421.*101·14.\text{Transp}.&\supset \vdash :xP_{\nu }y.\supset .\exists !P(x\vdash\dashv y).\\ +[*121·23] &\supset .xP_{*}y:\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p>If \(\nu\) is not a cardinal, or if \(\nu +_{c}1=\Lambda\), \(P_{\nu}=\dot{\Lambda}\).</p> + +<p class="nind"><b>*121·321.</b> \(\vdash :\nu >0.\supset .P_{\nu }\,\unicode{x2abd}\, P_{\text{po}}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*120·428.*121·11.\supset \vdash :\text{Hp}.xP_{\nu }y.&\supset .\text{Nc}ʻP(x\vdash\dashv y)>1.\\ +[*117·55.*52·181.*121·23] &\supset .(\exists z).z\in P(x\vdash\dashv y).z\neq x.\\ +[*121·103.*91·542] &\supset .(\exists z).xP_{\text{po}}z.zR_{*}y.\\ +[*91·574] &\supset .xP_{\text{po}}y:\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*121·322.</b> \(\vdash .CʻP_{\nu }\subset CʻP \quad[*121·32.*90·14]\)</p> + +<p><span class="pagenum" id="Page_240">[Pg 240]</span></p> + +<p class="nind"><b>*121·323.</b> \(\vdash :\nu >0.\supset .\text{D}ʻP_{\nu }\subset \text{D}ʻP.\text{ᗡ}ʻP_{\nu }\subset \text{ᗡ}ʻP \quad[*121·321.*91·504]\)</p> + +<p class="nind"><b>*121·324.</b> \(\vdash .\text{D}ʻP_{\nu +_{c}1}\subset \text{D}ʻP.\text{ᗡ}ʻP_{\nu +_{c}1}\subset \text{ᗡ}ʻP\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*121·273·323.\supset \vdash :\dot{\exists} !P_{\nu +_{c}1}.\supset .\text{D}ʻP_{\nu +_{c}1}\subset \text{D}ʻP.\text{ᗡ}ʻP_{\nu +_{c}1}\subset + \text{ᗡ}ʻP &\qquad \text{(1)}\\ +\vdash .(1).*33·241.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*121·325.</b> \(\vdash :\dot{\exists} !P_{\mu }\dot{\cap} P_{\nu }.\supset .\mu =\nu\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*121·11.&\supset \vdash :\text{Hp}.\supset .\exists !(\mu +_{c}1)\cap (\nu +_{c}1)\cap t_{0}ʻ\mu .\\ +[*100·42.*110·4] &\supset .\exists !(\mu +_{c}1)\cap t_{0}ʻ\mu .(\mu +_{c}1)\cap t_{0}ʻ\mu =\nu +_{c}1.\\ +[*120·311] &\supset .\mu =\nu :\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*121·326.</b> \(\vdash .\text{fin}ʻP\subset \text{finid}ʻP.\text{finid}ʻP-\iota ʻP_{0}\subset \text{fin}ʻP \quad[*121·12·121]\)</p> + +<p class="nind"><b>*121·327.</b> \(\vdash :\dot{\exists} !P_{0}.\supset .\text{fin}ʻP=\text{finid}ʻP-\iota ʻP_{0}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*121·325.\text{Transp}.*121·121.\supset \vdash \colon\ldotp \text{Hp}.\supset :R\in \text{fin}ʻP.\supset .R\neq P_{0} &\qquad \text{(1)}\\ +\vdash .(1).*121·326.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><a href="#*121·33">*121·33</a><a href="#*121·331">·331</a> are lemmas for <a href="#*121·332">*121·332</a>, which is a very useful proposition.</p> + +<p class="nind"><b><a id="*121·33">*121·33</a>.</b> \[\begin{align}\vdash \colon\ldotp P\in 1\rightarrow \text{Cls}.\supset :&z\in P(x-y).\equiv .z\in P(x\dashv Pʻy):\\ +&z\in P(x \unicode{x27dd} y).\equiv .z\in P(x\vdash\dashv Pʻy)\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*71·7.\supset \vdash \colon\ldotp \text{Hp}.\supset :zP_{*}(Pʻy).&\equiv .zP_*\mid Py.\\ +[*91·52] &\equiv .zP_{\text{po}}y &\qquad \text{(1)}\\ +\vdash .(1).*121·1·101·102·103.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p>From the above proposition it follows that +\[ +P\in 1\rightarrow \text{Cls}.y\in \text{ᗡ}ʻP.\supset .P(x-y)=P(x\dashv Pʻy).P(x \unicode{x27dd} y)=P(x\vdash\dashv Pʻy). +\]</p> + +<p>This does not follow unless \(y\in \text{ᗡ}ʻP\), because +\[ +\begin{aligned} +P&(x\unicode{x2013}y)=P(x\dashv Pʻy).\supset .\text{E}!Pʻy,\\ +\text{whereas}\qquad &z\in P(x\unicode{x2013}y).\equiv _{z}.z\in P(x\dashv Pʻy) +\end{aligned} +\] +will always be true if \(y{\sim}\in \text{ᗡ}ʻP\), and therefore (when +\(P\in 1\rightarrow \text{Cls}\)) if \({\sim}\text{E}!Pʻy\).</p> + +<p class="nind"><b><a id="*121·331">*121·331</a>.</b> \(\vdash \colon\ldotp P \in 1\rightarrow \text{Cls}.P_{\text{po}}\,\unicode{x2abd}\, J.\supset :xP_{\nu }(Pʻy).\equiv .xP_{\nu +_{c}1}y\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*121·324.*71·16. &\supset \vdash \colon\ldotp \text{Hp}.\supset :xP_{\nu +_{c}1}y.\supset .\text{E}!Pʻy &\qquad \text{(1)}\\ +\vdash .*121·33. &\supset \vdash :\text{Hp}.\text{E}!Pʻy.\supset .P(x \unicode{x27dd} y)=P(x\vdash\dashv Pʻy) &\qquad \text{(2)}\\ +\vdash .*121·242·32.(2). &\supset \vdash :\text{Hp}(2).xP_{*}y.\supset .P(x\vdash\dashv y)=P(x\vdash\dashv Pʻy)\cup \iota ʻy &\qquad \text{(3)}\\ +\vdash .*91·52. \supset \vdash :\text{Hp}.&\supset .{\sim}(yP_{*}\mid Py).\\ +[*71·7] &\supset .{\sim}\{yP_{*} (Pʻy)\} .\\ +[*121·103] &\supset .{\sim}\{y\in P(x\vdash\dashv Pʻy)\} &\qquad \text{(4)}\\ +\vdash .(3).(4).*110·63.&\supset \vdash :\text{Hp}(3).\supset .\text{Nc}ʻP(x\vdash\dashv y)=\text{Nc}ʻP(x\vdash\dashv Pʻy)+_{c}1 + &\qquad \text{(5)}\\ +\vdash .(1).(5).*121·11·32.\supset \\ +\vdash :\text{Hp}.xP_{\nu +_{c} 1}y.&\supset .(\nu +_{c} 1) +_{c} 1 = \text{Nc}ʻP(x\vdash\dashv Pʻy)+_{c} 1.\\ +[*120·311.*121·27] &\supset . \nu +_{c} 1 = \text{Nc}ʻP(x \vdash\dashv Pʻy).\\ +[*121·11] &\supset .xP_{\nu } (Pʻy) &\qquad \text{(6)}\\ +\vdash .(5).*14·21.*121·11·32. &\supset \vdash : \text{Hp}.xP_{\nu } (Pʻy).\supset .\text{Nc}ʻP(x \vdash\dashv y)=(\nu +_{c} 1) +_{c} 1.\\ +[*121·11] &\supset . xP_{\nu +_{c} 1}y &\qquad \text{(7)}\\ +\vdash .(6).(7).\supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_241">[Pg 241]</span></p> + +<p class="nind"><b><a id="*121·332">*121·332</a>.</b> \(\vdash : P \in 1 \rightarrow \text{Cls}. P_{\text{po}} \,\unicode{x2abd}\, J. \supset . P_{\nu +_{c} 1} = P_{\nu } \mid P \quad[*121·331]\)</p> + +<p class="nind"><b>*121·333.</b> \(\vdash : P \in \text{Cls} \rightarrow 1. P_{\text{po}} \,\unicode{x2abd}\, J. \supset . P_{\nu +_{c} 1} = P \mid P_{\nu }\)</p> + +<p class="nind"><b>*121·34.</b> \(\vdash : P \in 1 \rightarrow \text{Cls}. P_{\text{po}} \,\unicode{x2abd}\, J. \nu \in \text{NC induct}. \supset . P_{\nu } \in 1 \rightarrow \text{Cls}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *121·3. &\supset \vdash . P_{0} \in 1 \rightarrow \text{Cls} &\qquad \text{(1)}\\ +\vdash . *121·332. &\supset \vdash \colon\ldotp \text{Hp}. \supset : P_{\nu } \in 1 \rightarrow \text{Cls}. \supset . P_{\nu +_{c} + 1} \in 1 \rightarrow \text{Cls} &\qquad \text{(2)}\\ +\vdash . (1).(2). *120·11.\supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*121·341.</b> \(\vdash : P \in \text{Cls} \rightarrow 1. P_{\text{po}} \,\unicode{x2abd}\, J. \nu \in \text{NC induct}. \supset . P_{\nu } \in \text{Cls} \rightarrow 1\)</p> + +<p class="nind"><b>*121·342.</b> \(\vdash : P \in 1 \rightarrow 1. P_{\text{po}} \,\unicode{x2abd}\, J. \nu \in \text{NC induct}. \supset . P_{\nu } \in 1 \rightarrow 1 \quad[*121·34·341]\)</p> + +<p class="nind"><b>*121·35.</b> \(\vdash : P \in 1 \rightarrow \text{Cls}. P_{\text{po}} \,\unicode{x2abd}\, J. \mu , \nu \in \text{NC induct}. \supset . P_{\mu } \mid P_{\nu } + = P_{\mu +_{c} \nu }\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *50·62. *121·302·322. &\supset \vdash : \text{Hp}. \supset . P_{\mu } \mid P_{0} = P_{\mu +_{c} 0} &\qquad \text{(1)}\\ +\vdash . *121·332. &\supset \vdash \colon\ldotp \text{Hp}. \supset : \mu ,\nu \in \text{NC induct}. P_{\mu } \mid P_{\nu } = P_{\mu +_{c} \nu }. \supset .\\ +&P_{\mu } \mid P_{\nu +_{c} 1} = P_{\mu +_{c} \nu } \mid P\\ +[*121·332] &= P_{\mu +_{c} \nu +_{c} 1} &\qquad \text{(2)}\\ +\vdash . (1).(2). *120·13.\supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*121·351.</b> \(\vdash : P \in \text{Cls} \rightarrow 1. P_{\text{po}} \,\unicode{x2abd}\, J. \mu , \nu \in \text{NC induct}. \supset . P_{\mu } \mid P_{\nu } + = P_{\mu +_{c} \nu }\)</p> + +<p class="nind"><b>*121·352.</b> \[\begin{align}\vdash : P \in (1 \rightarrow \text{Cls}) \cup (\text{Cls} \rightarrow 1). &P_{\text{po}} + \,\unicode{x2abd}\, J. \mu , \nu \in \text{NC induct}. \supset .\\ +&P_{\mu } \mid P_{\nu } = P_{\nu } \mid P_{\mu } \quad[*121·35·351.*110·51]\end{align}\]</p> + +<p class="nind"><b>*121·36.</b> \[\begin{align}\vdash : P \in (1 \rightarrow \text{Cls}) \cup (\text{Cls} \rightarrow 1). &P_{\text{po}} + \,\unicode{x2abd}\, J. \mu , \nu \in \text{NC induct} -\iota ʻ0.\supset .\\ +&(P_{\mu })_{\nu } = P_{\mu \times _{c} \nu }\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *121·321.\supset \vdash : \text{Hp}. &\supset . P_{\mu }\,\unicode{x2abd}\, P_{\text{po}}.\\ +[*91·59·601] &\supset . (P_{\mu })_{\text{po}} \,\unicode{x2abd}\, J. &\qquad \text{(1)}\\ +[*121·31·34·341] &\supset .(P_{\mu })_1 = P_{\mu } &\qquad \text{(2)}\\ +\vdash . *121·332·333·352 . (1). \supset \\ +\vdash \colon\ldotp \text{Hp}. \supset : (P_{\mu })_{\nu +_{c} 1} &= (P_{\mu })_{\nu } \mid P_{\mu } :\\ +[*34·27] \supset : (P_{\mu })_{\nu } = P_{\mu \times _{c} \nu } . \supset . (P_{\mu })_{\nu +_{c} 1} &= P_{\mu \times _{c} + \nu } \mid P_{\mu }\\ +[*121·35·351] &= P_{(\mu \times _{c} \nu ) +_{c} \mu}\\ +[*113·671] &= P_{\mu \times _{c} (\nu +_{c} 1)} &\qquad \text{(3)}\\ +\vdash . (2). (3). *120·47. \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_242">[Pg 242]</span></p> + +<p class="nind"><b>*121·361.</b> \[\begin{align}\vdash :P\in (1\rightarrow \text{Cls})\cup (\text{Cls}\rightarrow 1).&P_{\text{po}}\,\unicode{x2abd}\, J.\mu ,\nu \in \text{NC induct}-\iota ʻ0.\supset .\\ +&(P_{\mu })_{\nu }=(P_{\nu })_{\mu } [*121·36.*113·27]\end{align}\]</p> + +<p class="nind"><b><a id="*121·37">*121·37</a>.</b> \(\vdash :P\in \text{Cls}\rightarrow 1.y\in P(x\vdash\dashv z).\supset .P(x\vdash\dashv z)=P(x\vdash\dashv y)\cup P(y\vdash\dashv z)\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*121·103.&\supset \vdash :\text{Hp}.\supset .xP_{*}y.yP_{*}z &\qquad \text{(1)}\\ +\vdash .(1).*121·103.\supset \\ +\vdash \colon\ldotp \text{Hp}.&\supset :w\in P(x\vdash\dashv z).\equiv .xP_{*}w.wP_{*}z.xP_{*}y.yP_{*}z &\qquad \text{(2)}\\ +\vdash .*96·302.&\supset \vdash \colon\colon \text{Hp}.\supset \colon\ldotp xP_{*}w.xP_{*}y.\supset :wP_{*}y.\lor.yP_{*}w &\qquad \text{(3)}\\ +\vdash .(2).(3).*4·73.\supset \\ +\vdash \colon\colon \text{Hp}.&\supset \colon\ldotp w\in P(x\vdash\dashv z).\equiv :xP_{*}w.wP_{*}z.xP_{*}y.yP_{*}z.wP_{*}y.\lor.\\ +&xP_{*}w.wP_{*}z.xP_{*}y.yP_{*}z.yP_{*}w &\qquad \text{(4)}\\ +\vdash .*90·17.*4·73.\supset \vdash :wP_{*}y.yP_{*}z.&\equiv .wP_{*}z.wP_{*}y.yP_{*}z:\\ +&yP_{*}w.wP_{*}z.\equiv .yP_{*}z.wP_{*}z.yP_{*}w &\qquad \text{(5)}\\ +\vdash .(4).(5).\supset \vdash \colon\colon \text{Hp}.&\supset \colon\ldotp w\in P(x\vdash\dashv z).\equiv :xP_{*}w.xP_{*}y.yP_{*}z.wP_{*}y.\lor.\\ +&xP_{*}w.wP_{*}z.xP_{*}y.yP_{*}w:\\ +[*90·17.*4·73] &\equiv:xP_{*}w.wP_{*}y.yP_{*}z.\lor.xP_{*}y.yP_{*}w.wP_{*}z:\\ +[(1).*4·73] &\equiv :xP_{*}w.wP_{*}y.\lor.yP_{*}w.wP_{*}z:\\ +[*121·103] &\equiv :w\in P(x\vdash\dashv y)\cup P(y\vdash\dashv z)\colon\colon \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*121·371.</b> \[\begin{align}\vdash :&P\in (\text{Cls}\rightarrow 1)\cup (1\rightarrow \text{Cls}).y\in P(x\vdash\dashv z).\supset .\\ +&P(x\vdash\dashv z)=P(x\vdash\dashv y)\cup P(y\vdash\dashv z)=P(x \unicode{x27dd} y)\cup P(y\vdash\dashv z)\\ +&=P(x\vdash\dashv y)\cup P(y\dashv z) \quad [\text{Proof as in *121·37}]\end{align}\]</p> + +<p class="nind"><b>*121·372.</b> \[\begin{align}\vdash :&P\in (\text{Cls}\rightarrow 1)\cup (1\rightarrow \text{Cls}).y\in P(x\dashv z).\supset .\\ +&P(x\dashv z)=P(x\dashv y)\cup P(y\dashv z)=P(x\dashv y)\cup P(y\vdash\dashv z)\end{align}\]</p> + +<p class="nind"><b>*121·373.</b> \[\begin{align}\vdash :&P\in (\text{Cls}\rightarrow 1)\cup (1\rightarrow \text{Cls}).y\in P(x \unicode{x27dd} z).\supset .\\ +&P(x \unicode{x27dd} z)=P(x \unicode{x27dd} y)\cup P(y \unicode{x27dd} z)= P(x\vdash\dashv y)\cup P(y \unicode{x27dd} z)\end{align}\]</p> + +<p class="nind"><b>*121·374.</b> \[\begin{align}\vdash :&P\in (\text{Cls}\rightarrow 1)\cup (1\rightarrow \text{Cls}).y\in P(x\unicode{x2013}z).\supset .\\ +&P(x\unicode{x2013}z)=P(x\dashv y)\cup P(y-z)=P(x\unicode{x2013}y)\cup P(y \unicode{x27dd} z)\\ +&=P(x\dashv y)\cup P(y \unicode{x27dd} z)\end{align}\]</p> + +<p>The proofs of these propositions are analogous to the proof of <a href="#*121·37">*121·37</a>.</p> + +<p class="nind"><b>*121·38.</b> \(\vdash :R\in \text{Cls}\rightarrow 1.xR_{\text{po}}x.\supset .R(x\vdash\dashv x)=\overleftarrow{R}_{*}ʻx \quad[*97·5]\)</p> + +<p class="nind"><b>*121·381.</b> \(\vdash :R\in 1\rightarrow \text{Cls}.xR_{\text{po}}x.\supset .R(x\vdash\dashv x)=\overrightarrow{R}_{*}ʻx \quad[*97·501]\)</p> + +<p class="nind"><b>*121·382.</b> \[\begin{align}\vdash :&R\in \text{Cls}\rightarrow 1.xR_{\text{po}}x.xR_{\text{po}}y.\supset .\\ +&R(x\vdash\dashv x)=R(x\vdash\dashv y)=\overleftarrow{R}_{*}ʻx= R(y\vdash\dashv y) \quad[*97·5.*91·56]\end{align}\]</p> + +<p class="nind"><b>*121·383.</b> \[\begin{align}\vdash :&R\in 1\rightarrow \text{Cls}.xR_{\text{po}}x.yR_{\text{po}}x.\supset .\\ +&R(x\vdash\dashv x)=R(y\vdash\dashv x)=\overrightarrow{R}_{*}ʻx=R(y\vdash\dashv y)\end{align}\]</p> + +<p class="nind"><b>*121·384.</b> \[\begin{align}\vdash :&R\in (\text{Cls}\rightarrow 1)\cup (1\rightarrow \text{Cls}).xR_{\text{po}}x.y\in R(x\vdash\dashv x).\supset .\\ +&R(x\vdash\dashv x)= R(x\vdash\dashv y)=R(y\vdash\dashv x)=R(y\vdash\dashv y) \quad[*121·382·383]\end{align}\]</p> + +<p><span class="pagenum" id="Page_243">[Pg 243]</span></p> + +<p class="nind"><b>*121·39.</b> \(\vdash \colon\ldotp R\in \text{Cls}\rightarrow 1.\supset :R(x\vdash\dashv y)\subset R(x\vdash\dashv z).\lor.R(x\vdash\dashv z)\subset R(x\vdash\dashv y)\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*96·302. &\supset \vdash \colon\ldotp +\text{Hp}.xR_{*}y.xR_{*}z.\supset :yR_{*}z.\lor.zR_{*}y &\qquad \text{(1)}\\ +\vdash .*121·37. &\supset \vdash :\text{Hp}.xR_{*}y.yR_{*}z.\supset .R(x\vdash\dashv y)\subset R(x\vdash\dashv z) &\qquad \text{(2)}\\ +\vdash .*121·37. &\supset \vdash :\text{Hp}.xR_{*}z.zR_{*}y.\supset .R(x\vdash\dashv z)\subset R(x\vdash\dashv y) &\qquad \text{(3)}\\ +\vdash .(1).(2).(3). &\supset \vdash \colon\ldotp \text{Hp}.xR_{*}y.xR_{*}z.\supset :\\ +&R(x\vdash\dashv y)\subset R(x\vdash\dashv z).\lor.R(x\vdash\dashv z)\subset R(x\vdash\dashv y) &\qquad \text{(4)}\\ +\vdash .*121·23. &\supset \vdash :{\sim}(xR_{*}y).\supset .R(x\vdash\dashv y)=\Lambda .\\ +[*24·12] &\supset .R(x\vdash\dashv y)\subset R(x\vdash\dashv z) &\qquad \text{(5)}\\ +\vdash .(5)\frac{z,\,y}{y,\,z}. &\supset \vdash :{\sim}(xR_{*}z).\supset .R(x\vdash\dashv z)\subset R(x\vdash\dashv y) &\qquad \text{(6)}\\ +\vdash .(4).(5).(6).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p>The following series of propositions are concerned with proving +<a href="#*121·47">*121·47</a>, <i>i.e.</i> +\[ +R\in (\text{Cls}\rightarrow 1)\cup (1\rightarrow \text{Cls}).\supset .R(x\vdash\dashv z)\in \text{Cls induct}. +\] +The proof for \(R\in 1\rightarrow \text{Cls}\) follows from that for +\(R\in \text{Cls}\rightarrow 1\) by <a href="#*121·143">*121·143</a>. Confining ourselves, +therefore, to \(R\in \text{Cls}\rightarrow 1\), we proceed as follows.</p> + +<p>We prove first that, starting from z and going backwards, each new +step adds only one term (which may not be distinct from all its +predecessors); <i>i.e.</i> we have +\[ +R\in \text{Cls}\rightarrow 1.xRy.yR_{*}z.\supset .R(x\vdash\dashv z)=\iota ʻx\cup R(y\vdash\dashv z). +\] +From this it follows by induction that if \(R(z\vdash\dashv z)\) is an +inductive class, so is \(R(x\vdash\dashv z)\). Thus we only have to +prove that \(R(z\vdash\dashv z)\) is an inductive class. Here we must +distinguish two cases, according as \({\sim}(zR_{\text{po}}z)\) or +\(zR_{\text{po}}z\). In the former case, we have +\[ +\exists !R(z\vdash\dashv z).\supset .R(z\vdash\dashv z)=\iota ʻz, +\] +whence \(R(z\vdash\dashv z)\) is an inductive class, and therefore so +is \(R(x\vdash\dashv z)\).</p> + +<p>But in the latter case, when \(zR_{\text{po}}z\), the matter is more +difficult. In this case, \(z\) is a member of a cycle, the cycle being +\(R(z\vdash\dashv z)\). We have to prove that this cycle must be an +inductive class. Given \(xR_{*}z\), \(x\) will be a member of this +cycle if \(xR_{\text{po}}x\), and may be at the end of the tail of a +\(Q\), if \({\sim}(xR_{\text{po}}x)\). (Cf. *96.)</p> + +<figure class="figcenter width500" id="i_243" style="width: 1200px;"> +<img src="images/i_243.jpg" width="1200" height="668" alt="Mathematical +diagram showing a sequence and cycle. Left side: linear progression +from x to R with arrows. Right side: circular cycle with arrows +connecting points labeled a and z, with R marking entry point. An +incoming arrow from above enters the cycle at top. Illustrates relation +R connecting linear and cyclical structures."> +</figure> + +<p>By *96·453, we know that \(R\) is \(1\rightarrow 1\) when confined to +\(R(z\vdash\dashv z)\). Hence<span class="pagenum" id="Page_244">[Pg 244]</span> in \(R(z\vdash\dashv z)\), \(z\) has +a unique predecessor, say \(a\). Assume \(a\neq z\). We then imagine +a barrier placed between \(a\) and \(z\), <i>i.e.</i> we construct a +relation \(S\) which is to hold between any two consecutive members +of \(R(z\vdash\dashv z)\) except \(a\) and \(z\). Putting \(\alpha=R(z\vdash\dashv z)-\iota ʻa\), +we have \(S=\alpha \upharpoonleft R\). Then the relation +\(S\) generates an open series consisting of all the terms of +\(R(z\vdash\dashv z)\); <i>i.e.</i> we have +\[ +{\sim}(aS_{\text{po}}a).S(z\vdash\dashv a)=R(z\vdash\dashv z). +\] +Hence, by our previous case, since \(S(z\vdash\dashv a)\) is an +inductive class, so is \(R(z\vdash\dashv z)\).</p> + +<p>If \(a = z\), then by *96·33 the cycle reduces to the single term +\(z\), and therefore \(R(z\vdash\dashv z)\) is still an inductive class.</p> + +<p>Hence \(R(z\vdash\dashv z)\), and therefore \(R(x\vdash\dashv z)\), is +always an inductive class when \(R\in \text{Cls}\rightarrow 1\), which +was to be proved.</p> + +<p class="nind"><b><a id="*121·4">*121·4</a>.</b> \(\vdash :R\in \text{Cls}\rightarrow 1.xRy.yR_{*}z.\supset .R(x\vdash\dashv z)=\iota ʻx\cup R(y\vdash\dashv z)\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*90·311.\supset \vdash \colon\colon \text{Hp}.\supset \colon\ldotp xR_{*}w.&\equiv :x=w.\lor.xR\mid R_{*}w:\\ +[*71·701.\text{Hp}] &\equiv :x=w.\lor.yR_{*}w &\qquad \text{(1)}\\ +\vdash .*90·172.&\supset \vdash \colon\ldotp \text{Hp}.\supset :x=w.\supset .wR_{*}z &\qquad \text{(2)}\\ +\vdash .(1).(2).\supset \vdash \colon\colon \text{Hp}.\supset \colon\ldotp xR_{*}w.wR_{*}z.&\equiv :x=w.\lor.yR_{*}w.wR_{*}z &\qquad \text{(3)}\\ +\vdash .(3).*121·103.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*121·41.</b> \(\vdash :R\in \text{Cls}\rightarrow 1.R(z\vdash\dashv z)\in \text{Cls induct}.\supset .R(x\vdash\dashv z)\in \text{Cls induct}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*121·4.*120·251.*90·172.&\supset \vdash \colon\ldotp \text{Hp}.\supset :\\ +&yR_{*}z.R(y\vdash\dashv z)\in \text{Cls induct}.xRy.\supset .xR_{*}z.R(x\vdash\dashv z)\in \text{Cls induct} &\qquad \text{(1)}\\ +\vdash .(1).*90·112 \frac{\breve{R}}{R}.&\supset \vdash :\text{Hp}.xR_{*}z.\supset .R(x\vdash\dashv z)\in \text{Cls induct} &\qquad \text{(2)}\\ +\vdash .*121·23.*120·212.&\supset \vdash :{\sim}(xR_{*}z).\supset .R(x\vdash\dashv z)\in \text{Cls induct} &\qquad \text{(3)}\\ +\vdash .(2).(3).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p>In virtue of this proposition, we have only to prove +\(R(z\vdash\dashv z)\in \text{Cls induct}\). This is obvious when +\({\sim}(zR_{\text{po}}z)\), for then either \(R(z\vdash\dashv z)=\iotaʻz\) +or \(R(z\vdash\dashv z=\Lambda\). But when \(zR_{\text{po}}z\), it +is more difficult.</p> + +<p class="nind"><b>*121·42.</b> \(\vdash :R\in \text{Cls}\rightarrow 1.{\sim}(zR_{\text{po}}z).\supset .R(x\vdash\dashv z)\in \text{Cls induct}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*121·303.\text{Transp}.*120·441.&\supset \vdash :\text{Hp}.\supset .\text{Nc}ʻR(z\vdash\dashv z)\leq 1.\\ +[*120·48] &\supset .\text{Nc}ʻR(z\vdash\dashv z)\in \text{NC induct}.\\ +[*120·211] & \supset .R(z\vdash\dashv z)\in \text{Cls induct} &\qquad \text{(1)}\\ +\vdash .(1).*121·41.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_245">[Pg 245]</span></p> + +<p class="nind"><b>*121·43.</b> \(\vdash :R\in \text{Cls}\rightarrow 1.zR_{\text{po}}z.\supset .\text{E}!\breve{\iota} ʻ(\overrightarrow{R}ʻz\cap \overleftarrow{R}_{*}ʻz)\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*91·52. \supset \vdash :\text{Hp}.&\supset .(\exists a).zR_{*}a.aRz &\qquad \text{(1)}\\ +\vdash .*96·453. &\supset \vdash :\text{Hp}.\supset .(\overleftarrow{R}_{*}ʻz)\upharpoonleft R\in 1\rightarrow 1.\\ +[*71·122] &\supset .\hat{a}(zR_{*}a.aRz)\in 1\cup \iota ʻ\Lambda &\qquad \text{(2)}\\ +\vdash .(1).(2). \supset \vdash :\text{Hp}.&\supset .\hat{a}(zR_{*}a.aRz)\in 1.\\ +[*52·15] &\supset .\text{E}!\breve{\iota} ʻ(\overrightarrow{R}ʻz\cap \overleftarrow{R}_{*}ʻz):\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*121·431.</b> \[\begin{align}\vdash :R\in \text{Cls}\rightarrow 1.zR_{\text{po}}z.a=\breve{\iota} ʻ(\overrightarrow{R}ʻz\cap \overleftarrow{R}_{*}ʻz).&\alpha + =\overleftarrow{R}_{*}ʻz-\iota ʻa.\\ +&S=\alpha \upharpoonleft R.\supset .{\sim}(aS_{\text{po}}a)\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*35·61. \supset \vdash :\text{Hp}.&\supset .a{\sim}\in \text{D}ʻS.\\ +[*91·504] &\supset .a{\sim}\in \text{D}ʻS_{\text{po}}.\\ +[*33·14] &\supset .{\sim}(aS_{\text{po}}a):\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*121·432.</b> \(\vdash :\text{Hp}*121·431.\supset .S(z\vdash\dashv a)\in \text{Cls induct}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*71·261.\supset \vdash :\text{Hp}.\supset .S\in \text{Cls}\rightarrow 1 &\qquad \text{(1)}\\ +\vdash .(1).*121·431·42.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*121·433.</b> \(\vdash :\text{Hp}*121·431.z\neq a.\supset .S(z\vdash\dashv a)=\overleftarrow{R}_{*}ʻz=R(z\vdash\dashv z)\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*96·11. \supset \vdash \colon\ldotp \text{Hp}.&\supset :zS_{*}w.\supset .zR_{*}w &\qquad \text{(1)}\\ +\vdash .*51·3.*91·504. \supset \vdash \colon\ldotp \text{Hp}.&\supset :z\in a.z\in \text{D}ʻR:\\ +[*35·61] &\supset :z\in \text{D}ʻS:\\ +[*90·12] &\supset :zS_{*}z &\qquad \text{(2)}\\ +\vdash .(1).*90·16. \supset \vdash \colon\colon \text{Hp}.&\supset \colon\ldotp zS_{*}w.wRy.\supset :w\in \alpha \cup \iota ʻa.wRy:\\ +[*35·1] &\supset :wSy.\lor.w=a.wRy:\\ +[\text{Hp}.*71·171] &\supset :wSy.\lor.y=z:\\ +[*90·16·17.(2)] &\supset :zS_{*}y &\qquad \text{(3)}\\ +\vdash .(2).(3).*90·112. \supset \vdash \colon\ldotp \text{Hp}.&\supset :zR_{*}w.\supset .zS_{*}w: &\qquad \text{(4)}\\ +[\text{Hp}] &\supset \vdash :zS_{*}a &\qquad \text{(5)}\\ +\vdash .*71·171. \supset \vdash :\text{Hp}.&aRy.\supset .y=z &\qquad \text{(6)}\\ +\vdash .*91·542·504.*35·61. &\supset \vdash \colon\ldotp \text{Hp}.\supset :wS_{*}a.w\neq a.wRy.\supset .wS_{\text{po}}a.wSy.\\ +[*92·111] &\supset .yS_{*}a &\qquad \text{(7)}\\ +\vdash .(5).(6).(7). &\supset \vdash \colon\ldotp \text{Hp}.\supset :wS_{*}a.wRy.\supset .yS_{*}a &\qquad \text{(8)}\\ +\vdash .(5).(8).*90·112. &\supset \vdash \colon\ldotp \text{Hp}.\supset :zR_{*}y.\supset .yS_{*}a &\qquad \text{(9)}\\ +\vdash .(4).(9). &\supset \vdash \colon\ldotp \text{Hp}.\supset :zR_{*}y.\supset .zS_{*}y.yS_{*}a &\qquad \text{(10)}\\ +\vdash .(1) \frac{y}{w}.(10). &\supset \vdash \colon\ldotp \text{Hp}.\supset :zS_{*}y.yS_{*}a.\equiv .zR_{*}y:\\ +[*121·103] \supset :S(z\vdash\dashv a)&=\overleftarrow{R}_{*}ʻz\\ +[*121·38] &=R(z\vdash\dashv z)\colon\ldotp \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_246">[Pg 246]</span></p> + +<p class="nind"><b>*121·434.</b> \(\vdash :\text{Hp}*121·431.z=a.\supset .\overleftarrow{R}_{*}ʻz=R(z\vdash\dashv z)=\iota ʻz\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*32·18.&\supset \vdash :\text{Hp}.\supset .zRz.\\ +[*96·33] &\supset .\overleftarrow{R}_{*}ʻz=\iota ʻz. &\qquad \text{(1)}\\ +[*121·38] &\supset .R(z\vdash\dashv z)=\iota ʻz &\qquad \text{(2)}\\ +\vdash .(1).(2).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*121·44.</b> \(\vdash :R\in \text{Cls}\rightarrow 1.zR_{\text{po}}z.\supset .R(z\vdash\dashv z)\in \text{Cls induct}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*121·43·432·433.\supset \\ +\vdash :\text{Hp}.z\neq \breve{\iota} ʻ(\overrightarrow{R}ʻz\cap \overleftarrow{R}_{*}ʻz).\supset .R(z\vdash\dashv z)\in \text{Cls induct} &\qquad \text{(1)}\\ +\vdash .*121·434.*120·213.\supset \\ +\vdash :\text{Hp}.z=\breve{\iota} ʻ(\overrightarrow{R}ʻz\cap \overleftarrow{R}_{*}ʻz).\supset .R(z\vdash\dashv z)\in \text{Cls induct} &\qquad \text{(2)}\\ +\vdash .(1).(2).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*121·441.</b> \(\vdash :R\in \text{Cls}\rightarrow 1.zR_{\text{po}}z.\supset .R(x\vdash\dashv z)\in \text{Cls induct} \quad[*121·44·41]\)</p> + +<p class="nind"><b>*121·45.</b> \(\vdash :R\in \text{Cls}\rightarrow 1.\supset .R(x\vdash\dashv z)\in \text{Cls induct} \quad[*121·42·441]\)</p> + +<p class="nind"><b>*121·46.</b> \(\vdash :R\in 1\rightarrow \text{Cls}.\supset .R(x\vdash\dashv z)\in \text{Cls induct} \quad[*121·45·143]\)</p> + +<p class="nind"><b><a id="*121·47">*121·47</a>.</b> \(\vdash :R\in (\text{Cls}\rightarrow 1)\cup (1\rightarrow \text{Cls}).\supset .R(x\vdash\dashv z)\in \text{Cls induct} \quad[*121·45·46]\)</p> + +<p class="nind"><b>*121·48.</b> \[\begin{align}\vdash \colon\ldotp &R\in \text{Cls}\rightarrow 1.\supset :\\ +&\text{Nc}ʻR(x\vdash\dashv y)<\text{Nc}ʻR(x\vdash\dashv z).\equiv .\exists !R(x\vdash\dashv z)-R(x\vdash\dashv y)\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*121·39.\supset \vdash \colon\ldotp \text{Hp}.&\supset :\exists !R(x\vdash\dashv z)-R(x\vdash\dashv y).\equiv .\\ +&R(x\vdash\dashv y)\subset R(x\vdash\dashv z).R(x\vdash\dashv y)\neq R(x\vdash\dashv z).\\ +[*120·7.*121·45] &\supset .\text{Nc}ʻR(x\vdash\dashv y)<\text{Nc}ʻR(x\vdash\dashv z) &\qquad \text{(1)}\\ +\vdash .*117·222·29.&\supset \vdash :\text{Nc}ʻR(x\vdash\dashv y)<\text{Nc}ʻR(x\vdash\dashv z).\supset .\\ +&{\sim}\{R(x\vdash\dashv z)\subset R(x\vdash\dashv y)\}.\\ +[*24·55] &\supset .\exists !R(x\vdash\dashv z)-R(x\vdash\dashv y) &\qquad \text{(2)}\\ +\vdash .(1).(2).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*121·481.</b> \[\begin{align}\vdash \colon\ldotp R\in \text{Cls}\rightarrow 1.\supset :\text{Nc}ʻR(x\vdash\dashv y)\leq \text{Nc}ʻ&R(x\vdash\dashv z).\equiv .\\ +&R(x\vdash\dashv y)\subset R(x\vdash\dashv z)\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*121·45.*120·441.\supset \\ +\vdash \colon\ldotp \text{Hp}.\supset :\text{Nc}ʻR(x\vdash\dashv y)\leq \text{Nc}ʻR(x\vdash\dashv z).&\equiv .{\sim}\{\text{Nc}ʻR(x\vdash\dashv z)<\text{Nc}ʻR(x\vdash\dashv y)\}.\\ +[*121·48] &\equiv .{\sim}\exists !R(x\vdash\dashv y)-R(x\vdash\dashv z).\\ +[*24·55] &\equiv .R(x\vdash\dashv y)\subset R(x\vdash\dashv z)\colon\ldotp \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p>The above proposition is used in the proof of <a href="#*122·35">*122·35</a>, which is an +important proposition in the theory of progressions.</p> + +<p><span class="pagenum" id="Page_247">[Pg 247]</span></p> + +<p>The following propositions are concerned with the identification of +such relations as \(P_{\nu }\) with powers of \(P\) in the sense of *91.</p> + +<p class="nind"><b><a id="*121·5">*121·5</a>.</b> \[\begin{align}\vdash :P\in (\text{Cls}\rightarrow 1)\cup (1\rightarrow \text{Cls}).&P_{\text{po}}\,\unicode{x2abd}\, J.\supset .\\ +&\text{finid}ʻP=\text{Potid}ʻP.\text{fin}ʻP=\text{Pot}ʻP\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*121·302·31. \supset \vdash :\text{Hp}.&\supset .P_{0}=I\upharpoonright CʻP.P_{1}=P &\qquad \text{(1)}\\ +\vdash .(1).*121·332·333·352.&\supset \vdash \colon\ldotp \text{Hp}.\nu \in \text{NC induct}.\supset :P_{\nu +_{c}1}=P_{\nu }\mid P: &\qquad \text{(2)}\\ +[*91·341] &\supset :P_{\nu }\in \text{Potid}ʻP.\supset .P_{\nu +_{c}1}\in \text{Potid}ʻP:P_{\nu }\in \text{Pot}ʻP.\supset .P_{\nu +_{c}1}\in + \text{Pot}ʻP &\qquad \text{(3)}\\ +\vdash .(1).*91·35.&\supset \vdash :\text{Hp}.\supset .P_{0}\in \text{Potid}ʻP.P_{1}\in \text{Pot}ʻP &\qquad \text{(4)}\\ +\vdash .(3).(4).*120·13·47.\supset \vdash \colon\ldotp \text{Hp}.&\supset :\nu \in \text{NC induct}.\supset .P_{\nu }\in \text{Potid}ʻP:\\ +&\nu \in \text{NC induct}-\iota ʻ0.\supset .P_{\nu }\in \text{Pot}ʻP:\\ +[*121·12·121] &\supset :\text{finid}ʻP\subset \text{Potid}ʻP.\text{fin}ʻP\subset \text{Pot}ʻP &\qquad \text{(5)}\\ +\vdash .(2).*121·121.\supset \vdash \colon\ldotp \text{Hp}.&\supset :\nu \in \text{NC induct}.\supset .P_{\nu }\mid P\in \text{fin}ʻP:\\ +[*121·12] & \supset :Q\in \text{finid}ʻP.\supset .Q\mid P\in \text{fin}ʻP:\\ +[(1).*91·17·171] & \supset :\text{Potid}ʻP\subset \text{finid}ʻP.\text{Pot}ʻP\subset \text{fin}ʻP &\qquad \text{(6)}\\ +\vdash .(5).(6).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*121·501">*121·501</a>.</b> \[\begin{align}\vdash :P\in (\text{Cls}\rightarrow 1)\cup (1\rightarrow \text{Cls}).P_{\text{po}}\,\unicode{x2abd}\, &J.\dot{\exists} !P.\supset .\\ +&\text{Pot}ʻP=\text{finid}ʻP-\iota ʻP_{0}=\text{fin}ʻP\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*121·302.\supset \vdash :\text{Hp}.\supset .\dot{\exists} !P_{0} &\qquad \text{(1)}\\ +\vdash .(1).*121·5·327.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*121·502">*121·502</a>.</b> \[\begin{align}\vdash :P\in (\text{Cls}\rightarrow 1)\cup (1\rightarrow \text{Cls}).P_{\text{po}}&\,\unicode{x2abd}\, J.\supset .\\ +&\dot{s} ʻ(\text{finid}ʻP-\iota ʻP_{0})=P_{\text{po}}=\dot{s} ʻ\text{fin}ʻP\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*91·504.*33·24.*121·5.\supset \vdash :P=\dot{\Lambda} .\supset .\dot{s} ʻ(\text{finid}ʻP-\iota ʻP_{0})=\dot{\Lambda} =P_{\text{po}} &\qquad \text{(1)}\\ +\vdash .(1).*121·501·5.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*121·51">*121·51</a>.</b> \(\vdash :P\in (\text{Cls}\rightarrow 1)\cup (1\rightarrow \text{Cls}).P_{\text{po}}\,\unicode{x2abd}\, + J.\supset .P_{2}=P^{2}.P_{3}=P^{3}.\text{etc}.\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*121·31. \supset \vdash :\text{Hp}.&\supset .P_{1}=P &\qquad \text{(1)}\\ +\vdash .*121·332·333. \supset \vdash :\text{Hp}.\supset .P_{2}&=P_{1}\mid P_{1}\\ +[(1)] & =P^{2} &\qquad \text{(2)}\\ +\vdash .*121·332·333·352. \supset \vdash :\text{Hp}.\supset .P_{3}&=P_{2}\mid P_{1}\\ +[(1).(2)] &=P^{3} &\qquad \text{(3)}\\ +\vdash .(2).(3). \text{etc.} .\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*121·52">*121·52</a>.</b> \[\begin{align}&\vdash :P\in (\text{Cls}\rightarrow 1)\cup (1\rightarrow \text{Cls}).P_{\text{po}}\,\unicode{x2abd}\, J.\supset .\dot{s} ʻ\text{finid}ʻP=P_{*}\\ +&[*121·5.*91·55]\end{align}\]</p> + +<p>We shall at a later stage (*301) give a general definition of +\(P^\nu\). When this definition has been introduced, we shall be able +to prove, with the hypothesis of *121·51, +\[ +\nu \in \text{NC induct}.\supset .P_{\nu }=P^\nu . +\] +The definition of \(P^\nu\) is postponed on account of various +complications which render a general definition of \(P^\nu\) difficult. +The chief difficulty arises when<span class="pagenum" id="Page_248">[Pg 248]</span> \(\dot{\exists} !P\dot{\cap} I\). +Thus suppose we have \(yPy\); we shall also have \(yP^{2}y\), \(yP^{3}y\), +etc. Hence if we have \(xPy\), we have +\[ +\nu \in \text{NC induct}-\iota ʻ0.\supset _{\nu }.xP^\nu y. +\] +Again, suppose this case excluded, but suppose +\[ +(\exists \mu ,y).\mu \in \text{NC induct}.y\in P(x\vdash\dashv z).yP^\mu y . +\] +Then we shall have +\[ +\nu \in \text{NC induct}-\iota ʻ0-\iota ʻ\Lambda .\supset .yP^{\mu \times _{c}\nu }y. +\] +Thus the general definition of \(P^\nu\) has to be complicated, except +when \(P_{\text{po}}\,\unicode{x2abd}\, J\).</p> + +<p>The following propositions are concerned with the series of relations +\(P_{\nu }\) and the series of terms \(\nu _{P}\). The relation +\(P_{\nu }\) holds between two terms (roughly speaking) when it +requires \(\nu\) steps to get from the first to the second; the term +\(\nu _{P}\) is the \(\nu\)th term starting from \(BʻP\), which, when +it exists, is \(1_{P}\). In order that \(\nu _{P}\) should exist, it is +necessary that \(BʻP\) should exist, and that there should be just one +term \(x\) in the field of \(P\) such that the interval from \(BʻP\) to +\(x\) (both included) consists of \(\nu\) terms. When this is the case +for all inductive cardinals from 1 to \(\nu\), we can say that \(P\) +generates a series starting from \(BʻP\) and having at least \(\nu\) +terms, each correlated with one of the cardinals in the interval from +1 to \(\nu\), both included; <i>i.e.</i> the series has a \(\mu\)th +term, whenever \(1\leq \mu \leq \nu\). If this holds for all inductive +values of \(\nu\), the family of \(BʻP\) is a progression<a id="FNanchor_9" href="#Footnote_9" class="fnanchor">[9]</a>. (It will +be observed that all such terms as \(\nu _{P}\) belong to the family of +\(BʻP\), which need not form the whole field of \(P\).)</p> + +<p class="nind"><b><a id="*121·6">*121·6</a>.</b> \(\vdash \colon\ldotp \nu \neq 0.\supset .f(\nu _{P}).\equiv .f[\breve{\iota} ʻ\hat{y} \{\text{N}_{0}\text{c}ʻP(BʻP\vdash\dashv y)=\nu\}]\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*121·11.*120·414·416.\supset \vdash \colon\ldotp \text{Hp}.\supset :\\ +f[\breve{\iota} ʻ\hat{y} \{\text{Nc}ʻP (BʻP \vdash\dashv y) = \nu\}] . &\equiv . f[\breve{\iota} ʻ\hat{y} \{(BʻP)P_{\nu -_{c}1}y\}].\\ +[*121·13] &\equiv.f(\nu _{P}):\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*121·601.</b> \(\vdash :\text{E}!BʻP.\supset .BʻP=1_{P}.{\sim}{(BʻP)P_{\text{po}}(BʻP)}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*91·504.*93·1.&\supset \vdash .{\sim}\{(BʻP)P_{\text{po}}(BʻP)\}. &\qquad \text{(1)}\\ +[*121·301] \supset \vdash \colon\ldotp \text{E}!BʻP.&\supset :(BʻP)P_{0}y.\equiv _{y}.BʻP=y:\\ +[*31·17] &\supset :BʻP=\breve{P} _{0}ʻBʻP:\\ +[*121·13] &\supset :BʻP=1_{P} &\qquad \text{(2)}\\ +\vdash .(1).(2).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*121·602.</b> \(\vdash :\text{E}!BʻP.P\in 1\rightarrow 1.\supset \breve{P} ʻBʻP=2_{P}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*121·306·601.&\supset \vdash :\text{Hp}.\supset .P(BʻP\vdash\dashv \breve{P} ʻBʻP)\in 2 &\qquad \text{(1)}\\ +\vdash .*121·23·601.&\supset \vdash \colon\colon \text{Hp}.\supset \colon\ldotp (BʻP)P_{\text{po}}y.\supset .BʻP,y\in P(BʻP\vdash\dashv y).BʻP\neq y\colon\ldotp \\ +[*54·53.*121·303] \supset \colon\ldotp P(BʻP\vdash\dashv y)\in 2.&\supset :P(BʻP\vdash\dashv y)=\iota ʻBʻP\cup \iota ʻy.(BʻP)P_{\text{po}}y:\\ +[*92·111] &\supset :(\breve{P} ʻBʻP)P_{*}y.P(BʻP\vdash\dashv y)=\iota ʻBʻP\cup \iota ʻy:\\ +[*121·103·601] &\supset :\breve{P} ʻBʻP\in \iota ʻBʻP\cup \iota ʻy.\breve{P} ʻBʻP\neq BʻP:\\ +[*51·232] &\supset :y=\breve{P} ʻBʻP &\qquad \text{(2)}\\ +\vdash .(1).(2).*121·6.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_249">[Pg 249]</span></p> + +<p class="nind"><b>*121·61.</b> \[\begin{align}\vdash :P\in 1\rightarrow \text{Cls}.P_{\text{po}}\,\unicode{x2abd}\, J.x\in &sʻ\text{gen}ʻP.\supset .\\ +&(\exists a,\nu ).a BP.\nu \in \text{NC induct}.a P_{\nu }x\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*93·36. &\supset \vdash \colon\ldotp P\in 1\rightarrow \text{Cls}.x\in sʻ\text{gen}ʻP.\supset .(\exists a).a BP.a P_{*}x &\qquad \text{(1)}\\ +\vdash .*121·52. &\supset \vdash \colon\ldotp P\in 1\rightarrow \text{Cls}.P_{\text{po}}\,\unicode{x2abd}\, J.\supset :a P_{*}x.\equiv + .a(\dot{s} ʻ\text{finid}ʻP)x &\qquad \text{(2)}\\ +\vdash .(1).(2).&\supset \vdash :\text{Hp}.\supset .(\exists a).\alpha BP.a (\dot{s} ʻ\text{finid}ʻP)x.\\ +[*121·12] &\supset .(\exists a,\nu ).a BP.\nu \in \text{NC induct}.a P_{\nu }x:\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*121·62">*121·62</a>.</b> \[\begin{align}\vdash :P\in \text{Cls}\rightarrow 1.P_{\text{po}}\,\unicode{x2abd}\, J.(BʻP)&P_{*}x.\supset .\\ +&(\exists \nu ).\nu \in \text{NC induct}-\iota ʻ0.x=\nu _{P}\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*121·52. \supset \vdash :\text{Hp}.&\supset .(BʻP)(\dot{s} ʻ\text{finid}ʻP)x.\\ +[*121·12] &\supset .(\exists \nu ).\nu \in \text{NC induct}.(BʻP)P_{\nu }x &\qquad \text{(1)}\\ +\vdash .*121·341.&\supset \vdash :\text{Hp}.\nu \in \text{NC induct}.\supset .P_{\nu }\in \text{Cls}\rightarrow 1 &\qquad \text{(2)}\\ +\vdash .(1).(2). \supset \vdash :\text{Hp}.&\supset .(\exists \nu ).\nu \in \text{NC induct}.x=\breve{P} _{\nu }ʻBʻP.\\ +[*121·13] &\supset .(\exists \nu ).\nu \in \text{NC induct}.x=(\nu +_{c}1)_P.\\ +[*120·471] &\supset .(\exists \mu ).\mu \in \text{NC induct}-\iota ʻ0.x=\mu _{P}:\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*121·63.</b> \(\vdash :\text{E}!\nu _{P}.\supset .\text{N}_{0}\text{c}ʻP(BʻP\vdash\dashv \nu _{P})=\nu\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*121·13·131.\supset \vdash :\text{Hp}.&\supset .(BʻP)P_{\nu -_{c}1}\nu _{P}.\\ +[*121·11] &\supset .\text{N}_{0}\text{c}ʻP(BʻP\vdash\dashv \nu _{P})=\nu :\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*121·631.</b> \[\begin{align}\vdash \colon\ldotp P\in \text{Cls}\rightarrow 1.&P_{\text{po}}\,\unicode{x2abd}\, J.\nu \in \text{NC induct}-\iota ʻ0.\supset :\\ +&\text{N}_{0}\text{c}ʻP(BʻP\vdash\dashv y)=\nu .\equiv .y=\nu _{P}.\equiv .(BʻP)P_{\nu -_{c}1}y\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*120·414·416.*121·11.\supset \\ +\vdash \colon\ldotp \text{Hp}.\supset :\text{N}_{0}\text{c}ʻP(BʻP\vdash\dashv y)=\nu .&\equiv .(BʻP)P_{\nu -{c}1}y. &\qquad \text{(1)}\\ +[*121·341] &\equiv .y=\breve{P} _{\nu -_{c}1}ʻBʻP.\\ +[*121·13] &\equiv .y=\nu _{P} &\qquad \text{(2)}\\ +\vdash .(1).(2).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><a href="#*121·632">*121·632</a><a href="#*121·633">·633</a> are required for proving <a href="#*121·634">*121·634</a>.</p> + +<p><span class="pagenum" id="Page_250">[Pg 250]</span></p> + +<p class="nind"><b><a id="*121·632">*121·632</a>.</b> \(\vdash :P\in \text{Cls}\rightarrow 1.P_{\text{po}}\,\unicode{x2abd}\, J.\nu \in \text{NC induct}-\iota ʻ0.y=\nu _{P}.yPz.\supset + .z=(\nu +_{c}1)_{P}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*121·13. \supset \vdash :\text{Hp}.&\supset .(BʻP)P_{\nu -_{c}1}y.yPz.\\ +[*121·333·352] &\supset .(BʻP)P_{\nu }z.\\ +[*121·631] &\supset .z=(\nu +_{c}1)_{P}:\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*121·633">*121·633</a>.</b> \[\begin{align}\vdash :P\in \text{Cls}\rightarrow 1.P_{\text{po}}\,\unicode{x2abd}\, J.\nu \in &\text{NC induct}-\iota ʻ0.\nu _{P}\in \text{D}ʻP.\supset .\\ +&\text{E}!(\nu +_{c}1)_{P}.(\nu +_{c}1)_{P}=\breve{P} ʻ\nu _{P}\\ +&[*121·632]\end{align}\]</p> + +<p class="nind"><b><a id="*121·634">*121·634</a>.</b> \[\begin{align}&\vdash \colon\ldotp P\in \text{Cls}\rightarrow 1.P_{\text{po}}\,\unicode{x2abd}\, J.\nu \in \text{NC induct}-\iota ʻ0.\supset :\nu _{P}\in + \text{D}ʻP.\equiv .\text{E}!(\nu +_{c}1)_{P}\\ +&[*121·633·631·333·352]\end{align}\]</p> + +<p class="nind"><b>*121·635.</b> \(\vdash :P\in \text{Cls}\rightarrow 1.P_{\text{po}}\,\unicode{x2abd}\, J.\text{E}!\nu _{P}.\supset .\nu \in \text{NC induct}-\iota ʻ0\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*121·63·45. &\supset \vdash :\text{Hp}.\supset .\nu \in \text{NC induct} &\qquad \text{(1)}\\ +\vdash .*121·13. \supset \vdash :\text{E}!\nu _{P}.&\supset .\dot{\exists} !P_{(\nu -_{c}1)}.\\ +[*121·272] &\supset .(\nu -_{c}1)+_{c}1>0.\\ +[*120·416] &\supset .\nu >0 &\qquad \text{(2)}\\ +\vdash .(1).(2).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*121·636.</b> \[\begin{align}\vdash :P\in \text{Cls}\rightarrow 1.P_{\text{po}}\,\unicode{x2abd}\, J.&\text{E}!\nu _{P}.{\sim}\text{E}!(\nu +_{c}1)_{P}.\supset + .\\ +&\overleftarrow{P}_{*}ʻBʻP=P(BʻP\vdash\dashv \nu _{P}).\text{N}_{0}\text{c}ʻ\overleftarrow{P}_{*}ʻBʻP=\nu\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*121·635. \supset \vdash :\text{Hp}.&\supset .\nu \in \text{NC induct}-\iota ʻ0. &\qquad \text{(1)}\\ +[*121·634.\text{Hp}] &\supset .\nu _{P}{\sim}\in \text{D}ʻP &\qquad \text{(2)}\\ +\vdash .(1).*121·63. \supset \vdash \colon\colon \text{Hp}.&\supset \colon\ldotp \exists !P(BʻP\vdash\dashv \nu _{P})\colon\ldotp \\ +[*121·23] &\supset \colon\ldotp (BʻP)P_{*}\nu _{P}\colon\ldotp \\ +[*96·302.*91·542] &\supset \colon\ldotp (BʻP)P_{*}z.\supset :zP_{*}\nu _{P}.\lor.\nu _{P}P_{\text{po}}z &\qquad \text{(3)}\\ +\vdash .(2).(3).*91·504. \supset \vdash \colon\ldotp \text{Hp}.&\supset :(BʻP)P_{*}z.\supset .zP_{*}\nu _{P}:\\ +[*4·71] &\supset :(BʻP)P_{*}z.\equiv .(BʻP)P_{*}z.zP_{*}v_{P}:\\ +[*121·103] &\supset :\overleftarrow{P}_{*}ʻBʻP=P(BʻP\vdash\dashv \nu _{P}): &\qquad \text{(4)}\\ +[*121·63] &\supset :\text{N}_{0}\text{c}ʻ\overleftarrow{P}_{*}ʻBʻP=\nu &\qquad \text{(5)}\\ +\vdash .(4).(5).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*121·637.</b> \(\vdash :\text{E}!\nu _{P}.\supset .\nu _{P}\in CʻP\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*121·13.*14·28. \supset \vdash :\text{E}!\nu _{P}.&\equiv .\nu _{P}.=\breve{P} _{\nu -_{c}1}ʻBʻP.\\ +[*121·322] &\supset .\nu _{P}\in CʻP:\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_251">[Pg 251]</span></p> + +<p class="nind"><b><a id="*121·638">*121·638</a>.</b> \(\vdash \colon\ldotp \text{E}!(\nu +_{c}1)_{P}.\supset :(BʻP)P_{\nu }x.\equiv .x=(\nu +_{c}1)_{P}:(\nu +_{c}1)-_{c}1=\nu + \)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*121·13.\supset \vdash :\text{E}!(\nu +_{c}1)_{P}.&\equiv .\text{E}!\breve{P}_\{(\nu +_{c}1)\}-_{c}1ʻBʻP. &\qquad \text{(1)}\\ +[*121·272] &\supset .(\nu +_{c}1)-_{c}1\geq 0.\\ +[*14·21] &\supset .\text{E}!(\nu +_{c}1)-_{c} 1.\\ +[*14·22.(*120·411)] &\supset .(\nu +_{c}1)-_{c}1=\nu &\qquad \text{(2)}\\ +\vdash .(2).\supset \vdash \colon\ldotp \text{Hp}.\supset :(BʻP)P_{\nu }x.&\equiv .(BʻP)P_{(\nu +_{c}1)-_{c}1}x.\\ +[(1).*30·4] &\equiv .x=\breve{P} _{(\nu +_{c}1)-_{c}1}ʻBʻP.\\ +[*121·13] &\equiv .x=(\nu +_{c}1)_{P} &\qquad \text{(3)}\\ +\vdash .(3).(2).\supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*121*64.</b> \(\vdash :P\in \text{Cls}\rightarrow 1.P_{\text{po}}\,\unicode{x2abd}\, J.\nu \in \text{NC induct}-\iota ʻ0.\text{Nc}ʻ\overleftarrow{P}_{*}ʻBʻP\geq + \nu .\supset .\text{E}!\nu _{P}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*121·636.&\supset \vdash \colon\ldotp \text{Hp}.\text{E}!\nu _{P}.\supset :{\sim}\text{E}!(\nu +_{c} 1)_{P}.\supset + .\text{N}_{0}\text{c}ʻ\overleftarrow{P}_{*}ʻBʻP=\nu &\qquad \text{(1)}\\ +\vdash .*120·428.\supset \vdash :\nu \in \text{NC induct}.\exists !\nu +_{c}1.&\supset .\nu +_{c}1>\nu .\\ +[*117·281] &\supset .{\sim}(\nu \geq \nu +_{c}1) &\qquad \text{(2)}\\ +\vdash .*117·15. &\supset \vdash ·{\sim} \exists !\nu +_{c}1.\supset .{\sim}(\nu \geq \nu +_{c}1) &\qquad \text{(3)}\\ +\vdash .(2).(3). &\supset \vdash :\nu \in \text{NC induct}.\supset .{\sim}(\nu \geq \nu +_{c}1)&\qquad \text{(4)}\\ +\vdash .(1).(4). &\supset \vdash \colon\ldotp \text{Hp}.\text{E}!\nu _{P}.\supset :\\ +&{\sim}\text{E}!(\nu +_{c}1)_{P}.\supset .{\sim}(\text{Nc}ʻ\overleftarrow{P}_{*}ʻBʻP\geq \nu +_{c}1):\\ +[\text{Transp}] &\supset :\text{Nc}ʻ\overleftarrow{P}_{*}ʻBʻP\geq \nu +_{c}1.\supset .\text{E}!(\nu +_{c}1)_{P} &\qquad \text{(5)}\\ +\vdash .(5).\text{Syll}.*117·6.\supset \vdash \colon\ldotp \text{Hp}:&\text{Nc}ʻ\overleftarrow{P}_{*}ʻBʻP\geq \nu .\supset .\text{E}!\nu _{p}:\supset :\\ +&\text{Nc}ʻ\overleftarrow{P}_{*}ʻBʻP\geq \nu +_{c}1.\supset .\text{E}!(\nu +_{c}1)_{P} &\qquad \text{(6)}\\ +\vdash .*14·21.*121·601.&\supset \vdash :\text{Nc}ʻ\overleftarrow{P}_{*}ʻBʻP\geq 1.\supset .\text{E}!1_{P} &\qquad \text{(7)}\\ +\vdash .(6).(7).*120·473.\supset \\ +\vdash \colon\ldotp \text{Hp}.\supset :\text{Nc}ʻ\overleftarrow{P}_{*}ʻBʻP\geq \nu .\supset .\text{E}!\nu _{P}\colon\ldotp \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*121·641.</b> \[\begin{align}\vdash \colon\ldotp P\in \text{Cls}\rightarrow 1.P_{\text{po}}\,\unicode{x2abd}\, J.\nu \in &\text{NC induct}-\iota ʻ0.\supset :\\ +&\text{Nc}ʻ\overleftarrow{P}_{*}ʻBʻP\geq \nu .\equiv .\text{E}!\nu _{P}\\ +&[*121·64·63·32]\end{align}\]</p> + +<p class="nind"><b>*121·65.</b> \(\vdash :P\in \text{Cls}\rightarrow 1.P_{\text{po}}\,\unicode{x2abd}\, J.\mu \neq 0.\text{E}!(\mu +_{c}\nu )_{P}.\supset .\mu _{P}P_{\nu }(\mu + +_{c}\nu )_{P}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*121·631·635·64.*120·452.\supset \\ +\vdash :\text{Hp}.&\supset .(BʻP)P_{\mu -_{c}1}\mu _{P}.(BʻP)P_{\mu +_{c}\nu -_{c}1}(\mu +_{c}\nu )_{P}.\\ +[*121·351.*120·424]&\supset .(BʻP)P_{\mu -_{c}1}\mu _{P}.(BʻP)(P_{\mu -_{c}1}\mid P_{\nu })(\mu +_{c}\nu )_{P}.\\ +[*121·341.*72·591] &\supset .\mu _{P}P_{\nu }(\mu +_{c}\nu )_{P}:\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_252">[Pg 252]</span></p> + +<p class="nind"><b>*121·66.</b> \(\vdash :P\in \text{Cls}\rightarrow 1.P_{\text{po}}\,\unicode{x2abd}\, J.\text{Nc}ʻP(BʻP\vdash\dashv x)>\nu .\supset .x\in \text{ᗡ}ʻP_{\nu }\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*121·45.*120·48.\supset \vdash :\text{Hp}.&\supset .\nu \in \text{NC induct}.\\ +[*120·429] &\supset .\text{Nc}ʻP(BʻP\vdash\dashv x)\geq \nu +_{c}1.\\ +[*117·31] &\supset .(\exists \mu ).\text{Nc}ʻP(BʻP\vdash\dashv x)=\nu +_{c}1+_{c}\mu .\\ +[*121·11] &\supset .(\exists \mu ).(BʻP)P_{\nu +_{c}\mu }x.\\ +[*121·351·352] &\supset .(\exists \mu ).(BʻP)(P_{\mu }\mid P_{\nu })x.\\ +[*34·36] &\supset .x\in \text{ᗡ}ʻP_{\nu }:\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p>The following proposition is used in <a href="#*122·38">*122·38</a><a href="#*122·381">·381</a>.</p> + +<p class="nind"><b><a id="*121·7">*121·7</a>.</b> \(\vdash :R\in 1\rightarrow 1.aBR.aR_{*}x.\supset .\overrightarrow{R}_{*}ʻx=R(a\vdash\dashv x).\overrightarrow{R}_{*}ʻx\in \text{Cls induct}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*96·25.\supset \vdash \colon\ldotp \text{Hp}.&\supset :yR_{*}x.\supset .aR_{*}y:\\ +[*4·71] &\supset :yR_{*}x.\equiv .aR_{*}y.yR_{*}x:\\ +[*121·103] &\supset :\overrightarrow{R}_{*}ʻx=R(a\vdash\dashv x) &\qquad \text{(1)}\\ +\vdash .(1).*121·45.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*121·71.</b> \[\begin{align}\vdash \colon\ldotp R\in 1\rightarrow 1:x\in sʻ\text{gen}ʻR.\lor.(\exists y).y\in &\overleftrightarrow{R}_{*}ʻx.yR_{\text{po}}y:\supset .\\ +&\overrightarrow{R}_{*}ʻx\in \text{Cls induct}\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*121·7.*93·36.&\supset \vdash :R\in 1\rightarrow 1.x\in sʻ\text{gen}ʻR.\supset .\overrightarrow{R}_{*}ʻx\in \text{Cls induct} &\qquad \text{(1)}\\ +\vdash .*97·55·111.\supset \vdash \colon\ldotp R\in 1\rightarrow 1:(\exists y).&y\in \overleftrightarrow{R}_{*}ʻx.yR_{\text{po}}y:\supset :\\ +&y\in \overleftrightarrow{R}_{*}ʻx .\supset _{y}.yR_{\text{po}}y:x\in \overleftrightarrow{R}_{*}ʻx:\\ +[*10·26] &\supset :xR_{\text{po}}x:\\ +[*121·381] &\supset :\overrightarrow{R}_{*}ʻx=R(x\vdash\dashv x):\\ +[*121·45] &\supset :\overrightarrow{R}_{*}ʻx\in \text{Cls induct} &\qquad \text{(2)}\\ +\vdash .(1).(2).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*121·72">*121·72</a>.</b> \[\begin{align}&\vdash :R\in 1\rightarrow 1.\overrightarrow{R}_{*}ʻx{\sim}\in + \text{Cls induct}.\supset .x\in pʻ\text{ᗡ}ʻʻ\text{Pot}ʻR.R_{\text{po}}\unicode{x0294f}\overleftrightarrow{R}_{*}ʻx\,\unicode{x2abd}\, J\\ +&[*121·71.\text{Transp}.*93·271.*120·212.*50·24]\end{align}\]</p> + + +<div class="footnotes"><h3>FOOTNOTES:</h3> + +<div class="footnote"> + +<p class="nind"> +<a id="Footnote_9" href="#FNanchor_9" class="label">[9]</a> +Cf. <a href="#*122">*122</a>, below.</p> + +</div> +</div> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_253">[Pg 253]</span></p> +<h2 class="nobreak" id="*122">*122. PROGRESSIONS.</h2> +</div> + + +<p class="nind"><i>Summary of</i> *122.</p> + +<p>By a "progression" we mean a series which is like the series of the +inductive cardinals in order of magnitude (assuming that all inductive +cardinals exist), <i>i.e.</i> a series whose terms can be called +\[ +1_{R}, 2_{R}, 3_{R}, ... \nu _{R}, ..., +\] +where every term of the series is correlated with some inductive +cardinal, and every inductive cardinal is correlated with some term +of the series. Such series belong to the relation-number (cf. <a href="#*152">*152</a> +and *263) which Cantor calls \(\omega\). Their generating relation +may be taken to be the transitive relation of earlier and later, or +the one-one relation of immediate predecessor to immediate successor. +We shall reserve the notation \(\omega\) for the <i>transitive</i> +generating relations of progressions; for the present, we are concerned +with the one-one relations which generate progressions. The class of +these relations we shall call "\(\text{Prog}\)."</p> + +<p>It is not convenient to <i>define</i> a progression as a series which +is ordinally similar to that of the inductive cardinals, both because +this definition only applies if we assume the axiom of infinity, +and because we have in any case to show that (assuming the axiom of +infinity) the series of inductive cardinals has certain properties, +which can be used to afford a direct definition of progressions. The +existence of progressions, however, is only obtainable by means of the +axiom of infinity, and is then most easily obtained from the fact that +the inductive cardinals form a progression. We shall not consider the +existence-theorem until the next number (<a href="#*123">*123</a>).</p> + +<p>From this number onwards convention \(\text{Infin}\,T\) of the Prefatory +Statement is used when relevant.</p> + +<p>The characteristics of the generating relation \(R\) of a progression, +which we employ in the definition, are the following:</p> + +<p>(1) \(R\) is a one-one relation;</p> + +<p>(2) there is a first term, <i>i.e.</i> \(\text{E}!BʻR\);</p> + +<p>(3) the whole field is contained in the posterity of the first term, +<i>i.e.</i> \(CʻR=\overleftarrow{R}_{*}ʻBʻR\). (If this failed, \(CʻR\) +would consist of two or more distinct families, of which, since we have +\(\text{E}!BʻR\), all but one would have to be cyclic families.)</p> + +<p>(4) every term of the field has a successor, <i>i.e.</i> the series is +endless. This is secured \(\text{ᗡ}ʻR\subset \text{D}ʻR\), or (what is +equivalent) \(CʻR=\text{D}ʻR\).</p> + +<p><span class="pagenum" id="Page_254">[Pg 254]</span></p> + +<p>These four properties suffice to define the one-one generating +relations of progressions. It will be observed that (2), (3) and (4) +are all secured by +\[ +\text{D}ʻR = \overleftarrow{R}_{*}ʻBʻR. +\]</p> + +<p>This secures \(\text{E}!BʻR\), by *14·21; it secures +\(\text{ᗡ}ʻR\subset \text{D}ʻR\), by *37·25 and *90·163; hence, by +*33·181, \(\text{D}ʻR=CʻR\), and therefore +\[ +CʻR=\overleftarrow{R}_{*}ʻBʻR. +\]</p> + +<p>Hence our definition of progressions is +\[ +\text{Prog}=(1\rightarrow 1)\cap \hat{R} (\text{D}ʻR=\overleftarrow{R}_{*}ʻBʻR) \quad\text{Df}. +\]</p> + +<p>Instead of stating in the definition that \(R\) is to +be a one-one relation, it is sufficient to put +\(R\in\text{Cls}\rightarrow 1.R_{\text{po}}\,\unicode{x2abd}\, J\), +which, with \(\text{D}ʻR=\overleftarrow{R}_{*}ʻBʻR\), implies +\(R\in 1\rightarrow 1\), and may be substituted for +\(R\in 1\rightarrow 1\) without altering the force of the definition +(<a href="#*122·17">*122·17</a>).</p> + +<p>In the present number we shall prove, among other propositions, that +every existent class contained in a progression has a first term +(<a href="#*122·23">*122·23</a>), <i>i.e.</i> that progressions are well-ordered series; +that in a progression \(R_{\text{po}}\,\unicode{x2abd}\, J\) (<a href="#*122·16">*122·16</a>), +which makes the propositions of <a href="#*121">*121</a> available; that if \(\nu\) is +any inductive cardinal other than 0, \(\nu _{R}\) exists (<a href="#*122·33">*122·33</a>), +<i>i.e.</i> the series has a \(\nu\)th term; that any class contained +in \(\text{D}ʻR\) and having a last term is an inductive class +(<a href="#*122·43">*122·43</a>), and that any class contained in \(\text{D}ʻR\) and not +having a last term is itself the domain of a progression (<a href="#*122·45">*122·45</a>), so +that every class contained in \(\text{D}ʻR\) is either inductive or +the domain of a progression (<a href="#*122·46">*122·46</a>); that if \(P\) is a many-one, +and \(x\) a member of its domain, and if the descendants of \(x\) +have no last term and are none of them descendants of themselves, +then \(P\) arranges these descendants in a progression (<a href="#*122·51">*122·51</a>); +and that the same holds if \(P\) is a one-one and \({\sim}(xPx)\) +(<a href="#*122·52">*122·52</a>); and that if \(P\in 1\rightarrow 1\) and \(x\) belongs to +one of the generations of \(P\), but not to one of the generations +of \(\breve{P}\), then \(P\) arranges the whole family of \(x\) in a +progression (<a href="#*122·54">*122·54</a>).</p> + +<p>The following general observations on the families of one-one relations +may serve to elucidate the bearing of the propositions of this section.</p> + +<p>Given any relation \(P\), we call \(\overleftrightarrow{P}_{*}ʻx\), +<i>i.e.</i> \(\overrightarrow{P}_{*}ʻx\cup \overleftarrow{P}_{*}ʻx\) +the family of \(x\). If \(P\) is a one-one, this family may be of four +different kinds. (1) It may be a closed series, like the angles of a +polygon. This occurs if \(xP_{\text{po}}x\). In this case the family +forms an inductive class. (2) It may be an open series with a beginning +and an end; this occurs if +\[ +{\sim}(xP_{\text{po}}x).\text{E}!\text{min}_{P}ʻ\overleftrightarrow{P}_{*}ʻx.\text{E}!\text{max}_{P}ʻ\overleftrightarrow{P}_{*}ʻx. +\]</p> + +<p>In this case also the family forms an inductive class. (3) It may +be an<span class="pagenum" id="Page_255">[Pg 255]</span> open series with a beginning and no end, or an end and no +beginning. This occurs if +\[ +{\sim}(xP_{\text{po}}x).\text{E}!\text{min}_{P}ʻ\overleftrightarrow{P}_{*}ʻx.{\sim}\text{E}!\text{max}_{P}ʻ\overleftrightarrow{P}_{*}ʻx, +\] +or if +\[ +{\sim}(xP_{\text{po}}x).{\sim}\text{E}!\text{min}_{P}ʻ\overleftrightarrow{P}_{*}ʻx.\text{E}!\text{max}_{P}ʻ\overleftrightarrow{P}_{*}ʻx. +\] +In this case, the series is of the type \(\omega\) or +\(\text{Cnv}ʻʻ\omega\), and is non-inductive and reflexive. (4) The +series may be open and have neither beginning nor end. This occurs if +\[ +{\sim}(xP_{\text{po}}x).{\sim}\text{E}!\text{min}_{P}ʻ\overleftrightarrow{P}_{*}ʻx.{\sim}\text{E}!\text{max}_{P}ʻ\overleftrightarrow{P}_{*}ʻx +\] +In this case we get a series whose relation-number is the sum (in the +sense of <a href="#*180">*180</a>) of \(\text{Cnv}ʻʻ\omega\) and \(\omega\), which again is +non-inductive and reflexive. In all four cases, if \(y\) and \(z\) be +any two members of the family of \(x\), the interval between \(y\) and +\(z\) is an inductive class.</p> + +<p>If \(x\) is a member of \(\overrightarrow{B}ʻP\), or if the family of +\(x\) contains a member of \(\overrightarrow{B}ʻP\), cases (1) and (4) +are excluded, since the series has a beginning. In this case the number +of predecessors of any term is an inductive number. It will be observed +that every family is either wholly contained in \(sʻ\text{gen}ʻP\) or +wholly contained in \(pʻ\text{ᗡ}ʻʻ\text{Pot}ʻP\); families of kinds (2) +and (3) (excluding, in (2), those which have an end but no beginning) +are contained in \(sʻ\text{gen}ʻP\), while families of kinds (1) and +(4), and those of (2) which have an end but no beginning, are contained +in \(pʻ\text{ᗡ}ʻʻ\text{Pot}ʻP\); families containing a member of +\(\overrightarrow{B}ʻP\) are contained in \(sʻ\text{gen}ʻP\), while all +others are contained in \(pʻ\text{ᗡ}ʻʻ\text{Pot}ʻP\).</p> + +<p>Thus a one-one relation in general gives rise to a number of wholly +disconnected series, some closed, others open and with or without a +beginning or an end. The condition that all the series should be open +is \(P_{\text{po}}\,\unicode{x2abd}\, J\).</p> + +<p>The case of a \(Q\)-shaped family, considered in *96, cannot arise when +\(P\in 1\rightarrow 1\), for in a \(Q\)-shaped family the term at the +junction of the tail and the circle has two predecessors, one in the +tail and one in the circle, so that the relation in question is not +\(1\rightarrow 1\). It follows that, when \(P\in 1\rightarrow 1\), if +\(\alpha\) is a family containing a member of \(\overrightarrow{B}ʻP\), +\(\alpha \upharpoonleft P_{\text{po}}\,\unicode{x2abd}\, J\) (cf. *96·23).</p> + +<p>When \(BʻP\) exists, there is only one family which has a beginning. In +this case, ignoring the other families (if any), we call the members +of the family of \(BʻP\) respectively \(1_{P}\), \(2_{P}\), \(3_{P}\), +.... If the family has \(\nu\) members, where \(\nu\) is an inductive +cardinal, its last member will be \(\nu_{P}\). If on the other hand the +number of members of the family is not an inductive cardinal, it must +be \(\aleph _{0}\); in this case, the family forms a progression, whose +members are \(1_{P}\), \(2_{P}\), \(3_{P}\), ..., \(\nu _{P}\), ..., +where \(\nu _{P}\) always exists when \(\nu\) is an inductive cardinal.</p> + +<p><span class="pagenum" id="Page_256">[Pg 256]</span></p> + +<p>In addition to the propositions already mentioned, the following are +important:</p> + +<p class="nind"><b>*122·21.</b> \(\vdash \colon\ldotp R\in \text{Prog}.x,y\in CʻR.\supset :xR_{\text{po}}y.\lor.x = y.\lor.yR_{\text{po}}x\)</p> + +<p>(Cf. note to <a href="#*122·21">*122·21</a>, below.)</p> + +<p class="nind"><b>*122·34.</b> \(\vdash \colon\ldotp R\in \text{Prog}.\supset :\nu \in \text{NC induct} - \iota ʻ0.\equiv .\text{E}!\nu _{R}\)</p> + +<p class="nind"><b>*122·341.</b> \(\vdash :R\in \text{Prog}.\supset .\text{D}ʻR = \hat{x} \{(\exists \nu ).\nu \in \text{NC induct} - \iota ʻ0.x = \nu _{R}\}\)</p> + +<p>In virtue of these two propositions, the terms of a progression are +\[ +1_{R},\,\, 2_{R},\,\, 3_{R},\,\, ... \nu _{R},\,\, ..., +\] +where every inductive cardinal occurs. This is the same fact as is +usually assumed when the terms are represented as +\[ +x_{1},\,\, x_{2},\,\, x_{3},\,\, ... x_{\nu }, .... +\]</p> + +<p class="nind"><b>*122·35.</b> \(\vdash :R\in \text{Prog}.\nu \in \text{NC induct} - \iota ʻ0.\supset .\overrightarrow{B}ʻR_{\nu } = R(1_{R}\vdash\dashv + \nu _{R}).\overrightarrow{B}ʻR_{\nu }\in \nu \)</p> + +<p class="nind"><b>*122·36.</b> \(\vdash :\exists !\text{Prog} \cap t^{11}ʻx.\supset .\text{Infin ax}(x)\)</p> + +<p class="nind"><b>*122·37.</b> \(\vdash :R\in \text{Prog}.\supset .\text{D}ʻR{\sim}\in \text{Cls induct}.\text{N}_{0}\text{c}ʻ\text{D}ʻR{\sim}\in \text{NC induct}\)</p> + +<p class="nind"><b>*122·38.</b> \(\vdash :R\in \text{Prog}.\supset .\overrightarrow{R}_{*}ʻx\in \text{Cls induct}\)</p> + +<p><i>I.e.</i> the number of terms up to any given point of a progression +is inductive.</p> + +<hr class="tb"> + +<p class="nind"><b>*122·01.</b> \(\text{Prog} = (1\rightarrow 1)\cap \hat{R} (\text{D}ʻR = \overleftarrow{R}_{*}ʻBʻR) \quad\text{Df}\)</p> + +<p class="nind"><b><a id="*122·1">*122·1</a>.</b> \(\vdash :R\in \text{Prog}.\equiv .R\in 1\rightarrow 1.\text{D}ʻR = \overleftarrow{R}_{*}ʻBʻR \quad[(*122·01)]\)</p> + +<p class="nind"><b><a id="*122·11">*122·11</a>.</b> \(\vdash \colon\ldotp R\in \text{Prog}.\equiv :R\in 1\rightarrow 1.\text{E}!BʻR:x\in \text{D}ʻR.\equiv _{x}.x\in \overleftarrow{R}_{*}ʻBʻR\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*122·1.*14·205.\supset \\ +\vdash \colon\colon R\in \text{Prog}.&\equiv \colon\ldotp R\in 1\rightarrow 1:(\exists a).a = BʻR.\text{D}ʻR = \overleftarrow{R}_{*}ʻa\colon\ldotp \\ +[*20·43] &\equiv \colon\ldotp R\in 1\rightarrow 1\colon\ldotp (\exists a):a = BʻR:x\in \text{D}ʻR.\equiv _{x}.x\in \overleftarrow{R}_{*}ʻa\colon\ldotp \\ +[*14·15] &\equiv \colon\ldotp R\in 1\rightarrow 1\colon\ldotp (\exists a):a = BʻR:x\in \text{D}ʻR.\equiv _{x}.x\in \overleftarrow{R}_{*}ʻBʻR\colon\ldotp \\ +[*14·204] &\equiv \colon\ldotp R\in 1\rightarrow 1.\text{E}!BʻR:x\in \text{D}ʻR.\equiv _{x}.x\in \overleftarrow{R}_{*}ʻBʻR\colon\colon \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p>Observe that, by the conventions as to descriptive symbols, +\(\text{D}ʻR = \overleftarrow{R}_{*}ʻBʻR\) involves the +existence of \(BʻR\), whereas \(x\in \text{D}ʻR.\equiv _{x}.x\in\overleftarrow{R}_{*}ʻBʻR\) +does not, since, if \(BʻR\) does not exist, +we have \((x).x{\sim}\in \overleftarrow{R}_{*}ʻBʻR\), and therefore +\((x).x{\sim}\in \text{D}ʻR\) will satisfy the equivalence, <i>i.e.</i> +\(\dot{\Lambda}\) will satisfy the equivalence although it has no first +term. This is the reason why \(\text{E}!BʻR\) appears explicitly in +<a href="#*122·11">*122·11</a>, though it was only implicit in <a href="#*122·1">*122·1</a>.</p> + +<p class="nind"><b>*122·12.</b> \[\begin{align}\vdash \colon\colon R\in \text{Prog}.&\equiv \colon\ldotp R\in 1\rightarrow 1.\text{E}!BʻR\colon\ldotp x\in \text{D}ʻR.\equiv _{x}:\\ +&BʻR\in \alpha .\breve{R} ʻʻ\alpha \subset \alpha .\supset _{\alpha }.x\in \alpha \quad[*122·11.*90·1]\end{align}\]</p> + +<p><span class="pagenum" id="Page_257">[Pg 257]</span></p> + +<p class="nind"><b>*122·14.</b> \(\vdash :R\in \text{Prog}.\supset .\overleftarrow{R}_{\text{po}}ʻBʻR=\text{ᗡ}ʻR\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*122·1.*37·25. \supset \vdash :\text{Hp}.\supset .\text{ᗡ}ʻR&=\breve{R} ʻʻ\overleftarrow{R}_{*}ʻBʻR\\ +[*91·52] & =\overleftarrow{R}_{\text{po}}ʻBʻR:\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*122·141.</b> \(\vdash :R\in \text{Prog}.\supset .\text{ᗡ}ʻR\subset \text{D}ʻR.CʻR=\text{D}ʻR\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*122·1.*37·25. \supset \vdash :\text{Hp}.&\supset .\text{ᗡ}ʻR=\breve{R} ʻʻ\overleftarrow{R}_{*}ʻBʻR.\\ +[*90·163] &\supset .\text{ᗡ}ʻR\subset \overleftarrow{R}_{*}ʻBʻR.\\ +[*122·1.*33·181] &\supset .\text{ᗡ}ʻR\subset \text{D}ʻR.CʻR=\text{D}ʻR:\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*122·142.</b> \(\vdash :R\in \text{Prog}.P\in \text{Pot}ʻR.\supset .\text{D}ʻP=\text{D}ʻR \quad[*122·141.*92·14]\)</p> + +<p class="nind"><b>*122·143.</b> \(\vdash :R\in \text{Prog}.P\in \text{Pot}ʻR.\supset .\text{ᗡ}ʻP\subset \text{D}ʻP \quad[*122·142·141.*91·271]\)</p> + +<p class="nind"><b>*122·15.</b> \(\vdash :R\in \text{Prog}.\supset .R=(\overleftarrow{R}_{*}ʻBʻR)\upharpoonleft R=R\upharpoonright (\overleftarrow{R}_{\text{po}}ʻBʻR)=R\upharpoonright + (\overleftarrow{R}_{*}ʻBʻR)\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*122·1.*35·63. \supset \vdash :\text{Hp}.\supset .R&=(\overleftarrow{R}_{*}ʻBʻR)\upharpoonleft R\\ +[*96·2] &=R\upharpoonright (\overleftarrow{R}_{\text{po}}ʻBʻR)\\ +[*96·21] &=R\upharpoonright (\overleftarrow{R}_{*}ʻBʻR):\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*122·151.</b> \[\begin{align}&\vdash :R\in \text{Prog}.\supset .R_{*}=(\overleftarrow{R}_{*}ʻBʻR)\upharpoonleft R_{*}=R_{*}\upharpoonright + (\overleftarrow{R}_{*}ʻBʻR)\\ +&[35·63·66.*90·14.*122·141·1]\end{align}\]</p> + +<p class="nind"><b>*122·152.</b> \[\begin{align}\vdash :R\in \text{Prog}.\supset .R_{\text{po}}=(\overleftarrow{R}_{*}ʻBʻR)\upharpoonleft R_{\text{po}}&=R_{\text{po}}\upharpoonright + (\overleftarrow{R}_{\text{po}}ʻBʻR)\\ +&=R_{\text{po}}\upharpoonright (\overleftarrow{R}_{*}ʻBʻR)\\ +[*35·63·66.*91·504.*121·1·14]\end{align}\]</p> + +<p class="nind"><b><a id="*122·16">*122·16</a>.</b> \(\vdash :R\in \text{Prog}.\supset .R_{\text{po}}\,\unicode{x2abd}\, J \quad[*96·23.*122·152]\)</p> + +<p>This proposition enables us to apply to progressions all the +propositions of <a href="#*121">*121</a> in which we have as hypothesis +\[ +R\in \text{Cls}\rightarrow 1.R_{\text{po}}\,\unicode{x2abd}\, J, or R\in 1\rightarrow \text{Cls}.R_{\text{po}}\,\unicode{x2abd}\, J. +\]</p> + +<p class="nind"><b><a id="*122·17">*122·17</a>.</b> \(\vdash :R\in \text{Prog}.\equiv .R\in \text{Cls}\rightarrow 1.R_{\text{po}}\,\unicode{x2abd}\, J.\text{D}ʻR=\overleftarrow{R}_{*}ʻBʻR\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*35·63. \supset \vdash :\text{D}ʻR=\overleftarrow{R}_{*}ʻBʻR.\supset .R=(\overleftarrow{R}_{*}ʻBʻR)\upharpoonleft R &\qquad \text{(1)}\\ +\vdash .*96·453. \supset \vdash :R\in \text{Cls}\rightarrow 1.(\overleftarrow{R}_{*}ʻBʻR)\upharpoonleft R_{\text{po}}\,\unicode{x2abd}\, J.\supset .(\overleftarrow{R}_{*}ʻBʻR)\upharpoonleft + R\in 1\rightarrow 1 &\qquad \text{(2)}\\ +\vdash .(1).(2).*122·1. \supset \vdash :\text{D}ʻR=\overleftarrow{R}_{*}ʻBʻR.R\in \text{Cls}\rightarrow 1.R_{\text{po}}\,\unicode{x2abd}\, + J.\supset .R\in \text{Prog} &\qquad \text{(3)}\\ +\vdash .(3).*122·1·16.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_258">[Pg 258]</span></p> + +<p>To illustrate this proposition, consider its application to the +inductive cardinals arranged in order of magnitude; <i>i.e.</i> take as +a value of \(R\) the relation +\[ +\hat{\mu} \hat{\nu} (\mu \in \text{NC induct}.\nu =\mu +_{c}1). +\]</p> + +<p>We then have \(R\in \text{Cls}\rightarrow 1.0=BʻR\); also +\[ +\text{NC induct}=\text{D}ʻR=\overleftarrow{R}_{*}ʻBʻR. +\]</p> + +<p>We have also +\[ +\exists !\mu +_{c}1.\mu +_{c}1=\nu +_{c}1.\supset .\mu =\nu , +\] +so that \(R\upharpoonright (-\iota ʻ\Lambda )\in 1\rightarrow \text{Cls}\).</p> + +<p>Again +\[ +\mu R_{\text{po}}\nu .\equiv .(\exists \varpi ).\varpi \in \text{NC induct}-\iota ʻ0.\nu =\mu +_{c}\varpi , +\] +whence +\[ +\mu R_{\text{po}}\nu .\exists !\mu .\supset .\mu \neq \nu , +\] +<i>i.e.</i> +\[ +(-\iota ʻ\Lambda )\upharpoonleft R_{\text{po}}\,\unicode{x2abd}\, J. +\]</p> + +<p>But we do not get \(R\in 1\rightarrow \text{Cls}\) or +\(R_{\text{po}}\,\unicode{x2abd}\, J\) unless we have +\[ +\Lambda {\sim}\in \text{NC induct}, +\] +which is the axiom of infinity. If this condition fails, we reach at +last an inductive cardinal which = \(\Lambda\), and we have +\[ +\Lambda =\Lambda +_{c}1, +\] +so that \(\Lambda\) has two immediate predecessors, namely itself +and the last existent cardinal. The posterity of 0, in this case, is +a \(Q\) in which the circle has narrowed to a single term, namely +\(\Lambda\).</p> + +<p>Thus we need the axiom of infinity in order to prove +\[ +\hat{\mu} \hat{\nu} (\mu \in \text{NC induct}.\nu =\mu +_{c}1)\in \text{Prog}. +\]</p> + +<p class="nind"><b>*122·2.</b> \(\vdash \colon\ldotp R\in \text{Prog}.x,y\in CʻR.\supset :xR_{*}y.\lor.yR_{*}x \quad[*96·302.*122·1·141]\)</p> + +<p class="nind"><b><a id="*122·21">*122·21</a>.</b> \[\begin{align}\vdash \colon\ldotp R\in \text{Prog}.x,y\in CʻR.\supset :xR_{\text{po}}y.&\lor.x=y.\lor.yR_{\text{po}}x\\ +&[*96·303.*122·1·141]\end{align}\]</p> + +<p>This proposition, together with <a href="#*122·16">*122·16</a> and *91·56, shows that if +\(R\in \text{Prog}\), \(R_{\text{po}}\) has the three properties by +which transitive serial relations are defined (cf. <a href="#*204">*204</a>), namely it is +(1) transitive, (2) contained in diversity, (3) connected, <i>i.e.</i> +such that it relates any two distinct members of its field. We shall at +a later stage define the ordinal number \(\omega\) as the class of such +relations as \(R_{\text{po}}\), where \(R\in \text{Prog}\).</p> + +<p class="nind"><b>*122·22.</b> \(\vdash :R\in \text{Prog}.\alpha \subset \text{D}ʻR.x,y\in \alpha -\breve{R} _{\text{po}}ʻʻ\alpha .\supset .x=y\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*122·21.&\supset \vdash \colon\ldotp \text{Hp}.\supset :xR_{\text{po}}y.\lor.x=y.\lor.yR_{\text{po}}x &\qquad \text{(1)}\\ +\vdash .*37·105.&\supset \vdash :x\in \alpha .xR_{\text{po}}y.\supset .y\in \breve{R} _{\text{po}}ʻʻ\alpha :\\ +[\text{Transp}] &\supset \vdash :x\in \alpha .y{\sim}\in \breve{R} _{\text{po}}ʻʻ\alpha .\supset .{\sim}(xR_{\text{po}}y) &\qquad \text{(2)}\\ +\vdash .(2). &\supset \vdash :\text{Hp}.\supset .{\sim}(xR_{\text{po}}y).{\sim}(yR_{\text{po}}x) &\qquad \text{(3)}\\ +\vdash .(1).(3).&\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_259">[Pg 259]</span></p> + +<p class="nind"><b><a id="*122·23">*122·23</a>.</b> \[\begin{align}\vdash :R\in \text{Prog}.\alpha \subset \text{D}ʻR.&\exists !\alpha .\supset .\\ +&\text{E}!\text{min}(R_{\text{po}})ʻ\alpha .\alpha -\breve{R} _{\text{po}}ʻʻ\alpha =\iota ʻ\text{min}(R_{\text{po}})ʻ\alpha\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*96·52. &\supset \vdash :\text{Hp}.\supset .\exists !\overrightarrow{\text{min}}(R_{\text{po}})ʻ\alpha &\qquad \text{(1)}\\ +\vdash .*93·111.*122·22. &\supset \vdash \colon\ldotp \text{Hp}.\supset :x,y\in \overrightarrow{\text{min}}(R_{\text{po}})ʻ\alpha .\supset _{x,y}.x=y &\qquad \text{(2)}\\ +\vdash .(1).(2).*32·4.*93·111.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p>This proposition shows that every existent class contained in a +progression has a first term, <i>i.e.</i> that a progression is a +well-ordered series (cf. *250).</p> + +<p class="nind"><b>*122·231.</b> \(\vdash :R\in \text{Prog}.\alpha \subset \breve{R} _{\text{po}}ʻʻ\alpha .\supset .\alpha =\Lambda\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*91·504.&\supset \vdash :\text{Hp}.\supset .\alpha \subset \text{ᗡ}ʻR &\qquad \text{(1)}\\ +\vdash .*93·11. &\supset \vdash :\text{Hp}.\supset.{\sim}\text{E}!\overrightarrow{\text{min}}(R_{\text{po}})ʻ\alpha &\qquad \text{(2)}\\ +\vdash .(1).(2).*122·23·141.\text{Transp}.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*122·24.</b> \(\vdash :R\in \text{Prog}.P\in PotʻR.\supset .\text{D}ʻP=\breve{P} _{*}ʻʻ\overrightarrow{B}ʻP=sʻ\text{gen}ʻP\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*122·1.*92·102.\supset \vdash :\text{Hp}.&\supset .P\in 1\rightarrow 1.\\ +[*93·42] &\supset .pʻ\text{ᗡ}ʻʻ\text{Pot}ʻP=\breve{P} ʻʻpʻ\text{ᗡ}ʻʻ\text{Pot}ʻP.\\ +[*91·581] &\supset .pʻ\text{ᗡ}ʻʻ\text{Pot}ʻP\subset \breve{R} _{\text{po}}ʻʻpʻ\text{ᗡ}ʻʻ\text{Pot}ʻP.\\ +[*122·231] &\supset .pʻ\text{ᗡ}ʻʻ\text{Pot}ʻP=\Lambda &\qquad \text{(1)}\\ +\vdash .(1).*93·37·36. \supset \vdash :\text{Hp}.&\supset .CʻP=\breve{P} _{*}ʻʻ\overrightarrow{B}ʻP=sʻ\text{gen}ʻP &\qquad \text{(2)}\\ +\vdash .(2).*122·143.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p>Except when \(P=R\), \(\overrightarrow{B}ʻP\) will not +reduce to a single term. In fact, if \(P=R_{\nu }\), +\(\overrightarrow{B}ʻP=R(1_{R}\vdash\dashv \nu _{R})\), <i>i.e.</i> +\(\overrightarrow{B}ʻP\) consists of the first \(\nu\) terms of the +progression.</p> + +<p class="nind"><b>*122·25.</b>\[\begin{align}\vdash :R\in \text{Prog}.P\in \text{Pot}ʻR.x\in &\text{D}ʻR.\supset .\\ +&(\overleftarrow{P}_{*}ʻx)\upharpoonleft P\in \text{Prog}.x=Bʻ\{(\overleftarrow{P}_{*}ʻx)\upharpoonleft P\}\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*122·1.*92·102.&\supset \vdash :\text{Hp}.\supset .(\overleftarrow{P}_{*}ʻx)\upharpoonleft P\in 1\rightarrow 1 &\qquad \text{(1)}\\ +\vdash .*122·143.& \supset \vdash :\text{Hp}.\supset .\overleftarrow{P}_{*}ʻx\subset \text{D}ʻP.\\ +[*35·62] &\supset .\text{D}ʻ\{(\overleftarrow{P}_{*}ʻx)\upharpoonleft P\}=\overleftarrow{P}_{*}ʻx &\qquad \text{(2)}\\ +\vdash .*37·4.*91·52. &\supset \vdash .\text{ᗡ}ʻ\{(\overleftarrow{P}_{*}ʻx)\upharpoonleft P\}=\overleftarrow{P}_{\text{po}}ʻx &\qquad \text{(3)}\\ +\vdash .*122·16.*91·6. &\supset \vdash :\text{Hp}.\supset .x{\sim}\in \overleftarrow{P}_{\text{po}}ʻx &\qquad \text{(4)}\\ +\vdash .*91·542. &\supset \vdash :y\in \overleftarrow{P}_{*}ʻx.y\neq x.\supset .y\in \overleftarrow{P}_{\text{po}}ʻx &\qquad \text{(5)}\\ +\vdash .(2).(3).(4).(5).&\supset \vdash :\text{Hp}.\supset .x=Bʻ\{(\overleftarrow{P}_{*}ʻx)\upharpoonleft P\} &\qquad \text{(6)}\\ +\vdash .(1).(2).(6).*96·131.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_260">[Pg 260]</span></p> + +<p>The above proposition shows that what we may call an "arithmetical +progression" in a progression is a progression, <i>i.e.</i> if, +starting from any term of a progression, we take every other term, or +every third term, or every \(\nu\)th term, we still have a progression.</p> + +<p class="nind"><b>*122·26.</b> \(\vdash :R\in \text{Prog}.\alpha \subset R_{\text{po}}ʻʻ\alpha .\exists !\alpha .\supset .\text{D}ʻR=R_{\text{po}}ʻʻ\alpha\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *22·1 . &\supset \vdash \colon\ldotp \text{Hp} . \supset : BʻR \in \alpha . \supset BʻR \in R_{\text{po}}ʻʻ\alpha &\qquad \text{(1)}\\ +\vdash .*91·542.*122·11.&\supset \vdash \colon\ldotp \text{Hp}.BʻR{\sim}\in \alpha .\supset :y\in \alpha \cap \text{D}ʻR.\supset _{y}.(BʻR)R_{\text{po}}y:\\ +[*91·504.*37·15] &\supset :y\in \alpha .\supset _{y}.(BʻR)R_{\text{po}}y:\\ +[*10·55.\text{Hp}] &\supset :(\exists y).y\in \alpha .(BʻR)R_{\text{po}}y:\\ +[*37·1] &\supset :BʻR\in R_{\text{po}}ʻʻ\alpha &\qquad \text{(2)}\\ +\vdash .(1).(2) .&\supset \vdash :\text{Hp}.\supset .BʻR\in R_{\text{po}}ʻʻ\alpha &\qquad \text{(3)}\\ +\vdash .*92·111. &\supset \vdash \colon\ldotp \text{Hp}.\supset :x\in R_{\text{po}}ʻʻ\alpha .xRy.\supset .y\in R_{*}ʻʻ\alpha .\\ +[*91·545] &\supset .y\in \alpha \cup R_{\text{po}}ʻʻ\alpha .\\ +[\text{Hp}] &\supset .y\in R_{\text{po}}ʻʻ\alpha &\qquad \text{(4)}\\ +\vdash .(3).(4).*90·112.&\supset \vdash \colon\ldotp \text{Hp}.\supset :(BʻR)R_{*}y.\supset .y\in R_{\text{po}}ʻʻ\alpha &\qquad \text{(5)}\\ +\vdash .(5).*122·1.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p>The above proposition shows that if an existent class contained in a +progression has no maximum, then any assigned member of the progression +is succeeded by members of the class.</p> + +<p>The following proposition states that if \(\alpha\) has members +belonging to a progression, and there are members of the progression +which do not precede any member of \(\alpha\), then there is in the +progression a last member of \(\alpha\).</p> + +<p class="nind"><b>*122·27.</b> \[\begin{align}\vdash :R\in \text{Prog}.\exists !\text{D}ʻR-R_{\text{po}}ʻʻ\alpha .&\exists !\alpha \cap \text{D}ʻR.\supset .\\ +&\text{E}!\text{max}(R_{\text{po}})ʻ\alpha .\exists !\alpha \cap \text{D}ʻR-R_{\text{po}}ʻʻ\alpha \end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*122·26.\text{Transp}.*37·265. &\supset \vdash :\text{Hp}.\supset .\exists !\alpha \cap CʻR-R_{\text{po}}ʻʻ\alpha &\qquad \text{(1)}\\ +\vdash .*122·21. &\supset \vdash :\text{Hp}.x,y\in \alpha \cap CʻR-R_{\text{po}}ʻʻ\alpha .\supset .x=y &\qquad \text{(2)}\\ +\vdash .(1).(2).*93·115.*122·141. &\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*122·28.</b> \(\vdash :R\in \text{Prog}.\alpha \subset \overrightarrow{R}_{*}ʻx.\exists !\alpha .\supset .\text{E}!\text{max}(R_{\text{po}})ʻ\alpha + .\exists !\alpha \cap \text{D}ʻR-R_{\text{po}}ʻʻ\alpha\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*90·13.*122·141. &\supset \vdash :\text{Hp}.\supset .\alpha \subset \text{D}ʻR &\qquad \text{(1)}\\ +\vdash .*90·14.*122·141. &\supset \vdash :\text{Hp}.\supset .x\in \text{D}ʻR.\\ +[*71·161.*122·16] &\supset .\breve{R} ʻx{\sim}\in R_{\text{po}}ʻʻ\alpha .\\ +[*122·1] &\supset .\exists !\text{D}ʻR-R_{\text{po}}ʻʻ\alpha &\qquad \text{(2)}\\ +\vdash .(1).(2).*122·27.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_261">[Pg 261]</span></p> + +<p class="nind"><b>*122·3.</b> \[\begin{align}&\vdash :R\in \text{Prog}.\supset .\text{D}ʻR=\hat{x} \{(\exists \nu ).\nu \in \text{NC induct}.(BʻR)R_{v}x\}\\ +&[*121·52.*122·1·16]\end{align}\]</p> + +<p class="nind"><b>*122·31.</b> \(\vdash :R\in \text{Prog}.\nu \in \text{NC induct}-\iota ʻ0.\supset .\text{ᗡ}ʻR_{\nu }=\hat{y} \{\text{Nc}ʻR(BʻR\vdash\dashv y)>\nu\}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*120·429.\supset \vdash \colon\ldotp \text{Hp}.\supset :&\text{Nc}ʻR(BʻR\vdash\dashv y)>\nu .\equiv .\text{Nc}ʻR(BʻR\vdash\dashv y)\geq \nu +_{c}1.\\ +[*117·31] &\equiv .(\exists \mu ).\mu \in \text{NC}.\text{Nc}ʻR(BʻR\vdash\dashv y)=\mu +_{c}\nu +_{c}1.\\ +&\nu +_{c}1,\mu +_{c}\nu +_{c}1\in \text{N}_{0}\text{C}.\\ +[*121·45.*120·452.*110·4] &\equiv .(\exists \mu ).\mu \in \text{NC induct}.\\ +&\text{Nc}ʻR(BʻR\vdash\dashv y)=\mu +_{c}\nu +_{c}1.\mu +_{c}\nu +_{c}1\in \text{N}_{0}\text{C}.\\ +[*121·11·35.*110·43.*100·3] &\equiv .(\exists \mu ).\mu \in \text{NC induct}.(BʻR)R_{\mu }\mid R_{\nu }y.\\ +[*34·1] &\equiv .(\exists \mu ,x).\mu \in \text{NC induct}.(BʻR)R_{\mu }x.xR_{\nu }y.\\ +[*122·3] &\equiv .(\exists x).x\in \text{D}ʻR.xR_{\nu }y.\\ +[*121·323] &\equiv .(\exists x).xR_{\nu }y.\\ +[*33·131] &\equiv .y\in \text{ᗡ}ʻR_{\nu }\colon\ldotp \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*122·32.</b> \[\begin{align}\vdash :R\in \text{Prog}.\nu \in &\text{NC induct}-\iota ʻ0.\supset .\\ +&\overrightarrow{B}ʻR_{\nu }=\text{D}ʻR\cap \hat{x} \{\text{Nc}ʻR(BʻR\vdash\dashv x)\leq \nu\}\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*122·142.*121·501.&\supset \vdash :\text{Hp}.\supset .\text{D}ʻR_{\nu }=\text{D}ʻR &\qquad \text{(1)}\\ +\vdash .*122·31.*120·442.&\supset \vdash :\text{Hp}.\supset .-\text{ᗡ}ʻR_{\nu }=\hat{x} \{\text{Nc}ʻR(BʻR\vdash\dashv x)\leq \nu\} &\qquad \text{(2)}\\ +\vdash .(1).(2).*93·101.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*122·33">*122·33</a>.</b> \(\vdash :R\in \text{Prog}.\nu \in \text{NC induct}-\iota ʻ0.\supset .\text{E}!\nu _{R}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*121·601.*122·11. &\supset \vdash :\text{Hp}.\supset .\text{E}!1_{R} &\qquad \text{(1)}\\ +\vdash .*121·634·637.*122·141.&\supset \vdash \colon\ldotp \text{Hp}.\supset :\text{E}!\nu _{R}.\supset .\text{E}!(\nu +_{c}1)_{R} &\qquad \text{(2)}\\ +\vdash .(1).(2).*120·473.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*122·34">*122·34</a>.</b> \(\vdash \colon\ldotp R\in \text{Prog}.\supset :\nu \in \text{NC induct}-\iota ʻ0.\equiv .\text{E}!\nu _{R} \quad[*122·33.*121·635]\)</p> + +<p class="nind"><b><a id="*122·341">*122·341</a>.</b> \(\vdash :R\in \text{Prog}.\supset .\text{D}ʻR=\hat{x} \{(\exists \nu ).\nu \in \text{NC induct}-\iota ʻ0.x=\nu _{R}\}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*122·3·34.*121·638.\supset \\ +\vdash :\text{Hp}.\supset .\text{D}ʻR&=\hat{x} \{(\exists \nu ).\nu \in \text{NC induct}.x=(\nu +_{c}1)_{R}\}\\ +[*120·471] &=\hat{x} \{(\exists \nu ).\nu \in \text{NC induct}-\iota ʻ0.x=\nu _{R}\}:\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_262">[Pg 262]</span></p> + +<p>In virtue of <a href="#*122·34">*122·34</a><a href="#*122·341">·341</a>, all the terms of a progression occur in the +series \(1_{R}\), \(2_{R}\), ... \(\nu _{R}\), ..., and every inductive +cardinal except 0 is used in forming this series.</p> + +<p class="nind"><b><a id="*122·35">*122·35</a>.</b> \(\vdash :R\in \text{Prog}.\nu \in \text{NC induct}-\iota ʻ0.\supset .\overrightarrow{B}ʻR_{\nu }=R(1_{R}\vdash\dashv + \nu _{R}).\overrightarrow{B}ʻR_{\nu }\in \nu\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*121·63.*122·33.&\supset \vdash :\text{Hp}.\supset .\text{Nc}ʻR(BʻR\vdash\dashv \nu _{R})=\nu . &\qquad \text{(1)}\\ +[*122·32] \supset .\overrightarrow{B}ʻR_{\nu }&=\text{D}ʻR\cap \hat{x} \{\text{Nc}ʻR(BʻR\vdash\dashv x)\leq \text{Nc}ʻR(BʻR\vdash\dashv \nu _{R})\}\\ +[*121·481] &=\text{D}ʻR\cap \hat{x} \{R(BʻR\vdash\dashv x)\subset R(BʻR\vdash\dashv \nu _{R})\}\\ +[*122·1.*121·103] &=\hat{x} \{(BʻR)R_{*}x:yR_{*}x.\supset _{y}.yR_{*}\nu _{R}\}\\ +[*90·17·13.*10·1] &=\hat{x} \{(BʻR)R_{*}x.xR_{*}\nu _{R}\}\\ +[*121·103] &=R(BʻR\vdash\dashv \nu _{R}) &\qquad \text{(2)}\\ +[*121·601.*122·11] &=R(1_{R}\vdash\dashv \nu _{R}) &\qquad \text{(3)}\\ +\vdash .(1).(2).(3).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*122·36.</b> \(\vdash :\exists !\text{Prog}\cap t^{11}ʻx.\supset .\text{Infin ax}(x)\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*122·35. &\supset \vdash :R\in \text{Prog}\cap t^{11}ʻx.\nu \in \text{NC induct}-\iota ʻ0.\supset .\exists !\nu (x) &\qquad \text{(1)}\\ +\vdash .(1).*101·12. &\supset \vdash \colon\ldotp R\in \text{Prog}\cap t^{11}ʻx.\supset :\nu \in \text{NC induct}.\supset _{\nu }.\exists !\nu (x):\\ +[*120·301] &\supset :\text{Infin ax}(x)\colon\ldotp \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*122·37.</b> \(\vdash :R\in \text{Prog}.\supset .\text{D}ʻR{\sim}\in \text{Cls induct}.\text{N}_{0}\text{c}ʻ\text{D}ʻR{\sim}\in \text{NC induct}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*122·35. \supset \vdash \colon\ldotp R\in \text{Prog}.&\supset :\nu \in \text{NC induct}.\supset _{\nu }.\exists !\text{Cl}ʻ\text{D}ʻR\cap (\nu +_{c}1).\\ +[*117·22·107] &\supset _{\nu }.\text{N}_{0}\text{c}ʻ\text{D}ʻR\geq \nu +_{c}1.\\ +[*120·429] &\supset _{\nu }.\text{N}_{0}\text{c}ʻ\text{D}ʻR>\nu .\\ +[*117·42] &\supset _{\nu }.\text{N}_{0}\text{c}ʻ\text{D}ʻR\neq \nu :\\ +[*13·196] &\supset :\text{N}_{0}\text{c}ʻ\text{D}ʻR{\sim}\in \text{NC induct} &\qquad \text{(1)}\\ +\vdash .(1).*120·21.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*122·38">*122·38</a>.</b> \(\vdash :R\in \text{Prog}.\supset .\overrightarrow{R}_{*}ʻx\in \text{Cls induct} \quad[*121·7.*90·13.*120·212]\)</p> + +<p class="nind"><b><a id="*122·381">*122·381</a>.</b> \[\begin{align}&\vdash :R\in \text{Prog}.\nu \in \text{NC induct}-\iota ʻ0.\supset .\overrightarrow{R}_{*}ʻ\nu _{R}=R(1_{R}\vdash\dashv + \nu _{R}).\overrightarrow{R}_{*}ʻ\nu _{R}\in \nu \\ +&[*121·7.*122·35]\end{align}\]</p> + +<p><span class="pagenum" id="Page_263">[Pg 263]</span></p> + +<p>The following series of propositions are concerned in proving that +any class contained in a progression is inductive if it has a last +term, and is a progression if it has no last term. In the latter case, +it is supposed arranged in the same order as it had in the original +progression. A certain complication is necessary in order to define +its one-one generating relation. If \(R\) is the generating relation +of the original progression, we proceed first to \(R_{\text{po}}\), +then to \(R_{\text{po}}\unicode{x0294f}\alpha\), where \(\alpha\) is +the class in question; this gives us a transitive generating relation +for \(\alpha\). Calling this relation \(P\), we then proceed to +\(P\dot{-} P^{2}\), <i>i.e.</i> the relation of consecutive members of +the series generated by \(P\). This relation turns out to be one-one, +and to arrange \(\alpha\) in a progression; hence our proposition is +proved. The reason for the necessity of this detour is that consecutive +members of \(\alpha\) may not be consecutive members of the original +progression.</p> + +<p class="nind"><b>*122*41.</b> \(\vdash :R\in \text{Prog}.\alpha \subset \text{D}ʻR.y\in \alpha -R_{\text{po}}ʻʻ\alpha .\supset .\alpha \subset R(BʻR\vdash\dashv y)\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*37·1.*10·51.&\supset \vdash \colon\ldotp \text{Hp}.\supset :z\in \alpha .\supset _{z}.{\sim}(yR_{\text{po}}z).\\ +[*122·21] &\supset _{z}.zR_{*}y &\qquad \text{(1)}\\ +\vdash .*122·1. &\supset \vdash \colon\ldotp \text{Hp}.\supset :z\in \alpha .\supset _{z}.(BʻR)R_{*}z &\qquad \text{(2)}\\ +\vdash .(1).(2).*121·103.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*122·42.</b> \(\vdash :R\in \text{Prog}.\alpha \subset R(BʻR\vdash\dashv y).y\in \alpha .\supset .y= \text{max}_{R}ʻ\alpha\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*121·103.&\supset \vdash \colon\ldotp \text{Hp}.\supset :z\in \alpha .\supset _{z}.zR_{*}y.\\ +[*91·574.*122·16] &\supset _{z}.{\sim}(yR_{\text{po}}z):\\ +[*37·1.*10·51] &\supset :y{\sim}\in R_{\text{po}}ʻʻ\alpha : &\qquad \text{(1)}\\ +[*96·303] &\supset :z\in \alpha -R_{\text{po}}ʻʻ\alpha .\supset _{z}.z=y &\qquad \text{(2)}\\ +\vdash .(1).(2).*93·115.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*122·43">*122·43</a>.</b> \[\begin{align}&\vdash :R\in \text{Prog}.\alpha \subset \text{D}ʻR.\exists !\alpha -R_{\text{po}}ʻʻ\alpha .\supset .\alpha \in \text{Cls induct}\\ +&[*122·41.*121·45.*120·481]\end{align}\]</p> + +<p>Thus every class which is contained in a progression and has a last +term is inductive. We have next to prove +\[ +R\in \text{Prog} .\alpha \subset \text{D}ʻR.\exists !\alpha .{\sim}\exists !\alpha -R_{\text{po}}ʻʻ\alpha .\supset .\alpha \in \text{D}ʻʻ\text{Prog}. +\] +This is effected in the following propositions.</p> + +<p class="nind"><b>*122·44.</b> \[\begin{align}\vdash :R\in \text{Prog} .\alpha \subset R_{\text{po}}ʻʻ\alpha .\exists !\alpha .P=R_{\text{po}}\unicode{x0294f}\alpha .Q&=P\dot{-} P^{2}.\supset + .\\ +&Q\in 1\rightarrow 1.Q\,\unicode{x2abd}\, R_{\text{po}}\end{align}\]</p> + +<p><i>Note</i>. The hypothesis here exceeds what is necessary for the +conclusion, but is the hypothesis required for <a href="#*122·45">*122·45</a>, for which the +present and the following propositions are lemmas.</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*23·43.*35·442.\supset \vdash :\text{Hp}.&\supset .Q\,\unicode{x2abd}\, R_{\text{po}} &\qquad \text{(1)}\\ +\vdash .*36·13. \supset \vdash \colon\ldotp \text{Hp}.&\supset :x,y,z\in \alpha .xR_{\text{po}}y.yR_{\text{po}}z.\supset .xP^{2}z:\\ +[\text{Transp}] &\supset :x,y,z\in \alpha .xR_{\text{po}}y.{\sim}(xP^{2}z).\supset .{\sim}(yR_{\text{po}}z):\\ +[*36·13] &\supset :xPy.{\sim}(xP^{2}z).\supset .{\sim}(yR_{\text{po}}z):\\ +[*3·47] &\supset :xQy.xQz.\supset .{\sim}(yR_{\text{po}}z).{\sim}(zR_{\text{po}}y).\\ +[*122·21.(1)] &\supset .y=z &\qquad \text{(2)}\\ +\text{Similarly}\quad \vdash \colon\ldotp \text{Hp}.&\supset :xQz.yQz.\supset .x=y &\qquad \text{(3)}\\ +\vdash .(1).(2).(3).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*122·441.</b> \(\vdash :\text{Hp}*122·44.\supset .\text{D}ʻQ=\alpha\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*37·41.\supset \vdash :\text{Hp}.&\supset .\text{D}ʻQ\subset \alpha &\qquad \text{(1)}\\ +\vdash .*37·1. &\supset \vdash \colon\ldotp \text{Hp}.\supset :x\in \alpha .\supset .(\exists y).y\in \alpha .xR_{\text{po}}y.\\ +[*36·13] &\supset .\exists !\overleftarrow{P}ʻx.\\ +[*122·23.*93·11] &\supset .\exists !\overleftarrow{P}ʻx-\breve{R} _{\text{po}}ʻʻ\overleftarrow{P}ʻx.\\ +[*35.442] &\supset .\exists !\overleftarrow{P}ʻx-\breve{P} ʻʻ\overleftarrow{P}ʻx.\\ +[*37·311.*32·31·35] &\supset .\exists !\overleftarrow{Q}ʻx &\qquad \text{(2)}\\ +\vdash .(1).(2).*33·4.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_264">[Pg 264]</span></p> + +<p class="nind"><b>*122*442.</b> \(\vdash :\text{Hp}*122·44.\supset .P=Q_{\text{po}}\)</p> + +<p>In proving \(P\,\unicode{x2abd}\, Q_{\text{po}}\) below, we assume \(xPz\) +and consider the maximum of \(\overrightarrow{R}_{\text{po}}ʻz\cap\overleftarrow{Q}_{*}ʻx\), +which is shown to exist and be \(Qʻz\), whence \(xQ_{\text{po}}z\).</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *23·43.&\supset \vdash :\text{Hp}.\supset .Q\,\unicode{x2abd}\, P &\qquad \text{(1)}\\ +\vdash . *91·56.&\supset \vdash \colon\ldotp \text{Hp}.\supset :P^{2}\,\unicode{x2abd}\, P:\\ +[(1)] &\supset :S\,\unicode{x2abd}\, P.\supset .S\mid Q\,\unicode{x2abd}\, P &\qquad \text{(2)}\\ +\vdash .(1).(2).*91·171.*41·151.&\supset \vdash \colon\ldotp \text{Hp}.\supset .Q_{\text{po}}\,\unicode{x2abd}\, P &\qquad \text{(3)}\\ +\vdash .*36·13.*121·1.&\supset \vdash \colon\ldotp \text{Hp}.\supset :xP^{2}z.\equiv .x,z\in \alpha .\exists !\alpha \cap R(x-z):\\ +[\text{Transp}.\text{Fact}] &\supset :xQz.\equiv .x,z\in \alpha .xR_{\text{po}}z.\alpha \cap R(x-z)=\Lambda &\qquad \text{(4)}\\ +\vdash .*122·441.&\supset \vdash \colon\ldotp \text{Hp}.xPz.\supset :x\in (\overrightarrow{R}_{\text{po}}ʻz\cap \overleftarrow{Q}_{*}ʻx):\\ +[*122·27] &\supset :\exists !\overrightarrow{R}_{\text{po}}ʻz\cap \overleftarrow{Q}_{*}ʻx-R_{\text{po}}ʻʻ(\overrightarrow{R}_{\text{po}}ʻz\cap + \overleftarrow{Q}_{*}ʻx):\\ +[*37·461] &\supset :(\exists y).y\in \overrightarrow{R}_{\text{po}}ʻz\cap \overleftarrow{Q}_{*}ʻx.\overleftarrow{R}_{\text{po}}ʻy\cap \overrightarrow{R}_{\text{po}}ʻz\cap + \overleftarrow{Q}_{*}ʻx=\Lambda :\\ +[*90·151] &\supset :(\exists y).y\in \overrightarrow{R}_{\text{po}}ʻz\cap \overleftarrow{Q}_{*}ʻx.\overleftarrow{R}_{\text{po}}ʻy\cap \overrightarrow{R}_{\text{po}}ʻz\cap + \overleftarrow{Q}ʻy=\Lambda :\\ +[(4)] &\supset :(\exists y):y\in \overrightarrow{R}_{\text{po}}ʻz\cap \overleftarrow{Q}_{*}ʻx:\\ +&{\sim}(\exists w).w\in \overleftarrow{R}_{\text{po}}ʻy\cap \overrightarrow{R}_{\text{po}}ʻz.\alpha \cap \overleftarrow{R}_{\text{po}}ʻy\cap + \overrightarrow{R}_{\text{po}}ʻw=\Lambda :\\ +[*22·43.*91·56] &\supset :(\exists y).y\in \overrightarrow{R}_{\text{po}}ʻz\cap \overleftarrow{Q}_{*}ʻx:\\ +&{\sim}(\exists w).w\in \alpha \cap \overleftarrow{R}_{\text{po}}ʻy\cap \overrightarrow{R}_{\text{po}}ʻz.\alpha \cap \overleftarrow{R}_{\text{po}}ʻy\cap + \overrightarrow{R}_{\text{po}}ʻz\cap \overrightarrow{R}_{\text{po}}ʻw=\Lambda :\\ +[*37·461] &\supset :(\exists y).y\in \overrightarrow{R}_{\text{po}}ʻz\cap \overleftarrow{Q}_{*}ʻx.\\ +&{\sim}\exists !\alpha \cap :\overleftarrow{R}_{\text{po}}ʻy\cap \overrightarrow{R}_{\text{po}}ʻz-\breve{R} _{\text{po}}ʻʻ(\alpha \cap \overleftarrow{R}_{\text{po}}ʻy\cap + \overrightarrow{R}_{\text{po}}ʻz):\\ +[*122*28.\text{Transp}] &\supset :(\exists y).y\in \overrightarrow{R}_{\text{po}}ʻz\cap \overleftarrow{Q}_{*}ʻx.\alpha \cap \overleftarrow{R}_{\text{po}}ʻy\cap + \overrightarrow{R}_{\text{po}}ʻz=\Lambda :\\ +[(4)] &\supset :(\exists y).y\in \overleftarrow{Q}_{*}ʻx.yQz:\\ +[*91·52] &\supset :xQ_{\text{po}}z &\qquad \text{(5)}\\ +\vdash .(3).(5).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*122·443.</b> \(\vdash :\text{Hp}*122·44.\supset .\text{min}(R_{\text{po}})ʻ\alpha =BʻQ.\text{ᗡ}ʻQ=\alpha \cap \breve{R} _{\text{po}}ʻʻ\alpha \)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *91·504.*122·442. \supset \vdash :\text{Hp}.\supset .\text{ᗡ}ʻQ &= \text{ᗡ}ʻP\\ +[*37·41] &=\alpha \cap \breve{R} _{\text{po}}ʻʻ\alpha &\qquad \text{(1)}\\ +\vdash .(1).*122·441. &\supset \vdash :\text{Hp}.\supset .\overrightarrow{B}ʻQ=\alpha -\breve{R} _{\text{po}}ʻʻ\alpha &\qquad \text{(2)}\\ +\vdash .(1).(2).*122·23. &\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*122·444.</b> \(\vdash :\text{Hp}*122·44.\supset .\text{D}ʻQ=\overleftarrow{Q}_{*}ʻBʻQ\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*122·443.*14·21.\supset \vdash :\text{Hp}.&\supset .\text{Hp}.\supset .\text{E}!BʻQ.\\ +[*90·13] &\supset .\overleftarrow{Q}_{*}ʻBʻQ\subset CʻQ.\\ +[*122·441·443] &\supset .\overleftarrow{Q}_{*}ʻBʻQʻ\subset \alpha &\qquad \text{(1)}\\ +\vdash .*122·443.*96·303.&\supset \\ +\vdash :\text{Hp}.x\in \alpha .x\neq BʻQ.&\supset .(BʻQ)R_{\text{po}}x.BʻQ.x\in \alpha .\\ +[\text{Hp}] &\supset .(BʻQ)Px.\\ +[*122·442] &\supset .(BʻQ)Q_{\text{po}}x &\qquad \text{(2)}\\ +\vdash .(2).*91·54.&\supset \vdash :\text{Hp}.x\in \alpha .\supset .(BʻQ)Q_{*}x &\qquad \text{(3)}\\ +\vdash .(1).(3). \supset \vdash :\text{Hp}.\supset .\overleftarrow{Q}_{*}ʻBʻQ&=\alpha \\ +[*122·441] &= \text{D}ʻQ:\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_265">[Pg 265]</span></p> + +<p class="nind"><b><a id="*122·45">*122·45</a>.</b> \[\begin{align}&\vdash :R\in \text{Prog}.\alpha \subset R_{\text{po}}ʻʻ\alpha .\exists !\alpha .P = R_{\text{po}}\unicode{x0294f}\alpha .Q = P\dot{-} P^{2}.\supset + .\\ +&Q\in \text{Prog}.\text{D}ʻQ = \alpha \quad[*122·44·444·441]\end{align}\]</p> + +<p>This proposition shows that every series extracted from a progression +and having no last term is a progression.</p> + +<p class="nind"><b><a id="*122·46">*122·46</a>.</b> \[\begin{align}&\vdash :R\in \text{Prog}.\alpha \subset \text{D}ʻR.\supset :\alpha \in \text{Cls induct} \cup \text{D}ʻʻ\text{Prog}\\ +&[*122·43·45.*120·212]\end{align}\]</p> + +<p>This proposition shows that any number less than the number of terms in +a progression is inductive. This result will be developed in the next +number (<a href="#*123">*123</a>).</p> + +<p class="nind"><b>*122·47.</b> \(\vdash \colon\ldotp R\in \text{Prog}.\alpha \subset \text{D}ʻR.\supset :\alpha \in \text{Cls induct} - \iota ʻ\Lambda .\equiv .\exists !\alpha -R_{\text{po}}ʻʻ\alpha + \)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*122·45.&\supset \vdash :\text{Hp}.\exists !\alpha .{\sim}\exists !\alpha -R_{\text{po}}ʻʻ\alpha .\supset .\alpha \in \text{D}ʻʻ\text{Prog}.\\ +[*122·37] &\supset .\alpha {\sim}\in \text{Cls induct} &\qquad \text{(1)}\\ +\vdash .(1).*122·43.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*122·48.</b> \(\vdash :R\in \text{Prog}.\alpha \subset \text{D}ʻR.\alpha \in \text{Cls induct}.\supset .\text{D}ʻR-\alpha {\sim}\in \text{Cls induct}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*120·71.&\supset \vdash :\alpha \subset \text{D}ʻR.\alpha ,\text{D}ʻR-\alpha \in \text{Cls induct}.\supset .\text{D}ʻR\in \text{Cls induct}:\\ +[\text{Transp}] &\supset \vdash :\alpha \subset \text{D}ʻR.\alpha \in \text{Cls induct}.\text{D}ʻR{\sim}\in \text{Cls induct}.\supset .\\ +&\text{D}ʻR-\alpha {\sim}\text{Cls induct} &\qquad \text{(1)}\\ +\vdash .(1).*122·37.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*122·49.</b> \[\begin{align}&\vdash :R\in \text{Prog}.\alpha \subset \text{D}ʻR.\alpha \in \text{Cls induct}.\supset .\text{D}ʻR-\alpha \in \text{D}ʻʻ\text{Prog}\\ +&[*122·46·48]\end{align}\]</p> + +<p>The following propositions are concerned with circumstances under which +the posterity or the family of a term forms a progression.</p> + +<p class="nind"><b><a id="*122·51">*122·51</a>.</b> \(\vdash :P\in \text{Cls}\rightarrow 1.I_{P}ʻx=\Lambda .x\in \text{D}ʻP.\overleftarrow{P}_{*}ʻx\subset \text{D}ʻP.\supset .(\overleftarrow{P}_{*}ʻx)\upharpoonleft + P\in \text{Prog}\)</p> + +<p>Here \(I_Pʻx\) has the meaning defined in *96.</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*71·261.*96·13.\supset \vdash :\text{Hp}.Q&=(\overleftarrow{P}_{*}ʻx)\upharpoonleft P.\supset .\\ +&Q\in \text{Cls}\rightarrow 1.Q_{\text{po}}=(\overleftarrow{P}_{*}ʻx)\upharpoonleft P_{\text{po}}. &\qquad \text{(1)}\\ +[*96·104] &\supset .Q_{\text{po}}\,\unicode{x2abd}\, J &\qquad \text{(2)}\\ +\vdash .*35·61.*37·4. \supset \vdash :\text{Hp}(1).\supset .\text{D}ʻQ=\overleftarrow{P}_{*}ʻx.\text{ᗡ}ʻQ&=\breve{P} ʻʻ\overleftarrow{P}_{*}ʻx &\qquad \text{(3)}\\ +[*91·52] &=\overleftarrow{P}_{\text{po}}ʻx\\ +[(1)] &=\overleftarrow{Q}_{\text{po}}ʻx\\ +[(2).*91·542] &=\overleftarrow{Q}_{*}ʻx-\iota ʻx &\qquad \text{(4)}\\ +\vdash .(1).(3).(4). &\supset \vdash :\text{Hp}(1).\supset .\text{D}ʻQ=\overleftarrow{Q}_{*}ʻx.\text{ᗡ}ʻQ=\overleftarrow{Q}_{*}ʻx-\iota ʻx.\\ +[*93·101] &\supset .\overrightarrow{B}ʻQ=\iota ʻx.\text{D}ʻQ=\overleftarrow{Q}_{*}ʻx &\qquad \text{(5)}\\ +\vdash .(1).(2).(5).\supset \\ +\vdash :\text{Hp}.Q&=(\overleftarrow{P}_{*}ʻx)\upharpoonleft P.\supset .Q\in \text{Cls} \rightarrow 1.Q_{\text{po}}\,\unicode{x2abd}\, J.\text{D}ʻQ=\overleftarrow{Q}_{*}ʻBʻQ.\\ +[*122·17] &\supset .Q\in \text{Prog} :\supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_266">[Pg 266]</span></p> + +<p>The following proposition (*122·52) is used in <a href="#*123·191">*123·191</a>, *261·4 and +*264·22.</p> + +<p class="nind"><b><a id="*122·52">*122·52</a>.</b> \(\vdash :P\in 1\rightarrow 1.x\in \text{D}ʻP.{\sim}(xP_{\text{po}}x).\overleftarrow{P}_{*}ʻx\subset \text{D}ʻP.\supset .(\overleftarrow{P}_{*}ʻx)\upharpoonleft + P\in \text{Prog}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*96·492.\supset \vdash :\text{Hp}.\supset .I_{P}ʻx=\Lambda &\qquad \text{(1)}\\ +\vdash .(1).*122·51.\supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p>The remaining propositions (<a href="#*122·53">*122·53</a><a href="#*122·54">·54</a><a href="#*122·55">·55</a>) are not used in the sequel.</p> + +<p class="nind"><b><a id="*122·53">*122·53</a>.</b> \(\vdash :P\in 1\rightarrow 1.x\in sʻ \text{gen}ʻP.\overleftarrow{P}_{*}ʻx\subset \text{D}ʻP.\supset .(\overleftrightarrow{P}_{*}ʻx)\upharpoonleft + P\in \text{Prog}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*97·21.&\supset \vdash :\text{Hp}.\supset .(\exists y).yBP.\overleftrightarrow{P}_{*}ʻx=\overleftarrow{P}_{*}ʻy.\\ +[*96·23.*93·1] &\supset .(\exists y).y\in \text{D}ʻP.\overleftrightarrow{P}_{*}ʻx=\overleftarrow{P}_{*}ʻy.I_{P}ʻy=\Lambda .\\ +[*97·17.*91·504.\text{Hp}] &\supset .(\exists y).y\in \text{D}ʻP.\overleftarrow{P}_{*}ʻy\subset \text{D}ʻP.I_{P}ʻy=\Lambda + .\overleftrightarrow{P}_{*}ʻx=\overleftarrow{P}_{*}ʻy.\\ +[*122·51] &\supset .(\overleftrightarrow{P}_{*}ʻx)\upharpoonleft P\in \text{Prog} :\supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*122·54">*122·54</a>.</b> \(\vdash :P\in 1\rightarrow 1.x\in sʻ \text{gen}ʻP-sʻ \text{gen}ʻ\breve{P} .\supset .(\overleftrightarrow{P}_{*}ʻx)\upharpoonleft P\in \text{Prog}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*93·27·272.\supset \vdash :\text{Hp}.&\supset .x\in sʻ \text{gen}ʻP\cap pʻ\text{ᗡ}ʻʻ \text{Pot} ʻ\breve{P} .\\ +[*93·381] &\supset .x\in sʻ \text{gen}ʻP.\overleftarrow{P}_{*}ʻx\subset \text{D}ʻP &\qquad \text{(1)}\\ +\vdash .(1).*122·53.\supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*122·55">*122·55</a>.</b> \(\vdash \colon\ldotp P\in 1\rightarrow 1.\supset :x\in sʻ \text{gen}ʻP-sʻ \text{gen}ʻ\breve{P} .\equiv .(\overleftrightarrow{P}_{*}ʻx)\upharpoonleft + P\in \text{Prog}\)</p> + +<p><span class="pagenum" id="Page_267">[Pg 267]</span></p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*35·61. &\supset \vdash :Q=(\overleftrightarrow{P}_{*}ʻx)\upharpoonleft P.\supset .\text{D}ʻQ=\overleftrightarrow{P}_{*}ʻx\cap + \text{D}ʻP &\qquad \text{(1)}\\ +\vdash .*37·4. &\supset \vdash \colon\ldotp Q=(\overleftrightarrow{P}_{*}ʻx)\upharpoonleft P.\supset :\text{ᗡ}ʻQ=\breve{P} ʻʻ\overleftrightarrow{P}_{*}ʻx:\\ +[*97·17.*92·111.*91·54·52] &\supset :Q\in 1\rightarrow 1.\supset .\text{ᗡ}ʻQ=\overleftrightarrow{P}_{*}ʻx\cap \text{ᗡ}ʻP &\qquad \text{(2)}\\ +\vdash .(1).(2). \supset \vdash \colon\ldotp \text{Hp}.\text{Hp}(1).&\supset :\exists !\overrightarrow{B}ʻQ.\supset .\exists !\overleftrightarrow{P}_{*}ʻx-\text{ᗡ}ʻP .\\ +[*97·17.*91·504] &\supset .\exists !\overrightarrow{P}_{*}ʻx-\text{ᗡ}ʻP.\\ +[*93·38·27] &\supset .x\in sʻ\text{gen}ʻP &\qquad \text{(3)}\\ +\vdash .(1).(2).\supset \vdash \colon\ldotp \text{Hp}(3).&\supset :\text{D}ʻQ=CʻQ.\supset .\overleftrightarrow{P}_{*}ʻx\cap \text{D}ʻP=\overleftrightarrow{P}_{*}ʻx.\\ +[*22·621] &\supset .\overleftrightarrow{P}_{*}ʻx\subset \text{D}ʻP.\\ +[*97·13] &\supset .\overleftarrow{P}_{*}ʻx\subset \text{D}ʻP.\\ +[*93·381·275] &\supset .x{\sim}\in sʻ\text{gen}ʻ\breve{P} &\qquad \text{(4)}\\ +\vdash .(3).(4).*122·11·141·54.\supset \vdash .\text{Prop} +\end{array} +\]</p> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_268">[Pg 268]</span></p> +<h2 class="nobreak" id="*123">*123. \(\aleph _{0}\).</h2> +</div> + + +<p class="nind"><i>Summary of</i> *123.</p> + +<p>In this number we are concerned with the arithmetical properties of +\(\aleph _{0}\), the smallest of Cantor's transfinite cardinals. +Cantor defines \(\aleph _{0}\) as the cardinal number of any class +which can be put into one-one relation with the inductive cardinals. +This definition assumes that \(\nu \neq \nu +_{c}1\), when \(\nu\) +is an inductive cardinal; in other words, it assumes the axiom of +infinity; for without this, the inductive cardinals would form a +finite series, with a last term, namely \(\Lambda\). For this reason +among others, we do not make similarity with the inductive cardinals +our <i>definition</i>. We define \(\aleph _{0}\) as the class of +those classes which can be arranged in progressions, <i>i.e.</i> as +\(\text{D}ʻʻ\text{Prog}\). We then have to prove that \(\aleph _{0}\) +so defined is a cardinal, and that if it is not null, it is the number +of the inductive numbers.</p> + +<p>For convenience we put for the moment \(N\) for the relation of \(\mu\) +to \(\mu +_{c}1\) when \(\mu\) is an inductive cardinal. We then easily +prove</p> + +<p class="nind"><b>*123·21·23.</b> \(\vdash .N\in \text{Cls}\rightarrow 1.\text{D}ʻN=\text{NC induct}.BʻN=0.\overleftarrow{N}_{*}ʻ0=\text{NC induct}\)</p> + +<p>The only thing further required to prove \(N\in \text{Prog}\) is \(N\in +1\rightarrow \text{Cls}\), <i>i.e.</i> +\[ +\mu ,\nu \in \text{NC induct}.\mu +_{c}1=\nu +_{c}1.\supset .\mu =\nu . +\]</p> + +<p>By <a href="#*120·311">*120·311</a>, this holds if \(\exists !\mu +_{c}1\), which holds if +\(\text{Infin ax}\) holds. Hence</p> + +<p class="nind"><b>*123·25·26.</b> \(\vdash :\text{Infin ax}(x).\supset .N\unicode{x0294f}t^{3}ʻx\in \text{Prog}.\text{NC induct}\cap t^{3}ʻx\in \aleph _{0}\)</p> + +<p class="nind">whence, by <a href="#*123·36">*123·36</a>,</p> + +<p class="nind"><b>*123·27.</b> \(\vdash :\exists !\aleph _{0}(x).\supset .\text{NC induct}\cap t^{3}ʻx\in \aleph _{0}\)</p> + +<p>Again it is obvious from <a href="#*122·34">*122·34</a><a href="#*122·341">·341</a> that if \(R\) is a progression, +\(\text{D}ʻR\) can always be put into a \(1\rightarrow 1\) relation +to the inductive cardinals (<a href="#*123·3">*123·3</a>) since \(\text{D}ʻR\) consists of +the terms \(1_{R}\), \(2_{R}\), ... \(\nu _{R}\), ..., and all the +inductive cardinals are used in putting \(\text{D}ʻR\) into this form. +Hence</p> + +<p class="nind"><b>*123·31.</b> \(\vdash :\alpha \in \aleph _{0}.\supset .\alpha \text{ sm }\text{NC induct}\)</p> + +<p class="nind">whence also</p> + +<p class="nind"><b>*123·311.</b> \(\vdash :\alpha ,\beta \in \aleph _{0}.\supset .\alpha \text{ sm }\beta\)</p> + +<p>It remains to prove that any class similar to the inductive cardinals +is an \(\aleph _{0}\); this can only be proved by assuming the axiom +of infinity. We prove<span class="pagenum" id="Page_269">[Pg 269]</span> first (<a href="#*120·32">*120·32</a>) that if \(R\) is a progression, +and \(S\) is a one-one whose converse domain is \(\text{D}ʻR\), +then \(S\mid R\mid \breve{S}\) is a progression whose domain is +\(\text{D}ʻS\). Hence</p> + +<p class="nind"><b>*123·321.</b> \(\vdash :\alpha \in \aleph _{0}.\alpha \text{ sm }\beta .\supset .\beta \in \aleph _{0}\)</p> + +<p>From this and \(\alpha\), \(\beta \in \aleph _{0}.\supset .\alpha\text{ sm }\beta\), +we obtain</p> + +<p class="nind"><b>*123·322.</b> \(\vdash :\alpha \in \aleph _{0}.\supset .\aleph _{0}=\text{Nc}ʻ\alpha\)</p> + +<p>Hence by our previous results</p> + +<p class="nind"><b>*123·34.</b> \(\vdash :\text{Infin ax}(x).\supset .\aleph _{0}=\text{Nc}ʻ(\text{NC induct}\cap t^{3}ʻx)\)</p> + +<p>Also we have, by <a href="#*120·322">*120·322</a> above, +\[ +\exists !\aleph _{0}.\supset .\aleph _{0}\in \text{NC}, +\] +whence, since \(\Lambda \in \text{NC}\), we obtain at last</p> + +<p class="nind"><b>*123·36.</b> \(\vdash .\aleph _{0}\in \text{NC}\)</p> + +<p>As to the existence of \(\aleph _{0}\) in various types, if +\(\text{Infin ax}(x)\) holds, <i>i.e.</i> if, given any inductive +cardinal \(\nu\), there are classes having \(\nu\) terms and composed +of terms of the same type as \(x\), then \(\text{NC induct}(tʻx)\in\aleph _{0}(t^{2}ʻx)\). +Thus</p> + +<p class="nind"><b>*123·37.</b> \(\vdash :\text{Infin ax}(x).\supset .\exists !\aleph _{0}(t^{2}ʻx).\aleph _{0}(t^{2}ʻx)\in \text{N}_{0}\text{C}\)</p> + +<p>The arithmetical properties of \(\aleph _{0}\) in regard to addition, +multiplication and exponentiation by an inductive cardinal are easily +proved. We have</p> + +<p class="nind"><b>*123·41.</b> \(\vdash :\nu \in \text{NC induct}.\supset .\aleph _{0}=\aleph _{0}+_{c}\nu\)</p> + +<p class="nind"><b>*123·421.</b> \(\vdash .\aleph _{0}=\aleph _{0}+_{c}\aleph _{0}=2\times _{c}\aleph _{0}\)</p> + +<p class="nind"><b>*123·422.</b> \(\vdash :\nu \in \text{NC induct}-\iota ʻ0.\supset .\nu \times _{c}\aleph _{0}=\aleph _{0}\)</p> + +<p class="nind"><b>*123·52.</b> \(\vdash .\aleph _{0}=\aleph _{0}\times _{c}\aleph _{0}=\aleph _{0}^{2}\)</p> + +<p class="nind"><b>*123·53.</b> \(\vdash :\nu \in \text{NC induct}-\iota ʻ0.\supset .\aleph _{0}^\nu =\aleph _{0}\)</p> + +<p>All these propositions are well known.</p> + +<p>The early propositions of the present number are for the most part +immediate consequences of propositions proved in <a href="#*122">*122</a>.</p> + +<hr class="tb"> + +<p class="nind"><b><a id="*123·01">*123·01</a>.</b> \(\aleph _{0}=\text{D}ʻʻ\text{Prog} \quad\text{Df}\)</p> + +<p class="nind"><b><a id="*123·02">*123·02</a>.</b> \(N=\hat{\mu} \hat{\nu} \{\mu \in \text{NC induct}.\nu =(\mu +_{c}1)\cap t_{0}ʻ\mu\} \quad\text{Dft} [*123—4]\)</p> + +<p class="nind"><b>*123·1.</b> \(\vdash :\alpha \in \aleph _{0}.\equiv .(\exists R).R\in \text{Prog}.\alpha =\text{D}ʻR \quad[*37·1.(*123·01)]\)</p> + +<p class="nind"><b>*123·101.</b> \(\vdash :R\in \text{Prog}.\supset .\text{D}ʻR\in \aleph _{0} \quad[*123·1]\)</p> + +<p class="nind"><b>*123·11.</b> \(\vdash :R\in 1\rightarrow 1.\text{D}ʻR=\overleftarrow{R}_{*}ʻBʻR.\supset .\text{D}ʻR\in \aleph _{0} \quad[*123·101.*122·1]\)</p> + +<p class="nind"><b>*123·12.</b> \[\begin{align}\vdash :\alpha \in \aleph _{0}.\supset .(\exists R).\text{D}ʻR=\alpha .R\in 1\rightarrow 1.\text{ᗡ}ʻR&\subset \text{D}ʻR.\overrightarrow{B}ʻR\in 1\\ +&[*123·1.*122·141·11]\end{align}\]</p> + +<p><span class="pagenum" id="Page_270">[Pg 270]</span></p> + +<p class="nind"><b>*123·13.</b> \(\vdash :\alpha \in \aleph _{0}.\supset .\text{Nc}ʻ\alpha =\text{Nc}ʻ\alpha +_{c}1\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*123·12.*110·32.&\supset \\ +\vdash :\alpha \in \aleph _{0}.&\supset .(\exists R).\text{D}ʻR=\alpha .R\in 1\rightarrow 1.\text{Nc}ʻ\text{D}ʻR=\text{Nc}ʻ\text{ᗡ}ʻR+_{c}1.\\ +[*100·321] &\supset .(\exists R).\text{D}ʻR=\alpha .\text{Nc}ʻ\text{D}ʻR=\text{Nc}ʻ\text{D}ʻR+_{c}1.\\ +[*35·94.*13·195] &\supset .\text{Nc}ʻ\alpha =\text{Nc}ʻ\alpha +_{c}1:\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*123·14.</b> \(\vdash :\alpha \in \aleph _{0}.\nu \in \text{NC induct}.\supset .\exists !\nu \cap \text{Cl}ʻ\alpha \quad[*122·35]\)</p> + +<p class="nind"><b>*123·15.</b> \(\vdash :\alpha \in \aleph _{0}.\supset .\alpha {\sim}\in \text{Cls induct} \quad[*122·37]\)</p> + +<p class="nind"><b>*123·16.</b> \(\vdash :\alpha \in \aleph _{0}.\supset .\text{Cl}ʻ\alpha \subset \text{Cls induct}\cup \aleph _{0} \quad[*122·46]\)</p> + +<p class="nind"><b>*123·17.</b> \(\vdash :\alpha \in \aleph _{0}.\beta \in \text{Cls induct}.\supset .\alpha -\beta \in \aleph _{0}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*120·481.\supset \vdash :\text{Hp}.&\supset .\alpha \cap \beta \in \text{Cls induct}.\\ +[*122·49] &\supset .\alpha -(\alpha \cap \beta )\in \aleph _{0}:\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*123·18.</b> \(\vdash :\exists !\aleph _{0}(x).\supset .\text{Infin ax}(x) \quad[*122·36]\)</p> + +<p class="nind"><b><a id="*123·19">*123·19</a>.</b> \(\vdash :R\in \text{Prog}.\exists !\alpha .\alpha \subset R_{\text{po}}ʻʻ\alpha .\supset .\alpha \in \aleph _{0} \quad[*122·45]\)</p> + +<p class="nind"><b><a id="*123·191">*123·191</a>.</b> \[\begin{align}&\vdash :R\in 1\rightarrow 1.x\in \text{D}ʻR.{\sim}(xR_{\text{po}}x).&\overleftarrow{R}_{*}ʻx\subset \text{D}ʻR.\supset .\overleftarrow{R}_{*}ʻx\in + \aleph _{0}\\ +&[*122·52]\end{align}\]</p> + +<p class="nind"><b>*123·192.</b> \(\vdash :R\in 1\rightarrow 1.\text{ᗡ}ʻR\subset \text{D}ʻR.\supset .\overleftarrow{R}_{*}ʻʻ\overrightarrow{B}ʻR\subset \aleph _{0}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*93·101. &\supset \vdash :x\in \overrightarrow{B}ʻR.\supset .x\in \text{D}ʻR &\qquad \text{(1)}\\ +\vdash .*91·504.*93·101. &\supset \vdash :x\in \overrightarrow{B}ʻR.\supset .{\sim}(xR_{\text{po}}x) &\qquad \text{(2)}\\ +\vdash .*90·13. &\supset \vdash :\text{ᗡ}ʻR\subset \text{D}ʻR.\supset .\overleftarrow{R}_{*}ʻx\subset \text{D}ʻR &\qquad \text{(3)}\\ +\vdash .(1).(2).(3).*123·191. &\supset \vdash :\text{Hp}.x\in \overrightarrow{B}ʻR.\supset .\overleftarrow{R}_{*}ʻx\in \aleph _{0}:\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*123·2.</b> \(\vdash :\mu N\nu .\equiv .\mu \in \text{NC induct}.\nu =(\mu +_{c}1)\cap t_{0}ʻ\mu \quad[(*123·02)]\)</p> + +<p class="nind"><b>*123·21.</b> \(\vdash .N\in \text{Cls}\rightarrow 1.\text{D}ʻN=\text{NC induct}.\text{ᗡ}ʻN=\text{NC induct}-\iota ʻ0.BʻN=0\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*123·2.*13·172. &\supset \vdash :\mu N\nu .\mu N\varpi .\supset .\nu =\varpi :\\ +[*71·171] &\supset \vdash .N\in \text{Cls}\rightarrow 1 &\qquad \text{(1)}\\ +\vdash .*123·2. &\supset \vdash .\text{D}ʻN=\text{NC induct} &\qquad \text{(2)}\\ +\vdash .*123·2. \supset \vdash .\text{ᗡ}ʻN&=\hat{\nu} \{(\exists \mu ).\mu \in \text{NC induct}.\nu =\mu +_{c}1\}\\ +[*120·423] &=\text{NC induct}-\iota ʻ0 &\qquad \text{(3)}\\ +\vdash .(2).(3).*93·101.&\supset \vdash .BʻN=0 &\qquad \text{(4)}\\ +\vdash .(1).(2).(3).(4).&\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_271">[Pg 271]</span></p> + +<p class="nind"><b>*123·22.</b> \(\vdash .\breve{N} =(+_{c}1)\upharpoonright \text{NC induct} \quad[*123·2]\)</p> + +<p class="nind"><b>*123·23.</b> \(\vdash .\overleftarrow{N}_{*}ʻ0=\text{NC induct}=\text{D}ʻN\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*123·22.\supset \vdash .\overleftarrow{N}_{*}ʻ0&=\hat{\mu} [\mu \{(+_{c}1)\upharpoonright \text{NC induct}\}_{*}0]\\ +[*120·1.*96·21·131]&=\hat{\mu} [\mu \{\text{NC induct}\upharpoonleft (+_{c}1)_{*}\}0]\\ +[*120·1] &=\text{NC induct} &\qquad \text{(1)}\\ +\vdash .(1).*123·21.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*123·24.</b> \(\vdash :\text{Infin ax}(x).\supset .N\unicode{x0294f}t^{3}ʻx\in 1\rightarrow 1\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*120·301·121.\supset \vdash \colon\colon \text{Hp}.\supset \colon\ldotp \mu \in &\text{NC induct}.\supset :\exists !(\mu +_{c}1)\cap t^{2}ʻx:\\ +[*120·311] &\supset :(\mu +_{c}1)\cap t^{2}ʻx=\nu +_{c}1.\supset .\mu =\nu :\\ +[*123·2.*71·17] &\supset :N\unicode{x0294f}t^{3}ʻx\in 1\rightarrow \text{Cls} &\qquad \text{(1)}\\ +\vdash .(1).*123·21.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*123·25.</b> \(\vdash :\text{Infin ax}(x).\supset .N\unicode{x0294f}t^{3}ʻx\in \text{Prog} \quad[*123·21·23·24.*122·1]\)</p> + +<p class="nind"><b>*123·26.</b> \(\vdash :\text{Infin ax}(x).\supset .\text{NC induct}\cap t^{3}ʻx\in \aleph _{0} \quad[*123·25·21·101]\)</p> + +<p class="nind"><b>*123·27.</b> \(\vdash :\exists !\aleph _{0}(x).\supset .\text{NC induct}\cap t^{3}ʻx\in \aleph _{0} \quad[*123·26·18]\)</p> + +<p class="nind"><b><a id="*123·3">*123·3</a>.</b> \[\begin{align}\vdash :R\in \text{Prog}.&S=\hat{x} \hat{\nu} \{\nu \in \text{NC induct}.x=(\nu +_{c}1)_{R}\}.\supset .\\ +&S\in 1\rightarrow 1.\text{D}ʻS=\text{D}ʻR.\text{ᗡ}ʻS=\text{NC induct}\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*120·423.\supset \vdash :\text{Hp}.&\supset .\text{D}ʻS=\hat{x} \{(\exists \mu ).\mu \in \text{NC induct}-\iota ʻ0.x=\mu _{R}\}\\ +[*122·341] &=\text{D}ʻR &\qquad \text{(1)}\\ +\vdash .*14·204.*122·34.&\supset \vdash :\text{Hp}.\supset .\text{ᗡ}ʻS=\hat{\nu} \{\text{E}!(\nu +_{c}1)_{R}\}\\ +[*122·34] &=\hat{\nu} \{\nu +_{c}1\in \text{NC induct}-\iota ʻ0\} &\qquad \text{(2)}\\ +\vdash .*122·36.*120·3. &\supset \vdash \colon\ldotp \text{Hp}.\supset :\nu +_{c}1\in \text{NC induct}.\supset .\exists !\nu +_{c}1.\\ +[*120·422] &\supset .\nu \in \text{NC induct} &\qquad \text{(3)}\\ +\vdash .(3).*120·421·121.&\supset \vdash \colon\ldotp \text{Hp}.\supset :\nu +_{c}1\in \text{NC induct}-\iota ʻ0.\equiv .\\ +&\nu \in \text{NC induct} &\qquad \text{(4)}\\ +\vdash .(2).(4). &\supset \vdash :\text{Hp}.\supset .\text{ᗡ}ʻS=\text{NC induct} &\qquad \text{(5)}\\ +\vdash .*13·172.*71·17. &\supset \vdash :\text{Hp}.\supset .S\in 1\rightarrow \text{Cls} &\qquad \text{(6)}\\ +\vdash .*121·631. &\supset \vdash \colon\ldotp \text{Hp}.\supset :xS\mu .xS\nu .\supset .\\ +&\text{Nc}ʻR(BʻR\vdash\dashv x)=\mu +_{c}1.\text{Nc}ʻR(BʻR\vdash\dashv x)=\nu +_{c}1.\\ +[*13·171] &\supset .\mu +_{c}1=\nu +_{c}1 &\qquad \text{(7)}\\ +\vdash .(5).*122·36.*120·3.&\supset \vdash \colon\ldotp \text{Hp}.\supset :xS\mu .\supset .\exists !\mu +_{c}1:\\ +[*120·41] &\supset :xS\mu .\mu +_{c}1=\nu +_{c}1.\supset .\mu =\nu :\\ +[(7)] &\supset :xS\mu .xS\nu .\supset .\mu =\nu :\\ +[*71·171] &\supset :S\in \text{Cls}\rightarrow 1 &\qquad \text{(8)}\\ +\vdash .(1).(5).(6).(8).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_272">[Pg 272]</span></p> + +<p class="nind"><b>*123·31.</b> \(\vdash :\alpha \in \aleph _{0}.\supset .\alpha \text{ sm }\text{NC induct} \quad[*123·3]\)</p> + +<p class="nind"><b>*123·311.</b> \(\vdash :\alpha ,\beta \in \aleph _{0}.\supset .\alpha \text{ sm }\beta \quad[*123·31.*73·31·32]\)</p> + +<p>It is not assumed here that \(\alpha\) and \(\beta\) are of the same +type.</p> + +<p class="nind"><b>*123·312.</b> \[\begin{align}\vdash :R\in &\text{Prog}.S\in 1\rightarrow 1.\text{ᗡ}ʻS=\text{D}ʻR.\supset .\\ +&S\mid R\mid \breve{S} \in 1\rightarrow 1.\text{D}ʻS=\text{D}ʻ(S\mid R\mid \breve{S} ).SʻBʻR=Bʻ(S\mid R\mid \breve{S} )\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*71·252.*122·1. &\supset \vdash :\text{Hp}.\supset .S\mid R\mid \breve{S} \in 1\rightarrow 1 &\qquad \text{(1)}\\ +\vdash .*122·141.*37·321.\supset \vdash :\text{Hp}.&\supset .\text{D}ʻ(R\mid \breve{S} )=\text{D}ʻR =\text{ᗡ}ʻS. &\qquad \text{(2)}\\ +[*37·323] &\supset .\text{D}ʻ(S\mid R\mid \breve{S} )=\text{D}ʻS &\qquad \text{(3)}\\ +\vdash .(2).*37·32. &\supset \vdash :\text{Hp}.\supset .\text{ᗡ}ʻ(S\mid R\mid \breve{S} ) = Sʻʻ\text{ᗡ}ʻR &\qquad \text{(4)}\\ +\vdash .(3).(4). \supset \vdash :\text{Hp}.\supset .\overrightarrow{B}ʻ(S\mid R\mid \breve{S} )&=\text{D}ʻS-Sʻʻ\text{ᗡ}ʻR\\ +[*37·25.\text{Hp}] &=Sʻʻ\text{D}ʻR-Sʻʻ\text{ᗡ}ʻR\\ +[*71·381] &=Sʻʻ\overrightarrow{B}ʻR\\ +[*122·11.*53·31] &=\iota ʻSʻBʻR &\qquad \text{(5)}\\ +\vdash .(1).(3).(5).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*123·313.</b> \(\vdash :R\in \text{Prog}.S\in 1\rightarrow 1.\text{ᗡ}ʻS=\text{D}ʻR.P= S\mid R\mid \breve{S} .\supset .\text{D}ʻP=\overleftarrow{P}_{*}ʻBʻP\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*34·36.*123·312.\supset \vdash :\text{Hp}.&\supset .\text{ᗡ}ʻP\subset \text{D}ʻP.\text{E}!BʻP.\\ +[*90·13] &\supset .\overleftarrow{P}_{*}ʻBʻP\subset \text{D}ʻP &\qquad \text{(1)}\\ +\vdash .*123·312. &\supset \vdash :\text{Hp}.\supset .SʻBʻR \in \overleftarrow{P}_{*}ʻBʻP &\qquad \text{(2)}\\ +\vdash .*33·14. &\supset \vdash :\text{Hp}.Sʻx\in \overleftarrow{P}_{*}ʻBʻP.xRy.\supset .y\in \text{ᗡ}ʻR.\\ +[*122·141.\text{Hp}] &\supset .y\in \text{ᗡ}ʻS.\\ +[*71·16] &\supset .\text{E}!Sʻy.\\ +[*30·32.*34·1] &\supset .Sʻx(S\mid R\mid \breve{S} )Sʻy.\\ +[\text{Hp}] &\supset .SʻxPSʻy.\\ +[*90·163] &\supset .Sʻy\in \overleftarrow{P}_{*}ʻBʻP &\qquad \text{(3)}\\ +\vdash .(2).(3).*90·112.\supset \vdash \colon\ldotp \text{Hp}.&\supset :(BʻR)R_{*}x.\supset .Sʻx\in \overleftarrow{P}_{*}ʻBʻP:\\ +[*37·63] &\supset :Sʻʻ\overleftarrow{R}_{*}ʻBʻR\subset \overleftarrow{P}_{*}ʻBʻP:\\ +[*122·1] &\supset :Sʻʻ\text{D}ʻR\subset \overleftarrow{P}_{*}ʻBʻP:\\ +[*37·25.\text{Hp}] &\supset :\text{D}ʻS\subset \overleftarrow{P}_{*}ʻBʻP:\\ +[*123·312] &\supset :\text{D}ʻP\subset \overleftarrow{P}_{*}ʻBʻP &\qquad \text{(4)}\\ +\vdash .(1).(4).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*123·32.</b> \[\begin{align}&\vdash :R\in \text{Prog}.S\in 1\rightarrow 1.\text{ᗡ}ʻS=\text{D}ʻR.\supset .\\ +&S\mid R\mid \breve{S} \in \text{Prog}.\text{D}ʻS=\text{D}ʻS\mid R\mid \breve{S} .SʻBʻR=Bʻ(S\mid R\mid \breve{S} ) \quad[*123·312·313]\end{align}\]</p> + +<p><span class="pagenum" id="Page_273">[Pg 273]</span></p> + +<p class="nind"><b>*123·321.</b> \(\vdash :\alpha \in \aleph _{0}.\alpha \text{ sm }\beta .\supset .\beta \in \aleph _{0} \quad[*123·32]\)</p> + +<p class="nind"><b>*123·322.</b> \(\vdash :\alpha \in \aleph _{0}.\supset .\aleph _{0}=\text{Nc}ʻ\alpha\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*123·311·321.&\supset \vdash \colon\ldotp \alpha \in \aleph _{0}.\supset :\beta \in \aleph _{0}.\equiv .\beta \text{ sm }\alpha &\qquad \text{(1)}\\ +\vdash .(1).*100·1.&\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*123·323.</b> \(\vdash :R\in \text{Prog}.\supset .\aleph _{0}=\text{Nc}ʻ\text{D}ʻR \quad[*123·322]\)</p> + +<p class="nind"><b>*123·33.</b> \(\vdash \colon\ldotp \text{Infin ax}(x).\supset :\alpha \in \aleph _{0}.\equiv .\alpha \text{ sm }(\text{NC induct}\cap t^{3}ʻx) \quad[*123·26·321·31]\)</p> + +<p class="nind"><b>*123·34.</b> \(\vdash :\text{Infin ax}(x).\supset .\aleph _{0}=\text{Nc}ʻ(\text{NC induct}\cap t^{3}ʻx) \quad[*123·33]\)</p> + +<p class="nind"><b>*123·35.</b> \(\vdash :\exists !\aleph _{0}(x).\supset .\aleph _{0}(x)=\text{Nc}ʻ(\text{NC induct}\cap t^{3}ʻx) \quad[*123·34·18]\)</p> + +<p class="nind"><b><a id="*123·36">*123·36</a>.</b> \(\vdash .\aleph _{0}\in \text{NC} \quad[*123·35.*102·74]\)</p> + +<p class="nind"><b>*123·361.</b> \(\vdash :\exists !\aleph _{0}.\supset .\aleph _{0}{\sim}\in \text{NC induct} \quad[*123·15·322.*120·211]\)</p> + +<p class="nind"><b>*123·37.</b> \(\vdash :\text{Infin ax}(x).\supset .\exists !\aleph _{0}(t^{2}ʻx).\aleph _{0}(t^{2}x)\in \text{N}_{0}\text{C}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*120·301.\supset \vdash \colon\ldotp \text{Hp}.&\supset :\nu \in \text{NC induct}.\supset _{\nu }.\exists !\nu (x):\\ +[*65·13] &\supset :\nu \in \text{NC induct}.\supset _{\nu }.\exists !\nu .\nu =\nu (x):\\ +[(*65·02)] &\supset :\nu \in \text{NC induct}.\supset _{\nu }.\exists !\nu :\text{NC induct}\subset t^{3}ʻx:\\ +[*123·34] &\supset :\text{NC induct}\in \aleph _{0}.\text{NC induct}\subset t^{3}ʻx:\\ +[(*65·02)] &\supset :\text{NC induct}\in \aleph _{0}(t^{2}ʻx) &\qquad \text{(1)}\\ +\vdash .(1).*103·34.*123·36.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*123·39.</b> \(\vdash .(\aleph _{0})_{\eta }=(\aleph _{0}+_{c}1)_{\eta }\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*118·12.*117·6.*123·322.&\supset \vdash :(\aleph _{0})_{\eta }=\Lambda .\supset .(\aleph _{0}+_{c}1)_{\eta }=\Lambda &\qquad \text{(1)}\\ +\vdash .*123·13·322. &\supset \vdash :\exists !(\aleph _{0})_{\eta }.\supset .(\aleph _{0})_{\eta }=(\aleph _{0}+_{c}1)_\eta + &\qquad \text{(2)}\\ +\vdash .(1).(2).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*123·4.</b> \(\vdash .\aleph _{0}=\aleph _{0}+_{c}1 \quad[*123·39]\)</p> + +<p class="nind"><b>*123·401.</b> \(\vdash :\exists !\aleph _{0}.\supset .\aleph _{0}=\aleph _{0}-_{c}1\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*120·124.*123·36·4.\supset \vdash :\exists !\aleph _{0}.&\supset. \aleph _{0}\in \text{NC}-\iota ʻ0.\\ +[*120·414·416] \supset .(\aleph _{0}-_{c}1)+_{c}1&=\aleph _{0}\\ +[*123·4] &=\aleph _{0}+_{c}1.\\ +[*120·311] &\supset .\aleph _{0}-_{c}1=\aleph _{0} &\qquad \text{(1)}\\ +\vdash .*119·11.&\supset \vdash :(\aleph _{0})_{\eta }=\Lambda .\supset .(\aleph _{0})_{\eta }=(\aleph _{0}-_{c}1)_{\eta } + &\qquad \text{(2)}\\ +\vdash .(1).(2).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*123·41.</b> \(\vdash :\nu \in \text{NC induct}.\supset .\aleph _{0}=\aleph _{0}+_{c}\nu \quad[*123·4.*120·11]\)</p> + +<p><span class="pagenum" id="Page_274">[Pg 274]</span></p> + +<p class="nind"><b>*123·411.</b> \(\vdash :\nu \in \text{NC induct}.\supset .\aleph _{0}=\aleph _{0}-_{c}\nu \quad[*123·401.*120·11]\)</p> + +<p class="nind"><b>*123·42.</b> \(\vdash :P\in \text{Prog}.Q=P^{2}.\supset .\overleftarrow{Q}_{*}ʻ1_{P},\overleftarrow{Q}_{*}ʻ2_{P}\in + \aleph _{0}.\overleftarrow{Q}_{*}ʻ1_{P}\cap \overleftarrow{Q}_{*}ʻ2_{P}=\Lambda\)</p> + +<p>Note that \(\overleftarrow{Q}_{*}ʻ1_{P}\) is the odd terms and \(\overleftarrow{Q}_{*}ʻ2_{P}\) the even terms of \(\text{D}ʻP\).</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*91·6. \supset \vdash \colon\ldotp \text{Hp}.&\supset :\overleftarrow{Q}_{*}ʻ1_{P}\subset \overleftarrow{P}_{*}ʻ1_{P}.\overleftarrow{Q}_{*}ʻ2_{P}\subset + \overleftarrow{P}_{*}ʻ2_{P}:\\ +[*122·1] &\supset :\overleftarrow{Q}_{*}ʻ1_{P}\subset \text{D}ʻP:\\ +[*33·13] \supset :y\in \overleftarrow{Q}_{*}ʻ1_{P}.&\supset .(\exists z).yPz.\\ +[*122·141] &\supset .(\exists z,w).yPz.zPw.\\ +[\text{Hp}.*90·163.*91·503] &\supset .(\exists w).yQw.w\in \overleftarrow{Q}_{*}ʻ1_{P}.yP_{\text{po}}w:\\ +[*37·1] &\supset :\overleftarrow{Q}_{*}ʻ1_{P}\subset P_{\text{po}}ʻʻ\overleftarrow{Q}_{*}ʻ1_{P}:\\ +[*123·19] &\supset :\overleftarrow{Q}_{*}ʻ1_{P}\in \aleph _{0} &\qquad \text{(1)}\\ +\text{Similarly}\quad \vdash :\text{Hp}.&\supset .\overleftarrow{Q}_{*}ʻ2_{P}\in \aleph _{0} &\qquad \text{(2)}\\ +\vdash .*121·601·602. \supset \vdash :\text{Hp}.&\supset .1_{P}P2_{P}.\\ +[*122·16.*91·52·6] &\supset .{\sim}(2_{P}Q_{*}1_{P}) &\qquad \text{(3)}\\ +\vdash .*121·602.*53·31.*93·1. &\supset \vdash \colon\ldotp \text{Hp}.\supset :\overrightarrow{Q}ʻ2_{P}=\overrightarrow{P}ʻ1_{P}=\Lambda :\\ +[*13·14] &\supset :yQz.\supset .z\neq 2_{P}:\\ +[*91·542] &\supset :2_{P}Q_{*}z.yQz.\supset .2_{P}Q_{\text{po}}z.yQz.\\ +[*92·11] &\supset .2_{P}Q_{*}y:\\ +[\text{Transp}] &\supset :{\sim}(2_{P}Q_{*}y).yQz.\supset .{\sim}(2_{P}Q_{*}z) &\qquad \text{(4)}\\ +\vdash .(3).(4).*90·112.&\supset \vdash \colon\ldotp \text{Hp}.\supset :1_{P}Q_{*}z.\supset .{\sim}(2_{P}Q_{*}z) &\qquad \text{(5)}\\ +\vdash .(1).(2).(5).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*123·421.</b> \(\vdash .\aleph _{0}=\aleph _{0}+_{c}\aleph _{0}=2\times _{c}\aleph _{0}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*123*42. \supset \vdash :\alpha \in \aleph _{0}.&\supset .(\exists \beta ,\gamma ).\beta ,\gamma \in \aleph _{0}.\beta + \cap \gamma =\Lambda .\beta \cup \gamma \subset \alpha .\\ +[*110·32.*117·22] &\supset .\text{Nc}ʻa\geq \aleph _{0}+_{c}\aleph _{0} &\qquad \text{(1)}\\ +\vdash .(1).*117·6·23. \supset \vdash :\exists !\aleph _{0}.&\supset .\aleph _{0}=\aleph _{0}+_{c}\aleph _{0} &\qquad \text{(2)}\\ +\vdash .(2).*118·12.*117·6.&\supset \vdash .\aleph _{0}=\aleph _{0}+_{c}\aleph _{0} &\qquad \text{(3)}\\ +\vdash .(3).*113·66.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*123·422.</b> \(\vdash :\nu \in \text{NC induct}-\iota ʻ0.\supset .\nu \times _{c}\aleph _{0}=\aleph _{0}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash :*113·671. \supset \vdash :\nu \times _{c}\aleph _{0}=\aleph _{0}.\supset .(\nu +_{c}1)\times \aleph _{0}&=\aleph _{0}+_{c}\aleph + _{0}\\ +[*123·421] &=\aleph _{0} &\qquad \text{(1)}\\ +\vdash .(1).*120·47. \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*123·43.</b> \(\vdash \colon\ldotp \exists !\aleph _{0}.\supset :\nu \in \text{NC induct}.\supset _{\nu }.\aleph _{0}>\nu\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*123·18·36·361. &\supset \vdash :\text{Hp}.\supset .\aleph _{0}\in \text{NC}-\text{NC induct}-\iota ʻ\Lambda .\\ +&\text{NC induct}\subset -\iota ʻ\Lambda &\qquad \text{(1)}\\ +\vdash .(1).*120·49.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_275">[Pg 275]</span></p> + +<p class="nind"><b>*123·44.</b> \(\vdash \colon\ldotp \exists !\aleph _{0}.\supset :\nu \in \text{NC induct} \cup \iota ʻ\aleph _{0}.\equiv .\aleph _{0}\geq \nu \)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*123·322. \supset \vdash \colon\ldotp \alpha \in \aleph _{0}.&\supset :\aleph _{0}\geq \nu .\supset .\text{Nc}ʻ\alpha \geq \nu .\\ +[*117·22·104·12] &\supset .\exists !\nu \cap \text{Cl}ʻ\alpha .\nu \in \text{N}_{0}\text{C}.\\ +[*123·16] &\supset .\exists !\nu \cap (\text{Cls induct} \cup \aleph _{0}).\nu \in \text{N}_{0}\text{C}.\\ +[*103·26] &\supset .(\exists \beta ).\nu =\text{N}_{0}\text{c}ʻ\beta .\beta \in \text{Cls induct} \cup \aleph _{0}.\\ +[*120·21.*103·26] &\supset .\nu \in \text{NC induct} \cup \iota ʻ\aleph _{0} &\qquad \text{(1)}\\ +\vdash .(1).*123·43. \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*123·45.</b> \(\vdash \colon\ldotp \exists !\aleph _{0}.\supset :\nu \in \text{NC induct} .\equiv .\aleph _{0}>\nu .\equiv .\nu <\aleph _{0} + \quad[*123·43·44]\)</p> + +<p class="nind"><b>*123·46.</b> \(\vdash :\alpha \in \text{Cls induct} .\beta \in \aleph _{0}.\supset .\alpha \cup \beta \in \aleph _{0}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*110·32.*22·91. &\supset \vdash .\text{Nc}ʻ(\alpha \cup \beta )=\text{Nc}ʻ\beta +_{c}\text{Nc}ʻ(\alpha -\beta ) &\qquad \text{(1)}\\ +\vdash .*120·481·21. &\supset \vdash :\text{Hp}.\supset .\text{N}_{0}\text{c}ʻ(\alpha -\beta )\in \text{NC induct} &\qquad \text{(2)}\\ +\vdash .*123·322. &\supset \vdash :\text{Hp}.\supset .\aleph _{0}=\text{Nc}ʻ\beta &\qquad \text{(3)}\\ +\vdash .(2).(3).(*110·04).*123·41. &\supset \vdash :\text{Hp}.\supset .\text{Nc}ʻ\beta +_{c}\text{Nc}ʻ(\alpha -\beta )=\aleph _{0} &\qquad \text{(4)}\\ +\vdash .(1).(4).*100·44.&\supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*123·47.</b> \[\begin{align}\vdash \colon\ldotp \exists !\aleph _{0}.\supset :\alpha \in \text{Cls induct} \cup \aleph _{0}.&\equiv + .(\exists \gamma ).\gamma \in \aleph _{0}.\alpha \subset \gamma .\\ +&\equiv .\text{Nc}ʻ\alpha \leq \aleph _{0}\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*123·46. &\supset \vdash \colon\ldotp \text{Hp}.\supset :\alpha \in \text{Cls induct} .\supset .(\exists \gamma ).\gamma \in \aleph _{0}.\alpha + \subset \gamma &\qquad \text{(1)}\\ +\vdash .*22·42. &\supset \vdash :\alpha \in \aleph _{0}.\supset .(\exists \gamma ).\gamma \in \aleph _{0}.\alpha \subset \gamma &\qquad \text{(2)}\\ +\vdash .*123·16. &\supset \vdash :(\exists \gamma ).\gamma \in \aleph _{0}.\alpha \subset \gamma .\supset .\alpha \in \text{Cls induct} \cup \aleph _{0} + &\qquad \text{(3)}\\ +\vdash .(1).(2).(3). &\supset \vdash \colon\ldotp \text{Hp}.\supset :\alpha \in \text{Cls induct} \cup \aleph _{0}.\equiv .(\exists \gamma ).\gamma \in \aleph _{0}.\alpha + \subset \gamma &\qquad \text{(4)}\\ +\vdash .*123·44·322. &\supset \vdash \colon\ldotp \beta \in \aleph _{0}.\supset :\text{N}_{0}\text{c}ʻ\alpha \in \text{NC induct} \cup \iota ʻ\aleph _{0}.\equiv + .\text{N}_{0}\text{c}ʻ\alpha \leq \text{N}_{0}\text{c}ʻ\beta :\\ +[*103·26.*120·21.*117·107] \supset :\alpha \in \text{Cls induct} \cup \aleph _{0}.&\equiv .\text{Nc}ʻ\alpha \leq \text{Nc}ʻ\beta .\\ +[*123·322] &\equiv .\text{Nc}ʻ\alpha \leq \aleph _{0} &\qquad \text{(5)}\\ +\vdash .(5).*10·11·23. &\supset \vdash \colon\ldotp \exists !\aleph _{0}.\supset :\alpha \in \text{Cls induct} \cup \aleph _{0}.\equiv .\text{Nc}ʻ\alpha \leq \aleph _{0} + &\qquad \text{(6)}\\ +\vdash .(4).(6).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_276">[Pg 276]</span></p> + +<p>The following propositions are concerned in proving +\(\aleph_{0}^{2}=\aleph _{0}\). The proof given is roughly Cantor's. It +consists in showing that the relation \(R\) defined in the hypothesis +of <a href="#*123·5">*123·5</a> is a progression.</p> + +<p class="nind"><b><a id="*123·5">*123·5</a>.</b> \[\begin{align}\vdash :P,Q\in \text{Prog}.\\ +R=\hat{X} \hat{Y} [(\exists \mu ,\nu ):&X=\mu _{P}\downarrow \nu _{Q}.Y=(\mu +_{c}1)_{P}\downarrow (\nu -_{c}1)_{Q}.\lor.\\ +&X=\mu _{P}\downarrow 1_{Q}.Y=1_{P}\downarrow (\mu +_{c}1)_{Q}].\supset .R\in 1\rightarrow 1\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*122·34. \supset \vdash \colon\ldotp \text{Hp}.\supset :X&=\mu _{P}\downarrow \nu _{Q}.Y=(\mu +_{c}1)_{P}\downarrow (\nu -_{c}1)_{Q}.\supset + .\\ +&\mu ,\nu \in \text{NC induct}-\iota ʻ0.\nu \neq 1 &\qquad \text{(1)}\\ +\vdash .(1). \supset \vdash \colon\ldotp \text{Hp}.\supset :&(\exists \mu ,\nu ).X=\mu _{P}\downarrow \nu _{Q}.Y=(\mu +_{c}1)_{P}\downarrow + (\nu -_{c}1)_{Q}.\supset .\\ +&{\sim}(\exists \mu ).X=\mu _{P}\downarrow 1_{Q}.Y=1_{P}\downarrow (\mu +_{c}1)_{Q} &\qquad \text{(2)}\\ +\vdash .(2).*123·3.\supset \\ +\vdash \colon\colon \text{Hp}.\supset \colon\ldotp (\exists \mu ,\nu ).&X=\mu _{P}\downarrow \nu _{Q}.Y=(\mu +_{c}1)_{Q}\downarrow (\nu -_{c}1)_{Q}:XRYʻ.XʻRY:\supset + .\\ +&X=Xʻ.Y=Yʻ &\qquad \text{(3)}\\ +\vdash .(2).\text{Transp}.*123·3.\supset \\ +\vdash \colon\colon \text{Hp}.\supset \colon\ldotp (\exists \mu ).&X=\mu _{P}\downarrow 1_{Q}.Y=1_{P}\downarrow (\mu +_{c}1)_{Q}:XRYʻ.XʻRY:\supset .\\ +&X=Xʻ.Y=Yʻ &\qquad \text{(4)}\\ +\vdash .(3).(4).\supset \vdash :\text{Hp}.\supset .R\in 1\rightarrow 1:\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*123·501.</b> \(\vdash :\text{Hp}*123·5.\supset .\text{D}ʻR=\text{D}ʻP\times \text{D}ʻQ\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*122·34.\supset \vdash \colon\ldotp \text{Hp}.\supset :\mu ,\nu \in &\text{NC induct}-\iota ʻ0.\nu \neq 1.\supset .\\ +&(\mu _{P}\downarrow \nu _{Q})R{(\mu +_{c}1)_{P}\downarrow (\nu -_{c}1)_{Q}} &\qquad \text{(1)}\\ +\vdash .*122·34. \supset \vdash \colon\ldotp \text{Hp}.\supset :\mu \in &\text{NC induct}-\iota ʻ0.\supset .\\ +&(\mu _{P}\downarrow 1_{Q})R{1_{P}\downarrow (\mu +_{c}1)_{Q}} &\qquad \text{(2)}\\ +\vdash .(1).(2). \supset \vdash :\text{Hp}.&\supset :\mu ,\nu \in \text{NC induct}-\iota ʻ0.\supset .\mu _{P}\downarrow \nu _{Q}\in \text{D}ʻR:\\ +[*122·341] &\supset :x\in \text{D}ʻP.y\in \text{D}ʻQ.\supset .x\downarrow y\in \text{D}ʻR &\qquad \text{(3)}\\ +\vdash .*21·33. \supset \vdash \colon\ldotp \text{Hp}.\supset :X\in \text{D}ʻR.&\supset .(\exists \mu ,\nu ).X=\mu _{P}\downarrow \nu _{Q}.\\ +[*122·341] &\supset .(\exists x,y).x\in \text{D}ʻP.y\in \text{D}ʻQ.X=x\downarrow y. &\qquad \text{(4)}\\ +\vdash .(3).(4).*113·101.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*123·502.</b> \(\vdash :\text{Hp}*123·5.\supset .\text{ᗡ}ʻR\subset \text{D}ʻR.\overleftarrow{R}_{*}ʻ(1_{P}\downarrow 1_{Q})\subset \text{D}ʻR\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*21·33.&\supset \vdash :\text{Hp}.Y=(\mu +_{c}1)_{P}\downarrow (\nu -_{c}1)_{Q}.\nu -_{c}1\neq 1.\supset .\\ +&YR\{(\mu +_{c}2)_{P}\downarrow (\nu -_{c}2)_{Q}\} &\qquad \text{(1)}\\ +\vdash .*21·33.&\supset \vdash :\text{Hp}.Y=(\mu +_{c}1)_{P}\downarrow 1_{Q}.\supset .YR\{1_{P}\downarrow (\mu +_{c}2)_{Q}\} + &\qquad \text{(2)}\\ +\vdash .*21·33.&\supset \vdash :\text{Hp}.Y=1_{P}\downarrow (\mu +_{c}1)_{Q}.\supset .YR2_{P}\downarrow \mu _{Q} &\qquad \text{(3)}\\ +\vdash .(1).(2).(3). &\supset \vdash :\text{Hp}.\supset .\text{ᗡ}ʻR\subset \text{D}ʻR:\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_277">[Pg 277]</span></p> + +<p class="nind"><b>*123·503.</b> \(\vdash : \text{Hp} *123·5 . \supset . \text{D}ʻR \subset \overleftarrow{R}_{*}ʻ(1_{P} \downarrow 1_{Q})\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *123·501 . *122·11 . &\supset \vdash : \text{Hp} . \supset . 1_{P} \downarrow 1_{Q} \in \overleftarrow{R}_{*}ʻ(1_{P} \downarrow 1_{Q}) + &\qquad \text{(1)}\\ +\vdash . *90·16 . \supset \vdash : \text{Hp} . (1_{P} \downarrow 1_{Q}) &R_{*} (\mu _{P} \downarrow \nu _{Q}) . \nu \neq 1 . \supset .\\ +&(1_p \downarrow 1_{Q}) R_{*} {(\mu +_{c} 1)_{P} \downarrow (\nu -_{c} 1)_{Q}} &\qquad \text{(2)}\\ +\vdash . (2) . *120·47 . \supset \\ +\vdash : \text{Hp} . (1_{P} \downarrow 1_{Q}) R_{*} (\mu _{P} \downarrow \nu _{Q}) . &\supset . (1_{P} \downarrow 1_{Q}) R_{*} + \{(\mu +_{c} \nu -_{c} 1)_{P} \downarrow 1_{Q}\} .\\ +[*90·16] &\supset . (1_{P} \downarrow 1_{Q}) R_{*} \{1_{P} \downarrow (\mu +_{c} \nu )_{Q}\} .\\ +[(2) . *120·47] &\supset . (1_{P} \downarrow 1_{Q}) R_{*} \{\mu _{P} \downarrow (\nu +_{c} 1)_{Q}\} . &\qquad \text{(3)}\\ +[*90·16] &\supset . (1_{P} \downarrow 1_{Q}) R_{*} \{(\mu +_{c} 1)_{P} \downarrow \nu _{Q}\} &\qquad \text{(4)}\\ +\vdash . (1) . (3) . (4) . *120·47 . \supset \\ +\vdash \colon\ldotp \text{Hp} . &\supset : \mu , \nu \in \text{NC induct} - \iota ʻ0 . \supset . (1_{P} \downarrow 1_{Q}) R_{*} (\mu _{P} \downarrow \nu _{Q}) + &\qquad \text{(5)}\\ +\vdash . (5) . *122·341 . \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*123·504.</b> \(\vdash : \text{Hp} *123·5 . \supset . BʻR = 1_{P} \downarrow 1_{Q} \quad[*123·34 . *120·414]\)</p> + +<p class="nind"><b>*123·51.</b> \[\begin{align}\vdash : \text{Hp} *123·5 . \supset . R \in \text{Prog} . \text{D}ʻR = &\text{D}ʻP \times \text{D}ʻQ\\ +&[*123·5·501·502·503·504]\end{align}\]</p> + +<p class="nind"><b>*123·52.</b> \(\vdash . \aleph _{0} = \aleph _{0} \times _{c} \aleph _{0} = \aleph _{0}^{2} \quad[*123·51 . *116·34 . *113·25·204]\)</p> + +<p class="nind"><b>*123·53.</b> \(\vdash : \nu \in \text{NC induct} - \iota ʻO . \supset . \aleph _{0}^\nu = \aleph _{0} \quad[*123·52 . *116·52]\)</p> + +<p class="nind"><b>*123·7.</b> \(\vdash : \text{Infin ax} (x) . \text{Mult ax} . \supset . \exists ! \aleph _{0} (tʻx)\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *123·34 . *120·301 . &\supset \vdash : \text{Hp} . \supset . \text{NC induct} (tʻx) \in \aleph _{0} &\qquad \text{(1)}\\ +\vdash . *100·43 . *120·301 . &\supset \vdash : \text{Hp} . \supset . \text{NC induct} (tʻx) \in \text{Cls ex}^{2}\,\text{excl}&\qquad \text{(2)}\\ +\vdash . (1) . (2) . *88·32 . &\supset \vdash : \text{Hp} . \supset . \exists ! \text{Prod}ʻ\text{NC induct} (tʻx) &\qquad \text{(3)}\\ +\vdash . (1) . (2) . *115·16 . &\supset \vdash : \text{Hp} . \supset . \text{Prod}ʻ\text{NC induct} (tʻx) \subset \aleph _{0} &\qquad \text{(4)}\\ +\vdash . *115·18 . (*65·02) . &\supset \vdash : \kappa \in \text{Prod}ʻ\text{NC induct} (tʻx) . \supset . \kappa \in tʻtʻtʻx &\qquad \text{(5)}\\ +\vdash . (3) . (4) . (5) . (*65·02) . \supset \vdash . \text{Prop} +\end{array} +\]</p> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_278">[Pg 278]</span></p> +<h2 class="nobreak" id="*124"> +*124. REFLEXIVE CLASSES AND CARDINALS.</h2> +</div> + + +<p class="nind"><i>Summary of</i> *124.</p> + +<p>In this number, we have to take up the second definition of +infinity mentioned in the introduction to this Section. A class +which is infinite according to this definition we propose to call a +reflexive class, because a class which is of this kind is capable +of <i>reflexion</i> into a part of itself. A class is called +<i>reflexive</i> when there is a one-one relation which correlates the +class with a proper part of itself. (A <i>proper part</i> is a part +not the whole.) A reflexive cardinal is the homogeneous cardinal of a +reflexive class.</p> + +<p>We prove easily that reflexive classes are not inductive (<a href="#*124·271">*124·271</a>), +that reflexive cardinals are such as are greater than or equal to +\(\aleph _{0}\) (<a href="#*124·23">*124·23</a>), and such as are unchanged by adding 1 +(excepting \(\Lambda\)) (<a href="#*124·25">*124·25</a>). To prove that classes which are +not inductive must be reflexive has not hitherto been found possible +without assuming the multiplicative axiom. We do not need, however, +to assume the axiom generally, but only as applied to products of +\(\aleph _{0}\) factors. With this assumption, the result follows by a +series of propositions explained below. Thus if a product of \(\aleph_{0}\) +factors, no one of which is zero, is never zero, then the two +definitions of the finite and the infinite coincide (<a href="#*124·56">*124·56</a>).</p> + +<p>We will call a cardinal \(\nu\) a "multiplicative cardinal" if a +product of \(\nu\) factors none of which are zero is never zero. +Thus all inductive cardinals are multiplicative cardinals; and the +assumption needed for identifying the two definitions of finite and +infinite is that \(\aleph _{0}\) should be a multiplicative cardinal.</p> + +<p>For a reflexive class we use the notation "\(\text{Cls refl}\)," and +for a reflexive cardinal we use "\(\text{NC refl}\)." We define a +reflexive cardinal as the <i>homogeneous</i> cardinal of a reflexive +class, <i>i.e.</i> we put +\[ +\text{NC refl} = \text{N}_{0}\text{c}ʻʻ\text{Cls refl} \quad\text{Df}. +\] +The only effect of this is to exclude \(\Lambda\) from reflexive +cardinals, which is convenient. We then need (on the analogy of +<a href="#*110·03">*110·03</a><a href="#*110·04">*110·04</a>) a definition of what is meant when an ambiguous symbol such +as \(\text{Nc}ʻ\alpha\) is said to be reflexive, and we therefore put +\[ +\text{Nc}ʻ\rho \in \text{NC refl} . = . \text{N}_{0}\text{c}ʻ\rho \in \text{NC refl} \quad\text{Df}. +\]</p> + +<p><span class="pagenum" id="Page_279">[Pg 279]</span></p> + +<p>For the class of multiplicative cardinals we use the notation +"\(\text{NC mult}\)." Thus we put +\[ +\text{NC mult} = \text{NC} \cap \hat{\alpha} \{\kappa \in \alpha \cap \text{Cls}^{2} \text{excl}. \supset _{\kappa } . \exists ! {\in}_{\Delta}ʻ\kappa + \} \quad\text{Df}, +\] +whence it follows that if \(\alpha \in \text{NC mult}\), a product of +\(\alpha\) factors, none of which is zero, will never be zero.</p> + +<p>We begin, in this number, with the more obvious properties of +\(\text{Cls refl}\), proving that a \(\text{Cls refl}\) is one which +contains sub-classes of \(\aleph _{0}\) terms (<a href="#*124·15">*124·15</a>), that it is one +whose number is unchanged when a single term is taken away (<a href="#*124·17">*124·17</a>), +and that it remains reflexive if any inductive class is taken away from +it (<a href="#*124·182">*124·182</a>).</p> + +<p>We then give corresponding propositions concerning \(\text{NC refl}\) +(<a href="#*124·23">*124·23</a><a href="#*124·25">·25</a><a href="#*124·252">·252</a>), proving, in addition to propositions already mentioned, +that a reflexive cardinal is greater than every inductive cardinal +(<a href="#*124·26">*124·26</a>), and that a class which is neither inductive nor reflexive +(if there be such) is one which neither contains nor is contained in +any progression (<a href="#*124·34">*124·34</a>). On such classes, see the remarks at the end +of this number.</p> + +<p>We then (<a href="#*124·4">*124·4</a><a href="#*124·41">·41</a>) give a proposition merely embodying the definition +of \(\text{NC mult}\), and show that all inductive cardinals are +multiplicative, which follows immediately from <a href="#*120·62">*120·62</a>.</p> + +<p>The following series of propositions (<a href="#*124·51">*124·51</a> ff.) are concerned with +the proof that, if \(\aleph _{0}\) is a multiplicative cardinal, then +the two definitions of finite and infinite coalesce. The proof, which +is somewhat complicated, proceeds as follows.</p> + +<p>To begin with, we know that if \(\rho\) is a class which is not +inductive, it contains classes having \(\nu\) terms, if \(\nu\) is any +inductive cardinal. Thus we have +\[ +\exists ! 0 \cap \text{Cl}ʻ\rho , \exists ! 1 \cap \text{Cl}ʻ\rho , ... \exists ! \nu \cap \text{Cl}ʻ\rho , .... +\] +The classes of classes \(0 \cap \text{Cl}ʻ\rho\), \(1 \cap\text{Cl}ʻ\rho\), +... \(\nu \cap \text{Cl}ʻ\rho\), ... thus form a progression, which is +contained in \(\text{Cl}ʻ\text{Cl}ʻ\rho\). Hence (<a href="#*124·511">*124·511</a>) +\[ +\vdash : \rho {\sim} \in \text{Cls induct} . \supset . \text{Cl}ʻ\text{Cl}ʻ\rho \in \text{Cls refl}. +\] +So far, the multiplicative axiom is not required.</p> + +<p>The above progression of classes of classes is +\[ +(\cap \text{Cl}ʻ\rho )ʻʻ\text{NC induct}. +\] +If \(P\) is a selective relation for this class of classes, +\(\text{D}ʻP\) is a progression contained in \(\text{Cl}ʻ\rho\). Hence</p> + +<p class="nind"><b>*124·513.</b> \(\vdash : \exists ! {\in}_{\Delta}ʻ(\cap \text{Cl}ʻ\rho )ʻʻ\text{NC induct} . \supset . \text{Cl}ʻ\rho \in \text{Cls refl}\)</p> + +<p class="nind">whence</p> + +<p class="nind"><b>*124·514.</b> \(\vdash \colon\ldotp \aleph _{0} \in \text{NC mult} . \supset : \rho {\sim} \in \text{Cls induct} . \supset . \text{Cl}ʻ\rho \in \text{Cls refl}\)</p> + +<p>To prove the next step, namely +\[ +\aleph _{0} \in \text{NC mult} . \exists ! \aleph _{0} \cap \text{Cl}ʻ\text{Cl}ʻ\rho . \supset . \exists ! \aleph _{0} \cap \text{Cl}ʻ\rho , +\]<span class="pagenum" id="Page_280">[Pg 280]</span> +we make a fresh start. We have, by hypothesis, a progression \(R\) +whose domain is contained in \(\text{Cl}ʻ\rho\); hence \(sʻ\text{D}ʻR\subset \rho\). +Thus it will suffice to prove +\[ +\aleph _{0} \in \text{NC mult} . R \in \text{Prog} . \text{D}ʻR \subset \text{Cls induct} . \supset . \exists ! \aleph _{0} \cap sʻ\text{D}ʻR, +\] +where the conditions of significance require that \(\text{D}ʻR\) should +consist of classes.</p> + +<p>For this purpose, we prove that no member of \(\text{D}ʻR\) can be the +last that has new members which have not occurred before. The proof +proceeds by showing that if this were not so, \(sʻ\text{D}ʻR\) would +be an inductive class, and therefore, by <a href="#*120·75">*120·75</a>, \(\text{D}ʻR\) would +be an inductive class. Hence (<a href="#*124·534">*124·534</a>) the members of \(\text{D}ʻR\) +which introduce new terms form an \(\aleph _{0}\), by <a href="#*123·19">*123·19</a>; +and so therefore do the classes of new terms which they introduce +(<a href="#*124·535">*124·535</a>). Hence (<a href="#*124·536">*124·536</a>) a selection from these classes of new +terms, which is a sub-class of \(sʻ\text{D}ʻR\), is also an \(\aleph_{0}\), +and therefore (<a href="#*124·54">*124·54</a>) there is a progression contained in +\(sʻ\text{D}ʻR\) if the selection in question exists. This completes +the proof.</p> + +<p>In virtue of <a href="#*124·511">*124·511</a> and <a href="#*120·74">*120·74</a>, we have, without the multiplicative +axiom,</p> + +<p class="nind"><b>*124·6.</b> \(\vdash : \rho {\sim} \in \text{Cls induct} . \equiv . \text{Cl}ʻ\text{Cl}ʻ\rho \in \text{Cls refl}\)</p> + +<p>Hence if it could be shown that \(\text{Cl}ʻ\rho\) cannot be reflexive +unless \(\rho\) is reflexive, a double application of this would enable +us, by means of <a href="#*124·6">*124·6</a>, to identify the two definitions of the finite +without the multiplicative axiom.</p> + +<hr class="tb"> + +<p class="nind"><b>*124·01.</b> + \(\text{Cls refl} = \hat{\rho} \{(\exists R) . R \in 1 \rightarrow 1 . \text{ᗡ}ʻR \subset \text{D}ʻR . \exists ! \overrightarrow{B}ʻR . \rho = \text{D}ʻR\} \quad\text{Df}\)</p> + +<p>An equivalent definition would be +\[ +\text{Cls refl} = \text{D}ʻʻ{(1 \rightarrow 1) \cap \text{ᗡ}ʻB - \text{Cnv}ʻʻ\text{ᗡ}ʻB} \text{Df}. +\]</p> + +<p class="nind"><b>*124·02.</b> \(\text{NC refl} = \text{N}_{0}\text{c}ʻʻ\text{Cls refl} \quad\text{Df}\)</p> + +<p class="nind"><b>*124·021.</b> \(\text{Nc}ʻ\rho \in \text{NC refl} . = . \text{N}_{0}\text{c}ʻ\rho \in \text{NC refl} \quad\text{Df}\)</p> + +<p class="nind"><b>*124·03.</b> \(\text{NC mult} = \text{NC} \cap \hat{\alpha} \{\kappa \in \alpha \cap \text{Cls ex}^{2}\,\text{excl}. \supset _{\kappa } . \exists ! {\in}_{\Delta}ʻ\kappa + \} \quad\text{Df}\)</p> + +<p class="nind"><b>*124·1.</b> + \[\begin{align}&\vdash : \rho \in \text{Cls refl} . \equiv . (\exists R). R \in 1 \rightarrow 1 . \text{ᗡ}ʻR \subset \text{D}ʻR . \exists ! \overrightarrow{B}ʻR . \rho = \text{D}ʻR\\ +&[(*124·01)]\end{align}\]</p> + +<p class="nind"><b>*124·11.</b> \(\vdash : R \in 1 \rightarrow 1 . \text{ᗡ}ʻR \subset \text{D}ʻR . \exists ! \overrightarrow{B}ʻR . \supset . \text{D}ʻR \in \text{Cls refl} \quad[*124·1]\)</p> + +<p class="nind"><b>*124·12.</b> \(\vdash . \aleph _{0} \subset \text{Cls refl} \quad[*123·12 . *124·1]\)</p> + +<p><span class="pagenum" id="Page_281">[Pg 281]</span></p> + +<p class="nind"><b>*124·13.</b> \(\vdash : \rho \in \text{Cls refl} . \supset . \exists ! \aleph _{0} \cap \text{Cl}ʻ\rho \quad[*124·1 . *123·192]\)</p> + +<p class="nind"><b>*124·14.</b> \(\vdash :\rho \in \text{Cls refl} .\supset .\rho \cup \sigma \in \text{Cls refl}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*71·242.*50·5·52.\supset \\ +\vdash :R\in \rightarrow 1.&\text{ᗡ}ʻR\subset \text{D}ʻR.\exists !\overrightarrow{B}ʻR.\text{D}ʻR=\rho .S=I\upharpoonright (\sigma -\rho ).\supset .\\ +&R\unicode{x228d} S\in 1\rightarrow 1.\text{D}ʻ(R\unicode{x228d} S)=\text{D}ʻR\cup \sigma .\text{ᗡ}ʻ(R\unicode{x228d} S)=\text{ᗡ}ʻR\cup (\sigma -\rho ).\\ +[\text{Hp}.*93·101] &\supset .R\unicode{x228d} S\in 1\rightarrow 1.\text{D}ʻ(R\unicode{x228d} S)=\rho \cup \sigma .\overrightarrow{B}ʻ(R\unicode{x228d} S)=\overrightarrow{B}ʻR.\\ +[\text{Hp}.*13·12] &\supset .R\unicode{x228d} S\in 1\rightarrow 1.\text{D}ʻ(R\unicode{x228d} S)=\rho \cup \sigma .\exists !\overrightarrow{B}ʻ(R\unicode{x228d} S).\\ +[*124·11] &\supset .\rho \cup \sigma \in \text{Cls refl} &\qquad \text{(1)}\\ +\vdash .(1).*124·1.\supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*124·141.</b> \(\vdash :\exists ! Cl ʻ\rho \cap \text{Cls refl} .\supset .\rho \in \text{Cls refl}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*124·14.&\supset \vdash :\mu \in \text{Cls refl} .\supset .\mu \cup (\rho -\mu )\in \text{Cls refl}.\\ +[*24·411] &\supset \vdash :\mu \subset \rho .\mu \in \text{Cls refl} .\supset .\rho \in \text{Cls refl} :\supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*124·15">*124·15</a>.</b> \(\vdash :\rho \in \text{Cls refl} .\equiv .\exists !\aleph _{0}\cap \text{Cl} ʻ\rho \)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*124·12.\supset \vdash :\exists !\aleph _{0}\cap \text{Cl} ʻ\rho .&\supset .\exists ! \text{Cls refl} \cap \text{Cl} ʻ\rho .\\ +[*124·141] &\supset .\rho \in \text{Cls refl} &\qquad \text{(1)}\\ +\vdash .(1).*124·13.\supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*124·151.</b> \(\vdash :\rho \in \text{Cls refl} .\equiv .\text{Nc}ʻ\rho \geq \aleph _{0} \quad[*124·15.*117·22]\)</p> + +<p class="nind"><b>*124·16.</b> \[\begin{align}\vdash :\rho \in \text{Cls refl} .&\equiv .(\exists \sigma ).\sigma \subset \rho .\exists !\rho -\sigma .\rho \text{ sm } \sigma.\\ +&\equiv .\exists ! \text{Nc} ʻ\rho \cap \text{Cl} ʻ\rho -\iota ʻ\rho \end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*73·1.\supset \vdash :&(\exists \sigma ).\sigma \subset \rho .\exists !\rho -\sigma .\rho \text{ sm } \sigma .\equiv .\\ +&(\exists R,\sigma ).\sigma \subset \rho .\exists !\rho -\sigma .R\in 1\rightarrow 1.\text{D}ʻR=\rho .\text{ᗡ}ʻR=\sigma .\\ +[*13·195] &\equiv .(\exists R).\text{ᗡ}ʻR\subset \rho .\exists !\rho -\text{ᗡ}ʻR.R\in 1\rightarrow 1.\text{D}ʻR=\rho .\\ +[*13·193] &\equiv .(\exists R).\text{ᗡ}ʻR\subset \text{D}ʻR.\exists !\text{D}ʻR-\text{ᗡ}ʻR.R\in 1\rightarrow 1.\text{D}ʻR=\rho .\\ +[93·101.*124·1] &\equiv .\rho \in \text{Cls refl} :\supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*124·17">*124·17</a>.</b> \(\vdash :\rho \in \text{Cls refl} .\equiv .(\exists x).x\in \rho .\rho -\iota ʻx \text{ sm } \rho\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*124·16.&\supset \vdash :(\exists x).x\in \rho .\rho -\iota ʻx \text{ sm } \rho .\supset .\rho \in \text{Cls refl} &\qquad \text{(1)}\\ +\vdash .*123·17·192·311.\supset \\ +\vdash :R\in 1\rightarrow 1.\text{ᗡ}ʻR&\subset \text{D}ʻR.x\in \overrightarrow{B}ʻR.\supset .\overleftarrow{R}_{*}ʻx \text{ sm } \overleftarrow{R}_{*}ʻx-\iota ʻx.\\ +[*73·7] &\supset .(\text{D}ʻR-\overleftarrow{R}_{*}ʻx)\cup \overleftarrow{R}_{*}ʻx \text{ sm } (\text{D}ʻR-\overleftarrow{R}_{*}ʻx)\cup (\overleftarrow{R}_{*}ʻx-\iota + ʻx).\\ +[*24·411·412] &\supset .\text{D}ʻR \text{ sm } \text{D}ʻR-\iota ʻx &\qquad \text{(2)}\\ +\vdash .(2).*124·1.&\supset \vdash :\rho \in \text{Cls refl} .\supset .(\exists x).x\in \rho .\rho \text{ sm } \rho -\iota ʻx &\qquad \text{(3)}\\ +\vdash .(1).(3).\supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_282">[Pg 282]</span></p> + +<p class="nind"><b>*124·18.</b> \(\vdash :\rho \in \text{Cls refl}.\rho \text{ sm }\sigma .\supset .\sigma \in \text{Cls refl} \quad[*124·151.*100·321]\)</p> + +<p class="nind"><b>*124·181.</b> \(\vdash :\rho \in \text{Cls refl}.\supset .\rho -\iota ʻx\in \text{Cls refl}.\rho -\iota ʻx\text{ sm }\rho \)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*124·17·18.*73·72.\supset \\ +\vdash :\rho \in \text{Cls refl}.x\in \rho .\supset .\rho -\iota ʻx\text{ sm }\rho .\rho -\iota ʻx\in \text{Cls refl} &\qquad \text{(1)}\\ +\vdash .(1).*51·222.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*124·182">*124·182</a>.</b> \[\begin{align}&\vdash :\rho \in \text{Cls refl}.\sigma \in \text{Cls induct}.\supset .\rho -\sigma \in \text{Cls refl}.\rho -\sigma \text{ sm }\rho \\ +&[*124·181.*120·26]\end{align}\]</p> + +<p class="nind"><b>*124·2.</b> \(\vdash :\mu \in \text{NC refl}.\equiv .(\exists \rho ).\rho \in \text{Cls refl}.\mu =\text{N}_{0}\text{c}ʻ\rho \quad[(*124·02)]\)</p> + +<p class="nind"><b>*124·21.</b> \[\begin{align}&\vdash :\mu \in \text{NC refl}.\equiv .\\ +&(\exists R).R\in 1\rightarrow 1.\text{ᗡ}ʻR\subset \text{D}ʻR.\exists !.\overrightarrow{B}ʻR.\mu =\text{N}_{0}\text{c}ʻ\text{D}ʻR \quad[*124·2·1]\end{align}\]</p> + +<p class="nind"><b><a id="*124·23">*124·23</a>.</b> \(\vdash :\mu \in \text{NC refl}.\equiv .\mu \geq \aleph _{0}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*117·241. \supset \vdash :\mu \geq \aleph _{0}.&\equiv .(\exists \alpha ,\beta ).\mu =\text{N}_{0}\text{c}ʻ\alpha .\aleph _{0}=\text{N}_{0}\text{c}ʻ\beta + .\exists !\text{Cl}ʻ\alpha \cap \text{Nc}ʻ\beta .\\ +[*123·36·322.*103·26] &\equiv .(\exists \alpha ,\beta ).\mu =\text{N}_{0}\text{c}ʻ\alpha .\beta \in \aleph _{0}.\exists !\text{Cl}ʻ\alpha \cap \aleph _{0}.\\ +[*10·35] &\equiv .(\exists \alpha ).\mu =\text{N}_{0}\text{c}ʻ\alpha .\exists !\text{Cl}ʻ\alpha \cap \aleph _{0}.\\ +[*124·15] &\equiv .(\exists \alpha ).\mu =\text{N}_{0}\text{c}ʻ\alpha .\alpha \in \text{Cls refl}.\\ +[*124·2] &\equiv .\mu \in \text{NC refl}:\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*124·231.</b> \(\vdash :\exists !\text{NC refl}.\equiv .\exists !\text{Cls refl}.\equiv .\exists !\aleph _{0} \quad[*124·2·12·13]\)</p> + +<p class="nind"><b>*124·232.</b> \(\vdash :\exists !\text{NC refl}.\supset .\text{Infin ax} \quad[*124·231.*123·18]\)</p> + +<p class="nind"><b>*124·24.</b> \(\vdash \colon\ldotp \mu \in \text{NC refl}.\equiv :\mu \in \text{N}_{0}\text{C}:(\exists \nu ).\mu =\aleph _{0}+_{c}\nu .\nu \in \text{NC}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*124·23.*117·31.\supset \\ +\vdash \colon\ldotp \mu \in \text{NC refl}.\equiv :\mu ,\aleph _{0}\in \text{N}_{0}\text{C}:(\exists \nu ).\nu \in \text{NC}.\mu =\aleph _{0}+_{c}\nu + &\qquad \text{(1)}\\ +\vdash .*110·4.\supset \vdash :\mu =\aleph _{0}+_{c}\nu .\mu \in \text{N}_{0}\text{C}.\supset .\aleph _{0}\in \text{N}_{0}\text{C} &\qquad \text{(2)}\\ +\vdash .(1).(2).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*124·25">*124·25</a>.</b> \[\begin{align}&\vdash :\mu \in \text{NC refl}.\equiv .\mu \in \text{N}_{0}\text{C}.\mu =\mu +_{c}1.\equiv .\exists !\mu .\mu =\mu +_{c}1 \\ +&[*124·17·2]\end{align}\]</p> + +<p class="nind"><b>*124·251.</b> \(\vdash :\mu \in \text{NC refl}.\supset .\mu =\mu +_{c}1 \quad[*124·25]\)</p> + +<p class="nind"><b><a id="*124·252">*124·252</a>.</b> \(\vdash :\mu \in \text{NC refl}.\nu \in \text{NC induct}.\supset .\mu =\mu +_{c}\nu\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*124·251.\supset \vdash :\mu \in \text{NC refl}.\mu =\mu +_{c}\nu .\supset .\mu =\mu +_{c}\nu +_{c}1 &\qquad \text{(1)}\\ +\vdash .(1).*120·11.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*124·253.</b> \(\vdash :\mu \in \text{NC refl}.\supset .\mu =\mu +_{c}\aleph _{0}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*124·24. \supset \vdash :\text{Hp}.&\supset .(\exists \nu ).\mu =\aleph _{0}+_{c}\nu .\\ +[*123·421] &\supset .(\exists \nu ).\mu =\aleph _{0}+_{c}\aleph _{0}+_{c}\nu .\mu =\aleph _{0}+_{c}\nu .\\ +[*13·13] &\supset .\mu =\aleph _{0}+_{c}\mu :\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_283">[Pg 283]</span></p> + +<p class="nind"><b><a id="*124·26">*124·26</a>.</b> \(\vdash \colon\ldotp \mu \in \text{NC refl}.\supset :\nu \in \text{NC induct}.\supset _{\nu }.\mu >\nu\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*124·231. \supset \vdash \colon\ldotp \text{Hp}.&\supset :\exists !\aleph _{0}:\\ +[*123·43] &\supset :\nu \in \text{NC induct}.\supset _{\nu }.\aleph _{0}>\nu &\qquad \text{(1)}\\ +\vdash .(1).*124·23.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*124·27.</b> \(\vdash .\text{NC refl}\cap \text{NC induct}=\Lambda \quad[*124·26.*117·42]\)</p> + +<p class="nind"><b><a id="*124·271">*124·271</a>.</b> \(\vdash .\text{Cls refl} \cap \text{Cls induct}=\Lambda\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*124·2. \supset \vdash :\rho \in \text{Cls refl}.&\supset .\text{N}_{0}\text{c}ʻ\rho \in \text{NC refl}.\\ +[*124·27] &\supset .\text{N}_{0}\text{c}ʻ\rho {\sim}\in \text{NC induct}.\\ +[*120·21] &\supset .\rho {\sim}\in \text{Cls induct}:\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*124·28.</b> \(\vdash :\rho \in \text{Cls refl}.\equiv .\text{N}_{0}\text{c}ʻ\rho \in \text{NC refl}.\equiv .\text{Nc}ʻ\rho \in \text{NC refl}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*4·2.(*124·021). \supset \vdash :\text{Nc}ʻ\rho \in \text{NC refl}.&\equiv .\text{N}_{0}\text{c}ʻ\rho \in \text{NC refl}.\\ +[*124·2] &\equiv .(\exists \sigma ).\sigma \in \text{Cls refl}.\text{N}_{0}\text{c}ʻ\rho =\text{N}_{0}\text{c}ʻ\sigma .\\ +[*103·14] &\equiv .(\exists \sigma ).\sigma \in \text{Cls refl}.\rho \text{ sm }\sigma .\rho \in tʻ\sigma .\\ +[*124·18.*73·3.*63·103] &\equiv .\rho \in \text{Cls refl}:\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*124·29.</b> \(\vdash .sʻ\text{NC refl}=\text{Cls refl}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*40·11. \supset \vdash :\rho \in sʻ\text{NC refl}.&\equiv .(\exists \mu ).\mu \in \text{NC refl}.\rho \in \mu .\\ +[*103·26] &\equiv .(\exists \mu ).\mu \in \text{NC refl}.\mu =\text{N}_{0}\text{c}ʻ\rho .\\ +[*13·195] &\equiv .\text{N}_{0}\text{c}ʻ\rho \in \text{NC refl}.\\ +[*124·28] &\equiv .\rho \in \text{Cls refl}:\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*124·3.</b> \[\begin{align}\vdash \colon\colon \exists !\aleph _0.\supset \colon\ldotp \mu <\aleph _0.\lor.\mu \geq \aleph _0:\equiv .\mu \in &\text{NC induct}\cup \text{NC refl}\\ +&[*123·45.*124·23]\end{align}\]</p> + +<p class="nind"><b>*124·31.</b> \(\vdash :\exists !\aleph _{0}.\supset .\text{spec}ʻ\aleph _{0}=\text{NC induct}\cup \text{NC refl} \quad[*124·3.*120·431]\)</p> + +<p>In virtue of the above proposition, if there are any numbers which are +neither inductive nor reflexive, they are such as are neither greater +than, less than, nor equal to \(\aleph _{0}\). (The existence of +\(\aleph _{0}\) in a suitable type can be deduced from the existence +of numbers which are neither inductive nor reflexive; cf. <a href="#*124·6">*124·6</a>.) Two +further propositions (<a href="#*124·33">*124·33</a><a href="#*124·34">·34</a>) are given below on non-inductive +non-reflexive classes and cardinals. The subject is resumed in the +remarks at the end of the number.</p> + +<p class="nind"><b><a id="*124·33">*124·33</a>.</b> \[\begin{align}\vdash \colon\ldotp \exists !\aleph _{0}.\supset :&\mu \in \text{NC}-\text{NC induct}-\text{NC refl}.\equiv .\\ +&\mu \in \text{NC}.{\sim}(\mu <\aleph _{0}).{\sim}(\mu \geq \aleph _{0}) \quad[*124·3.\text{Transp}]\end{align}\]<span class="pagenum" id="Page_284">[Pg 284]</span></p> + +<p class="nind"><b><a id="*124·34">*124·34</a>.</b> \[\begin{align}\vdash \colon\colon \exists !\aleph _{0}.\supset \colon\ldotp \alpha {\sim}\in &(\text{Cls induct} \cup \text{Cls refl}).\equiv :\\ +&{\sim}(\exists \gamma ):\gamma \in \aleph _{0}:\alpha \subset \gamma .\lor.\gamma \subset \alpha\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*120·21.*124·28.&\supset \vdash :\alpha {\sim}\in (\text{Cls induct} \cup \text{Cls refl}).\equiv .\\ +&\text{N}_{0}\text{c}ʻ\alpha {\sim}\in (\text{NC induct} \cup \text{NC refl}) &\qquad \text{(1)}\\ +\vdash .*123·36.*103·26.&\supset \vdash :\beta \in \aleph _{0}.\supset .\aleph _{0}=\text{N}_{0}\text{c}ʻ\beta &\qquad \text{(2)}\\ +\vdash .(1).(2).*124·31.&\supset \vdash \colon\colon \beta \in \aleph _{0}.\supset \colon\ldotp \alpha {\sim}\in (\text{Cls induct} \cup \text{Cls refl}).\equiv :\\ +&\text{N}_{0}\text{c}ʻ\alpha {\sim}\in \text{spec}ʻ\text{N}_{0}\text{c}ʻ\beta :\\ +[*120·432] &\equiv :{\sim}(\text{N}_{0}\text{c}ʻ\alpha \leq \text{N}_{0}\text{c}ʻ\beta ).{\sim}(\text{N}_{0}\text{c}ʻ\alpha \geq \text{N}_{0}\text{c}ʻ\beta ):\\ +[*117·107·22] &\equiv :{\sim}(\text{Nc}ʻ\alpha \leq \text{Nc}ʻ\beta ):{\sim}(\exists \gamma ).\gamma \in \text{Nc}ʻ\beta .\gamma \subset \alpha :\\ +[*123·322] &\equiv :{\sim}(\text{Nc}ʻ\alpha \leq \aleph _{0}):{\sim}(\exists \gamma ).\gamma \in \aleph _{0}.\gamma \subset \alpha :\\ +[*123·47] &\equiv :{\sim}(\exists \gamma ).\gamma \in \aleph _{0}.\alpha \subset \gamma :{\sim}(\exists \gamma ).\gamma \in \aleph _{0}.\gamma + \subset \alpha &\qquad \text{(3)}\\ +\vdash .(3).*10·11·21.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*124·4">*124·4</a>.</b> \[\begin{align}&\vdash \colon\ldotp \mu \in \text{NC mult}.\equiv :\mu \in \text{NC}:\kappa \in \mu \cap \text{Cls ex}^{2}\, \text{excl}.\supset _{\kappa }.\exists + !{\in}_{\Delta}ʻ\kappa\\ &[(*124·03)]\end{align}\]</p> + +<p class="nind"><b><a id="*124·41">*124·41</a>.</b> \(\vdash .\text{NC induct}\subset \text{NC mult}\quad[*120·62 . *124·4]\)</p> + +<p>The following propositions give the proof of <a href="#*124·56">*124·56</a>, which identifies +the two definitions of the finite, on the assumption that \(\aleph_{0}\) +is a multiplicative cardinal. (<a href="#*124·513">*124·513</a>, however, is only +used in proving <a href="#*124·514">*124·514</a>, and *124·514 is not used in the proof. It +is retained as marking a stage in the argument, although the actual +propositions subsequently used are not it, but the lemmas which lead to +it.)</p> + +<p class="nind"><b><a id="*124·51">*124·51</a>.</b> \[\begin{align}\vdash :\rho {\sim}\in &\text{Cls induct}.Q=(\cap \text{Cl}ʻ\rho )\mid N\mid \text{Cnv}ʻ(\cap \text{Cl}ʻ\rho ).\supset .\\ +&Q\in \text{Prog}.\text{D}ʻQ\subset \text{Cl}ʻ\text{Cl}ʻ\rho .\text{D}ʻQ=(\cap \text{Cl}ʻ\rho )ʻʻ\text{NC induct}\end{align}\]</p> + +<p>\(N\) here has the meaning defined in <a href="#*123·02">*123·02</a>.</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*120·61·21.*123·25.&\supset \vdash :\text{Hp}.\supset .N\in \text{Prog} &\qquad \text{(1)}\\ +\vdash .*120·491.\supset \vdash \colon\ldotp \text{Hp}.&\supset :\mu ,\nu \in \text{NC induct}.\supset _{\mu ,\nu }.\exists !\mu \cap \text{Cl}ʻ\rho .\exists !\nu \cap \text{Cl}ʻ\rho :\\ +[*22·5] &\supset :\mu ,\nu \in \text{NC induct}.\mu \cap \text{Cl}ʻ\rho =\nu \cap \text{Cl}ʻ\rho .\supset _{\mu ,\nu }.\\ +&\exists !\mu \cap \nu \cap \text{Cl}ʻ\rho .\\ +[*100·43] &\supset {\mu ,\nu }.\mu =\nu \\ +[*71*55] &\supset :(\cap \text{Cl}ʻ\rho )\upharpoonright \text{NC induct}\in 1\rightarrow 1 &\qquad \text{(2)}\\ +\vdash .(1).(2).*123·32.&\supset \vdash :\text{Hp}.\supset .Q\in \text{Prog} &\qquad \text{(3)}\\ +\vdash .*22·43.&\supset \vdash :\alpha \in \text{D}ʻQ.\supset .\alpha \subset \text{Cl}ʻ\rho &\qquad \text{(4)}\\ +\vdash .(3).(4).*37·32·321.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*124·511">*124·511</a>.</b> \[\begin{align}\vdash :\rho {\sim}\in &\text{Cls induct}.\supset .\\ +&\text{Cl}ʻ\text{Cl}ʻ\rho \in \text{Cls refl}.(\cap \text{Cl}ʻ\rho )ʻʻ\text{NC induct}\in \aleph _{0}\cap \text{Cls}^{2} \text{excl}\\ +&[*124·51·15. *120·491.*100·43]\end{align}\]</p> + +<p><span class="pagenum" id="Page_285">[Pg 285]</span></p> + +<p class="nind"><b><a id="*124·512">*124·512</a>.</b> \[\begin{align}\vdash :P\in {\in}_{\Delta}ʻ(\cap \text{Cl}ʻ\rho )ʻʻ&\text{NC induct}.\supset .\\ +&\text{D}ʻP\in \aleph _{0}\cap \text{Cl}ʻ\text{Cl}ʻ\rho .\text{D}ʻP\subset \text{Cls induct}\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*83·11.\text{Transp}.&\supset \vdash \colon\ldotp \text{Hp}.\supset :\nu \in \text{NC induct}.\supset _{\nu }.\exists !\nu \cap \text{Cl}ʻ\rho &\qquad \text{(1)}\\ +\vdash .*115·16.(1).*124·511.*120·491.\supset \\ +\vdash :\text{Hp}.&\supset .\text{D}ʻP\in \text{Nc}ʻ(\cap \text{Cl}ʻ\rho )ʻʻ\text{NC induct}.p{\sim}\in \text{Cls induct}.\\ +[*124·511] &\supset .\text{D}ʻP\in\aleph_{0} &\qquad \text{(2)}\\ +\vdash .*83·21.&\supset \vdash \colon\ldotp \text{Hp}.\supset :\alpha \in \text{D}ʻP.\supset .(\exists \nu ).\nu \in \text{NC induct}.\alpha \in \nu \cap \text{Cl}ʻ\rho .\\ +[*10·5.*120·2] &\supset .\alpha \in \text{Cls induct}.\alpha \in \nu \text{Cl}ʻ\rho . &\qquad \text{(3)}\\ +\vdash .(2).(3).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*124·513">*124·513</a>.</b> \(\vdash :\exists !{\in}_{\Delta}ʻ(\cap \text{Cl}ʻ\rho )ʻʻ\text{NC induct}.\supset .\text{Cl}ʻ\rho \in \text{Cls refl} \quad[*124·512·15]\)</p> + +<p class="nind"><b><a id="*124·514">*124·514</a>.</b> \[\begin{align}\vdash \colon\ldotp \aleph _{0}\in \text{NC mult}.\supset :\rho {\sim}\in &\text{Cls induct}.\supset .\text{Cl}ʻ\rho \in \text{Cls refl}\\ +&[*124·511·513·4]\end{align}\]</p> + +<p>The following propositions are concerned in proving that, if +\(\aleph _{0}\) is a multiplicative cardinal, then a class such as +\(\text{D}ʻP\) in <a href="#*124·512">*124·512</a> must be such that a progression is contained +in \(sʻ\text{D}ʻP\). The characteristics of \(\text{D}ʻP\) which are +used in the proof are \(\text{D}ʻP\in \aleph _{0}.\text{D}ʻP\subset\text{Cls induct}\). +Since \(\text{D}ʻP\in \aleph _{0}\), we have +\((\exists R).R\in \text{Prog}.\text{D}ʻP = \text{D}ʻR\). Hence the +hypothesis with which the following series of propositions is concerned +is +\[ +R\in \text{Prog}.\text{D}ʻR\subset \text{Cls induct}, +\] +but the earlier propositions do not need the full hypothesis.</p> + +<p>In what follows, note that if \(\gamma \in \text{D}ʻR,\gamma-sʻ\overrightarrow{R}_{\text{po}}ʻ\gamma\) +is the class of those terms which occur in \(\gamma\) and have never +occurred before in any earlier member of \(\text{D}ʻR\). We prove that, +with our hypothesis, members of \(\text{D}ʻR\) for which this class +of new terms is not null form a class which has no last member, and +therefore form a progression.</p> + +<p class="nind"><b>*124·52.</b> \[\begin{align}&\vdash \colon\ldotp R\in \text{Prog}.\sigma = \hat{\beta} \{(\exists \gamma ).\gamma \in \text{D}ʻR.\beta = \gamma -sʻ\overrightarrow{R}_{\text{po}}ʻ\gamma + .\exists !\beta \}.\supset :\\ +&\sigma \in \text{Cls ex}^{2} \text{excl}:\gamma ,\delta \in \text{D}ʻR.\gamma \neq \delta .\supset .(\gamma -sʻ\overrightarrow{R}_{\text{po}}ʻ\gamma + )\cap (\delta -sʻ\overrightarrow{R}_{\text{po}}ʻ\delta ) = \Lambda \end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*20·33. & \supset \vdash \colon\ldotp \text{Hp}.\supset :\beta \in \sigma .\supset _{\beta }.\exists !\beta &\qquad \text{(1)}\\ +\vdash .*122·21. & \supset \vdash \colon\ldotp \text{Hp}.\gamma ,\delta \in \text{D}ʻR.\gamma \neq \delta .\supset :\gamma R_{\text{po}}\delta .\lor.\delta R_{\text{po}}\gamma + &\qquad \text{(2)}\\ +\vdash .*40·13. \supset \vdash \colon\ldotp \text{Hp}.\gamma R_{\text{po}}\delta .&\supset :\gamma \subset sʻ\overrightarrow{R}_{\text{po}}ʻ\delta :\\ +[*24·3] &\supset :(\gamma -sʻ\overrightarrow{R}_{\text{po}}ʻ\gamma )\cap (\delta -sʻ\overrightarrow{R}_{\text{po}}ʻ\delta = \Lambda &\qquad \text{(3)}\\ +\text{Similarly}\quad &\vdash \colon\ldotp \text{Hp}.\delta R_{\text{po}}\gamma .\supset :(\gamma -sʻ\overrightarrow{R}_{\text{po}}ʻ\gamma + )\cap (\delta -sʻ\overrightarrow{R}_{\text{po}}ʻ\delta ) = \Lambda &\qquad \text{(4)}\\ +\vdash .(2).(3).(4).&\supset \vdash :\text{Hp}.\gamma ,\delta \in \text{D}ʻR.\gamma \neq \delta .\supset .\\ +&(\gamma -sʻ\overrightarrow{R}_{\text{po}})\cap (\delta -sʻ\overrightarrow{R}_{\text{po}}ʻ\delta ) = \Lambda &\qquad \text{(5)}\\ +\vdash .(5).*20·33. &\supset \vdash :\text{Hp}.\beta ,\beta'\in \sigma .\beta \neq \beta'.\supset .\beta \cap \beta' = \Lambda &\qquad \text{(6)}\\ +\vdash .(1).(6).(5).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_286">[Pg 286]</span></p> + +<p class="nind"><b>*124·521.</b> \(\vdash :\text{Hp}*124·52.\pi =\hat{\gamma}\{\gamma \in \text{D}ʻR.\exists !\gamma -sʻ\overrightarrow{R}_{\text{po}}ʻ\gamma \}.\supset .\sigma \text{ sm }\pi\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*124·52.*24·57.\supset \\ +\vdash :\text{Hp}.\gamma ,\delta \in \pi .\gamma \neq \delta .&\supset .\gamma -sʻ\overrightarrow{R}_{\text{po}}ʻ\gamma \neq \delta -sʻ\overrightarrow{R}_{\text{po}}ʻ\delta + &\qquad \text{(1)}\\ +\vdash .(1).\supset \vdash :\text{Hp}.&S=\hat{\beta} \hat{\gamma}\{\gamma \in \text{D}ʻR.\beta =\gamma -sʻ\overrightarrow{R}_{\text{po}}ʻ\gamma .\exists !\beta\}.\supset .\\ +&S\in 1\rightarrow 1.\text{D}ʻS=\sigma .\text{ᗡ}ʻS=\pi \colon\ldotp \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*124·53.</b> \(\vdash :R\in \text{Prog}.\supset .sʻ\text{D}ʻR{\sim}\in \text{Cls induct} \quad[*120·75.*122·37]\)</p> + +<p class="nind"><b>*124*531.</b> \(\vdash :R\in \text{Prog}.\text{D}ʻR\subset \text{Cls induct}.\supset .sʻ\overrightarrow{R}_{*}ʻ\gamma \in \text{Cls induct}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*122·38.\supset \vdash :\text{Hp}.\supset .\overrightarrow{R}_{*}ʻ\gamma \in \text{Cls induct} &\qquad \text{(1)}\\ +\vdash .(1).*120·75.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*124·532.</b> \[\begin{align}&\vdash :R\in \text{Prog}.\text{D}ʻR\subset \text{Cls induct}.\supset .\exists !sʻ\text{D}ʻR-sʻ\overrightarrow{R}_{*}ʻ\gamma \\ +&[*124·53·531.*120·481.\text{Transp}]\end{align}\]</p> + +<p class="nind"><b>*124·533.</b> \[\begin{align}\vdash :R\in \text{Prog}.\text{D}ʻR\subset \text{Cls induct}.\gamma \in &\text{D}ʻR.\supset .\\ +&(\exists \beta ).\gamma R_{\text{po}}\beta .\exists !\beta -sʻ\overrightarrow{R}_{\text{po}}ʻ\beta\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*124·532.&\supset \vdash :\text{Hp}.\supset .(\exists \beta ).\beta \in \text{D}ʻR.\exists !\beta -sʻ\overrightarrow{R}_{*}ʻ\gamma &\qquad \text{(1)}\\ +\vdash .*40·13. &\supset \vdash :\beta R_{*}\gamma .\supset .\beta \subset sʻ\overrightarrow{R}_{*}\gamma :\\ +[\text{Transp}] &\supset \vdash :\exists !\beta -sʻ\overrightarrow{R}_{*}\gamma .\supset .{\sim}(\beta R_{*}\gamma ):\\ +[*122·21] &\supset \vdash :\text{Hp}.\beta \in \text{D}ʻR.\exists !\beta -sʻ\overrightarrow{R}_{*}ʻ\gamma .\supset .\gamma R_{\text{po}}\beta &\qquad \text{(2)}\\ +\vdash .(1).(2).\supset \\ +\vdash \colon\ldotp \text{Hp}.&\supset :(\exists \beta ).\gamma R_{\text{po}}\beta .\exists !\beta -sʻ\overrightarrow{R}_{*}ʻ\gamma :\\ +[*122·23] &\supset :\text{E}!\text{min}(R_{\text{po}})ʻ\hat{\beta}\{\gamma R_{\text{po}}\beta .\exists !\beta -sʻ\overrightarrow{R}_{\text{po}}ʻ\beta \}:\\ +[*93·111] &\supset :(\exists \beta ):\gamma R_{\text{po}}\beta .\exists !\beta -sʻ\overrightarrow{R}_{*}\gamma :\delta R_{\text{po}}\beta .\supset _{\delta }.\delta + \subset sʻ\overrightarrow{R}_{*}ʻ\gamma :\\ +[*40*151] &\supset :(\exists \beta ).\gamma R_{\text{po}}\beta .\exists !\beta -sʻ\overrightarrow{R}_{*}\gamma .sʻ\overrightarrow{R}_{\text{po}}ʻ\beta + \subset sʻ\overrightarrow{R}_{*}ʻ\gamma :\\ +[*22*81] &\supset :(\exists \beta ).\gamma R_{\text{po}}\beta .\exists !\beta -sʻ\overrightarrow{R}_{\text{po}}\beta \colon\ldotp \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*124·534">*124·534</a>.</b> \[\begin{align}\vdash :R\in \text{Prog}.\text{D}ʻR\subset &\text{Cls induct}.\\ +&\pi =\hat{\gamma}\{\gamma \in \text{D}ʻR.\exists !\gamma -sʻ\overrightarrow{R}_{\text{po}}ʻ\gamma\}.\supset .\pi \in \aleph _{0}\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*124·533. \supset \vdash :\text{Hp}.\supset .\exists !\pi .\pi \subset R_{\text{po}}ʻʻ\pi &\qquad \text{(1)}\\ +\vdash .(1).*123·19.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*124·535">*124·535</a>.</b> \[\begin{align}\vdash :R\in \text{Prog}.&\text{D}ʻR\subset \text{Cls induct}.\\ +&\sigma =\hat{\beta}\{(\exists y).\gamma \in \text{D}ʻR.\beta =\gamma -sʻ\overrightarrow{R}_{\text{po}}ʻ\gamma .\exists !\beta\}.\supset .\sigma \in \aleph _{0}\\ +&[*124·534·521.*123·321]\end{align}\]</p> + +<p><span class="pagenum" id="Page_287">[Pg 287]</span></p> + +<p class="nind"><b><a id="*124·536">*124·536</a>.</b> \[\begin{align}\vdash :R\in \text{Prog}.\text{D}ʻR\subset &\text{Cls induct}.\\ +&\sigma =\hat{\beta} \{(\exists \gamma ).\gamma \in \text{D}ʻR.\beta =\gamma -sʻ\overrightarrow{R}_{\text{po}}ʻ\gamma .\exists !\beta \}.\\ +&S\in {\in}_{\Delta}ʻ\sigma .\supset .\text{D}ʻS\in \aleph _{0}.\text{D}ʻS\subset sʻ\text{D}ʻR\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*115·16.*124·52·535. &\supset \vdash :\text{Hp}.\supset .\text{D}ʻS\in \aleph _{0} &\qquad \text{(1)}\\ +\vdash .*83·21. \supset \vdash \colon\ldotp \text{Hp}.&\supset :\text{D}ʻS\subset sʻ\sigma :\\ +[*40·11] \supset :x\in \text{D}ʻS.&\supset .(\exists \beta ,\gamma ).\gamma \in \text{D}ʻR.\beta =\gamma -sʻ\overrightarrow{R}_{\text{po}}ʻ\gamma .\exists !\beta .x\in \beta .\\ +[*13·195] &\supset .(\exists \gamma ).\gamma \in \text{D}ʻR.x\in \gamma -sʻ\overrightarrow{R}_{\text{po}}ʻ\gamma .\\ +[*22·43] &\supset .(\exists \gamma ).\gamma \in \text{D}ʻR.x\in \gamma\\ +[*40·11] &\supset .x\in sʻ\text{D}ʻR &\qquad \text{(2)}\\ +\vdash .(1).(2).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*124·54">*124·54</a>.</b> \(\vdash :\aleph _{0}\in \text{NC mult}.R\in \text{Prog}.\text{D}ʻR\subset \text{Cls induct}.\supset .\exists !\aleph _0\cap \text{Cl}ʻsʻ\text{D}ʻR\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*124·52·535·4.\supset \\ +\vdash \colon\ldotp \text{Hp}.\supset :\sigma =\hat{\beta} \{(\exists \gamma ).\gamma \in \text{D}ʻR.\beta &=\gamma -sʻ\overrightarrow{R}_{\text{po}}ʻ\gamma + .\exists !\beta\}.\supset .\exists !\in _{\Delta }ʻ\sigma .\\ +[*124·536] &\supset .\exists !\aleph _{0}\cap \text{Cl}ʻsʻ\text{D}ʻR\colon\ldotp \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*124·541.</b> \[\begin{align}\vdash :\aleph _{0}\in \text{NC mult}.P\in {\in}_{\Delta}ʻ(\cap \text{Cl}ʻ\rho )ʻʻ&\text{NC induct}.\supset .\\ +&\exists !\aleph _{0}\cap \text{Cl}ʻsʻ\text{D}ʻP.sʻ\text{D}ʻP\subset \rho\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*124·512. \supset \vdash :\text{Hp}.&\supset .\text{D}ʻP\in \aleph _{0}.\text{D}ʻP\subset \text{Cls induct}.\\ +[*123·1] &\supset .(\exists R).\text{D}ʻP=\text{D}ʻR.R\in \text{Prog}.\text{D}ʻR\subset \text{Cls induct}.\\ +[*124·54] &\supset .(\exists R).\text{D}ʻP=\text{D}ʻR.\exists !\aleph _{0}\cap \text{Cl}ʻsʻ\text{D}ʻR.\\ +[*13·193.*10·35] &\supset .\exists !\aleph _{0}\cap \text{Cl}ʻsʻ\text{D}ʻP &\qquad \text{(1)}\\ +\vdash .*124·512. \supset \vdash :\text{Hp}.&\supset .\text{D}ʻP\in \text{Cl}ʻ\text{Cl}ʻ\rho .\\ +[*60·2] &\supset .\text{D}ʻP\subset \text{Cl}ʻ\rho .\\ +[*60·52] &\supset .sʻ\text{D}ʻP\subset \rho &\qquad \text{(2)}\\ +\vdash .(1).(2).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*124·55.</b> \(\vdash :\aleph _{0}\in \text{NC mult}.\rho {\sim}\in \text{Cls induct}.\supset .\exists !\aleph _{0}\cap \text{Cl}ʻ\rho\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*124·511·4.\supset \vdash :\text{Hp}.&\supset .\exists !{\in}_{\Delta}ʻ(\cap \text{Cl}ʻ\rho )ʻʻ\text{NC induct}.\\ +[*124·541.*60·4] &\supset .\exists !\aleph _{0}\cap \text{Cl}ʻ\rho :\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*124·56">*124·56</a>.</b> \(\vdash :\aleph _{0}\in \text{NC mult}.\supset .-\text{Cls induct}=\text{Cls refl}.\text{N}_{0}\text{C}-\text{NC induct}=\text{NC refl}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*124·55·15. \supset \vdash :\text{Hp}.&\supset .-\text{Cls induct}\subset \text{Cls refl} &\qquad \text{(1)}\\ +\vdash .*124·271. \supset \vdash :\text{Hp}.&\supset .\text{Cls refl}\subset -\text{Cls induct} &\qquad \text{(2)}\\ +\vdash .(1).(2). \supset \vdash \colon\ldotp \text{Hp}.&\supset :-\text{Cls induct}=\text{Cls refl}: &\qquad \text{(3)}\\ +[*120·21.*124·28] &\supset :\text{N}_{0}\text{c}ʻ\rho {\sim}\in \text{NC induct}.\equiv .\text{N}_{0}\text{c}ʻ\rho \in \text{NC refl}:\\ +[*103·2.*124·2] &\supset :\alpha \in \text{N}_{0}\text{C}-\text{NC induct}.\equiv .\alpha \in \text{NC refl} &\qquad \text{(4)}\\ +\vdash .(3).(4).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_288">[Pg 288]</span></p> + +<p>The above proposition identifies the two definitions of the finite, on +the hypothesis \(\aleph _{0}\in \text{NC mult}\).</p> + +<p class="nind"><b>*124·57.</b> \(\vdash :\mu \in \text{N}_{0}\text{C}-\text{NC induct}.\supset .2^{2^\mu }\in \text{NC refl} \quad[*124·511.*116·72]\)</p> + +<p class="nind"><b>*124·58.</b> \(\vdash \colon\ldotp 2^\mu \in \text{NC refl}.\supset _{\mu }.\mu \in \text{NC refl}:\supset .\text{N}_{0}\text{C}-\text{NC induct}=\text{NC refl}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*124·57.\supset \vdash \colon\ldotp \text{Hp}.&\supset :\mu \in \text{N}_{0}\text{C}-\text{NC induct}.\supset .2^\mu \in \text{NC refl}.\\ +[\text{Hp}] &\supset .\mu \in \text{NC refl} &\qquad \text{(1)}\\ +\vdash .(1).*124·2·27.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p>The above proposition gives another hypothesis which would enable us to +identify the two definitions of the finite if it could be proved, namely +\[ +2^\mu \in \text{NC refl}.\supset _{\mu }.\mu \in \text{NC refl}, +\] +or, what comes to the same thing, +\[ +\text{Cl}ʻ\rho \in \text{Cls refl}.\supset .\rho \in \text{Cls refl}. +\]</p> + +<p class="nind"><b><a id="*124·6">*124·6</a>.</b> \(\vdash :\rho {\sim}\in \text{Cls induct}.\equiv .\text{Cl}ʻ\text{Cl}ʻ\rho \in \text{Cls refl}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*124·511.&\supset \vdash :\rho {\sim}\in \text{Cls induct}.\supset .\text{Cl}ʻ\text{Cl}ʻ\rho \in \text{Cls refl} &\qquad \text{(1)}\\ +\vdash .*120·74. &\supset \vdash :\rho \in \text{Cls induct}.\supset .\text{Cl}ʻ\text{Cl}ʻ\rho \in \text{Cls induct}.\\ +[*124·271] &\supset .\text{Cl}ʻ\text{Cl}ʻ\rho {\sim}\in \text{Cls refl} &\qquad \text{(2)}\\ +\vdash .(1).(2).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*124·61.</b> \(\vdash \colon\ldotp \aleph _{0}\in + \text{NC mult}.\supset :\rho \in \text{Cls refl}.\equiv .\text{Cl}ʻ\rho \in \text{Cls refl}.\equiv .\text{Cl}ʻ\text{Cl}ʻ\rho \in \text{Cls refl}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*124·6·271.&\supset \vdash :\rho \in \text{Cls refl}.\supset .\text{Cl}ʻ\rho \in \text{Cls refl}.\supset .\text{Cl}ʻ\text{Cl}ʻ\rho \in \text{Cls refl} &\qquad \text{(1)}\\ +\vdash .*124·6·56. &\supset \vdash \colon\ldotp \aleph _{0}\in + \text{NC mult}.\supset :\text{Cl}ʻ\text{Cl}ʻ\rho \in \text{Cls refl}.\supset .\rho \in \text{Cls refl}. &\qquad \text{(2)}\\ +[(1)] &\supset .\text{Cl}ʻ\rho \in \text{Cls refl} &\qquad \text{(3)}\\ +\vdash .(1).(2).(3).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p>The following properties of cardinals which are neither inductive nor +reflexive (supposing there are such) are easily proved. Let us put +\[ +\begin{aligned} +\text{NC med}&=\text{N}_{0}\text{C}-\text{NC induct}-\text{NC refl} \quad\text{Df},\\ +\text{Cls med}&=-\text{Cls induct}-\text{Cls refl} \quad\text{Df}, +\end{aligned} +\]</p> + +<p class="nind">where "\(\text{med}\)" stands for "mediate." Then +\[ +\mu \in \text{NC med}.\supset .\mu +_{c}1\in \text{NC med}.\mu -_{c}1\in \text{NC med}.\mu \neq \mu +_{c}1.\mu \neq \mu -_{c}1. +\] +Hence mediate cardinals have no maximum or minimum. +\[ +\begin{aligned} +\mu ,\nu \in &\text{NC med}.\supset .\mu +_{c}\nu \in &\quad\text{NC med},\\ +\mu \in &\text{NC med}.\nu \in \text{NC med}\cup \text{NC induct}-\iota ʻ0.\supset .\mu \times _{c}\nu \in &\quad\text{NC med}, +\end{aligned} +\] +whence +\[ +\begin{aligned} +&\mu \in \text{NC med}.\supset .\mu ^{2},\mu ^{3},...\in \text{NC med},\\ +&\mu ^\nu \in \text{NC med}.\supset :\mu \in \text{NC med}.\lor.\nu \in \text{NC med},\\ +&\mu \in \text{NC med}.\supset .2^{2^\mu }\in \text{NC refl},\\ +&\text{whence}\qquad \exists !\text{NC med}.\supset .(\exists \nu ).\nu \in \text{NC med}.2^\nu \in \text{NC refl}, +\end{aligned} +\] +since we have either \(\mu \in \text{NC med}.2^\mu \in \text{NC refl}\) +or \(2^\mu \in \text{NC med}.2^{2^\mu }\in \text{NC refl}\).</p> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_289">[Pg 289]</span></p> +<h2 class="nobreak" id="*125"> +*125. THE AXIOM OF INFINITY.</h2> +</div> + + +<p class="nind"><i>Summary of</i> *125.</p> + +<p>The present number is merely concerned to give a few equivalent +forms of the axiom of infinity, and of the kindred assumption of the +existence of \(\aleph _{0}\).</p> + +<p>In virtue of <a href="#*125·24">*125·24</a><a href="#*125·25">·25</a> below, if the axiom of infinity holds in any +one type, then it holds in any other type which can be derived from +this one, or from any type from which this one can be derived. Hence if +we assume, as it seems natural to do, that all extensional types are +derived from a first type, namely that of individuals, then the axiom +of infinity in any such type is equivalent to the assumption that the +number of individuals is not inductive.</p> + +<p>We deal, in this number, first with equivalent forms of \(\text{Infin ax}\), +then with equivalent forms of \(\text{Infin ax} (x)\), then +with equivalent forms of \(\exists !\aleph _{0}\) or \(\exists !\aleph_{0}(x)\). +When "\(\text{Infin ax}\)" or "\(\exists !\aleph _{0}\)" +occurs in this number without typical definition, it and all other +typically ambiguous symbols are to be taken in the lowest logically +possible types, or with the same relative types as if this had been +done. The propositions of this number are often not referred to in the +sequel, but are here collected together on account of their intrinsic +interest.</p> + +<hr class="tb"> + +<p class="nind"><b>*125·1.</b> \(\vdash \colon\ldotp \text{Infin ax}.\equiv :\alpha \in \text{NC induct}.\supset _{\alpha } .\exists !\alpha \quad[*120·3]\)</p> + +<p class="nind"><b>*125·11.</b> \(\vdash \colon\ldotp \text{Infin ax}.\equiv :\alpha \in \text{NC induct}.\supset _{\alpha }.\alpha \neq \alpha +_{c}1 \quad[*120·33]\)</p> + +<p class="nind"><b>*125·12.</b> \(\vdash \colon\ldotp \text{Infin ax}.\equiv :\alpha \in \text{NC induct}.\supset _{\alpha }.\exists !\alpha +_{c}1\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *101·12. *125·1.\supset \\ +\vdash \colon\ldotp \text{Infin ax}.&\equiv :\alpha \in \text{NC induct}-\iota ʻ0.\supset_\alpha.\exists !\alpha :\\ +[*120·423]&\equiv :\alpha \in \text{NC induct}.\supset_\alpha.\exists !\alpha +_c 1\colon\ldotp \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*125·13.</b> \(\vdash :\text{Infin ax}.\equiv .\Lambda {\sim}\in \text{NC induct} \quad[*125·1. *24·63]\)</p> + +<p class="nind"><b>*125·14.</b> \(\vdash :\text{Infin ax}.\equiv .(+_{c}1)\upharpoonright \text{NC induct}\in 1\rightarrow 1\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*123·22·24.&\supset \vdash :\text{Infin ax}.\supset .(+_{c}1)\upharpoonright \text{NC induct}\in 1\rightarrow 1 &\qquad \text{(1)}\\ +\vdash .*71·55. &\supset \vdash \colon\ldotp (+_{c}1)\upharpoonright \text{NC induct}\in 1\rightarrow 1.\supset :\\ +&\alpha ,\beta \in \text{NC induct}.\alpha +_{c}1=\beta +_{c}1.\supset _{\alpha ,\beta }.\alpha =\beta :\\ +[\text{Transp}] &\supset :\alpha ,\beta \in \text{NC induct}.\alpha \neq \beta .\supset _{\alpha ,\beta }.\alpha +_{c}1\neq \beta +_{c}1:\\ +[*10·1] &\supset :\Lambda ,\beta \in \text{NC induct}.\Lambda \neq \beta .\supset _{\beta }.\Lambda +_{c}1\neq + \beta +_{c}1:\\ +[*110·4.\text{Transp}]&\supset : \Lambda \in \text{NC induct}. \beta \in \text{NC induct}. \exists !\beta .\supset _{\beta }.\exists ! (\beta +_{c}1) &\qquad \text{(2)}\\ +\vdash .(2).*101·12.*120·13.\supset \\ +\vdash \colon\ldotp (+_{c}1) \upharpoonright \text{NC induct} \in 1 \rightarrow 1 . \Lambda \in \text{NC induct}.&\supset : \beta \in \text{NC induct}.\supset _{\beta }.\exists + !\beta :\\ +[*24·63] &\supset : \Lambda {\sim}\in \text{NC induct} &\qquad \text{(3)}\\ +\vdash .(3).*2·01.*125·13.&\supset \vdash :(+_{c}1) \upharpoonright \text{NC induct} \in 1 \rightarrow 1.\supset . \text{Infin ax} &\qquad \text{(4)}\\ +\vdash .(1).(4).\supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_290">[Pg 290]</span></p> + +<p class="nind"><b>*125·15.</b> \(\vdash \colon\ldotp \text{Infin ax}.\equiv :\rho \in \text{Cls induct}.\supset _{p}.\exists !-\rho\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*110·63.&\supset \vdash :x{\sim}\in \rho .\supset _{x}.\rho \cup \iota ʻx\in \text{Nc}ʻ\rho +_{c}1:\\ +[*10·28]&\supset \vdash :\exists !-\rho .\supset .\exists ! \text{Nc}ʻ\rho +_{c}1:\\ +[\text{Syll}] &\supset \vdash \colon\ldotp \rho \in \text{Cls induct}.\supset _{\rho }.\exists !-\rho :\supset :\\ +&\rho \in \text{Cls induct}.\supset _{\rho }.\exists ! \text{Nc}ʻ\rho +_{c}1:\\ +[*120·2] &\supset :\alpha \in \text{NC induct} .\rho \in \alpha .\supset _{\alpha ,\rho }.\exists ! \text{Nc}ʻ\rho +_{c}1 :\\ +[*100·45]&\supset :\alpha \in \text{NC induct}.\exists !\alpha .\supset _{\alpha } .\exists !\alpha +_{c}1:\\ +[*120·13.*101·12] &\supset :\alpha \in \text{NC induct}.\supset _{\alpha }.\exists !\alpha &\qquad \text{(1)}\\ +\vdash .*13·12.&\supset \vdash \colon\ldotp \alpha \in \text{NC induct}.\supset _{\alpha }.\exists !(\alpha +_{c}1):\supset :\\ +&\alpha \in \text{NC induct}. \text{N}_{0}\text{c}ʻ\rho = \alpha .\supset _{\alpha ,\rho }.\exists !(\text{N}_{0}\text{c}ʻ\rho +_{c}1) :\\ +[*120·21] &\supset :\rho \in \text{Cls induct}.\supset _{\rho }.\exists ! (\text{N}_{0}\text{c}ʻ\rho +_{c}1).\\ +[*103·11.*63·101.*110·63] &\supset _{\rho }. (\exists \gamma ,z) . \gamma \text{ sm } \rho .z{\sim} \in \gamma .\gamma \in \iota ʻ\rho \cup -\iota ʻ\rho &\qquad \text{(2)}\\ +\vdash .*13·12.*10·24. &\supset \vdash :\gamma =\rho .z{\sim} \in \gamma .\supset .\exists !-\rho &\qquad \text{(3)}\\ +\vdash .*120·426.*24·6. &\supset \vdash :\rho \in \text{Cls induct} .\gamma \neq \rho .\gamma \subset \rho .\supset .{\sim}(\gamma \text{ sm } \rho ) :\\ +[\text{Transp}] &\supset \vdash : \rho \in \text{Cls induct}.\gamma \text{ sm } \rho .\gamma \neq \rho .\supset .\exists !\gamma -\rho .\\ +[*24·561] &\supset .\exists !-\rho &\qquad \text{(4)}\\ +\vdash .(3).(4).&\supset \vdash \colon\ldotp \rho \in \text{Cls induct}: (\exists \gamma , z). \gamma \text{ sm } \rho . z {\sim} \in \gamma .\gamma \in \iota ʻ\rho \cup -\iota ʻ\rho :\supset . +\exists !-\rho &\qquad \text{(5)}\\ +\vdash .(2).(5).&\supset \vdash \colon\ldotp \alpha \in \text{NC induct}.\supset _{\alpha } . \exists ! (\alpha +_{c} 1) : \supset :\\ +&\rho \in \text{Cls induct}. \supset _{\rho } . \exists ! - \rho &\qquad \text{(6)}\\ +\vdash .(1).(6).*125·12·1.\supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*125·16.</b> \[\begin{align}\vdash : \text{Infin ax} . \equiv . \exists ! \text{Cls} - \text{Cls induct}. \equiv . &\exists ! \text{N}_{0}\text{C} + - \text{NC induct}. \equiv .\\ +&\text{V} {\sim} \in \text{Cls induct}\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *125·15 . \supset \vdash \colon\ldotp \text{Infin ax} . &\equiv : \rho \in \text{Cls induct}. \supset _{\rho }. \rho \neq \text{V} :\\ +[*13·196] &\equiv :\text{V} {\sim} \in \text{Cls induct} &\qquad \text{(1)}\\ +\vdash . *120·481 . \text{Transp} . &\supset \vdash : \exists ! \text{Cls} - \text{Cls induct}. \supset . \text{V} {\sim} \in \text{Cls induct} &\qquad \text{(2)}\\ +\vdash . (1). (2). *120·21. \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_291">[Pg 291]</span></p> + +<p class="nind"><b>*125·2.</b> \(\vdash \colon\ldotp \text{Infin ax} (x). \equiv : \alpha \in \text{NC induct}. \supset _{\alpha } . \exists ! \alpha (x) \quad[*120·301]\)</p> + +<p class="nind"><b>*125·21.</b> \(\vdash :\text{Infin ax}(x).\equiv .tʻx{\sim}\in \text{Cls induct}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*125·15.\supset \vdash \colon\ldotp \text{Infin ax}(x).&\equiv :\rho \in \text{Cls induct}\cap \text{Cl}ʻtʻx.\supset _{\rho }.\exists !-\rho :\\ +[*63·102] &\equiv :\rho \in \text{Cls induct}\cap \text{Cl}ʻtʻx.\supset _{\rho }.\rho \neq tʻx:\\ +[*13·196] &\equiv :tʻx{\sim}\in \text{Cls induct}\colon\ldotp \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*125·22.</b> \(\vdash :\text{Infin ax}(x).\equiv .t^{3}ʻx\in \text{Cls refl} \quad[*125·21.*63·66.*124·6]\)</p> + +<p class="nind"><b>*125·23.</b> \(\vdash :\text{Infin ax}(x).\equiv .\exists !\aleph _{0}(t^{2}ʻx) \quad[*125·22.*124·15]\)</p> + +<p class="nind"><b><a id="*125·24">*125·24</a>.</b> \(\vdash :\text{Infin ax}(x).\equiv .\text{Infin ax}(tʻx).\equiv .\text{Infin ax}(t^{2}ʻx).\equiv .\text{etc}.\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*125·21. \supset \vdash :\text{Infin ax}(x).&\equiv .tʻx{\sim}\in \text{Cls induct}.\\ +[*120·74] &\equiv .\text{Cl}ʻtʻx{\sim}\in \text{Cls induct}.\\ +[*63·66] &\equiv .t^{2}ʻx{\sim}\in \text{Cls induct}.\\ +[*125·21] &\equiv .\text{Infin ax}(tʻx):\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*125·25">*125·25</a>.</b> \[\begin{align}\vdash :\text{Infin ax}(\alpha ).\equiv .\text{Infin ax}(t_{00}ʻ\alpha ).\equiv .&\text{Infin ax}(t_{0}^{1}ʻ\alpha ).\equiv .\\ +&\text{Infin ax}(t^{11}ʻ\alpha ).\equiv .\text{etc}.\\ +[*116·91·92.*120·56·52.*125·21]\end{align}\]</p> + +<p class="nind"><b>*125·3.</b> \(\vdash :\exists !\aleph _{0}.\equiv .\exists !(1\rightarrow 1)\cap \hat{R} (\exists !\overrightarrow{B}ʻR.{\sim}\exists !\overrightarrow{B}ʻ\breve{R} )\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*123·1. \supset \vdash :\exists !\aleph _{0}.&\equiv .\exists !\text{Prog}.\\ +[*122·11·141] &\supset .\exists !(1\rightarrow 1)\cap \hat{R} (\exists !\overrightarrow{B}ʻR.{\sim}\exists !\overrightarrow{B}ʻ\breve{R} ) &\qquad \text{(1)}\\ +\vdash .*123·192. &\supset \vdash :\exists !(1\rightarrow 1)\cap \hat{R} (\exists !\overrightarrow{B}ʻR.{\sim}\exists !\overrightarrow{B}ʻ\breve{R} ).\supset .\exists !\aleph _{0} + &\qquad \text{(2)}\\ +\vdash .(1).(2).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*125·31.</b> \(\vdash :\exists !\aleph _{0}(x).\equiv .tʻx\in \text{Cls refl} \quad[*124·15]\)</p> + +<p class="nind"><b>*125·32.</b> \(\vdash :\exists !\aleph _{0}(x).\equiv \exists !(1\rightarrow 1)\cap \overleftarrow{\text{D}}ʻtʻx-\overleftarrow{\text{ᗡ}}ʻtʻx\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*63·102. \supset \vdash :\exists !&(1\rightarrow 1)\cap -\overleftarrow{\text{D}}ʻtʻx-\overleftarrow{\text{ᗡ}}ʻtʻx.\equiv .\\ +&(\exists R).R\in 1\rightarrow 1.\text{D}ʻR=tʻx.\text{ᗡ}ʻR\subset tʻx.\exists !tʻx-\text{ᗡ}ʻR.\\ +[*124·1] &\equiv .tʻx\in \text{Cls refl} &\qquad \text{(1)}\\ +\vdash .(1).*125·31.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*125·33.</b> \(\vdash \colon\ldotp \exists !\aleph _{0}(x).\equiv :\alpha \subset tʻx.\exists !\alpha .\supset _{\alpha }.\exists + !(1\rightarrow 1)\cap \overleftarrow{\text{D}}ʻ\alpha -\overleftarrow{\text{ᗡ}}ʻ\alpha\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*73·7.51·222. &\supset \vdash :\alpha \subset tʻx.y\in \alpha .z\in tʻx-\alpha .\supset .\alpha \text{ sm }(\alpha -\iota ʻy)\cup \iota ʻz.\\ +[*73·1] &\supset .(\exists R).R\in 1\rightarrow 1.\text{D}ʻR=\alpha .\text{ᗡ}ʻR=(\alpha -\iota ʻy)\cup \iota ʻz.\\ +[*33·6·61] &\supset .\exists !(1\rightarrow 1)\cap \overleftarrow{\text{D}}ʻ\alpha -\overleftarrow{\text{ᗡ}}ʻ\alpha &\qquad \text{(1)}\\ +\vdash .(1). &\supset \vdash :\exists !\alpha .\exists !tʻx-\alpha .\supset .\exists !(1\rightarrow 1)\cap \overleftarrow{\text{D}}ʻ\alpha -\overleftarrow{\text{ᗡ}}ʻ\alpha :\\ +[*63·102]& \supset \vdash :\exists !\alpha .\alpha \subset tʻx.\alpha \neq tʻx.\supset .\exists !(1\rightarrow 1)\cap \overleftarrow{\text{D}}ʻ\alpha -\overleftarrow{\text{ᗡ}}ʻ\alpha &\qquad \text{(2)}\\ +\vdash .(2).*125·32.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_292">[Pg 292]</span></p> + +<p class="nind"><b>*125·34.</b> \(\vdash :\exists !\aleph _{0}(x).\equiv .\iota ʻtʻx{\sim}\in \text{NC}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*125·32.\supset \vdash \colon\ldotp {\sim}\exists !\aleph _{0}(x).&\equiv :R\in 1\rightarrow 1.\text{D}ʻR=tʻx.\supset _{R}.\text{ᗡ}ʻR=tʻx:\\ +[*100·13] &\equiv :\text{Nc}ʻtʻx=\iota ʻtʻx.\\ +[*100·41·45] &\equiv :\iota ʻtʻx\in \text{NC} &\qquad \text{(1)}\\ +\vdash .(1).\text{Transp}.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*125·35.</b> \(\vdash \colon\ldotp \aleph _{0}\text{NC mult}.\supset :\exists !\aleph _{0}(x).\equiv .\text{Infin ax}(x)\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*125·21.*124·56.\supset \vdash \colon\ldotp \text{Hp}.\supset :\text{Infin ax}(x).&\equiv .tʻx\in \text{Cls refl}.\\ +[*125·31] &\equiv .\exists !\aleph _{0}(x)\colon\ldotp \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*125·36.</b> \(\vdash :\text{Infin ax}(\text{Cls}).\equiv .\exists !\aleph _{0}(\text{Cls})\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*24·14.*63·102·66.\supset \vdash .t^{2}ʻx&=\text{Cl}ʻ\text{V} +[*24·11] &=\text{Cls} &\qquad \text{(1)}\\ +\vdash .(1).*125·24.\supset \vdash :\text{Infin ax}(\text{Cls}).&\equiv .\text{Infin ax}(x).\\ +[*125·23.(1)] &\equiv .\exists !\aleph _{0}(\text{Cls}):\supset \vdash .\text{Prop} +\end{array} +\]</p> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_293">[Pg 293]</span></p> +<h2 class="nobreak" id="*126"> +*126. ON TYPICALLY INDEFINITE INDUCTIVE CARDINALS.</h2> +</div> + + +<p class="nindc"><i>Recapitulation of Conventions and Summary of</i> *126.</p> + +<p>We have now arrived at the stage where we can adopt the standpoint of +ordinary arithmetic, and can for the future in arithmetical operations +with cardinals ignore differences of type. In order to understand how +this is so, it will be necessary briefly to recall the line of thought +of some of the previous numbers and the conventions upon which the +symbolism is based.</p> + +<p>The symbolism of <a href="#*102">*102</a>, though perfectly precise as to the typical +relations of the various symbols, is in fact too complex for use, +except in cases of absolute necessity. It is better to use the +typically ambiguous symbols \(\text{Nc}\) and \(\text{sm}\), combined +with some simple rules of interpretation of the symbolism, so as to +secure that the various occurrences of the same symbols are in their +proper relationships of type. This is the course followed in <a href="#*100">*100</a>, +<a href="#*101">*101</a>, and in every number from <a href="#*110">*110</a> onwards.</p> + +<p>The important symbols which involve an explicit or implicit use of +\(\text{Nc}\) or \(\text{ sm }\) are called 'formal numbers,' and it is +only necessary to make the rules of interpretation apply to them.</p> + +<p>A constant formal number is any symbol representing a typically +ambiguous constant such that there is a constant \(\alpha\) such that, +however the ambiguities of type may be determined, the former constant +is identical with \(\text{Nc}ʻ\alpha\). The variable formal numbers are +defined by enumeration. They are divided into three Sets, the Primary +Set, the Argumental Set, and the Arithmetical Set.</p> + +<p>The Primary Set consists of \(\text{Nc}ʻ\alpha\), \(\Sigma\text{Nc}ʻ\kappa\), +\(\Pi \text{Nc}ʻ\kappa\), where \(\alpha\) is a variable \(\text{Cls}\) +of any type and \(\kappa\) is a variable \(\text{Cls}^{2}\) of any type. +Also \(\alpha\) and \(\kappa\) may themselves be complex symbols which +in some way involve variables.</p> + +<p>The Argumental Set has only one member \(\text{ sm }ʻʻ\mu\), where +\(\mu\) is a variable \(\text{Cls}^{2}\) of any type. In its capacity +of a formal number \(\text{ sm }ʻʻ\mu\) is only interesting when +\(\mu\) is an \(\text{NC}\); then \(\text{ sm }ʻʻ\mu\) gives the +corresponding \(\text{NC}\) in another type, provided that \(\mu\) is +not \(\Lambda\). Also \(\mu\) may be a complex symbol which in some +way involves a variable, <i>e.g.</i> \(\text{ sm }ʻʻ\text{Nc}ʻ\alpha\) +is a formal number of the Argumental Set: \(\mu\) is called the +<i>argument</i> of \(\text{ sm }ʻʻ\mu\).</p> + +<p><span class="pagenum" id="Page_294">[Pg 294]</span></p> + +<p>The Arithmetical Set consists of \(\mu +_{c}\nu\), \(\mu \times_{c}\nu\), +\(\mu ^\nu\), \(\mu -_{c}\nu\). These formal numbers +are only interesting when \(\mu\) and \(\nu\) are also members of +\(\text{NC}\). Also \(\mu\) and \(\nu\) may be complex symbols, so long +as one of them at least involves a variable. For example \(2^{3+_{c}\nu}\) +is a formal number, and so is \(\alpha +_{c}(3+_{c}\nu)\).</p> + +<p>The Primary and Argumental and Arithmetical Sets of Formal Numbers are +derived from the corresponding sets of <i>variable</i> formal numbers, +by adding to them the constant formal numbers obtained by substituting +constants for the variables occurring in the expressions for the +members of the variable set in question.</p> + +<p>In the formal numbers of the arithmetical set as written above, \(\mu\) +and \(\nu\) are called the <i>first components</i>. Thus every formal +number of this set has two first components. The first components (if +any) of the first components are also called <i>components</i> of the +original formal number, and so on; so that components of components are +components of the original symbol.</p> + +<p>A formal number of the arithmetical set, whose components are all +formal numbers, either constant or variable but <i>not</i> belonging to +the argumental set, is called a <i>pure arithmetical</i> formal number. +These are the formal numbers which it is important in arithmetic to +secure from assuming the value \(\Lambda\) owing to lowness of type.</p> + +<p>The logical investigation of <a href="#*100">*100</a> and <a href="#*101">*101</a>, where typically ambiguous +formal numbers are used, is directly concerned in investigating the +premisses necessary to secure various propositions from fluctuating +truth-values owing to the intrusion of null-values among the cardinals. +The convention, necessary to avoid determinations of type which we +never wish to consider, is as follows, where the terms used are +explained fully in the prefatory statement:</p> + +<p>\(\text{IT}\). Argumental occurrences are bound to logical and +attributive occurrences; and, if there are no argumental occurrences, +equational occurrences are bound to logical occurrences. This rule only +applies so far as meaning permits after the assignment of types to the +real variables.</p> + +<p>In <a href="#*110">*110</a>, <a href="#*113">*113</a>, <a href="#*116">*116</a>, <a href="#*119">*119</a> we consider the arithmetical operations of +addition, multiplication, exponentiation, and subtraction. Also in +<a href="#*117">*117</a> we consider the comparison of cardinal numbers in respect to the +relation of greater and less.</p> + +<p>There is no interest in complicating our theorems by allowing for the +cases when a pure arithmetical formal number, whose components are +ambiguous as to type, becomes equal to \(\Lambda\) owing to the low +type of one of its <i>components</i>. Also in the theory of greater and +less the possibility of null-values in low types has no real interest. +Accordingly these are excluded from any consideration by the definitions</p> +<p class="nindc"> +<a href="#*110·03">*110·03</a><a href="#*110·04">·04</a>, <a href="#*113·04">*113·04</a><a href="#*113·05">·05</a>, <a href="#*116·03">*116·03</a><a href="#*116·04">*116·04</a>, <a href="#*117·02">*117·02</a><a href="#*117·03">·03</a>, +</p><p><span class="pagenum" id="Page_295">[Pg 295]</span> +as far as members of the primary set of formal numbers are concerned; +and for other formal numbers by the following convention:</p> + +<p>\(\text{IIT}\). Whenever a formal number \(\sigma\) occurs, so that, +if it were replaced by \(\text{Nc}ʻ\alpha\), the dominant type of +\(\text{Nc}ʻ\alpha\) would by definition have to be adequate, then the +dominant type of \(\sigma\) is also to be adequate.</p> + +<p>When \(\sigma\) is a pure arithmetical formal number, this convention +secures that the type of every component is adequate.</p> + +<p>But in arithmetic we also wish to avoid the intrusion of null-values +into the consideration of equations, so far as this avoidance can be +attained by the use of high types. Accordingly when we are concerned +with the purely arithmetical point of view, we add also the following +definition and convention (\(\text{AT}\)).</p> + +<p><i>Definition</i>. An <i>arithmetical equation</i> is an equation +between pure arithmetical formal numbers whose dominant types are both +determined adequately.</p> + +<p>\(\text{AT}\). All equations involving pure arithmetical formal numbers +are to be arithmetical.</p> + +<p>This convention is used in <a href="#*117">*117</a> and in some earlier propositions which +are noted in the prefatory statement.</p> + +<p>Its effect is to render the statement of hypotheses often unnecessary. +Examples of its application to the numbers where it is not used in the +symbolism are also considered in the prefatory statement.</p> + +<p>In the case of the inductive numbers we cannot logically prove, apart +from \(\text{Infin ax}\), that one type exists which is adequate for +all the formal numbers 0, 1, 2, 3, etc. But we can prove that for any +particular inductive number, say 521, a type exists for which 521 is +not equal to \(\Lambda\). Accordingly for a given symbolic form, in +which the symbolism necessarily has only finite complexity, when the +types of variables which by hypothesis represent inductive classes or +inductive numbers, not \(\Lambda\), have been settled, it is always +possible to fix on a type which will be adequate for all the pure +arithmetical formal numbers produced by the symbolism of the form, and +also at the same time (and here the peculiar properties of inductive +numbers come in) to have chosen the original types of the variables so +that any of the variables can assume the value of any assigned constant +inductive number, say 521, without being null.</p> + +<p>The result is that we may assume that the symbols representing +inductive numbers are never null, and thereby obtain the stable +truth-values of propositions about them.</p> + +<p>Accordingly we proceed as follows: we put</p> + +<p class="nind"><b>*126·01.</b> \(\text{NC ind}=\text{NC induct}-\iota ʻ\Lambda \quad\text{Df}\).</p> + +<p><span class="pagenum" id="Page_296">[Pg 296]</span></p> + +<p>We make the rule that when \(\text{NC ind}\) appears, convention +\(\text{AT}\) is always applied. The result is that when a formal +number is an \(\text{NC ind}\) we need never think about its type, and +accordingly all the conventions vanish from the mind, as far as pure +arithmetical indefinite inductive cardinals are concerned. We supersede +all other conventions by the single one that, if it has been proved +or assumed that a formal number represents an inductive cardinal, +the types are so arranged that that formal number is not equal to +\(\Lambda\). The proofs of propositions in this number consist largely +of the production of a definite type in which this result is attained.</p> + +<p>The important propositions are</p> + +<p class="nind"><b>*126·12.</b> \(\vdash :\nu \in \text{NC ind}.\supset .(\nu +_{c}1)\cap tʻ\nu \in \text{NC ind}\)</p> + +<p class="nind"><b>*126·121.</b> \(\vdash .1,2,3, ... \in \text{NC ind}\)</p> + +<p class="nind"><b>*126·13·14·15.</b> \(\vdash :\alpha ,\beta \in \text{NC ind}.\supset .\alpha +_{c}\beta ,\alpha \times _{c}\beta ,\alpha ^\beta \in \text{NC ind}\)</p> + +<p class="nind"><b>*126·141.</b> \(\vdash :\alpha ,\beta \in \text{NC ind}-\iota ʻ0.\equiv .\alpha \times _{c}\beta \in \text{NC ind}-\iota ʻ0\)</p> + +<p class="nind"><b>*126·151.</b> \(\vdash :\alpha ,\beta \in \text{NC ind}-\iota ʻ0.\alpha \neq 1.\equiv .\alpha ^\beta \in \text{NC ind}-\iota ʻ0-\iota ʻ1\)</p> + +<p>Also <a href="#*126·4">*126·4</a><a href="#*126·42">·42</a><a href="#*126·43">·43</a> give the fundamental propositions for subtraction, +division, and "inverse exponentiation"; and <a href="#*126·5">*126·5</a><a href="#*126·51">·51</a><a href="#*126·52">·52</a><a href="#*126·53">·53</a> the +fundamental propositions for the relations of greater and less.</p> + +<hr class="tb"> + +<p class="nind"><b>*126·01.</b> \(\text{NC ind}=\text{Nc induct}-\iota ʻ\Lambda \quad\text{Df}\)</p> + +<p>Whenever the symbol \(\text{NC ind}\) is used the <i>Rule of Indefinite +Numbers</i> is adhered to, so that all consideration of distinctions +in type among inductive cardinals can be laid aside (cf. Prefatory +Statement and also the Summary of this number).</p> + +<p class="nind"><b>*126·011.</b> \(\vdash :\nu \in \text{NC ind}.\equiv .\nu \in \text{NC induct}-\iota ʻ\Lambda \quad[(*126·01)]\)</p> + +<p class="nind"><b>*126·1.</b> \(\vdash :\nu \in \text{NC ind}.\equiv .(\exists \alpha ).\alpha \in \text{Cls induct}.\nu =\text{Nc}ʻ\alpha .\exists !\nu\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*120·14.*100·4.*126·011.\supset \\ +\vdash :\nu \in \text{NC ind}.&\supset .(\exists \alpha ).\nu =\text{Nc}ʻ\alpha .\nu \in \text{NC induct}-\iota ʻ\Lambda .\\ +[*118·01] &\supset .(\exists \alpha ).\nu =\text{Nc}ʻ\alpha .\text{Nc}ʻ\alpha \in \text{NC induct}-\iota ʻ\Lambda .\exists !\nu .\\ +[*120·211] &\supset .(\exists \alpha ).\alpha \in \text{Cls induct}.\nu =\text{Nc}ʻ\alpha .\exists !\nu &\qquad \text{(1)}\\ +\vdash .*120·21. &\supset \vdash :(\exists \alpha ).\alpha \in \text{Cls induct}.\nu =\text{Nc}ʻ\alpha .\exists !\nu .\\ +&\supset .(\exists \alpha ).N _{0}cʻ\alpha \in \text{Nc induct}.\nu =\text{Nc}ʻ\alpha .\exists !\nu .\\ +[*120·15.*100·511] &\supset .\nu \in \text{NC induct}-\iota ʻ\Lambda &\qquad \text{(2)}\\ +\vdash .(1).(2).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*126·101.</b> \[\begin{align}&\vdash \colon\ldotp \mu ,\nu \in \text{NC ind}.\exists !\mu _{\lambda }.\supset :\mu _{\lambda }=\nu _{\lambda }.\equiv + .\mu _{\lambda }=\nu .\equiv .\mu =\nu \\ +&[*126·1.*103·16]\end{align}\]</p> + +<p class="nind"><b><a id="*126·11">*126·11</a>.</b> \(\vdash .0\in \text{NC ind} \quad[*120·12.*101·12]\)</p> + +<p><span class="pagenum" id="Page_297">[Pg 297]</span></p> + +<p class="nind"><b><a id="*126·12">*126·12</a>.</b> \(\vdash :\nu \in \text{NC ind}.\supset .(\nu +_{c}1)\cap tʻ\nu \in \text{NC ind}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*120·151. &\supset \vdash :\nu \in \text{NC ind}.\supset .\nu +_{c}1\in \text{NC induct} &\qquad \text{(1)}\\ +\vdash .*117·66.*118·01. &\supset \vdash :\alpha \in \text{Cls induct}.\nu =\text{N}_{0}\text{c}ʻ\alpha .\exists !\nu .\supset .\text{Nc}ʻ\text{Cl}ʻ\alpha >\nu .\\ +[*126·1.*120·429] &\supset .\text{Nc}ʻ\text{Cl}ʻ\alpha \geq \nu +_{c}1.\\ +[*103·13.*117·32] &\supset .\exists !(\nu +_{c}1)\cap tʻ\text{Cl}ʻ\alpha .\\ +[*103·12.*60·34] &\supset .\exists !(\nu +_{c}1)\cap tʻ\nu &\qquad \text{(2)}\\ +\vdash .(1).(2).*126·1.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*126·121">*126·121</a>.</b> \(\vdash .1,2,3, ... \in \text{NC ind} \quad[*126·11·12]\)</p> + +<p>This proposition, taken in connection with <a href="#*120·4232">*120·4232</a>, embodies the +convention named the Rule of Indefinite Numbers and its justification. +The convention is that 1, 2, 3, ... are always in future to be used in +existential types. In other words whenever any particular inductive +number is employed, it is determined in a type in which it is not +\(\Lambda\). The justification is that by <a href="#*126·11">*126·11</a><a href="#*126·12">·12</a> such a type can +always be found for each particular inductive number.</p> + +<p>The convention is also applied to arithmetical formal numbers in +<a href="#*126·13">*126·13</a><a href="#*126·14">·14</a><a href="#*126·15">·15</a>.</p> + +<p>For all arithmetical and equational occurrences this convention is +really the outcome of \(\text{IT}\), \(\text{IIT}\), and \(\text{AT}\).</p> + +<p class="nind"><b><a id="*126·13">*126·13</a>.</b> \[\begin{align}&\vdash :\alpha ,\beta \in \text{NC ind}.\equiv .\alpha +_{c}\beta \in \text{NC ind}\\ +&[*120·71.*126·1.*110·3.*103·13]\end{align}\]</p> + +<p class="nind"><b><a id="*126·14">*126·14</a>.</b> \[\begin{align}&\vdash :\alpha ,\beta \in \text{NC ind}.\supset .\alpha \times _{c}\beta \in \text{NC ind}\\ +&[*120·72.*126·1.*113·25.*103·13]\end{align}\]</p> + +<p class="nind"><b>*126·141.</b> \[\begin{align}&\vdash :\alpha ,\beta \in \text{NC ind}-\iota ʻ0.\equiv .\alpha \times _{c}\beta \in \text{NC ind}-\iota ʻ0\\ +&[*120·721.*113·114]\end{align}\]</p> + +<p class="nind"><b><a id="*126·15">*126·15</a>.</b> \(\vdash :\alpha ,\beta \in \text{NC ind}.\supset .\alpha ^\beta \in \text{NC ind} \quad[*120·73.*116·25.*103·13]\)</p> + +<p class="nind"><b>*126·151.</b> \[\begin{align}&\vdash :\alpha ,\beta \in \text{NC ind}-\iota ʻ0.\alpha \neq 1.\equiv .\alpha ^\beta \in \text{NC ind}-\iota ʻ0-\iota ʻ1\\ +&[*120·731.*116·35.*117·592]\end{align}\]</p> + +<p class="nind"><b>*126·23.</b> \(\vdash :\mu \in \text{NC}.\exists !\mu \cap tʻ\alpha .\supset .\exists !2^\mu \cap tʻtʻ\alpha .\exists !(\mu +_{c}1)\cap tʻtʻ\alpha\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*63·661.*116·72.\supset \\ +\vdash :\mu \in \text{NC}.\beta \in \mu \cap tʻ\alpha .\supset .\text{Cl}ʻ\beta \in 2^\mu \cap tʻtʻ\alpha &\qquad \text{(1)}\\ +\vdash .(1).*117·32.\supset \\ +\vdash :\text{Hp}(1).2^\mu \geq \nu .\supset .\exists !\text{ sm }ʻʻ\nu \cap tʻtʻ\alpha &\qquad \text{(2)}\\ +\vdash .*117·661·31.\supset \vdash :\text{Hp}(1).\supset .2^\mu \geq \mu +_{c}1 &\qquad \text{(3)}\\ +\vdash .(1).(2).(3).*100·511.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_298">[Pg 298]</span></p> + +<p class="nind"><b>*126·31.</b> \(\vdash :\alpha +_{c}1\in \text{NC ind}.\equiv .\alpha \in \text{NC ind} \quad[*126·12·13·121.*120·452]\)</p> + +<p>Note that the specification of the type of \(\alpha +_{c}1\) is omitted +in accordance with the convention. The reference to <a href="#*126·12">*126·12</a> shows that +it is always possible to apply the convention.</p> + +<p class="nind"><b>*126·32.</b> \(\vdash :\alpha \in \text{NC}-\iota ʻ0-\iota ʻ\Lambda .\nu \in \text{NC ind}.\supset .\alpha +_{c}\nu >\nu \quad[*120·428.*110·3]\)</p> + +<p class="nind"><b>*126·33.</b> \(\vdash \colon\ldotp \alpha \in \text{NC ind}.\beta \in \text{NC}-\iota ʻ\Lambda .\supset :\alpha <\beta .\lor.\alpha =\beta .\lor.\alpha >\beta \quad[*120·441]\)</p> + +<p class="nind"><b><a id="*126·4">*126·4</a>.</b> \[\begin{align}&\vdash \colon\ldotp \mu ,\nu ,\varpi \in \text{NC ind}.\supset :\mu +_{c}\varpi =\nu +_{c}\varpi .\equiv .\mu =\nu \\ +&[*126·13.*120·41]\end{align}\]</p> + +<p class="nind"><b>*126·41.</b> \[\begin{align}&\vdash \colon\ldotp \mu ,\nu ,\varpi \in \text{NC ind}.\varpi \neq 0.\supset :\mu \times _{c}\varpi =\nu \times _{c}\varpi .\equiv .\mu =\nu \\ +&[*120·51.*126·14]\end{align}\]</p> + +<p class="nind"><b><a id="*126·42">*126·42</a>.</b> \[\begin{align}&\vdash \colon\ldotp \mu \nu ,\varpi \in \text{NC ind}.\varpi \neq 0.\supset :\mu ^{\varpi }=\nu ^{\varpi }.\equiv .\mu =\nu \\ +&[*120·55.*126·15]\end{align}\]</p> + +<p class="nind"><b><a id="*126·43">*126·43</a>.</b> \[\begin{align}&\vdash \colon\ldotp \mu ,\nu ,\varpi \in \text{NC ind}.\varpi \neq 0.\omega \neq 1·\supset :\varpi ^\mu =\varpi ^\nu .\equiv .\mu =\nu \\ +&[*120·53.*126·15]\end{align}\]</p> + +<p class="nind"><b><a id="*126·5">*126·5</a>.</b> \(\vdash \colon\ldotp \mu ,\nu ,\varpi \in \text{NC ind}.\supset :\mu +_{c}\varpi >\nu +_{c}\varpi .\equiv .\mu >\nu\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*117·561. &\supset \vdash :\text{Hp}.\mu >\nu .\supset .\mu +_{c}\varpi \geq \nu +_{c}\varpi &\qquad \text{(1)}\\ +\vdash .*126·4. &\supset \vdash :\text{Hp}.\mu >\nu .\supset .\mu +_{c}\varpi \neq \nu +_{c}\varpi &\qquad \text{(2)}\\ +\vdash .(1).(2).*117·26. &\supset \vdash :\text{Hp}.\mu >\nu .\supset .\mu +_{c}\varpi >\nu +_{c}\varpi &\qquad \text{(3)}\\ +\vdash .*117·561.\text{Transp}.*117·281.\supset\\ +\vdash :\text{Hp}.\mu +_{c}\varpi >\nu +_{c}\varpi .&\supset .{\sim}(\nu \geq \mu ).\\ +[*126·33] &\supset .\mu >\nu &\qquad \text{(4)}\\ +\vdash .(3).(4).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_299">[Pg 299]</span></p> + +<p class="nind"><b><a id="*126·51">*126·51</a>.</b> \[\begin{align}&\vdash \colon\ldotp \mu ,\nu ,\varpi \in \text{NC ind}.\varpi \neq 0.\supset :\mu \times _{c}\varpi >\nu \times _{c}\varpi + .\equiv .\mu >\nu \\ +&[*117·571.*126·41]\end{align}\]</p> + +<p>The proof proceeds as in <a href="#*126·5">*126·5</a>.</p> + +<p class="nind"><b><a id="*126·52">*126·52</a>.</b> \[\begin{align}&\vdash \colon\ldotp \mu ,\nu ,\varpi \in \text{NC ind}.\varpi \neq 0.\supset :\mu ^{\varpi }>\nu ^{\varpi }.\equiv .\mu >\nu \\ +&[*117·581.*126·42]\end{align}\]</p> + +<p class="nind"><b><a id="*126·53">*126·53</a>.</b> \[\begin{align}&\vdash \colon\ldotp \mu ,\nu ,\varpi \in \text{NC ind}.\varpi \neq 0.\varpi \neq 1.\supset :\varpi ^\mu >\varpi ^\nu + .\equiv .\mu >\nu \\ +&[*117·591.*126·43]\end{align}\]</p> + + +<hr class="chap x-ebookmaker-drop"> + + +<div class="chapter"> +<h2 class="nobreak" id="PART_IV">PART IV.<br> +<br> +RELATION-ARITHMETIC.</h2> +</div> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_301">[Pg 301]</span></p> +<h2 class="nobreak" id="SUMMARY_OF_PART_IV">SUMMARY OF PART IV.</h2> +</div> + + +<p><span class="allsmcap">The</span> subject to be treated in this Part is a general kind of +arithmetic of which ordinal arithmetic is a particular application. +The form of arithmetic to be treated in this Part is applicable to all +relations, though its chief importance is in regard to such relations +as generate series. The analogy with cardinal arithmetic is very close, +and the reader will find that what follows is much facilitated by +bearing the analogy in mind.</p> + +<p>The outlines of relation-arithmetic are as follows. We first define +a relation between relations, which we shall call <i>ordinal +similarity</i> or <i>likeness</i>, and which plays the same part for +relations as similarity plays for classes. Likeness between \(P\) and +\(Q\) is constituted by the fact that the fields of \(P\) and \(Q\) +can be so correlated by a one-one relation that if any two terms have +the relation \(P\), their correlates have the relation \(Q\), and vice +versa. If \(P\) and \(Q\) generate series, we may express this by +saying that \(P\) and \(Q\) are like if their fields can be correlated +without change of order. Having defined likeness, our next step is to +define the <i>relation-number</i> of a relation \(P\) as the class of +relations which are like \(P\), just as the cardinal number of a class +\(\alpha\) is the class of classes which are similar to \(\alpha\). We +then proceed to addition. The ordinal sum of two relations \(P\) and +\(Q\) is defined as the relation which holds between \(x\) and \(y\) +when \(x\) and \(y\) have the relation \(P\) or the relation \(Q\), or +when \(x\) is a member of \(CʻP\) and \(y\) is a member of \(CʻQ\). +If \(P\) and \(Q\) generate series, it will be seen that this defines +the sum of \(P\) and \(Q\) as the series resulting from adding the +\(Q\)-series after the end of the \(P\)-series. The sum is thus not +commutative. The sum of the relation-numbers of \(P\) and \(Q\) is of +course the relation-number of their sum, provided \(CʻP\) and \(CʻQ\) +have no common terms.</p> + +<p>The ordinal product of two relations \(P\) and \(Q\) is the relation +between two couples \(z\downarrow x\), \(w\downarrow y\), when \(x\), +\(y\) belong to \(CʻP\) and \(z\), \(w\) belong to \(CʻQ\) and either +\(xPy\) or \(x=y.zQw\). Thus, for example, if the field of \(P\) +consists of \(1_{P}\), \(2_{P}\), \(3_{P}\), and the field of \(Q\) +consists of \(1_{Q}\), \(2_{Q}\), the relation \(P \times Q\) will hold +from any earlier to any later term of the following series: +\[ +1_{Q}\downarrow 1_{P},\, 2_{Q}\downarrow 1_{P},\, 1_{Q}\downarrow 2_{P},\, 2_{Q}\downarrow 2_{P},\, 1_{Q}\downarrow 3_{P},\, + 2_{Q}\downarrow 3_{P}. +\] +It is plain that, denoting the ordinal product of \(P\) and \(Q\) by +\(P \times Q\), we have +\[ +Cʻ(P\times Q)=CʻP\times CʻQ, +\]<span class="pagenum" id="Page_302">[Pg 302]</span> +where the second "\(\times\)" as standing between classes has the +meaning defined in *113·01.</p> + +<p>Infinite ordinal sums and products will also be defined, but the +definitions are somewhat complicated.</p> + +<p>The arithmetic which results from the above definitions satisfies all +those of the formal laws which are satisfied in ordinal arithmetic, +when this is not confined to finite ordinals; that is to say, +relation-numbers satisfy the associative law for addition and for +multiplication<a id="FNanchor_10" href="#Footnote_10" class="fnanchor">[10]</a>, they satisfy the distributive law in the shape +(where the + and \(\times\) are those appropriate to relation-numbers) +\[ +(\beta +\gamma )\times \alpha =(\beta \times \alpha )+(\gamma \times \alpha ), +\] +and they satisfy the exponential laws +\[ +\begin{aligned} +\alpha ^\beta \times \alpha ^\gamma &=\alpha ^{(\beta +\gamma )},\\ +(\alpha ^\beta )^\gamma &=\alpha ^{\beta \times \gamma }. +\end{aligned} +\] +They do not in general satisfy the commutative law either in addition +or in multiplication, nor do they satisfy the distributive law in the +form +\[ +\alpha \times (\beta +\gamma )=(\alpha \times \beta )+(\alpha \times \gamma ), +\] +nor the exponential law +\[ +\alpha ^\gamma \times \beta ^\gamma =(\alpha \times \beta )^\gamma . +\] +But in the particular case in which the relations concerned are finite +serial relations, the corresponding relation-numbers do satisfy these +additional formal laws; hence the arithmetic of <i>finite</i> ordinals +is exactly analogous to that of inductive cardinals (cf. Part V, +Section E).</p> + +<p>If the relations concerned are limited to well-ordered relations, +relation-arithmetic becomes ordinal arithmetic as developed by Cantor; +but many of Cantor's propositions, as we shall see in this Part, do not +require the limitation to well-ordered relations.</p> + + +<div class="footnotes"><h3>FOOTNOTES:</h3> + +<div class="footnote"> + +<p class="nind"> +<a id="Footnote_10" href="#FNanchor_10" class="label">[10]</a> +For the associative law of multiplication, a hypothesis +is required as to the kind of relation concerned. Cf. <a href="#*174·241">*174·241</a><a href="#*174·25">·25</a>.</p> + +</div> +</div> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_303">[Pg 303]</span></p> +<h2 class="nobreak" id="SECTION_A_II">SECTION A.<br> +<span class="allsmcap">ORDINAL SIMILARITY AND RELATION-NUMBERS.</span></h2> +</div> + + +<p class="nind"><i>Summary of Section A.</i></p> + +<p>Two series generated by the relations \(P\) and \(Q\) respectively +are said to be ordinally similar when their terms can be correlated +as they stand, without change of order. In the accompanying figure, +the relation \(S\) correlates the members of \(CʻP\) and \(CʻQ\) in +such a way that if \(xPy\), then \((\breve{S} ʻx)Q(\breve{S} ʻy)\), +and if \(zQw\), then \((Sʻz)P(Sʻw)\). It is evident that the journey +from \(x\) to \(y\) (where \(xPy\)) may, in such a case, be taken +by going first to \(\breve{S} ʻx\), thence to \(\breve{S} ʻy\), and +thence back to \(y\), so that \(xPy.\equiv .x(S\mid Q\mid \breve{S})y\), +<i>i.e.</i> \(P=S\mid Q\mid \breve{S}\). Hence to say that \(P\) +and \(Q\) are ordinally similar is equivalent to saying that there is +a one-one relation \(S\) which has \(CʻQ\) for its converse domain +and gives \(P=S\mid Q\mid \breve{S}\). In this case we call \(S\) a +<i>correlator</i> of \(Q\) and \(P\).</p> + +<figure class="figcenter width500" id="i_303" style="width: 1200px;"> +<img src="images/i_303.jpg" width="1200" height="575" alt="Commutative +diagram showing relations P, Q, S, and Š connecting elements x and y. +Top row shows P:x→y extending to P=s|Q|Š. Bottom row shows Q:Š'x→Š'y +extending to Q. Vertical arrows show S:x↓ and Š:y↑, demonstrating P +decomposes as composition s|Q|Š in relational algebra."> +</figure> + +<p>We denote the relation of ordinal similarity by "\(\text{smor}\)," +which is short for "similar ordinally." Thus +\[ +P\,\text{smor}\,Q.\equiv .(\exists S).S\in 1\rightarrow 1.CʻQ=\text{ᗡ}ʻS.P=S\mid Q\mid \breve{S} . +\]</p> + +<p>It will be found that the relation \(S\mid Q\mid \breve{S}\) plays the +same part in relation to \(Q\) in relation-arithmetic as \(Sʻʻ\beta\) +plays in relation to \(\beta\) in cardinal arithmetic. It is therefore +desirable to have a simpler notation for \(S\mid Q\mid \breve{S}\). We +put +\[ +S^{;}Q=S\mid Q\mid \breve{S} \quad\text{Df}. +\]<span class="pagenum" id="Page_304">[Pg 304]</span> +We shall find that the semi-colon so defined has the same kind of +properties in relation-arithmetic as the two inverted commas have in +cardinal arithmetic. Corresponding to the notation \(S_{\in }ʻ\beta\), +we put +\[ +S\dagger Q = S\mid Q\breve{S} \quad\text{Df}. +\] +We shall thus have \(S\dagger =S\parallel\breve{S}\). It will +appear that \(S\dagger\) has ordinal properties analogous to +the cardinal properties of \(S_{\in }\). Thus <i>e.g.</i> where +\(S\parallel\breve{S} _{\in }\) appears as a cardinal correlator, +\(S\parallel \text{Cnv}ʻS\dagger\) will appear as an ordinal correlator +(in each case with the converse domain suitably limited).</p> + +<p>The elementary properties of \(S^{;}Q\) will be considered in <a href="#*150">*150</a>. We +shall then, in <a href="#*151">*151</a>, be able to study ordinal similarity, taking as our +definition of an ordinal correlator +\[ +P\,\overline{\text{smor}}\,Q=\hat{S}\{S\in 1\rightarrow 1.CʻQ=\text{ᗡ}ʻS.P=S^{;}Q\} \quad\text{Df}, +\] +and defining two relations as ordinally similar when they have at least +one ordinal correlator, <i>i.e.</i> putting (on the analogy of *73) +\[ +\text{smor}=\hat{P} \hat{Q}\{\exists !P\,\overline{\text{smor}}\,Q\} \quad\text{Df}. +\]</p> + +<p>There is no need to confine the notion of ordinal similarity (or +likeness, as we shall also call it) to <i>serial</i> relations. When +two relations have ordinal similarity, their internal structures are +analogous, and they therefore have many common properties. Whenever +similarity has been proved between two classes \(\alpha\) and +\(\beta\), then if \(\beta\) is given as the field of some relation +\(Q\), and \(S\) is the correlating relation, \(S^{;}Q\) is like \(Q\), +and has \(\alpha\) for its field. Hence similar classes are the +fields of like relations. It must not be supposed, however, that like +relations are coextensive with relations whose fields are similar. +This does not hold even when we confine ourselves to serial relations, +except in the special case of <i>finite</i> serial relations.</p> + +<p>The definition of relation-numbers (<a href="#*152">*152</a>) is as follows: The +relation-number of \(P\), which we call \(\text{Nr}ʻP\), is the class +of relations which are ordinally similar to \(P\); and the class of +relation-numbers, which we denote by \(\text{N}R\), is the class of +all classes of the form \(\text{Nr}ʻP\). The elementary properties of +relation-numbers, treated in *152, are closely analogous to those of +cardinal numbers treated in <a href="#*100">*100</a>.</p> + +<p>After a few propositions about the ordinal 0 and the ordinal 2, which +we call \(0_{r}\) and \(2_{r}\) (<a href="#*153">*153</a>), we pass to the consideration +of relation-numbers of various types. It will be observed that +"\(\text{smor}\)," like "\(\text{sm}\)," is a relation which is +ambiguous as to the type both of its domain and of its converse +domain. Thus "\(P \,\text{smor} \,Q\)" only has an unambiguous meaning +when the types of \(P\) and \(Q\) are determined. \(P\) and \(Q\) may +or may not be of the same type; the only restriction upon the type +of either is that both must be "homogeneous" relations, <i>i.e.</i> +relations whose domain and converse domain<span class="pagenum" id="Page_305">[Pg 305]</span> are of the same type. This +restriction results from the fact that \(CʻQ\) occurs in the definition +of "\(P\,\text{smor}\,Q\)," and a relation does not have a field unless +it is homogeneous; hence \(Q\) must be homogeneous, and therefore, +whatever \(S\) may be, \(S\mid Q\mid \breve{S}\) must be homogeneous, +<i>i.e.</i> \(P\) must be homogeneous. Thus <i>e.g.</i> such relations +as \(\text{D}\), \(\iota\), or \(\in\) are not ordinally similar either +to themselves or to anything else. Whenever "\(P\,\text{smor}\,Q\)" +is <i>significant</i> for a suitable \(Q\), we have \(P\,\text{smor}\,P\); +but if \(P\) is not homogeneous, "\(P\,\text{smor}\,Q\)" is never +significant. Hence throughout the theory of ordinal similarity, the +relations of which ordinal similarity is affirmed or denied must be +homogeneous. The correlators, on the contrary, need not be homogeneous.</p> + +<p>Owing to the homogeneity of our relations, the types of +relation-numbers are much more easily dealt with than they otherwise +would be; for the type of a homogeneous relation is determined by that +of a single class, namely its field, whereas the type of a relation +in general depends upon the types of <i>two</i> classes, namely its +domain and its converse domain. Since, where likeness is concerned, the +type of the field determines the type of the relation, propositions +concerning the relations between different typical determinations of +a given relation-number are, for the most part, exactly analogous to +and deducible from those for cardinals. In fact, a relation ordinally +similar to \(Q\) exists in the type of \(P\) when, and only when, a +class similar to \(CʻQ\) exists in the type of \(CʻP\), <i>i.e.</i> +\[ +\exists !\text{Nr}(P)ʻQ.\equiv .\exists !\text{Nc}(CʻP)ʻCʻQ. +\] +The half of this proposition follows from the fact that, if \(P\) is +like \(Q\), \(CʻP\) is similar to \(CʻQ\). The other half follows +from the fact, mentioned above, that if \(\beta =CʻQ\) and \(\alpha\text{ sm } \beta\), +then there is a relation like \(Q\) and having +\(\alpha\) for its field. Now if \(\alpha\) belongs to the type of +\(CʻP\), any relation having \(\alpha\) for its field is contained +in \(t_{0}ʻCʻP\uparrow t_{0}ʻCʻP\). Hence in the case supposed +there is a relation like \(Q\) and contained in \(t_{0}ʻCʻP\uparrow t_{0}ʻCʻP\). +But the relations contained in \(t_{0}ʻCʻP\uparrow t_{0}ʻCʻP\) +constitute \(tʻP\). Hence there is a relation which is like \(Q\) +and is a member of \(tʻP\), whence our proposition results. By +means of this proposition and those of <a href="#*102">*102</a>—<a href="#*106">6</a>, the properties +of relation-numbers with respect to types follow easily. The +conventions \(\text{IT}\), \(\text{IIT}\) and \(\text{AT}\) apply to +relation-numbers as to cardinals; they are to be applied in the same +way as in the analogous propositions of Part III, Section A.</p> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_306">[Pg 306]</span></p> +<h2 class="nobreak" id="*150"> +*150. INTERNAL TRANSFORMATION OF A RELATION.</h2> +</div> + + +<p class="nind"><i>Summary of</i> *150.</p> + +<p>In this number we introduce two notations which have uses in regard to +relations closely analogous to the uses of \(Rʻʻ\alpha\) and \(R_{\in}\) in regard to classes. These two notations are defined as follows: +\[ +\begin{aligned} +S^{;}Q&=S\mid Q\mid \breve{S} \quad\text{Df},\\ +S\dagger Q&=S\mid Q\mid \breve{S} \quad\text{Df}. +\end{aligned} +\] +We then have \(\vdash .S\dagger Q=S^{;}Q=S\mid Q\mid \breve{S}=(S\parallel \breve{S})ʻQ\).</p> + +<p>\(S\dagger Q\) is merely an alternative to \(S^{;}Q\), just as \(R_{\in}ʻ\alpha\) +is an alternative to \(Rʻʻ\alpha\). Also \(S\dagger +=S\parallel \breve{S}\), in virtue of *38·01 and *43·01.</p> + +<p>The uses of \(S^{;}Q\) occur chiefly when \(S\) is a one-one relation +and \(CʻQ\subset \text{ᗡ}ʻS\). This case is illustrated in the figure +in the introduction to this section. Here if \(Q\) relates \(x\) +and \(y\), \(S^{;}Q\) relates \(Sʻx\) and \(Sʻy\). Thus given a class +\(\alpha\) similar to \(CʻQ\), if \(S\) is the correlating relation, +\(S^{;}Q\) has \(\alpha\) for its field, and has, in very many respects, +properties analogous to those of \(Q\).</p> + +<p>\(S^{;}Q\) is important for many special values of \(S\). For example, +let \(Q\) be a relation between relations; then \(C^{;}Q\) will be the +corresponding relation of the fields of these relations. If \(Q\) be +any relation, \(\downarrow x^{;}Q\) will be the corresponding relation +between ordered couples of which \(x\) is the relatum; <i>i.e.</i> +if \(yQz\), the relation \(\downarrow x^{;}Q\) will hold between +\(y\downarrow x\) and \(z\downarrow x\). If \(Q\) is a relation between +classes, and we have \(\beta Q\gamma\), then the relation \(\alpha\cup ^{;}Q\) +will hold between \(\alpha \cup \beta\) and \(\alpha\cup \gamma\). +In short, whenever \(S\) is a one-many relation, and therefore gives +rise to a descriptive function, then \(S^{;}Q\) is the relation which +holds between \(Sʻy\) and \(Sʻz\) whenever \(Q\) holds between \(y\) +and \(z\).</p> + +<p>We introduce one other new notation in this number, corresponding to +\(\alpha \unicode{x2640}_{,,}y\) in *38. This notation is thus defined: +\[ +Q\unicode{x2640}_{.,}y=\unicode{x2640}y^{;}Q \quad\text{Df}. +\] +The purpose of this notation is to enable us to proceed to +\(Q\unicode{x2640}_{.,}^{;}P\) and other similar notations; or, otherwise +stated, to enable us to treat \(\unicode{x2640}y^{;}Q as\) a function +of y rather than of \(Q\). Take for example the case of \(x\downarrow^{;}Q\). +<span class="pagenum" id="Page_307">[Pg 307]</span>We may wish to consider various relations \(x\downarrow ^{;}Q\), +\(y\downarrow ^{;}Q\), where we are to have (say) \(xPy\). To express the +relation of \(x\downarrow ^{;}Q\) to \(y\downarrow ^{;}Q\) resulting from +\(xPy\), we need the above notation. By its help, we have +\[ +\begin{aligned} +x\downarrow ^{;}Q&=Q\downarrow_{.,} ʻx.y\downarrow ^{;}Q=Q\downarrow_{.,} ʻy.\\ +\text{Hence}\qquad xPy.&\equiv .(Q\downarrow_{.,} ʻx)(Q\downarrow_{.,} ^{;}P)(Q\downarrow_{.,} ʻy). +\end{aligned} +\] +Thus \(Q\downarrow_{.,} ^{;}P\) is the relation between \(x\downarrow^{;}Q\) +and \(y\downarrow ^{;}Q\) corresponding to the relation \(P\) +between \(x\) and \(y\). \(Q\downarrow_{.,} ^{;}P\) plays the same part +in relation-arithmetic as is played by \(\alpha\downarrow_{,,}ʻʻ\beta\) +in cardinal arithmetic.</p> + +<p>The notations of this number are capable of occasional uses in cardinal +arithmetic<a id="FNanchor_11" href="#Footnote_11" class="fnanchor">[11]</a>, but their chief utility is in relation-arithmetic, in +which they are fundamental.</p> + +<p>In order to minimize the use of brackets, we put +\[ +\begin{aligned} +RʻS^{;}Q&=Rʻ(S^{;}Q) &\quad\text{Df},\\ +R^{;}S^{;}Q&=R^{;}(S^{;}Q) &\quad\text{Df}. +\end{aligned} +\]</p> + +<p>As an immediate result of the definition of \(S^{;}Q\) we have</p> + +<p class="nind"><b>*150·11.</b> \(\vdash :x(S^{;}Q)y.\equiv .(\exists z,w).xSz.ySw.zQw\)</p> + +<p>We have also</p> + +<p class="nind"><b>*150·12.</b> \(\vdash .\text{Cnv}ʻS^{;}Q=S^{;}\breve{Q}\)</p> + +<p class="nind"><b>*150·13.</b> \(\vdash .R^{;}S^{;}Q=(R\mid S)^{;}Q\)</p> + +<p>This proposition, which is the analogue of \((P\mid Q)ʻʻ\gamma=PʻʻQʻʻy\) +(*37·33), is very often used. We have also</p> + +<p class="nind"><b>*150·3.</b> \(\vdash .S^{;}(Q\unicode{x228d} R)=S^{;}Q\unicode{x228d} S^{;}R\)</p> + +<p class="nind"><b>*150·42.</b> \(\vdash .S^{;}\dot{\Lambda} =\dot{\Lambda}\)</p> + +<p>The remaining propositions of this number (with a few exceptions) may +be thus classified:</p> + +<p>(1) Propositions concerning the domain, converse domain, and field of +\(S^{;}Q\) (<a href="#*150·2">*150·2</a>—<a href="#*150·23">·23</a>). Owing to the fact that the chief applications +of this subject are to cases where \(Q\) and \(S^{;}Q\) are serial, the +<i>field</i> of \(S^{;}Q\) is more important than its domain or converse +domain. Thus the chief propositions here are</p> + +<p class="nind"><b>*150·22.</b> \(\vdash :CʻQ\subset \text{ᗡ}ʻS.\supset .CʻS^{;}Q=SʻʻCʻQ\)</p> + +<p class="nind"><b>*150·23.</b> \(\vdash :CʻQ=\text{ᗡ}ʻS.\supset .CʻS^{;}Q=\text{D}ʻS\)</p> + +<p>The hypothesis \(CʻQ\subset \text{ᗡ}ʻS\) is verified in almost all +applications of \(S^{;}Q\). When it is not verified, the part of \(CʻQ\) +not contained in \(\text{ᗡ}ʻS\) is irrelevant to the value of \(S^{;}Q\). +The hypothesis \(CʻQ=\text{ᗡ}ʻS\) is very often verified in practice, +<span class="pagenum" id="Page_308">[Pg 308]</span>since it is verified when \(S\) is a correlator of \(S^{;}Q\) and \(Q\).</p> + +<p>(2) Propositions concerning relations with limited domains, converse +domains, or fields (<a href="#*150·32">*150·32</a>—<a href="#*150·38">·38</a>). Broadly speaking, a limitation +on the <i>field</i> of \(Q\) is equivalent to a limitation on the +<i>converse domain</i> of \(S\), and both are equivalent to a +corresponding limitation on the field of \(S^{;}Q\) provided \(S\in\text{Cls}\rightarrow 1\). +The limitations that occur in practice are limitations on the converse +domain of \(S\), with consequent limitations on the fields of \(Q\) and +\(S^{;}Q\).</p> + +<p>The chief propositions on this subject are</p> + +<p class="nind"><b>*150·32.</b> \(\vdash .(S\upharpoonright CʻQ)^{;}Q=S^{;}Q\)</p> + +<p class="nind"><b>*150·35.</b> \(\vdash \colon\ldotp y\in CʻQ.\supset _{y}.Rʻy=Sʻy:\supset .R^{;}Q=S^{;}Q\)</p> + +<p>(This follows from *150·32 and *35·71.)</p> + +<p class="nind"><b>*150·36.</b> \(\vdash .(S\upharpoonright \beta )^{;}Q=S^{;}(Q\unicode{x0294f}\beta )\)</p> + +<p class="nind"><b>*150·37.</b> \(\vdash :S\in \text{Cls}\rightarrow 1.\supset .S^{;}(Q\unicode{x0294f}\beta )=(S^{;}Q)\unicode{x0294f}Sʻʻ\beta =(S\upharpoonright \beta )^{;}Q=\{(Sʻʻ\beta + )\upharpoonleft S\}^{;}Q\)</p> + +<p>(3) Propositions on \(S^{;}Q\) when \(S\) is one-many or many-one +(<a href="#*150·4">*150·4</a>—<a href="#*150·56">·56</a>). We have</p> + +<p class="nind"><b>*150·4.</b> \(\vdash \colon\ldotp S\in 1\rightarrow \text{Cls}.\supset :x(S^{;}Q)y.\equiv .(\exists z,w).x=Sʻz.y=Sʻw.zQw\)</p> + +<p>This proposition is used constantly. Only slightly less useful is</p> + +<p class="nind"><b>*150·41.</b> \(\vdash \colon\ldotp S\in \text{Cls}\rightarrow 1.\supset :x(S^{;}Q)y.\equiv .(\breve{S} ʻx)Q(\breve{S} ʻy)\)</p> + +<p>The remaining propositions of this set are chiefly applications of +<a href="#*150·4">*150·4</a><a href="#*150·41">·41</a> to special cases.</p> + +<p>(4) A few propositions on \(Q\unicode{x2640}_{.,}y\) (<a href="#*150·6">*150·6</a>—<a href="#*150·62">·62</a>). +These are immediate consequences of the definition.</p> + +<p>(5) A set of propositions on couples and matters connected with them +(<a href="#*150·7">*150·7</a>—<a href="#*150·75">·75</a>). The chief of these is</p> + +<p class="nind"><b>*150·71.</b> \(\vdash :S\in 1\rightarrow \text{Cls}.z,w\in \text{ᗡ}ʻS.\supset .S^{;}(z\downarrow w)=(Sʻz)\downarrow (Sʻw)\)</p> + +<p>This proposition is very often used in relation-arithmetic. Useful also +is</p> + +<p class="nind"><b>*150·73.</b> \(\vdash .S^{;}(\alpha \uparrow \beta )=Sʻʻ\alpha \uparrow Sʻ\beta\)</p> + +<p>(6) We next have four propositions (<a href="#*150·8">*150·8</a>—<a href="#*150·83">·83</a>) on \(S^{;}P\) when \(P\) +is a power of \(Q\). These belong with the propositions of *92; they +are useful in the ordinal theory of finite and infinite. We have</p> + +<p class="nind"><b>*150·82·83.</b> \[\begin{align}\vdash \colon\ldotp S\in \text{Cls}\rightarrow 1:\text{D}ʻQ\subset &\text{ᗡ}ʻS.\lor.\text{ᗡ}ʻQ\subset \text{ᗡ}ʻS:\supset .\\ +\text{Pot}ʻS^{;}Q=S\daggerʻʻ\text{Pot}ʻQ.(S^{;}Q)_{\text{po}}=S^{;}Q_{\text{po}}\end{align}\]</p> + +<p>It follows that, in the hypothesis supposed, if \(S\) is a correlator +of \(P\) and \(Q\), it is also a correlator of \(P_{\text{po}}\) and +\(Q_{\text{po}}\).</p> + +<p><span class="pagenum" id="Page_309">[Pg 309]</span></p> + +<p>(7) Propositions concerning the relation \(S\dagger\) (<a href="#*150·14">*150·14</a>—<a href="#*150·171">·171</a> +and <a href="#*150·9">*150·9</a>—<a href="#*150·94">·94</a>). These have uses analogous to those of propositions +concerning \(S_{\in }\). The most important are</p> + +<p class="nind"><b>*150·14.</b> \(\vdash .R\dagger \mid S\dagger =(R\mid S)\dagger\)</p> + +<p>(This follows immediately from <a href="#*150·13">*150·13</a>, above.)</p> + +<p class="nind"><b>*150·141.</b> \(\vdash .S\dagger =S\parallel \breve{S} \)</p> + +<p>(This follows immediately from the definition.)</p> + +<p class="nind"><b>*150·16.</b> \(\vdash .\dot{s} ʻR\dagger ʻʻ\lambda =R\dagger (\dot{s} ʻ\lambda )=R^{;}\dot{s} ʻ\lambda\)</p> + +<p>This proposition is analogous to \(sʻRʻʻʻ\kappa =Rʻʻsʻ\kappa\) +(*40·38), <i>i.e.</i> to +\[ +sʻR_{\in }ʻʻ\kappa =R_{\in }ʻsʻ\kappa =Rʻʻsʻ\kappa , +\] +as appears on substituting \(\dot{s}\) and \(R\dagger\) for \(s\) and +\(R_{\in }\) in this variant of *40·38.</p> + +<p>The remaining propositions are mainly of the nature of lemmas, to be +used once or twice each in relation-arithmetic.</p> + +<hr class="tb"> + +<p class="nind"><b>*150·01.</b> \(S^{;}Q=S\mid Q\mid \breve{S} \quad\text{Df}\)</p> + +<p class="nind"><b>*150·02.</b> \(S\dagger Q=S\mid Q\mid \breve{S} \quad \text{Df}\)</p> + +<p class="nind"><b><a id="*150·03">*150·03</a>.</b> \(Q\unicode{x2640}_{.,}y=\unicode{x2640}y^{;}Q \quad\text{Df}\)</p> + +<p>Here, as in *38, "\(\unicode{x2640}\)" stands for any sign which, +when placed between two letters, defines a descriptive function +of the arguments represented by those letters. Thus for example +"\(\unicode{x2640}\)" may represent any of the following: +\[ +\cap ,\, \cup ,\, \dot{\cap} ,\, \unicode{x228d} ,\, \mid ,\, \upharpoonleft ,\, \upharpoonright ,\, \unicode{x0294f},\, \uparrow ,\, \downarrow . +\]</p> + +<p>The two following definitions serve merely for the avoidance of +brackets.</p> + +<p class="nind"><b>*150·04.</b> \(RʻS^{;}Q=Rʻ(S^{;}Q) \quad\text{Df}\)</p> + +<p class="nind"><b>*150·05.</b> \(R^{;}S^{;}Q=R^{;}(S^{;}Q) \quad\text{Df}\)</p> + +<p class="nind"><b>*150·1.</b> \[\begin{align}&\vdash .S^{;}Q=S\mid Q\mid \breve{S} =(S\parallel \breve{S} )ʻQ=S\dagger Q=S\dagger ʻQ\\ +&[*43·112.*38·11.(*150·01·02)]\end{align}\]</p> + +<p class="nind"><b>*150·11.</b> \(\vdash :x(S^{;}Q)y.\equiv .(\exists z,w).xSz.ySw.zQw \quad[*34·1.*31·11]\)</p> + +<p class="nind"><b>*150·12.</b> \(\vdash .\text{Cnv}ʻS^{;}Q=S^{;}\breve{Q} \quad [*34·2.*31·33]\)</p> + +<p class="nind"><b><a id="*150·13">*150·13</a>.</b> \(\vdash .R^{;}S^{;}Q=(R\mid S)^{;}Q\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*150·1.(*150·05). \supset \vdash .R^{;}S^{;}Q&=R^{;}(S\mid Q\mid \breve{S} )\\ +[*150·1] &=R\mid S\mid Q\mid \breve{S} \mid \breve{R}\\ +[*34·2] &=R\mid S\mid Q\mid \text{Cnv}ʻ(R\mid S)\\ +[*150·1] &=(R\mid S)^{;}Q.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_310">[Pg 310]</span></p> + +<p class="nind"><b>*150·131.</b> \(\vdash .(R^{;}S)^{;}Q=R^{;}(S\mid \breve{R} )^{;}Q\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*150·13. \supset \vdash .R^{;}(S\mid \breve{R} )^{;}Q&=(R\mid S\mid \breve{R} )^{;}Q\\ +[*150·1] & =(R^{;}S)^{;}Q.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p>Observe that we do not have \((R^{;}S)^{;}Q=R^{;}(S^{;}Q)\).</p> + +<p class="nind"><b><a id="*150·14">*150·14</a>.</b> \(\vdash .R\dagger \mid S\dagger =(R\mid S)\dagger\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*150·1·13. \supset \vdash .R\dagger ʻS\dagger ʻQ=(R\mid S)\dagger ʻQ &\qquad \text{(1)}\\ +\vdash .(1).*34·42.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p>This proposition is the relational analogue of *37·34.</p> + +<p class="nind"><b>*150·141.</b> \(\vdash .S\dagger =S\parallel \breve{S} \quad[*150·1.*30·41]\)</p> + +<p class="nind"><b>*150·15.</b> \(\vdash .S\dagger \in 1\rightarrow \text{Cls} \quad[*72·14]\)</p> + +<p class="nind"><b>*150·151.</b> \(\vdash \colon\ldotp (x).\text{E}!Sʻx: S\in \text{Cls}\rightarrow 1:\supset .S\dagger \in 1\rightarrow 1 \quad[*74·772.*150·141]\)</p> + +<p>The following proposition is used in the theory of double ordinal +similarity (<a href="#*164·13">*164·13</a>).</p> + +<p class="nind"><b>*150·152.</b> \(\vdash :S\upharpoonright sʻCʻʻ\lambda \in \text{Cls}\rightarrow 1.sʻCʻʻ\lambda \subset \text{ᗡ}ʻS.\supset .(S\dagger )\upharpoonright \lambda \in 1\rightarrow 1\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*74·775 \frac{S,S}{Q,R}.\supset \\ +\vdash :S\upharpoonright sʻCʻʻ\lambda \in \text{Cls}\rightarrow 1.sʻ\text{D}ʻʻ\lambda \subset \text{ᗡ}ʻS.sʻ\text{ᗡ}ʻʻ\lambda &\subset \text{ᗡ}ʻS.\supset .\\ +&(S\parallel \breve{S} )\upharpoonright \in 1\rightarrow 1 &\qquad \text{(1)}\\ +\vdash .(1).*40·57.*150·141.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*150·153.</b> \[\begin{align}\vdash \colon\ldotp S\upharpoonright sʻCʻʻ\lambda \in \text{Cls}\rightarrow 1.sʻCʻʻ\lambda \subset \text{ᗡ}ʻS.Q,&R\in \lambda .\supset :\\ +&S^{;}Q=S^{;}R.\supset .Q=R\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*150·152.*71·55.\supset \vdash \colon\ldotp \text{Hp}.&\supset :S\dagger ʻQ=S\dagger ʻR.\supset .Q=R:\\ +[*150·1] &\supset :S^{;}Q=S^{;}R.\supset .Q=R\colon\ldotp \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p>The above proposition is used in dealing with relations of relations of +couples (<a href="#*165·23">*165·23</a>).</p> + +<p class="nind"><b>*150·16.</b> \(\vdash .\dot{s} ʻR\dagger ʻʻ\lambda =R\dagger (\dot{s} ʻ\lambda )=R^{;}\dot{s} ʻ\lambda \left[*43·43\frac{\breve{R}}{S}.*150·141·1\right]\)</p> + +<p>The following proposition is a lemma for <a href="#*150·171">*150·171</a>.</p> + +<p class="nind"><b>*150·17.</b> \(\vdash .(R\upharpoonright \lambda )\dagger =R\dagger \mid \unicode{x0294f}\lambda\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*150·1.\supset \vdash .(R\upharpoonright \lambda )\dagger ʻP&=(R\upharpoonright \lambda )\mid P\mid (\lambda \upharpoonleft \breve{R} )\\ +[*35·354] &=R\mid \lambda \upharpoonleft P\upharpoonright \lambda \mid \breve{R} \\ +[*150·1.*36·11] &=R\dagger (P\unicode{x0294f}\lambda )\\ +[*38·11] & =R\dagger ʻ\unicode{x0294f}\lambda ʻP &\qquad \text{(1)}\\ +\vdash .(1).*34·42.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_311">[Pg 311]</span></p> + +<p class="nind"><b><a id="*150·171">*150·171</a>.</b> \(\vdash :sʻCʻʻCʻQ\subset \lambda .\supset .(R\upharpoonright \lambda )\dagger ^{;}Q=R\dagger ^{;}Q.\unicode{x0294f}\lambda ^{;}Q=Q\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*150·17·13.&\supset \vdash .(R\upharpoonright \lambda )\dagger ^{;}Q=R\dagger ^{;}\unicode{x0294f}\lambda ^{;}Q &\qquad \text{(1)}\\ +\vdash . *150·11. &\supset \vdash :M(\unicode{x0294f}\lambda ^{;}Q)N.\equiv .(\exists S,T).M=S\unicode{x0294f}\lambda .N=T\unicode{x0294f}\lambda .SQT &\qquad \text{(2)}\\ +\vdash .*33·17. \supset \vdash :SQT.&\supset .S,T\in CʻQ.\\ +[*37·62] &\supset .CʻS,CʻT\in CʻʻCʻQ.\\ +[*40·13] &\supset .CʻS\subset sʻCʻʻCʻQ.CʻT\subset sʻCʻʻCʻQ &\qquad \text{(3)}\\ +\vdash .(3). \supset \vdash \colon\ldotp \text{Hp}.&\supset :SQT.\supset .CʻS\subset \lambda .CʻT\subset \lambda .\\ +[*36·25] &\supset .S\unicode{x0294f}\lambda =S.T\unicode{x0294f}\lambda =T &\qquad \text{(4)}\\ +\vdash .(2).(4).\supset \vdash \colon\ldotp \text{Hp}.\supset :M(\unicode{x0294f}\lambda ^{;}Q)N.&\equiv .(\exists S,T).M=S.N=T.SQT.\\ +[*13·22] &\equiv .MQT:\\ +[*21·43] &\supset :\unicode{x0294f}\lambda ^{;}Q=Q &\qquad \text{(5)}\\ +\vdash .(1).(5).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p>The above proposition is required in the theory of double ordinal +similarity. It is used in proving <a href="#*164·141">*164·141</a>, which is used in <a href="#*164·18">*164·18</a>, +which is a fundamental proposition in the theory of double ordinal +similarity.</p> + +<p>The following propositions, on the domain, converse domain and field of +\(S^{;}Q\), are much used, especially <a href="#*150·202">*150·202</a><a href="#*150·22">·22</a><a href="#*150·23">·23</a>. <a href="#*150·201">*150·201</a> is hardly +ever used, but is inserted in order that the general case may not +remain unconsidered.</p> + +<p class="nind"><b><a id="*150·2">*150·2</a>.</b> \(\vdash .\text{D}ʻS^{;}Q=SʻʻQʻʻ\text{ᗡ}ʻS.\text{ᗡ}ʻS^{;}Q=Sʻʻ\breve{Q} ʻʻ\text{ᗡ}ʻS \quad[*37·32.*150·1]\)</p> + +<p class="nind"><b><a id="*150·201">*150·201</a>.</b> \(\vdash .CʻS^{;}Q=Sʻʻ(Q\unicode{x228d} \breve{Q} )ʻʻ\text{ᗡ}ʻS=\text{D}ʻS^{;}(Q\unicode{x228d} \breve{Q} )\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*150·2.*37·22.\supset \vdash .CʻS^{;}Q&=Sʻʻ(Qʻʻ\text{ᗡ}ʻS\cup \breve{Q} ʻʻ\text{ᗡ}ʻS)\\ +[*37·221] &=Sʻʻ(Q\unicode{x228d} \breve{Q} )ʻʻ\text{ᗡ}ʻS\\ +[*150·2] &=\text{D}ʻS^{;}(Q\unicode{x228d} \breve{Q} ).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*150·202">*150·202</a>.</b> \(\vdash .\text{D}ʻS^{;}Q\subset Sʻʻ\text{D}ʻQ.\text{ᗡ}ʻS^{;}Q\subset Sʻʻ\text{ᗡ}ʻQ.CʻS^{;}Q\subset SʻʻCʻQ\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*37·15·16. &\supset \vdash .Qʻʻ\text{ᗡ}ʻS\subset \text{D}ʻQ.\breve{Q} ʻʻ\text{ᗡ}ʻS\subset \text{ᗡ}ʻQ.\\ +[*37·2.*150·2] &\supset \vdash .\text{D}ʻS^{;}Q\subset Sʻʻ\text{D}ʻQ.\text{ᗡ}ʻS^{;}Q\subset Sʻʻ\text{ᗡ}ʻQ &\qquad \text{(1)}\\ +[*37·22] &\supset \vdash .CʻS^{;}Q\subset SʻʻCʻQ &\qquad \text{(2)}\\ +\vdash .(1).(2). &\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*150·203.</b> \(\vdash .CʻS^{;}Q\subset \text{D}ʻS \quad[*150·202.*37·15]\)</p> + +<p class="nind"><b>*150·21.</b> \(\vdash :\text{ᗡ}ʻQ\subset \text{ᗡ}ʻS.\supset .\text{D}ʻS^{;}Q=Sʻʻ\text{D}ʻQ=\text{D}ʻ(S\mid Q) \quad[*150·2.*37·27·32]\)</p> + +<p class="nind"><b>*150·211.</b> \[\begin{align}&\vdash :\text{D}ʻQ\subset \text{ᗡ}ʻS.\supset .\text{ᗡ}ʻS^{;}Q=Sʻʻ\text{ᗡ}ʻQ=\text{D}ʻ(S\mid \breve{Q} )\\ +&[*150·2.*37·271·32]\end{align}\]</p> + +<p><span class="pagenum" id="Page_312">[Pg 312]</span></p> + +<p class="nind"><b><a id="*150·22">*150·22</a>.</b> \(\vdash :CʻQ\subset \text{ᗡ}ʻS.\supset .CʻS^{;}Q=SʻʻCʻQ \quad[*150·21·211.*37·22]\)</p> + +<p>In practice, when \(S^{;}Q\) is used, we almost always have \(CʻQ\subset\text{ᗡ}ʻS\). +For the use of \(S^{;}Q\) is to obtain a relation analogous +to \(Q\) and having a different field; now \(S^{;}Q\) is analogous +to \(Q\unicode{x0294f}\text{ᗡ}ʻS\), for the part of \(CʻQ\) which +lies outside \(\text{ᗡ}ʻS\) is unaffected by \(S\). Hence if we +have, to start with, a relation \(Q\) whose field is not contained +in \(\text{ᗡ}ʻS\), we shall usually find it profitable to limit the +field to \(\text{ᗡ}ʻS\), and consider the transformed relation rather +as \(S^{;}(Q\unicode{x0294f}\text{ᗡ}ʻS)\) than as \(S^{;}Q\). Thus the +hypothesis \(CʻQ\subset \text{ᗡ}ʻS\) will be verified in almost all +useful applications of the notion of \(S^{;}Q\).</p> + +<p class="nind"><b><a id="*150·23">*150·23</a>.</b> \(\vdash :CʻQ=\text{ᗡ}ʻS.\supset .CʻS^{;}Q=\text{D}ʻS \quad[*150·22.*37·25]\)</p> + +<p class="nind"><b>*150·24.</b> \(\vdash \colon\ldotp CʻQ\subset \text{ᗡ}ʻS.\supset :_{\dot{\exists} }!S^{;}Q.\equiv ._{\dot{\exists} }!Q\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*37·.*150·22.\supset \vdash \colon\ldotp \text{Hp} .&\supset :_{\exists }!CʻS^{;}Q.\equiv ._{\exists }!CʻQ:\\ +[*33·24] &\supset :_{\dot{\exists} }!S^{;}Q.\equiv ._{\dot{\exists} }!Q\colon\ldotp \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*150·25</b> \(\vdash \colon\ldotp (y).\text{E}!Sʻy.\supset :_{\dot{\exists} }!S^{;}Q.\equiv ._{\dot{\exists} }!Q \quad[*150·24.*33·431]\)</p> + +<p class="nind"><b>*150·3.</b> \(\vdash .S^{;}(Q\unicode{x228d} R)=S^{;}Q\unicode{x228d} S^{;}R \quad[*34·25·26.*150·1]\)</p> + +<p class="nind"><b>*150·301.</b> \(\vdash .S^{;}(Q\dot{\cap} R)\,\unicode{x2abd}\, (S^{;}Q)\dot{\cap} (S^{;}R) \quad[*34·23·24.*150·1]\)</p> + +<p class="nind"><b>*150·31.</b> \(\vdash :P\,\unicode{x2abd}\, Q.R\,\unicode{x2abd}\, S.\supset .R^{;}P\,\unicode{x2abd}\, S^{;}Q \quad[*34·34.*150·1]\)</p> + +<p>The following propositions are frequently useful when we have to deal +with correlators of the form \(S\upharpoonright CʻQ\), which often +happens.</p> + +<p class="nind"><b><a id="*150·32">*150·32</a>.</b> \(\vdash .(S\upharpoonright CʻQ)^{;}Q=S^{;}Q \left[*43.5\frac{S,\,\breve{S},\,CʻQ,\,CʻQ}{Q,\,R,\,\alpha,\,\beta }.*150·141\right]\)</p> + +<p class="nind"><b>*150·33.</b> \(\vdash :CʻQ\subset \beta .\supset .(S\upharpoonright \beta )^{;}Q=S^{;}Q \left[*43·5\frac{S,\,\breve{S}}{Q,\,R}.*150·141\right]\)</p> + +<p class="nind"><b>*150·34.</b> \[\begin{align}&\vdash :\text{D}ʻQ\subset \alpha .\text{ᗡ}ʻQ\subset \beta .\supset .S\upharpoonright \alpha \mid Q\mid \beta \upharpoonleft \breve{S} =S^{;}\alpha + \upharpoonleft Q\upharpoonright \beta =S^{;}Q\\ +&[*43·5.*35·354.*150·141·1]\end{align}\]</p> + +<p class="nind"><b>*150·35.</b> \(\vdash \colon\ldotp y\in CʻQ.\supset _{y}.Rʻy=Sʻy:\supset .R^{;}Q=S^{;}Q\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*35·71.\supset \vdash : \text{Hp} .&\supset .R\upharpoonright CʻQ=S\upharpoonright CʻQ.\\ +[*34·27·28.*150·1] &\supset .(R\upharpoonright CʻQ)^{;}Q=(S\upharpoonright CʻQ)^{;}Q.\\ +[*150·32] &\supset .R^{;}Q=S^{;}Q:\supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p>The above proposition, which is the analogue of *37·69, is much used in +relation-arithmetic.</p> + +<p><span class="pagenum" id="Page_313">[Pg 313]</span></p> + +<p>The following proposition is much used after we reach the theory of +well-ordered series, but not before (except in <a href="#*150·37">*150·37</a>).</p> + +<p class="nind"><b>*150·36.</b> \(\vdash .(S\upharpoonright \beta )^{;}Q=S^{;}(Q\unicode{x0294f}\beta )\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*150·11.*35·101.\supset \\ +\vdash :x{(S\upharpoonright \beta )^{;}Q}w.&\equiv .(\exists y,z).xSy.y\in \beta .yQz.z\in \beta .wSz.\\ +[*36·13] &\equiv .(\exists y,z).xSy.y(Q\unicode{x0294f}\beta )z.wSz.\\ +[*150·11] &\equiv .x{S^{;}(Q\unicode{x0294f}\beta )}w:\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*150·361.</b> \(\vdash .(\alpha \upharpoonleft S)^{;}Q=(S^{;}Q)\unicode{x0294f}\alpha \quad[\text{Proof as in *150·36}]\)</p> + +<p class="nind"><b><a id="*150·37">*150·37</a>.</b> \[\begin{align}\vdash :S\in \text{Cls}\rightarrow 1.\supset .S^{;}(Q\unicode{x0294f}\beta )&=(S^{;}Q)\unicode{x0294f}Sʻʻ\beta \\ +&=(S\upharpoonright \beta )^{;}Q={(Sʻʻ\beta )\upharpoonleft S}^{;}Q\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*74·141.\supset \vdash :\text{Hp}.\supset .(S\upharpoonright \beta )^;Q={(Sʻʻ\beta )\upharpoonleft S}^;Q &\qquad \text{(1)}\\ +\vdash .(1).*150·36·361.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p>The above proposition is not used until we reach the theory of series.</p> + +<p class="nind"><b><a id="*150·38">*150·38</a>.</b> \(\vdash :S\in 1\rightarrow 1.\supset .S^{;}\breve{S} ^{;}Q=Q\unicode{x0294f}\text{D}ʻS\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*150·1. \supset \vdash .S^{;}\breve{S} ^{;}Q=S\mid \breve{S} \mid Q\mid S\mid \breve{S} &\qquad \text{(1)}\\ +\vdash .(1).*72·59·591.\supset \vdash :\text{Hp}.\supset .S^{;}\breve{S} ^{;}Q=(\text{D}ʻS)\upharpoonleft Q\upharpoonright \text{D}ʻS\\ +[*36·11] =Q\unicode{x0294f}\text{D}ʻS:\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p>The above proposition is used in dealing with the correlation of series +(<a href="#*208·2">*208·2</a>).</p> + +<p class="nind"><b><a id="*150·4">*150·4</a>.</b> \[\begin{align}&\vdash \colon\ldotp S\in 1\rightarrow \text{Cls}.\supset :x(S^{;}Q)y.\equiv .(\exists z,w).x=Sʻz.y=Sʻw.zQw\\ +&[*150·11.*71·36]\end{align}\]</p> + +<p>This proposition is fundamental in the theory of \(S^{;}Q\), because in +most of the uses of this notion \(S\) is one-many. The proposition +states that when \(S\) is one-many, \(S^{;}Q\) is the relation between +the \(Sʻs\) of terms related by \(Q\). Thus if \(S\) is the relation +of wife to husband, and \(Q\) is the relation of brother to brother, +\(S^{;}Q\) is the relation between wives of brothers. If \(Q\) is a +relation between relations, \(C^{;}P\) will be the corresponding relation +of their fields; and so on.</p> + +<p class="nind"><b><a id="*150·41">*150·41</a>.</b> \(\vdash \colon\ldotp S\in \text{Cls}\rightarrow 1.\supset :x(S^{;}Q)y.\equiv .(\breve{S} ʻx)Q(\breve{S} ʻy) \quad[*150·11.*71·331]\)</p> + +<p class="nind"><b>*150·42.</b> \(\vdash .S^{;}\dot{\Lambda} =\dot{\Lambda} \quad[*150·1.*34·32]\)</p> + +<p>The following propositions, down to <a href="#*150·56">*150·56</a>, are, with the exception of +<a href="#*150·52">*150·52</a>—<a href="#*150·535">·535</a>, all illustrations of <a href="#*150·4">*150·4</a><a href="#*150·41">·41</a>.</p> + +<p class="nind"><b>*150·5.</b> \(\vdash :\alpha (\overrightarrow{R}^{;}P)\beta .\equiv .(\exists x,y).\alpha =\overrightarrow{R}ʻx.\beta =\overrightarrow{R}ʻy.xPy\)</p> + +<p class="nind"><b>*150·51.</b> \(\vdash :\alpha (\text{D}^{;}R)\beta .\equiv .(\exists P,Q).\alpha =\text{D}ʻP.\beta =\text{D}ʻQ.PRQ\)</p> + +<p><span class="pagenum" id="Page_314">[Pg 314]</span></p> + +<p class="nind"><b>*150·511.</b> \(\vdash :\alpha (\text{ᗡ}^{;}R)\beta .\equiv .(\exists P,Q).\alpha =\text{ᗡ}ʻP.\beta =\text{ᗡ}ʻQ.PRQ\)</p> + +<p class="nind"><b>*150·512.</b> \(\vdash :\alpha (C^{;}R)\beta .\equiv .(\exists P,Q).\alpha =CʻR.\beta =CʻQ.PRQ\)</p> + +<p class="nind"><b><a id="*150·52">*150·52</a>.</b> \[\begin{align}&\vdash :x(F^{;}R)y.\equiv .(\exists P,Q).x\in CʻP.y\in CʻQ.PRQ\\ +&[*150·11.*33·51]\end{align}\]</p> + +<p>\(F^{;}R\) is a relation which plays a great part in relation-arithmetic.</p> + +<p class="nind"><b>*150·53.</b> \(\vdash .I^{;}P=P \quad[*50·4]\)</p> + +<p class="nind"><b>*150·531.</b> \(\vdash .P^{;}I=P\mid \breve{P} \quad[*50·4]\)</p> + +<p class="nind"><b>*150·532.</b> \(\vdash .P^{;}I^{;}Q=P^{;}Q \quad[*150·13.*50·4]\)</p> + +<p class="nind"><b>*150·534.</b> \(\vdash .(I\upharpoonright CʻP)^{;}P=P \quad[*150·53·32]\)</p> + +<p class="nind"><b><a id="*150·535">*150·535</a>.</b> \(\vdash :CʻP\subset \alpha .\supset .(I\upharpoonright \alpha )^{;}P=P \quad[*150·53·33]\)</p> + +<p class="nind"><b>*150·54.</b> \(\vdash :\alpha (\iota ^{;}R)\beta .\equiv .(\breve{\iota} ʻ\alpha )R(\breve{\iota} ʻ\beta )\)</p> + +<p class="nind"><b>*150·541.</b> \(\vdash :x(\breve{\iota} ^{;}R)y.\equiv .(\iota ʻx)R(\iota ʻy)\)</p> + +<p class="nind"><b>*150·55.</b> \(\vdash :Q(\downarrow z^{;}P)R.\equiv .(\exists u,v).Q=u\downarrow z.R=v\downarrow z.uPv\)</p> + +<p class="nind"><b><a id="*150·56">*150·56</a>.</b> \[\begin{align}&\vdash :M(S\dagger ^{;}Q)N.\equiv .(\exists X,Y).XQY.M=S^{;}X.N=S^{;}Y\\ +&[*150·4·15·1]\end{align}\]</p> + +<p class="nind"><b><a id="*150·6">*150·6</a>.</b> \(\vdash .P\unicode{x2640}_{.,}y=\unicode{x2640}y^{;}P \quad[(*150·03)]\)</p> + +<p class="nind"><b>*150·601.</b> \(\vdash .P\unicode{x2640}_{.,}\in 1\rightarrow \text{Cls} \quad[*150·6.*14·21.*71·166]\)</p> + +<p class="nind"><b>*150·61.</b> \[\begin{align}&\vdash :z(P\unicode{x2640}_{.,}y)w.\equiv .(\exists u,v).z=u\unicode{x2640}y.w=v\unicode{x2640}y.uPv\\ +&[*150·11.*38·101.*150·6]\end{align}\]</p> + +<p class="nind"><b><a id="*150·62">*150·62</a>.</b> \[\begin{align}&\vdash :R(P\unicode{x2640}_{.,}^{;}Q)S.\equiv .(\exists z,w).R=\unicode{x2640}z^{;}P.S=\unicode{x2640}w^{;}P.zQw\\ +&[*150·4·601·6]\end{align}\]</p> + +<p>Relations of the form \(P\unicode{x2640}_{.,}^{;}Q\) are frequently +useful in relation-arithmetic, especially in the particular case of +\(P\downarrow_{.,} ^{;}Q\), which takes the place taken by \(\alpha +\downarrow_{,,}ʻʻ\beta\) in cardinal arithmetic. Relations of the form +\(P\downarrow_{.,} ^{;}Q\) will be considered in *165.</p> + +<p>The following propositions are chiefly concerned with correlations of +couples. They are of great utility in relation-arithmetic. <a href="#*150·71">*150·71</a>, in +particular, is fundamental.</p> + +<p class="nind"><b><a id="*150·7">*150·7</a>.</b> \(\vdash .S^{;}(z\downarrow w)=\overrightarrow{S}ʻz\uparrow \overrightarrow{S}ʻw \quad [*55·6]\)</p> + +<p class="nind"><b><a id="*150·71">*150·71</a>.</b> \(\vdash :S\in 1\rightarrow \text{Cls}.z,w\in \text{ᗡ}ʻS.\supset .S^{;}(z\downarrow w)=(Sʻz)\downarrow (Sʻw) \quad[*55·61]\)</p> + +<p class="nind"><b>*150·72.</b> \(\vdash :z\neq w.S=x\downarrow z\unicode{x228d} y\downarrow w.\supset .S^{;}(z\downarrow w)=x\downarrow y \quad[*55·62·61]\)</p> + +<p class="nind"><b>*150·73.</b> \(\vdash .S^{;}(\alpha \uparrow \beta )=Sʻʻ\alpha \uparrow Sʻʻ\beta \quad\left[*37·82 \frac{S,\,\breve{S}}{R,\,S}\right]\)</p> + +<p><span class="pagenum" id="Page_315">[Pg 315]</span></p> + +<p class="nind"><b>*150·74.</b> \(\vdash .(S\unicode{x228d} T)^{;}Q=S^{;}Q\unicode{x228d} T^{;}Q\unicode{x228d} + S\mid Q\mid \breve{T} \unicode{x228d} T\mid Q\mid \breve{S} \quad[*150·1]\)</p> + +<p class="nind"><b><a id="*150·75">*150·75</a>.</b> \(\vdash :{\sim}(yQy).\supset .(S\unicode{x228d} x\downarrow y)^{;}Q=S^{;}Q\unicode{x228d} + Sʻʻ\overrightarrow{Q}ʻy\uparrow \iota ʻx\unicode{x228d} \iota ʻx\uparrow Sʻʻ\overleftarrow{Q}ʻy\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*150·1. \supset \vdash :\text{Hp}.&\supset .(x\downarrow y)^{;}Q=\dot{\Lambda} .\\ +[*150·74] \supset .(S\unicode{x228d} x\downarrow y)^{;}Q&=S^{;}Q\unicode{x228d} S\mid Q\mid y\downarrow x\unicode{x228d} x\downarrow y\mid Q\mid \breve{S}\\ +[*55·57·571] &=S^{;}Q\unicode{x228d} Sʻʻ\overrightarrow{Q}ʻy\uparrow \iota ʻx\unicode{x228d} \iota ʻx\uparrow Sʻʻ\overleftarrow{Q}ʻy:\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p>The four following propositions belong to the subject of *92, but +could not be given in that number owing to the fact that they involve +the notations of <a href="#*150">*150</a>. They are required for proving that, if \(S\) +is a correlator of \(P\) and \(Q\), it is also a correlator of +\(P_{\text{po}}\) and \(Q_{\text{po}}\) (*151·45), and for one of +the fundamental propositions in the ordinal theory of progressions +(*263·17).</p> + +<p class="nind"><b><a id="*150·8">*150·8</a>.</b> \[\begin{align}\vdash \colon\ldotp S\in \text{Cls}\rightarrow 1:\text{D}ʻQ\subset &\text{ᗡ}ʻS.\lor.\text{ᗡ}ʻQ\subset \text{ᗡ}ʻS:P\in \text{Pot}ʻQ:\supset .\\ +&S^{;}P\in \text{Pot}ʻ(S^{;}Q).(S^{;}P)\mid (S^{;}Q)=S^{;}(P\mid Q)\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*91·351. &\supset \vdash .S^{;}Q\in \text{Pot}ʻ(S^{;}Q) &\qquad \text{(1)}\\ +\vdash .*150·1. &\supset \vdash .(S^{;}P)\mid (S^{;}Q)=S\mid P\mid \breve{S} \mid S\mid Q\mid \breve{S} &\qquad \text{(2)}\\ +\vdash .(2).*71·191. &\supset \vdash :\text{Hp}.\supset .(S^{;}P)\mid (S^{;}Q)=S\mid P\mid I\upharpoonright \text{ᗡ}ʻS\mid Q\mid \breve{S} &\qquad \text{(3)}\\ +\vdash .*50·63. &\supset \vdash :\text{D}ʻQ\subset \text{ᗡ}ʻS.\supset .I\upharpoonright \text{ᗡ}ʻS\mid Q=Q &\qquad \text{(4)}\\ +\vdash .*50·62.*91·271. &\supset \vdash :P\in \text{Pot}ʻQ.\text{ᗡ}ʻQ\subset \text{ᗡ}ʻS.\supset .P\mid I\upharpoonright \text{ᗡ}ʻS=P &\qquad \text{(5)}\\ +\vdash .(3).(4).(5). \supset \vdash :\text{Hp}.P\in \text{Pot}ʻQ.\supset .(S^{;}P)\mid (S^{;}Q)&=S\mid P\mid Q\mid \breve{S} \\ +[*150·1] &=S^{;}(P\mid Q) &\qquad \text{(6)}\\ +\vdash .*91·282. &\supset \vdash :S^{;}P\in \text{Pot}ʻS^{;}Q.\supset .(S^{;}P)\mid (S^{;}Q)\in \text{Pot}ʻS^{;}Q &\qquad \text{(7)}\\ +\vdash .(6).(7). \supset \vdash \colon\ldotp \text{Hp}.P\in \text{Pot}ʻQ.\supset :&S^{;}P\in \text{Pot}ʻS^{;}Q.\supset .\\ +&S^{;}(P\mid Q)\in \text{Pot}ʻS^{;}Q &\qquad \text{(5)}\\ +\vdash .(1).(8).*91·373 \frac{S^{;}P\in \text{Pot}ʻS^{;}Q}{\phi P} .\supset \\ +\vdash \colon\ldotp \text{Hp}.\supset :P\in \text{Pot}ʻQ.&\supset _{P}.S^{;}P\in \text{Pot}ʻ(S^{;}Q) &\qquad \text{(9)}\\ +\vdash .(6).(9).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*150·81.</b> \[\begin{align}\vdash \colon\ldotp S\in \text{Cls}\rightarrow 1:\text{D}ʻQ\subset \text{ᗡ}ʻS.\lor.\text{ᗡ}ʻQ\subset &\text{ᗡ}ʻS:T\in \text{Pot}ʻS^{;}Q:\supset .\\ +&(\exists P).P\in \text{Pot}ʻQ.T=S^{;}P\end{align}\]</p> + + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*91·351.&\supset \vdash .(\exists P).P\in \text{Pot}ʻQ.S^{;}Q=S^{;}P &\qquad \text{(1)}\\ +\vdash .*150·8. &\supset \vdash :\text{Hp}.P\in \text{Pot}ʻQ.T=S^{;}P.\supset .T\mid (S^{;}Q)=S^{;}(P\mid Q).\\ +[*91·282] &\supset .(\exists R).R\in \text{Pot}ʻQ.T\mid (S^{;}Q)=S^{;}R &\qquad \text{(2)}\\ +\vdash .(2).*10·23.\supset \\ +\vdash \colon\ldotp \text{Hp}.&\supset :(\exists P).P\in \text{Pot}ʻQ.T=S^{;}P.\supset .\\ +&(\exists R).R\in \text{Pot}ʻQ.T\mid (S^{;}Q)=S^{;}R &\qquad \text{(3)}\\ +\vdash .(1).(3).*91·171 \frac{S^{;}Q,T,(\exists P).P\in \text{Pot}ʻQ.T=S^{;}P}{R,\,S,\,\phi T} .\supset \\ +\vdash :\text{Hp}.T\in \text{Pot}ʻS^{;}Q.\supset .(\exists P).P\in \text{Pot}ʻQ.T=S^{;}P:\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_316">[Pg 316]</span></p> + +<p class="nind"><b>*150·82.</b> \(\vdash \colon\ldotp S \in \text{Cls} \rightarrow 1 : \text{D}ʻQ \subset \text{ᗡ}ʻS . \lor . \text{ᗡ}ʻQ \subset \text{ᗡ}ʻS : \supset . \text{Pot}ʻS^{;}Q + = S\dagger ʻʻ\text{Pot}ʻQ\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *150·8·81 . \supset \\ +\vdash : \text{Hp} . \supset . \text{Pot}ʻS^{;}Q &= \hat{T}\{(\exists P) . P \in \text{Pot}ʻQ . T = S^{;}P\}\\ +[*150·1] &= S\dagger ʻʻ\text{Pot}ʻQ : \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*150·83">*150·83</a>.</b> \(\vdash \colon\ldotp S\in \text{Cls} \rightarrow 1 : \text{D}ʻQ \subset \text{ᗡ}ʻS . \lor . \text{ᗡ}ʻQ \subset \text{ᗡ}ʻS : \supset . (S^{;}Q)_{\text{po}} + = S^{;}Q_{\text{po}}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *150·82 . (*91·05) . \supset \vdash : \text{Hp} . \dot{\supset} . (S^{;}Q)_{\text{po}} &= \dot{s} ʻS\dagger ʻʻ\text{Pot}ʻQ\\ +[*150·16 . (*91·05)] &= S^{;}Q_{\text{po}} : \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p>The following propositions, down to <a href="#*150·94">*150·94</a> inclusive, resume the +subject of the relation \(S\dagger\), which has already been treated in +<a href="#*150·14">*150·14</a>—<a href="#*150·171">·171</a>.</p> + +<p class="nind"><b><a id="*150·9">*150·9</a>.</b> \(\vdash . (I\dagger )^{;}Q = Q\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *150·56. \supset \vdash : M (I\dagger ^{;}Q) N . &\equiv . (\exists X, Y) . XQY . M = I^{;}X . N = I^{;}Y .\\ +[*150·53] &\equiv . (\exists X, Y) . XQY. M = X . N = Y .\\ +[*13·22] &\equiv . MQY : \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p>The following propositions lead up to <a href="#*150·931">*150·931</a><a href="#*150·94">·94</a>, which are used in +the theory of double ordinal similarity (<a href="#*164·3">*164·3</a><a href="#*164·21">·21</a>).</p> + +<p class="nind"><b>*150·91.</b> \(\vdash : sʻCʻʻCʻQ \subset \alpha .\supset . (I \upharpoonright \alpha )\dagger ^{;}Q = Q\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *150·535 . &\supset \vdash \colon\ldotp \text{Hp} . \supset : X \in CʻQ . \supset . (I \upharpoonright \alpha )^{;}X = X &\qquad \text{(1)}\\ +\vdash . (1) . *150·56 . &\supset \vdash \colon\ldotp \text{Hp} . \supset :\\ +M \{(I \upharpoonright \alpha )\dagger ^{;}Q\} N . &\equiv . (\exists X,Y) . XQY . M = X . N = Y .\\ +[*13·22] &\equiv . MQN \colon\ldotp \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*150·92.</b> \(\vdash : S \in \text{Cls} \rightarrow 1 . sʻCʻʻCʻQ \subset \text{ᗡ}ʻS. \supset . \breve{S} \dagger ^{;}S\dagger ^{;}Q = Q\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *150·13·14. \supset \vdash . \breve{S} \dagger ^{;}S\dagger ^{;}Q = (\breve{S} \mid S)\dagger ^{;}Q &&\qquad \text{(1)}\\ +\vdash . (1) . *71·191 . \supset \vdash : \text{Hp} . \supset . \breve{S} \dagger ^{;}S\dagger ^{;}Q &= (I \upharpoonright \text{ᗡ}ʻS)\dagger ^{;}Q\\ +[*150·91] &= Q : \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*150·921.</b> \(\vdash : S \in 1 \rightarrow \text{Cls} . sʻCʻʻCʻP \subset \text{D}ʻS . \supset . S\dagger ^{;} \breve{S} \dagger ^{;}P = P\)</p> + +<p class="nind"><b>*150·93.</b> \[\begin{align}]\vdash \colon\ldotp S \in 1 \rightarrow 1 . sʻCʻʻCʻP \subset &\text{D}ʻS . sʻCʻʻCʻQ \subset \text{ᗡ}ʻS . \supset :\\ +&P = S\dagger ^{;}Q . \equiv . Q = \breve{S} \dagger ^{;}P \quad[*150·92·921]\end{align}\]</p> + +<p class="nind"><b><a id="*150·931">*150·931</a>.</b> \(\vdash : sʻCʻʻCʻQ \subset \text{ᗡ}ʻS . \supset . CʻʻCʻS\dagger ^{;}Q = S_{\in}ʻʻCʻʻCʻQ\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *150·22 . &\supset \vdash . CʻS\dagger ^{;}Q = S\dagger ʻʻCʻQ &\qquad \text{(1)}\\ +\vdash . *150·22·1 . \supset \vdash \colon\ldotp \text{Hp} . &\supset : M \in CʻQ . \supset . CʻS\dagger ʻM = SʻʻCʻM :\\ +[*37·68·11] &\supset : CʻʻS\dagger ʻʻCʻQ = S_{\in}ʻʻCʻʻCʻQ &\qquad \text{(2)}\\ +\vdash . (1) . (2) . \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_317">[Pg 317]</span></p> + +<p class="nind"><b>*150·932.</b> \[\begin{align}&\vdash :sʻCʻʻCʻQ\subset \text{ᗡ}ʻS.\supset .sʻCʻʻCʻS{\dagger}^{;}Q = SʻʻsʻCʻʻCʻQ\\ +&[*150·931.*37·11.*40·38]\end{align}\]</p> + +<p class="nind"><b>*150·933.</b> \(\vdash :sʻCʻʻCʻQ\subset \text{ᗡ}ʻS.\supset .sʻCʻʻCʻ{\dagger}^{;}Q\subset \text{D}ʻS \quad[*150·932.*37·15]\)</p> + +<p class="nind"><b><a id="*150·94">*150·94</a>.</b> \[\begin{align}\vdash \colon\ldotp S\in 1\rightarrow 1.&\supset :sʻCʻʻCʻQ\subset \text{ᗡ}ʻS.P = S{\dagger}^{;}Q. \equiv .\\ +&sʻCʻʻCʻP\subset \text{D}ʻS.Q = \breve{S} \dagger ^{;}P\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*150·933. \supset \vdash :sʻCʻʻCʻQ&\subset \text{ᗡ}ʻS.P = S{\dagger}^{;}Q. \equiv .\\ +&sʻCʻʻCʻP\subset \text{D}ʻS.sʻCʻʻCʻQ\subset \text{ᗡ}ʻS.P = S{\dagger}^{;}Q &\qquad \text{(1)}\\ +\vdash .*150·933 \frac{\breve{S}}{S}.\supset \vdash :sʻCʻʻCʻP&\subset \text{D}ʻS.Q = \breve{S} {\dagger}^{;}P. \equiv .\\ +&sʻCʻʻCʻP\subset \text{D}ʻS.sʻCʻʻCʻQ\subset \text{ᗡ}ʻS.Q = \breve{S} {\dagger}^{;}P &\qquad \text{(2)}\\ +\vdash .*150·93.*5·32.\supset \\ +\vdash \colon\ldotp S\in 1\rightarrow 1.&\supset :sʻCʻʻCʻP\subset \text{D}ʻS.sʻCʻʻCʻQ\subset \text{ᗡ}ʻS.P = S{\dagger}^{;}Q. \equiv .\\ +&sʻCʻʻCʻP\subset \text{D}ʻS.sʻCʻʻCʻQ\subset \text{ᗡ}ʻS.Q = \breve{S} {\dagger}^{;}P &\qquad \text{(3)}\\ +\vdash .(1).(2).(3).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p>The above proposition is the analogue of *74·61, which (with a few +trivial transformations) may be written +\[ +\vdash \colon\ldotp S\in 1\rightarrow 1.\supset :sʻ\lambda \subset \text{ᗡ}ʻS.\kappa = S_{\in}ʻʻ\lambda . \equiv .sʻ\kappa \subset \text{D}ʻS.\lambda = (\breve{S} )_{\in}ʻʻ\kappa . +\]</p> + +<p>In obtaining ordinal analogues of such propositions, \(S_{\in }\) +will be replaced by \(S\dagger\), and the two inverted commas will +be replaced by the semi-colon; a class of classes \(\kappa\) will be +replaced, in most of its occurrences, by a relation of relations \(P\), +but will sometimes be replaced by \(CʻʻCʻP\).</p> + +<p>The above proposition (<a href="#*150·94">*150·94</a>) is used in proving that the converse of +a double correlator of \(P\) and \(Q\) is a double correlator of \(Q\) +and \(P\) (<a href="#*164·21">*164·21</a>). The corresponding cardinal proposition (<a href="#*111·131">*111·131</a>) +uses *74·6, which is practically the same proposition as *74·61, which +is the analogue of *150·94.</p> + +<p class="nind"><b>*150·95.</b> \(\vdash :CʻR\subset \text{Cl}ʻ\alpha .\supset .(S\upharpoonright \alpha )_{\in }^{;}R = S^{;}R\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*37·421.\supset \vdash \colon\ldotp \text{Hp}.\supset :\beta \in CʻR.\supset .(S\upharpoonright \alpha )ʻʻ\beta = Sʻʻ\beta .\\ +[*37·11] \supset .(S\upharpoonright \alpha )_{\in }ʻ\beta = S_{\in }ʻ\beta &\qquad \text{(1)}\\ +\vdash .(1).*150·35.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_318">[Pg 318]</span></p> + +<p>The above proposition is used in the theory of "first differences" +(<a href="#*170·41">*170·41</a>).</p> + +<p class="nind"><b>*150·96.</b> \(\vdash :\text{ᗡ}ʻ\dot{s} ʻ\lambda \subset \text{ᗡ}ʻS.\supset .\text{D}^{;}(T\parallel \breve{S} \upharpoonright \lambda =T_{\in }\upharpoonright + \text{D}ʻʻ\lambda \)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*150·51.&\supset \vdash :\alpha {\text{D}^{;}(T\parallel \breve{S} )\upharpoonright \lambda }\beta .\equiv .\\ +&(\exists M,N).N\in \lambda .M=T\mid N\mid \breve{S} .\alpha =\text{D}ʻM.\beta =\text{D}ʻN &\qquad \text{(1)}\\ +\vdash .*41·13.\supset \vdash \colon\ldotp \text{Hp}.&\supset :N\in \lambda .\supset .\text{ᗡ}ʻN\subset \text{ᗡ}ʻS.\\ +[*37·321] &\supset .\text{D}ʻ(N\mid \breve{S} )=\text{D}ʻN.\\ +[*37.32] &\supset .\text{D}ʻ(T\mid N\mid \breve{S} )=Tʻʻ\text{D}ʻN &\qquad \text{(2)}\\ +\vdash .(1).(2).*37·6.\supset \\ +\vdash \colon\ldotp \text{Hp}.\supset .\alpha \{\text{D}^{;}(T\parallel \breve{S} )\upharpoonright \lambda\}\beta .&\equiv .\beta \in \text{D}ʻʻ\lambda .\alpha =Tʻʻ\beta .\\ +[*37·101] &\equiv .\alpha (T_{\in }\upharpoonright \text{D}ʻʻ\lambda )\beta \colon\ldotp \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*150·961.</b> \(\vdash .\dot{s} ^{;}(U\parallel \breve{W} )_{\in }\upharpoonright \lambda =(U\parallel \breve{W} )\upharpoonright \dot{s}ʻʻ\lambda \)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*150·4.\supset \vdash :R{\dot{s} ^{;}(U\parallel \breve{W} _{\in }\upharpoonright + \lambda }S.&\equiv .(\exists \beta ).\beta \in \lambda .S=\dot{s} ʻ\beta .R=\dot{s} ʻ(U\parallel \breve{W} ʻʻ\beta .\\ +[*43·43] &\equiv .(\exists \beta ).\beta \in \lambda .S=\dot{s} ʻ\beta .R=(U\parallel \breve{W} )ʻ\dot{s} ʻ\beta .\\ +[*13·193.*37·6] &=.S\in \dot{s}ʻʻ\lambda .R=(U\parallel \breve{W} )ʻS:\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p>The above proposition is used in the theory of ordinal exponentiation +(<a href="#*176·21">*176·21</a>).</p> + + +<div class="footnotes"><h3>FOOTNOTES:</h3> + +<div class="footnote"> + +<p class="nind"> +<a id="Footnote_11" href="#FNanchor_11" class="label">[11]</a> +<i>E.g.</i> in <a href="#*116·53">*116·53</a> and following propositions, where +the notation \(S\dagger\) was introduced by a temporary definition.</p> + +</div> +</div> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_319">[Pg 319]</span></p> +<h2 class="nobreak" id="*151"> +*151. ORDINAL SIMILARITY.</h2> +</div> + + +<p class="nind"><i>Summary of</i> *151.</p> + +<p>In this number, we give the definition of ordinal similarity, and +various equivalent forms; we prove that ordinal similarity is reflexive +(<a href="#*151·13">*151·13</a>), symmetrical (<a href="#*151·14">*151·14</a>) and transitive (<a href="#*151·15">*151·15</a>), and we give +some particular cases of ordinal similarity (<a href="#*151·6">*151·6</a> ff.). Propositions +in this number should be compared with those in *73, to which they are +analogous.</p> + +<p>The class of ordinal correlators of \(P\) and \(Q\) is written +\(P\,\overline{\text{smor}}\,Q\), where "\(\text{smor}\)" stands for +"similar ordinally." We put +\[ +P\,\overline{\text{smor}}\,Q=\hat{S}\{S\in 1\rightarrow 1.CʻQ=\text{ᗡ}ʻS.P=S^{;}Q\} \quad\text{Df}. +\]</p> + +<p>(We might equally well put +\[ +P\,\overline{\text{smor}}\,Q=(1\rightarrow 1)\cap \overleftarrow{\text{ᗡ}}ʻCʻQ\cap \dagger \overleftarrow{Q}ʻP \quad\text{Df}, +\] +which is an equivalent but more condensed form of the definition.) We +then define "\(P\) is ordinally similar to \(Q\)" as meaning that there +is at least one ordinal correlator of \(P\) and \(Q\), <i>i.e.</i> +\[ +\text{smor}=\hat{P} \hat{Q} (\exists !P\,\overline{\text{smor}}\,Q) \quad\text{Df}. +\]</p> + +<p>We shall find that if \(P\) and \(Q\) generate well-ordered series, +they have at most one correlator (*250·6), but this does not hold in +general for other series.</p> + +<p>After giving the elementary properties of ordinal similarity, we +have three important propositions on its connection with cardinal +similarity, namely: (<a href="#*151·18">*151·18</a>) if \(P\) is similar to \(Q\), the field +of \(P\) is similar to the field of \(Q\) (the converse does not hold +in general, but holds if \(P\) and \(Q\) are finite serial relations); +(<a href="#*151·19">*151·19</a>) if \(CʻP\) is similar to \(CʻQ\), there is a relation \(R\) +similar to \(Q\) and having \(CʻP\) for its field, and vice versa; +(<a href="#*151·191">*151·191</a>) \(S\) is an ordinal correlator of \(P\) and \(Q\) when, +and only when, it is a cardinal correlator of \(CʻP\) and \(CʻQ\) and +\(P=S^{;}Q\).</p> + +<p><span class="pagenum" id="Page_320">[Pg 320]</span></p> + +<p>We then have a set of propositions on correlators of the form +\(S\upharpoonright CʻQ\) (<a href="#*151·2">*151·2</a>—<a href="#*151·243">·243</a>). Most of the correlators +with which we shall be concerned are of this form. The most useful +proposition here is</p> + +<p class="nind"><b>*151·22.</b> \(\vdash :S\upharpoonright CʻQ\in 1\rightarrow 1.CʻQ\subset \text{ᗡ}ʻS.P=S^{;}Q.\equiv .S\upharpoonright CʻQ\in P\,\overline{\text{smor}}\,Q\)</p> + +<p>A useful consequence of this proposition is</p> + +<p class="nind"><b>*151·231.</b> \(\vdash \colon\ldotp (y).\text{E}!Sʻy:S\upharpoonright CʻQ\in 1\rightarrow 1.P=S^{;}Q:\supset .S\upharpoonright CʻQ\in P\,\overline{\text{smor}}\,Q\)</p> + +<p>This consequence is useful because the hypothesis \((y).\text{E}!Sʻy\) +is satisfied by most of the relations which occur as correlators.</p> + +<p>We have next a number of propositions on the inferribility of +\(Q=\breve{S} ^{;}P\) or \(Q\,\unicode{x2abd}\, \breve{S} ^{;}P\) from +\(P=S^{;}Q\) or \(P\,\unicode{x2abd}\, S^{;}Q\), and connected matters +(<a href="#*151·25">*151·25</a>—<a href="#*151·29">·29</a>).</p> + +<p>We have</p> + +<p class="nind"><b>*151·25.</b> \(\vdash :S\in \text{Cls}\rightarrow 1.CʻQ\subset \text{ᗡ}ʻS.P=S^{;}Q.\supset .Q=\breve{S} ^{;}P\)</p> + +<p class="nind"><b>*151·26.</b> \[\begin{align}\vdash \colon\ldotp S\in \text{Cls}\rightarrow 1.CʻQ\subset \text{ᗡ}ʻS.\supset :&P\,\unicode{x2abd}\, S^{;}Q.\supset .\breve{S} ^{;}P\,\unicode{x2abd}\, + Q:\\ +&S^{;}Q\,\unicode{x2abd}\, P.\supset .Q\,\unicode{x2abd}\, \breve{S} ^{;}P\end{align}\]</p> + +<p class="nind"><b>*151·29.</b> \[\begin{align}\vdash \colon\ldotp P\,\text{smor}\,Q.\equiv :(\exists S):&xPy.\supset _{x,y}.(\breve{S} ʻx)Q(\breve{S} ʻy):\\ +&zQw.\supset _{z,w}.(Sʻz)P(Sʻw)\end{align}\]</p> + +<p><a href="#*151·29">*151·29</a> is never used, but is inserted in order to show that our +definition of "ordinal similarity" agrees with what is commonly +understood by that term. If \(P\) and \(Q\) are regarded as serial, so +that "\(xPy\)" means "\(x\) precedes \(y\) in the \(P\)-series," and +"\(zQw\)" means "\(z\) precedes \(w\) in the \(Q\)-series," then our +proposition states that two series are ordinally similar when their +terms can be so correlated that predecessors in either are correlated +with predecessors in the other, and successors with successors, +<i>i.e.</i> when the two series can be correlated without change of +order.</p> + +<p>We have next (<a href="#*151·31">*151·31</a>—<a href="#*151·52">·52</a>) a set of miscellaneous propositions, of +which the most useful are</p> + +<p class="nind"><b><a id="*151·401">*151·401</a>.</b> \[\begin{align}\vdash :T\upharpoonright CʻP\in X,\overline{\text{smor}}\,P.T\upharpoonright CʻQ\in Y\overline{\text{smor}} Q.S\in &P\,\overline{\text{smor}}\,Q.\supset .\\ +&T^{;}S\in X\,\overline{\text{smor}} \,Y\end{align}\]</p> + +<p class="nind"><b><a id="*151·5">*151·5</a>.</b> + \[\begin{align}\vdash :S\upharpoonright CʻQ\in P\,\overline{\text{smor}}\,Q.\supset .\text{D}ʻP=Sʻʻ\text{D}ʻQ.\text{ᗡ}ʻP=Sʻʻ\text{ᗡ}ʻQ.&\overrightarrow{B}ʻP=Sʻʻ\overrightarrow{B}ʻQ.\\ +&\overrightarrow{B}ʻ\breve{P} =Sʻʻ\overrightarrow{B}ʻ\breve{Q}\end{align}\]</p> + +<p><a href="#*151·401">*151·401</a> will be useful in such cases as the following: Let \(P\) +and \(Q\) be relations between relations, then \(\text{D}^{;}P\) and +\(\text{D}^{;}Q\) will be the corresponding relations of their domains. +Suppose \(\text{D}\upharpoonright CʻP\), \(\text{D}\upharpoonright CʻQ\in 1\rightarrow 1\). +Then, by *151·401, if \(S\) is a correlator of \(P\) and \(Q\), +\(\text{D}^{;}S\) is a correlator of \(\text{D}^{;}P\) and \(\text{D}^{;}Q\).</p> + +<p><a href="#*151·5">*151·5</a> shows that if \(S\) is a correlator of \(P\) and \(Q\), it correlates \(\text{D}ʻP\) with +\(\text{D}ʻQ\), \(\text{ᗡ}ʻP\) with \(\text{ᗡ}ʻQ\), \(\overrightarrow{B}ʻP\) with \(\overrightarrow{B}ʻQ\), and \(\overrightarrow{B}ʻ\breve{P}\) with \(\overrightarrow{B}ʻ\breve{Q}\).</p> + +<p>Our next set of propositions (<a href="#*151·53">*151·53</a>—<a href="#*151·59">·59</a>) is concerned with the +correlation of powers of \(P\) and \(Q\) and kindred matters. We show +(<a href="#*151·55">*151·55</a>) that a<span class="pagenum" id="Page_321">[Pg 321]</span> correlator of \(P\) and \(Q\) is also a correlator +of \(P_{\text{po}}\) and \(Q_{\text{po}}\), and therefore if \(P\) +and \(Q\) are similar, so are \(P_{\text{po}}\) and \(Q_{\text{po}}\) +(<a href="#*151·56">*151·56</a>); we show also (<a href="#*151·59">*151·59</a>) that if \(P\) and \(Q\) are similar, +so are \(P_{\nu }\) and \(Q_{\nu }\). These propositions are used in +the theory of progressions (*263·17).</p> + +<p>The remaining propositions (<a href="#*151·6">*151·6</a> to the end) are concerned with +applications to particular cases. The most useful of these are</p> + +<p class="nind"><b>*151·61.</b> \(\vdash .\iota ^{;}P\,\text{smor}\,P\)</p> + +<p class="nind">which shows how to raise the type of a relation without changing its +relation-number;</p> + +<p class="nind"><b>*151·64.</b> \(\vdash .x\downarrow ^{;}P\,\text{smor}\,P.(x\downarrow )\upharpoonright CʻP\in (x\downarrow ^{;}P)\,\overline{\text{smor}}\,P\)</p> + +<p class="nind"><b>*151·65.</b> \(\vdash .\downarrow x^{;}P\,\text{smor}\,P.(\downarrow x)\upharpoonright CʻP\in (\downarrow x^{;}P)\,\overline{\text{smor}}\,P\)</p> + +<p>We prove also that all members of \(2_{r}\) (<i>i.e.</i> all relations +of the form \(x\downarrow y\), where \(x\neq y\)) are similar +<a href="#*151·63">*151·63</a>), and that all relations of the form \(x\downarrow x\) are +similar (<a href="#*151·631">*151·631</a>).</p> + +<hr class="tb"> + +<p class="nind"><b>*151·01.</b> \(P\,\overline{\text{smor}}\,Q=\hat{S}\{S\in 1\rightarrow 1.CʻQ=\text{ᗡ}ʻS.P=S^{;}Q\} \quad\text{Df}\)</p> + +<p class="nind"><b>*151·02.</b> \(\text{smor}=\hat{P} \hat{Q}\{\exists !P\,\overline{\text{smor}}\,Q\} \quad\text{Df}\)</p> + +<p class="nind"><b>*151·1.</b> \(\vdash :P\,\text{smor}\,Q.\equiv .(\exists S).S\in 1\rightarrow 1.CʻQ=\text{ᗡ}ʻS.P=S^{;}Q \quad[(*151·02)]\)</p> + +<p class="nind"><b>*151·11.</b> \(\vdash :S\in P\,\overline{\text{smor}}\,Q.\equiv .S\in 1\rightarrow 1.CʻQ=\text{ᗡ}ʻS.P=S^{;}Q \quad[(*151·01)]\)</p> + +<p class="nind"><b>*151·12.</b> \(\vdash :P\,\text{smor}\,Q.\equiv .\exists !P\,\overline{\text{smor}}\,Q \quad[(*151·02)]\)</p> + +<p class="nind"><b>*151·121.</b> \(\vdash .I\upharpoonright CʻQ\in (Q\overline{\text{smor}} Q)\quad[*72·17.*50·5·52.*150·534.*151·11]\)</p> + +<p class="nind"><b><a id="*151·13">*151·13</a>.</b> \(\vdash .Q\text{smor}Q \quad[*151·121·12]\)</p> + +<p class="nind"><b>*151·131.</b> \(\vdash :S\in P\,\overline{\text{smor}}\,Q.=.\breve{S} \in Q\,\overline{\text{smor}}\,P\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*71·212.&\supset \vdash :S\in 1\rightarrow 1.\equiv .\breve{S} \in 1\rightarrow 1 &\qquad \text{(1)}\\ +\vdash .*150·13.&\supset \vdash :P=S^{;}Q.\supset .\breve{S} ^{;}P=(\breve{S} \mid S)^{;}Q:\\ +[*71·192] &\supset \vdash :S\in 1\rightarrow 1.P=S^{;}Q.\supset .\breve{S} ^{;}P=(I\upharpoonright \text{ᗡ}ʻS)^{;}Q:\\ +[*150·534] &\supset \vdash :S\in 1\rightarrow 1.CʻQ=\text{ᗡ}ʻS.P=S^{;}Q.\supset .\breve{S} ^{;}P=Q &\qquad \text{(2)}\\ +\vdash .*150·23.&\supset \vdash :CʻQ=\text{ᗡ}ʻS.P=S^{;}Q.\supset .CʻP=\text{D}ʻS &\qquad \text{(3)}\\ +\vdash .(1).(2).(3).*33·21.\supset \\ +\vdash :S\in 1\rightarrow 1.&CʻQ =\text{ᗡ}ʻS.P=S^{;}Q.\supset .\breve{S} \in 1\rightarrow 1.CʻP=\text{ᗡ}ʻ\breve{S} .Q=\breve{S} ^{;}P &\qquad \text{(4)}\\ +\vdash .(4) \frac{\breve{S},\,Q,\,P}{S,\,P,\,Q}.*31·33.\supset \\ +\vdash :\breve{S} \in 1\rightarrow 1.&CʻP=\text{ᗡ}ʻS.Q=\breve{S} ^{;}P.\supset .S\in 1\rightarrow 1.CʻQ=\text{ᗡ}ʻS.P=S^{;}Q &\qquad \text{(5)}\\ +\vdash .(4).(5).*151·11.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*151·14">*151·14</a>.</b> \(\vdash :P\,\text{smor}\,Q.\equiv .Q\,\text{smor}\,P \quad[*151·131·12.*31·52]\)</p> + +<p><span class="pagenum" id="Page_322">[Pg 322]</span></p> + +<p class="nind"><b>*151·141.</b> \(\vdash :S\in P\,\overline{\text{smor}}\,Q.T\in Q\,\overline{\text{smor}}\,R.\supset .S\mid T\in P\,\overline{\text{smor}}\,R\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*151·11.*71·252. \supset \vdash :\text{Hp}.&\supset .S\mid T\in 1\rightarrow 1 &\qquad \text{(1)}\\ +\vdash .*151·11.*150·23. \supset \vdash :\text{Hp}.&\supset .\text{ᗡ}ʻS=CʻQ.\text{D}ʻT=CʻQ.\text{ᗡ}ʻT=CʻR.\\ +[*37·323] &\supset .\text{ᗡ}ʻ(S\mid T)=\text{ᗡ}ʻT.\text{ᗡ}ʻT=CʻR.\\ +[*13·17] &\supset .\text{ᗡ}ʻ(S\mid T)=CʻR &\qquad \text{(2)}\\ +\vdash .*151·11. \supset \vdash :\text{Hp}.\supset .P&=S^{;}T^{;}R\\ +[*150·13] &=(S\mid T)^{;}R\\ +\vdash .(1).(2).(3).*151·11.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*151·15">*151·15</a>.</b> \(\vdash :P\,\text{smor}\,Q.Q\,\text{smor}\,R.\supset .P\,\text{smor}\,R \quad[*151·141]\)</p> + +<p class="nind"><b>*151·16.</b> \(\vdash .I\,\unicode{x2abd}\, \text{smor} \quad[*151·13]\)</p> + +<p class="nind"><b>*151·161.</b> \(\vdash .\text{smor}=\text{Cnv}ʻ\text{smor} \quad[*151·14]\)</p> + +<p class="nind"><b>*151·162.</b> \(\vdash .(\text{smor})^{2}=\text{smor} \quad[*151·15·161.*34·81]\)</p> + +<p class="nind"><b>*151·17.</b> \(\vdash \colon\ldotp P\,\text{smor}\,Q.\supset :R\,\text{smor}\,P.\equiv .R\,\text{smor}\,Q \quad[*151·14·15]\)</p> + +<p class="nind"><b><a id="*151·18">*151·18</a>.</b> \(\vdash:P\,\text{smor}\,Q.\supset .CʻP\text{ sm }CʻQ\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*151·11.*150·23.\supset \vdash :\text{Hp}.&\supset .(\exists S).S\in 1\rightarrow 1.\text{D}ʻS=CʻP.\text{ᗡ}ʻS=CʻQ.\\ +[*73·1] &\supset .CʻP\text{ sm }CʻQ:\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*151·19">*151·19</a>.</b> \(\vdash :CʻP\text{ sm }CʻQ.\equiv .(\exists R).CʻR=CʻP.R\,\text{smor}\,Q\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*73·1.\supset \vdash :CʻP\text{ sm }CʻQ.&\equiv .(\exists S).S\in 1\rightarrow 1.\text{D}ʻS=CʻP.\text{ᗡ}ʻS=CʻQ.\\ +[*150·23] &\equiv .(\exists S).S\in 1\rightarrow 1.\text{ᗡ}ʻS=CʻQ.CʻS^{;}Q=CʻP.\\ +[*13·195] &\equiv .(\exists R,S).S\in 1\rightarrow 1.\text{ᗡ}ʻS=CʻQ.R=S^{;}Q.CʻR=CʻP.\\ +[*151·1] &\equiv .(\exists R).CʻR=CʻP.R\,\text{smor}\,Q:\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*151·191">*151·191</a>.</b> \(\vdash :S\in P\,\overline{\text{smor}}\,Q.\equiv .S\in (CʻP)\,\overline{\text{ sm }}\, (CʻQ).P=S^{;}Q\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*151·131·11.\supset \vdash :S\in P\,\overline{\text{smor}}\,Q.&\supset .CʻP=\text{D}ʻS:\\ +[*4·71.*151·11]\supset \vdash :S\in P\,\overline{\text{smor}}\,Q.&\equiv .S\in 1\rightarrow 1.CʻQ=\text{ᗡ}ʻS.P=S^{;}Q.CʻP=\text{D}ʻS.\\ +[*73·03] &\equiv .S\in (CʻP)\,\overline{\text{ sm }}\, (CʻQ).P=S^{;}Q:\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*151·2">*151·2</a>.</b> \(\vdash :S\in 1\rightarrow 1.CʻQ\subset \text{ᗡ}ʻS.P=S^{;}Q.\supset .S\upharpoonright CʻQ\in P\,\overline{\text{smor}}\,Q\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*71·29. &\supset \vdash :\text{Hp}.\supset .S\upharpoonright CʻQ\in 1\rightarrow 1 &\qquad \text{(1)}\\ +\vdash .*35·65. &\supset \vdash :\text{Hp}.\supset .\text{ᗡ}ʻS\upharpoonright CʻQ=CʻQ &\qquad \text{(2)}\\ +\vdash .*150·32.&\supset \vdash :\text{Hp}.\supset .P=S\upharpoonright CʻQ^{;}Q &\qquad \text{(3)}\\ +\vdash .(1).(2).(3).*151·11.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_323">[Pg 323]</span></p> + +<p class="nind"><b>*151·21.</b> \(\vdash :P\,\text{smor}\,Q.\equiv .(\exists S).S\in 1\rightarrow 1.CʻQ\subset \text{ᗡ}ʻS.P=S^{;}Q \quad[*151·2]\)</p> + +<p class="nind"><b>*151·22.</b> \(\vdash :S\upharpoonright CʻQ\in 1\rightarrow 1.CʻQ\subset \text{ᗡ}ʻS.P=S^{;}Q.\equiv .S\upharpoonright CʻQ\in P\,\overline{\text{smor}}\,Q\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*35·65.*150·32. &\supset \vdash :S\upharpoonright CʻQ\in 1\rightarrow 1.CʻQ\subset \text{ᗡ}ʻS.P=S^{;}Q.\supset .\\ +&S\upharpoonright CʻQ\in P\,\overline{\text{smor}}\,Q &\qquad \text{(1)}\\ +\vdash .*151·11.*150·32.&\supset \vdash :S\upharpoonright CʻQ\in P\,\overline{\text{smor}}\,Q.\supset .S\upharpoonright CʻQ\in 1\rightarrow l.P=S^{;}Q &\qquad \text{(2)}\\ +\vdash .*151·11. \supset \vdash :S\upharpoonright CʻQ\in P\,\overline{\text{smor}}\,Q.\supset .CʻQ&=\text{ᗡ}ʻ(S\upharpoonright CʻQ)\\ +[*35·64] &=CʻQ\cap \text{ᗡ}ʻS.\\ +[*22·621] &\supset .CʻQ\supset \text{ᗡ}ʻS &\qquad \text{(3)}\\ +\vdash .(1).(2).(3).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*151·23.</b> \(\vdash :P\,\text{smor}\,Q.\equiv .(\exists S).S\upharpoonright CʻQ\in 1\rightarrow 1.CʻQ\subset \text{ᗡ}ʻS.P=S^{;}Q \quad[*151·22]\)</p> + +<p>The above proposition (*151·23) is very useful. It is the analogue of +*73·15. (It should be observed that, in all propositions concerning +likeness, \(S^{;}Q\) plays the same part as \(Sʻʻ\beta\) plays in +propositions concerning similarity.) By means of *151·23, we can +establish likeness in all those numerous cases in which a relation +which is not usually one-one becomes one-one when confined to a +certain converse domain, as for example if we have to deal with +\(\text{D}\upharpoonright {\in}_{\Delta}ʻ\kappa\), where \(\kappa +\in \text{Cls}^{2} \text{excl}\), or with \(\text{D}\upharpoonright P_{\Delta }ʻ\kappa\), +where \(P\upharpoonright \kappa \in \text{Cls}\rightarrow 1\). Thus +<i>e.g.</i> by the above proposition, if \(Q\) is any relation whose +field is \(P_{\Delta }ʻ\kappa\), where \(P\upharpoonright \kappa \in\text{Cls}\rightarrow 1\), +\(\text{D}^{;}Q\) will be an ordinally similar +relation whose field is \(\text{D}ʻʻP_{\Delta }ʻ\kappa\).</p> + +<p class="nind"><b>*151·231.</b> \[\begin{align}&\vdash \colon\ldotp (y).\text{E}!Sʻy:S\upharpoonright CʻQ\in 1\rightarrow 1.P=S^{;}Q:\supset .S\upharpoonright CʻQ\in P\,\overline{\text{smor}}\,Q\\ +&[*151·22.*33·431]\end{align}\]</p> + +<p class="nind"><b>*151·232.</b> \[\begin{align}&\vdash \colon\ldotp (\exists S):(y).\text{E}!Sʻy:S\upharpoonright CʻQ\in 1\rightarrow 1.P=S^{;}Q:\supset .P\,\text{smor}\,Q\\ +&[*151·231·12]\end{align}\]</p> + +<p class="nind"><b>*151·24.</b> \[\begin{align}\vdash \colon\ldotp &(y).\text{E}!Sʻy:y,z\in CʻQ.Sʻy=Sʻz.\supset _{y,z}.y=z:P=S^{;}Q:\supset .\\ +&S\upharpoonright CʻQ\in P\,\overline{\text{smor}}\,Q.P\,\text{smor}\,Q \quad[*71·166·55.*33·431.*151·22·23]\end{align}\]</p> + +<p class="nind"><b>*151·241.</b> \[\begin{align}\vdash \colon\ldotp S\in 1\rightarrow \text{Cls}.&CʻQ\subset \text{ᗡ}ʻS:y,z\in CʻQ.Sʻy=Sʻz.\supset _{y,z}.y=z:P=S^{;}Q:\supset .\\ +&S\upharpoonright CʻQ\in P\,\overline{\text{smor}}\,Q.P\,\text{smor}\,Q \quad[*71·55.*151·22·23]\end{align}\]</p> + +<p class="nind"><b>*151·242.</b> \[\begin{align}&\vdash \colon\colon y,z\in CʻQ.\supset _{y,z}:Sʻy=Sʻz.\equiv .y=z\colon\ldotp P=S^{;}Q\colon\ldotp + \equiv .S\upharpoonright CʻQ\in P\,\overline{\text{smor}}\,Q\\ +&[*71·59.*151·22]\end{align}\]</p> + +<p class="nind"><b><a id="*151·243">*151·243</a>.</b> \[\begin{align}&\vdash \colon\colon y,z\in CʻQ.\supset _{y,z}:Sʻy=Sʻz.\equiv .y=z\colon\ldotp P=S^{;}Q\colon\ldotp \supset .P\,\text{smor}\,Q\\ +&[*151·242·12]\end{align}\]</p> + +<p class="nind"><b><a id="*151·25">*151·25</a>.</b> \(\vdash :S\in \text{Cls}\rightarrow 1.CʻQ\subset \text{ᗡ}ʻS.P=S^{;}Q.\supset .Q=\breve{S} ^{;}P\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*150·13.\supset \vdash :\text{Hp}.\supset .\breve{S} ^{;}P&=(\breve{S} \mid S)^{;}Q\\ +[*71·191] & =(I\upharpoonright \text{ᗡ}ʻS)^{;}Q\\ +[*150·535] & =Q:\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_324">[Pg 324]</span></p> + +<p class="nind"><b>*151·251.</b> \[\begin{align}&\vdash \colon\ldotp S\in 1\rightarrow 1.\supset :CʻQ\subset \text{ᗡ}ʻS.P=S^{;}Q.\equiv .CʻP\subset \text{D}ʻS.Q=\breve{S} ^{;}P\\ +&[*151·25.*150·22.*37·15]\end{align}\]</p> + +<p class="nind"><b>*151·252.</b> \(\vdash :S\in \text{Cls}\rightarrow 1.CʻQ\subset \text{ᗡ}ʻS.\supset .Q=\breve{S} ^{;}S^{;}Q \quad[*151·25]\)</p> + +<p class="nind"><b>*151·253.</b> \(\vdash :S\in 1\rightarrow \text{Cls}.CʻP\subset \text{D}ʻS.\supset .P=S^{;}\breve{S} ^{;}P \quad\left[*151·252\frac{\breve{S}}{S}\right]\)</p> + +<p class="nind"><b>*151·254.</b> + \(\vdash :S\in 1\rightarrow 1.\supset .S\dagger \upharpoonright \breve{C} ʻʻ\text{Cl}ʻ\text{ᗡ}ʻS=\text{Cnv}ʻ{\breve{S} \dagger \upharpoonright \breve{C} ʻʻ\text{Cl}ʻ\text{D}ʻS}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*151·251. \supset \vdash \colon\ldotp \text{Hp}.\supset :&CʻQ\in \text{Cl}ʻ\text{ᗡ}ʻS.P(S\dagger )Q.\equiv .\\ +&CʻP\in \text{Cl}ʻ\text{D}ʻS.Q(\breve{S} \dagger )P\colon\ldotp \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p>This proposition is the analogue of *72·54. "\(\breve{S} \dagger\)" +means "\((\text{Cnv}ʻS)\dagger\)," not "\(\text{Cnv}ʻ(S\dagger)\)."</p> + +<p class="nind"><b>*151·26.</b> \[\begin{align}\vdash \colon\ldotp S\in \text{Cls}\rightarrow 1.CʻQ\subset &\text{ᗡ}ʻS.\supset :\\ +&P\,\unicode{x2abd}\, S^{;}Q.\supset .\breve{S} ^{;}P\,\unicode{x2abd}\, Q:S^{;}Q\,\unicode{x2abd}\, P.\supset .Q\,\unicode{x2abd}\, \breve{S} ^{;}P\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*150·31. &\supset \vdash :P\,\unicode{x2abd}\, S^{;}Q.\supset .\breve{S} ^{;}P\,\unicode{x2abd}\, \breve{S} ^{;}S^{;}Q:\\ +[*151·252] &\supset \vdash \colon\ldotp \text{Hp}.\supset :P\,\unicode{x2abd}\, S^{;}Q.\supset .\breve{S} ^{;}P\,\unicode{x2abd}\, Q &\qquad \text{(1)}\\ +\text{Similarly}\quad &\vdash \colon\ldotp \text{Hp}.\supset :S^{;}Q\,\unicode{x2abd}\, P.\supset .Q\,\unicode{x2abd}\, \breve{S} ^{;}P &\qquad \text{(2)}\\ +\vdash .(1).(2). \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*151·261.</b> \[\begin{align}&\vdash \colon\ldotp S\in 1\rightarrow \text{Cls}.CʻP\subset \text{D}ʻS.\supset :\\ +&Q\,\unicode{x2abd}\, \breve{S} ^{;}P.\supset .S^{;}Q\,\unicode{x2abd}\, P:\breve{S} ^{;}P\,\unicode{x2abd}\, Q.\supset .P\,\unicode{x2abd}\, S^{;}Q + \quad\left[*151·26\frac{\breve{S},\,Q,\,P}{S,\,P,\,Q}\right]\end{align}\]</p> + +<p class="nind"><b>*151·262.</b> \[\begin{align}&\vdash \colon\ldotp S\in 1\rightarrow 1.CʻP\subset \text{D}ʻS.CʻQ\subset \text{ᗡ}ʻS.\supset :\\ +&P\,\unicode{x2abd}\, S^{;}Q.\equiv .\breve{S} ^{;}P\,\unicode{x2abd}\, Q:Q\,\unicode{x2abd}\, \breve{S} ^{;}P.\equiv .S^{;}Q\,\unicode{x2abd}\, + P \\ &[*151·26·261]\end{align}\]</p> + +<p class="nind"><b>*151·263.</b> \[\begin{align}&\vdash \colon\ldotp S\in 1\rightarrow 1.CʻP\subset \text{D}ʻS.CʻQ\subset \text{ᗡ}ʻS.\supset :\\ +&P\,\unicode{x2abd}\, S^{;}Q.Q\,\unicode{x2abd}\, \breve{S} ^{;}P.\equiv .\breve{S} ^{;}P\,\unicode{x2abd}\, Q.S^{;}Q\,\unicode{x2abd}\, P.\equiv .P=S^{;}Q.\equiv + .Q=\breve{S} ^{;}P\\ +&[*151·262]\end{align}\]</p> + +<p class="nind"><b>*151·264.</b> \(\vdash \colon\ldotp S\upharpoonright CʻQ\in 1\rightarrow 1.\supset :P\,\unicode{x2abd}\, S^{;}Q.Q\,\unicode{x2abd}\, \breve{S} ^{;}P.\equiv .P=S^{;}Q\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash *150·202.*37·401. &\supset \vdash :P\,\unicode{x2abd}\, S^{;}Q.\supset .CʻP\subset \text{D}ʻS\upharpoonright CʻQ &\qquad \text{(1)}\\ +\vdash .(1).*151·262\frac{S\upharpoonright CʻQ)}{Q}.&\supset \vdash :\text{Hp}.P\,\unicode{x2abd}\, S^{;}Q.\supset :\\ +&Q\,\unicode{x2abd}\, (CʻQ\upharpoonleft \breve{S} )^{;}P.\equiv .(S\upharpoonright CʻQ)^{;}Q\,\unicode{x2abd}\, P:\\ +[*150·361·32] &\supset :Q\,\unicode{x2abd}\, (\breve{S} ^{;}P)\unicode{x0294f}CʻQ.\equiv .S^{;}Q\,\unicode{x2abd}\, P:\\ +[*35·9.*36·29] &\supset :Q\,\unicode{x2abd}\, \breve{S} ^{;}P.\equiv .S^{;}Q\,\unicode{x2abd}\, P &\qquad \text{(2)}\\ +\vdash .(2).*5·32.&\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_325">[Pg 325]</span></p> + +<p class="nind"><b>*151·27.</b> \[\begin{align}&\vdash :S\in 1\rightarrow 1.P\,\unicode{x2abd}\, S^{;}Q.Q\,\unicode{x2abd}\, \breve{S} ^{;}P.\\ +&\equiv .S\in 1\rightarrow 1.CʻP\subset \text{D}ʻS.CʻQ\subset \text{ᗡ}ʻS.\breve{S} ^{;}P\,\unicode{x2abd}\, Q.S^{;}Q\,\unicode{x2abd}\, P.\\ +&\equiv .S\in 1\rightarrow 1.CʻQ\subset \text{ᗡ}ʻS.P=S^{;}Q.\\ +&\equiv .S\in 1\rightarrow 1.CʻP\subset \text{D}ʻS.Q=\breve{S} ^{;}P\\ +&[*151·263.*5·32.*150·203.*4·73]\end{align}\]</p> + +<p class="nind"><b>*151·271.</b> \[\begin{align}&\vdash :(\exists S).S\in 1\rightarrow 1.P\,\unicode{x2abd}\, S^{;}Q.Q\,\unicode{x2abd}\, \breve{S} ^{;}P.\\ +&\equiv .(\exists S).S\in 1\rightarrow 1.CʻP\subset \text{D}ʻS.CʻQ\subset \text{ᗡ}ʻS.\breve{S} ^{;}P\,\unicode{x2abd}\, Q.S^{;}Q\,\unicode{x2abd}\, P.\\ +&\equiv .P\,\text{smor}\,Q \quad[*151·27·21]\end{align}\]</p> + +<p class="nind"><b><a id="*151·28">*151·28</a>.</b> \[\begin{align}\vdash \colon\ldotp P\,\text{smor}\,Q.\equiv :(\exists S):S\in 1\rightarrow 1:&xPy.\supset _{x,y}.(\breve{S} ʻx)Q(\breve{S} ʻy):\\ +&zQw.\supset _{z,w}.(Sʻz)P(Sʻw)\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*150·41. &\supset \vdash \colon\colon S\in 1\rightarrow 1.\supset \colon\ldotp (\breve{S} ʻx)Q(\breve{S} ʻy).\equiv .xS^{;}Qy:(Sʻz)P(Sʻw).\equiv .z\breve{S} ^{;}Pw\colon\ldotp + \\ +[*23·1] &\supset \colon\ldotp xPy.\supset _{x,y}.(\breve{S} ʻx)Q(\breve{S} ʻy):\equiv .P\,\unicode{x2abd}\, S^{;}Q:\\ +&zQw.\supset _{z,w}.(Sʻz)P(Sʻw):\equiv .Q\,\unicode{x2abd}\, \breve{S} ^{;}P\colon\ldotp \\ +[*151·27] &\supset \colon\ldotp xPy.\supset _{x,y}.(\breve{S} ʻx)Q(\breve{S} ʻy):zQw.\supset _{z,w}.(Sʻz)P(Sʻw):\equiv .\\ +&CʻQ\subset \text{ᗡ}ʻS.P=S^{;}Q &\qquad \text{(1)}\\ +\vdash .(1).*5·32.*151·21.&\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p>The above proposition shows that ordinal similarity as we have defined +it has the properties which are commonly associated with the term +"ordinal similarity," namely that \(P\) and \(Q\) are ordinally similar +when their fields can be so correlated that two terms having the +relation \(P\) are always correlated with two terms having the relation +\(Q\), and vice versa.</p> + +<p>The hypothesis \(S\in 1\rightarrow 1\) is redundant in <a href="#*151·28">*151·28</a>; this is +shown in the following proposition.</p> + +<p class="nind"><b>*151·281.</b> \[\begin{align}\vdash \colon\ldotp xPy.&\supset _{x,y}.(\breve{S} ʻx)Q(\breve{S} ʻy):zQw.\supset _{z,w}.(Sʻz)P(Sʻw):\\ +&\supset .CʻP\upharpoonleft S=S\upharpoonright CʻQ.S\upharpoonright CʻQ\in P\,\overline{\text{smor}}\,Q\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*14·21.\supset \vdash \colon\ldotp \text{Hp}.&\supset :xPy.\supset .\text{E}!\breve{S} ʻx.\text{E}!\breve{S} ʻy:\\ +[*33·352] &\supset :x\in CʻP.\supset .\text{E}!\breve{S} ʻx:\\ +[*71·571] &\supset :(CʻP)\upharpoonleft S\in \text{Cls}\rightarrow 1.CʻP\subset \text{D}ʻS &\qquad \text{(1)}\\ +\text{Similarly}\quad \vdash :\text{Hp}.&\supset .S\upharpoonright CʻQ\in 1\rightarrow \text{Cls}.CʻQ\subset \text{ᗡ}ʻS &\qquad \text{(2)}\\ +\vdash .*33·17. \supset \vdash \colon\ldotp \text{Hp}.&\supset :xPy.\supset .\breve{S} ʻx,\breve{S} ʻy\in CʻQ:\\ +[*33·352] &\supset :x\in CʻP.\supset .\breve{S} ʻx\in CʻQ:\\ +[*14·21·26] &\supset :x\in CʻP.xSz.\supset .z\in CʻQ:\\ +[*4·71] &\supset :x\in CʻP.xSz.\equiv .x\in CʻP.xSz.z\in CʻQ:\\ +[*35·1·102] &\supset :(CʻP)\upharpoonleft S=(CʻP)\upharpoonleft S\upharpoonright CʻQ &\qquad \text{(3)}\\ +\text{Similarly}\quad \vdash :\text{Hp}.&\supset .S\upharpoonright CʻQ=(CʻP)\upharpoonleft S\upharpoonright CʻQ &\qquad \text{(4)}\\ +\vdash .(3).(4).\supset \vdash :\text{Hp}.&\supset .(CʻP)\upharpoonleft S=S\upharpoonright CʻQ. &\qquad \text{(5)}\\ +[(1).(2)] &\supset .S\upharpoonright CʻQ\in 1\rightarrow 1.CʻQ\subset \text{ᗡ}ʻS &\qquad \text{(6)}\\ +\vdash .(6).*35·7.(1).(2).*150·4·41.&\supset \vdash :\text{Hp}.\supset .P\,\unicode{x2abd}\, S^{;}Q.Q\,\unicode{x2abd}\, \breve{S} ^{;}P.\\ +[*151·264.(6)] &\supset .P=S^{;}Q &\qquad \text{(7)}\\ +\vdash .(5).(6).(7).*151·22.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_326">[Pg 326]</span></p> + +<p class="nind"><b><a id="*151·29">*151·29</a>.</b> \[\begin{align}&\vdash \colon\ldotp P\,\text{smor}\,Q.\equiv :(\exists S):xPy.\supset _{x,y}.(\breve{S} ʻx)Q(\breve{S} ʻy):zQw.\supset _{z,w}.(Sʻz)P(Sʻw)\\ +&[*151·28·281]\end{align}\]</p> + +<p class="nind"><b><a id="*151·31">*151·31</a>.</b> \(\vdash :S\in \text{Cls}\rightarrow 1.S^{;}Q=S^{;}R.CʻQ\subset \text{ᗡ}ʻS.CʻR\subset \text{ᗡ}ʻS.\supset .Q=R\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*151·252.\supset \vdash :\text{Hp}.\supset .Q&=\breve{S} ^{;}S^{;}Q\\ +[\text{Hp}] &=\breve{S} ^{;}S^{;}R\\ +[*151·252] &=R:\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*151·32.</b> \(\vdash \colon\ldotp P\,\text{smor}\,Q.\supset :\dot{\exists} !P.\equiv .\dot{\exists} !Q \quad[*151·18.*73·36.*33·24]\)</p> + +<p class="nind"><b>*151·33.</b> \(\vdash :S\in P\,\overline{\text{smor}}\,Q.\supset .P\mid S=S\mid Q.\breve{S} \mid P=Q\mid \breve{S}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*151·11.\supset \vdash :\text{Hp}.&\supset .P\mid S=S\mid Q\mid \breve{S} \mid S.S\in 1\rightarrow 1.CʻQ=\text{ᗡ}ʻS.\\ +[*72·601] &\supset .P\mid S=S\mid Q &\qquad \text{(1)}\\ +\text{Similarly}\quad \vdash :\text{Hp}.&\supset .\breve{S} \mid P=Q\mid \breve{S} &\qquad \text{(2)}\\ +\vdash .(1).(2).\supset \vdash .\text{Prop} +\end{array} +\] +</p> + +<p class="nind"><b>*151·4.</b> \(\vdash :T\upharpoonright CʻQ\in 1\rightarrow 1.CʻP=TʻʻCʻQ.Q=\breve{T} ^{;}P.\supset .T\upharpoonright CʻQ\in P\,\overline{\text{smor}}\,Q\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*35·52.*37·4.&\supset \vdash :\text{Hp}.\supset .(CʻQ)\upharpoonleft \breve{T} \in 1\rightarrow 1.\text{ᗡ}ʻ{(CʻQ)\upharpoonleft \breve{T} }=CʻP &\qquad \text{(1)}\\ +\vdash .*36·33. \supset \vdash :\text{Hp}.\supset .Q&=(\breve{T} ^{;}P)\unicode{x0294f}CʻQ\\ +[*150·361] &=\{(CʻQ)\upharpoonleft \breve{T}\}^{;}P &\qquad \text{(2)}\\ +\vdash .(1).(2).*151·11.&\supset \vdash :\text{Hp}.\supset .(CʻQ)\upharpoonleft \breve{T} \in Q\,\overline{\text{smor}}\,P.\\ +[*151·131] &\supset .T\upharpoonright CʻQ\in P\,\overline{\text{smor}}\,Q:\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*151·401.</b> \[\begin{align}\vdash :T\upharpoonright CʻP\in X,\overline{\text{smor}}\,P.T\upharpoonright CʻQ\in Y\overline{\text{smor}} &Q.S\in P\,\overline{\text{smor}}\,Q.\supset .\\ +&T^{;}S\in X\,\overline{\text{smor}} \,Y\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*151·131·141.\supset \vdash :\text{Hp}.&\supset .T\upharpoonright CʻP\mid S\mid (CʻQ)\upharpoonleft \breve{T} \in X\,\overline{\text{smor}} \,Y &\qquad \text{(1)}\\ +\vdash .*151·11·131. \supset \vdash :\text{Hp}.&\supset .\text{D}ʻS=CʻP.\text{ᗡ}ʻS=CʻQ.\\ +[*150·34] &\supset .T\upharpoonright CʻP\mid S\mid (CʻQ)\upharpoonleft \breve{T} =T^{;}S &\qquad \text{(2)}\\ +\vdash .(1).(2).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_327">[Pg 327]</span></p> + +<p class="nind"><b>*151·41.</b> \[\begin{align}\vdash :S\in P\,\overline{\text{smor}}\,Q.T\upharpoonright CʻP,T\upharpoonright &CʻQ\in 1\rightarrow 1.CʻP\cup CʻQ\subset \text{ᗡ}ʻT.\supset .\\ +&T^{;}S\in (T^{;}P)\overline{\text{smor}} (T^{;}Q) \quad[*151·401·22]\end{align}\]</p> + +<p>This proposition is the analogue of *73·63.</p> + +<p>The following proposition is used frequently both in +relation-arithmetic and in the theory of series.</p> + +<p class="nind"><b>*151·5.</b> \[\begin{align}\vdash :S\upharpoonright &CʻQ\in P\,\overline{\text{smor}}\,Q.\supset .\\ +&\text{D}ʻP=Sʻʻ\text{D}ʻQ.\text{ᗡ}ʻP=Sʻʻ\text{ᗡ}ʻQ.\overrightarrow{B}ʻP=Sʻʻ\overrightarrow{B}ʻQ.\overrightarrow{B}ʻ\breve{P} =Sʻʻ\overrightarrow{B}ʻ\breve{Q}\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*151·22.*150·21·211. \supset \vdash :\text{Hp}.&\supset .\text{D}ʻP=Sʻʻ\text{D}ʻQ.\text{ᗡ}ʻP=Sʻʻ\text{ᗡ}ʻQ. &\qquad \text{(1)}\\ +[*93·101]& \supset .\overrightarrow{B}ʻP=Sʻʻ\text{D}ʻQ-Sʻʻ\text{ᗡ}ʻQ.\\ +[*37·421.*151·22] \supset .\overrightarrow{B}ʻP&=(S\upharpoonright CʻQ)ʻʻ\text{D}ʻQ-(S\upharpoonright CʻQ)ʻʻ\text{ᗡ}ʻQ\\ +[*71·381.*151·22] &=(S\upharpoonright CʻQ)ʻʻ(\text{D}ʻQ-\text{ᗡ}ʻQ)\\ +[*93·101.*37·421] &=Sʻʻ\overrightarrow{B}ʻQ &\qquad \text{(2)}\\ +\text{Similarly}\quad \vdash :\text{Hp}.&\supset .\overrightarrow{B}ʻ\breve{P} =Sʻʻ\overrightarrow{B}ʻ\breve{Q} &\qquad \text{(3)}\\ +\vdash .(1).(2).(3).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*151·51.</b> \(\vdash :S\upharpoonright CʻQ\in P\,\overline{\text{smor}}\,Q.R\,\unicode{x2abd}\, Q.\supset .S\upharpoonright CʻR\in (S^{;}R)\overline{\text{smor}} R.S^{;}R\,\unicode{x2abd}\, + P\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*151·22.*33·265. &\supset \vdash :\text{Hp}.\supset .CʻR\subset \text{ᗡ}ʻS &\qquad \text{(1)}\\ +\vdash .*151·22.*71·222. &\supset \vdash :\text{Hp}.\supset .S\upharpoonright CʻR\in 1\rightarrow 1 &\qquad \text{(2)}\\ +\vdash .*150·31.*151·22. &\supset \vdash :\text{Hp}.\supset .S^{;}R\,\unicode{x2abd}\, P &\qquad \text{(3)}\\ +\vdash .(1).(2).(3).*151·22.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*151·52">*151·52</a>.</b> \(\vdash :P\,\text{smor}\,Q.\supset .\text{Rl}ʻQ\subset \text{smor}ʻʻ\text{Rl}ʻP \quad[*151·51·12·14]\)</p> + +<p class="nind"><b><a id="*151·53">*151·53</a>.</b> \[\begin{align}\vdash :S\upharpoonright CʻQ\in P\,\overline{\text{smor}}\,Q.T\in &\text{Pot}ʻQ.\supset .\\ +&S\upharpoonright CʻT\in (S^{;}T)\overline{\text{smor}} T.S^{;}T\in \text{Pot}ʻP\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*150·8. &\supset \vdash :\text{Hp}.\supset .S^{;}T\in \text{Pot}ʻP &\qquad \text{(1)}\\ +\vdash .*91·27. &\supset \vdash :\text{Hp}.\supset .CʻT\subset \text{ᗡ}ʻS &\qquad \text{(2)}\\ +\vdash .(1).(2).*151·22.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*151·54.</b> \(\vdash :S\upharpoonright CʻQ\in P\,\overline{\text{smor}}\,Q.\supset .S\upharpoonright CʻQ\in P_{\text{po}}\overline{\text{smor}} Q_{\text{po}}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*91·504.*151·22. &\supset \vdash :\text{Hp}.\supset .S\upharpoonright CʻQ=S\upharpoonright CʻQ_{\text{po}}.CʻQ_{\text{po}}\subset \text{ᗡ}ʻS &\qquad \text{(1)}\\ +\vdash .*150·83.*151·22. &\supset \vdash :\text{Hp}.\supset .P_{\text{po}}=S^{;}Q_{\text{po}} &\qquad \text{(2)}\\ +\vdash .(1).(2).*151·22. &\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*151·55">*151·55</a>.</b> \(\vdash :S\in P\,\overline{\text{smor}}\,Q.\supset .S\in P_{\text{po}}\overline{\text{smor}} Q_{\text{po}} \quad[*151·54]\)</p> + +<p class="nind"><b><a id="*151·56">*151·56</a>.</b> \(\vdash :P\,\text{smor}\,Q.\supset .P_{\text{po}}\text{smor}Q_{\text{po}} \quad[*151·55]\)</p> + +<p>*151·56 is used in *263·17.</p> + +<p><span class="pagenum" id="Page_328">[Pg 328]</span></p> + +<p>The two following propositions are lemmas for *<a href="#*151·59">*151·59</a>, which is used in +*263·17.</p> + +<p class="nind"><b>*151·57.</b> \(\vdash :S\in P\,\overline{\text{smor}}\,Q.z,w\in CʻQ.\supset .P(Sʻz\vdash\dashv Sʻw)=SʻʻQ(z\vdash\dashv w)\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*151·33·55.\supset \vdash :\text{Hp}.&\supset .\overleftarrow{P}_{\text{po}}ʻSʻz=Sʻʻ\overleftarrow{Q}_{\text{po}}ʻz.\overrightarrow{P}_{\text{po}}ʻSʻw=Sʻʻ\overrightarrow{Q}_{\text{po}}ʻw.\\ +[*91·54] &\supset .\overleftarrow{P}_{*}ʻSʻz=Sʻʻ\overleftarrow{Q}_{\text{po}}ʻz\cup \iota ʻSʻz.\\ +&\overrightarrow{P}_{*}ʻSʻw=Sʻʻ\overrightarrow{Q}_{\text{po}}ʻw\cup \iota ʻSʻw.\\ +[*53·31.*91·54] &\supset .\overleftarrow{P}_{*}ʻSʻz=Sʻʻ\overleftarrow{Q}_{*}ʻz.\overrightarrow{P}_{*}ʻSʻw=Sʻʻ\overrightarrow{Q}_{*}ʻw.\\ +[(*121·103)] &\supset .P(Sʻz\vdash\dashv Sʻw)=SʻʻQ(z\vdash\dashv w):\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*151·58.</b> \(\vdash :S\in P\,\overline{\text{smor}}\,Q.\supset .S\upharpoonright CʻQ_{\nu }\in P_{\nu }\overline{\text{smor}} Q_{\nu }\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*151·57.*73·22.\supset \vdash \colon\ldotp \text{Hp}.\supset :z,w\in &CʻQ.\supset .\\ +&\text{Nc}ʻP(Sʻz\vdash\dashv Sʻw)=\text{Nc}ʻQ(z\vdash\dashv w) &\qquad \text{(1)}\\ +\vdash .(1).*121·11. &\supset \vdash \colon\ldotp \text{Hp}.z,w\in CʻQ.\supset :zQ_{\nu }w.\equiv .(Sʻz)P_{\nu }(Sʻw) &\qquad \text{(2)}\\ +\vdash .(2).*150·41. \supset \vdash :\text{Hp}.&\supset .Q_{\nu }=\breve{S} ^{;}P_{\nu }.\\ +[*151·253.*121·322] &\supset .S^{;}Q_{\nu }=P_{\nu }.CʻQ_{\nu }\subset \text{ᗡ}ʻS &\qquad \text{(3)}\\ +\vdash .(3).*151·22.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*151·59">*151·59</a>.</b> \(\vdash :P\,\text{smor}\,Q.\supset .P_{\nu }\text{smor}Q_{\nu } \quad[*151·58]\)</p> + +<p>The remaining propositions of this number consist of applications to +particular cases.</p> + +<p class="nind"><b><a id="*151·6">*151·6</a>.</b> \[\begin{align}&\vdash .\text{Cnv}^{;}P\,\text{smor}\,P.\text{Cnv}\upharpoonright CʻP\in (\text{Cnv}^{;}P)\,\overline{\text{smor}}\,P\\ +&[*151·231.*31·13.*72·11]\end{align}\]</p> + +<p>This proposition is only significant when \(P\) is a relation between +relations.</p> + +<p class="nind"><b>*151·61.</b> \(\vdash .\iota ^{;}P\,\text{smor}\,P \quad[*151·232.*51·12.*72·18]\)</p> + +<p class="nind"><b>*151·62.</b> \(\vdash :CʻP\subset 1.\supset .\breve{\iota} ^{;}P\,\text{smor}\,P \quad[*52·62 . *151·243]\)</p> + +<p class="nind"><b><a id="*151·63">*151·63</a>.</b> \(\vdash :x\neq y.z\neq w.\supset .x\downarrow y\text{smor}z\downarrow w.x\downarrow z\unicode{x228d} y\downarrow w\in (x\downarrow y)\overline{\text{smor}} (z\downarrow w)\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*150·72.&\supset \vdash :S=x\downarrow z\unicode{x228d} y\downarrow w.z\neq w.\supset .S^{;}(z\downarrow w)=x\downarrow y &\qquad \text{(1)}\\ +\vdash .*72·182.*71·242.&\supset \vdash :\text{Hp}.\text{Hp}(1).\supset .S\in 1\rightarrow 1 &\qquad \text{(2)}\\ +\vdash .*55·15. &\supset \vdash :\text{Hp}(1).\supset .\text{ᗡ}ʻS=Cʻ(z\downarrow w) &\qquad \text{(3)}\\ +\vdash .(1).(2).(3).*151·1·11.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_329">[Pg 329]</span></p> + +<p>The above proposition shows that all ordinal couples (<i>i.e.</i> all +members of \(2_{r})\) are ordinally similar. The following proposition +shows the same for couples whose referent and relatum are identical.</p> + +<p class="nind"><b><a id="*151·631">*151·631</a>.</b> \(\vdash .x\downarrow x \text{smor} z\downarrow z\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*72·182.*55·15.&\supset \vdash .x\downarrow z\in 1\rightarrow 1.\text{ᗡ}ʻ(x\downarrow z) = Cʻ(z\downarrow z) &\qquad \text{(1)}\\ +\vdash .*55·13.&\supset \vdash :u\{x\downarrow z\mid z\downarrow z\mid \text{Cnv}ʻ(x\downarrow z)\}u'. \equiv .\\ +&u(x\downarrow z)z.u'(x\downarrow z)z.\\ +[*55·13] &\equiv .u = x.u' = x.\\ +[*55·13] &\equiv .u(x\downarrow x)u' &\qquad \text{(2)}\\ +\vdash .(2).*150·1. &\supset \vdash .(x\downarrow z)^{;}(z\downarrow z) = x\downarrow x &\qquad \text{(3)}\\ +\vdash .(1).(3).*151·1. &\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*151·64.</b> \[\begin{align}&\vdash .x\downarrow ^{;}P \,\text{smor} P.(x\downarrow )\upharpoonright CʻP\in (x\downarrow ^{;}P) \overline{\text{smor}} P\\ +&[*72·184.*55·12.*151·231]\end{align}\]</p> + +<p>The following proposition is frequently used in relation-arithmetic.</p> + +<p class="nind"><b>*151·65.</b> \[\begin{align}&\vdash .\downarrow x^{;}P \,\text{smor} P.(\downarrow x)\upharpoonright CʻP\in (\downarrow x^{;}P)\overline{\text{smor}} P\\ +&[*72·184.*55·121.*151·231]\end{align}\]</p> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_330">[Pg 330]</span></p> +<h2 class="nobreak" id="*152"> +*152. DEFINITION AND ELEMENTARY PROPERTIES +OF RELATION-NUMBERS.</h2> +</div> + + +<p class="nind"><i>Summary of</i> *152.</p> + +<p>The relation-number of \(P\), which we denote by \(\text{Nr}ʻP\), is +defined as the class of relations which are ordinally similar to \(P\), +<i>i.e.</i> +\[ +\text{Nr}ʻP = \overrightarrow{\text{smor}}ʻP. +\] +Hence our definition is +\[ +\text{Nr} = \overrightarrow{\text{smor}} \quad\text{Df}. +\] +The class of relation-numbers consists of all such classes as +\(\text{Nr}ʻP\), <i>i.e.</i> +\[ +\text{NR} = \text{D}ʻ\text{Nr} \quad\text{Df}. +\] +These two definitions are analogous to those of <a href="#*100">*100</a>, merely +substituting "\(\text{smor}\)" for "\(\text{sm}\)." They are justified +by similar considerations, and lead to similar results. With the +exception of <a href="#*152·7">*152·7</a><a href="#*152·71">·71</a><a href="#*152·72">·72</a>, the propositions of this number are the +analogues of those of *100, and call for no remarks other than those in +the introduction to *100 (<i>mutatis mutandis</i>).</p> + +<p><a href="#*152·7">*152·7</a><a href="#*152·71">·71</a><a href="#*152·72">·72</a> give relations between relation-numbers and cardinals. +*152·7, which is constantly used, states that the cardinal number +of \(CʻQ\) consists of the fields of the relation-number of \(Q\), +<i>i.e.</i> the classes similar to \(CʻQ\) are the fields of the +relations similar to \(Q\); in symbols,</p> + +<p class="nind"><b>*152·7.</b> \(\vdash . \text{Nc}ʻCʻQ = Cʻʻ\text{Nr}ʻQ\)</p> + +<p>Hence it follows that the fields of a relation-number form a cardinal +number, <i>i.e.</i></p> + +<p class="nind"><b>*152·71.</b> \(\vdash : \mu \in \text{NR} . \supset . Cʻʻ\mu \in \text{NC}\)</p> + +<p>Hence also it follows that cardinals other than \(\Lambda\) consist +of classes of the form \(Cʻʻ\mu\), where \(\mu\) is a relation-number +other than \(\Lambda\), <i>i.e.</i></p> + +<p class="nind"><b>*152·72.</b> \(\vdash . \text{NC} - \iota ʻ\Lambda = Cʻʻʻ(\text{NR} - \iota ʻ\Lambda )\)</p> + +<p>In <a href="#*154·9">*154·9</a>, we shall show how to remove the restriction to numbers other +than \(\Lambda\), thus arriving at +\[ +\vdash . \text{NC} = Cʻʻʻ\text{NR}. +\]</p> + +<hr class="tb"> + +<p class="nind"><b>*152·01.</b> \(\text{Nr} = \overrightarrow{\text{smor}} \quad\text{Df}\)</p> + +<p class="nind"><b>*152·02.</b> \(\text{NR}= \text{D}ʻ\text{Nr} \quad\text{Df}\)</p> + +<p><span class="pagenum" id="Page_331">[Pg 331]</span></p> + +<p class="nind"><b>*152·1.</b> \[\begin{align}&\vdash .\text{Nr}ʻP=\hat{Q} (Q\,\text{smor}\,P)=\hat{Q} (P\,\text{smor}\,Q)\\ +&[*32·11.(*152·01).*151·14]\end{align}\]</p> + +<p class="nind"><b>*152·11.</b> \(\vdash :Q\in \text{Nr}ʻP.\equiv .Q\,\text{smor}\,P.\equiv .P\,\text{smor}\,Q \quad[*152·1]\)</p> + +<p class="nind"><b>*152·2.</b> \(\vdash .\text{E}!\text{Nr}ʻP \quad[*152·1.*14·21]\)</p> + +<p class="nind"><b>*152·21.</b> \(\vdash .\text{ᗡ}ʻ\text{Nr}=\text{Rel} \quad[*152·2.*33·432]\)</p> + +<p class="nind"><b>*152·22.</b> \(\vdash .\text{Nr}\in 1\rightarrow \text{Cls} \quad[*152·2.*71·166]\)</p> + +<p class="nind"><b>*152·3.</b> \(\vdash .P\in \text{Nr}ʻP \quad[*151·13.*152·11]\)</p> + +<p class="nind"><b>*152·31.</b> \(\vdash :P\in \text{Nr}ʻQ.\equiv .Q\in \text{Nr}ʻP \quad[*152·11]\)</p> + +<p class="nind"><b>*152·32.</b> \(\vdash :P\in \text{Nr}ʻQ.Q\in \text{Nr}ʻR.\supset .P\in \text{Nr}ʻR \quad [*151·15.*152·11]\)</p> + +<p class="nind"><b>*152·321.</b> \(\vdash :P\,\text{smor}\,Q.\supset .\text{Nr}ʻP=\text{Nr}ʻQ \quad[*151·17.*152·1]\)</p> + +<p class="nind"><b>*152·33.</b> \(\vdash :\exists !\text{Nr}ʻP\cap \text{Nr}ʻQ.\supset .P\,\text{smor}\,Q.\text{Nr}ʻP=\text{Nr}ʻQ\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*152·11.*151·14. \supset \vdash :\text{Hp}.&\supset .(\exists R).P\,\text{smor}\,R.R\,\text{smor}\,Q.\\ +[*151·15] &\supset .P\,\text{smor}\,Q &\qquad \text{(1)}\\ +\vdash .(1).*152·321.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*152·35.</b> \[\begin{align}\vdash \colon\ldotp \exists !&\text{Nr}ʻP.\lor.\exists !\text{Nr}ʻQ:\supset :\\ +&\text{Nr}ʻP=\text{Nr}ʻQ.\equiv .P\in \text{Nr}ʻQ.\equiv .Q\in \text{Nr}ʻP.\equiv .P\,\text{smor}\,Q\end{align}\]</p> + + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*24·571. \supset \vdash \colon\ldotp \text{Hp}.&\supset :\text{Nr}ʻP=\text{Nr}ʻQ.\supset .\exists !\text{Nr}ʻP\cap \text{Nr}ʻQ.\\ +[*152·33] &\supset .P\,\text{smor}\,Q &\qquad \text{(1)}\\ +\vdash .(1).*152·321. &\supset \vdash \colon\ldotp \text{Hp}.\supset :\text{Nr}ʻP=\text{Nr}ʻQ.\equiv .P\,\text{smor}\,Q &\qquad \text{(2)}\\ +\vdash .(2).*152·11.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p>In the above proposition, the same remarks as to types are to be +made as in the case of <a href="#*100·35">*100·35</a>. If in a certain type \(\text{Nr}ʻP\) +and \(\text{Nr}ʻQ\) are both null, we have in that type +\(\text{Nr}ʻP=\text{Nr}ʻQ\), but we need not have \(P\,\text{smor}\,Q\). +Thus for example we shall find that, in the type of \(x\downarrow x\), +\[ +\text{Nr}ʻ(t^{2}ʻx\uparrow t^{2}ʻx)=\Lambda =\text{Nr}ʻʻ(t^{3}ʻx\uparrow t^{3}ʻx). +\] +But we do not have +\[ +(t^{2}ʻx\uparrow t^{2}ʻx)\text{smor}(t^{3}ʻx\uparrow t^{3}ʻx). +\]</p> + +<p class="nind"><b>*152·4.</b> \(\vdash :\mu \in \text{NR}.\equiv .(\exists P).\mu =\text{Nr}ʻP \quad[*37·78·79.(*152·02·01)]\)</p> + +<p>Note that "\(\text{Nr}ʻP\)," like "\(\text{Nc}ʻ\alpha\)," is a formal +number, and may be subjected to the conventions \(\text{IT}\), +\(\text{IIT}\), \(\text{AT}\).</p> + +<p class="nind"><b>*152·41.</b> \(\vdash .\text{Nr}ʻP\in \text{NR} \quad[*152·4·2]\)</p> + +<p class="nind"><b>*152·42.</b> \(\vdash :\mu ,\nu \in \text{NR}.\exists !\mu \cap \nu .\supset .\mu =\nu \quad[*152·33·4]\)</p> + +<p><span class="pagenum" id="Page_332">[Pg 332]</span></p> + +<p class="nind"><b>*152·43.</b> \(\vdash .\text{NR}\in \text{Cls}^{2} \text{excl} \quad[*152·42]\)</p> + +<p class="nind"><b>*152·44.</b> \[\begin{align}&\vdash \colon\ldotp \mu \in \text{NR}:\exists !\mu .\lor.\exists !\text{Nr}ʻP.\supset :P\in \mu .\equiv .\text{Nr}ʻP=\mu \\ +&[*152·35·4]\end{align}\]</p> + +<p class="nind"><b>*152·45.</b> \(\vdash :\mu \in \text{NR}.P\in \mu .\supset .\text{Nr}ʻP=\mu \quad[*152·44.*10·24]\)</p> + +<p class="nind"><b>*152·5.</b> \(\vdash :\mu \in \text{NR}.P,Q\in \mu .\supset .P\,\text{smor}\,Q \quad[*152·31·32·4]\)</p> + +<p class="nind"><b>*152·51.</b> \(\vdash :\mu \in \text{NR}.P\in \mu .\supset .\text{smor}ʻʻ\mu =\text{Nr}ʻP\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*37·1. &\supset \vdash :R\in \text{smor}ʻʻ\mu .\equiv .(\exists Q).Q\in \mu .P\,\text{smor}\,Q &\qquad \text{(1)}\\ +\vdash .*152·5.&\supset \vdash \colon\ldotp \mu \in \text{NR}.P\in \mu .\supset :Q\in \mu .P\,\text{smor}\,Q.\equiv .Q\in \mu &\qquad \text{(2)}\\ +\vdash .(1).(2).\supset \vdash \colon\ldotp \text{Hp}.\supset :R\in \text{smor}ʻʻ\mu .&\equiv .(\exists Q).Q\in \mu .P\,\text{smor}\,Q.R\,\text{smor}\,Q.\\ +[*151·17] &\equiv .(\exists Q).Q\in \mu .P\,\text{smor}\,Q.R\,\text{smor}\,P.\\ +[(2)] &\equiv .(\exists Q).Q\in \mu .R\,\text{smor}\,P.\\ +[*10·35] &\equiv .\exists !\mu .R\,\text{smor}\,P &\qquad \text{(3)}\\ +\vdash .*10·24. \supset \vdash \colon\ldotp \text{Hp}.&\supset :\exists !\mu :\\ +[*4·73] \supset :R\,\text{smor}\,P.&\equiv .\exists !\mu .R\,\text{smor}\,P &\qquad \text{(4)}\\ +\vdash .(3).(4).\supset \vdash \colon\ldotp \text{Hp}.\supset :R\in \text{smor}ʻʻ\mu .&\equiv .R\,\text{smor}\,P.\\ +[*152·11] &\equiv .R\in \text{Nr}ʻP\colon\ldotp \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*152·52.</b> \(\vdash :\mu \in \text{NR}.\exists !\mu .\supset .\text{smor}ʻʻ\mu \in \text{NR} \quad[*152·51·4]\)</p> + +<p>The restriction involved in \(\exists ! \mu\) is, as we shall see +later, not necessary, since \(\Lambda \in \text{NR}\) in any assigned +type.</p> + +<p class="nind"><b>*152·53.</b> \(\vdash :\exists !\text{Nr}ʻQ.\supset .\text{smor}ʻʻ\text{Nr}ʻQ=\text{Nr}ʻQ\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*152·51. \supset \vdash :P\in \text{Nr}ʻQ.\supset .\text{smor}ʻʻ\text{Nr}ʻQ=\text{Nr}ʻP &\qquad \text{(1)}\\ +\vdash .*152·321.\supset \vdash :P\in \text{Nr}ʻQ.\supset .\text{Nr}ʻP=\text{Nr}ʻQ &\qquad \text{(2)}\\ +\vdash .(1).(2).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*152·54.</b> \[\begin{align}&\vdash \colon\ldotp \exists !\mu .\exists !\nu .\supset :\mu \in \text{NR}.\nu =\text{smor}ʻʻ\mu .\equiv .\nu \in \text{NR}.\mu =\text{smor}ʻʻ\nu \\ +&[\text{Proof as in *100·53}]\end{align}\]</p> + +<p class="nind"><b><a id="*152·6">*152·6</a>.</b> \(\vdash .\iota ^{;}P\in \text{Nr}ʻP \quad[*151·61]\)</p> + +<p class="nind"><b><a id="*152·62">*152·62</a>.</b> \(\vdash .x\downarrow ^{;}P\in \text{Nr}ʻP \quad[*151·64]\)</p> + +<p class="nind"><b><a id="*152·63">*152·63</a>.</b> \(\vdash .\downarrow x^{;}P\in \text{Nr}ʻP \quad[*151·65]\)</p> + +<p>The utility of <a href="#*152·6">*152·6</a><a href="#*152·62">·62</a><a href="#*152·63">·63</a> is that they enable us to raise the type of +a relation-number to any required extent. Thus \(\iota ^{;}P\) gives a +relation whose field is a class of the next type above that of \(CʻP\), +<i>i.e.</i> of the type \(t^{2}ʻCʻP\); while \(x\downarrow ^{;}P\) gives +a relation whose field is \(x\downarrowʻʻCʻP\), which is of the type +\(tʻtʻ(\iota ʻx\uparrow CʻP)\). If \(x\in CʻP\), or, more generally, +if \(x\in t_{0}ʻCʻP\), this is the type \(t^{2}ʻP\). Thus if we put +\(Q=x\downarrow ^{;}P\), we have +\[ +tʻQ=tʻ(CʻQ\uparrow CʻQ)=tʻ(tʻP\uparrow tʻP)=tʻ(P\downarrow P). +\] +Thus \(x\downarrow ^{;}P\) is a relation whose field consists of terms of +the same type as \(P\).</p> + +<p><span class="pagenum" id="Page_333">[Pg 333]</span></p> + +<p>The following propositions on the relations of cardinals and +relation-numbers are very important.</p> + +<p class="nind"><b><a id="*152·7">*152·7</a>.</b> \(\vdash .\text{Nc}ʻCʻQ=Cʻʻ\text{Nr}ʻQ\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*151·19.*35·942. \supset \vdash :\alpha \in \text{Nc}ʻCʻQ.\supset .(\exists R).CʻR=\alpha .R\in \text{Nr}ʻQ.\\ +[*37·6] \supset .\alpha \in Cʻʻ\text{Nr}ʻQ &\qquad \text{(1)}\\ +\vdash .*151·18. \supset \vdash :P\in \text{Nr}ʻQ.\supset .CʻP\in \text{Nc}ʻCʻQ\\ +[*37·61] \supset \vdash .Cʻʻ\text{Nr}ʻQ\subset \text{Nc}ʻCʻQ &\qquad \text{(2)}\\ +\vdash .(1).(2).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*152·71">*152·71</a>.</b> \(\vdash :\mu \in \text{NR}.\supset .Cʻʻ\mu \in \text{NC} \quad[*152·7]\)</p> + +<p class="nind"><b><a id="*152·72">*152·72</a>.</b> \(\vdash .\text{NC}-\iota ʻ\Lambda =Cʻʻʻ(\text{NR}-\iota ʻ\Lambda )\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*152·71. &\supset \vdash .Cʻʻʻ\text{NR}\subset \text{NC} &\qquad \text{(1)}\\ +\vdash .*152·7.*50·5·52. &\supset \vdash :\mu \in \text{NC}.\alpha \in \mu .\supset .Cʻʻ\text{Nr}ʻ(I\upharpoonright \alpha )=\text{NC}ʻ\alpha .\\ +[*100·45] &\supset .Cʻʻ\text{Nr}ʻ(I\upharpoonright \alpha )=\mu .\\ +[*37·103] &\supset .\mu \in Cʻʻʻ\text{NR} &\qquad \text{(2)}\\ +\vdash .(2).*10·11·23·35. &\supset \vdash :\mu \in \text{NC}.\exists !\mu .\supset .\mu \in Cʻʻʻ\text{NR} &\qquad \text{(3)}\\ +\vdash .*37·45. &\supset \vdash :\mu =Cʻʻ\nu .\exists !\mu .\supset .\exists !\nu :\\ +[*37·103] &\supset \vdash :\mu \in Cʻʻʻ\text{NR}.\exists !\mu .\supset .\mu \in Cʻʻʻ(\text{NR}-\iota ʻ\Lambda ) &\qquad \text{(4)}\\ +\vdash .(1).(3).(4).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p>We shall show in <a href="#*154·9">*154·9</a> that the exclusion of \(\Lambda\) in <a href="#*152·72">*152·72</a> is +unnecessary.</p> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_334">[Pg 334]</span></p> +<h2 class="nobreak" id="*153"> +*153. THE RELATION-NUMBERS \(0_{r}\), \(2_{r}\) AND \(1_{s}\).</h2> +</div> + + +<p class="nind"><i>Summary of</i> *153.</p> + +<p>The relation-numbers \(0_{r}\) and \(2_{r}\) have already been defined +(in *56), though it remains for the present number to show that they +are relation-numbers. They are the ordinal 0 and 2 respectively, +<i>i.e.</i> they are the ordinal numbers of well-ordered series of no terms and +series of two terms respectively. But there is no means of introducing +an ordinal 1 which shall be analogous to the cardinal 1 as completely +as \(0_{r}\) and \(2_{r}\) are analogous to 0 and 2. The only relations +whose fields are unit classes are relations of the form \(x\downarrow x\). We therefore put</p> + +<p class="nind"><b>*153·01.</b> \(1_{s}=\hat{R} \{(\exists x).R=x\downarrow x\} \quad\text{Df}\)</p> + +<p>The above definition gives the nearest possible approach to an +ordinal 1. \(1_{s}\) so defined is a relation-number, and is the +relation-number corresponding to 1 in the sense that it is the +relation-number of all such relations as have a field consisting of +one term. But \(1_{s}\) is not what is called an "ordinal number," +because this term is confined by usage to the relation-numbers of +well-ordered series, and \(x\downarrow x\) is not a serial relation. It +is essential to a serial relation to be contained in diversity; and if, +by definition, we include \(x\downarrow x\) among series, we introduce +more exceptions than we avoid. Moreover \(1_{s}\) does not have the +kind of properties which we wish 1 to have; <i>e.g.</i> \(1_{s}\dot{+}1_{s}\) +is not \(2_{r}\).</p> + +<p>We do not use \(1_{r}\), because we shall at a later stage define +\(\nu _{r}\) as the class of those <i>series</i> whose fields have +\(\nu\) terms, so that \(1_{r}=\Lambda\), while \(0_{r}\) and \(2_{r}\) +have the values \(\iota ʻ\dot{\Lambda}\) and \(\hat{R} \{(\exists x,y).x\neq y.R=x\downarrow y\}\), +as already defined. On account of this general definition of \(\nu_{r}\), +we choose a different symbol for the relation-number 1, and +\(1_{s}\) has the merit of being as like \(1_{r}\) as possible.</p> + +<p>To illustrate, by anticipation, the way in which \(1_{s}\) differs from +proper ordinal numbers, we may point out that if \(1_{s}\) is added to +\(2_{r}\), we do not obtain \(3_{r}\). We shall define \(3_{r}\) as the +class of series which consist of three terms, <i>i.e.</i> the class of +relations of the form +\[ +x\downarrow y\unicode{x228d} x\downarrow z\unicode{x228d} y\downarrow z, +\] +where \(x\neq y.x\neq z.y\neq z\). We shall define the sum of two +ordinal numbers<span class="pagenum" id="Page_335">[Pg 335]</span> as the ordinal number of the sum of two relations +having these ordinal numbers (cf. <a href="#*180">*180</a>), and it will appear that if +\(P\) and \(Q\) are relations whose fields have no members in common, +then +\[ +P\unicode{x228d} Q\unicode{x228d} CʻP\uparrow CʻQ +\] +has a relation-number which is the sum of those of \(P\) and \(Q\). +Suppose now \(P=x\downarrow y\) and \(Q=z\downarrow z\), where +\(x\neq y.x\neq z.y\neq z\). Then +\[ +P\unicode{x228d} Q\unicode{x228d} CʻP\uparrow CʻQ=x\downarrow y\unicode{x228d} x\downarrow z\unicode{x228d} y\downarrow z\unicode{x228d} z\downarrow z. +\] +This is not a member of \(3_{r}\), because of the additional term +\(z\downarrow z\). Thus the addition of one term to a series \(P\) does +not give the same number as results from the addition of \(1_{s}\) to +\(\text{Nr}ʻP\). Hence the addition of 1 to an ordinal number has to be +separately treated<a id="FNanchor_12" href="#Footnote_12" class="fnanchor">[12]</a>.</p> + +<p>We prove in this number that \(0_{r} = \text{Nr}ʻ\dot{\Lambda}\) +(<a href="#*153·11">*153·11</a>), that \(2_{r}=\text{Nr}ʻ(\Lambda \downarrow \iotaʻx)\) +(<a href="#*153·24">*153·24</a>; observe that we have to take a couple of +classes (or relations) in order to be sure of the existence +of two different objects of the class in question), and that +\(1_{s}=\text{Nr}ʻ(y\downarrow y\)) (<a href="#*153·32">*153·32</a>). We prove \(Cʻʻ0_{r} =0\) +(<a href="#*153·18">*153·18</a>), \(Cʻʻ2_{r}=2\) (<a href="#*153·212">*153·212</a>), and \(Cʻʻ1_{s}=1\) (<a href="#*153·36">*153·36</a>). +We have also \(\breve{C} ʻʻ0=0_{r}\) (not proved) and \(\breve{C}ʻʻ1=1_{s}\) +(<a href="#*153·301">*153·301</a>). But we do not have \(\breve{C} ʻʻ2=2_{r}\); +<i>e.g.</i> \((x\downarrow y\unicode{x228d} y\downarrow x)\in \breve{C}ʻʻ2\) +if \(x\neq y\), but \((x\downarrow y\unicode{x228d} y\downarrow x){\sim}\in 2_{r}\). +We have \(\exists !0_{r}\) (<a href="#*153·12">*153·12</a>) and \(\exists!1_{s}\) (<a href="#*153·34">*153·34</a>), +but from our primitive propositions we cannot deduce \(\exists !2_{r}\) +unless we rise above the lowest type of relations. The case is exactly +analogous to that of \(\exists !2\) (cf. <a href="#*101">*101</a>); we have</p> + +<p class="nind"><b>*153·26·262.</b> \(\vdash .\exists !2_{r}\cap \text{Rl}ʻ(\text{Cls}\uparrow \text{Cls}).\exists !2_{r}\cap \text{Rel}^{2}\)</p> + +<p>But if, as monists aver, there is only one individual, we shall not +have \(\exists !2_{r}\) in the type of relations of individuals to +individuals. Our primitive propositions do not suffice to disprove this +supposition.</p> + +<hr class="tb"> + +<p class="nind"><b>*153·01.</b> \(1_{s}=\hat{R}\{(\exists x).R=x\downarrow x\} \quad\text{Df}\)</p> + +<p class="nind"><b>*153·1.</b> \(\vdash :P\in 0_{r}.\equiv .P=\dot{\Lambda} \quad[*56·104]\)</p> + +<p class="nind"><b>*153·101.</b> \(\vdash :P\text{smor}\dot{\Lambda} .\equiv .P=\dot{\Lambda}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*151·32.\text{Transp}.&\supset \vdash :P\text{smor}\dot{\Lambda} .\supset .{\sim}\dot{\exists} !P &\qquad \text{(1)}\\ +\vdash .*151·13. &\supset \vdash :P=\dot{\Lambda} .\supset .P\text{smor}\dot{\Lambda} &\qquad \text{(2)}\\ +\vdash .(1).(2).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*153·11">*153·11</a>.</b> \(\vdash .0_{r}=\text{Nr}ʻ\dot{\Lambda} \quad[*153·1·101.*152·1]\)</p> + +<p class="nind"><b>*153·111.</b> \(\vdash .0_{r}\in \text{NR} \quad[*152·41.*153·11]\)</p> + +<p class="nind"><b><a id="*153·12">*153·12</a>.</b> \(\vdash .\exists !0_{r} \quad[*51·161]\)</p> + +<p><span class="pagenum" id="Page_336">[Pg 336]</span></p> + +<p class="nind"><b>*153·13.</b> \(\vdash .\exists !0_{r}\cap \text{Rl}ʻR.\Lambda \in 0_{r}\cap \text{Rl}ʻR \quad[*61·3]\)</p> + +<p class="nind"><b>*153·14.</b> \(\vdash :\text{Nr}ʻP = 0_{r}. \equiv .P = \dot{\Lambda}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*152·44.*153·111·12.\supset \vdash :\text{Nr}ʻP = 0_{r}. &\equiv .P\in 0_{r}.\\ +[*152·1] &\equiv .P = \dot{\Lambda} :\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*153·15.</b> \(\vdash .\text{smor}ʻʻ0_{r} = 0_{r}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*152·51.*153·111·13.\supset \vdash .\text{smor}ʻʻ0_{r}. &= \text{Nr}ʻ\dot{\Lambda} \\ +[*153·11] &= 0_{r}.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*153·16.</b> \(\vdash \colon\ldotp \mu \in \text{NR}-\iota ʻ0_{r}.\supset :P\in \mu .\supset _{P}.\dot{\exists} !P\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*153·13.*152·42.\supset \vdash \colon\ldotp \mu \in \text{NR}.&\supset :\dot{\Lambda} \in \mu .\supset .\mu = 0_{r}:\\ +[\text{Transp}] & \supset :\mu \neq 0_{r}.\supset .\dot{\Lambda} {\sim} \in \mu &\qquad \text{(1)}\\ +\vdash .(1).*25·63.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*153·17.</b> \[\begin{align}&\vdash :\dot{\Lambda} \in \text{Nr}ʻP. \equiv .\text{Nr}ʻP = 0_r. \equiv .\text{Nr}ʻP = \text{Nr}ʻ\dot{\Lambda} . \equiv .P = \dot{\Lambda} \\ +&[*152·35.*153·11·14]\end{align}\]</p> + +<p class="nind"><b><a id="*153·18">*153·18</a>.</b> \(\vdash .Cʻʻ0_{r} = 0\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*53·31.\supset \vdash .Cʻʻ\iota ʻ\dot{\Lambda} = \iota ʻCʻ\dot{\Lambda} &\qquad \text{(1)}\\ +\vdash .(1).*33·241.(*56·03.*54.01).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*153·2.</b> \(\vdash :P\in 2_{r}. \equiv .(\exists x,y).x \neq y.P = x\downarrow y \quad[*56·11]\)</p> + +<p class="nind"><b>*153·201.</b> \(\vdash :x \neq y. \equiv .x\downarrow y\in 2_{r} \quad[*56·17]\)</p> + +<p class="nind"><b>*153·202.</b> \(\vdash :P,Q\in 2_{r}.\supset .P \,\text{smor} \,Q \quad[*151·63.*153·2]\)</p> + +<p class="nind"><b>*153·203.</b> \(\vdash :Q\in 2_{r}.P \,\text{smor} \,Q.\supset .P\in 2_{r}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*113·123.&\supset \vdash :S\in 1\rightarrow \text{Cls}.z,w\in \text{ᗡ}ʻS.\supset .S^{;}(z\downarrow w) = (Sʻz)\downarrow (Sʻw):\\ +[*55·15] & \supset \vdash :S\in 1\rightarrow \text{Cls}.Cʻ(z\downarrow w) = \text{ᗡ}ʻS.\supset .\\ +&S^{;}(z\downarrow w) = (Sʻz)\downarrow (Sʻw) &\qquad \text{(1)}\\ +\vdash .*71·56. \supset \vdash \colon\ldotp S\in 1\rightarrow 1.Cʻ(z\downarrow w) = \text{ᗡ}ʻS.&\supset :z = w. \equiv .Sʻz = Sʻw:\\ +[\text{Transp}] &\supset :z \neq w. \equiv .Sʻz \neq Sʻw &\qquad \text{(2)}\\ +\vdash .(1).(2).*153·201.\supset \\ +&\vdash :S\in 1\rightarrow 1.z \neq w.Cʻ(z\downarrow w) = \text{ᗡ}ʻS.P = S^{;}(z\downarrow w).\supset .P\in 2_{r}:\\ +[*151·1] &\supset \vdash :z \neq w. P \,\text{smor} (z\downarrow w).\supset .P\in 2_{r}:\\ +[*153·2] &\supset \vdash :Q\in 2_{r}.P \,\text{smor} \,Q.\supset .P\in 2_{r}:\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*153·21.</b> \(\vdash :P\in 2_{r}.\supset .2_{r} = \text{Nr}ʻP \quad[*153·202·203]\)</p> + +<p class="nind"><b>*153·211.</b> \(\vdash :x \neq y.\supset .2_{r} = \text{Nr}ʻ(x\downarrow y) \quad[*153·21·201]\)</p> + +<p><span class="pagenum" id="Page_337">[Pg 337]</span></p> + +<p class="nind"><b><a id="*153·212">*153·212</a>.</b> \(\vdash .Cʻʻ2_{r} = 2 \quad[*55·15.*56·11.*54·101]\)</p> + +<p class="nind"><b>*153·22.</b> \(\vdash :\exists !2_{r}\cap t''ʻz.\equiv .\exists !2(z).\equiv .(\exists x,y).x\neq yx\in tʻz \quad[*153·211.*101·4]\)</p> + +<p class="nind"><b>*153·23.</b> \(\vdash :P\in 2_{r}.\supset .\text{Rl}ʻP\subset 0_{r}\cup 2_{r} \quad[*56·261]\)</p> + +<p>This proposition illustrates the reasons for not putting +\[ +1_{r}=\hat{P} \{(\exists x).P=x\downarrow x\} \quad\text{Df}. +\] +We want the inductive ordinals, like the inductive cardinals, to +form a series in order of magnitude; but, as the above proposition +illustrates, the relation-number of such relations as \(x\downarrow x\) +is not in the same series with \(0_{r}\) and \(2_{r}\). The above +proposition should be contrasted with *51·411.</p> + +<p class="nind"><b><a id="*153·24">*153·24</a>.</b> \(\vdash .2_{r}=\text{Nr}ʻ(\Lambda \downarrow \iota ʻx) \quad[*153·211.*51·161]\)</p> + +<p class="nind"><b>*153·25.</b> \(\vdash .2_{r}\in \text{NR} \quad[*153·24.*152·41]\)</p> + +<p class="nind"><b>*153·251.</b> \(\vdash .2_{r}\neq 0_{r}.2_{r}\cap 0_{r}=\Lambda\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*153·212·18.*101·34·35.&\supset \vdash .Cʻʻ2_{r}\neq Cʻʻ0_{r}.Cʻʻ2_{r}\cap Cʻʻ0_{r}=\Lambda .\\ +[*13·12.\text{Transp}.*37·21] &\supset \vdash .2_{r}\neq 0_{r}.Cʻʻ(2_{r}\cap 0_{r})=\Lambda .\\ +[*37·45] &\supset \vdash .2_{r}\neq 0_{r}.2_{r}\cap 0_{r}=\Lambda +\end{array} +\]</p> + +<p class="nind"><b>*153·26.</b> \(\vdash .\exists !2_{r}\cap \text{Rl}ʻ(\text{Cls}\uparrow \text{Cls}) \quad[*153·24.*152·3]\)</p> + +<p class="nind"><b>*153·261.</b> \(\vdash .\dot{\Lambda} \downarrow (x\downarrow x)\in 2_{r} \quad[*55·134.*56·11]\)</p> + +<p class="nind"><b>*153·262.</b> \(\vdash .\exists !2_{r}\cap \text{Rel}^{2} \quad[*153·261.(*61·03)]\)</p> + +<p class="nind"><b>*153·27.</b> \[\begin{align}&\vdash .2_r=\text{smor}ʻʻ(2_r\cap \text{Rl}ʻ\text{Cls})=\text{smor}ʻʻ(2_r\cap \text{Rel}^2)\\ +&[*152·53.*153·26·262·24]\end{align}\]</p> + +<p class="nind"><b>*153·28.</b> \(\vdash :x\neq y.\supset .Bʻ(x\downarrow y)=x.Bʻ\text{Cnv}ʻ(x\downarrow y)=y\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*93·101.*55·15.\supset \vdash :\text{Hp}.\supset .\overrightarrow{B}ʻ(x\downarrow y)=\iota ʻx.\overrightarrow{B}ʻ\text{Cnv}ʻ(x\downarrow y)=\iota ʻy:\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*153·281.</b> \(\vdash :P\in 2_{r}.\supset .BʻP=\breve{\iota} ʻ\text{D}ʻP.Bʻ\breve{P} =\breve{\iota} ʻ\text{ᗡ}ʻP \quad[*153·28.*55·15]\)</p> + +<p>The above proposition is used in the theory of series (*204·48).</p> + +<p class="nind"><b>*153·3.</b> \(\vdash .1_{s}=\dot{2} -2_{r}=\hat{R} \{(\exists x).R=x\downarrow x\} \quad[*56·13.(*153·01)]\)</p> + +<p class="nind"><b><a id="*153·301">*153·301</a>.</b> \(\vdash .1_{s}=\breve{C} ʻʻ1 \quad[*153·3.*56·39]\)</p> + +<p class="nind"><b>*153·31.</b> \(\vdash .x\downarrow y\in (x\downarrow x)\overline{\text{smor}} (y\downarrow y)\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*72·182.*55·15.&\supset \vdash .x\downarrow y\in 1\rightarrow 1.\text{ᗡ}ʻ(x\downarrow y)=Cʻ(y\downarrow y) &\qquad \text{(1)}\\ +\vdash .*35·89.*55·1. &\supset \vdash .x\downarrow y\mid y\downarrow y=x\downarrow y.\\ +[*150·1.*55·14] \supset \vdash .(x\downarrow y)^{;}(y\downarrow y)&=x\downarrow y\mid y\downarrow x\\ +[*35·89.*55·1] &=x\downarrow x &\qquad \text{(2)}\\ +\vdash .(1).(2).*151·11.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_338">[Pg 338]</span></p> + +<p class="nind"><b>*153·311.</b> \(\vdash :Q\in 1_{s}.P\,\text{smor}\,Q.\supset .P\in 1_{s}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*153·3.*151·1.\supset \vdash :\text{Hp}.&\supset .\\ +&(\exists S,y).Q=y\downarrow y.S\in 1\rightarrow 1.\text{ᗡ}ʻS=\iota ʻy.P=S^{;}Q.\\ +[*150·71] &\supset .(\exists S,y).P=(Sʻy)\downarrow (Sʻy).\\ +[*153·3] &\supset .P\in 1_{s}:\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*153·32">*153·32</a>.</b> \(\vdash .1_{s}=\text{Nr}ʻ(y\downarrow y) \quad[*153·31·311]\)</p> + +<p class="nind"><b>*153·33.</b> \(\vdash .1_{s}\in \text{NR} \quad[*153·32]\)</p> + +<p class="nind"><b><a id="*153·34">*153·34</a>.</b> \(\vdash .\exists !1_{s}.1_{s}\neq 0_{r}.1_{s}\neq 2_{r}.1_{s}\cap 0_{r}=\Lambda .1_{s}\cap + 2_{r}=\Lambda\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*153·3. &\supset \vdash .x\downarrow x\in 1_{s}.\\ +[*10·24] &\supset \vdash .\exists !1_{s} &\qquad \text{(1)}\\ +\vdash .*56·103·104. &\supset \vdash .1_{s}\cap 0_{r}=\Lambda &\qquad \text{(2)}\\ +\vdash .(1).(2). &\supset \vdash .1_{s}\neq 0_{r} &\qquad \text{(3)}\\ +\vdash .*153·301.*56·113.&\supset \vdash .1_{s}\cap 2_{r}\subset \breve{C} ʻʻ1\cap \breve{C} ʻʻ2.\\ +[*72·41.*101·35] &\supset \vdash .1_{s}\cap 2_{r}=\Lambda &\qquad \text{(4)}\\ +[(1)] &\supset \vdash .1_{s}\neq 2_{r} &\qquad \text{(5)}\\ +\vdash .(1).(2).(3).(4).(5).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*153·341.</b> \(\vdash :R\in 1_{s}.\equiv .\text{Nr}ʻR=1_{s} \quad[*153·33·34.*152·44]\)</p> + +<p class="nind"><b>*153·35.</b> \(\vdash :R\in 1_{s}.\supset .\text{Nc}ʻCʻR=Cʻʻ\text{Nr}ʻR=1\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*55·15.*153·3.\supset \vdash :\text{Hp}.\supset .\text{Nc}ʻCʻR=1 &\qquad \text{(1)}\\ +\vdash .(1).*152·7.\supset \vdash .\text{Prop} +\end{array}\ +\]</p> + +<p class="nind"><b><a id="*153·36">*153·36</a>.</b> \(\vdash .Cʻʻ1_{s}=1\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*153·301.\supset \vdash .Cʻʻ1_{s}&=Cʻʻ\breve{C} ʻʻ1\\ +[*72·502] &=1.\supset \vdash .\text{Prop}\ +\end{array} +\]</p> + + +<div class="footnotes"><h3>FOOTNOTES:</h3> + +<div class="footnote"> + +<p class="nind"> +<a id="Footnote_12" href="#FNanchor_12" class="label">[12]</a> +Cf. <a href="#*161">*161</a> and <a href="#*181">*181</a>, where this point is more fully +elucidated.</p> + +</div> +</div> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_339">[Pg 339]</span></p> +<h2 class="nobreak" id="154_RELATION-NUMBERS_OF_ASSIGNED_TYPES"> +*154. RELATION-NUMBERS OF ASSIGNED TYPES.</h2> +</div> + + +<p class="nind"><i>Summary of</i> *154.</p> + +<p>This number gives propositions analogous to those of *102. In +accordance with our general notations for typical definiteness, +"\(\text{Nr}(P)ʻQ\)" means "the class of relations like \(Q\) and of +the same type as \(P\)," "\(\text{Nr} (P_{Q})\)" means "the relation to +a relation of the type of \(Q\) of the class of relations like it and +of the type of \(P\)." By a special definition, "\(\text{NR}^{Q}(P)\)" +is to mean all typically definite relation-numbers of the form +"\(\text{Nr}(P_{Q})ʻR\)," <i>i.e.</i> all relation-numbers generated +by the relation \(\text{Nr}(P_{Q})\), <i>i.e.</i> the domain of +\(\text{Nr}(P_{Q})\).</p> + +<p>Existence-theorems in this subject can be proved by means of <a href="#*154·14">*154·14</a>, +which states that relations like \(Q\) exist in the type of \(P\) +when, and only when, classes similar to \(CʻQ\) exist in the type of +\(CʻP\). In virtue of this proposition, the existence-theorems of our +present topic are deducible from those for cardinals. In symbols, this +proposition is</p> + +<p class="nind"><b>*154·14.</b> \(\vdash :\exists !\text{Nr}(P)ʻQ.\equiv .\exists !\text{Nc}(CʻP)ʻCʻQ\)</p> + +<p>Hence by <a href="#*102·73">*102·73</a> we deduce</p> + +<p class="nind"><b>*154·242.</b> \(\vdash . \Lambda \in \text{NR}^{\iota ^{;}P}(P)\)</p> + +<p class="nind">whence, by <a href="#*152·72">*152·72</a>,</p> + +<p class="nind"><b>*154·9.</b> \(\vdash .\text{NC}=Cʻʻʻ\text{NR}\)</p> + +<p>The remaining propositions are chiefly analogues of those in <a href="#*102">*102</a>. Very +few of them are subsequently referred to.</p> + +<hr class="tb"> + +<p class="nind"><b>*154·01.</b> \(\text{NR}^{Y}(X) = \text{D}ʻ\text{Nr}(X_{Y}) \quad\text{Df}\)</p> + +<p class="nind"><b>*154·1.</b> \(\vdash : \exists ! \text{Rl}ʻP \cap \text{Nr}ʻQ . \supset . \exists ! \text{Cl}ʻCʻP \cap \text{Nc}ʻCʻQ\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*152·1.\supset \vdash :\text{Hp}.&\supset .(\exists R).R\,\unicode{x2abd}\, P.R\,\text{smor}\,Q.\\ +[*151·18] &\supset .(\exists R).R\,\unicode{x2abd}\, P.CʻR\text{ sm }CʻQ.\\ +[*33·265] &\supset .(\exists R).CʻR\subset CʻP.CʻR\text{ sm }CʻQ.\\ +[*100·1] &\supset .\exists !\text{Cl}ʻCʻP\cap \text{Nc}ʻCʻQ:\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_340">[Pg 340]</span></p> + +<p class="nind"><b>*154·11.</b> \(\vdash :\exists !\text{Cl}ʻCʻP\cap \text{Nc}ʻCʻQ.\supset .(\exists R).R\,\text{smor}\,Q.CʻR\subset CʻP\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*100·1.*73·1.\supset \vdash :\text{Hp}.&\supset .(\exists S).S\in 1\rightarrow 1.\text{D}ʻS\subset CʻP.\text{ᗡ}ʻS=CʻQ.\\ +[*151·1] &\supset .(\exists S).\text{D}ʻS\subset CʻP.S^{;}Q\text{smor}Q.\\ +[*150·203] & \supset .(\exists S).CʻS^{;}Q\subset CʻP.S^{;}Q\text{smor}Q:\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*154·12.</b> \(\vdash :\exists !\text{Rl}ʻ(\alpha \uparrow \alpha )\cap \text{Nr}ʻQ.\equiv .\exists !\text{Cl}ʻ\alpha \cap \text{Nc}ʻCʻQ.\equiv .\text{Nc}ʻ\alpha \geq \text{Nc}ʻCʻQ\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*154·1.*35·9. &\supset \vdash :\exists !\text{Rl}ʻ(\alpha \uparrow \alpha )\cap \text{Nr}ʻQ.\supset .\exists !\text{Cl}ʻ\alpha \cap \text{Nc}ʻCʻQ &\qquad \text{(1)}\\ +\vdash .*154·11.*35·92.&\supset \vdash :\exists !\text{Cl}ʻ\alpha \cap \text{Nc}ʻCʻQ.\supset .\exists !\text{Rl}ʻ(\alpha \uparrow \alpha )\cap \text{Nr}ʻQ &\qquad \text{(2)}\\ +\vdash .(1).(2).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*154·121.</b> \(\vdash .\text{Rl}ʻ(t_{0}ʻCʻP\uparrow t_{0}ʻCʻP)=tʻP=t_{00}ʻCʻP\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*64·5. &\supset \vdash .\text{Rl}ʻ(t_{0}ʻCʻP\uparrow t_{0}ʻCʻP)=tʻ(CʻP\uparrow CʻP) &\qquad \text{(1)}\\ +\vdash .*64·201.&\supset \vdash .tʻ(CʻP\uparrow CʻP)=tʻP &\qquad \text{(2)}\\ +\vdash .(1).(2).*64·54.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*154·13.</b> \(\vdash :\exists !tʻP\cap \text{Nr}ʻQ.\equiv .\exists !tʻCʻP\cap \text{Nc}ʻCʻQ.\equiv .\text{Nc}ʻt_{0}ʻCʻP\geq \text{Nc}ʻCʻQ\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*154·12 \frac{t_{0}ʻCʻP}{\alpha}.*154·121.\supset \\ +\vdash :\exists !tʻP\cap \text{Nr}ʻQ.\equiv .\exists !\text{Cl}ʻt_{0}ʻCʻP\cap \text{Nc}ʻCʻQ &\qquad \text{(1)}\\ +\vdash .(1).*63·65.*117·22\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*154·14">*154·14</a>.</b> \(\vdash :\exists !\text{Nr}(P)ʻQ.\equiv .\exists !\text{Nc}(CʻP)ʻCʻQ \quad[*154·13.(*65·04)]\)</p> + +<p>In virtue of *154·14 and the propositions of <a href="#*102">*102</a>, <a href="#*103">*103</a>, <a href="#*104">*104</a>, <a href="#*105">*105</a>, +*106, we see that all homogeneous or ascending relation-numbers +exist, while \(\Lambda\) is a member of every descending type of +relation-numbers. Remembering that the relations concerned must be +homogeneous, we see that there are two kinds of steps by which their +types may be raised, namely (1) from \(P\) to relations of the type of +\(tʻCʻP\uparrow tʻCʻP\), <i>i.e.</i> from \(P\) to relations of the +type of \(CʻP\downarrow CʻP\), or of \(\iota ^{;}P\); (2) from \(P\) to +relations of the type of \(tʻP\uparrow tʻP\), <i>i.e.</i> from \(P\) to +relations of the type of \(P\downarrow P\), or of \(\downarrow x^{;}P\) +if \(x\in t_{0}ʻCʻP\). Thus repetitions of the two steps from \(P\) to +\(\iota ^{;}P\), and from \(P\) to \(\downarrow x^{;}P\), where \(x \in t_{0}ʻCʻP\), +will enable us, without changing the relation-number, to +raise its type indefinitely. It will be observed that, in accordance +with our general definitions for relative types, the type of \(\iota^{;}P\) +is \(t^{11}ʻCʻP\), and the type of \(\downarrow x^{;}P\) (where +\(x\in t_{0}ʻCʻP\)) is \(t^{11}ʻP\).</p> + +<p class="nind"><b>*154·2.</b> \(\vdash .\text{Nr}(X_{Y})ʻQ=\hat{P} \{P\text{smor}_{(X,Y)}Q\} \quad[*65·2.(*152·01)]\)</p> + +<p class="nind"><b>*154·201.</b> \(\vdash .\text{Nr}(X)ʻQ=\text{Nr}ʻQ\cap tʻX \quad[\text{Proof as in *102·6}]\)</p> + +<p><span class="pagenum" id="Page_341">[Pg 341]</span></p> + +<p class="nind"><b>*154·202.</b> \[\begin{align}&\vdash :P\in \text{Nr}(X_{Y})ʻQ.\equiv .P\in \text{Nr}(X)ʻQ.Q\in tʻY.\equiv .\\ +&P\in \text{Nr}ʻQ.P\in tʻX.Q\in tʻY \quad[*152·2·201. (*65·1)]\end{align}\]</p> + +<p class="nind"><b>*154·203.</b> \(\vdash :Q\in tʻY.\supset .\text{Nr}(X_{Y})ʻQ=\text{Nr}(X)ʻQ \quad[*154·202]\)</p> + +<p>When \(Q\) belongs to any other type than \(tʻY\), +\(\text{Nr}(X_{Y})ʻQ\) is meaningless.</p> + +<p class="nind"><b>*154·21.</b> \(\vdash .\text{NR}^{Y}(X)=\hat{\lambda} \{(\exists Q).\lambda =\text{Nr}(X_{Y})ʻQ\} \quad[(*154·01)]\)</p> + +<p class="nind"><b>*154·22.</b> \(\vdash .\text{NR}^{Y}(X)=\text{Nr}(X)ʻʻtʻY=(\cap tʻX)ʻʻ\text{Nr}ʻʻtʻY\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*154·21·202.\supset \\ +\vdash \colon\ldotp \lambda \in \text{NR}^{Y}(X).&\equiv :(\exists Q):P\in \lambda .\equiv _{P}.P\in \text{Nr}(X)ʻQ.Q\in tʻY:\\ +[*63·108.*4·73] &\equiv :(\exists Q):Q\in tʻY:P\in \lambda .\equiv _{P}.P\in \text{Nr}(X)ʻQ:\\ +[*20·43] &\equiv :(\exists Q):Q\in tʻY.\lambda =\text{Nr}(X)ʻQ:\\ +[*37·6] &\equiv :\lambda \in \text{Nr}(X)ʻʻtʻY &\qquad \text{(1)}\\ +\vdash .(1).*154·201.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*154·23.</b> \(\vdash :\Lambda \in \text{NR}^{Q}(P).\equiv .\Lambda \in \text{NC}^{CʻQ}(CʻP).\equiv .\Lambda \in \text{NC}(CʻP)ʻʻtʻCʻQ\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*154·22.\supset \vdash :\Lambda \in \text{NR}^{Q}(P).&\equiv .\Lambda \in \text{Nr}(P)ʻʻtʻQ.\\ +[*37·6] &\equiv .(\exists R).R\in tʻQ.\Lambda =\text{Nr}(P)ʻR.\\ +[*154·14.\text{Transp}] &\equiv .(\exists R).R\in tʻQ.\Lambda =\text{Nc}(CʻP)ʻCʻR.\\ +[*64·24] &\equiv .(\exists R).CʻR\in tʻCʻQ.\Lambda =\text{Nc}(CʻP)ʻCʻR.\\ +[*35·942] &\equiv .(\exists \alpha ).\alpha \in tʻCʻQ.\Lambda =\text{Nc}(CʻP)ʻ\alpha .\\ +[*37·6] &\equiv .\Lambda \in \text{Nc}(CʻP)ʻʻtʻCʻQ. &\qquad \text{(1)}\\ +[*102·62] &\equiv .\Lambda \in \text{NC}^{CʻQ}(CʻP) &\qquad \text{(2)}\\ +\vdash .(1).(2).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*154·24">*154·24</a>.</b> \(\vdash :CʻQ=tʻCʻP.\supset .\text{Nr}(P)ʻQ=\Lambda \quad[*102·73.*154·14]\)</p> + +<p class="nind"><b><a id="*154·241">*154·241</a>.</b> \(\vdash .\text{Nr}(P)ʻI\upharpoonright tʻCʻP=\Lambda \quad[*154·24]\)</p> + +<p class="nind"><b><a id="*154·242">*154·242</a>.</b> \(\vdash .\Lambda \in \text{NR}^{\iota ^{;}P}(P)\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*35·91.\supset \vdash .I\upharpoonright tʻCʻP&\,\unicode{x2abd}\, tʻCʻP\uparrow tʻCʻP\\ +[*63·64] &\,\unicode{x2abd}\, t_{0}ʻ\iotaʻʻCʻP\uparrow t_{0}ʻ\iotaʻʻCʻP\\ +[*150·22] &\,\unicode{x2abd}\, t_{0}ʻCʻ\iota ^{;}P\uparrow t_{0}ʻCʻ\iota ^{;}P &\qquad \text{(1)}\\ +\vdash .(1).*154·121.&\supset \vdash .I\upharpoonright tʻCʻP\in tʻ\iota ^{;}P &\qquad \text{(2)}\\ +\vdash .(2).*154·22·241.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*154·25">*154·25</a>.</b> \(\vdash :CʻQ=t_{00}ʻCʻP.\supset .\text{Nr}(P)ʻQ=\Lambda \quad[*106·53.*154·14]\)</p> + +<p class="nind"><b><a id="*154·251">*154·251</a>.</b> \(\vdash .\Lambda \in \text{NR}^{P\downarrow P}(P)\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*154·23.\supset \vdash :\Lambda \in \text{NR}^{P\downarrow P}(P).&\equiv .\Lambda \in \text{Nc}(CʻP)ʻʻtʻCʻ(P\downarrow P).\\ +[*55·15] &\equiv .\Lambda \in \text{Nc}(CʻP)ʻʻtʻ\iota ʻP.\\ +[*63·61] &\equiv .\Lambda \in \text{Nc}(CʻP)ʻʻtʻtʻP.\\ +[*154·121] &\equiv .\Lambda \in \text{Nc}(CʻP)ʻʻtʻt_{00}ʻCʻP &\qquad \text{(1)}\\ +\vdash .(1).*106·53.*104·264.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_342">[Pg 342]</span></p> + +<p class="nind"><b><a id="*154·26">*154·26</a>.</b> \(\vdash :P\in tʻQ.\supset .\exists !\text{Nr}(P)ʻQ \quad[*64·231.*103·3·13.*154·14]\)</p> + +<p class="nind"><b><a id="*154·261">*154·261</a>.</b> \(\vdash :CʻP\in t^{2}ʻCʻQ.\supset .\exists !\text{Nr}(P)ʻQ \quad[*104·21·1.*154·14]\)</p> + +<p class="nind"><b><a id="*154·262">*154·262</a>.</b> \(\vdash :CʻP\in t_{00}ʻCʻQ.\supset .\exists !\text{Nr}(P)ʻQ \quad[*106·21·1.*154·14]\)</p> + +<p>The following propositions are concerned with the two particular +transformations from \(P\) to \(\iota ^{;}P\) and from \(P\) to +\(x\downarrow ^{;}P\), which are useful in raising the type of a +relation-number.</p> + +<p class="nind"><b><a id="*154·31">*154·31</a>.</b> \(\vdash .tʻ\iota ^{;}P=t^{11}ʻCʻP\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*154·121.*150·22.\supset \vdash .tʻ\iota ^{;}P&=\text{Rl}ʻ(t_{0}ʻ\iotaʻʻCʻP\uparrow t_{0}ʻ\iotaʻʻCʻP)\\ +[*63·64] &=\text{Rl}ʻ(tʻCʻP\uparrow tʻCʻP)\\ +[*64·56] &=t^{11}ʻCʻP.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*154·311">*154·311</a>.</b> \(\vdash .\exists !\text{Nr}(t^{11}ʻCʻP)ʻP \quad[*154·31.*152·6]\)</p> + +<p class="nind"><b><a id="*154·32">*154·32</a>.</b> \(\vdash :x\in t_{0}ʻCʻP.\supset .tʻx\downarrow ^{;}P=t^{11}ʻP.t_{0}ʻx\downarrowʻʻCʻP=tʻP\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*154·121.*150·22.&\supset \vdash .tʻx\downarrow ^{;}P=\text{Rl}ʻ\{(t_{0}ʻx\downarrowʻʻCʻP)\uparrow (t_{0}ʻx\downarrowʻʻCʻP)\} &\qquad \text{(1)}\\ +\vdash .*64·52. &\supset \vdash :x,y\in t_{0}ʻCʻP.\supset .x\downarrow y\in tʻ(t_{0}ʻCʻP\uparrow t_{0}ʻCʻP).\\ +[*154·121] &\supset .x\downarrow y\in tʻP &\qquad \text{(2)}\\ +\vdash .(2). &\supset \vdash :x\in t_{0}ʻCʻP.\supset .x\downarrowʻʻCʻP\subset tʻP.\\ +[*63·21] &\supset .t_{0}ʻx\downarrowʻʻCʻP=tʻP &\qquad \text{(3)}\\ +\vdash .(1).(3). \supset \vdash :\text{Hp}.\supset .tʻx\downarrow ^{;}P&=\text{Rl}ʻ(tʻP\uparrow tʻP)\\ +[*64·56] &= t^{11}ʻP &\qquad \text{(4)}\\ +\vdash .(3).(4).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*154·321">*154·321</a>.</b> \(\vdash .\exists !\text{Nr}(t^{11}ʻP)ʻP \quad[*154·32.*152·62.*63·18]\)</p> + +<p class="nind"><b><a id="*154·322">*154·322</a>.</b> \(\vdash :x\in t_{0}ʻCʻP.\supset .tʻ\downarrow x^{;}P=t^{11}ʻP \quad[\text{Proof as in *154·32}]\)</p> + +<p class="nind"><b><a id="*154·33">*154·33</a>.</b> \(\vdash :x\in t_{0}ʻCʻP.\supset .tʻP\downarrow ^{;}x\downarrow ^{;}P=t^{11}ʻ\dot{s} ʻt^{11}ʻP\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*154·32. \supset \vdash :\text{Hp}.&\supset .P\in t_{0}ʻx\downarrowʻʻCʻP.\\ +[*150·22] &\supset .P\in t_{0}ʻCʻx\downarrow ^{;}P.\\ +[*154·32] \supset .tʻP\downarrow ^{;}x\downarrow ^{;}P&=t^{11}ʻx\downarrow ^{;}P\\ +[*64·23] & =t^{11}ʻ\dot{s} ʻtʻx\downarrow ^{;}P\\ +[*154·32] & =t^{11}ʻ\dot{s} ʻt^{11}ʻP:\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*154·331">*154·331</a>.</b> \(\vdash .\exists !\text{Nr}(t^{11}ʻ\dot{s} ʻt^{11}ʻP) \quad[*154·33.*152·62.*63·18]\)</p> + +<p class="nind"><b>*154·4.</b> \[\begin{align}\vdash .\text{Nr}(X_{Y})ʻQ=\hat{P} \{(\exists S).S\in 1\rightarrow 1.&\text{ᗡ}ʻS=CʻQ.P=S^{;}Q.\\ +&\text{D}ʻS\in tʻCʻX.\text{ᗡ}ʻS\in tʻCʻY\}\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*154·202.*152·1.\supset \\ +\vdash \colon\ldotp P\in \text{Nr}(X_{Y})ʻQ.&\equiv :(\exists S).S\in 1\rightarrow 1.\text{ᗡ}ʻS=CʻQ.P=S^{;}Q:P\in tʻX.Q\in tʻY:\\ +[*64·24] & \equiv :(\exists S).S\in 1\rightarrow 1.\text{ᗡ}ʻS=CʻQ.P=S^{;}Q.CʻP\in tʻCʻX.\\ +&CʻQ\in tʻCʻY:\\ +[*13·193.*150·23] &\equiv :(\exists S).S\in 1\rightarrow 1.\text{ᗡ}ʻS=CʻQ.P=S^{;}Q.\\ +&\text{D}ʻS\in tʻCʻX.\text{ᗡ}ʻS\in tʻCʻY\colon\ldotp \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_343">[Pg 343]</span></p> + +<p class="nind"><b>*154·401.</b> \[\begin{align}&\vdash .\text{Nr}(X_{Y})ʻQ=\hat{P} \{\exists !(P\,\overline{\text{smor}}\,Q)\cap tʻ(CʻX\uparrow CʻY)\}\\ +&[*154·4.*151·11.*64·63]\end{align}\]</p> + +<p>The remaining propositions of this number (except <a href="#*154·9">*154·9</a>) are the +analogues of those whose numbers have the same decimal part in <a href="#*102">*102</a>. +They are here given without proof, because the proofs are, step by +step, analogous to the proofs of the corresponding propositions in *102.</p> + +<p class="nind"><b>*154·41.</b> \(\vdash :P\in \text{Nr}(X_{Z})ʻR.Q\in \text{Nr}(Y_{Z})ʻR.\supset .P\in \text{Nr}(X_{Y})ʻQ.Q\in \text{Nr}(Y_{X})ʻP\)</p> + +<p class="nind"><b>*154·42.</b> \(\vdash .P\in \text{Nr}(P_{P})ʻP\)</p> + +<p class="nind"><b>*154·43.</b> \(\vdash .\exists !\text{Nr}(P_{P})ʻP\)</p> + +<p class="nind"><b>*154·46.</b> \(\vdash :P\in \text{Nr}(X_{Y})ʻQ.\equiv .Q\in \text{Nr}(Y_{X})ʻP.\equiv .P\,\text{smor}\,Q.P\in tʻX.Q\in tʻY\)</p> + +<p class="nind"><b>*154·52.</b> \(\vdash :\exists !\text{Nr}(X_{Y})ʻQ.\supset .\text{Nr}(X_{Y})ʻQ\in \text{NR}^{X}(X)\)</p> + +<p class="nind"><b>*154·53.</b> \(\vdash :\text{NR}^{Y}(X)-\iota ʻ\Lambda \subset \text{NR}^{X}(X)\)</p> + +<p class="nind"><b>*154·55.</b> \(\vdash :\Lambda {\sim}\in \text{NR}^{X}(Y).\supset .\text{NR}^{Y}(X)-\iota ʻ\Lambda =\text{NR}^{X}(X)\)</p> + +<p class="nind"><b>*154·64.</b> \(\vdash :\mu \in \text{NR}.\exists !\mu .\supset .(\exists P,Q).\mu =\text{Nr}(P)ʻQ\)</p> + +<p class="nind"><b>*154·641.</b> \(\vdash :\mu \in \text{NR}.\supset .(\exists P,Q).\mu =\text{Nr}(P)ʻQ \quad[*154·64·241]\)</p> + +<p class="nind"><b>*154·8.</b> \(\vdash :P\in \text{Nr}(X_{Y})ʻQ.P\,\text{smor}\,R.R\in tʻS.\supset .R\in \text{Nr}(S_{Y})ʻQ.R\in \text{Nr}(S_{X})ʻP\)</p> + +<p class="nind"><b>*154·81.</b> \(\vdash :P\in \text{Nr}(X_{Y})ʻQ.\supset .\text{smor}ʻʻ\text{Nr}(X_{Y})ʻQ\cap tʻS=\text{Nr}(S_{Y})ʻQ=\text{Nr}(S_{X})ʻP\)</p> + +<p class="nind"><b>*154·82.</b> \(\vdash :\mu \in \text{NR}^{Y}(X).\exists !\mu .\supset .\text{smor}ʻʻ\mu \cap tʻS\in \text{NR}^{Y}(S)\)</p> + +<p class="nind"><b>*154·83.</b> \[\begin{align}\vdash :\mu \in \text{NR}^{Y}(X).\nu =&\text{smor}ʻʻ\mu \cap tʻS.\exists !\nu .\supset .\\ +&\text{smor}ʻʻ\mu \cap tʻS=\text{smor}ʻʻ\nu \cap tʻS.\mu =\text{smor}ʻʻ\nu \cap tʻX\end{align}\]</p> + +<p class="nind"><b>*154·84.</b> \(\vdash :(\exists P).P\,\text{smor}\,X.P\in tʻX.Q\,\text{smor}\,P.\equiv .Q\,\text{smor}\,X\)</p> + +<p class="nind"><b>*154·85.</b> \(\vdash .\text{smor}ʻʻ\mu \cap tʻY=\text{smor}_{Y}ʻʻ\mu\)</p> + +<p class="nind"><b>*154·86.</b> \(\vdash :\mu =\text{Nr}(X)ʻQ.\exists !\mu .\supset .\text{smor}_{Y}ʻʻ\mu =\text{Nr}(Y)ʻQ\)</p> + +<p class="nind"><b>*154·861.</b> \(\vdash .\text{smor}_{X}ʻʻ\text{smor}_{Y}ʻʻ\mu \subset \text{smor}_{X}ʻʻ\mu\)</p> + +<p class="nind"><b>*154·87.</b> \(\vdash :\mu =\text{Nr}(Y)ʻQ.\exists !\text{Nr}(X)ʻQ.\supset .\text{smor}_{P}ʻʻ\mu =\text{smor}_{P}ʻʻ\text{smor}_{X}ʻʻ\mu\)</p> + +<p class="nind"><b>*154·88.</b> \[\begin{align}\vdash :\mu =\text{Nr}(Y)ʻQ.\exists !&\text{smor}_{P}ʻʻ\mu .\supset .\\ +&\text{smor}_{P}ʻʻ\mu =\text{Nr}(P)ʻQ.\text{smor}_{X}ʻʻ\mu =\text{Nr}(X)ʻQ.\\ +&\text{smor}_{X}ʻʻ\mu =\text{smor}_{X}ʻʻ\text{smor}_{P}ʻʻ\mu =\text{smor}_{X}ʻʻ\text{Nr}(P)ʻQ\end{align}\]</p> + +<p class="nind"><b><a id="*154·9">*154·9</a>.</b> \(\vdash.\text{NC}=Cʻʻʻ\text{NR}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*37·29.\supset \vdash :\mu =\Lambda .\supset .\mu =Cʻʻ\Lambda . +[*154·241] \supset .\mu \in Cʻʻʻ\text{NR} &\qquad \text{(1)}\\ +\vdash .*37·29.\supset \vdash :\nu =\Lambda .\supset .Cʻʻ\nu =\Lambda . +[*102·73] \supset .Cʻʻ\nu \in \text{NC} &\qquad \text{(2)}\\ +\vdash .(1).(2).*152·72.\supset \vdash .\text{Prop} +\end{array} +\]</p> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_344">[Pg 344]</span></p> +<h2 class="nobreak" id="155_HOMOGENEOUS_RELATION-NUMBERS"> +*155. HOMOGENEOUS RELATION-NUMBERS.</h2> +</div> + + +<p class="nind"><i>Summary of</i> *155.</p> + +<p>A relation-number is called homogeneous when it is generated by +a homogeneous relation of likeness, <i>i.e.</i> when it consists +of all relations which are like a given relation \(P\) and of +the same type as \(P\). For the homogeneous relation-number +of \(P\) we write "\(\text{N}_{0}\text{r}ʻP\)"; thus +\(\text{N}_{0}\text{r}ʻP=\text{Nr}ʻP\cap tʻP\). When \(P\) is given, +\(\text{N}_{0}\text{r}ʻP\) is typically definite. We have always \(P\in \text{N}_{0}\text{r}ʻP\), +hence \(\exists !\text{N}_{0}\text{r}ʻP\). Conversely, if a +typically definite relation-number is not null, it is a homogeneous +relation-number; in fact, if \(P\) is a member of it, it is +\(\text{N}_{0}\text{r}ʻP\). Thus the homogeneous relation-numbers are +all the relation-numbers except \(\Lambda\).</p> + +<p>Homogeneous relation-numbers play the same part in relation-arithmetic +as homogeneous cardinals play in cardinal arithmetic. The propositions +of this number (except <a href="#*155·6">*155·6</a><a href="#*155·61">·61</a>) are the analogues of those with the +same decimal part in <a href="#*103">*103</a>. Their proofs are exactly analogous to the +proofs of their analogues in *103, and are therefore omitted.</p> + +<p>The following propositions are the most useful in this number.</p> + +<p class="nind"><b>*155·11.</b> \(\vdash :Q\in \text{N}_{0}\text{r}ʻP.\equiv .Q\,\text{smor}\,P.Q\in tʻP.\equiv .Q\in \text{Nr}ʻP.Q\in tʻP\)</p> + +<p>This merely embodies the definition.</p> + +<p class="nind"><b>*155·12.</b> \(\vdash .P\in \text{N}_{0}\text{r}ʻP\)</p> + +<p class="nind">whence</p> + +<p class="nind"><b>*155·13.</b> \(\vdash .\exists !\text{N}_{0}\text{r}ʻP\)</p> + +<p class="nind"><b>*155·16.</b> \(\vdash :\text{N}_{0}\text{r}ʻP=\text{Nr}ʻQ.\equiv .\text{Nr}ʻP=\text{Nr}ʻQ\)</p> + +<p>This proposition is used in the theory of well-ordered series (*253 and +*255). It requires that the equation "\(\text{Nr}ʻP=\text{Nr}ʻQ\)" on +the right-hand side should be subject to the convention \(\text{AT}\). +Otherwise, the typical ambiguities might be so determined as to +give \(\text{Nr}ʻP=\text{Nr}ʻQ=\Lambda\), which would not imply +\(\text{N}_{0}\text{r}ʻP=\text{Nr}ʻQ\).</p> + +<p class="nind"><b>*155·2.</b> \(\vdash :\mu \in \text{N}_{0}\text{R}.\equiv .(\exists P).\mu =\text{Nr}ʻP\cap tʻP.\equiv .(\exists P).\mu =\text{N}_{0}\text{r}ʻP\)</p> + +<p>This merely embodies the definition of \(\text{N}_{0}\text{R}\).</p> + +<p><span class="pagenum" id="Page_345">[Pg 345]</span></p> + +<p class="nind"><b>*155·22.</b> \(\vdash :\mu \in \text{N}_{0}\text{R}.\supset .\exists !\mu\)</p> + +<p class="nind"><b>*155·26.</b> \(\vdash \colon\ldotp \mu \in \text{NR}.\supset :P\in \mu .\equiv .\text{N}_{0}\text{r}ʻP=\mu\)</p> + +<p class="nind"><b>*155·27.</b> \(\vdash :\mu =\text{N}_{0}\text{r}ʻP.\equiv .\mu \in \text{NR}.P\in \mu\)</p> + +<p class="nind"><b>*155·34.</b> \(\vdash .\text{NR}-\iota ʻ\Lambda \subset \text{N}_{0}\text{R}\)</p> + +<p class="nind"><b>*155·4.</b> \(\vdash .\text{smor}ʻʻ\text{N}_{0}\text{r}ʻP=\text{Nr}ʻP\)</p> + +<p class="nind"><b>*155·5.</b> \(\vdash .0_{r}\in \text{N}_{0}\text{R}\)</p> + +<p class="nind"><b>*155·6.</b> \(\vdash .Cʻʻ\text{N}_{0}\text{r}ʻP=\text{N}_{0}\text{c}ʻCʻP\)</p> + +<p>This last proposition connects homogeneous relation-numbers with +homogeneous cardinals.</p> + +<hr class="tb"> + +<p class="nind"><b>*155·01.</b> \(\text{N}_{0}\text{r}ʻP=\text{Nr}ʻP\cap tʻP \quad\text{Df}\)</p> + +<p class="nind"><b>*155·02.</b> \(\text{N}_{0}\text{R}=\text{D}ʻ\text{N}_{0}\text{r} \quad\text{Df}\)</p> + +<p class="nind"><b>*155·11.</b> \(\vdash :Q\in \text{N}_{0}\text{r}ʻP. =.Q\,\text{smor}\,P.Q\in tʻP.\equiv .Q\in \text{Nr}ʻP.Q\in tʻP\)</p> + +<p class="nind"><b>*155·12.</b> \(\vdash .P\in \text{N}_{0}\text{r}ʻP\)</p> + +<p class="nind"><b>*155·13.</b> \(\vdash .\exists !\text{N}_{0}\text{r}ʻP\)</p> + +<p class="nind"><b>*155·14.</b> \(\vdash :\text{N}_{0}\text{r}ʻP=\text{N}_{0}\text{r}ʻQ.\equiv .P\in \text{N}_{0}\text{r}ʻQ.\equiv .Q\in \text{N}_{0}\text{r}ʻP.\equiv + .P\,\text{smor}\,Q.Q\in tʻP\)</p> + +<p class="nind"><b>*155·15.</b> \(\vdash :\exists !\text{N}_{0}\text{r}ʻP\cap \text{N}_{0}\text{r}ʻQ.\equiv .\text{N}_{0}\text{r}ʻP=\text{N}_{0}\text{r}ʻQ\)</p> + +<p class="nind"><b>*155·16.</b> \(\vdash :\text{N}_{0}\text{r}ʻP=\text{Nr}ʻQ.\equiv .\text{Nr}ʻP=\text{Nr}ʻQ\)</p> + +<p class="nind"><b>*155·2.</b> \(\vdash :\mu \in \text{N}_{0}\text{R}.\equiv .(\exists P).\mu =\text{Nr}ʻP\cap tʻP.\equiv .(\exists P).\mu =\text{N}_{0}\text{r}ʻP\)</p> + +<p class="nind"><b>*155·21.</b> \(\vdash .\text{N}_{0}\text{r}ʻP\in \text{N}_{0}\text{R}.\text{N}_{0}\text{r}ʻP\in \text{NR}\)</p> + +<p class="nind"><b>*155·22.</b> \(\vdash :\mu \in \text{N}_{0}\text{R}.\supset .\exists !\mu\)</p> + +<p class="nind"><b><a id="*155·23">*155·23</a>.</b> \(\vdash .\Lambda {\sim}\in \text{N}_{0}\text{R}\)</p> + +<p class="nind"><b>*155·24.</b> \(\vdash .\text{N}_{0}\text{R}\in \text{Cls ex}^{2}\text{excl}\)</p> + +<p class="nind"><b>*155·25.</b> \(\vdash \colon\ldotp \mu ,\nu \in \text{N}_{0}\text{R}.\supset :\exists !\mu \cap \nu .\equiv .\mu =\nu\)</p> + +<p class="nind"><b>*155*26.</b> \(\vdash \colon\ldotp \mu \in \text{NR}.\supset :P\in \mu .\equiv .\text{N}_{0}\text{r}ʻP=\mu\)</p> + +<p class="nind"><b>*155*27.</b> \(\vdash :\mu =\text{N}_{0}\text{r}ʻP.\equiv .\mu \in \text{NR}.P\in \mu\)</p> + +<p class="nind"><b>*155·28.</b> \(\vdash :(\exists R).R\,\text{smor}\,P.\mu =\text{N}_{0}\text{r}ʻR.\equiv .\exists !\mu .\mu = \text{Nr}ʻP\)</p> + +<p class="nind"><b>*155·3.</b> \(\vdash :Q\in tʻP.\supset .\text{N}_{0}\text{r}ʻQ=\text{Nr}(P)ʻQ=\text{Nr}(P_{P})ʻQ=\text{Nr}ʻQ\cap tʻP\)</p> + +<p class="nind"><b>*155·301.</b> \(\vdash .\text{NR}^{P}(P)=\text{N}_{0}\text{R}(P)\)</p> + +<p class="nind"><b>*155·31.</b> \(\vdash :\exists !\text{Nr}(X_{Y})ʻQ.\supset .\text{Nr}(X_{Y})ʻQ\in \text{N}_{0}\text{R}(X)\)</p> + +<p class="nind"><b>*155·32.</b> \(\vdash .\text{NR}^{Y}(X)-\iota ʻ\Lambda \subset \text{N}_{0}\text{R}(X)\)</p> + +<p class="nind"><b>*155·33.</b> \(\vdash .\text{NR}(X)-\iota ʻ\Lambda \subset \text{N}_{0}\text{R}(X)\)</p> + +<p class="nind"><b><a id="*155·34">*155·34</a>.</b> \(\vdash .\text{NR}-\iota ʻ\Lambda \subset \text{N}_{0}\text{R}\)</p> + +<p class="nind"><b>*155·35.</b> \(\vdash :\Lambda {\sim}\in \text{NR}^{X}(Y).\supset .\text{NR}^{Y}(X)-\iota ʻ\Lambda =\text{N}_{0}\text{R}(X)\)</p> + +<p class="nind"><b>*155·4.</b> \(\vdash .\text{smor}ʻʻ\text{N}_{0}\text{r}ʻP=\text{Nr}ʻP\)</p> + +<p><span class="pagenum" id="Page_346">[Pg 346]</span></p> + +<p class="nind"><b>*155·41.</b> \(\vdash .\text{smor}ʻʻ\text{N}_{0}\text{r}ʻP\cap tʻQ=\text{Nr}(Q)ʻP\)</p> + +<p class="nind"><b>*155·42.</b> \(\vdash :Q\,\text{smor}\,P.\equiv .\text{Nr}(Q)ʻP=\text{N}_{0}\text{r}ʻQ\)</p> + +<p class="nind"><b>*155·43.</b> \(\vdash :\mu \in \text{NR}.\supset .\text{smor}ʻʻ\mu \cap t_{0}ʻ\mu =\mu\)</p> + +<p class="nind"><b>*155·44.</b> \(\vdash \colon\ldotp \mu ,\nu \in \text{N}_{0}\text{R}.\supset :\mu =\text{smor}ʻʻ\nu .\equiv .\nu =\text{smor}ʻʻ\mu\)</p> + +<p class="nind"><b>*155·5.</b> \(\vdash .0_{r}\in \text{N}_{0}\text{R}\)</p> + +<p class="nind"><b>*155·51.</b> \(\vdash .2_{r}\cap \text{Rl}ʻ\text{Cls}\in \text{N}_{0}\text{R}\)</p> + +<p class="nind"><b>*155·52.</b> \(\vdash .2_{r}\cap \text{Rel}^{2}\in \text{N}_{0}\text{R}\)</p> + +<p>The following propositions have no analogue in <a href="#*103">*103</a>.</p> + +<p class="nind"><b><a id="*155·6">*155·6</a>.</b> \(\vdash .Cʻʻ\text{N}_{0}\text{r}ʻP=\text{N}_{0}\text{c}ʻCʻP\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*100·11.*103·11.&\supset \vdash \colon\ldotp \alpha \in \text{N}_{0}\text{c}ʻCʻP.\equiv :\\ +&\alpha \in tʻCʻP:(\exists S).S\in 1\rightarrow 1.\text{D}ʻS=\alpha .\text{ᗡ}ʻS=CʻP:\\ +[*150·23] &\equiv :\alpha \in tʻCʻP:(\exists S).S\in 1\rightarrow 1.CʻS^{;}P=\alpha .\text{ᗡ}ʻS=CʻP:\\ +[*151·11] &\equiv :\alpha \in tʻCʻP:(\exists Q).Q\,\text{smor}\,P.\alpha =CʻQ:\\ +[*64·24] &\equiv :(\exists Q).Q\,\text{smor}\,P.Q\in tʻP.\alpha =CʻQ:\\ +[*152·11.*155·11] &\equiv :\alpha \in Cʻʻ\text{N}_{0}\text{r}ʻP\colon\ldotp \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*155·61">*155·61</a>.</b> \(\vdash .Cʻʻʻ\text{N}_{0}\text{R}=\text{N}_{0}\text{C} \quad[*155·6]\)</p> + +<p>On ascending and descending relation-numbers, propositions +analogous to those of <a href="#*104">*104</a>, <a href="#*105">*105</a>, and <a href="#*106">*106</a> might be proved by +proofs analogous to those given in those numbers. It is, however, +scarcely necessary to add anything to the propositions already +proved, namely <a href="#*154·24">*154·24</a><a href="#*154·241">·241</a><a href="#*154·242">·242</a><a href="#*154·25">·25</a><a href="#*154·251">·251</a> on descending relation-numbers, +<a href="#*154·26">*154·26</a><a href="#*154·261">·261</a><a href="#*154·262">·262</a><a href="#*154·31">·31</a><a href="#*154·311">·311</a><a href="#*154·32">·32</a><a href="#*154·321">·321</a><a href="#*154·322">·322</a><a href="#*154·33">·33</a><a href="#*154·331">·331</a> on ascending relation-numbers, +and <a href="#*155·23">*155·23</a><a href="#*155·34">·34</a> giving the relations of non-homogeneous to homogeneous +relation-numbers. Ascending relation-numbers all exist, and those +that start from the type of \(P\), wherever they end<a id="FNanchor_13" href="#Footnote_13" class="fnanchor">[13]</a>, are the +correspondents<a id="FNanchor_14" href="#Footnote_14" class="fnanchor">[14]</a> of the homogeneous relation-numbers of the type +of \(P\), and are only some of the homogeneous relation-numbers of +the type in which they end. Descending relation-numbers consist of +\(\Lambda\) together with the homogeneous relation-numbers of the type +in which they end: they are the correspondents of only some of the type +in which they begin, or rather, \(\Lambda\) is the common correspondent +of all those relation-numbers in the initial type which are not +correspondents of any homogeneous relation-number in the end-type. +These properties are exactly the same as in the case of cardinals, as +might be foreseen by <a href="#*154·14">*154·14</a>.</p> + + +<div class="footnotes"><h3>FOOTNOTES:</h3> + +<div class="footnote"> + +<p class="nind"> +<a id="Footnote_13" href="#FNanchor_13" class="label">[13]</a> +We say that \(\text{Nr}(P)ʻQ\) starts from the type of +\(Q\) and ends in the type of \(P\).</p> + +</div> + +<div class="footnote"> + +<p class="nind"> +<a id="Footnote_14" href="#FNanchor_14" class="label">[14]</a> +We call two typically definite relation-numbers +<i>correspondents</i> when they only differ as to the typical +determination, <i>i.e.</i> \(\text{Nr}(X)ʻP\) and \(\text{Nr}(Y)ʻP\) +are correspondents.</p> + +</div> +</div> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_347">[Pg 347]</span></p> +<h2 class="nobreak" id="SECTION_B_II">SECTION B.<br> +<span class="allsmcap">ADDITION OF RELATIONS, AND THE PRODUCT OF TWO RELATIONS.</span></h2> +</div> + + +<p class="nind"><i>Summary of Section B.</i></p> + +<p>In the present section, we have to consider the kind of addition +of relations which is required in ordinal arithmetic. In cardinal +arithmetic, if \(\kappa\) is a class of mutually exclusive classes, +\(sʻ\kappa\) has the properties required of their sum, and thus we +do not require a new kind of logical addition before dealing with +arithmetical addition. But in ordinal arithmetic this is not so. +Suppose \(P\) and \(Q\) are the generating relations of two series, +and we wish to add the \(Q\)-series at the end of the \(P\)-series. +Then we wish every term of the \(P\)-series to precede every term of +the \(Q\)-series; thus \(P\unicode{x228d} Q\) is not the generating +relation of the new series, since \(P\unicode{x228d} Q\) gives no +relation between the terms of the \(P\)-series and the terms of the +\(Q\)-series. The relation we want is +\[ +P\unicode{x228d} Q\unicode{x228d} CʻP\uparrow CʻQ, +\] +since this makes every term of the \(P\)-series precede every term of +the \(Q\)-series. Hence we put +\[ +P\unicode{x2909}Q = P\unicode{x228d} Q\unicode{x228d} CʻP\uparrow CʻQ \quad\text{Df}. +\] +It will be seen that \(P\unicode{x2909}Q\) is in general different from +\(Q\unicode{x2909}P\).</p> + +<p>If \(CʻP\) and \(CʻQ\) have no common terms, the sum of the +relation-numbers of \(P\) and \(Q\) is the relation-number of +\(P\unicode{x2909}Q\) (cf. <a href="#*180">*180</a>).</p> + +<p>The addition of a single term to a series requires a new definition, +and cannot be dealt with as a particular case of the addition of two +relations. It might be thought that, just as \(\alpha \cup \iota ʻx\) +gives the result of adding the one term \(x\) to the class \(\alpha\), +so \(P\unicode{x2909}(x\downarrow x)\) would give the result of +adding the one term \(x\) to the series \(P\). But this is not the +case, since, when we add a term to a series, we do not want this term +to precede itself, whereas \(P\unicode{x2909}(x\downarrow x)\) is a +relation which \(x\) has to itself. What we want is a relation which +every member of \(CʻP\) has to \(x\) but which \(x\) does not have to +itself; thus we take \(P\unicode{x228d} CʻP\uparrow \iota ʻx\) as our +relation, and put +\[ +P\unicode{x21f8}x=P\unicode{x228d} CʻP\uparrow \iota ʻx \quad\text{Df}. +\]<span class="pagenum" id="Page_348">[Pg 348]</span> +This definition defines the generating relation of the series obtained +by adding \(x\) at the end of the \(P\)-series; similarly for adding +\(x\) at the beginning we put +\[ +x\unicode{x21f7}P=\iota ʻx\uparrow CʻP\unicode{x228d} P \quad\text{Df}. +\] +If \(x\) is not a member of \(CʻP\), the relation-number of +\(P\unicode{x21f8}x\) is the sum of the relation-number of \(P\) and +the ordinal 1, which we represent by \(\dot{1}\). (The ordinal 1 has no +meaning by itself, but only as a summand.)</p> + +<p>The sum of a series of series is defined in the same way as the sum +of two series was defined. Let \(P\) be a serial relation whose +field consists of serial relations. Then the sum of all the series +generated by members of \(CʻP\), when these series are taken in the +order generated by \(P\), must be a relation which holds between \(x\) +and \(y\) whenever either (1) \(x\) and \(y\) both belong to the +field of one of the series, and \(x\) precedes \(y\) in this series, +or (2) \(x\) belongs to the field of an earlier series than that to +which \(y\) belongs. In the first case, we have \((\exists Q) . Q\in CʻP.xQy\), +<i>i.e.</i> \(x (\dot{s} ʻCʻP)y\). In the second case, we +have \((\exists Q,R).QPR.x\in CʻQ.y\in CʻR\), <i>i.e.</i> \((\exists Q,R).QPR.xFQ.yFR\), +<i>i.e.</i> \(x(F^{;}P)y\). Hence the generating relation of the sum of +all the series is \(\dot{s} ʻCʻP\unicode{x228d} F^{;}P\). Hence we put +\[ +\Sigma ʻP=\dot{s} ʻCʻP\unicode{x228d} F^{;}P \quad\text{Df}. +\] +The relation \(\Sigma ʻP\) has all the properties which we should +expect of the sum of a series of series.</p> + +<p>If a series is to result from the addition of a series of series, it is +necessary that no two of the series should have any common terms. For +if we have +\[ +QPR.x\in CʻQ\cap CʻR, +\] +we shall also have \(x(\Sigma ʻP)x\).</p> + +<p>Hence instead of a series, we shall have cycles; for it is essential to +a series that no term should precede itself. (What seem to be series +in which there is repetition are always the result of a one-many +correlation with series in which there is no repetition, so that a +term can be counted once as the correlate of one term, and again +as the correlate of a later term.) For this reason, as well as for +many others, it is important to consider relations between mutually +exclusive relations, <i>i.e.</i> between relations whose fields have no +common terms. We put +\[ +\text{Rel}^{2}\text{excl}=\hat{P} \{Q,R\in CʻP.Q\neq R.\supset _{Q,R}.CʻQ\cap CʻR=\Lambda\} \quad\text{Df}. +\] +Then \(\text{Rel}^{2}\text{excl}\) has much the same utility in +relation-arithmetic as \(\text{Cls}^{2}\text{excl}\) has in cardinal +arithmetic. We have +\[ +\vdash :R\in \text{Rel}^{2}\text{excl}.\equiv .F\upharpoonright CʻP\in \text{Cls}\rightarrow 1, +\] +which is analogous to the proposition (*84·14) +\[ +\vdash :\kappa \in \text{Cls}^{2} \text{excl}.\equiv .\in \upharpoonright \kappa \in \text{Cls}\rightarrow 1. +\]<span class="pagenum" id="Page_349">[Pg 349]</span> +It will be found that in relation-arithmetic the relation \(F\) often +appears where \({\in}\) appears in the analogous proposition of +cardinal arithmetic.</p> + +<p>Analogous to "\(\text{sm sm}\)" is the relation of double ordinal +similarity. This holds between two relations \(P\) and \(Q\) when they +are ordinally similar relations between ordinally similar relations +with known correlators, <i>i.e.</i> when, if \(T\) is an ordinal +correlator of \(P\) and \(Q\), so that \(P = T^{;}Q\), then if \(X\) is +a member of \(CʻP\), and \(Y\) is the corresponding member of \(CʻQ\), +so that \(XTY\), we shall have \(X\,\text{smor}\,Y\), and shall be able +to specify a member of \(X\,\overline{\text{smor}}\,Y\). But as in +cardinals, so here, we have to frame our definition of double ordinal +similarity in such a way as to minimize the use of the multiplicative +axiom. We therefore take as our definition the following: \(P\) and +\(Q\) are said to have double ordinal similarity when there is a +one-one relation \(S\) which has \(Cʻ\Sigma ʻQ\) for its converse +domain, and is such that \(P = S\dagger ^{;}Q\). A relation \(S\) which +has these properties is called a <i>double correlator</i> of \(P\) and +\(Q\), <i>i.e.</i> we put +\[ +P \,\overline{\text{smor}}\,\overline{\text{smor}}\, Q = (1 \rightarrow 1) \cap \overleftarrow{\text{ᗡ}}ʻCʻ\Sigma ʻQ \cap \hat{S} (P = S\dagger ^{;}Q) \quad\text{Df}, +\] +a definition which, as will be perceived, is closely analogous to +that of \(\kappa \,\overline{\text{ sm }}\, \,\overline{\text{ sm }}\, \Lambda\) +in <a href="#*111">*111</a>. Two relations have double similarity when they have a double +correlator, <i>i.e.</i> +\[ +\text{smor} \text{smor} = \hat{P} \hat{Q} \{\exists ! P \,\overline{\text{smor}}\,\overline{\text{smor}}\, Q\} \quad\text{Df}. +\] +\(S\) is a double correlator of \(P\) and \(Q\) when \(S\) is +a correlator of \(\Sigma ʻP\) and \(\Sigma ʻQ\) and \(S\dagger +\upharpoonright CʻQ\) is a correlator of \(P\) and \(Q\). This might be +taken as the definition of a double correlator, since it is equivalent +to the above definition.</p> + +<p>If we assume the multiplicative axiom, we can prove that double +similarity holds between similar relations of mutually exclusive +similar relations, <i>i.e.</i> between two relations of mutually +exclusive relations \(P\) and \(Q\) which have a correlator \(S\) +such that, if \(Y \in CʻQ\), then \(Y\) and \(SʻY\) are always +similar. In this case, \(S \,\unicode{x2abd}\, \text{smor}\). Thus +if we assume the multiplicative axiom we have, if \(P\), \(Q \in\text{Rel}^{2}\text{excl}\), +\[ +P \,\text{smor}\,\text{smor}\,Q . \equiv . \exists ! P\,\overline{\text{smor}}\,Q \cap \text{Rl}ʻ\text{smor}. +\] +In the particular case in which the fields of \(P\) and \(Q\) +consist of well-ordered relations (<i>i.e.</i> relations generating +<i>well-ordered</i> series), this equivalence can be proved without +the use of the multiplicative axiom, because two similar well-ordered +relations have only one correlator, so that the difficulty of selecting +among correlators does not arise.</p> + +<p>Double ordinal correlators have the same importance in proving the +formal laws of relation-arithmetic that double cardinal correlators +have in cardinal arithmetic. The construction of double correlators in +various cases constitutes a large part of relation-arithmetic.</p> + +<p>In defining the ordinal product of two relation-numbers, and in +defining exponentiation, we use a relation which has properties +<span class="pagenum" id="Page_350">[Pg 350]</span>analogous to those of \(\alpha \downarrow_{,,}ʻʻ\beta\). This relation +is \(P\downarrow_{.,} ^{;}Q\), of which the structure is as follows: +Let \(z\), \(w\) be two terms having the relation \(Q\); then form +the two relations \(\downarrow z^{;}P\), \(\downarrow w^{;}P\). The +relation \(\downarrow z^{;}P\) holds between two couples \(x\downarrow z\) +and \(y\downarrow z\) whenever \(xPy\); thus it arranges couples +whose referents are members of \(CʻP\), and whose relata are \(z\), +in an order similar to \(P\). The relations \(\downarrow z^{;}P\) and +\(\downarrow w^{;}P\) are (by <a href="#*150·03">*150·03</a>) the same as \(P\downarrow_{.,}z\) +and \(P\downarrow_{.,} w\). Thus \(P\downarrow_{.,} ^{;}Q\) arranges +such relations as \(\downarrow z^{;}P\) in an order similar to \(Q\). +Thus \(P\downarrow_{.,} ^{;}Q\) is similar to \(Q\), and every +member of its field is similar to \(P\). Thus the relation-number +of \(P\downarrow_{.,} ^{;}Q\) is \(\text{Nr}ʻQ\), and every member +of its field has the relation-number \(\text{Nr}ʻP\). Moreover +\(P\downarrow_{.,} ^{;}Q\), as it is easy to see, is a relation of +mutually exclusive relations. Hence it is suitable for defining the +product of \(Q\) and \(P\), and we put +\[ +Q\times P = \Sigma ʻP\downarrow_{.,} ^{;}Q \quad\text{Df}. +\] +In the next section, after we have defined the product of a relation of +relations, we shall use the same relation \(P\downarrow_{.,} ^{;}Q\) for +the definition of exponentiation, putting +\[ +P \,\,\text{exp}\,\, Q = \text{Prod}ʻP\downarrow_{.,} ^{;}Q \quad\text{Df}. +\] +These two definitions should be compared with those in <a href="#*113">*113</a> and <a href="#*116">*116</a>.</p> + +<p>In virtue of the definition of \(\Sigma\), the relation +\(\SigmaʻP\downarrow_{.,} ^{;}Q\) holds between terms which either have one +of the relations of the form \(P\downarrow_{.,} z\), or belong +respectively to the fields of two relations \(P\downarrow_{.,}z\), +\(P\downarrow_{.,} w\), where \(zQw\). Thus the relation +\(\SigmaʻP\downarrow_{.,} ^{;}Q\) holds between \(x\downarrow z\) and +\(y\downarrow z\) whenever \(xPy\) and \(z\in CʻQ\), and also between +\(x\downarrow z\) and \(y\downarrow w\) whenever \(x,y\in CʻP.zQw\). +Thus if, for the sake of illustration, \(P\) and \(Q\) generate finite +series, so that their fields are +\[ +\begin{aligned} +1_{P}, 2_{P} ..., \mu _{P},\\ +1_{Q}, 2_{Q} ..., \nu _{Q}, +\end{aligned} +\] +then the field of \(\Sigma ʻP\downarrow_{.,} ^{;}Q\) will consist of the +couples +\[ +\begin{aligned} +1_{P}\downarrow 1_{Q}, 2_{P}\downarrow 1_{Q}, ..., \mu _{P}\downarrow 1_{Q};\\ +1_{P}\downarrow 2_{Q}, 2_{P}\downarrow 2_{Q}, ..., \mu _{P}\downarrow 2_{Q};\\ +\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\\ +1_{P}\downarrow \nu _{Q}, 2_{P}\downarrow \nu _{Q}, ..., \mu _{P}\downarrow \nu _{Q}; +\end{aligned} +\] +and their order as arranged by \(\Sigma ʻP\downarrow_{.,} ^{;}Q\) is that +in which they are written above. Thus the above couples in the above +order constitute the series \(Q\times P\), and it is evident that this +series has \(\nu \times \mu\) terms.</p> + +<p>When the factors of a product are not enumerated, but are given as the +field of a relation, a new definition of multiplication is required. +This definition, which has the advantage of being applicable to +infinite products, will be dealt with in the following section.</p> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_351">[Pg 351]</span></p> +<h2 class="nobreak" id="*160"> +*160. THE SUM OF TWO RELATIONS.</h2> +</div> + + +<p class="nind"><i>Summary of</i> *160.</p> + +<p>In this number, we introduce the definition +\[ +P\unicode{x2909}Q=P\unicode{x228d} Q\unicode{x228d} CʻP\uparrow CʻQ \quad\text{Df}, +\] +which was explained in the introduction to this section. Although the +propositions of this and other numbers in this Part do not require +that \(P\) and \(Q\) should be such as to generate series, yet the +reader will find it convenient to imagine them to be such, since the +important applications of the ideas of this Part are to series. Thus we +may regard the sum of \(P\) and \(Q\) as a relation which holds between +\(x\) and \(y\) when either \(x\) precedes \(y\) in the \(P\)-series, +or \(x\) precedes \(y\) in the \(Q\)-series, or \(x\) belongs to the +\(P\)-series and \(y\) belongs to the \(Q\)-series.</p> + +<p>The most important propositions of this number are :</p> + +<p class="nind"><b>*160·14.</b> \(\vdash .Cʻ(P\unicode{x2909}Q)=CʻP\cup CʻQ\)</p> + +<p class="nind"><b>*160·21.</b> \(\vdash .P\unicode{x2909}\dot{\Lambda} =P\)</p> + +<p class="nind"><b>*160·22.</b> \(\vdash .\dot{\Lambda} \unicode{x2909}Q=Q\)</p> + +<p class="nind"><b>*160·31.</b> \(\vdash .(P\unicode{x2909}Q)\unicode{x2909}R=P\unicode{x2909}(Q\unicode{x2909}R)\)</p> + +<p class="nind">which is the associative law, and</p> + +<p class="nind"><b>*160·4.</b> \(\vdash .(P\unicode{x228d} Q)\unicode{x2909}R=(P\unicode{x2909}R)\unicode{x228d} (Q\unicode{x2909}R)\)</p> + +<p class="nind">which is the distributive law for logical and arithmetical addition;</p> + +<p class="nind"><b>*160·44.</b> \(\vdash :CʻP\subset \text{ᗡ}ʻS.CʻQ\subset \text{ᗡ}ʻS.\supset .S^{;}(P\unicode{x2909}Q)=S^{;}P\unicode{x2909}S^{;}Q\)</p> + +<p class="nind">which is also a kind of distributive law;</p> + +<p class="nind"><b>*160·47.</b> \[\begin{align}\vdash :CʻP\cap CʻQ=\Lambda .CʻP'\cap CʻQ'=\Lambda .&S\in P\,\overline{\text{smor}}\,P'.T\in Q\overline{\text{smor}} Q'.\supset .\\ +&S\unicode{x228d} T\in (P\unicode{x2909}Q)\overline{\text{smor}} (P'\unicode{x2909}Q')\end{align}\]</p> + +<p class="nind">whence</p> + +<p class="nind"><b>*160·48.</b> \[\begin{align}\vdash :CʻP\cap CʻQ=\Lambda .CʻP'\cap CʻQ'=\Lambda .P\,\text{smor}\,P'.&Q\,\text{smor}\,Q'.\supset .\\ +&P\unicode{x2909}Q\,\text{smor}\,P'\unicode{x2909}Q'\end{align}\]</p> + +<p class="nind"> +whence it follows that if \(P\) and \(Q\) are mutually exclusive, the +relation-number of their sum depends only upon the relation-numbers of +\(P\) and \(Q\);</p> + +<p class="nind"><b>*160·5.</b> \(\vdash :CʻP\cap CʻQ=\Lambda .\supset .(P\unicode{x2909}Q)\unicode{x0294f}CʻP=P.(P\unicode{x2909}Q)\unicode{x0294f}CʻQ=Q\)</p> + +<p class="nind"><b>*160·52.</b> \(\vdash :CʻP\cap CʻQ=\Lambda .CʻP\cap CʻR=\Lambda .P\unicode{x2909}Q=P\unicode{x2909}R.\supset .Q=R\)</p> + +<p><span class="pagenum" id="Page_352">[Pg 352]</span></p> + +<hr class="tb"> + +<p class="nind"><b>*160·01.</b> \(P\unicode{x2909}Q=P\unicode{x228d} Q\unicode{x228d} CʻP\uparrow CʻQ \quad\text{Df}\)</p> + +<p class="nind"><b>*160·1.</b> \(\vdash .P\unicode{x2909}Q=P\unicode{x228d} Q\unicode{x228d} CʻP\uparrow CʻQ \quad[(*160·01)]\)</p> + +<p class="nind"><b>*160·11.</b> \(\vdash \colon\ldotp x(P\unicode{x2909}Q)y.\equiv :xPy.\lor.xQy.\lor.x\in CʻP.y\in CʻQ \quad[*160·1]\)</p> + +<p class="nind"><b>*160·111.</b> \(\vdash \colon\ldotp x(P\unicode{x2909}Q)y.\equiv :xPy.\lor.xQy.\lor.xFP.yFQ \quad[*160·11.*33·51]\)</p> + +<p class="nind"><b>*160·12.</b> \(\vdash :\dot{\exists} !Q.\supset .\text{D}ʻ(P\unicode{x2909}Q)=CʻP\cup \text{D}ʻQ \quad[*33·26.*35·85.*160·1]\)</p> + +<p class="nind"><b>*160·13.</b> \(\vdash :\dot{\exists} !P.\supset .\text{ᗡ}ʻ(P\unicode{x2909}Q)=\text{ᗡ}ʻP\cup CʻQ\)</p> + +<p class="nind"><b>*160·14.</b> \(\vdash .Cʻ(P\unicode{x2909}Q)=CʻP\cup CʻQ\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*33·262.*160·1.&\supset \vdash .Cʻ(P\unicode{x2909}Q)=CʻP\cup CʻQ\cup Cʻ(CʻP\uparrow CʻQ) &\qquad \text{(1)}\\ +\vdash .*35·85·86·88. &\supset \vdash .Cʻ(CʻP\uparrow CʻQ)\subset CʻP\cup CʻQ &\qquad \text{(2)}\\ +\vdash .(1).(2).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p>The above proposition is constantly used. The following propositions +(<a href="#*160·15">*160·15</a>—<a href="#*160·161">·161</a>) are not used, but are inserted to show that +\(P\unicode{x2909}Q\) has the kind of structure that we should expect +of a sum.</p> + +<p class="nind"><b><a id="*160·15">*160·15</a>.</b> \(\vdash :\dot{\exists} !P.\supset .\overrightarrow{B}ʻ(P\unicode{x2909}Q)=\overrightarrow{B}ʻP-CʻQ\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*160·12·13.\supset \vdash :\dot{\exists} !P.\dot{\exists} !Q.\supset .\overrightarrow{B}ʻ(P\unicode{x2909}Q)&=(CʻP\cup \text{D}ʻQ)-(\text{ᗡ}ʻP\cup CʻQ)\\ +[*93·101.*33·161] &=\overrightarrow{B}ʻP-CʻQ &\qquad \text{(1)}\\ +\vdash .*160·1.\supset \vdash :Q=\dot{\Lambda} .&\supset .P\unicode{x2909}Q=P.\\ +[*30·37] \supset .\overrightarrow{B}ʻ(P\unicode{x2909}Q)&=\overrightarrow{B}ʻP\\ +[*33·241] &=\overrightarrow{B}ʻP-CʻQ &\qquad \text{(2)}\\ +\vdash .(1).(2).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*160·151.</b> \(\vdash :\dot{\exists} !Q.\supset .\overrightarrow{B}ʻ\text{Cnv}ʻ(P\unicode{x2909}Q)=\overrightarrow{B}ʻ\breve{Q} -CʻP\)</p> + +<p class="nind"><b>*160·16.</b> \(\vdash :\dot{\exists} !P.\overrightarrow{B}ʻP\cap CʻQ=\Lambda .\supset .\overrightarrow{B}ʻ(P\unicode{x2909}Q)=\overrightarrow{B}ʻP \quad[*160·15]\)</p> + +<p class="nind"><b><a id="*160·161">*160·161</a>.</b> \(\vdash :\dot{\exists} !Q.\overrightarrow{B}ʻ\breve{Q} \cap CʻP=\Lambda .\supset .\overrightarrow{B}ʻ\text{Cnv}ʻ(P\unicode{x2909}Q)=\overrightarrow{B}ʻ\breve{Q} \)</p> + +<p class="nind"><b>*160·2.</b> \(\vdash .\text{Cnv}ʻ(P\unicode{x2909}Q)=\breve{Q} \unicode{x2909}\breve{P} \quad[*31·15.*35·84]\)</p> + +<p class="nind"><b>*160·21.</b> \(\vdash .P\unicode{x2909}\dot{\Lambda} =P \quad[*35·88.*25·24]\)</p> + +<p class="nind"><b>*160·22.</b> \(\vdash .\dot{\Lambda} \unicode{x2909}Q=Q\)</p> + +<p class="nind"><b>*160·3.</b> \(\vdash .(P\unicode{x2909}Q)\unicode{x2909}R=P\unicode{x228d} Q\unicode{x228d} R\unicode{x228d} CʻP\uparrow CʻQ\unicode{x228d} CʻP\uparrow CʻR\unicode{x228d} CʻQ\uparrow CʻR\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*160·14·1.\supset \\ +\vdash.(P\unicode{x2909}Q)\unicode{x2909}R&=(P\unicode{x2909}Q)\unicode{x228d} R\unicode{x228d} (CʻP\cup CʻQ)\uparrow CʻR\\ +[*160·1.*35·41·82]&=P\unicode{x228d} Q\unicode{x228d} CʻP\uparrow CʻQ\unicode{x228d} R\unicode{x228d} CʻP\uparrow CʻR\unicode{x228d} CʻQ\uparrow CʻR.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_353">[Pg 353]</span></p> + +<p class="nind"><b>*160·31.</b> \(\vdash .(P\unicode{x2909}Q)\unicode{x2909}R=P\unicode{x2909}(Q\unicode{x2909}R)\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*160·14·1.\supset \\ +\vdash .P\unicode{x2909}(Q\unicode{x2909}R)=P\unicode{x228d} Q\unicode{x228d} R\unicode{x228d} CʻP\uparrow CʻQ\unicode{x228d} CʻP\uparrow CʻR\unicode{x228d} CʻQ\uparrow CʻR &\qquad \text{(1)}\\ +\vdash .(1).*160·3.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*160·32.</b> \(P\unicode{x2909}Q\unicode{x2909}R=(P\unicode{x2909}Q)\unicode{x2909}R \quad\text{Df}\)</p> + +<p>This definition serves merely for the avoidance of brackets.</p> + +<p class="nind"><b>*160·33.</b> \(\vdash :P\,\unicode{x2abd}\, Q.\supset .P\unicode{x2909}R\,\unicode{x2abd}\, Q\unicode{x2909}R \quad[*33·265.*160·1]\)</p> + +<p class="nind"><b>*160*34.</b> \(\vdash :R\,\unicode{x2abd}\, S.\supset .Q\unicode{x2909}R\,\unicode{x2abd}\, Q\unicode{x2909}S \quad [*33·265.*160·1]\)</p> + +<p class="nind"><b>*160·35.</b> \(\vdash :P\,\unicode{x2abd}\, Q.R\,\unicode{x2abd}\, S.\supset .P\unicode{x2909}Q\,\unicode{x2abd}\, R\unicode{x2909}S \quad[*160*33*34]\)</p> + +<p class="nind"><b>*160·4.</b> \(\vdash .(P\unicode{x228d} Q)\unicode{x2909}R=(P\unicode{x2909}R)\unicode{x228d} (Q\unicode{x2909}R)\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*160·1. \supset \vdash .(P\unicode{x228d} Q)\unicode{x2909}R&=P\unicode{x228d} Q\unicode{x228d} R\unicode{x228d} Cʻ(P\unicode{x228d} Q)\uparrow CʻR\\ +[*33·262.*23·56] & = P\unicode{x228d} R\unicode{x228d} Q\unicode{x228d} R\unicode{x228d} (CʻP\cup CʻQ)\uparrow CʻR\\ +[*35·41·82] & =P\unicode{x228d} R\unicode{x228d} Q\unicode{x228d} R\unicode{x228d} CʻP\uparrow CʻR\unicode{x228d} CʻQ\uparrow CʻR\\ +[*160·1] &=(P\unicode{x2909}R)\unicode{x228d} (Q\unicode{x2909}R).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*160·401.</b> \(\vdash .P\unicode{x2909}(Q\unicode{x228d} R)=(P\unicode{x2909}Q)\unicode{x228d} (P\unicode{x2909}R)\)</p> + +<p>The above two propositions state the distributive law for logical +and arithmetical addition. The three following propositions give the +generalized form of this law, when \(\dot{s} ʻ\lambda\) replaces +\(P\unicode{x228d} Q\); these propositions are not subsequently used +but are inserted for the sake of their intrinsic interest.</p> + +<p class="nind"><b>*160·41.</b> \(\vdash :\exists !\lambda .\supset .\dot{s} ʻ\lambda \unicode{x2909}R=\dot{s} ʻ\unicode{x2909}Rʻʻ\lambda =\dot{s} ʻ(\lambda \unicode{x2909}R)\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*41·11.\supset \vdash \colon\ldotp &x(\dot{s} ʻ\unicode{x2909}Rʻʻ\lambda )y.\equiv :(\exists P).P\in \lambda .x(P\unicode{x2909}R)y:\\ +[*160·11] &\equiv :(\exists P):P\in \lambda :xPy.\lor.xRy.\lor.x\in CʻP.y\in CʻR: \\ +[*10·42] &\equiv :(\exists P).P\in \lambda .xPy.\lor.(\exists P).P\in \lambda .xRy.\lor.\\ +&(\exists P).P\in \lambda .x\in CʻP.y\in CʻR:\\ +[*41·11.*10·35.*41·45] &\equiv :x(\dot{s} ʻ\lambda )y.\lor.\exists !\lambda .xRy.\lor.x\in Cʻ\dot{s} ʻ\lambda .y\in CʻR &\qquad \text{(1)}\\ +\vdash .(1)\supset \vdash \colon\colon \text{Hp}.\supset \colon\ldotp x(\dot{s}ʻ\unicode{x2909}Rʻʻ\lambda ).&\equiv :x(\dot{s} ʻ\lambda )y.\lor.xRy.\lor.x\in Cʻ\dot{s} ʻ\lambda .y\in CʻR:\\ +[*160·11] & \equiv :x(\dot{s} ʻ\lambda \unicode{x2909}R)y\colon\colon \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*160·411.</b> \(\vdash :\exists !\lambda .\supset .P\unicode{x2909}\dot{s} ʻ\lambda =\dot{s} ʻP\unicode{x2909}ʻʻ\lambda \quad[\text{Proof as in *160*41}]\)</p> + +<p class="nind"><b>*160·412.</b> \(\vdash :\exists !\lambda .\exists !\mu .\supset .\dot{s} ʻ\lambda \unicode{x2909}\dot{s} ʻ\mu = \dot{s} ʻsʻ\lambda \unicode{x2909}_{,,}ʻʻ\mu\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*160·411. &\supset \vdash :\exists !\mu .\supset .\dot{s} ʻ\lambda \unicode{x2909}\dot{s} ʻ\mu =\dot{s} ʻ(\dot{s} ʻ\lambda )\unicode{x2909}ʻʻ\mu &\qquad \text{(1)}\\ +\vdash .*160·41. &\supset \vdash :\exists !\lambda .\supset .(\dot{s} ʻ\lambda )\unicode{x2909}ʻʻ\mu =\dot{s}ʻʻ\lambda \unicode{x2909}_{,,}ʻʻ\mu &\qquad \text{(2)}\\ +\vdash .(1).(2). \supset \vdash :\exists !\lambda .\exists !\mu .\supset .\dot{s} ʻ\lambda \unicode{x2909}\dot{s} ʻ\mu &=\dot{s} ʻ\dot{s}ʻʻ\unicode{x2909}_{,,}ʻʻ\mu \\ +[*42·12] & =\dot{s} ʻsʻ\lambda \unicode{x2909}_{,,}ʻʻ\mu :\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_354">[Pg 354]</span></p> + +<p>The following propositions lead up to <a href="#*160·44">*160·44</a>, which is frequently used.</p> + +<p class="nind"><b>*160·42.</b> \(\vdash .(P\unicode{x2909}Q)\mid S=P\mid S\unicode{x228d} Q\mid S\unicode{x228d} CʻP\uparrow \breve{S} ʻʻCʻQ\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*160·1. \supset \vdash .(P\unicode{x2909}Q)\mid S&=P\mid S\unicode{x228d} Q\mid S\unicode{x228d} (CʻP\uparrow CʻQ)\mid S\\ +[*37·8] &=P\mid S\unicode{x228d} Q\mid S\unicode{x228d} CʻP\uparrow \breve{S} ʻʻCʻQ.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*160·421.</b> \(\vdash .S\mid (P\unicode{x2909}Q)=S\mid P\unicode{x228d} S\mid Q\unicode{x228d} SʻʻCʻP\uparrow CʻQ\)</p> + +<p class="nind"><b>*160·43.</b> \(\vdash .S^{;}(P\unicode{x2909}Q)=S^{;}P\unicode{x228d} S^{;}Q\unicode{x228d} SʻʻCʻP\uparrow SʻʻCʻQ\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*150·1.*160·421.\supset \\ +\vdash .S^{;}(P\unicode{x2909}Q)&=(S\mid P\unicode{x228d} S\mid Q\unicode{x228d} SʻʻCʻP\uparrow CʻQ)\mid \breve{S} \\ +[*150·1.*37·8]&=S^{;}P\unicode{x228d} S^{;}Q\unicode{x228d} SʻʻCʻP\uparrow SʻʻCʻQ.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*160·44">*160·44</a>.</b> \(\vdash :CʻP\subset \text{ᗡ}ʻS.CʻQ\subset \text{ᗡ}ʻS.\supset .S^{;}(P\unicode{x2909}Q)=S^{;}P\unicode{x2909}S^{;}Q\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*160·43.*150·22.\supset \\ +\vdash :\text{Hp}.\supset .S^{;}(P\unicode{x2909}Q)&=S^{;}P\unicode{x228d} S^{;}Q\unicode{x228d} (CʻS^{;}P)\uparrow (CʻS^{;}Q)\\ +[*160·1] &=S^{;}P\unicode{x2909}S^{;}Q:\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*160·45.</b> + \[\begin{align}\vdash :S\upharpoonright (CʻP'\cup CʻQ')\in 1\rightarrow 1.&S\upharpoonright CʻP'\in P\,\overline{\text{smor}}\,P'.S\upharpoonright CʻQ'\in Q\overline{\text{smor}} Q'.\supset .\\ +&S\upharpoonright Cʻ(P'\unicode{x2909}Q')\in (P\unicode{x2909}Q)\overline{\text{smor}} (P'\unicode{x2909}Q')\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*151·22.\supset \vdash :\text{Hp}.&\supset .CʻP'\subset \text{ᗡ}ʻS.CʻQ'\subset \text{ᗡ}ʻS.P=S^{;}P'.Q=S^{;}Q'. &\qquad \text{(1)}\\ +[*160·44] &\supset .P\unicode{x2909}Q=S^{;}(P'\unicode{x2909}Q') &\qquad \text{(2)}\\ +\vdash .(1).*160·14.&\supset \vdash :\text{Hp}.\supset .Cʻ(P'\unicode{x2909}Q')\subset \text{ᗡ}ʻS &\qquad \text{(3)}\\ +\vdash .*160·14. &\supset \vdash :\text{Hp}.\supset .S\upharpoonright Cʻ(P'\unicode{x2909}Q')\in 1\rightarrow 1 &\qquad \text{(4)}\\ +\vdash .(2).(3).(4).*151·22.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*160·451.</b> \[\begin{align}\vdash :S\upharpoonright CʻP'\in P\overline{\text{smor}} &P'.S\upharpoonright CʻQ'\in Q\overline{\text{smor}} Q'.Sʻʻ(CʻP'-CʻQ')\cap CʻQ=\Lambda .\\ +&\supset .S\upharpoonright Cʻ(P'\unicode{x2909}Q')\in (P\unicode{x2909}Q)\overline{\text{smor}} (P'\unicode{x2909}Q')\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*151·22.*150·22.\supset \vdash :\text{Hp}.&\supset .CʻQ=SʻʻCʻQ'.\\ +[*71·381.*37·421] &\supset .Sʻʻ(CʻP'-CʻQ')\cap SʻʻCʻQ'=\Lambda .\\ +[*74·823] &\supset .S\upharpoonright (CʻP'\cup CʻQ')\in 1\rightarrow 1 &\qquad \text{(1)}\\ +\vdash .(1).*160·45.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*160·452.</b> \[\begin{align}\vdash :S\upharpoonright CʻP'\in P\,\overline{\text{smor}}\,P'.&S\upharpoonright CʻQ'\in Q\overline{\text{smor}} Q'.CʻP\cap CʻQ=\Lambda .\supset .\\ +&S\upharpoonright Cʻ(P'\unicode{x2909}Q')\in (P\unicode{x2909}Q)\overline{\text{smor}} (P'\unicode{x2909}Q')\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*151·22.*150·22.\supset \vdash :\text{Hp}.&\supset .CʻP-SʻʻCʻP'.CʻQ=SʻʻCʻQ'.\\ +[\text{Hp}] &\supset .SʻʻCʻP'\cap SʻʻCʻQ'=\Lambda .\\ +[*74·833] &\supset .S\upharpoonright Cʻ(P'\unicode{x2909}Q')\in 1\rightarrow 1 &\qquad \text{(1)}\\ +\vdash .(1).*160·45.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_355">[Pg 355]</span></p> + +<p class="nind"><b>*160·46.</b> \[\begin{align}\vdash :CʻP=\text{ᗡ}ʻS.CʻQ=\text{ᗡ}ʻT.CʻP\cap Cʻ&Q=\Lambda .\supset .\\ +&(S\unicode{x228d} T)^{;}(P\unicode{x2909}Q)=S^{;}P\unicode{x2909}T^{;}Q\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*160·44.\supset \vdash :\text{Hp}.\supset .&(S\unicode{x228d} T)^{;}(P\unicode{x2909}Q)=(S\unicode{x228d} T)^{;}P\unicode{x2909}(S\unicode{x228d}T)^{;}Q\\ +[*150·32] &=\{(S\unicode{x228d} T)\upharpoonright{CʻP}\}^{;}P\unicode{x2909}\{(S\unicode{x228d} T)\upharpoonright CʻQ\}^{;}Q\\ +[*35·644.\text{Hp}] &=(S\upharpoonright CʻP)^{;}P\unicode{x2909}(T\upharpoonright CʻQ)^{;}Q\\ +[*150·32] &=S^{;}P\unicode{x2909}T^{;}Q:\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*160·47.</b> \[\begin{align}\vdash :CʻP\cap CʻQ=\Lambda .CʻP'\cap CʻQ'=\Lambda .&S\in P\,\overline{\text{smor}}\,P'.T\in Q\overline{\text{smor}} Q'.\supset .\\ +&S\unicode{x228d} T\in (P\unicode{x2909}Q)\overline{\text{smor}} (P'\unicode{x2909}Q')\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*151·11·131.\supset \vdash :\text{Hp}.&\supset .\text{D}ʻS=CʻP.\text{D}ʻT=CʻQ.\text{ᗡ}ʻS=CʻP'.\\ +&\text{ᗡ}ʻT=CʻQ'. &\qquad \text{(1)}\\ +[\text{Hp}] &\supset .\text{D}ʻS\cap \text{D}ʻT=\Lambda .\text{ᗡ}ʻS\cap \text{ᗡ}ʻT=\Lambda .\\ +[*151·11.*71·242] &\supset .S\unicode{x228d} T\in 1\rightarrow 1 &\qquad \text{(2)}\\ +\vdash .(1).*160·14.\supset \vdash :\text{Hp}.\supset .Cʻ(P'\unicode{x2909}Q')&=\text{ᗡ}ʻS\cup \text{ᗡ}ʻT\\ +[*33·261] &=\text{ᗡ}ʻ(S\unicode{x228d} T) &\qquad \text{(3)}\\ +\vdash .*160·46.*151·11.\supset \vdash :\text{Hp}.\supset .P\unicode{x2909}Q&=(S\unicode{x228d} T)^{;}(P'\unicode{x2909}Q') &\qquad \text{(4)}\\ +\vdash .(2).(3).(4).*151·11.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*160·48.</b> \[\begin{align}\vdash :CʻP\cap CʻQ=\Lambda .CʻP'\cap &CʻQ'=\Lambda .P\,\text{smor}\,P'.Q\text{smor}Q'.\supset .\\ +&P\unicode{x2909}Q\,\text{smor}\,P'\unicode{x2909}Q' \quad[*160·47.*151·12]\end{align}\]</p> + +<p class="nind"><b>*160·5.</b> \(\vdash :CʻP\cap CʻQ=\Lambda .\supset .(P\unicode{x2909}Q)\unicode{x0294f}CʻP=P.(P\unicode{x2909}Q)\unicode{x0294f}CʻQ=Q\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*160·1.*36·23.\supset \\ +\vdash.(P\unicode{x2909}Q)\unicode{x0294f}CʻP&=P\unicode{x0294f}CʻP\unicode{x228d} (CʻP\uparrow CʻQ)\unicode{x0294f}CʻP\unicode{x228d} Q\unicode{x0294f}CʻP\\ +[*36·29·33] &=P\unicode{x228d} \{(CʻP\uparrow CʻQ)\dot{\cap} (CʻP\uparrow CʻP)\}\unicode{x228d} Q\unicode{x0294f}CʻP &\qquad \text{(1)}\\ +\vdash .*36·31. &\supset \vdash :\text{Hp}.\supset .Q\unicode{x0294f}CʻP=\dot{\Lambda} &\qquad \text{(2)}\\ +\vdash .*35·834·88.&\supset \vdash :\text{Hp}.\supset .\{(CʻP\uparrow CʻQ)\dot{\cap} (CʻP\uparrow CʻP)\}=\dot{\Lambda} &\qquad \text{(3)}\\ +\vdash .(1).(2).(3).&\supset \vdash :\text{Hp}.\supset .(P\unicode{x2909}Q)\unicode{x0294f}CʻP=P &\qquad \text{(4)}\\ +\text{Similarly}\quad &\vdash :\text{Hp}.\supset .(P\unicode{x2909}Q)\unicode{x0294f}CʻQ=Q &\qquad \text{(5)}\\ +\vdash .(4).(5).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*160·51.</b> \(\vdash :CʻP\cap CʻQ=\Lambda .\supset .(P\unicode{x2909}Q)^{2}=P^{2}\unicode{x228d} Q^{2}\unicode{x228d} + \text{D}ʻP\uparrow CʻQ\unicode{x228d} CʻP\uparrow \text{ᗡ}ʻQ\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*34·73. &\supset \vdash :\text{Hp}.\supset .(P\unicode{x228d} Q)^{2}=P^{2}\unicode{x228d} Q^{2} &\qquad \text{(1)}\\ +\vdash .*35·895.&\supset \vdash :\text{Hp}.\supset .(CʻP\uparrow CʻQ)^{2}=\dot{\Lambda} &\qquad \text{(2)}\\ +\vdash .*34·62. &\supset \vdash .(P\unicode{x2909}Q)^{2}=(P\unicode{x228d} Q)^{2}\unicode{x228d} (CʻP\uparrow CʻQ)^{2}\\ +&\unicode{x228d} (P\unicode{x228d} Q)\mid (CʻP\uparrow CʻQ)\unicode{x228d} (CʻP\uparrow CʻQ)\mid (P\unicode{x228d} Q)\\ +[(1).(2)] &=P^{2}\unicode{x228d} Q^{2}\unicode{x228d} (P\unicode{x228d} Q)\mid (CʻP\uparrow CʻQ)\unicode{x228d} (CʻP\uparrow CʻQ)\mid (P\unicode{x228d} Q) &\qquad \text{(3)}\\ +\vdash .*37·81. &\supset \vdash :\text{Hp}.\supset .(P\unicode{x228d} Q)\mid (CʻP\uparrow CʻQ)=\text{D}ʻP\uparrow CʻQ &\qquad \text{(4)}\\ +\vdash .*37·8. &\supset \vdash :\text{Hp}.\supset .(CʻP\uparrow CʻQ)\mid (P\unicode{x228d} Q)=CʻP\uparrow \text{ᗡ}ʻQ &\qquad \text{(5)}\\ +\vdash .(3).(4).(5).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_356">[Pg 356]</span></p> + +<p>The above proposition is useful in proving that, if \(CʻP\cap CʻQ=\Lambda\), +\(P\unicode{x2909}Q\) is transitive when \(P\) and \(Q\) +are transitive (cf. <a href="#*201·4">*201·4</a>).</p> + +<p class="nind"><b>*160·52.</b> \(\vdash :CʻP\cap CʻQ=\Lambda .CʻP\cap CʻR=\Lambda .P\unicode{x2909}Q=P\unicode{x2909}R.\supset .Q=R\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*160·14.\supset \vdash :\text{Hp}.&\supset .(P\unicode{x2909}Q)\unicode{x0294f}(-CʻP)=(P\unicode{x2909}Q)\unicode{x0294f}CʻQ.\\ +&(P\unicode{x2909}Q)\unicode{x0294f}(-CʻP)=(P\unicode{x2909}R)\unicode{x0294f}CʻR.\\ +[*160·5] &\supset .(P\unicode{x2909}Q)\unicode{x0294f}(-CʻP)=Q.(P\unicode{x2909}R)\unicode{x0294f}(-CʻP)=R.\\ +[\text{Hp}] &\supset .Q=R:\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p>The above proposition is used in dealing with the series of segments of +a series (<a href="#*213·561">*213·561</a>).</p> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_357">[Pg 357]</span></p> +<h2 class="nobreak" id="*161"> +*161. ADDITION OF A TERM TO A RELATION.</h2> +</div> + + +<p class="nind"><i>Summary of</i> *161.</p> + +<p>The addition of a term has two forms, according as it occurs at the +beginning or end of the field of the relation in question. If we add +first \(x\) and then \(y\) at the end, the result is the same as if we +added \(x\downarrow y\) (<a href="#*161·22">*161·22</a>); if at the beginning, it is the same +as if we added \(y\downarrow x\) (<a href="#*161·221">*161·221</a>). The propositions of the +present number are all obvious, and offer no difficulties of any kind. +As explained in the introduction to this section, we put +\[ +\begin{aligned} +P\unicode{x21f8} x &= P\unicode{x228d} CʻP\uparrow {℩}ʻx \quad\text{Df},\\ +x\unicode{x21f7}P &= {℩}ʻx\uparrow CʻP\unicode{x228d} P \quad\text{Df}. +\end{aligned} +\] +Most of the propositions of this number require the hypothesis +\(\dot{\exists} !P\), because if \(P = \dot{\Lambda}\), +\(P\unicode{x21f8} x = x\unicode{x21f7}P = \dot{\Lambda}\) +(<a href="#*161·2">*161·2</a><a href="#*161·201">·201</a>). This is connected with the fact that there is no ordinal +number 1. Apart from propositions already mentioned, the chief +propositions of this number are the following (we omit propositions +about \(x\unicode{x21f7}P\) when they are merely analogues of propositions +about \(P\unicode{x21f8} x\)):</p> + +<p class="nind"><b>*161·12.</b> \(\vdash .x\unicode{x21f7}P = \text{Cnv}ʻ(\breve{P} \unicode{x21f8} x)\)</p> + +<p class="nind"><b>*161·14.</b> \(\vdash :\dot{\exists} !P.\supset .Cʻ(P\unicode{x21f8} x) = CʻP\cup {℩}ʻx = Cʻ(x\unicode{x21f7}P)\)</p> + +<p class="nind"><b>*161·15.</b> \[\begin{align}\vdash :\dot{\exists} !P.x &{\sim} \in CʻP.\supset .\\ +&\overrightarrow{B}ʻ(P\unicode{x21f8} x) = \overrightarrow{B}ʻP.\overrightarrow{B}ʻ\text{Cnv}ʻ(P\unicode{x21f8} x) = {℩}ʻx.Bʻ(x\unicode{x21f7}\breve{P} ) = x\end{align}\]</p> + +<p class="nind"><b>*161·211.</b> \(\vdash .x\unicode{x21f7}(y\downarrow z) = x\downarrow y\unicode{x228d} x\downarrow z\unicode{x228d} y\downarrow z = (x\downarrow y)\unicode{x21f8} z\)</p> + +<p class="nind"><b>*161·31.</b> \[\begin{align}\vdash :P \,\text{smor} \,Q.x &{\sim} \in CʻP.y {\sim} \in CʻQ.\supset .\\ +&P\unicode{x21f8} x \,\text{smor}\, Q\unicode{x21f8} y.x\unicode{x21f7}P \,\text{smor}\, y\unicode{x21f7}Q\end{align}\]</p> + +<p class="nind"><b>*161·4.</b> \(\vdash :CʻQ\subset \text{ᗡ}ʻS.x\in \text{ᗡ}ʻS.S\in 1\rightarrow \text{Cls}.\supset S^{;}(Q\unicode{x21f8} x) = S^{;}Q\unicode{x21f8} Sʻx\)</p> + +<hr class="tb"> + +<p class="nind"><b>*161·01.</b> \(P\unicode{x21f8} x = P\unicode{x228d} CʻP\uparrow {℩}ʻx \quad\text{Df}\)</p> + +<p class="nind"><b>*161·02.</b> \(x\unicode{x21f7}P = {℩}ʻx\uparrow CʻP\unicode{x228d} P \quad\text{Df}\)</p> + +<p class="nind"><b>*161·1.</b> \(\vdash .P\unicode{x21f8} x = P\unicode{x228d} CʻP\uparrow {℩}ʻx \quad[(*161·01)]\)</p> + +<p class="nind"><b>*161·101.</b> \(\vdash .x\unicode{x21f7}P = {℩}ʻx\uparrow CʻP\unicode{x228d} P \quad[(*161·02)]\)</p> + +<p class="nind"><b>*161·11.</b> \(\vdash \colon\ldotp y(P\unicode{x21f8} x)z.\equiv :yPz.\lor.y\in CʻP.z = x \quad[*161·1]\)</p> + +<p><span class="pagenum" id="Page_358">[Pg 358]</span></p> + +<p class="nind"><b>*161·111.</b> \(\vdash \colon\ldotp y(x\unicode{x21f7}P)z.\equiv :y=x.z\in CʻP.\lor.yPz \quad[*161·101]\)</p> + +<p class="nind"><b>*161·12.</b> \(\vdash .x\unicode{x21f7}P=\text{Cnv}ʻ(\breve{P} \unicode{x21f8} x) \quad[*161·1·101.*35·84.*33·22]\)</p> + +<p class="nind"><b>*161·13.</b> \(\vdash .\text{D}ʻ(P\unicode{x21f8} x)=CʻP.\text{ᗡ}ʻ(x\unicode{x21f7}P)=CʻP\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*161·1.\supset \vdash .\text{D}ʻ(P\unicode{x21f8} x)&=\text{D}ʻP\cup \text{D}ʻ(CʻP\uparrow \iota ʻx)\\ +[*35·85] &=\text{D}ʻP\cup CʻP\\ +[*33·161] &=CʻP &\qquad \text{(1)}\\ +\text{Similarly}\quad \vdash .\text{ᗡ}ʻ(x\unicode{x21f7}P)&=CʻP &\qquad \text{(2)}\\ +\vdash .(1).(2).\supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*161·131.</b> \[\begin{align}&\vdash :\dot{\exists} !P.\supset .\text{ᗡ}ʻ(P\unicode{x21f8} x)=\text{ᗡ}ʻP\cup \iota ʻx.\text{D}ʻ(x\unicode{x21f7}P)=\text{D}ʻP\cup \iota ʻx\\ +&[*35·86.*161·1]\end{align}\]</p> + +<p class="nind"><b>*161·14.</b> \(\vdash :\dot{\exists} !P.\supset .Cʻ(P\unicode{x21f8} x)=CʻP\cup \iota ʻx=Cʻ(x\unicode{x21f7}P) \quad[*161·13·131]\)</p> + +<p>The hypothesis \(\dot{\exists} !P\) is necessary in this proposition, +since without it we have \(P\unicode{x21f8} x=\Lambda\).</p> + +<p class="nind"><b>*161·141.</b> + \[\begin{align}&\vdash :\dot{\exists} !P.\supset .\overrightarrow{B}ʻ(P\unicode{x21f8} x)=\overrightarrow{B}ʻP-\iota ʻx.\overrightarrow{B}ʻ\text{Cnv}ʻ(P\unicode{x21f8} x)=\iota ʻx-CʻP\\ +&[*161·13·131.*93·101]\end{align}\]</p> + +<p class="nind"><b>*161·15.</b> \[\begin{align}&\vdash :\dot{\exists} !P.x{\sim}\in CʻP.\supset .\\ +&\overrightarrow{B}ʻ(P\unicode{x21f8} x)=\overrightarrow{B}ʻP.\overrightarrow{B}ʻ\text{Cnv}ʻ(P\unicode{x21f8} x)=\iota ʻx.Bʻ(x\unicode{x21f7}\breve{P} )=x \quad[*161·141]\end{align}\]</p> + +<p class="nind"><b>*161·16.</b> \(\vdash :x{\sim}\in CʻP.\supset .(P\unicode{x21f8} x)\unicode{x0294f}CʻP=(P\unicode{x21f8} x)\unicode{x0294f}(-\iota ʻx)=P \quad[*161·1]\)</p> + +<p>The above proposition is used in the theory of connected relations +(<a href="#*202·412">*202·412</a>).</p> + +<p class="nind"><b>*161·161.</b> \(\vdash :x{\sim}\in CʻP.\supset .(x\unicode{x21f7}P)\unicode{x0294f}CʻP=(x\unicode{x21f7}P)\unicode{x0294f}(-\iota ʻx)=P\)</p> + +<p>The two following propositions are frequently used.</p> + +<p class="nind"><b><a id="*161·2">*161·2</a>.</b> \(\vdash .\dot{\Lambda} \unicode{x21f8} x=\dot{\Lambda} \quad[*35·75·82.*161·1]\)</p> + +<p class="nind"><b><a id="*161·201">*161·201</a>.</b> \(\vdash .x\unicode{x21f7}\dot{\Lambda} =\dot{\Lambda}\)</p> + +<p class="nind"><b>*161·21.</b> \(\vdash .(x\downarrow y)\unicode{x21f8} z=x\downarrow y\unicode{x228d} x\downarrow z\unicode{x228d} y\downarrow z\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*161·1.*55·15.\supset \vdash .(x\downarrow y)\unicode{x21f8} z&=x\downarrow y\unicode{x228d} (\iota ʻx\cup \iota ʻy)\uparrow \iota ʻz\\ +[*35·82·41.*55·1] &=x\downarrow y\unicode{x228d} x\downarrow z\unicode{x228d} y\downarrow z.\supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p>Note that \(x\downarrow y\unicode{x228d} x\downarrow z\unicode{x228d}y\downarrow z\) +is the relation which orders \(x\) and \(y\) and \(z\) in the order +\(x\), \(y\), \(z\).</p> + +<p class="nind"><b>*161·211.</b> \[\begin{align}&\vdash .x\unicode{x21f7}(y\downarrow z)=x\downarrow y\unicode{x228d} x\downarrow z\unicode{x228d} y\downarrow z=(x\downarrow y)\unicode{x21f8} z\\ +&[\text{Proof as in *161·21}]\end{align}\]</p> + +<p class="nind"><b>*161·212.</b> \(P\unicode{x21f8} x\unicode{x21f8} y=(P\unicode{x21f8} x)\unicode{x21f8} y \quad\text{Df}\)</p> + +<p class="nind"><b>*161·213.</b> \(x\unicode{x21f7}y\unicode{x21f7}P=x\unicode{x21f7}(y\unicode{x21f7}P) \quad\text{Df}\)</p> + +<p>These definitions serve merely for the avoidance of brackets.</p> + +<p><span class="pagenum" id="Page_359">[Pg 359]</span></p> + +<p class="nind"><b><a id="*161·22">*161·22</a>.</b> \(\vdash :\dot{\exists} !P.\supset .(P\unicode{x21f8} x)\unicode{x21f8} y=P\unicode{x2909}(x\downarrow y)\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*161·14·1.\supset \vdash :\text{Hp}.\supset .(P\unicode{x21f8} x)\unicode{x21f8} y&=P\unicode{x228d} CʻP\uparrow {℩}ʻx\unicode{x228d} (CʻP\cup {℩}ʻx)\uparrow {℩}ʻy\\ +[*35·82·41] &=P\unicode{x228d} CʻP\uparrow {℩}ʻx\unicode{x228d} CʻP\uparrow {℩}ʻy\unicode{x228d} {℩}ʻx\uparrow {℩}ʻy\\ +[*35·82·412] &=P\unicode{x228d} CʻP\uparrow ({℩}ʻx\cup {℩}ʻy)\unicode{x228d} {℩}ʻx\uparrow {℩}ʻy\\ +[*55·1·15] &=P\unicode{x228d} CʻP\uparrow Cʻ(x\downarrow y)\unicode{x228d} x\downarrow y\\ +[*160·1] &=P\unicode{x2909}(x\downarrow y):\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*161·221">*161·221</a>.</b> \(\vdash :\dot{\exists} !P.\supset .x\unicode{x21f7}(y\unicode{x21f7}P)=(x\downarrow y)\unicode{x2909}P\)</p> + +<p class="nind"><b>*161·23.</b> \(\vdash :\dot{\exists} !Q.\supset .(P\unicode{x2909}Q)\unicode{x21f8} y=P\unicode{x2909}(Q\unicode{x21f8} y)\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*161·14·1.*160·1.&\supset \vdash :\text{Hp}.\supset .\\ +&P\unicode{x2909}(Q\unicode{x21f8} y)=P\unicode{x228d} Q\unicode{x228d} CʻQ\uparrow {℩}ʻy\unicode{x228d} CʻP\uparrow (CʻQ\cup {℩}ʻy)\\ +[*35·82·412] &=P\unicode{x228d} Q\unicode{x228d} CʻP\uparrow CʻQ\unicode{x228d} CʻP\uparrow {℩}ʻy\unicode{x228d} CʻQ\uparrow {℩}ʻy\\ +[*160·1] &=P\unicode{x2909}Q\unicode{x228d} CʻP\uparrow {℩}ʻy\unicode{x228d} CʻQ\uparrow {℩}ʻy\\ +[*35·82·41.*160·14] &=P\unicode{x2909}Q\unicode{x228d} Cʻ(P\unicode{x2909}Q)\uparrow {℩}ʻy\\ +[*161·1] &=(P\unicode{x2909}Q)\unicode{x21f8} y:\supset \vdash :\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*161·231.</b> \(\vdash :\dot{\exists} !P.\supset .x\unicode{x21f7}(P\unicode{x2909}Q)=(x\unicode{x21f7}P)\unicode{x2909}Q\)</p> + +<p class="nind"><b>*161·232.</b> \(\vdash :\dot{\exists} !P.\dot{\exists} !Q.\supset .P\unicode{x2909}(x\unicode{x21f7}Q)=(P\unicode{x21f8} x)\unicode{x2909}Q\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*161·14·101.*160·1.\supset \vdash :\text{Hp}.\supset .\\ +P \unicode{x2909}(x\unicode{x21f7}Q)&=P\unicode{x228d} {℩}ʻx\uparrow CʻQ\unicode{x228d} Q\unicode{x228d} CʻP\uparrow ({℩}ʻx\cup CʻQ)\\ +[*35·82·412] &=P\unicode{x228d} CʻP\uparrow {℩}ʻx\unicode{x228d} Q\unicode{x228d} CʻP\uparrow CʻQ\unicode{x228d} {℩}ʻx\uparrow CʻQ\\ +[*161·1·14.*35·82·41] &=(P\unicode{x21f8} x)\unicode{x228d} Q\unicode{x228d} Cʻ(P\unicode{x21f8} x)\uparrow CʻQ\\ +[*160·1] &=(P\unicode{x21f8} x)\unicode{x2909}Q:\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*161·24.</b> \(\vdash .x\unicode{x21f7}(P\unicode{x21f8} y)=(x\unicode{x21f7}P)\unicode{x21f8} y\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*161·101·14.\supset \vdash :\dot{\exists} !P.\supset .\\ +x\unicode{x21f7}(P\unicode{x21f8} y)&={℩}ʻx\uparrow (CʻP\cup {℩}ʻy)\unicode{x228d} P\unicode{x228d} CʻP\uparrow {℩}ʻy\\ +[*35·82·412] &={℩}ʻx\uparrow CʻP\unicode{x228d} P\unicode{x228d} {℩}ʻx\uparrow {℩}ʻy\unicode{x228d} CʻP\uparrow {℩}ʻy\\ +[*35·82·41.*161·101·14] &=(x\unicode{x21f7}P)\unicode{x228d} Cʻ(x\unicode{x21f7}P)\uparrow {℩}ʻy\\ +[*161·1] &=(x\unicode{x21f7}P)\unicode{x21f8} y &\qquad \text{(1)}\\ +\vdash .*161·2·201.\supset \vdash :P &=\dot{\Lambda} .\supset .x\unicode{x21f7}(P\unicode{x21f8} y)=\dot{\Lambda} .(x\unicode{x21f7}P )\unicode{x21f8} y=\dot{\Lambda} &\qquad \text{(2)}\\ +\vdash .(1).(2).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_360">[Pg 360]</span></p> + +<p class="nind"><b>*161·25.</b> \(\vdash :\dot{\exists} !P.\dot{\exists} !Q.\supset .(P \unicode{x21f8} x) \unicode{x2909} (y \unicode{x21f7} Q) = P \unicode{x2909} (x\downarrow y) \unicode{x2909} Q\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*161·14.*160·1.\supset \\ +\vdash :\text{Hp}. \supset .(P \unicode{x21f8} x) \unicode{x2909} (y \unicode{x21f7} Q) &= (P \unicode{x21f8} x) \unicode{x228d} (y \unicode{x21f7} Q) \unicode{x228d} (CʻP \cup \iota ʻx)\uparrow (CʻQ \cup \iota ʻy)\\ +[*161·1·101] &= P \unicode{x228d} CʻP\uparrow \iota ʻx \unicode{x228d} \iota ʻy\uparrow CʻQ \unicode{x228d} Q\\ +&\unicode{x228d} (CʻP \cup \iota ʻx)\uparrow (CʻQ \cup \iota ʻy)\\ +[*35·82·41·412] & = P \unicode{x228d} CʻP\uparrow (\iota ʻx \cup \iota ʻy)\unicode{x228d} \iota ʻx\uparrow \iota ʻy \unicode{x228d} Q\\ +&\unicode{x228d} (CʻP \cup \iota ʻx \cup \iota ʻy)\uparrow CʻQ\\ +[*55·15·1.*160·14·1] & = {P \unicode{x2909} (x\downarrow y)} \unicode{x228d} Q \unicode{x228d} Cʻ{P \unicode{x2909} (x\downarrow y)}\uparrow CʻQ\\ +[*160·1.(*160·32)] &= P \unicode{x2909} (x\downarrow y) \unicode{x2909} Q : \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*161·26.</b> + \[\begin{align}\vdash . x \unicode{x21f7} \{y \unicode{x21f7} (z\downarrow w)\} &= (x\downarrow y) \unicode{x2909} (z\downarrow w) = \{(x\downarrow y) \unicode{x21f8} z\} \unicode{x21f8} w\\ +&= \{x \unicode{x21f7} (y\downarrow z)\} \unicode{x21f8} w\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*161·221.*55·134. \supset \vdash . x \unicode{x21f7} \{y \unicode{x21f7} (z\downarrow w)\} &= (x\downarrow y) \unicode{x2909} (z\downarrow w)\\ +[*161·22.*55·134] &= \{(x\downarrow y) \unicode{x21f8} z\} \unicode{x21f8} w\\ +[*161·211] &= \{x \unicode{x21f7} (y\downarrow z)\} \unicode{x21f8} w. \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p>The following propositions lead up to <a href="#*161·33">*161·33</a>.</p> + +<p class="nind"><b>*161·3.</b> \[\begin{align}\vdash : \dot{\exists} !Q . S\in P \overline{\text{smor}} Q . x{\sim}\in CʻP . &y{\sim}\in CʻQ. \supset .\\ +&S \unicode{x228d} x\downarrow y\in (P \unicode{x21f8} x) \overline{\text{smor}} (Q \unicode{x21f8} y)\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*151·11·131. &\supset \vdash : \text{Hp}. \supset . S\in 1\rightarrow 1. CʻQ = \text{ᗡ}ʻS. P = S^{;}Q. CʻP = \text{D}ʻS &\qquad \text{(1)}\\ +\vdash .(1).*55·15. \supset \vdash : \text{Hp}. &\supset . \text{D}ʻS \cap \text{D}ʻ(x\downarrow y) = \Lambda . \text{ᗡ}ʻS \cup \text{ᗡ}ʻ(x\downarrow y) = \Lambda . &\qquad \text{(2)}\\ +[*72·182.*71·242] &\supset . S \unicode{x228d} x\downarrow y\in 1\rightarrow 1 &\qquad \text{(3)}\\ +\vdash .*55·15.*151·11. \supset \vdash : \text{Hp}. \supset . \text{ᗡ}ʻ(S \unicode{x228d} x\downarrow y) &= CʻQ \cup \iota ʻy\\ +[*161·14] & = Cʻ(Q \unicode{x21f8} y) &\qquad \text{(4)}\\ +\vdash .(1).(2).*34·301. \supset \vdash : \text{Hp}. \supset . &(x\downarrow y)\mid Q = \dot{\Lambda} . Q\mid (y\downarrow x) = \dot{\Lambda} .\\ +&(x\downarrow y)\mid (CʻQ\uparrow \iota ʻy) = \dot{\Lambda} . (CʻQ\uparrow \iota ʻy)\mid \breve{S} = \dot{\Lambda} .\\ +[*34·25·26] &\supset . (S \unicode{x228d} x\downarrow y)\mid (Q \unicode{x228d} CʻQ\uparrow \iota ʻy) = S\mid (Q \unicode{x228d} CʻQ\uparrow \iota ʻy).\\ +&(Q \unicode{x228d} CʻQ\uparrow \iota ʻy)\mid (y\downarrow x \unicode{x228d} \breve{S} ) = Q\mid \breve{S} \unicode{x228d} (CʻQ\uparrow \iota ʻy)\mid (y\downarrow x)\\ +[*35·89.*55·1] & = Q\mid \breve{S} \unicode{x228d} CʻQ\uparrow \iota ʻx &\qquad \text{(5)}\\ +\vdash .(5).*150·1. \supset \vdash : \text{Hp}. \supset . (S \unicode{x228d} x\downarrow y)^{;}(Q + \unicode{x228d} CʻQ\uparrow \iota ʻy) &= S\mid \{Q\mid \breve{S} \unicode{x228d} CʻQ\uparrow \iota ʻx\}\\ +[*150·1] &=S^{;}Q \unicode{x228d} S\mid CʻQ\uparrow \iota ʻx\\ +[*37·81.(1).*150·23] & =P \unicode{x228d} CʻP\uparrow \iota ʻx &\qquad \text{(6)}\\ +\vdash .(6).*161·1. &\supset \vdash : \text{Hp}. \supset . (S \unicode{x228d} x\downarrow y)^{;}(Q \unicode{x21f8} y) = P \unicode{x21f8} x &\qquad \text{(7)}\\ +\vdash .(3).(4).(7).*151·11. \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_361">[Pg 361]</span></p> + +<p class="nind"><b>*161·301.</b> \[\begin{align}\vdash : \dot{\exists} !Q. S\in P \overline{\text{smor}} Q. x{\sim}\in CʻP. &y{\sim}\in CʻQ. \supset .\\ +&x\downarrow y \unicode{x228d} S\in (x \unicode{x21f7} P) \overline{\text{smor}} (y \unicode{x21f7} Q)\end{align}\]</p> + +<p class="nind"><b>*161·31.</b> \[\begin{align}\vdash : P \,\text{smor} \,Q. x{\sim}\in CʻP. &y{\sim}\in CʻQ. \supset .\\ +&P \unicode{x21f8} x \,\text{smor}\, Q \unicode{x21f8} y. x \unicode{x21f7} P \,\text{smor}\, y \unicode{x21f7} Q\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*161·3·301.*151·12. \supset \\ +\vdash :\text{Hp}. \dot{\exists} !Q. &\supset . P \unicode{x21f8} x \,\text{smor}\, Q \unicode{x21f8} y. x \unicode{x21f7} P \,\text{smor}\, y \unicode{x21f7} Q &\qquad \text{(1)}\\ +\vdash .*151·32.*161·2·201. \supset \\ +\vdash : \text{Hp}. Q = \dot{\Lambda} . &\supset . P \unicode{x21f8} x = \dot{\Lambda} . Q \unicode{x21f8} y = \dot{\Lambda} . x \unicode{x21f7} P = \dot{\Lambda} . y \unicode{x21f7} Q = \dot{\Lambda} .\\ +[*153·101] & \supset . P \unicode{x21f8} x\, \text{smor}\, Q \unicode{x21f8} y. x \unicode{x21f7} P \,\text{smor}\, y \unicode{x21f7} Q &\qquad \text{(2)}\\ +\vdash .(1).(2). \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*161·32.</b> \[\begin{align}\vdash : \dot{\exists} !Q. x{\sim}\in CʻP. y{\sim}\in CʻQ. &S\in (P \unicode{x21f8} x) \overline{\text{smor}} (Q \unicode{x21f8} y). \supset .\\ +&S\upharpoonright (-\iota ʻy)\in P \overline{\text{smor}} Q. xSy\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*151·5.*161.15. &\supset \vdash : \text{Hp}. \supset . xSy &\qquad \text{(1)}\\ +\vdash .(1).*150·1. \supset \colon\ldotp \text{Hp}. &\supset : u{S\upharpoonright (-\iota ʻy)^{;}Q}v.\\ +&\equiv . (\exists z,w). z(Q \unicode{x21f8} y)w. u\neq x. v\neq x. uSz. vSw.\\ +[*151·11] & \equiv . u\neq x. v\neq x. u(P \unicode{x21f8} x)v.\\ +[*161·11] & \equiv . uPv &\qquad \text{(2)}\\ +\vdash .*35·64. \supset \vdash : \text{Hp}. \supset . \text{ᗡ}ʻS\upharpoonright (-\iota ʻy) &= Cʻ(Q \unicode{x21f8} y) - \iota ʻy\\ +[*161·14·2] & = CʻQ &\qquad \text{(3)}\\ +\vdash .(1).(2).(3). \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*161·321.</b> \[\begin{align}\vdash : \dot{\exists} !Q. x{\sim}\in CʻP. y{\sim}\in CʻQ. &S\in (x \unicode{x21f7} P) \overline{\text{smor}} (y \unicode{x21f7} Q). \supset .\\ +&S\upharpoonright (-\iota ʻy)\in P \overline{\text{smor}} Q. xSy\end{align}\]</p> + +<p class="nind"><b><a id="*161·33">*161·33</a>.</b> \[\begin{align}\vdash \colon\ldotp x&{\sim}\in CʻP. y{\sim}\in CʻQ. \supset :\\ +&P \,\text{smor} \,Q. \equiv . (P \unicode{x21f8} x) \text{smor} (Q \unicode{x21f8} y). \equiv . (x \unicode{x21f7} P) \text{smor} (y \unicode{x21f7} Q)\\ +&[*161·31·32·321·2·201. *153·101]\end{align}\]</p> + +<p>The above proposition justifies addition of 1 or subtraction of 1 in +ordinal arithmetic.</p> + +<p>The following proposition (<a href="#*161·4">*161·4</a>) is much used.</p> + +<p class="nind"><b><a id="*161·4">*161·4</a>.</b> \(\vdash : CʻQ \subset \text{ᗡ}ʻS. x\in \text{ᗡ}ʻS. S\in 1\rightarrow \text{Cls}. \supset . S^{;}(Q \unicode{x21f8} x) = S^{;}Q \unicode{x21f8} Sʻx\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *161·1.*150·3. \supset \vdash . S^{;}(Q \unicode{x21f8} x) &= S^{;}Q \unicode{x228d} S^{;}(CʻQ\uparrow \iota ʻx)\\ +[*150·73] & = S^{;}Q \unicode{x228d} (SʻʻCʻQ)\uparrow (Sʻʻ\iota ʻx) &\qquad \text{(1)}\\ +\vdash .(1).*150·22.*53·31. \supset \vdash : \text{Hp}. \supset . S^{;}(Q \unicode{x21f8} x) &= S^{;}Q \unicode{x228d} (CʻS^{;}Q)\uparrow (\iota ʻSʻx)\\ +[*161·1] & = S^{;}Q \unicode{x21f8} Sʻx: \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*161·41.</b> \(\vdash : CʻQ \subset \text{ᗡ}ʻS. x\in \text{ᗡ}ʻS. S\in 1\rightarrow \text{Cls}. \supset . S^{;}(x \unicode{x21f7} Q) = Sʻx \unicode{x21f7} S^{;}Q\)</p> + +<p class="nind"><b>*161·42.</b> \(\vdash . \downarrow y^{;}(Q \unicode{x21f8} x) = \downarrow y^{;}Q \unicode{x21f8} (x\downarrow y) \quad[*161·4.*55·21.*72·184]\)</p> + +<p class="nind"><b>*161·43.</b> \(\vdash . \downarrow y^{;}(x \unicode{x21f7} Q) = (x\downarrow y) \unicode{x21f7} y^{;}Q\)</p> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_362">[Pg 362]</span></p> +<h2 class="nobreak" id="*162"> +*162. THE SUM OF THE RELATIONS OF A FIELD.</h2> +</div> + + +<p class="nind"><i>Summary of</i> *162.</p> + +<p>The form of summation defined in <a href="#*160">*160</a> cannot be extended beyond a +finite number of summands, since it involves explicit mention of all +the summands. In the present number, we shall be concerned with a +form of summation which is not subject to this restriction. It will +be observed that, since relational summation is not permutative, we +cannot define the sum of a <i>class</i> of relations, for this would +not determine the order in which the summation is to be effected. Our +relations must be given as the field of some relation which orders +them; thus the sum appears not as the sum of a class, but as the sum +of a relation, namely of a relation whose field is the relations to +be summed. In the case of two relations \(Q\) and \(R\), the sum of +\(Q\downarrow R\), as defined in the present number, will be equal to +\(Q\unicode{x2909}R\); similarly for three, the sum of +\(Q\downarrow R\unicode{x228d} Q\downarrow S\unicode{x228d} R\downarrow S\) +will be equal to \(Q\unicode{x2909}R\unicode{x2909}S\), and so on for +any finite number of summands.</p> + +<p>As explained in the introduction to this Section, if P is a relation +between relations, we put</p> + +<p>\[ +\Sigma ʻP=\dot{s} ʻCʻP\unicode{x228d} F^{;}P \quad\text{Df}. +\] +It is convenient to suppose that \(P\) is serial, and that every member +of \(CʻP\) is also serial. Then \(\Sigma ʻP\) holds between \(x\) +and \(y\) if either (1) there is a series, in the field of \(P\), in +which \(x\) precedes \(y\), or (2) \(x\) belongs to a series which is +earlier, in the \(P\)-series, than the series to which y belongs. The +following are the chief propositions of this number:</p> + +<p class="nind"><b>*162·22·23.</b> \(\vdash .Cʻ\Sigma ʻP=sʻCʻʻCʻP=Cʻ\dot{s} ʻCʻP=FʻʻCʻP=\overrightarrow{F}^{2}ʻP\)</p> + +<p class="nind"><b>*162·26.</b> \(\vdash .\Sigma ʻ(P\unicode{x228d} Q)=\Sigma ʻP\unicode{x228d} \Sigma ʻQ\)</p> + +<p class="nind"><b>*162·3.</b> \(\vdash .\Sigma ʻ(Q\downarrow R)=Q\unicode{x2909}R\)</p> + +<p class="nind"><b>*162·31.</b> \(\vdash .\Sigma ʻQ\unicode{x2909}\Sigma ʻR=\Sigma ʻ(Q\unicode{x2909}R)\)</p> + +<p class="nind"><b>*162·34.</b> \(\vdash .\Sigma ʻ\Sigma ^{;}P=\Sigma ʻ\Sigma ʻP \quad[\text{Associative Law. Cf.*42·1}]\)</p> + +<p class="nind"><b>*162·35.</b> \(\vdash :Cʻ\Sigma ʻQ\subset \text{ᗡ}ʻR.\supset .\Sigma ʻR\dagger ^{;}Q=R^{;}\Sigma ʻQ\)</p> + +<p>This is the analogue of *40·38. (Cf. note to <a href="#*162·35">*162·35</a>, below.)</p> + +<p class="nind"><b>*162·4.</b> \(\vdash .\Sigma ʻ\dot{\Lambda} =\dot{\Lambda} \)</p> + +<p class="nind"><b>*162·42.</b> \(\vdash :\dot{\exists} !\Sigma ʻP.\equiv .\dot{\exists} !\dot{s} ʻCʻP.\equiv .\exists !CʻP-{℩}ʻ\dot{\Lambda} \)</p> + +<p><span class="pagenum" id="Page_363">[Pg 363]</span></p> + +<p class="nind"><b>*162·43.</b> \(\vdash :\dot{\exists} !P.\supset .\Sigma ʻ(P\unicode{x21f8} R)=\Sigma ʻP\unicode{x2909}R\)</p> + +<p>It should be observed that the ordinal analogues of propositions about +classes of classes often involve the substitution of \(\Sigma\) (not +\(\dot{s}\)) for \(s\). Examples are afforded by <a href="#*162·34">*162·34</a><a href="#*162·35">·35</a>, quoted +above.</p> + +<hr class="tb"> + +<p class="nind"><b>*162·01.</b> \(\Sigma ʻP=\dot{s} ʻCʻP\unicode{x228d} F^{;}P \quad\text{Df}\)</p> + +<p class="nind"><b>*162·1.</b> \(\vdash .\Sigma ʻP=\dot{s} ʻCʻP\unicode{x228d} F^{;}P \quad[(*162·01)]\)</p> + +<p class="nind"><b>*162·11.</b> \(\vdash \colon\ldotp x(\Sigma ʻP)y.\equiv :x(\dot{s} ʻCʻP)y.\lor.x(F^{;}P)y \quad[*162·1]\)</p> + +<p class="nind"><b>*162·12.</b> \[\begin{align}&\vdash \colon\ldotp x(\Sigma ʻP)y.\equiv :(\exists Q).Q\,\unicode{x2abd}\, CʻP.xQy.\lor.(\exists Q,R).xFQ.yFR.QPR\\ +&[*162·1.*41·11.*150·11]\end{align}\]</p> + +<p class="nind"><b>*162·13.</b> \[\begin{align}\vdash \colon\ldotp x(\Sigma ʻP)&y.\equiv :(\exists Q).Q\in CʻP.xQy.\\ +&\lor.(\exists Q,R).x\in CʻQ.y\in CʻR.QPR \quad[*161·12.*33·51]\end{align}\]</p> + +<p class="nind"><b>*162·14.</b> \[\begin{align}&\vdash \colon\ldotp x(\Sigma ʻP)y.=:(\exists Q).QFP.xQy.\lor.(\exists Q,R).xFQ.yFR.QPR\\ +&[*161·12.*33·51]\end{align}\]</p> + +<p class="nind"><b>*162·2.</b> \(\vdash .\text{Cnv}ʻ\Sigma ʻP=\Sigma ʻ\text{Cnv}^{;}\breve{P}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*162·13.\supset \vdash \colon\ldotp x(\Sigma ʻ\text{Cnv}^{;}\breve{P} )y.&\equiv :(\exists Q).Q\in Cʻ\text{Cnv}^{;}\breve{P} .xQy .\\ +&\lor.(\exists Q,R).Q(\text{Cnv}^{;}\breve{P} )R.x\in CʻQ.y\in CʻR:\\ +[*150·22·41] &\equiv :(\exists Q).Q\in \text{Cnv}ʻʻCʻP.xQy.\\ +&\lor.(\exists Q,R).\breve{Q} \breve{P} \breve{R} .x\in CʻQ.y\in CʻR:\\ +[*37·64.*33·22] &\equiv :(\exists Q).Q\in CʻP.yQx.\lor.(\exists Q,R).RPQ.x\in CʻQ.y\in CʻR:\\ +[*162·13] &\equiv :y(\Sigma ʻP)x\colon\ldotp \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*162·21.</b> \(\vdash .\text{D}ʻ\Sigma ʻP=sʻ\text{D}ʻʻCʻP\cup sʻCʻʻ\text{D}ʻ(P\upharpoonright -{℩}ʻ\dot{\Lambda} )\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*162·13.\supset \vdash \colon\ldotp x\in \text{D}ʻ\Sigma ʻP.&\equiv :(\exists Q,y).Q\in CʻP.xQy.\\ +&\lor.(\exists Q,R,y).QPR.x\in CʻQ.y\in CʻR:\\ +[*33·13·24] &\equiv :(\exists Q).Q\in CʻP.x\in \text{D}ʻQ.\lor.(\exists Q,R).QPR.x\in CʻQ.\dot{\exists} !R:\\ +[*40·4.*35·101] &\equiv :x\in sʻ\text{D}ʻʻCʻP.\lor.x\in sʻCʻʻ\text{D}ʻ(P\upharpoonright -{℩}ʻ\dot{\Lambda} )\colon\ldotp \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*162·211.</b> \(\vdash .\text{ᗡ}ʻ\Sigma ʻP=sʻ\text{ᗡ}ʻʻCʻP\cup sʻCʻʻ\text{ᗡ}ʻ(-{℩}ʻ\dot{\Lambda} )\upharpoonleft P\)</p> + +<p class="nind"><b>*162·212.</b> \(\vdash :\dot{\Lambda} {\sim}\in \text{ᗡ}ʻP.\supset .\text{D}ʻ\Sigma ʻP=sʻCʻʻ\text{D}ʻP\cup sʻ\text{D}ʻʻ\overrightarrow{B}ʻ\breve{P} \)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*162·21.\supset \vdash :\text{Hp}.\supset .\text{D}ʻ\Sigma ʻP&=sʻ\text{D}ʻʻCʻP\cup sʻCʻʻ\text{D}ʻP\\ +[*40·31.*93·12] &=sʻ\text{D}ʻʻ\text{D}ʻP\cup sʻ\text{D}ʻʻ\overrightarrow{B}ʻ\breve{P} \cup sʻCʻʻ\text{D}ʻP\\ +[*40·57] &=sʻ\text{D}ʻʻ\overrightarrow{B}ʻ\breve{P} \cup sʻCʻʻ\text{D}ʻP:\supset \vdash .\text{Prop}\ +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_364">[Pg 364]</span></p> + +<p class="nind"><b>*162·213.</b> \(\vdash :\dot{\Lambda} {\sim} \in \text{D}ʻP.\supset .\text{ᗡ}ʻ\Sigma ʻP = sʻCʻʻ\text{ᗡ}ʻP\cup sʻ\text{ᗡ}ʻʻ\overrightarrow{B}ʻP\)</p> + +<p>The above proposition is used in <a href="#*163·22">*163·22</a>.</p> + +<p>The two following propositions are used very often.</p> + +<p class="nind"><b><a id="*162·22">*162·22</a>.</b> \(\vdash .Cʻ\Sigma ʻP = sʻCʻʻCʻP\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*162·21·211.*40·57.\supset \\ +\vdash .Cʻ\Sigma ʻP &= sʻCʻʻCʻP\cup sʻCʻʻ\text{D}ʻ(P\upharpoonright -\iota ʻ\dot{\Lambda} )\cup sʻCʻʻ\text{ᗡ}ʻ(-\iota ʻ\dot{\Lambda} )\upharpoonleft P\\ +[*40·161] &= sʻCʻʻCʻP.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*162·23.</b> \(\vdash .Cʻ\Sigma ʻP = Cʻ\dot{s} ʻCʻP = FʻʻCʻP = \overrightarrow{F}^{2}ʻP \quad[*162·22.*42·2]\)</p> + +<p class="nind"><b>*162·26.</b> \(\vdash .\Sigma ʻ(P\unicode{x228d} Q) = \Sigma ʻP\unicode{x228d} \Sigma ʻQ\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*162·1.\supset \vdash .\Sigma ʻ(P\unicode{x228d} Q) &= \dot{s} ʻCʻ(P\unicode{x228d} Q)\unicode{x228d} F^{;}(P\unicode{x228d} Q)\\ +[*33·262.*41·171.*150·3] &= \dot{s} ʻCʻP\unicode{x228d} \dot{s} ʻCʻQ\unicode{x228d} F^{;}P\unicode{x228d} F^{;}Q\\ +[*162·1] & = \Sigma ʻP\unicode{x228d} \Sigma ʻQ.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*162·27.</b> \(\vdash .\Sigma ʻS^{;}(P\unicode{x228d} Q) = \Sigma ʻS^{;}P\unicode{x228d} \Sigma ʻS^{;}Q \quad[*162·26.*150·3]\)</p> + +<p class="nind"><b>*162·3.</b> \(\vdash .\Sigma ʻ(Q\downarrow R) = Q\unicode{x2909}R\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*160·1.\supset \vdash .\Sigma ʻ(Q\downarrow R) &= \dot{s} ʻCʻ(Q\downarrow R)\unicode{x228d} F^{;}(Q\downarrow R)\\ +[*55·15.*150·7] & = \dot{s} ʻ(\iota ʻQ\cup \iota ʻR)\unicode{x228d} \overrightarrow{F}ʻQ\uparrow \overrightarrow{F}ʻR\\ +[*53·13.*33·5] & = Q\unicode{x228d} R\unicode{x228d} CʻQ\uparrow CʻR\\ +[*160·1] &= Q\unicode{x2909}R.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p>This proposition establishes the connection between the two kinds of +arithmetical addition of relations.</p> + +<p class="nind"><b>*162·31.</b> \(\vdash .\Sigma ʻQ\unicode{x2909}\Sigma ʻR = \Sigma ʻ(Q\unicode{x2909}R)\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*160.1.\supset \vdash .\Sigma ʻQ\unicode{x2909}\Sigma ʻR &= \Sigma ʻQ\unicode{x228d} \Sigma ʻR\unicode{x228d} Cʻ\Sigma ʻQ\uparrow Cʻ\Sigma ʻR\\ +[*162·123] & = \dot{s} ʻCʻQ\unicode{x228d} F^{;}Q\unicode{x228d} \dot{s} ʻCʻR\unicode{x228d} F^{;}R\unicode{x228d} (FʻʻCʻQ)\uparrow (FʻʻCʻR)\\ +[*150·73] &= \dot{s} ʻCʻQ\unicode{x228d} \dot{s} ʻCʻR\unicode{x228d} F^{;}Q\unicode{x228d} F^{;}R\unicode{x228d} F^{;}(CʻQ\uparrow CʻR)\\ +[*41.171.*160·14.*150·3.*160·1] &= \dot{s} ʻCʻ(Q\unicode{x2909}R)\unicode{x228d} F^{;}(Q\unicode{x2909}R)\\ +[*162·1] & = \Sigma ʻ(Q\unicode{x2909}R)\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p>The following propositions lead up to <a href="#*162·34">*162·34</a>.</p> + +<p class="nind"><b>*162·32.</b> \(\vdash .\Sigma ʻ\dot{s} ʻ\kappa = \dot{s} ʻ\Sigma ʻʻ\kappa\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*41·6.*162·1.*150·1.\supset \vdash .\dot{s} ʻ\Sigma ʻʻ\kappa & = \dot{s} ʻ\dot{s} ʻʻCʻʻ\kappa \unicode{x228d} \dot{s} ʻF\daggerʻʻ\kappa \\ +[*42·12.*150·16] & = \dot{s} ʻsʻCʻʻ\kappa \unicode{x228d} F^{;}\dot{s} ʻ\kappa \\ +[*41·45]& = \dot{s} ʻCʻ\dot{s} ʻ\kappa \unicode{x228d} F^{;}\dot{s} ʻ\kappa \\ +[*162·1] &= \Sigma ʻ\dot{s} ʻ\kappa .\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_365">[Pg 365]</span></p> + +<p class="nind"><b>*162·33.</b> \(\vdash .\Sigma ʻ\Sigma ʻP = \dot{s} ʻCʻ\dot{s} ʻCʻP\unicode{x228d} F^{;}\dot{s} ʻCʻP\unicode{x228d} F^{2;}P\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*162·1.\supset \vdash .\Sigma ʻ\Sigma ʻP&=\dot{s} ʻCʻ\Sigma ʻP\unicode{x228d} F^{;}\Sigma ʻP\\ +[*162·23] &= \dot{s} ʻCʻ\dot{s} ʻCʻP\unicode{x228d} F^{;}(\dot{s} ʻCʻP\unicode{x228d} F^{;}P)\\ +[*150·3·13] &= \dot{s} ʻCʻ\dot{s} ʻCʻP\unicode{x228d} F^{;}\dot{s} ʻCʻP\unicode{x228d} F^{2;}P.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*162·331.</b> \(\vdash .F\mid \Sigma =F\mid \dot{s} \mid C=F^{2}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*71·7.\supset \vdash :x(F\mid \Sigma )P.&\equiv .xF(\Sigma ʻP).\\ +[*33.51] &\equiv .x\in Cʻ\Sigma ʻP.\\ +[*162·23] &\equiv .xF^{2}P &\qquad \text{(1)}\\ +\vdash .*71·7.\supset \vdash :x(F\mid \dot{s} \mid C)P.&\equiv .xF(\dot{s} ʻCʻP).\\ +[*33·51] &\equiv .x\in Cʻ\dot{s} ʻCʻP.\\ +[*42·2] &\equiv .xF^{2}P &\qquad \text{(2)}\\ +\vdash .(1).(2).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*162·332.</b> \(\vdash .\Sigma ʻ\Sigma ^{;}P= \dot{s} ʻCʻ\dot{s} ʻP\unicode{x228d} F^{;}\dot{s} ʻCʻP\unicode{x228d} F^{2;}P\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*162·1.\supset \vdash .\Sigma ʻ\Sigma ^{;}P&=\dot{s} ʻCʻ\Sigma ^{;}P\unicode{x228d} F^{;}\Sigma ^{;}P\\ +[*150·22·13] &= \dot{s} ʻ\Sigma ʻʻCʻP\unicode{x228d} (F\mid \Sigma )^{;}P\\ +[*162·32·331] &= \Sigma ʻ\dot{s} ʻCʻP\unicode{x228d} F^{2;}P\\ +[*162·1] &= \dot{s} ʻCʻ\dot{s} ʻCʻP\unicode{x228d} F^{;}\dot{s} ʻCʻP\unicode{x228d} F^{2;}P.\supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*162·34">*162·34</a>.</b> \(\vdash .\Sigma ʻ\Sigma ^{;}=\Sigma ʻ\Sigma ʻP \quad[*162·33·332]\)</p> + +<p>This is the associative law for arithmetical sums of relations.</p> + +<p>The following propositions lead up to <a href="#*162·35">*162·35</a>.</p> + +<p class="nind"><b>*162·341.</b> \(\vdash \colon\ldotp CʻQ\subset \text{ᗡ}ʻR.\supset :x(F\mid R\dagger )Q.\equiv .x(R\mid F)Q\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *71·7.*150·1.\supset \vdash :x(F\mid R\dagger )Q.&\equiv .xF(R^{;}Q).\\ +[*33·51] &\equiv .x\in CʻR^{;}Q &\qquad \text{(1)}\\ +\vdash .(1).*150·22. \supset \vdash \colon\ldotp \text{Hp}.\supset :x(F\mid R\dagger )Q.&\equiv .x\in RʻʻCʻQ.\\ +[*33·5] &\equiv .x\in Rʻʻ\overrightarrow{F}ʻQ.\\ +[*37·3.*32·18] &\equiv .x(R\mid F)Q\colon\ldotp \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*162·342.</b> \(\vdash :Cʻ\dot{s} ʻ\lambda \subset \text{ᗡ}ʻR.\supset .(F\mid R\dagger )\upharpoonright \lambda =(R\mid F)\upharpoonright \lambda\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*41·13.\supset \vdash \colon\ldotp \text{Hp}.&\supset :Q\in \lambda .\supset CʻQ\subset \text{ᗡ}ʻR:\\ +[*162·341] &\supset :Q\in \lambda .x(F\mid R\dagger )Q.\equiv .Q\in \lambda .x(R\mid F)Q\colon\ldotp \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*162·343.</b> \(\vdash : Cʻ\Sigma ʻP\subset \text{ᗡ}ʻR.\supset .F^{;}R\dagger ^{;}P=R^{;}F^{;}P\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*162·23.\supset \vdash :\text{Hp}.&\supset .Cʻ\dot{s} ʻCP\subset \text{ᗡ}ʻR.\\ +[*162·342]&\supset .(F\mid R\dagger )\upharpoonright (CʻP)^{;}P=(R\mid F)\upharpoonright (CʻP)^{;}P.\\ +[*150·32] &\supset .(F\mid R\dagger )^{;}P=(R\mid F)^{;}P.\\ +[*150·13]&\supset .F^{;}R\dagger ^{;}P=R^{;}F^{;}P:\supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_366">[Pg 366]</span></p> + +<p class="nind"><b><a id="*162·35">*162·35</a>.</b> \(\vdash :Cʻ\Sigma ʻQ\subset \text{ᗡ}ʻR.\supset .\Sigma ʻR\dagger ^{;}Q=R^{;}\Sigma ʻQ\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*162·1.*150·22. \supset \vdash .\Sigma ʻR\dagger ^{;}Q&=\dot{s} ʻR\dagger ʻʻCʻQ\unicode{x228d} F^{;}R\dagger ^{;}Q\\ +[*150·16] &=R^{;}\dot{s} ʻCʻQ\unicode{x228d} F^{;}R\dagger ^{;}Q &\qquad \text{(1)}\\ +\vdash .(1).*162·343. \supset \vdash :\text{Hp}.\supset .\Sigma ʻR\dagger ^{;}Q&=R^{;}\dot{s} ʻCʻQ\unicode{x228d} R^{;}F^{;}Q\\ +[*150·3.*162·1] & =R^{;}\Sigma ʻQ:\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p>This proposition is important, since it enables us to infer (with a +suitable hypothesis) that if \(R^{;}M\) is always like \(M\) when \(M\in CʻQ\), +then the arithmetical sum of all such relations as \(R^{;}M\) is +like \(\Sigma ʻQ\), being in fact \(R^{;}\Sigma ʻQ\). In other words, +if, whenever \(M\in CʻQ\), \(R\upharpoonright CʻM\) is a correlator of +\(R^{;}M\) and \(M\), then \(R\upharpoonright \Sigma ʻQ\) is a correlator +if \(\Sigma ʻR\dagger ^{;}Q\) and \(\Sigma ʻQ\). This proposition is +analogous in its uses to the proposition +\[ +sʻR_{\in}ʻʻ\kappa =Rʻʻsʻ\kappa , +\] +which is *40·38. In general, in obtaining relational analogues of +cardinal propositions, \(Rʻʻ\kappa\) is to be replaced by \(R^{;}Q\), +\(R_{\in }\) by \(R\dagger\), and \(s\) by \(\Sigma\). When these +substitutions are made in \(sʻR_{\in}ʻʻ\kappa =Rʻʻsʻ\kappa\), <a href="#*162·35">*162·35</a> +results, except for its hypothesis.</p> + +<p>If we regard \(R^{;}Q\) as a kind of product of \(R\) and \(Q\), *162·35 becomes +a distributive law. For it asserts that if we multiply each member of \(CʻQ\) +by \(R\), and then sum the resulting products, we get the same relation as +if we first sum \(CʻQ\), and then multiply by \(R\). The following application +of *162·35 to the sum of two relations makes its distributive character +more evident.</p> + +<p class="nind"><b>*162·36.</b> \(\vdash :CʻP\cup CʻQ\subset \text{ᗡ}ʻR.\supset .R^{;}P\unicode{x2909}R^{;}Q=R^{;}(P\unicode{x2909}Q)\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*162·3. \supset \vdash .R^{;}P\unicode{x2909}R^{;}Q&=\Sigma ʻ\{(R^{;}P)\downarrow (R^{;}Q)\}\\ +[*150·1·71] &=\Sigma ʻR\dagger ^{;}(P\downarrow Q) &\qquad \text{(1)}\\ +\vdash .(1).*162·35. \supset \vdash :\text{Hp}.\supset .R^{;}P\unicode{x2909}R^{;}Q&=R^{;}\Sigma ʻ(P\downarrow Q)\\ +[*162·3] &=R^{;}(P\unicode{x2909}Q):\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p>This proposition can be extended to any finite number of summands.</p> + +<p class="nind"><b>*162·37.</b> \(\vdash :\exists !\lambda .\exists !\mu .\supset .\Sigma ʻ(\lambda \uparrow \mu )=\dot{s} ʻ\lambda \unicode{x2909}\dot{s} ʻ\mu\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*35·85·86. \supset \vdash :\text{Hp}.\supset .Cʻ(\lambda \uparrow \mu )&=\lambda \cup \mu .\\ +[*162·1] \supset .\Sigma ʻ(\lambda \uparrow \mu )&=\dot{s} ʻ(\lambda \cup \mu )\unicode{x228d} F^{;}(\lambda \uparrow \mu )\\ +[*41·171.*150·73] &=\dot{s} ʻ\lambda \unicode{x228d} \dot{s} ʻ\mu \unicode{x228d} (Fʻʻ\lambda )\uparrow (Fʻʻ\mu )\\ +[*41·45.*40·56] &=\dot{s} ʻ\lambda \unicode{x228d} \dot{s} ʻ\mu \unicode{x228d} (Cʻ\dot{s} ʻ\lambda )\uparrow (Cʻ\dot{s} ʻ\mu )\\ +[*160·1] &=\dot{s} ʻ\lambda \unicode{x2909}\dot{s} ʻ\mu :\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*162·371.</b> \(\vdash :\exists !\alpha .\supset .\Sigma ʻ(\alpha \uparrow \iota ʻQ)=\dot{s} ʻ\alpha \unicode{x2909}Q \quad[*162·37.*53·04]\)</p> + +<p class="nind"><b>*162·372.</b> \(\vdash :\exists !\beta .\supset .\Sigma ʻ(\iota ʻP)\uparrow \beta =P\unicode{x2909}\dot{s} ʻ\beta \)</p> + +<p><span class="pagenum" id="Page_367">[Pg 367]</span></p> + +<p class="nind"><b><a id="*162·4">*162·4</a>.</b> \(\vdash . \Sigma ʻ\dot{\Lambda} = \dot{\Lambda}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *33·241.*41·21 .&\supset \vdash . \dot{s} ʻCʻ\dot{\Lambda} = \dot{\Lambda} &\qquad \text{(1)}\\ +\vdash . *150·42 . &\supset \vdash . F^{;}\dot{\Lambda} = \dot{\Lambda} &\qquad \text{(2)}\\ +\vdash . (1).(2).*162·1 .\supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*162·41.</b> \(\vdash . \Sigma ʻ(\dot{\Lambda} \downarrow \dot{\Lambda} ) = \dot{\Lambda} \)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *162·3 .\supset \vdash . \Sigma ʻ(\dot{\Lambda} \downarrow \dot{\Lambda} ) &= \dot{\Lambda} \unicode{x2909}\dot{\Lambda} \\ +[*160·21] &=\dot{\Lambda} .\supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*162·42.</b> \(\vdash : \dot{\exists} !\Sigma ʻP .\equiv . \dot{\exists} !\dot{s} ʻCʻP .\equiv . .\exists !CʻP-\iota ʻ\dot{\Lambda}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *162·23.*33·24 .\supset \vdash : \dot{\exists} !\Sigma ʻP .&\equiv . \dot{\exists} !\dot{s} ʻCʻP.\\ +[*41·26] &\equiv . \exists !CʻP-\iota ʻ\dot{\Lambda} +\end{array} +\]</p> + +<p class="nind"><b><a id="*162·43">*162·43</a>.</b> \(\vdash : \dot{\exists} !P .\supset . \Sigma ʻ(P\unicode{x21f8} R) = \Sigma ʻP\unicode{x2909}R\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *162·26.*161·1 .&\supset \vdash . \Sigma ʻ(P\unicode{x21f8} R) = \Sigma ʻP\unicode{x228d} \Sigma ʻ(CʻP\uparrow \iota ʻR) &\qquad \text{(1)}\\ +\vdash . *162·371.*33·24 .&\supset \vdash : \dot{\exists} !P .\supset . \Sigma ʻ(CʻP\uparrow \iota ʻR) = \dot{s} ʻCʻP\unicode{x2909}R &\qquad \text{(2)}\\ +\vdash . (1).(2).*160·1 .\supset \\ +\vdash : \text{Hp} .\supset . \Sigma ʻ(P\unicode{x21f8}R) &= \Sigma ʻP\unicode{x228d} \dot{s} ʻCʻP\unicode{x228d} R\unicode{x228d} (Cʻ\dot{s} ʻCʻP)\uparrow CʻR\\ +[*162·1·23] & = \Sigma ʻP\unicode{x228d} R\unicode{x228d} (Cʻ\Sigma ʻP)\uparrow CʻR\\ +[*160·1] & = \Sigma ʻP\unicode{x2909}R :\supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*162·431">*162·431</a>.</b> \(\vdash : \dot{\exists} !P .\supset . \Sigma ʻ(R\unicode{x21f7}P) = R\unicode{x2909}\Sigma ʻP \quad[\text{Proof as in *162·43}]\)</p> + +<p>Observe that in <a href="#*162·43">*162·43</a><a href="#*162·431">·431</a>, \(P\) and \(R\) must be of different +types, in fact \(R\) must be of the type to which members of \(CʻP\) +belong. *162·43·431 are often useful.</p> + +<p class="nind"><b>*162.44.</b> \(\vdash . \Sigma ʻ(P\unicode{x21f8} \dot{\Lambda} ) = \Sigma ʻ(\dot{\Lambda} \unicode{x21f7}P) = \Sigma ʻP\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *162·43 . \supset \vdash : \dot{\exists} !P .\supset . \Sigma ʻ(P\unicode{x21f8} \dot{\Lambda} ) &= \Sigma ʻP\unicode{x2909}\dot{\Lambda} +[*160·21] = \Sigma ʻP &\qquad \text{(1)}\\ +\vdash . *33·241.*35·88 .\supset \vdash : P = \dot{\Lambda} .&\supset . CʻP\uparrow \iota ʻ\dot{\Lambda} = \dot{\Lambda} .\\ +[*162·4] &\supset . \Sigma ʻ(CʻP\uparrow \iota ʻ\Lambda ) = \dot{\Lambda}\\ . +[*25·24] \supset . \Sigma ʻP &= \Sigma ʻP\unicode{x228d} \Sigma ʻ(CʻP\uparrow \iota ʻ\dot{\Lambda} )\\ +[*162·26] &= \Sigma ʻ(P\unicode{x228d} CʻP\uparrow \iota ʻ\dot{\Lambda} )\\ +[*161·1] & = \Sigma ʻ(P\unicode{x21f8} \dot{\Lambda} ) &\qquad \text{(2)}\\ +\vdash . (1).(2). .&\supset \vdash . \Sigma ʻ(P\unicode{x21f8} \dot{\Lambda} ) = \Sigma ʻP &\qquad \text{(3)}\\ +\text{Similarly}\quad &\vdash . \Sigma ʻ(\dot{\Lambda} \unicode{x21f7}P)=\Sigma ʻP &\qquad \text{(4)}\\ +\vdash . (3).(4) .\supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_368">[Pg 368]</span></p> + +<p class="nind"><b><a id="*162·45">*162·45</a>.</b> \(\vdash :\dot{\exists} !P.\Sigma ʻP=\dot{\Lambda} .\equiv .P=\dot{\Lambda} \downarrow \dot{\Lambda}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*162·42.\supset \vdash :\Sigma ʻP=\dot{\Lambda} .&\equiv .CʻP\subset {℩}ʻ\dot{\Lambda} .\\ +[*33·16] &\equiv .\text{D}ʻP\subset {℩}ʻ\dot{\Lambda} .\text{ᗡ}ʻP\subset {℩}ʻ\Lambda &\qquad \text{(1)}\\ +\vdash .*33·24. \supset \vdash :\dot{\exists} !P.&\equiv .\exists !\text{D}ʻP.\exists !\text{ᗡ}ʻP &\qquad \text{(2)}\\ +\vdash .(1).(2).*51·4.\supset \\ +\vdash :\dot{\exists} !P.\Sigma ʻP=\dot{\Lambda} .&\equiv .\text{D}ʻP={℩}ʻ\dot{\Lambda} .\text{ᗡ}ʻP={℩}ʻ\dot{\Lambda} .\\ +[*55·16] &\equiv .P=\dot{\Lambda} \downarrow \dot{\Lambda} :\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p>The above proposition is used in <a href="#*174·162">*174·162</a>.</p> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_369">[Pg 369]</span></p> +<h2 class="nobreak" id="163_RELATIONS_OF_MUTUALLY_EXCLUSIVE_RELATIONS"> +*163. RELATIONS OF MUTUALLY EXCLUSIVE RELATIONS.</h2> +</div> + + +<p class="nind"><i>Summary of</i> *163.</p> + +<p>In the present number we have to define mutually exclusive relations, +and to give a few of their properties. Mutually exclusive relations +play much the same part in relation-arithmetic as mutually exclusive +classes play in cardinal arithmetic. <i>Prima facie</i>, there are +various ways in which we might define them. We might define \(P\) as a +relation of mutually exclusive relations when +\[ +\begin{aligned} +QPR . Q &\neq R . \supset _{Q,R} . Q \dot{\cap} R = \dot{\Lambda} ,\\ +\text{or when}\quad Q,R \in CʻP . Q &\neq R . \supset _{Q,R} . Q \dot{\cap} R = \dot{\Lambda} , +\end{aligned} +\] +or when +\[ +Q,R \in CʻP . Q \neq R . \supset _{Q,R} . \text{D}ʻQ \cap \text{D}ʻR = \Lambda . \text{ᗡ}ʻQ \cap \text{ᗡ}ʻR = \Lambda , +\] +or in several other ways. But in fact the most useful property to +choose is the property that any two members of the field have mutually +exclusive fields, <i>i.e.</i> +\[ +Q,R \in CʻP . Q \neq R . \supset _{Q,R} . CʻQ \cap CʻR = \Lambda . +\]</p> + +<p>The principal applications of the subjects studied in this Part are to +series, and in series it is always the <i>fields</i> of the relations +that are important. We want, for instance, to define relations of +mutually exclusive relations in such a way that, if \(P\) is a serial +relation, and every member of \(CʻP\) is a serial relation, then +\(\Sigma ʻP\) is a serial relation. For this purpose it is necessary +that \(\Sigma ʻP\) should be contained in diversity, which requires +that \(F^{;}P\) should be contained in diversity, <i>i.e.</i> that +\[ +QPR . \supset _{Q,R} . CʻQ \cap CʻR = \Lambda . +\] +If \(P\) is a serial relation, as we are supposing, this is equivalent +to +\[ +Q,R \in CʻP . Q \neq P . \supset _{Q,R} . CʻQ \cap CʻR = \Lambda . +\]</p> + +<p>Again we want to define relations of mutually exclusive relations in +such a way that, if \(P\) and \(Q\) are two such relations, and \(P\) +and \(Q\) have double likeness (cf. <a href="#*164">*164</a>), then \(\Sigma ʻP\) is like +\(\Sigma ʻQ\); <i>i.e.</i> if we are given a correlator \(S\) of \(P\) +and \(Q\), and for every \(M\) and \(N\) which \(S\) correlates, we are +again given a correlator, then \(\Sigma ʻP\) is to be like \(\SigmaʻQ\). +That is, if \(\lambda\) is the class of relations which correlate +pairs of relations \(M\) and \(N\), where \(N\in CʻQ.MSN\), we want<span class="pagenum" id="Page_370">[Pg 370]</span> +\(\dot{s} ʻ\lambda\) to be a correlator of \(P\) and \(Q\). Now this +requires that \(\dot{s} ʻ\lambda\) should be a one-one relation, which +requires +\[ +M,M' \in CʻP . M \neq M' . \supset _{M,M'} . \text{D}ʻM \cap \text{D}ʻM' = \Lambda . \text{ᗡ}ʻM \cap \text{ᗡ}ʻM' = \Lambda . +\] +This is secured by +\[ +M,M' \in CʻP . M \neq M' . \supset _{M,M'} . CʻM \cap CʻM' = \Lambda , +\] +but except for special classes of relations it is not secured by +\[ +MPM' . \supset _{M,M'} . CʻM \cap CʻM' = \Lambda , +\] +since there may be two relations \(M\) and \(M'\) which both belong to +the field of \(P\), but of which neither has the relation \(P\) to the +other. Again, the analogy with cardinal arithmetic fails at many points +unless, when \(P\) is a relation of mutually exclusive relations, +\(CʻʻCʻP\) is a class of mutually exclusive classes. But this is +not secured by any of the other possible definitions we have been +considering. There are further reasons, connected with the arithmetical +product of a relation of relations, for choosing as the definition +\[ +Q,R \in CʻP . Q \neq R . \supset _{Q,R} . CʻQ \cap CʻR = \Lambda . +\]</p> + +<p>From a technical point of view, the properties of a +\(\text{Cls}^{2} \text{excl}\) depend mainly upon the fact that +when \(\kappa\) is such a class, \(\in \upharpoonright \kappa \in +\text{Cls}\rightarrow 1\) (*84·14); in like manner the properties of a +\(\text{Rel}^{2}\text{excl}\) depend upon +\[ +F\upharpoonright CʻP \in \text{Cls} \rightarrow 1 , +\] +which requires our definition, and is equivalent to it (<a href="#*163·12">*163·12</a>). We +thus become able to use the propositions of *81 on selections from +many-one relations, which would not otherwise be the case.</p> + +<p>It should be observed that +\[ +Q,R \in CʻP . Q \neq R . \supset _{Q,R} . CʻQ \cap CʻR = \Lambda +\] +is not equivalent to +\[ +CʻʻCʻP \in \text{Cls}^{2} \text{excl} , +\] +though it implies this. The converse implication will fail if \(CʻP\) +contains two different relations with the same field. <i>E.g.</i> +take a relation \(P\) whose field consists of the four relations +\(S\), \(\breve{S}\), \(T\), \(\breve{T}\), and suppose \(CʻS\cap CʻT=\Lambda\). +Then \(CʻʻCʻP=\iota ʻCʻS\cup \iota ʻCʻT\), and +\(CʻʻCʻP\in \text{Cls}^{2} \text{excl}\). But unless \(S = \breve{S}\) +and \(T=\breve{T}\) we shall not have +\[ +Q,R \in CʻP . Q \neq R . \supset _{Q,R} . CʻQ \cap CʻR = \Lambda . +\]</p> + +<p>The property by which we define relations of mutually exclusive +relations is a property which only depends on the field, so that we +might equally well put +\[ +(\text{Cl}ʻ\text{Rel})\text{excl} = \hat{\lambda} \{Q,R \in \lambda . Q \neq R . \supset _{Q,R} . CʻQ \cap CʻR = \Lambda \} \quad\text{Df}. +\] +<span class="pagenum" id="Page_371">[Pg 371]</span>But for our purposes this would be less convenient than the definition +of \(\text{Rel}^{2}\text{excl}\).</p> + +<p>We thus put</p> + +<p class="nind"><b>*163·01.</b> \(\text{Rel}^{2}\text{excl}=\hat{P} \{Q,R\in CʻP.Q\neq R.\supset _{Q,R}.CʻQ\cap CʻR=\Lambda\} \quad\text{Df}\)</p> + +<p>We have</p> + +<p class="nind"><b>*163·11.</b> \(\vdash \colon\ldotp P\in \text{Rel}^{2}\text{excl}.\equiv :Q,R\in CʻP.\exists !CʻQ\cap CʻR.\supset _{Q,R}.Q=R\)</p> + +<p class="nind"><b>*163·12.</b> \(\vdash :P\in \text{Rel}^{2}\text{excl}.\equiv .F\upharpoonright CʻP\in \text{Cls}\rightarrow 1\)</p> + +<p class="nind"><b>*163·17.</b> \(\vdash :P\in \text{Rel}^{2}\text{excl}.\equiv .C\upharpoonright CʻP\in 1\rightarrow 1.CʻʻCʻP\in \text{Cls}^{2} \text{excl}\)</p> + +<p>Any of the above might have been used to define +\(\text{Rel}^{2}\text{excl}\). The following propositions are important.</p> + +<p class="nind"><b>*163·3.</b> \(\vdash :Q\in \text{Rel}^{2}\text{excl}.S\in \text{Cls}\rightarrow 1.\supset .S\dagger ^{;}Q\in \text{Rel}^{2}\text{excl}\)</p> + +<p>This is the analogue of *84·53.</p> + +<p class="nind"><b>*163·4·41.</b> \(\vdash .\dot{\Lambda} ,P\downarrow P\in \text{Rel}^{2}\text{excl}\)</p> + +<p class="nind"><b>*163·441.</b> \(\vdash :P,Q\in \text{Rel}^{2}\text{excl}.Cʻ\Sigma ʻP\cap Cʻ\Sigma ʻQ=\Lambda .\supset .P\unicode{x2909}Q\in \text{Rel}^{2}\text{excl}\)</p> + +<p class="nind"><b>*163·451.</b> \(\vdash :P\in \text{Rel}^{2}\text{excl}.Cʻ\Sigma ʻP\cap CʻR=\Lambda .\supset .P\unicode{x21f8} R\in \text{Rel}^{2}\text{excl}\)</p> + +<hr class="tb"> + +<p class="nind"><b>*163·01.</b> \(\text{Rel}^{2}\text{excl}=\hat{P} \{Q,R\in CʻP.Q\neq R.\supset _{Q,R}.CʻQ\cap CʻR=\Lambda\} \quad\text{Df}\)</p> + +<p class="nind"><b>*163·1.</b> \[\begin{align}&\vdash \colon\ldotp P\in \text{Rel}^{2}\text{excl}.\equiv :Q,R\in CʻP.Q\neq R.\supset _{Q,R}.CʻQ\cap CʻR=\Lambda \\ +&[(*163·01)]\end{align}\]</p> + +<p class="nind"><b>*163·11.</b> \[\begin{align}&\vdash \colon\ldotp P\in \text{Rel}^{2}\text{excl}.\equiv :Q,R\in CʻP.\exists !CʻQ\cap CʻR.\supset _{Q,R}.Q=R\\ +&[*163·1.\text{Transp}]\end{align}\]</p> + +<p class="nind"><b><a id="*163·12">*163·12</a>.</b> \(\vdash :P\in \text{Rel}^{2}\text{excl}.\equiv .F\upharpoonright CʻP\in \text{Cls}\rightarrow 1 \quad[*163·1.*74·632]\)</p> + +<p>For many purposes, this proposition gives the most useful equivalent of +\(P\in \text{Rel}^{2}\text{excl}\).</p> + +<p>Instead of the above proof, we may use *74·62, which gives us the +result in virtue of *33·5.</p> + +<p class="nind"><b>*163·13.</b> \[\begin{align}\vdash \colon\ldotp P&\in \text{Rel}^{2}\text{excl}.\supset :\\ +&Q,R\in CʻP.Q\neq R.\supset _{Q,R}.\text{D}ʻQ\cap \text{D}ʻR=\Lambda .\text{ᗡ}ʻQ\cap \text{ᗡ}ʻR=\Lambda \\ +&[*24·402 .*163·1]\end{align}\]</p> + +<p class="nind"><b>*163·14.</b> \(\vdash :P\in \text{Rel}^{2}\text{excl}.\supset .C\upharpoonright CʻP\in 1\rightarrow 1 \quad[*163·12.*74·32.*33·5]\)</p> + +<p class="nind"><b>*163·15.</b> \(\vdash :P\in \text{Rel}^{2}\text{excl}.\supset .\text{D}\upharpoonright CʻP,\text{ᗡ}\upharpoonright CʻP\in 1\rightarrow 1\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*74·63.*163·13.\supset \vdash :\text{Hp}.&\supset .(\in \mid \text{D})\upharpoonright CʻP\in \text{Cls}\rightarrow 1.\\ +[*74·32] &\supset .\overrightarrow{\in \mid \text{D}} \upharpoonright CʻP\in 1\rightarrow 1.\\ +[*72·27] &\supset .\text{D}\upharpoonright CʻP\in 1\rightarrow 1 &\qquad \text{(1)}\\ +\text{Similarly}\quad \vdash :\text{Hp}.&\supset .\text{ᗡ}\upharpoonright CʻP\in 1\rightarrow 1 &\qquad \text{(2)}\\ +\vdash .(1).(2).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_372">[Pg 372]</span></p> + +<p class="nind"><b>*163·16.</b> \(\vdash :P\in \text{Rel}^{2}\text{excl}.\supset .CʻʻCʻP\in \text{Cls}^{2} \text{excl} \quad[*84·51.*33·5.*163·12]\)</p> + +<p class="nind"><b>*163·17.</b> \[\begin{align}&\vdash :P\in \text{Rel}^{2}\text{excl}.\equiv .C\upharpoonright CʻP\in 1\rightarrow 1.CʻʻCʻP\in \text{Cls}^{2} \text{excl}\\ +&[*163·12.*84·522.*33·5]\end{align}\]</p> + +<p class="nind"><b>*163·2.</b> \[\begin{align}&\vdash :P\in \text{Rel}^{2}\text{excl}.\supset .\text{D}\upharpoonright F_{\Delta }ʻCʻP\in 1\rightarrow 1.F_{\Delta }ʻCʻP\subset + 1\rightarrow 1\\ +&[*81·21·1.*163·12]\end{align}\]</p> + +<p class="nind"><b>*163·21.</b> \(\vdash :P\in \text{Rel}^{2}\text{excl}.\supset .\text{D}ʻʻF_{\Delta }ʻCʻP=\text{Prod}ʻCʻʻCʻP\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*85·1.\frac{F,\,CʻP}{Q,\,\lambda}.*163·12.\supset \vdash :\text{Hp}.\supset .\text{D}ʻʻF_{\Delta }ʻCʻP&=\text{D}ʻʻ{\in}_{\Delta}ʻ\overrightarrow{F}ʻʻCʻP\\ +[*115·1.*33·5] & =\text{Prod}ʻCʻʻCʻP:\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p>This proposition is important in connection with the multiplication +of relations, for we shall define as the product of a relation +\(P\) (whose field consists of relations) a relation whose field is +\(\text{D}ʻʻF_{\Delta }ʻCʻP\). Thus by the above proposition, whenever +\(P\) is a \(\text{Rel}^{2}\text{excl}\), the field of its product is the +product (in the cardinal sense) of the fields of its field, just as the +field of its sum is (by <a href="#*162·22">*162·22</a>) the sum of the fields of its field.</p> + +<p class="nind"><b><a id="*163·22">*163·22</a>.</b> \[\begin{align}\vdash :P\in \text{Rel}^{2}\text{excl}.\dot{\Lambda} &{\sim}\in CʻP.\supset .\\ +&\overrightarrow{B}ʻ\Sigma ʻP=Bʻʻ\overrightarrow{B}ʻP.\overrightarrow{B}ʻ\text{Cnv}ʻ\Sigma ʻP=Bʻʻ\text{Cnv}ʻʻ\overrightarrow{B}ʻ\breve{P}\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*162·23·213.*93·103.\supset \vdash :\text{Hp}.\supset .\overrightarrow{B}ʻ\Sigma ʻP&=FʻʻCʻP-sʻCʻʻ\text{ᗡ}ʻP-sʻ\text{ᗡ}ʻʻ\overrightarrow{B}ʻP\\ +[*40·56] &=FʻʻCʻP-Fʻʻ\text{ᗡ}ʻP-sʻ\text{ᗡ}ʻʻ\overrightarrow{B}ʻP\\ +[*71·381.*37·421.*163·12]&=Fʻʻ(CʻP-\text{ᗡ}ʻP)-sʻ\text{ᗡ}ʻʻ\overrightarrow{B}ʻP\\ +[*40·56.*93·103] &=sʻCʻʻ\overrightarrow{B}ʻP-sʻ\text{ᗡ}ʻʻ\overrightarrow{B}ʻP &\qquad \text{(1)}\\ +\vdash .*163·11.\supset \vdash \colon\colon \text{Hp}.\supset \colon\ldotp Q\in \overrightarrow{B}ʻP.x\in CʻQ.&\supset :R\in \overrightarrow{B}ʻP.x\in \text{ᗡ}ʻR.\supset .R=Q.\\ +[*13·12] &\supset .x\in \text{ᗡ}ʻQ:\\ +[*40·4] &\supset :x\in sʻ\text{ᗡ}ʻʻ\overrightarrow{B}ʻP.\supset .x\in \text{ᗡ}ʻQ:\\ +[*40·13] &\supset :x\in sʻ\text{ᗡ}ʻʻ\overrightarrow{B}ʻP.\equiv .x\in \text{ᗡ}ʻQ &\qquad \text{(2)}\\ +\vdash .(2).*5·32.\supset \vdash \colon\ldotp \text{Hp}.\supset :Q\in \overrightarrow{B}ʻP.x&\in CʻQ.x{\sim}\in sʻ\text{ᗡ}ʻʻ\overrightarrow{B}ʻP.\equiv .\\ +&Q\in \overrightarrow{B}ʻP.x\in CʻQ.x{\sim}\in \text{ᗡ}ʻQ:\\ +[*10·281.*40·4.*93·103] \supset :x\in sʻCʻʻ\overrightarrow{B}ʻP-sʻ\text{ᗡ}ʻʻ\overrightarrow{B}ʻP.&\equiv .(\exists Q).Q\in \overrightarrow{B}ʻP.xBQ.\\ +[*37·1] &\equiv .x\in Bʻʻ\overrightarrow{B}ʻP &\qquad \text{(3)}\\ +\vdash .(1).(3).&\supset \vdash :\text{Hp}.\supset .\overrightarrow{B}ʻ\Sigma ʻP=Bʻʻ\overrightarrow{B}ʻP &\qquad \text{(4)}\\ +\vdash .(4).*162·2.*33·22.*163·1.&\supset \vdash :\text{Hp}.\supset .\overrightarrow{B}ʻ\text{Cnv}\\ʻ\Sigma ʻP&=Bʻʻ\overrightarrow{B}ʻ\text{Cnv}^{;}\breve{P} \\ +[*151·6·5] & =Bʻʻ\text{Cnv}ʻʻ\overrightarrow{B}ʻ\breve{P} &\qquad \text{(5)}\\ +\vdash .(4).(5).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_373">[Pg 373]</span></p> + +<p class="nind"><b>*163·3.</b> \(\vdash :Q\in \text{Rel}^{2}\text{excl}.S\in \text{Cls}\rightarrow 1.\supset .S\dagger ^{;}Q\in \text{Rel}^{2}\text{excl}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*72·421.&\supset \vdash \colon\ldotp \text{Hp}.\supset :M,N\in CʻQ.\exists !SʻʻCʻM\cap SʻʻCʻN.\supset .\exists !CʻM\cap CʻN.\\ +[*163·11] &\supset .M=N.\\ +[*30·37] &\supset .S\dagger M=S\dagger N &\qquad \text{(1)}\\ +\vdash .(1).*150·202.&\supset \vdash \colon\ldotp \text{Hp}.\supset :\\ +&M,N\in CʻQ.\exists !Cʻ(S\dagger M)\cap Cʻ(S\dagger N).\supset .S\dagger M=S\dagger N &\qquad \text{(2)}\\ +\vdash .(2).*163·11.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*163·31.</b> \(\vdash \colon\ldotp CʻP=CʻQ.\supset :P\in \text{Rel}^{2}\text{excl}.\equiv .Q\in \text{Rel}^{2}\text{excl} \quad[*163·1.*13·12]\)</p> + +<p class="nind"><b>*163·311.</b> \(\vdash \colon\ldotp CʻQ=\text{Cnv}ʻʻCʻP.\supset :P\in \text{Rel}^{2}\text{excl}.\equiv .Q\in \text{Rel}^{2}\text{excl}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*72·513.\supset \vdash \colon\colon \text{Hp}.&\supset \colon\ldotp M,N\in CʻP.\equiv .\breve{M} ,\breve{N} \in CʻQ\colon\ldotp \\ +[*31·32] &\supset \colon\ldotp M,N\in CʻP.M\neq N.\equiv .\breve{M} ,\breve{N} \in CʻQ.\breve{M} \neq \breve{N} \colon\ldotp \\ +[*33·22] &\supset \colon\ldotp M,N\in CʻP.M\neq N.\supset .CʻM\cap CʻN=\Lambda :\equiv :\\ +&\breve{M} ,\breve{N} \in CʻQ.\breve{M} \neq \breve{N} .\supset .Cʻ\breve{M} \cap Cʻ\breve{N} =\Lambda \colon\ldotp \\ +[*11·33.*163·1]&\supset \colon\ldotp P\in \text{Rel}^{2}\text{excl}.\equiv :\\ +&\breve{M} ,\breve{N} \in CʻQ.\breve{M} \neq \breve{N} .\supset _{M,N}.Cʻ\breve{M} \cap Cʻ\breve{N} =\Lambda :\\ +[*31·51] &\equiv :M,N\in CʻQ.M\neq N.\supset _{M,N}.CʻM\cap CʻN=\Lambda :\\ +[*163·1] &\equiv :Q\in \text{Rel}^{2}\text{excl}\colon\colon \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*163·32.</b> \[\begin{align}\vdash :P\in \text{Rel}^{2}\text{excl}.&\equiv .\breve{P} \in \text{Rel}^{2}\text{excl}.\equiv .\text{Cnv}^{;}P\in \text{Rel}^{2}\text{excl}.\equiv + .\\ +&\text{Cnv}^{;}\breve{P} \in \text{Rel}^{2}\text{excl}\quad[*163·31·311.*33·22.*150·22·12]\end{align}\]</p> + +<p class="nind"><b>*163·33.</b> \(\vdash :P\unicode{x2909}Q\in \text{Rel}^{2}\text{excl}.\equiv .Q\unicode{x2909}P\in \text{Rel}^{2}\text{excl} \quad[*163·31.*160·14]\)</p> + +<p class="nind"><b>*163·331.</b> \[\begin{align}&\vdash :P\unicode{x21f8} R\in \text{Rel}^{2}\text{excl}.\equiv .R\unicode{x21f7}P\in \text{Rel}^{2}\text{excl}\\ +&[*163·31.*161·14·2·201]\end{align}\]</p> + +<p class="nind"><b>*163·4.</b> \(\vdash .\dot{\Lambda} \in \text{Rel}^{2}\text{excl}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*33·241.*24·105.&\supset \vdash .(Q).Q{\sim}\in Cʻ\dot{\Lambda} .\\ +[*11·57] &\supset \vdash .(Q,R).Q,R{\sim}\in Cʻ\dot{\Lambda} .\\ +[*11·63] &\supset \vdash :Q,R\in Cʻ\dot{\Lambda} .Q\neq R.\supset _{Q,R}.CʻQ\cap CʻR=\Lambda &\qquad \text{(1)}\\ +\vdash .(1).*163·1.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*163·41.</b> \(\vdash .P\downarrow P\in \text{Rel}^{2}\text{excl}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*54·25.*55·15.&\supset \vdash .Cʻ(P\downarrow P)\in 1.\\ +[*52·41.\text{Transp}] &\supset \vdash .{\sim}(\exists Q,R).Q,R\in Cʻ(P\downarrow P).Q\neq R.\\ +[*11·63] &\supset \vdash :Q,R\in Cʻ(P\downarrow P).Q\neq R.\supset _{Q,R}.CʻQ\cap CʻR=\Lambda &\qquad \text{(1)}\\ +\vdash .(1).*163·1.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_374">[Pg 374]</span></p> + +<p class="nind"><b>*163·42.</b> \(\vdash \colon\ldotp P\downarrow Q\in \text{Rel}^{2}\text{excl}.\equiv :P=Q.\lor.CʻP\cap CʻQ=\Lambda \)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*163·1.*55·15.\supset \\ +\vdash \colon\ldotp P\downarrow Q\in \text{Rel}^{2}\text{excl}.&\equiv :M,N\in \iota ʻP\cup \iota ʻQ.M\neq N.\supset _{M,N}.CʻM\cap CʻN=\Lambda :\\ +[*54·441] & \equiv :P=Q.\lor.CʻP\cap CʻQ=\Lambda .CʻQ\cap CʻP=\Lambda :\\ +[*22·51] &\equiv :P=Q.\lor.CʻP\cap CʻQ=\Lambda \colon\ldotp \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p>The above proposition is used in *251·22.</p> + +<p class="nind"><b>*163·43.</b> \(\vdash :P\in \text{Rel}^{2}\text{excl}.Q\,\unicode{x2abd}\, P.\supset .Q\in \text{Rel}^{2}\text{excl}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*33·265. \supset \vdash \colon\ldotp \text{Hp}.&\supset :M,N\in CʻQ.\supset .M,N\in CʻP:\\ +[\text{Fact}] &\supset :M,N\in CʻQ.M\neq N.\supset .M,N\in CʻP.M\neq N:\\ +[*163·1.\text{Hp}] &\supset .CʻM\cap CʻN=\Lambda &\qquad \text{(1)}\\ +\vdash .(1).*163·1.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*163·431.</b> \(\vdash :P\in \text{Rel}^{2}\text{excl}.\supset .\text{Rl}ʻP\subset \text{Rel}^{2}\text{excl} \quad[*163·43]\)</p> + +<p class="nind"><b>*163·44.</b> \[\begin{align}\vdash :P\unicode{x2909}Q\in \text{Rel}^{2}&\text{excl}.\equiv .\\ +&P,Q\in \text{Rel}^{2}\text{excl}.sʻCʻʻCʻP\cap sʻCʻʻ(CʻQ-CʻP)=\Lambda\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*163·12.*160·14.&\supset \vdash :P\unicode{x2909}Q\in \text{Rel}^{2}\text{excl}.\equiv .F\upharpoonright (CʻP\cup CʻQ)\in \text{Cls}\rightarrow 1.\\ +[*74·821] &\equiv .F\upharpoonright CʻP,F\upharpoonright CʻQ\in \text{Cls}\rightarrow 1.FʻʻCʻP\cap Fʻʻ(CʻQ-CʻP)=\Lambda .\\ +[*163·12.*40·56] &\equiv .P,Q\in \text{Rel}^{2}\text{excl}.sʻCʻʻCʻP\cap sʻCʻʻ(CʻQ-CʻP)=\Lambda :\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*163·441.</b> \[\begin{align}&\vdash :P,Q\in \text{Rel}^{2}\text{excl}.Cʻ\Sigma ʻP\cap Cʻ\Sigma ʻQ=\Lambda .\supset .P\unicode{x2909}Q\in \text{Rel}^{2}\text{excl}\\ +&[*163·44.*162·22]\end{align}\]</p> + +<p>The above proposition is used in <a href="#*173·26">*173·26</a>.</p> + +<p class="nind"><b>*163·442.</b> \[\begin{align}\vdash \colon\ldotp CʻP&\cap CʻQ=\Lambda .\supset :\\ +&P\unicode{x2909}Q\in \text{Rel}^{2}\text{excl}.\equiv .P,Q\in \text{Rel}^{2}\text{excl}.Cʻ\Sigma ʻP\cap Cʻ\Sigma ʻQ=\Lambda \end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*24·313. \supset \vdash :\text{Hp}.\supset .CʻQ-CʻP=CʻQ &\qquad \text{(1)}\\ +\vdash .(1).*163·44.*162·22.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*163·45.</b> \(\vdash :P\unicode{x2909}R\in \text{Rel}^{2}\text{excl}.\equiv .P\in \text{Rel}^{2}\text{excl}.sʻCʻʻ(CʻP-\iota ʻR)\cap CʻR=\Lambda\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*161·14.*163·12.\supset \\ +\vdash \colon\ldotp \dot{\exists} !P.\supset :P\unicode{x21f8} R\in \text{Rel}^{2}\text{excl}.&\equiv .F\upharpoonright (CʻP\cup \iota ʻR)\in \text{Cls}\rightarrow 1.\\ +[*74·821.*53·301.*33·5] +&\equiv .F\upharpoonright CʻP,F\upharpoonright \iota ʻR\in \text{Cls}\rightarrow 1.Fʻʻ(CʻP-\iota ʻR)\cap CʻR=\Lambda .\\ +[*35·101.*71·171] &\equiv .F\upharpoonright CʻP\in \text{Cls}\rightarrow 1.Fʻʻ(CʻP-\iota ʻR)\cap CʻR=\Lambda .\\ +[*163·12.*40·56] &\equiv :P\in \text{Rel}^{2}\text{excl}.sʻCʻʻ(CʻP-\iota ʻR)\cap CʻR=\Lambda &\qquad \text{(1)}\\ +\vdash .*161·2.*163·4.&\supset \vdash :P=\dot{\Lambda} .\supset .P\unicode{x21f8} R\in \text{Rel}^{2}\text{excl}.P\in \text{Rel}^{2}\text{excl} &\qquad \text{(2)}\\ +\vdash .*33·241.*37·29.*40·21.&\supset \vdash :P=\dot{\Lambda} .\supset .sʻCʻʻ(CʻP-{℩}ʻR)\cap CʻR=\Lambda &\qquad \text{(3)}\\ +\vdash .(2).(3).\text{Comp}.*5·1.&\supset \vdash \colon\ldotp P=\dot{\Lambda} .\supset :\\ +&P\unicode{x21f8} R\in \text{Rel}^{2}\text{excl}.\equiv .P\in \text{Rel}^{2}\text{excl}.sʻCʻʻ(CʻP-{℩}ʻR)\cap CʻR=\Lambda &\qquad \text{(4)}\\ +\vdash .(1).(4).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_375">[Pg 375]</span></p> + +<p class="nind"><b>*163·451.</b> \[\begin{align}&\vdash :P\in \text{Rel}^{2}\text{excl}.Cʻ\Sigma ʻP\cap CʻR=\Lambda .\supset .P\unicode{x21f8} R\in \text{Rel}^{2}\text{excl}\\ +&[*163·45.*162·22]\end{align}\]</p> + +<p>The above proposition is used in <a href="#*173·25">*173·25</a>.</p> + +<p class="nind"><b>*163·452.</b> \[\begin{align}&\vdash \colon\ldotp R{\sim}\in CʻP.\supset :P\unicode{x21f8} R\in \text{Rel}^{2}\text{excl}.\equiv .P\in \text{Rel}^{2}\text{excl}.Cʻ\Sigma + ʻP\cap CʻR=\Lambda \\ +&[*51·222.*163·45.*162·22]\end{align}\]</p> + +<p class="nind"><b>*163·46.</b> \[\begin{align}&\vdash :R\unicode{x21f7}P\in \text{Rel}^{2}\text{excl}.\equiv .P\in \text{Rel}^{2}\text{excl}.sʻCʻʻ(CʻP-{℩}ʻR)\cap CʻR=\Lambda \\ +&[*163·45·331]\end{align}\]</p> + +<p class="nind"><b>*163·461.</b> \[\begin{align}&\vdash :P\in \text{Rel}^{2}\text{excl}.Cʻ\Sigma ʻP\cap CʻR=\Lambda .\supset .R\unicode{x21f7}P\in \text{Rel}^{2}\text{excl}\\ +&[*163·451·331]\end{align}\]</p> + +<p class="nind"><b>*163·462.</b> \[\begin{align}&\vdash \colon\ldotp R{\sim}\in CʻP.\supset :R\unicode{x21f7}P\in \text{Rel}^{2}\text{excl}.\equiv .P\in \text{Rel}^{2}\text{excl}.Cʻ\Sigma + ʻP\cap CʻR=\Lambda \\ +&[*163·452·331]\end{align}\]</p> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_376">[Pg 376]</span></p> +<h2 class="nobreak" id="*164">*164. DOUBLE LIKENESS.</h2> +</div> + + +<p class="nind"><i>Summary of</i> *164.</p> + +<p>The subject of this number is of great importance throughout +relation-arithmetic and its applications. Double likeness, or double +ordinal similarity, is a relation which is to hold between \(P\) and +\(Q\) when (1) \(P\) and \(Q\) are like, (2) correlated members of the +fields of \(P\) and \(Q\) are like, with a specific given correlator +in each case. (It is necessary, in general, to have a given correlator +in each case, to avoid the necessity of the multiplicative axiom +for selecting among correlators.) This definition can be somewhat +simplified by starting from a relation correlating \(\Sigma ʻP\) and +\(\Sigma ʻQ\). If \(S\) is such a correlator, so that +\[ +S\in 1\rightarrow 1.\text{ᗡ}ʻS=Cʻ\Sigma ʻQ.\Sigma ʻP=S^{;}\Sigma ʻQ, +\] +we want \(S\) to be such that it not only correlates the whole of +\(\Sigma ʻP\) with the whole of \(\Sigma ʻQ\), but also correlates +each member of \(CʻP\) with the corresponding member of \(CʻQ\), +<i>i.e.</i> such that, if \(N\) is any member of \(CʻQ\), \(S^{;}N\) is +the corresponding member of \(CʻP\). This requires +\[ +NQN'.\equiv .(S^{;}N)P(S^{;}N'), +\] +<i>i.e.</i> writing \(S\dagger ʻN\), \(S\dagger ʻN'\) in place of +\(S^{;}N, S^{;}N'\), it requires +\[ +P=S\dagger ^{;}Q. +\] +When \(P=S\dagger ^{;}Q\) and \(\text{ᗡ}ʻS=Cʻ\Sigma ʻQ\), we have +\(\Sigma ʻP=S^{;}\Sigma ʻQ\) by <a href="#*162·35">*162·35</a>. Hence double likeness will +subsist if there is a relation \(S\) such that +\[ +S\in 1\rightarrow 1.\text{ᗡ}ʻS=Cʻ\Sigma ʻQ.P=S\dagger ^{;}Q. +\]</p> + +<p>A relation \(S\) fulfilling this condition will be called a double +correlator of \(P\) and \(Q\). Thus two relations \(P\) and \(Q\) have +double likeness when there exists a double correlator of \(P\) and +\(Q\), <i>i.e.</i> when +\[ +(\exists S).S\in 1\rightarrow 1.\text{ᗡ}ʻS=Cʻ\Sigma ʻQ.P=S\dagger ^{;}Q. +\] +A double correlator of \(P\) and \(Q\) is a relation \(S\) which is +a correlator of \(\Sigma ʻP\) and \(\Sigma ʻQ\) and is such that +\(S\dagger \upharpoonright CʻQ\) is a correlator of \(P\) and \(Q\).</p> + +<p><span class="pagenum" id="Page_377">[Pg 377]</span></p> + +<p>It will be seen that this definition has the usual analogy to the +corresponding definition in cardinals (<a href="#*111·01">*111·01</a>). The two inverted +commas of the cardinal definition are replaced by the semi-colon, and +\(S_{\in }\) is replaced by \(S\dagger\), and \(sʻ\lambda\) is replaced +by \(\Sigma ʻQ\) or \(Cʻ\Sigma ʻQ\). The propositions of the present +number consist largely of analogues of the propositions of <a href="#*111">*111</a>, in +accordance with the above substitutions.</p> + +<p>If it were not for the difficulty of choice among correlators, we could +define two relations as having double likeness when they are like +relations of like relations, <i>i.e.</i> when, if \(P\) and \(Q\) are +the two relations, they have a correlator \(S\) such that, if \(MSN\), +then \(M \text{smor} N\). In this case, \(S\in P\overline{\text{smor}}Q\cap \text{Rl}ʻ\text{smor}\). +Thus we have to consider the relations of the class +\(P\,\overline{\text{smor}}\,Q\cap \text{Rl}ʻ\text{smor}\) to the +class of double correlators, and we have to consider the relation +of the relation "\(\exists !P\,\overline{\text{smor}}\,Q\cap\text{Rl}ʻ\text{smor}\)" +to the relation of double likeness. The propositions to be proved on +this subject in the present number are analogous to the propositions of +<a href="#*111">*111</a>. But at a later stage (*251·61) we shall show that if the field of +\(P\) consists entirely of relations which generate <i>well-ordered</i> +series, then the use of the multiplicative axiom ceases to be +necessary in identifying double likeness with the relation +\(\exists!P\,\overline{\text{smor}}\,Q\cap \text{Rl}ʻ\text{smor}\), the +reason being that two well-ordered series can never be correlated in +more than one way.</p> + +<p>Our definitions are</p> + +<p class="nind"><b>*164·01.</b> \(P\,\overline{\text{smor}}\,\overline{\text{smor}}\,Q=(1\rightarrow 1)\cap \overleftarrow{\text{ᗡ}}ʻCʻ\Sigma ʻQ\cap \hat{S} (P=S\dagger ^{;}Q) \quad\text{Df}\)</p> + +<p class="nind"><b>*164·02.</b> \(\text{smor} \text{smor}=\hat{P} \hat{Q} (\exists !P\,\overline{\text{smor}}\,\overline{\text{smor}}\,Q) \quad\text{Df}\)</p> + +<p>The principal propositions of this number are</p> + +<p class="nind"><b>*164·15.</b> + \(\vdash :S\in P\,\overline{\text{smor}}\,\overline{\text{smor}}\,Q.\equiv .S\in \Sigma ʻP\,\overline{\text{smor}}\,\Sigma ʻQ.(S\dagger )\upharpoonright CʻQ\in P\,\overline{\text{smor}}\,Q\)</p> + +<p class="nind">whence</p> + +<p class="nind"><b>*164*151.</b> \(\vdash :P \,\text{smor}\,\text{smor}\,Q.\supset .\Sigma ʻP\,\text{smor}\,\Sigma ʻQ.P\,\text{smor}\,Q\)</p> + +<p class="nind"><b>*164*18.</b> \[\begin{align}\vdash :S\upharpoonright Cʻ\Sigma ʻQ\in P\,\overline{\text{smor}}\,&\overline{\text{smor}}\,Q.\equiv .\\ +&S\upharpoonright Cʻ\Sigma ʻQ\in 1\rightarrow 1.Cʻ\Sigma ʻQ\subset \text{ᗡ}ʻS.P=S\dagger ^{;}Q\end{align}\]</p> + +<p>This is usually the most convenient proposition when a double +correlation has to be proved.</p> + +<p class="nind"><b>*164·201·211·221.</b> Double likeness is reflexive, symmetrical and transitive.</p> + +<p class="nind"><b>*164·31.</b> \(\vdash :S\in P\,\overline{\text{smor}}\,\overline{\text{smor}}\,Q.\equiv .S\in (CʻʻCʻP)\,\overline{\text{ sm }}\, \,\overline{\text{ sm }}\, (CʻʻCʻQ).P=S\dagger ^{;}Q\)</p> + +<p>(Cf. note to *164·31, below.)</p> + +<p>We then have a set of propositions (<a href="#*164·4">*164·4</a> to the end) on the +identification of \(\exists !P\,\overline{\text{smor}}\,Q\cap\text{Rl}ʻ\text{smor}\) +with double likeness by means of the multiplicative axiom. We have</p> + +<p class="nind"><b>*164·43.</b> \[\begin{align}\vdash \colon\ldotp P,Q&\in \text{Rel}^{2}\text{excl}.S\in P\,\overline{\text{smor}}\,Q.\\ +&\mu =\hat{\lambda}\{(\exists N).N\in CʻQ.\lambda =(SʻN)\,\overline{\text{smor}}\,N\}.\supset :\\ +&R\in {\in}_{\Delta}ʻ\mu .\supset .\dot{s} ʻ\text{D}ʻR\in P\,\overline{\text{smor}}\,\overline{\text{smor}}\,Q.S=(\dot{s} ʻ\text{D}ʻR)\dagger \upharpoonright CʻQ\end{align}\]</p> + +<p><span class="pagenum" id="Page_378">[Pg 378]</span></p> + +<p>That is to say, given that \(P\) and \(Q\) are like relations of +like mutually exclusive relations, if we can pick out one correlator +for each pair of correlated members of \(CʻP\) and \(CʻQ\), then the +sum \((\dot{s})\) of such selected correlators is a double correlator +of \(P\) and \(Q\). Hence, observing that if \(S\) is a double +correlator of \(P\) and \(Q\), \((S\dagger)\upharpoonright CʻQ\in +P\,\overline{\text{smor}}\,Q\cap \text{Rl}ʻ\text{smor}\) (<a href="#*164·15">*164·15</a><a href="#*164·16">·16</a>), we +arrive at</p> + +<p class="nind"><b>*164·45.</b> \[\begin{align}\vdash \colon\colon &\text{Mult ax}.\supset \colon\ldotp \\ +&P,Q\in \text{Rel}^{2}\text{excl}.\supset :\exists !P\,\overline{\text{smor}}\,Q\cap \text{Rl}ʻ\text{smor}.\equiv .P \,\text{smor}\,\text{smor}\,Q\end{align}\]</p> + +<p>From <a href="#*164·43">*164·43</a> we deduce also</p> + +<p class="nind"><b>*164·46.</b> \[\begin{align}\vdash \colon\ldotp &\text{Mult ax}.\supset :\\ +&P,Q\in \text{Rel}^{2}\text{excl}.\exists !P\,\overline{\text{smor}}\,Q\cap \text{Rl}ʻ\text{smor}.\supset .\Sigma ʻP \,\text{smor}\,\Sigma ʻQ\end{align}\]</p> + +<p class="nind"><b>*164·48.</b> \[\begin{align}\vdash \colon\ldotp \text{Mult ax}.\supset :R,S\in \text{Rel}^{2}\text{excl}\cap &\text{Nr}ʻQ.CʻR,CʻS\in \text{Cl}ʻ\text{Nr}ʻP.\supset .\\ +&R\,\text{smor}\,\text{smor}\,S.\Sigma ʻR\,\text{smor}\,\Sigma ʻS\end{align}\]</p> + +<p><i>I.e.</i> in effect, assuming the multiplicative axiom, if two +series \((\Sigma ʻR\) and \(\Sigma ʻS\)) can each be divided into +\(\beta\) sets of \(\alpha\) terms (\(\alpha\), \(\beta\) being +relation-numbers), then the two series are ordinally similar, and +the \(\beta\) sets in the one case have double similarity with the +\(\beta\) sets in the other. (Here we have written \(\alpha\), +\(\beta\) in place of the \(\text{Nr}ʻP\) and \(\text{Nr}ʻQ\) of the +enunciation.)</p> + +<p>It is by means of the above propositions that ordinal addition and +multiplication are connected, as will appear in <a href="#*166">*166</a>.</p> + +<hr class="tb"> + +<p class="nind"><b>*164·01.</b> \(P\,\overline{\text{smor}} \,\overline{\text{smor}}\, Q=(1\rightarrow 1)\cap \overleftarrow{\text{ᗡ}}ʻCʻ\Sigma ʻQ\cap \hat{S} (P=S\dagger ^{;}Q) \quad\text{Df}\)</p> + +<p class="nind"><b>*164·02.</b> \(\text{smor} \text{smor} =\hat{P} \hat{Q} (\exists !P\,\overline{\text{smor}} \,\overline{\text{smor}}\, Q) \quad\text{Df}\)</p> + +<p class="nind"><b>*164·1.</b> \[\begin{align}&\vdash :S\in P\,\overline{\text{smor}} \,\overline{\text{smor}}\, Q.\equiv .S\in 1\rightarrow 1.\text{ᗡ}ʻS=Cʻ\Sigma ʻQ.P=S\dagger ^{;}Q\\ +&[(*164·01)]\end{align}\]</p> + +<p class="nind"><b>*164·11.</b> \(\vdash :P \,\text{smor} \text{smor} Q.\equiv .\exists !P\,\overline{\text{smor}} \,\overline{\text{smor}}\, Q \quad[(*164·02)]\)</p> + +<p class="nind"><b>*164·12.</b> \[\begin{align}&\vdash :P \,\text{smor} \text{smor} Q.\equiv .(\exists S).S\in 1\rightarrow 1.\text{ᗡ}ʻS=Cʻ\Sigma ʻQ.P=S\dagger ^{;}Q\\ +&[*164·1·11]\end{align}\]</p> + +<p class="nind"><b><a id="*164·13">*164·13</a>.</b> \[\begin{align}&\vdash :S\upharpoonright Cʻ\Sigma ʻQ\in 1\rightarrow 1.Cʻ\Sigma ʻQ\subset \text{ᗡ}ʻS.\supset .(S\dagger )\upharpoonright CʻQ\in 1\rightarrow 1\\ +&[*150·152.*162·22]\end{align}\]</p> + +<p class="nind"><b>*164·131.</b> \(\vdash :\text{ᗡ}ʻS=Cʻ\Sigma ʻQ.P=S\dagger ^{;}Q.\supset .\text{D}ʻS=Cʻ\Sigma ʻP.\Sigma ʻP=S^{;}\Sigma ʻQ\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*162·35.\supset \vdash :\text{Hp}.&\supset .\Sigma ʻP=S^{;}\Sigma ʻQ &\qquad \text{(1)}\\ +[*150·23.\text{Hp}] &\supset .Cʻ\Sigma ʻP=\text{D}ʻS &\qquad \text{(2)}\\ +\vdash .(1).(2).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*164·14.</b> \(\vdash :S\in P\,\overline{\text{smor}} \,\overline{\text{smor}}\, Q.\supset .S\in \Sigma ʻP\overline{\text{smor}} \Sigma ʻQ \quad[*164·1·131.*151·11]\)</p> + +<p>The two following propositions are required for proving <a href="#*164·18">*164·18</a>.</p> + +<p class="nind"><b><a id="*164·141">*164·141</a>.</b> \(\vdash :Cʻ\Sigma ʻQ\subset \alpha .\supset .(T\upharpoonright \alpha )\dagger ^{;}Q=T\dagger ^{;}Q \quad[*150·171.*162·22]\)</p> + +<p class="nind"><b>*164·142.</b> \(\vdash .(T\upharpoonright Cʻ\Sigma ʻQ)\dagger ^{;}Q=T\dagger ^{;}Q=\{(T\dagger )\upharpoonright CʻQ\}^{;}Q \quad[*164·141.*150·32]\)</p> + +<p class="nind"><b>*164·143.</b> \(\vdash :S\in P\,\overline{\text{smor}} \,\overline{\text{smor}}\, Q.\supset .(S\dagger )\upharpoonright CʻQ\in P\,\overline{\text{smor}}\,Q\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*164·1·13. &\supset \vdash :\text{Hp}.\supset .(S\dagger )\upharpoonright CʻQ\in 1\rightarrow 1 &\qquad \text{(1)}\\ +\vdash .*35·65. &\supset \vdash .\text{ᗡ}ʻ(S\dagger )\upharpoonright CʻQ=CʻQ &\qquad \text{(2)}\\ +\vdash .*164·1.*150·32. &\supset \vdash :\text{Hp}.\supset .P={(S\dagger )\upharpoonright CʻQ}^{;}Q &\qquad \text{(3)}\\ +\vdash .(1).(2).(3).*151·11.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_379">[Pg 379]</span></p> + +<p class="nind"><b><a id="*164·15">*164·15</a>.</b> + \(\vdash :S\in P\,\overline{\text{smor}} \,\overline{\text{smor}}\, Q.\equiv .S\in \Sigma ʻP\overline{\text{smor}} \Sigma ʻQ.(S\dagger )\upharpoonright CʻQ\in P\,\overline{\text{smor}}\,Q\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*164·14·143.\supset \\ +\vdash :S\in P\,\overline{\text{smor}} \,\overline{\text{smor}}\, Q.\supset .S\in \Sigma ʻP\overline{\text{smor}} \Sigma ʻQ.(S\dagger )\upharpoonright &CʻQ\in P\,\overline{\text{smor}}\,Q &\qquad \text{(1)}\\ +\vdash .*151·11.\supset \\ +\vdash :S\in \Sigma ʻP\overline{\text{smor}} \Sigma ʻQ.(S\dagger )\upharpoonright &CʻQ\in P\,\overline{\text{smor}}\,Q.\supset .\\ +&S\in 1\rightarrow 1.\text{ᗡ}ʻS=Cʻ\Sigma ʻQ.P=\{(S\dagger )\upharpoonright CʻQ\}^{;}Q &\qquad \text{(2)}\\ +\vdash .(2).*150·32.*164·1.\supset \\ +\vdash :S\in \Sigma ʻP\overline{\text{smor}} \Sigma ʻQ.(S\dagger )\upharpoonright &CʻQ\in P\,\overline{\text{smor}}\,Q.\supset .S\in P\,\overline{\text{smor}} \,\overline{\text{smor}}\, Q &\qquad \text{(3)}\\ +\vdash .(1).(3).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*164·151.</b> \(\vdash :P\,\text{smor}\,\text{smor}\,Q.\supset .\Sigma ʻP\,\text{smor}\,\Sigma ʻQ.P\,\text{smor}\,Q \quad[*164·15·11]\)</p> + +<p class="nind"><b><a id="*164·16">*164·16</a>.</b> \(\vdash :S\in P\,\overline{\text{smor}} \,\overline{\text{smor}}\, Q.\supset .(S\dagger )\upharpoonright CʻQ\,\unicode{x2abd}\, \text{smor}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*35·101.*150·1.&\supset \vdash :M{(S\dagger )\upharpoonright CʻQ}N.\equiv .N\in CʻQ.M=S^{;}N &\qquad \text{(1)}\\ +\vdash .*164·1.*162·22.\supset \vdash \colon\ldotp \text{Hp}.&\supset :S\in 1\rightarrow 1:N\in CʻQ.\supset _{N}.CʻN\subset \text{ᗡ}ʻS:\\ +[*151·23] &\supset :N\in CʻQ.M=S^{;}N.\supset _{M,N}.M\,\text{smor}\,N\\ +[(1)] &\supset :M{(S\dagger )\upharpoonright CʻQ}N.\supset _{M,N}.M\,\text{smor}\,N\colon\ldotp \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*164·17.</b> \(\vdash :P\,\text{smor}\,\text{smor}\,Q.\supset .\exists !P\,\overline{\text{smor}}\,Q\cap \text{Rl}ʻ\text{smor} \quad[*164·143·16]\)</p> + +<p>This proposition states that when \(P\) and \(Q\) have double likeness, +there is a correlator of \(P\) and \(Q\) which couples like with +like relations; <i>i.e.</i> if \(S\) is the correlator, then, if +\(MSN\), \(M\) and \(N\) are ordinally similar. The converse of this +proposition, namely, that if \(P\) and \(Q\) have a correlator which +couples ordinally similar relations, then \(P\) and \(Q\) have double +likeness, can be proved if the multiplicative axiom is assumed, but not +otherwise, except in special cases, such as that of well-ordered series.</p> + +<p>The following proposition is used frequently, owing to the fact that, +in the cases we are concerned with, double correlators generally have +the form \(S\upharpoonright Cʻ\Sigma ʻQ\), where \(S\) is some relation +for which we have \((y).\text{E}!Sʻy\).</p> + +<p class="nind"><b><a id="*164·18">*164·18</a>.</b> \[\begin{align}\vdash :S\upharpoonright Cʻ\Sigma ʻQ\in P\,\overline{\text{smor}}\,&\overline{\text{smor}}\,Q.\equiv .\\ +&S\upharpoonright Cʻ\SigmaʻQ \in 1\rightarrow 1.Cʻ\Sigma ʻQ\subset \text{ᗡ}ʻS.P=S\dagger ^{;}Q\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*35·64.*22·621.&\supset \vdash :\text{ᗡ}ʻ(S\upharpoonright Cʻ\Sigma ʻQ)=Cʻ\Sigma ʻQ.\equiv .Cʻ\Sigma ʻQ\subset \text{ᗡ}ʻS &\qquad \text{(1)}\\ +\vdash .*164·142. &\supset \vdash :P=(S\upharpoonright Cʻ\Sigma ʻQ)\dagger Q.\equiv .P=S\dagger ^{;}Q &\qquad \text{(2)}\\ +\vdash .*164·1. &\supset \vdash :S\upharpoonright Cʻ\Sigma ʻQ\in P\,\overline{\text{smor}} \,\overline{\text{smor}}\, Q.\equiv .\\ +&S\upharpoonright Cʻ\Sigma ʻQ\in 1\rightarrow 1.\text{ᗡ}ʻ(S\upharpoonright Cʻ\Sigma ʻQ)=Cʻ\Sigma ʻQ.P=(S\upharpoonright Cʻ\Sigma ʻQ)\dagger ^{;}Q &\qquad \text{(3)}\\ +\vdash .(1).(2).(3).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_380">[Pg 380]</span></p> + +<p class="nind"><b>*164·181.</b> \(\vdash : P \,\text{smor} \text{smor} Q. \equiv . (\exists S). S\upharpoonright Cʻ\Sigma ʻQ\in 1\rightarrow 1. Cʻ\Sigma ʻQ \subset \text{ᗡ}ʻS. P = S\dagger ^{;} Q\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *35·66. *164·1. \supset \\ +\vdash : S\in P \,\overline{\text{smor}}\,\overline{\text{smor}}\, Q. \supset . S\upharpoonright Cʻ\Sigma ʻQ\in 1\rightarrow 1. Cʻ\Sigma ʻQ \subset \text{ᗡ}ʻS .P = S\dagger ^{;} + Q &\qquad \text{(1)}\\ +\vdash . (1). *164·11. \supset \\ +\vdash : P \,\text{smor}\,\text{smor}\,Q. \supset . (\exists S). S\upharpoonright Cʻ\Sigma ʻQ\in 1\rightarrow 1. Cʻ\Sigma ʻQ \subset \text{ᗡ}ʻS. P = S\dagger ^{;} Q &\qquad \text{(2)}\\ +\vdash . *164·18·11. \supset \\ +\vdash : (\exists S). S\upharpoonright Cʻ\Sigma ʻQ\in 1\rightarrow 1. Cʻ\Sigma ʻQ \subset \text{ᗡ}ʻS. P = S\dagger ^{;} Q. \supset . P \,\text{smor} \text{smor} Q &\qquad \text{(3)}\\ +\vdash . (2). (3). \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p>The following propositions are concerned in proving that double +likeness is reflexive, symmetrical, and transitive.</p> + +<p class="nind"><b>*164·2.</b> \(\vdash . I\upharpoonright Cʻ\Sigma ʻP\in P \,\overline{\text{smor}}\,\overline{\text{smor}}\, P\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *151·121. &\supset \vdash . I\upharpoonright Cʻ\Sigma ʻP\in \Sigma ʻP\,\overline{\text{smor}}\,\Sigma ʻP. I\upharpoonright CʻP\in P\,\overline{\text{smor}}\,P &\qquad \text{(1)}\\ +\vdash . *35·101. *150·1.\supset \\ +\vdash : M{(I\upharpoonright Cʻ\Sigma ʻP)\dagger \upharpoonright CʻP}N. &\equiv . N\in CʻP. M = (I\upharpoonright Cʻ\Sigma ʻP)^{;} N.\\ +[*150·33. *162·22] & \equiv .N\in CʻP. M = I^{;} N.\\ +[*150·53] &\equiv . M(I\upharpoonright CʻP)N &\qquad \text{(2)}\\ +\vdash . (1). (2). &\supset \vdash . I\upharpoonright Cʻ\Sigma ʻP\in \Sigma ʻP\,\overline{\text{smor}}\,\Sigma ʻP. (I\upharpoonright Cʻ\Sigma ʻP)\dagger \upharpoonright CʻP\in P \overline{\text{smor}} P.\\ +[*164·15] &\supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*164·201">*164·201</a>.</b> \(\vdash . P \,\text{smor smor}\, P \quad[*164·2·11]\)</p> + +<p class="nind"><b><a id="*164·21">*164·21</a>.</b> \(\vdash : S\in P \,\overline{\text{smor}}\,\overline{\text{smor}}\, Q. \equiv . \breve{S} \in Q \,\overline{\text{smor}}\,\overline{\text{smor}}\, P\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *164·1. *71·212. &\supset \vdash : S\in P \,\overline{\text{smor}}\,\overline{\text{smor}}\, Q. \supset . \breve{S} \in 1\rightarrow 1 &\qquad \text{(1)}\\ +\vdash . *164·131·1 . &\supset \vdash : S\in P \,\overline{\text{smor}}\,\overline{\text{smor}}\, Q. \supset . \text{ᗡ}ʻ\breve{S} = Cʻ\Sigma ʻP &\qquad \text{(2)}\\ +\vdash . *150·94. *164·1. *162·22. &\supset \vdash : S\in P \,\overline{\text{smor}}\,\overline{\text{smor}}\, Q. \supset . Q = \breve{S} \dagger ^{;} P &\qquad \text{(3)}\\ +\vdash . (1). (2). (3). *164·1. &\supset \vdash : S\in P \,\overline{\text{smor}}\,\overline{\text{smor}}\, Q. \supset . \breve{S} \in Q \,\overline{\text{smor}}\,\overline{\text{smor}}\, P &\qquad \text{(4)}\\ +\vdash . (4) \frac{\breve{S},\,Q,\,P}{S,\,P,\,Q}. &\supset \vdash : \breve{S} \in Q \,\overline{\text{smor}}\,\overline{\text{smor}}\, P. \supset . S\in P \,\overline{\text{smor}}\,\overline{\text{smor}}\, Q &\qquad \text{(5)}\\ +\vdash . (4). (5). \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*164·211.</b> \(\vdash : P \,\text{smor}\,\text{smor}\,Q. \equiv . Q\,\text{smor}\,\text{smor}\,P \quad[*164·21·11]\)</p> + +<p class="nind"><b>*164·22.</b> + \(\vdash :S\in P \,\overline{\text{smor}}\,\overline{\text{smor}}\, Q. T\in Q \,\overline{\text{smor}}\,\overline{\text{smor}}\, R. \supset . S\mid T\in P \,\overline{\text{smor}}\,\overline{\text{smor}}\, R\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *164·1. \supset \vdash : \text{Hp}. &\supset . S,T\in 1\rightarrow 1.\\ +[*71·252] &\supset . S\mid T\in 1\rightarrow 1 &\qquad \text{(1)}\\ +\vdash . *164·1·131. \supset \vdash : \text{Hp}. &\supset . \text{ᗡ}ʻS = Cʻ\Sigma ʻQ. \text{D}ʻT = Cʻ\Sigma ʻQ.\\ +[*37·323] & \supset . \text{ᗡ}ʻ(S\mid T) = \text{ᗡ}ʻT.\\ +[*164·1] &\supset . \text{ᗡ}ʻ(S\mid T) = Cʻ\Sigma ʻR &\qquad \text{(2)}\\ +\vdash .*150·13·14.&\supset \vdash .(S\mid T)\dagger ^{;}R = S\dagger ^{;}T\dagger ^{;}R &\qquad \text{(3)}\\ +\vdash .*164·1.&\supset \vdash :\text{Hp}.\supset .T\dagger ^{;}R = Q.S\dagger ^{;}Q = P &\qquad \text{(4)}\\ +\vdash .(3).(4).&\supset \vdash :\text{Hp}.\supset .(S\mid T)\dagger ^{;}R = P &\qquad \text{(5)}\\ +\vdash .(1).(2).(5).*164·1.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_381">[Pg 381]</span></p> + +<p class="nind"><b>*164·221.</b> \(\vdash :P \,\text{smor}\,\text{smor}\,Q.Q\,\text{smor}\,\text{smor}\,R.\supset .P \,\text{smor} \text{smor} R \quad[*164·22·11]\)</p> + +<p class="nind"><b><a id="*164·23">*164·23</a>.</b> \(\vdash \colon\ldotp P \,\text{smor}\,\text{smor}\,Q.\supset :P \in \text{Rel}^{2}\text{excl}.\equiv .Q \in \text{Rel}^{2}\text{excl}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*164·12. \supset \vdash \colon\ldotp \text{Hp}.&\supset :(\exists T).T \in 1\rightarrow 1.\text{ᗡ}ʻT=Cʻ\Sigma ʻQ.P = T\dagger ^{;}Q:\\ +[*163·3] &\supset :Q \in \text{Rel}^{2}\text{excl}.\supset .P \in \text{Rel}^{2}\text{excl} &\qquad \text{(1)}\\ +\vdash .(1).*164·211.\supset \vdash \colon\ldotp \text{Hp}.&\supset :P \in \text{Rel}^{2}\text{excl}. \supset .Q \in \text{Rel}^{2}\text{excl} &\qquad \text{(2)}\\ +\vdash .(1).(2).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*164·3">*164·3</a>.</b> \(\vdash :S \in P \,\overline{\text{smor}} \,\overline{\text{smor}}\, Q.\supset .S \in (CʻʻCʻP)\,\overline{\text{ sm }}\, \,\overline{\text{ sm }}\, (CʻʻCʻQ)\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*164·1.*162·22.\supset \vdash : \text{Hp}.&\supset .S \in 1 \rightarrow 1.\text{ᗡ}ʻS = sʻCʻʻCʻQ.P = S\dagger ^{;}Q. &\qquad \text{(1)}\\ +[*150·931] &\supset .CʻʻCʻP = S_{\in}ʻʻCʻʻCʻQ &\qquad \text{(2)}\\ +\vdash .(1).(2).*111·1.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*164·301.</b> \(\vdash :P \,\text{smor}\,\text{smor}\,Q.\supset .CʻʻCʻP \text{sm sm} CʻʻCʻQ \quad[*164·3·11.*111·4]\)</p> + +<p class="nind"><b><a id="*164·31">*164·31</a>.</b> \(\vdash :S \in P \,\overline{\text{smor}}\,\overline{\text{smor}}\, Q. \equiv .S \in (CʻʻCʻP) \,\overline{\text{ sm }}\, \,\overline{\text{ sm }}\, (CʻʻCʻQ).P=S\dagger ^{;}Q\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*164·3·1.\supset \\ +\vdash :S \in P \,\overline{\text{smor}} \,\overline{\text{smor}}\, Q.\supset .S \in (CʻʻCʻP)\,\overline{\text{ sm }}\, \,\overline{\text{ sm }}\, (CʻʻCʻQ).P = S\dagger ^{;}Q &\qquad \text{(1)}\\ +\vdash .*111·1.*162·22.\supset \\ +\vdash :S \in (CʻʻCʻP)\,\overline{\text{ sm }}\, \,\overline{\text{ sm }}\, (CʻʻCʻQ).\supset .S \in 1 \rightarrow 1.\text{ᗡ}ʻS = Cʻ\Sigma ʻQ &\qquad \text{(2)}\\ +\vdash .(2).\text{Fact}.*164·1.\supset \\ +\vdash :S \in (CʻʻCʻP)\,\overline{\text{ sm }}\, \,\overline{\text{ sm }}\, (CʻʻCʻQ).P = S\dagger ^{;}Q.\supset .S \in P \,\overline{\text{smor}} \,\overline{\text{smor}}\, Q &\qquad \text{(3)}\\ +\vdash .(1).(3).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p>This proposition has the merit of reducing the ordinal element in +double likeness to a minimum. The proof of +\[ +S \in (CʻʻCʻP)\,\overline{\text{ sm }}\, \,\overline{\text{ sm }}\, (CʻʻCʻQ) +\] +is a cardinal problem, and what has to be added for ordinal purposes is +merely \(P = S\dagger ^{;}Q\).</p> + +<p class="nind"><b>*164·32.</b> \(\vdash .\dot{\Lambda} \in (\dot{\Lambda} \,\overline{\text{smor}} \,\overline{\text{smor}}\, \dot{\Lambda} ).\dot{\Lambda} \text{smor} \text{smor} \dot{\Lambda}\)</p> + +<p>In this proposition, the various \(\dot{\Lambda}\)'s need not be +of the same type. Hence "\(\dot{\Lambda}\,\text{smor}\,\text{smor}\,\dot{\Lambda}\)" +is not an immediate consequence of <a href="#*164·201">*164·201</a>.</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*72·1.*162·4. &\supset \vdash .\dot{\Lambda} \in 1\rightarrow 1.\text{ᗡ}ʻ\dot{\Lambda} =Cʻ\Sigma ʻ\dot{\Lambda} &\qquad \text{(1)}\\ +\vdash .*150·42. &\supset \vdash .\dot{\Lambda} = \dot{\Lambda} ^{;}\dot{\Lambda} &\qquad \text{(2)}\\ +\vdash .(1).(2).*164·1.&\supset \vdash .\dot{\Lambda} \in (\dot{\Lambda} \,\overline{\text{smor}} \,\overline{\text{smor}}\, \dot{\Lambda} ). &\qquad \text{(3)}\\ +[*164·11] &\supset .\dot{\Lambda}\,\text{smor smor}\,\dot{\Lambda} &\qquad \text{(4)}\\ +\vdash .(3).(4).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_382">[Pg 382]</span></p> + +<p class="nind"><b>*164·33.</b> \[\begin{align}\vdash :M \in P\,\overline{\text{smor}}\,R.N \in Q \,\overline{\text{smor}}\,S.Cʻ&P \cap CʻQ = \Lambda .CʻR \cap CʻS = \Lambda .\supset .\\ +&M \unicode{x228d} N \in (P \downarrow Q)\,\overline{\text{smor}} \,\overline{\text{smor}}\, (R \downarrow S)\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*160·47. \supset \vdash :\text{Hp}.&\supset .M \unicode{x228d} N \in (P\unicode{x2909}Q)\,\overline{\text{smor}}\,(R\unicode{x2909}S).\\ +[*162·3.*151·11] &\supset .M \unicode{x228d} N \in 1 \rightarrow 1.\text{ᗡ}ʻ(M \unicode{x228d} N) = Cʻ\Sigma ʻ(R \downarrow S) &\qquad \text{(1)}\\ +\vdash .*150·32. \supset \vdash : \text{Hp}.\supset .(M \unicode{x228d} N)^{;}R &= \{(M \unicode{x228d} N)\upharpoonright CʻR\}^{;}R\\ +[*35·644.*150·32] & = M^{;}R\\ +[*151·11] & = P &\qquad \text{(2)}\\ +\text{Similarly}\quad &\vdash :\text{Hp}.\supset .(M \unicode{x228d} N)^{;}S = Q &\qquad \text{(3)}\\ +\vdash .*150·71·1.\supset \vdash :\text{Hp} .\supset .(M \unicode{x228d} N)\dagger ^{;}(R \downarrow S) &= \{(M \unicode{x228d} N)^{;}R\}\downarrow + {(M \unicode{x228d} N)^{;}S}\\ +[(2).(3)] &= P \downarrow Q &\qquad \text{(4)}\\ +\vdash .(1).(4).*164·1.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*164·34.</b> \[\begin{align}\vdash :P \,\text{smor} R.Q\,\text{smor}\,S.CʻP \cap CʻQ = \Lambda .&CʻR \cap CʻS =\Lambda .\supset .\\ +&P \downarrow Q\,\text{smor smor}\,R \downarrow S\\ +[*164·33·11.*151·12]\end{align}\]</p> + +<p>The following propositions are concerned in showing that, if \(P\) +and \(Q\) are like relations, and the correlator of \(P\) and \(Q\) +is contained in likeness (<i>i.e.</i> correlates relations which +have the relation of likeness), a correlator being given for each pair +of relations coupled by the correlator of \(P\) and \(Q\), then the +logical sum of such correlators is a double correlator of \(P\) and +\(Q\), provided \(P\) and \(Q\) are relations of mutually exclusive +relations. That is, assuming \(S\) to be the correlator of \(P\) and +\(Q\), and assuming that \(SʻN \,\text{smor}\, N\) whenever \(N \in CʻQ\), +let it be possible to choose one correlator out of the class of +correlators \((SʻN)\,\overline{\text{smor}}\,N\), for every \(N\) which +belongs to \(CʻQ\). That is, assume that it is possible to make a +selection from the class of classes of correlators. If \(\mu\) is such +a selection, then \(\dot{s} ʻ\mu\) will be a double correlator of \(P\) +and \(Q \), if \(P\), \(Q \in \text{Rel}^{2}\text{excl}\).</p> + +<p>The following propositions, down to <a href="#*164·421">*164·421</a>, are lemmas for <a href="#*164·43">*164·43</a>.</p> + +<p class="nind"><b><a id="*164·4">*164·4</a>.</b> \(\vdash \colon\ldotp N \in CʻQ.\supset _{N}.RʻN \in (SʻN)\,\overline{\text{smor}}\,N:\supset .\text{ᗡ}ʻ\dot{s} ʻRʻʻCʻQ = Cʻ\Sigma ʻQ\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*41·44. &\supset \vdash .\text{ᗡ}ʻ\dot{s} ʻRʻʻCʻQ = sʻ\text{ᗡ}ʻʻRʻʻCʻQ &\qquad \text{(1)}\\ +\vdash .*151·11.\supset \vdash \colon\ldotp \text{Hp}.&\supset :N \in CʻQ.\supset .\text{ᗡ}ʻRʻN = CʻN:\\ +[*37·68] &\supset :\text{ᗡ}ʻʻRʻʻCʻQ = CʻʻCʻQ &\qquad \text{(2)}\\ +\vdash .(1).(2).\supset \vdash \colon\ldotp\text{Hp}.\supset :\text{ᗡ}ʻ\dot{s} ʻRʻʻCʻQ &= sʻCʻʻCʻQ\\ +[*162·22] &= Cʻ\Sigma ʻQ:\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*164·41.</b> \[\begin{align}\vdash \colon\ldotp Q \in \text{Rel}^{2}\text{excl}: N \in CʻQ.\supset _{N}.RʻN \in (SʻN)\,&\overline{\text{smor}}\,N:\supset .\\ +&\dot{s} ʻRʻʻCʻQ \in 1 \rightarrow \text{Cls}\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*151·11.\supset \vdash \colon\ldotp \text{Hp}.\supset :M,N \in CʻQ.&\exists !\text{D}ʻRʻM \cap \text{ᗡ}ʻRʻN.\supset .\\ +&\exists !CʻM \cap CʻN.\\ +[*163·11] &\supset .M = N.\\ +[*30·37] &\supset .RʻM = RʻN &\qquad \text{(1)}\\ +\vdash .*151·11.&\supset \vdash \colon\ldotp \text{Hp}.\supset :M \in CʻQ.\supset .RʻM \in 1 \rightarrow 1 &\qquad \text{(2)}\\ +\vdash .(1).(2).*72·32.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_383">[Pg 383]</span></p> + +<p class="nind"><b>*164·411.</b> \(\vdash :S^{;}Q \in \text{Rel}^{2}\text{excl}.S\upharpoonright + CʻQ \in 1 \rightarrow 1.\text{Hp}*164·4.\supset .\dot{s} ʻRʻʻCʻQ \in \text{Cls} \rightarrow 1\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*151·11.\supset \vdash \colon\ldotp \text{Hp}.\supset :M,N \in CʻQ.&\exists !\text{D}ʻRʻM \cap \text{D}ʻRʻN.\supset .\\ +&\exists !CʻSʻM \cap CʻSʻN.\\ +[*163·11.*150·22] &\supset .SʻM = SʻN.\\ +[*71·532] & \supset .M = N.\\ +[*30·37] &\supset .RʻM = RʻN &\qquad \text{(1)}\\ +\vdash .*151·11.&\supset \vdash \colon\ldotp \text{Hp}.\supset :M \in CʻQ.\supset .RʻM \in 1 \rightarrow 1 &\qquad \text{(2)}\\ +\vdash . (1). (2). *72·321 .\supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*164·412.</b> \[\begin{align}\vdash \colon\ldotp S^{;}Q,&Q \in \text{Rel}^{2}\text{excl}.S\upharpoonright CʻQ \in 1 \rightarrow 1:\\ +&N \in CʻQ.\supset _N.RʻN \in (SʻN)\,\overline{\text{smor}}\, N:\supset .\dot{s} ʻRʻʻCʻQ \in 1 \rightarrow 1\\ +&[*164·41·411]\end{align}\]</p> + +<p class="nind"><b>*164·413.</b> \[\begin{align}\vdash \colon\ldotp \text{Hp}&*164·41.\supset :\\ +&N \in CʻQ.\supset .RʻN = (\dot{s} ʻRʻʻCʻQ)\upharpoonright CʻN.SʻN = (\dot{s} ʻRʻʻCʻQ)^{;}N\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*41·13. \supset \vdash :\text{Hp}.N \in CʻQ.&\supset .RʻN \,\unicode{x2abd}\, \dot{s} ʻRʻʻCʻQ.\\ +[*72·92.*164·41] \supset .RʻN &= (\dot{s} ʻRʻʻCʻQ) \upharpoonright \text{ᗡ}ʻRʻN\\ +[*151·11.\text{Hp}] &= (\dot{s} ʻRʻʻCʻQ) \upharpoonright CʻN &\qquad \text{(1)}\\ +\vdash .*151·11.\supset \vdash :\text{Hp}.N \in CʻQ.\supset .SʻN &= (RʻN)^{;}N\\ +[(1).*150·32] &= (\dot{s} ʻRʻʻCʻQ)^{;}N &\qquad \text{(2)}\\ +\vdash .(1).(2).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*164·414.</b> \(\vdash :\text{Hp}*164·41.\supset .S^{;}Q = (\dot{s} ʻRʻʻCʻQ)\dagger ^{;}Q \quad[*164·413.*150·1·35]\)</p> + +<p class="nind"><b>*164·42.</b> \[\begin{align}\vdash \colon\ldotp Q,S^{;}Q \in &\text{Rel}^{2}\text{excl}.S \upharpoonright CʻQ \in 1 \rightarrow 1:\\ +&N \in CʻQ.\supset _N.RʻN \in (SʻN)\,\overline{\text{smor}}\,N:\supset .\\ +&\dot{s} ʻRʻʻCʻQ \in (S^{;}Q) \,\overline{\text{smor}}\,\overline{\text{smor}}\, Q \quad[*164·4·412·414·1]\end{align}\]</p> + +<p class="nind"><b><a id="*164·421">*164·421</a>.</b> \[\begin{align}\vdash \colon\ldotp P,Q \in &\text{Rel}^{2}\text{excl}.S \upharpoonright CʻQ \in P\,\overline{\text{smor}}\,Q:\\ +&N \in CʻQ.\supset _N.RʻN \in (SʻN)\,\overline{\text{smor}}\,N:\supset .\\ +&\dot{s} ʻRʻʻCʻQ \in P \,\overline{\text{smor}}\,\overline{\text{smor}}\, Q \quad[*164·42]\end{align}\]</p> + +<p><span class="pagenum" id="Page_384">[Pg 384]</span></p> + +<p>The following proposition, besides being used in proving all subsequent +propositions of this number (except <a href="#*164·432">*164·432</a><a href="#*164·432">·433</a>, which are mere lemmas +for <a href="#*164·44">*164·44</a>), is used in *251·6, in the theory of ordinal numbers.</p> + +<p class="nind"><b><a id="*164·43">*164·43</a>.</b> \[\begin{align}\vdash \colon\ldotp P,Q\in &\text{Rel}^{2}\text{excl}.S\in P\,\overline{\text{smor}}\,Q.\\ +&\mu =\hat{\lambda}\{(\exists N).N\in CʻQ.\lambda =(SʻN)\,\overline{\text{smor}}\,N\}.\supset :\\ +&R\in {\in}_{\Delta}ʻ\mu .\supset .\dot{s} ʻ\text{D}ʻR\in P\,\overline{\text{smor}} \,\overline{\text{smor}}\, Q.S=(\dot{s} ʻ\text{D}ʻR)]\dagger \upharpoonright CʻQ\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*83·2·22.&\supset \vdash \colon\ldotp \text{Hp}.R\in {\in}_{\Delta}ʻ\mu .\supset :\\ +&N\in CʻQ.\supset .Rʻ{(SʻN)\overline{\text{smor}} N}\in (SʻN)\overline{\text{smor}} N:\dot{s} ʻ\text{D}ʻR=Rʻʻ\mu &\qquad \text{(1)}\\ +\vdash .(1).&\supset \vdash \colon\ldotp \text{Hp}(1).T=\hat{\lambda} \hat{N} \{N\in CʻQ.\lambda =(SʻN)\overline{\text{smor}} N\}.\supset :\\ +&N\in CʻQ.\supset .RʻTʻN\in (SʻN)\overline{\text{smor}} N:\dot{s} ʻ\text{D}ʻR= RʻʻTʻʻCʻQ: &\qquad \text{(2)}\\ +\left[*164·42 \frac{R\mid T}{R}\right]&\supset :\dot{s} ʻ\text{D}ʻR\in P\,\overline{\text{smor}} \,\overline{\text{smor}}\, Q &\qquad \text{(3)}\\ +\vdash .(2).*164·413 \frac{R\mid T}{R}.*151·11.*35·71.&\supset \vdash :\text{Hp}(2).\supset .S=(\dot{s} ʻ\text{D}ʻR)\dagger \upharpoonright CʻQ &\qquad \text{(4)}\\ +\vdash .(3).(4).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*164·431.</b> \[\begin{align}\vdash \colon\ldotp &P,Q\in \text{Rel}^{2}\text{excl}:(\exists S).S\in P\,\overline{\text{smor}}\,Q.\\ +&\exists !{\in}_{\Delta}ʻ\hat{\lambda}\{(\exists N).N\in CʻQ.\lambda =(SʻN)\,\overline{\text{smor}}\,N\}:\supset .P \,\text{smor} \text{smor} Q\\ +&[*163·43·11]\end{align}\]</p> + +<p class="nind"><b><a id="*164·432">*164·432</a>.</b> \[\begin{align}\vdash :S\in P\,\overline{\text{smor}}\,Q\cap \text{Rl}ʻ&\text{smor}.\supset .\\ +&\Lambda {\sim}\in \hat{\lambda} \{(\exists N).N\in CʻQ.\lambda =(SʻN)\overline{\text{smor}} N\}\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*151·11.\supset \vdash \colon\ldotp \text{Hp}.&\supset :N\in CʻQ.\supset .N\in \text{ᗡ}ʻS.\\ +[*71·31] &\supset .(SʻN)SN.\\ +[\text{Hp}] &\supset .(SʻN)\text{smor}N.\\ +[*151·12] &\supset .\exists !(SʻN)\overline{\text{smor}} N\colon\ldotp \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*164·433.</b> \[\begin{align}\vdash \colon\ldotp \text{Mult ax}.\supset :S\in &P\,\overline{\text{smor}}\,Q\cap \text{Rl}ʻ\text{smor}.\supset .\\ +&\exists !{\in}_{\Delta}ʻ\hat{\lambda} \{(\exists N).N\in CʻQ.\lambda =(SʻN)\overline{\text{smor}} N\}\\ +&[*164·432.*88·37]\end{align}\]</p> + +<p>All the remaining propositions of the number are important.</p> + +<p class="nind"><b><a id="*164·44">*164·44</a>.</b> \[\begin{align}\vdash \colon\ldotp \text{Mult ax}.\supset :P,Q\in \text{Rel}^{2}\text{excl}.\exists !&P\,\overline{\text{smor}}\,Q\cap \text{Rl}ʻ\text{smor}.\supset .\\ +&P \,\text{smor smor}\,Q \quad[*164·433·431]\end{align}\]</p> + +<p class="nind"><b>*164·45.</b> \[\begin{align}\vdash \colon\colon &\text{Mult ax}.\supset \colon\ldotp P,Q\in \text{Rel}^{2}\text{excl}.\supset :\\ +&\exists !P\,\overline{\text{smor}}\,Q\cap \text{Rl}ʻ\text{smor}.\equiv .P \,\text{smor smor}\,Q \quad[*164·44·17]\end{align}\]</p> + +<p class="nind"><b>*164·46.</b> \[\begin{align}\vdash \colon\ldotp \text{Mult ax}.\supset :P,Q\in \text{Rel}^{2}\text{excl}.\exists !&P\,\overline{\text{smor}}\,Q\cap \text{Rl}ʻ\text{smor}.\supset .\\ +&\Sigma ʻP\text{smor}\Sigma ʻQ \quad[*164·44·151]\end{align}\]</p> + +<p><span class="pagenum" id="Page_385">[Pg 385]</span></p> + +<p class="nind"><b>*164·47.</b> \(\vdash : R, S \in \text{Nr}ʻQ . CʻR, CʻS \in \text{Cl}ʻ\text{Nr}ʻP . \supset . \exists ! R \overline{\text{smor}} S \cap \text{Rl}ʻ\text{smor}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *152·5·4 . \supset \vdash : \text{Hp} . &\supset . R \text{smor} S .\\ +[*151·12] &\supset . \exists ! R \,\overline{\text{smor}}\, S &\qquad \text{(1)}\\ +\vdash . *60·2 . &\supset \vdash \colon\ldotp \text{Hp} . \supset : M \in CʻR . N \in CʻS . \supset . M, N \in \text{Nr}ʻP .\\ +[*152·5·4] &\supset . M \,\text{smor}\, N &\qquad \text{(2)}\\ +\vdash . *151·1·131 . &\supset \vdash \colon\ldotp T \in R \,\overline{\text{smor}}\,S . \supset : MTN . \supset . M \in CʻR . N \in CʻS &\qquad \text{(3)}\\ +\vdash . (2) . (3) . &\supset \vdash \colon\ldotp \text{Hp} . \supset : T \in R \, \overline{\text{smor}}\,S . \supset . T \,\unicode{x2abd}\, \text{smor} &\qquad \text{(4)}\\ +\vdash . (1) . (4) . \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*164·48">*164·48</a>.</b> \[\begin{align}\vdash \colon\ldotp \text{Mult ax} . \supset : R, S &\in \text{Rel}^{2}\text{excl} + \cap \text{Nr}ʻQ . CʻR, CʻS \in \text{Cl}ʻ\text{Nr}ʻP . \supset .\\ +&R\, \text{smor smor}\, S . \Sigma ʻR \,\text{smor}\, \Sigma ʻS \quad[*164·47·44·46]\end{align}\]</p> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_386">[Pg 386]</span></p> +<h2 class="nobreak" id="*165"> +*165. RELATIONS OF RELATIONS OF COUPLES.</h2> +</div> + + +<p class="nind"><i>Summary of</i> *165.</p> + +<p>In the present number, we shall give various propositions concerning +the relation \(P\downarrow_{.,} ^{;}Q\), which has the same uses in +relation-arithmetic \(\alpha \downarrow_{,,}ʻʻ\beta\) has in cardinal +arithmetic. The propositions of this number will be used in the +next number to establish the properties of the arithmetical product +of two relations \(Q\) and \(P\), which is defined as \(\SigmaʻP\downarrow_{.,} ^{;}Q\). +Again in connection with exponentiation the propositions of the present +number will be useful, since, after the product of a relation of +relations has been defined (<a href="#*172">*172</a>), we shall define exponentiation by +means of the definition +\[ +P \,\,\text{exp}\,\, Q = \text{Prod}ʻP\downarrow_{.,} ^{;}Q \text{Df}. \quad(Cf. *176.) +\] +There will also be occasional uses of the propositions of +this number throughout the theory of series. The relation +\(P\downarrow_{.,} ^{;}Q\) is important because its structure is +thoroughly known. It is a \(\text{Rel}^{2}\text{excl}\) which consists +of \(\text{Nr}ʻQ\) relations, each like \(P\) (<a href="#*165·27">*165·27</a>); and if +\(P\,\text{smor}\,P'.Q\,\text{smor}\,Q'\), we can construct a double correlator +of \(P\downarrow_{.,} ^{;}Q\) and \(Pʻ\downarrow_{.,} ^{;}Q'\) without invoking +the multiplicative axiom. In fact we have</p> + +<p class="nind"><b>*165·362. </b> \[\begin{align}\vdash :R\upharpoonright CʻP'\in &P\,\overline{\text{smor}}\,Pʻ.S\upharpoonright CʻQ'\in Q\,\overline{\text{smor}}\,Q'.\supset .\\ +&(R\parallel \breve{S} )\upharpoonright Cʻ\Sigma ʻP'\downarrow_{.,} ^{;}Q'\in (P\downarrow_{.,} ^{;}Q)\,\overline{\text{smor}} + \,\overline{\text{smor}}\,(Pʻ\downarrow_{.,} ^{;}Q')\end{align}\]</p> + +<p>This proposition should be compared with <a href="#*113·127">*113·127</a>. In virtue of +<a href="#*164·31">*164·31</a>, together with various propositions of <a href="#*165">*165</a> and <a href="#*166">*166</a>, it will +appear that <a href="#*165·362">*165·362</a> includes *113·127 as part of what it asserts.</p> + +<p>In the present number, we begin with a set of propositions on fields. +We have</p> + +<p class="nind"><b>*165·12.</b> \(\vdash .CʻP\downarrow_{.,} ^{;}Q=P\downarrow_{.,} ʻʻCʻQ\)</p> + +<p class="nind"><b>*165·13.</b> \(\vdash .CʻP\downarrow_{.,} z=\downarrow zʻʻCʻP=(CʻP)\downarrow_{,,} z\)</p> + +<p class="nind">whence</p> + +<p><span class="pagenum" id="Page_387">[Pg 387]</span></p> + +<p class="nind"><b>*165·14.</b> \(\vdash .CʻʻCʻP\downarrow_{.,} ^{;}Q=(CʻP)\downarrow_{,,}ʻʻCʻQ\)</p> + +<p class="nind">which connects the theory of \(P\downarrow_{.,} ^{;}Q\) with that of +\(\alpha \downarrow_{,,}ʻʻ\beta\) (<a href="#*113">*113</a> and <a href="#*116">*116</a>). Hence</p> + +<p class="nind"><b>*165·16.</b> \(\vdash .Cʻ\Sigma ʻP\downarrow_{.,} ^{;}Q=CʻQ\times CʻP\)</p> + +<p>In <a href="#*166">*166</a>, we shall define \(Q\times P\) as \(\Sigma ʻP\downarrow_{.,}^{;}Q\); +thus the above will become +\[ +\vdash .Cʻ(Q\times P)=CʻQ\times CʻP. +\]</p> + +<p>We next have a set of propositions concerned with \(P\downarrow_{.,}\) +as a relation, and with the circumstances under which we can infer +\(x=y\) or \(P=Q\) from data as to \(P\downarrow_{.,} x\) and +\(Q\downarrow_{.,} y\). We have</p> + +<p class="nind"><b>*165·21.</b> \(\vdash .P\downarrow_{.,} ^{;}Q\in \text{Rel}^{2}\text{excl}\)</p> + +<p class="nind"><b>*165·211.</b> \(\vdash :\exists !CʻP\downarrow_{.,} x\cap CʻP\downarrow_{.,} y.\supset .x=y\)</p> + +<p class="nind"><b>*165·22.</b> \(\vdash :\dot{\exists} !P.\supset .P\downarrow_{.,} \in 1\rightarrow 1\)</p> + +<p>We then have various propositions concerning \(\dot{\Lambda}\), of +which the chief are</p> + +<p class="nind"><b>*165·241.</b> \(\vdash :Q=\dot{\Lambda} .\supset .P\downarrow_{.,} ^{;}Q=\dot{\Lambda}\)</p> + +<p class="nind"><b>*165·242.</b> \(\vdash :P=\dot{\Lambda} .\dot{\exists} !Q.\supset .P\downarrow_{.,} ^{;}Q=\dot{\Lambda} \downarrow \dot{\Lambda}\)</p> + +<p>We have next four propositions which are constantly used, proving that +\(P\downarrow_{.,} ^{;}Q\) consists of \(\text{Nr}ʻQ\) relations each +like \(P\). These propositions are</p> + +<p class="nind"><b>*165·25.</b> \(\vdash :\dot{\exists} !P.\supset .P\downarrow_{.,} ^{;}Q\text{smor}Q.(P\downarrow_{.,} )\upharpoonright CʻQ\in (P\downarrow_{.,} ^{;}Q)\overline{\text{smor}} + Q\)</p> + +<p class="nind"><b>*165·251.</b> \(\vdash .P\downarrow_{.,} x\,\text{smor}\,P.(\downarrow x)\upharpoonright CʻP\in (P\downarrow_{.,} x)\,\overline{\text{smor}}\,P\)</p> + +<p class="nind"><b>*165·26.</b> \(\vdash .CʻP\downarrow_{.,} ^{;}Q\subset \text{Nr}ʻP\)</p> + +<p class="nind"><b>*165·27.</b> \(\vdash :\dot{\exists} !P.\supset .P\downarrow_{.,} ^{;}Q\in \text{Rel}^{2}\text{excl}\cap \text{Nr}ʻQ.CʻP\downarrow_{.,} ^{;}Q\in + \text{Cl}ʻ\text{Nr}ʻP\)</p> + +<p>From <a href="#*165·3">*165·3</a> to *165·372, we are concerned with constructing a double +correlator of \(P\downarrow_{.,} ^{;}Q\) and \(Pʻ\downarrow_{.,} ^{;}Q'\) +when we are given simple correlators of \(P\) with \(P'\) and of \(Q\) +with \(Q'\). The result (<a href="#*165·362">*165·362</a>) has already been given. Hence we have</p> + +<p class="nind"><b>*165·37.</b> \(\vdash : P \,\text{smor}\,P'.Q \text{smor} Q'.\supset .P\downarrow_{.,} ^{;}Q\,\text{smor smor}\,P'\downarrow_{.,} ^{;}Q'\)</p> + +<p class="nind">and by <a href="#*164·48">*164·48</a> and <a href="#*165·27">*165·27</a> we have</p> + +<p class="nind"><b>*165·38.</b> \[\begin{align}\vdash \colon\ldotp &\text{Mult ax}.\supset :\\ +&R\in \text{Rel}^{2}\text{excl}\cap \text{Nr}ʻQ.CʻR\subset \text{Nr}ʻP.\supset .R\, \text{smor smor}\, P\downarrow_{.,} ^{;}Q\end{align}\]</p> + +<p>Hence propositions concerning a series of \(\beta\) series, each +containing \(\alpha\) terms (where \(\alpha\) and \(\beta\) are +relation-numbers), which in general require the multiplicative +axiom, can be deduced, assuming that axiom, from propositions<span class="pagenum" id="Page_388">[Pg 388]</span> (not +requiring the axiom) concerning \(P\downarrow_{.,} ^{;}Q\), where +\(\text{Nr}ʻP=\alpha\) and \(\text{Nr}ʻQ=\beta\). Thus the use of +\(P\downarrow_{.,} ^{;}Q\) enables us to minimize the use of the +multiplicative axiom.</p> + +<hr class="tb"> + +<p class="nind"><b>*165·01.</b> \(\vdash .P\downarrow_{.,} z=\downarrow z^{;}P \quad[*150·6]\)</p> + +<p class="nind"><b>*165·1.</b> \(\vdash :R(P\downarrow_{.,} ^{;}Q)S.\equiv .(\exists z,w).zQw.R=\downarrow z^{;}P.S=\downarrow w^{;}P \quad[*150·62]\)</p> + +<p class="nind"><b>*165·11.</b> \(\vdash :X(\downarrow z^{;}P)Y.\equiv .(\exists x,y).xPy.X=x\downarrow z.Y=y\downarrow z \quad [*150·55]\)</p> + +<p class="nind"><b>*165·12.</b> \(\vdash .CʻP\downarrow_{.,} ^{;}Q=P\downarrow_{.,} ʻʻCʻQ \quad[*150·22]\)</p> + +<p class="nind"><b>*165·13.</b> \(\vdash .CʻP\downarrow_{.,} z=\downarrow zʻʻCʻP=(CʻP)\downarrow_{,,} z \quad[*165·01.*150·22.*38·2]\)</p> + +<p class="nind"><b>*165·131.</b> \(\vdash .CʻʻP\downarrow_{.,} ʻʻ\beta =(CʻP)\downarrow_{,,}ʻʻ\beta \quad[*165·13.*38·11.*37·68]\)</p> + +<p class="nind"><b>*165·14.</b> \(\vdash .CʻʻCʻP\downarrow_{.,} ^{;}Q=(CʻP)\downarrow_{,,}ʻʻCʻQ \quad[*165·12·131]\)</p> + +<p class="nind"><b>*165·15.</b> \(\vdash .sʻCʻʻCʻP\downarrow_{.,} ^{;}Q=CʻQ\times CʻP \quad[*165·14.*113·1]\)</p> + +<p class="nind"><b>*165·16.</b> \(\vdash .Cʻ\Sigma ʻP\downarrow_{.,} ^{;}Q=CʻQ\times CʻP \quad[*165·15.*162·22]\)</p> + +<p class="nind"><b>*165·161.</b> \[\begin{align}\vdash :M(F^{;}P\downarrow_{.,} ^{;}Q)&N.\equiv .\\ +&(\exists x,y,z,w).x,y\in CʻP.zQw.M=x\downarrow z.N=y\downarrow w\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*150·52.\supset \\ +\vdash \colon\ldotp M(F^{;}P\downarrow_{.,} ^{;}Q)N.&\equiv :(\exists R,S).R(P\downarrow_{.,} ^{;}Q)S.M\in CʻR.N\in CʻS.\\ +[*165·1] &\equiv :(\exists R,S,z,w).zQw.R=\downarrow z^{;}P.S=\downarrow w^{;}P.M\in CʻR.N\in CʻS.\\ +[*165·01·13] &\equiv :(\exists R,S,z,w).zQw.R=\downarrow z^{;}P.S=\downarrow w^{;}P.\\ +&M\in \downarrow zʻʻCʻP.N\in \downarrow wʻʻCʻP.\\ +[*21·151] &\equiv :(\exists z,w).zQw.M\in \downarrow zʻʻCʻP.N\in \downarrow wʻʻCʻP.\\ +[*38·131] &\equiv :(\exists x,y,z,w).zQw.x,y\in CʻP.M=x\downarrow z.N=y\downarrow w\colon\ldotp \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*165·162.</b> \(\vdash :M(\dot{s} ʻCʻP\downarrow_{.,} ^{;}Q)N.\equiv .(\exists x,y,z).xPy.z\in CʻQ.M=x\downarrow z.N=y\downarrow z\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*165·12.*41·11.\supset \\ +\vdash :M(\dot{s} ʻCʻP\downarrow_{.,} ^{;}Q)N.&\equiv .(\exists R).R\in P\downarrow_{.,} ʻʻCʻQ.MRN.\\ +[*38·13] &\equiv .(\exists z).z\in CʻQ.M(P\downarrow_{.,} z)N.\\ +[*165·01·11] & \equiv .(\exists x,y,z).xPy.z\in CʻQ.M=x\downarrow z.N=y\downarrow z:\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_389">[Pg 389]</span></p> + +<p class="nind"><b>*165·17.</b> \[\begin{align}\vdash \colon\ldotp &M(\Sigma ʻP\downarrow_{.,} ^{;}Q)N.\equiv :(\exists x,y,z,w):\\ +&x,y\in CʻP.z,w\in CʻQ:zQw.\lor.z=w.xPy:M=x\downarrow z.N=y\downarrow w\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*165·161·162.*162·11.\supset \\ +\vdash \colon\ldotp M(\Sigma ʻP\downarrow_{.,} ^{;}Q)N.&\equiv :(\exists x,y,z,w).x,y\in CʻP.zQw.M=x\downarrow z.N=y\downarrow w.\lor.\\ +&(\exists x,y,z).xPy.z\in CʻQ.M=x\downarrow z.N=y\downarrow w:\\ +[*13·195] &\equiv :(\exists x,y,z,w).x,y\in CʻP.zQw.M=x\downarrow z.N=y\downarrow w.\lor.\\ +&(\exists x,y,z,w).xPy.z,w\in CʻQ.z=w.M=x\downarrow z.N=y\downarrow w:\\ +[*33·17.*4·71] &\equiv:(\exists x,y,z,w).x,y\in CʻP.z,w\in CʻQ.zQw.M=x\downarrow z.N=y\downarrow w.\lor.\\ +&(\exists x,y,z,w).x,y\in CʻP.z,w\in CʻQ.xPy.z=w.M=x\downarrow z.N=y\downarrow w:\\ +[*11·41.*4·4] &\equiv :(\exists x,y,z,w):x,y\in CʻP.z,w\in CʻQ:zQw.\lor.z=w.xPy:\\ +&M=x\downarrow z.N=y\downarrow w\colon\ldotp \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*165·18.</b> \(\vdash .\text{Cnv}ʻP\downarrow_{.,} ^{;}Q=P\downarrow_{.,} ^{;}\breve{Q} \quad[*150·12]\)</p> + +<p class="nind"><b>*165·181.</b> \(\vdash .\text{Cnv}ʻP\downarrow_{.,} z=\breve{P} \downarrow_{.,} z \quad[*165·01.*150·12]\)</p> + +<p class="nind"><b>*165·182.</b> \(\vdash .\text{Cnv}^{;}P\downarrow_{.,} ^{;}Q=\breve{P} \downarrow_{.,} ^{;}Q \quad[*165·181.*150·35]\)</p> + +<p class="nind"><b>*165·19.</b> \(\vdash .\text{Cnv}ʻ\text{Cnv}^{;}P\downarrow_{.,} ^{;}Q=\breve{P} \downarrow_{.,} ^{;}\breve{Q} + =\text{Cnv}^{;}\text{Cnv}ʻP\downarrow_{.,} ^{;}Q \quad[*165·18·182]\)</p> + +<p class="nind"><b>*165·2.</b> \(\vdash .P\downarrow_{.,} \in 1\rightarrow \text{Cls} \quad[*72·14]\)</p> + +<p class="nind"><b>*165·201.</b> \(\vdash .Cʻ(P\downarrow_{.,} z)=(CʻP\uparrow \iota ʻz)_{\Delta }ʻ\iota ʻz\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*35·103.&\supset \vdash :y(CʻP\uparrow \iota ʻz)z.\equiv .y\in CʻP:\\ +[*85·51] \supset \vdash .(CʻP\uparrow \iota ʻz)_{\Delta }ʻ\iota ʻz&=\downarrow zʻʻCʻP\\ +[*165·13] &=Cʻ(P\downarrow_{.,} z).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*165·202.</b> \(\vdash .CʻʻCʻP\downarrow_{.,} ^{;}Q=(CʻP\uparrow CʻQ)_{\Delta }ʻʻ\iotaʻʻCʻQ \quad[*165·14.*113·103]\)</p> + +<p class="nind"><b>*165·203.</b> \(\vdash .CʻʻCʻP\downarrow_{.,} ^{;}Q\in \text{Cls}^{2} \text{excl} \quad[*84·55.*165·202]\)</p> + +<p class="nind"><b>*165·204.</b> \(\vdash :CʻP\downarrow_{.,} x=CʻP\downarrow_{.,} y.\equiv .P\downarrow_{.,} x=P\downarrow_{.,} y\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*165·13.*55·232.\supset \\ +\vdash :CʻP\downarrow_{.,} x=CʻP\downarrow_{.,} y.\exists !CʻP\downarrow_{.,} x.&\supset .x=y.\\ +[*30·37] &\supset .P\downarrow_{.,} x=P\downarrow_{.,} y &\qquad \text{(1)}\\ +\vdash .*33·241.&\supset \vdash :CʻP\downarrow_{.,} x=CʻP\downarrow_{.,} y.CʻP\downarrow_{.,} x=\Lambda .\supset .P\downarrow_{.,} + x=\dot{\Lambda} .P \downarrow_{.,} y=\dot{\Lambda} &\qquad \text{(2)}\\ +\vdash .(1).(2).&\supset \vdash :CʻP\downarrow_{.,} x=CʻP\downarrow_{.,} y.\supset .P\downarrow_{.,} x=P\downarrow_{.,} y &\qquad \text{(3)}\\ +\vdash .(3).*30·37.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_390">[Pg 390]</span></p> + +<p class="nind"><b>*165·205.</b> \(\vdash .C\upharpoonright \text{D}ʻP\downarrow_{.,} \in 1\rightarrow 1 \quad[*165·204.*71·58]\)</p> + +<p class="nind"><b>*165·206.</b> \(\vdash :(x).\text{E}!P\downarrow_{.,} ʻx:(\alpha ).\alpha \subset \text{ᗡ}ʻP\downarrow_{.,} \quad[*38·12.*33·431]\)</p> + +<p class="nind"><b>*165·21.</b> \(\vdash .P\downarrow_{.,} ^{;}Q\in \text{Rel}^{2}\text{excl}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*165·205.*150·203.\supset \vdash .C\upharpoonright CʻP\downarrow_{.,} ^{;}Q\in 1\rightarrow 1 &\qquad \text{(1)}\\ +\vdash .(1).*165·203.*163·17.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*165·211.</b> \(\vdash :\exists !CʻP\downarrow_{.,} x\cap CʻP\downarrow_{.,} y.\supset .x=y \quad[*165·13.*55·232]\)</p> + +<p class="nind"><b>*165·212.</b> \(\vdash :\dot{\exists} !P.\equiv .\dot{\exists} !P\downarrow_{.,} x\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *165·11·01.\supset \vdash \dot{\exists} !P\downarrow_{.,} x.&\equiv .(\exists X,Y,x,y).xPy.X=x\downarrow z.Y=y\downarrow z.\\ +[*13·19] &\equiv .(\exists x,y).xPy:\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*165·22.</b> \(\vdash :\dot{\exists} !P.\supset .P\downarrow_{.,} \in 1\rightarrow 1\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*165·212.\supset \vdash \colon\ldotp \text{Hp}.\supset :\dot{\exists} !P\downarrow_{.,} x:\\ +[*30·37.*24·571.*33·24]&\supset :P\downarrow_{.,} x=P\downarrow_{.,} y.\supset .\exists !CʻP\downarrow_{.,} x\cap CʻP\downarrow_{.,} y.\\ +[*165·211] &\supset .x=y &\qquad \text{(1)}\\ +\vdash .(1).*71·54.*165·2.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*165·221.</b> \(\vdash \colon\ldotp \dot{\exists} !P.\supset :\dot{\exists} !P\downarrow_{.,} x\dot{\cap} P\downarrow_{.,} y.\equiv .P\downarrow_{.,} + x=P\downarrow_{.,} y.\equiv .x=y\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*33·252.\supset \vdash :\dot{\exists} !P\downarrow_{.,} x\dot{\cap} P\downarrow_{.,} y.&\supset .\exists !CʻP\downarrow_{.,} x\cap CʻP\downarrow_{.,} y.\\ +[*165·211] &\supset .x=y &\qquad \text{(1)}\\ +\vdash .*165·212.*25·571.\supset \vdash \colon\ldotp \dot{\exists} !P.&\supset :x=y.\supset .\dot{\exists} !P\downarrow_{.,} x\dot{\cap} P\downarrow_{.,} + y &\qquad \text{(2)}\\ +\vdash .(1).(2).*165·212.*30·37.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*165·222.</b> \[\begin{align}&\vdash \colon\ldotp \dot{\exists} !P.\supset :\exists !CʻP\downarrow_{.,} x\cap CʻP\downarrow_{.,} y.\equiv .CʻP \downarrow_{.,} + x=CʻP\downarrow_{.,} y.\equiv .x=y\\ +&[\text{Proof as in *165·221}]\end{align}\]</p> + +<p class="nind"><b>*165·223.</b> \(\vdash \colon\ldotp \dot{\exists} !P.\supset :P\downarrow_{.,} ^{;}Q=P\downarrow_{.,} ^{;}R.\equiv .Q=R\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*151·31.*165·22.&\supset \vdash \colon\ldotp \text{Hp}.\supset :P\downarrow_{.,} ^{;}Q=P\downarrow_{.,} ^{;}R.\supset .Q=R &\qquad \text{(1)}\\ +\vdash .*34·29.*150·1. &\supset \vdash :Q=R.\supset .P\downarrow_{.,} ^{;}Q=P\downarrow_{.,} ^{;}R &\qquad \text{(2)}\\ +\vdash .(1).(2).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*165·23">*165·23</a>.</b> \(\vdash :P\downarrow_{.,} x=Q\downarrow_{.,} y.\supset .P=Q\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*72·184.*150·153.\supset \vdash :\downarrow x^{;}P=\downarrow x^{;}Q.\supset .P=Q &\qquad \text{(1)}\\ +\vdash .(1).*165·01.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_391">[Pg 391]</span></p> + +<p class="nind"><b>*165·231.</b> \(\vdash :P\downarrow_{.,} x=Q\downarrow_{.,} x.\equiv .P=Q \quad[*165·23.*30·37]\)</p> + +<p class="nind"><b>*165·232.</b> \(\vdash \colon\ldotp \dot{\exists} !P.\lor.\dot{\exists} !Q:\supset :P\downarrow_{.,} x=Q\downarrow_{.,} y.\equiv .P=Q.x=y\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*165·23.\supset \vdash \colon\ldotp P\downarrow_{.,} x=Q\downarrow_{.,} y.&\supset :P=Q: &\qquad \text{(1)}\\ +[*13·12.\text{Hp}(1)] &\supset :P\downarrow_{.,} x=P\downarrow_{.,} y.Q\downarrow_{.,} x=Q\downarrow_{.,} y:\\ +[*165·221] &\supset :\dot{\exists} !P.\supset .x=y:\dot{\exists} !Q.\supset .x=y &\qquad \text{(2)}\\ +\vdash .(1).(2).&\supset \vdash \colon\ldotp \dot{\exists} !P.\lor.\dot{\exists} !Q:\supset :P\downarrow_{.,} x=Q\downarrow_{.,} y.\supset .P=Q.x=y &\qquad \text{(3)}\\ +\vdash .(3).*13·12·15.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*165·233.</b> \[\begin{align}&\vdash :\exists !CʻP\downarrow_{.,} x\cap CʻQ\downarrow_{.,} y.\equiv .x=y.\exists !CʻP\cap CʻQ\\ +&[*55·232.*165·13]\end{align}\]</p> + +<p class="nind"><b>*165·24.</b> \(\vdash :P=\dot{\Lambda} .\supset .P\downarrow_{.,} x=\dot{\Lambda} .P\downarrow_{.,} =\iota ʻ\dot{\Lambda} \uparrow \text{V}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*165·212.\text{Transp}.\supset \vdash :P=\dot{\Lambda} .\supset .P\downarrow_{.,} x=\dot{\Lambda} &\qquad \text{(1)}\\ +\vdash .(1).*38·1. \supset \vdash \colon\ldotp P=\dot{\Lambda} .\supset :R(P\downarrow_{.,} )x.\equiv .R=\dot{\Lambda} .\\ +[*51·15.*24·104] \equiv .R\in \iota ʻ\dot{\Lambda} .x\in \text{V}.\\ +[*35·103] \equiv .R(\iota ʻ\dot{\Lambda} \uparrow \text{V})x &\qquad \text{(2)}\\ +\vdash .(1).(2).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*165·241.</b> \(\vdash :Q=\dot{\Lambda} .\supset .P\downarrow_{.,} ^{;}Q=\dot{\Lambda} \quad[*150·42]\)</p> + +<p class="nind"><b>*165·242.</b> \(\vdash :P=\dot{\Lambda} .\dot{\exists} !Q.\supset .P\downarrow_{.,} ^{;}Q=\dot{\Lambda} \downarrow \dot{\Lambda}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*165·1·24.\supset \vdash \colon\ldotp P=\dot{\Lambda} .\supset :R(P\downarrow_{.,} ^{;}Q)S.&\equiv .(\exists z,w).zQw.R=\dot{\Lambda} .S=\dot{\Lambda} .\\ +[*10·35] &\equiv .\dot{\exists} !Q.R=\dot{\Lambda} .S=\dot{\Lambda} &\qquad \text{(1)}\\ +\vdash .(1).*55·13.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*165·243.</b> \(\vdash :\dot{\exists} !Q.\equiv .\dot{\exists} !P\downarrow_{.,} ^{;}Q\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*165·1.\supset \vdash :\dot{\exists} !P\downarrow_{.,} ^{;}Q.\equiv .(\exists x,y,R,S).xQy.R=P\downarrow_{.,} x.S=P\downarrow_{.,} y.\\ +[*13·19] \equiv .(\exists x,y).xQy:\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*165·244.</b> \(\vdash :\dot{\Lambda} \in CʻP\downarrow_{.,} ^{;}Q.\equiv .P=\dot{\Lambda} .\dot{\exists} !Q.\equiv .P\downarrow_{.,} ^{;}Q=\dot{\Lambda} + \downarrow \dot{\Lambda}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*165·212·12. &\supset \vdash :\dot{\Lambda} \in CʻP\downarrow_{.,} ^{;}Q.\supset .P=\dot{\Lambda} &\qquad \text{(1)}\\ +\vdash .*10·24.*33·24.&\supset \vdash :\dot{\Lambda} \in CʻP\downarrow_{.,} ^{;}Q.\supset .\dot{\exists} !P\downarrow_{.,} ^{;}Q.\\ +[*165·243] &\supset .\dot{\exists} !Q &\qquad \text{(2)}\\ +\vdash .*165·242.*55·15. &\supset \vdash :P = \dot{\Lambda} .\dot{\exists} !Q.\supset .\dot{\Lambda} \in CʻP\downarrow_{.,} ^{;}Q &\qquad \text{(3)}\\ +\vdash .(1).(2).(3). &\supset \vdash :\dot{\Lambda} \in CʻP\downarrow_{.,} ^{;}Q. \equiv .P = \dot{\Lambda} .\dot{\exists} !Q &\qquad \text{(4)}\\ +\vdash .*55·15. &\supset \vdash :P\downarrow_{.,} ^{;}Q. = \dot{\Lambda} \downarrow \dot{\Lambda} .\supset .\dot{\Lambda} \in CʻP\downarrow_{.,} ^{;}Q.\\ +[(4)] &\supset .P = \dot{\Lambda} .\dot{\exists} !Q &\qquad \text{(5)}\\ +\vdash .(5).*165·242. & \supset \vdash :P = \dot{\Lambda} .\dot{\exists} !Q. \equiv .P\downarrow_{.,} ^{;}Q = \dot{\Lambda} \downarrow \dot{\Lambda} &\qquad \text{(6)}\\ +\vdash .(4).(6).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_392">[Pg 392]</span></p> + +<p class="nind"><b>*165·245.</b> \[\begin{align}&\vdash \colon\ldotp \dot{\exists} !P.\lor.Q = \dot{\Lambda} : \equiv .\dot{\Lambda} {\sim} \in CʻP\downarrow_{.,} ^{;}Q. + \equiv .\dot{\Lambda} {\sim} \in (CʻP)\downarrow_{,,}ʻʻCʻQ\\ +&[*165·244.\text{Transp}.*33·241.*165·14]\end{align}\]</p> + +<p class="nind"><b>*165·25.</b> \[\begin{align}&\vdash :\dot{\exists} !P.\supset .P\downarrow_{.,} ^{;}Q\,\text{smor}\,Q.(P\downarrow_{.,} )\upharpoonright CʻQ\in (P\downarrow_{.,} + ^{;}Q)\,\overline{\text{smor}}\,Q\\ +&[*165·22·206.*151·231]\end{align}\]</p> + +<p class="nind"><b>*165·251.</b> \[\begin{align}&\vdash .P\downarrow_{.,} x\,\text{smor}\,P.(\downarrow x)\upharpoonright CʻP\in (P\downarrow_{.,} x)\overline{\text{smor}} P\\ +&[*72·184.*55·21.*151·22]\end{align}\]</p> + +<p class="nind"><b>*165·26.</b> \(\vdash .CʻP\downarrow_{.,} ^{;}Q\subset \text{Nr}ʻP \quad[*165·251·12.*152·11]\)</p> + +<p class="nind"><b><a id="*165·27">*165·27</a>.</b> \[\begin{align}&\vdash :\dot{\exists} !P.\supset .P\downarrow_{.,} ^{;}Q\in \text{Rel}^{2}\text{excl}\cap \text{Nr}ʻQ.CʻP\downarrow_{.,} ^{;}Q\in + \text{Cl}ʻ\text{Nr}ʻP\\ +&[*165·21·25.*152·11.*165·26]\end{align}\]</p> + +<p>The following propositions are concerned in proving that, if \(R\) is +a correlator of \(P\) and \(P'\) and \(S\) is a correlator of \(Q\) +and \(Q'\) then \(R\parallel \breve{S}\) (with its converse domain +limited) is a double correlator of \(P\downarrow_{.,} ^{;}Q\) and +\(P'\downarrow_{.,} ^{;}Q'\).</p> + +<p>This proposition is required subsequently in establishing likenesses.</p> + +<p class="nind"><b><a id="*165·3">*165·3</a>.</b> \(\vdash :\text{E}!Rʻy.\supset .\downarrow zʻRʻy = R\mid ʻ\downarrow zʻy\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*34·1.*38·11.\supset \vdash :u\{R\mid ʻ\downarrow zʻy\}w. &\equiv .(\exists v).uRv.v(y\downarrow z)w.\\ +[*55·13] &\equiv .uRy.w = z &\qquad \text{(1)}\\ +\vdash .(1).*30·4. \supset \vdash \colon\ldotp \text{Hp}.\supset :u{R\mid ʻ\downarrow zʻy}w. &\equiv .u = Rʻy.w = z.\\ +[*55·13.*38·11] &\equiv .u(\downarrow zʻRʻy)w\colon\ldotp \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*165·301.</b> \(\vdash :R\in 1\rightarrow \text{Cls}.\supset .\downarrow z\mid R = (R\mid )\mid (\downarrow z)\upharpoonright \text{ᗡ}ʻR\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*165·3. &\supset \vdash \colon\ldotp \text{E}!Rʻy.\supset :M\{(\downarrow z)\mid R\}y. \equiv .M\{(R\mid )\mid \downarrow z\}y\colon\ldotp \\ +[*71·16.*34·36]&\supset \vdash \colon\ldotp \text{Hp}.\supset :M\{(\downarrow z)\mid R\}y. \equiv .\\ +&M\{(R\mid )\mid \downarrow z\}y.y\in \text{ᗡ}ʻR\colon\ldotp \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*165·302.</b> \(\vdash :\text{E}!!RʻʻCʻP.\supset .\downarrow z^{;}R^{;}P = R\mid ^{;}\downarrow z^{;}P\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*165·3.\supset \vdash \colon\ldotp \text{Hp}.\supset :y\in CʻP.\supset .\downarrow zʻRʻy = R\mid ʻ\downarrow zʻy &\qquad \text{(1)}\\ +\vdash .(1).*150·35·13.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_393">[Pg 393]</span></p> + +<p class="nind"><b>*165·31.</b> \(\vdash :\text{E}!!RʻʻCʻP.\supset .(R^{;}P)\downarrow_{.,} z=R\mid ^{;}P\downarrow_{.,} z.(R^{;}P)\downarrow_{.,} ^{;}Q=(R\mid + )\dagger ^{;}P\downarrow_{.,} ^{;}Q\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*165·302·01.\supset \vdash :\text{Hp}.\supset .(R^{;}P)\downarrow_{.,} z&=R\mid ^{;}P\downarrow_{.,} z &\qquad \text{(1)}\\ +[*150·1] &=(R\mid )\dagger ʻP\downarrow_{.,} z &\qquad \text{(2)}\\ +\vdash .(1).(2).*150·35.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*165·311.</b> \[\begin{align}\vdash :R\upharpoonright CʻP\in 1\rightarrow \text{Cls}.&CʻP\subset \text{ᗡ}ʻR.\supset .\\ +&(R^{;}P)\downarrow_{.,} z=R\mid ^{;}P\downarrow_{.,} z.(R^{;}P)\downarrow_{.,} ^{;}Q=(R\mid )\dagger ^{;}P\downarrow_{.,} + ^{;}Q\\ +[*165·31.*71·571]\end{align}\]</p> + +<p class="nind"><b>*165·32.</b> \(\vdash :\text{E}!Sʻz.\supset .\downarrow (Sʻz)=(\mid \breve{S} )\mid \downarrow z.\downarrow (Sʻz)^{;}P=\mid \breve{S} ^{;}\downarrow z^{;}P\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*34·1.*43·101.*38·101.\supset \\ +\vdash :M\{(\mid \breve{S} )\mid \downarrow z\}x.\equiv .(\exists N).M=N\mid \breve{S} .N=x\downarrow z.\\ +[*13·195] \equiv .M=(x\downarrow z)\mid \breve{S} &\qquad \text{(1)}\\ +\vdash .(1).*55·581.\supset \\ +\vdash \colon\ldotp \text{Hp}.\supset :M\{(\mid \breve{S} )\mid \downarrow z\}x.\equiv .M=x\downarrow (Sʻz).\\ +[*38·101] \equiv .M\{\downarrow (Sʻz)\}x &\qquad \text{(2)}\\ +\vdash .(2).*21·43.\supset \vdash :\text{Hp}.\supset .\downarrow (Sʻz)=(\mid \breve{S} )\mid \downarrow z.\\ +[*150·13] \supset .\downarrow (Sʻz)^{;}P=\mid \breve{S} ^{;}\downarrow z^{;}P:\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*165·321.</b> \(\vdash :\text{E}!Sʻz.\supset .P\downarrow_{.,} (Sʻz)=\mid \breve{S} ^{;}P\downarrow_{.,} z \quad[*165·32·01]\)</p> + +<p class="nind"><b>*165·33.</b> \(\vdash :\text{E}!!SʻʻCʻQ.\supset .P\downarrow_{.,} ^{;}S^{;}Q=(\mid \breve{S} )\dagger ^{;}P\downarrow_{.,} ^{;}Q\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*165·321.*38·11.*150·1.\supset \\ +\vdash \colon\ldotp \text{Hp}.\supset :z\in CʻQ.\supset .P\downarrow_{.,} ʻSʻz=(\mid \breve{S} )\dagger ʻP\downarrow_{.,} ʻz &\qquad \text{(1)}\\ +\vdash .(1).*150·35.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*165·331.</b> \[\begin{align}&\vdash :S\upharpoonright CʻQ\in 1\rightarrow \text{Cls}.CʻQ\subset \text{ᗡ}ʻS.\supset .P\downarrow_{.,} ^{;}S^{;}Q=(\mid + \breve{S} )\dagger ^{;}P\downarrow_{.,} ^{;}Q\\ +&[*165·33.*71·571]\end{align}\]</p> + +<p class="nind"><b>*165·34.</b> \(\vdash :\text{E}!!RʻʻCʻP.\text{E}!!SʻʻCʻQ.\supset .(R^{;}P)\downarrow_{.,} ^{;}(S^{;}Q)=(R\parallel + \breve{S} )\dagger ^{;}(P\downarrow_{.,} ^{;}Q)\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*165·31.\supset \vdash :\text{Hp}.\supset .(R^{;}P)\downarrow_{.,} ^{;}(S^{;}Q)&=(R\mid )\dagger ^{;}P\downarrow_{.,} + ^{;}(S^{;}Q)\\ +[*165·33] &=(R\mid )\dagger ^{;}(\mid \breve{S} )\dagger ^{;}P\downarrow_{.,} ^{;}Q\\ +[*150·13·14.(*43·01)] &=(R\parallel \breve{S} )\dagger ^{;}(P\downarrow_{.,} ^{;}Q):\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_394">[Pg 394]</span></p> + +<p class="nind"><b>*165·341.</b> \[\begin{align}\vdash :R \upharpoonright CʻP,S &\upharpoonright CʻQ \in 1 \rightarrow \text{Cls}.CʻP \subset \text{ᗡ}ʻR.CʻQ \subset \text{ᗡ}ʻS.\supset .\\ +&(R^{;}P) \downarrow_{.,} ^{;}(S^{;}Q) = (R \parallel \breve{S} )\dagger ^{;}P\downarrow_{.,} ^{;}Q + \quad[*165·34.*71·571]\end{align}\]</p> + +<p class="nind"><b>*165·35.</b> \(\vdash :R \upharpoonright CʻP \in \text{Cls} \rightarrow 1.CʻP \subset \text{ᗡ}ʻR.\supset .(R\mid )\upharpoonright Cʻ\Sigma ʻP \downarrow_{.,} ^{;}Q + \in 1 \rightarrow 1\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*113·118.*165·16.\supset \vdash .sʻ\text{D}ʻʻCʻ\Sigma ʻP\downarrow_{.,} ^{;}Q \subset CʻP &\qquad \text{(1)}\\ +\vdash .(1).*74·751 \frac{Cʻ\Sigma ʻP\downarrow_{.,} ^{;}Q,\,CʻP}{\lambda,\,\alpha}.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*165·351.</b> \(\vdash :S \upharpoonright CʻQ \in \text{Cls} \rightarrow 1.CʻQ \subset \text{ᗡ}ʻS.\supset .(\mid \breve{S} ) \upharpoonright Cʻ\Sigma ʻP \downarrow_{.,} ^{;}Q + \in 1 \rightarrow 1\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*113·118.*165·16.&\supset \vdash .sʻ\text{ᗡ}ʻʻCʻ\Sigma ʻP \downarrow_{.,} ^{;}Q \subset CʻQ.\\ +\left[*74·75 \frac{\breve{S}}{Q}\right] &\supset \vdash :\text{Hp}.\supset .(\mid \breve{S} ) \upharpoonright Cʻ\Sigma ʻP\downarrow_{.,} ^{;}Q + \in 1 \rightarrow 1:\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*165·352.</b> \[\begin{align}\vdash :R \upharpoonright CʻP,S \upharpoonright CʻQ \in \text{Cls} \rightarrow 1.CʻP \subset \text{ᗡ}ʻR.&CʻQ \subset \text{ᗡ}ʻS.\supset .\\ +&(R\parallel \breve{S} ) \upharpoonright Cʻ\Sigma ʻP\downarrow_{.,} ^{;}Q \in 1 \rightarrow 1\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*113·118.*165·16.\supset \\ +\vdash :\text{Hp}.\supset .sʻ\text{D}ʻʻCʻ\Sigma ʻP\downarrow_{.,} ^{;}Q \subset CʻP.sʻ\text{ᗡ}ʻʻCʻ\Sigma ʻP\downarrow_{.,} ^{;}Q \subset CʻQ.\\ +[*74·773] \supset .(R \parallel \breve{S} ) \upharpoonright Cʻ\Sigma ʻP\downarrow_{.,} ^{;}Q \in 1 \rightarrow 1:\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*165·36">*165·36</a>.</b> \[\begin{align}\vdash :R \upharpoonright CʻP' \in \,P\overline{\text{smor}}\, &P'.\supset .\\ +&(R\mid ) \upharpoonright Cʻ\Sigma ʻP'\downarrow_{.,} ^{;}Q \in (P\downarrow_{.,} ^{;}Q)\,\overline{\text{smor}} \,\overline{\text{smor}}\, (P'\downarrow_{.,} + ^{;}Q)\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*151·22.*165·35. &\supset \vdash :\text{Hp}.\supset .(R\mid ) \upharpoonright Cʻ\Sigma ʻPʻ\downarrow_{.,} ^{;}Q \in 1 \rightarrow 1 &\qquad \text{(1)}\\ +\vdash .*43·3. &\supset \vdash .Cʻ\Sigma ʻPʻ\downarrow_{.,} ^{;}Q \subset \text{ᗡ}ʻR\mid &\qquad \text{(2)}\\ +\vdash .*151·22.*165·311.&\supset \vdash :\text{Hp}.\supset .P\downarrow_{.,} ^{;}Q = (\mid R)\dagger ^{;}Pʻ\downarrow_{.,} ^{;}Q &\qquad \text{(3)}\\ +\vdash .(1).(2).(3).*164·18.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*165·361">*165·361</a>.</b> \[\begin{align}\vdash :S\upharpoonright CʻQ' \in Q&\overline{\text{smor}} Q'.\supset .\\ +&(\mid \breve{S} )\upharpoonright Cʻ\Sigma ʻP\downarrow_{.,} ^{;}Q' \in (P\downarrow_{.,} ^{;}Q)\,\overline{\text{smor}} \,\overline{\text{smor}}\, (P\downarrow_{.,} + ^{;}Q')\\ +[*165·351·331]\end{align}\]</p> + +<p>The proof proceeds as in <a href="#*165·36">*165·36</a>.</p> + +<p class="nind"><b><a id="*165·362">*165·362</a>.</b> \[\begin{align}\vdash :R \upharpoonright CʻP' \in P\,&\overline{\text{smor}}\,P'.S \upharpoonright CʻQ' \in Q\,\overline{\text{smor}}\,Q'.\supset .\\ +&(R \parallel \breve{S} ) \upharpoonright Cʻ\Sigma ʻP'\downarrow_{.,} ^{;}Q' \in (P\downarrow_{.,} ^{;}Q) + \,\overline{\text{smor}} \,\overline{\text{smor}}\, (P'\downarrow_{.,} ^{;}Q')\\ +[*165·352·341]\end{align}\]</p> + +<p>The above three propositions are of great utility in relation-arithmetic.</p> + +<p><span class="pagenum" id="Page_395">[Pg 395]</span></p> + +<p class="nind"><b>*165·37.</b> \[\begin{align}&\vdash :P\,\text{smor}\,P'.Q\text{smor}Q'.\supset .P\downarrow_{.,} ^{;}Q\,\text{smor smor}\,P'\downarrow_{.,} ^{;}Q'\\ +&[*165·362.*164·11.*151·12]\end{align}\]</p> + +<p class="nind"><b>*165·38.</b> \[\begin{align}\vdash \colon\ldotp \text{Mult ax}.\supset :R\in \text{Rel}^{2}\text{excl}\cap &\text{Nr}ʻQ.CʻR\subset \text{Nr}ʻP.\supset .\\ +&R\,\text{smor smor}\,P\downarrow_{.,} ^{;}Q\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*164·48.*165·27.&\supset \vdash :\text{Hp}.\dot{\exists} !P.\supset .R\,\text{smor smor}\,P\downarrow_{.,} ^{;}Q &\qquad \text{(1)}\\ +\vdash .*153·17.*165·241.&\supset \\ +\vdash :Q=\dot{\Lambda} .R\in \text{Rel}^{2}\text{excl}\cap \text{Nr}ʻQ.&\supset .R=\dot{\Lambda} .P\downarrow_{.,} ^{;}Q=\dot{\Lambda} .\\ +[*164·32] &\supset .R\,\text{smor smor}\,P\downarrow_{.,} ^{;}Q &\qquad \text{(2)}\\ +\vdash .*165*242&.\supset \vdash :P=\dot{\Lambda} .\dot{\exists} !Q.\supset .P\downarrow_{.,} ^{;}Q=\dot{\Lambda} \downarrow \dot{\Lambda} &\qquad \text{(3)}\\ +\vdash .*153·17.*51·4.*151·32.\supset \\ +\vdash :R\in \text{Nr}ʻQ.CʻR\subset \text{Nr}ʻP.P=\dot{\Lambda} .\dot{\exists} !Q.&\supset .CʻR={℩}ʻ\dot{\Lambda} .\\ +[*56·381] &\supset .R=\dot{\Lambda} \downarrow \dot{\Lambda} &\qquad \text{(4)}\\ +\vdash .(3).(4).*153·101.*164·34.\supset \\ +\vdash :R\in \text{Nr}ʻQ.CʻR\subset \text{Nr}ʻP.P=\dot{\Lambda} .\dot{\exists} !Q.&\supset .R\,\text{smor smor}\,P\downarrow_{.,} ^{;}Q &\qquad \text{(5)}\\ +\vdash .(1).(2).(5).\supset \vdash .\text{Prop} +\end{array} +\]</p> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_396">[Pg 396]</span></p> +<h2 class="nobreak" id="*166"> +*166. THE PRODUCT OF TWO RELATIONS.</h2> +</div> + + +<p class="nind"><i>Summary of</i> *166.</p> + +<p>The product \(Q \times P\) is defined as \(\Sigma ʻP\downarrow_{.,}^{;}Q\). +This is a relation which has for its field all the couples that +can be formed by choosing the referent in \(CʻP\) and the relatum in +\(CʻQ\). These couples are arranged by \(Q \times P\) on the following +principle: If the relatum of the one couple has the relation \(Q\) to +the relatum of the other, we put the one before the other, and if the +relata of the two couples are equal while the referent of the one has +the relation \(P\) to the referent of the other, we put the one before +the other. Thus in advancing from any term \(x\downarrow y\) in the +field of \(Q \times P\), we first keep \(y\) fixed and alter \(x\) into +later terms as long as possible; then we alter \(y\) into a later term, +move \(x\) back to the beginning, and so on. Thus with a given \(y\), +we get a series which is like \(P\), and this series is wholly followed +or wholly preceded by the series with the referent \(y'\), where \(y'\) +follows or precedes \(y\).</p> + +<p>The propositions of this number are for the most part immediate +consequences of those of <a href="#*165">*165</a>. The most important of them are:</p> + +<p class="nind"><b>*166·12.</b> \(\vdash .Cʻ(P\times Q)=CʻP\times CʻQ\)</p> + +<p class="nind"><b>*166·13.</b> \(\vdash \colon\ldotp P\times Q=\dot{\Lambda} .\equiv :P=\dot{\Lambda} .\lor.Q=\dot{\Lambda}\)</p> + +<p>Hence it follows that an ordinal product of a finite number of factors +vanishes when, and only when, one of its factors vanishes.</p> + +<p class="nind"><b>*166·16.</b> + \(\vdash .\overrightarrow{B}ʻ(P\times Q)=\overrightarrow{B}ʻP\times \overrightarrow{B}ʻQ.\overrightarrow{B}ʻ\text{Cnv}ʻ(P\times Q)=\overrightarrow{B}ʻ\breve{P} \times \overrightarrow{B}ʻ\breve{Q}\)</p> + +<p class="nind"><b>*166·23.</b> \(\vdash :P\,\text{smor}\,P'.Q\text{smor}Q'.\supset .Q\times P\,\text{smor}\,Q'\times P'\)</p> + +<p>This proposition shows that the relation-number of a product \(Q \times P\) +depends only upon the relation-numbers of its factors.</p> + +<p class="nind"><b>*166·24.</b> \[\begin{align}\vdash \colon\ldotp \text{Mult ax}.\supset :R\in \text{Rel}^{2}\text{excl}\cap \text{Nr}ʻQ.CʻR\subset \text{Nr}ʻ&P.\supset .\\ +&\Sigma ʻR\,\text{smor}\,Q\times P\end{align}\]</p> + +<p>This proposition connects addition and multiplication (cf. note to +*166·24, below).</p> + +<p class="nind"><b>*166·42.</b> \(\vdash .(P\times Q)\times R\,\text{smor}\,P\times (Q\times R)\)</p> + +<p>This is the associative law. The distributive law has two forms:</p> + +<p><span class="pagenum" id="Page_397">[Pg 397]</span></p> + +<p class="nind"><b>*166·44.</b> \(\vdash .\Sigma ʻ\times P^{;}Q=(\Sigma ʻQ)\times P\)</p> + +<p class="nind"><b>*166·45.</b> \(\vdash .(Q\unicode{x2909}R)\times P=(Q\times P)\unicode{x2909}(R\times P)\)</p> + +<p>We do not have in general (cf. note before *166·44, below) +\[ +P\times (Q\unicode{x2909}R)=(P\times Q)\unicode{x2909}(P\times R). +\]</p> + +<p>We have also a distributive law for the addition of a single term, +<i>i.e.</i></p> + +<p class="nind"><b>*166·53.</b> \(\vdash :\dot{\exists} !Q.\supset .(Q\unicode{x21f8} y)\times P=(Q\times P)\unicode{x2909}(P\downarrow_{.,} y)\)</p> + +<p class="nind"><b>*166·531.</b> \(\vdash :\dot{\exists} !Q.\supset .(y\unicode{x21f7}Q)\times P=(P\downarrow_{.,} y)\unicode{x2909}(Q\times P)\)</p> + +<p>Here again the law does not hold in general for \(P\times(Q\unicode{x21f8} y)\) +or \(P\times (y\unicode{x21f7}Q)\).</p> + +<hr class="tb"> + +<p class="nind"><b>*166·01.</b> \(Q\times P=\Sigma ʻP\downarrow_{.,} ^{;}Q \quad\text{Df}\)</p> + +<p class="nind"><b>*166·1.</b> \(\vdash .Q\times P=\Sigma ʻP\downarrow_{.,} ^{;}Q \quad[(*166·01)]\)</p> + +<p class="nind"><b>*166·11.</b> \[\begin{align}\vdash \colon\ldotp M(Q\times P)&N.\equiv :(\exists x,y,z,w):x,y\in CʻP.z,w\in CʻQ:zQw.\lor.\\ +&z=w.xPy:M=x\downarrow z.N=y\downarrow w \quad[*165·17.*166·1]\end{align}\]</p> + +<p class="nind"><b>*166·111.</b> \[\begin{align}\vdash \colon\ldotp M(P\times Q)&N.\equiv :(\exists x,y,z,w):x,y\in CʻP.z,w\in CʻQ:xPy.\lor.\\ +&x=y.zQw:M=z\downarrow x.N=w\downarrow y \quad[*165·17.*166·1]\end{align}\]</p> + +<p class="nind"><b>*166·112.</b> \[\begin{align}\vdash \colon\ldotp (x\downarrow z)(Q\times P)(y\downarrow w).&\equiv :x,y\in CʻP.z,w\in CʻQ:zQw.\lor.\\ +&z=w.xPy \quad[*166·11.*55·202.*13·22]\end{align}\]</p> + +<p class="nind"><b>*166·113.</b> \[\begin{align}\vdash \colon\colon x,y\in &CʻP.z,w\in CʻQ.\supset \colon\ldotp \\ +&(x\downarrow z)(Q\times P)(y\downarrow w).\equiv :zQw.\lor.z=w.xPy \quad[*166·112]\end{align}\]</p> + +<p class="nind"><b>*166·12.</b> \(\vdash .Cʻ(P\times Q)=CʻP\times CʻQ \quad[*165·16.*166·1]\)</p> + +<p class="nind"><b>*166·13.</b> \(\vdash \colon\ldotp P\times Q=\dot{\Lambda} .=:P=\dot{\Lambda} .\lor.Q=\dot{\Lambda} \quad[*166·12.*113·114.*33·241]\)</p> + +<p class="nind"><b>*166·14.</b> \(\vdash :\dot{\exists} !P\times Q.\equiv .\dot{\exists} !P.\dot{\exists} !Q \quad[*166·13]\)</p> + +<p class="nind"><b>*166·15.</b> \(\vdash .\text{Cnv}ʻ(P\times Q)=\breve{P} \times \breve{Q} \quad[*165·19 . *162·2]\)</p> + +<p class="nind"><b>*166·16.</b> + \(\vdash .\overrightarrow{B}ʻ(P\times Q)=\overrightarrow{B}ʻP\times \overrightarrow{B}ʻQ.\overrightarrow{B}ʻ\text{Cnv}ʻ(P\times Q)=\overrightarrow{B}ʻ\breve{P} \times \overrightarrow{B}ʻ\breve{Q}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*166·111.*93·103.\supset \\ +\vdash \colon\ldotp M\in \overrightarrow{B}ʻ(P\times Q).&\equiv :(\exists x,z):x\in CʻP.z\in CʻQ.M=z\downarrow x:\\ +&{\sim}(\exists y).yPx:{\sim}(\exists w).wQz:\\ +[*93·103] &\equiv :(\exists x,z).x\in \overrightarrow{B}ʻP.y\in \overrightarrow{B}ʻQ.M=z\downarrow x:\\ +[*113·101] &\equiv :M\in \overrightarrow{B}ʻP\times \overrightarrow{B}ʻQ &\qquad \text{(1)}\\ +\vdash .(1).*166·15.&\supset \vdash .\overrightarrow{B}ʻ\text{Cnv}ʻ(P\times Q)=\overrightarrow{B}ʻ\breve{P} \times \overrightarrow{B}ʻ\breve{Q} &\qquad \text{(2)}\\ +\vdash .(1).(2).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_398">[Pg 398]</span></p> + +<p>The above proposition is used in the ordinal theory of progressions +(*263·62·65).</p> + +<p class="nind"><b>*166·2.</b> \[\begin{align}&\vdash :R\upharpoonright CʻPʻ\in P\,\overline{\text{smor}}\,Pʻ.\supset .(R\mid )\upharpoonright Cʻ(Q\times Pʻ)\in (Q\times P)\overline{\text{smor}} (Q\times Pʻ)\\ +&[*165·36.*166·1.*164·14]\end{align}\]</p> + +<p class="nind"><b>*166·21.</b> + \[\begin{align}&\vdash :S\upharpoonright CʻQʻ\in Q\,\overline{\text{smor}}\,Qʻ.\supset .(\mid \breve{S} )\upharpoonright Cʻ(Qʻ\times P)\in (Q\times P)\overline{\text{smor}} (Qʻ\times P)\\ +&[*165·361.*166·1.*164·14]\end{align}\]</p> + +<p class="nind"><b>*166·22.</b> \[\begin{align}\vdash :R\upharpoonright CʻP'\in P\,\overline{\text{smor}}\,P'.&S\upharpoonright CʻQ'\in Q\overline{\text{smor}} Q'.\supset .\\ +&(R\parallel \breve{S} )\upharpoonright Cʻ(Q'\times P')\in (Q\times P)\,\overline{\text{smor}}\,(Q'\times P')\\ +[*165·362.*166·1.*164·14]\end{align}\]</p> + +<p>This proposition gives the correlator for the product when correlators +are given for the factors.</p> + +<p class="nind"><b>*166·23.</b> \[\begin{align}&\vdash :P \,\text{smor} P'.Q \text{smor} Q'.\supset .Q\times P \,\text{smor} \,Q'\times P'\\ +&[*166·22.*151·12]\end{align}\]</p> + +<p>This proposition enables us to use \(Q\times P\) to define the +product of the relation-numbers of \(Q\) and \(P\), for it shows +that the relation-number of \(Q\times P\) is determinate when the +relation-numbers of \(Q\) and \(P\) are given. We shall therefore (in +Section D of this part) define the product of two relation-numbers +\(\nu\) and \(\mu\) as the relation-number of \(Q\times P\) when +\(\text{N}_{0}\text{r}ʻQ=\nu\) and \(\text{N}_{0}\text{r}ʻP=\mu\).</p> + +<p class="nind"><b>*166·24.</b> \[\begin{align}&\vdash \colon\ldotp \text{Mult ax}.\supset :R\in \text{Rel}^{2}\text{excl}\cap \text{Nr}ʻQ.CʻR\subset \text{Nr}ʻP.\supset .\\ +&\Sigma ʻR\,\text{smor}\,Q\times P \quad[*165·38.*164·151.*166·1]\end{align}\]</p> + +<p>This proposition exhibits the connection of addition and +multiplication. If we put \(\text{Nr}ʻP=\mu\) and \(\text{Nr}ʻQ=\nu\), +then \(\Sigma ʻR\) in the above proposition is the sum of \(\nu\) +relations of which each is a \(\mu\). In virtue of the above +proposition, it follows that (if the multiplicative axiom is assumed) +\(\text{Nr}ʻ\Sigma ʻR=\nu \times \mu\). In other words, assuming the +multiplicative axiom, the sum of \(\nu\) series (or other relations), +each of which has \(\mu\) terms, has \(\nu \times \mu\) terms.</p> + +<p class="nind"><b>*166·3.</b> \[\begin{align}&\vdash :\exists !Cʻ(P\times Q)\cap Cʻ(Pʻ\times Qʻ).\equiv .\exists !CʻP\cap CʻPʻ.\exists !CʻQ\cap CʻQʻ\\ +&[*166·12.*113·19]\end{align}\]</p> + +<p>The analogous proposition +\[ +\begin{aligned} +\dot{\exists} !(P\times Q)\dot{\cap} (Pʻ\times Qʻ).&\equiv :\\ +\dot{\exists} !(P\dot{\cap} Pʻ).&\exists !CʻQ\cap CʻQʻ.\lor.\dot{\exists} !(Q\dot{\cap} Qʻ).\exists !CʻP\cap CʻPʻ +\end{aligned} +\] +is only true in general if \(P\,\unicode{x2abd}\, J.Pʻ\,\unicode{x2abd}\, J\)</p> + +<p class="nind"><b>*166·31.</b> \(\vdash .\dot{s} ʻCʻ(Q\times P)=CʻP\uparrow CʻQ \quad[*113·115.*166·12]\)</p> + +<p class="nind"><b>*166·311.</b> \[\begin{align}&\vdash :\dot{\exists} !Q.\supset .sʻ\text{D}ʻʻCʻ(Q\times P)=CʻP:\dot{\exists} !P.\supset .sʻ\text{ᗡ}ʻʻCʻ(Q\times P)=CʻQ\\ +&[*113·116.*166·12.*33·24]\end{align}\]</p> + +<p class="nind"><b>*166·312.</b> \[\begin{align}&\vdash .sʻ\text{D}ʻʻCʻ(Q\times P)\subset CʻP.sʻ\text{ᗡ}ʻʻCʻ(Q\times P)\subset CʻQ\\ +&[*113·118.*166·12]\end{align}\]</p> + +<p><span class="pagenum" id="Page_399">[Pg 399]</span></p> + +<p>The following propositions are lemmas for the associative law (<a href="#*166·42">*166·42</a>).</p> + +<p class="nind"><b>*166·4.</b> \[\begin{align}\vdash \colon\ldotp M \{(P \times Q) \times R\} &M'. \equiv : (\exists x, y, z, x', y', z') :\\ +&x, x' \in CʻP . y, y' \in CʻQ . z, z' \in CʻR :\\ +&xPx' . \lor . x = x' . yQy'. \lor . x = x' . y = yʻ . zRz' :\\ +&M = z \downarrow (y \downarrow x) . M' = z' \downarrow (y' \downarrow x')\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *116·111 . &\supset \vdash \colon\ldotp M \{(P \times Q) \times R\} M' . \equiv : (\exists N, N', z, z') :\\ +&N, N' \in Cʻ(P \times Q) . z, z' \in CʻR : N (P \times Q) N' . \lor . N = N' . zRz' :\\ +&M = z \downarrow N . M' = z' \downarrow N' :\\ +[*116·12 . *113·101] &\equiv : (\exists N, N', x, x', y, y', z, z') :\\ +&x, x' \in CʻP . y, y' \in CʻQ . z, z' \in CʻR . N = y \downarrow x . N' = y' \downarrow x' :\\ +&N (P \times Q) N' . \lor . N = N' . zRz' : M = z \downarrow N . M' = z' \downarrow N' :\\ +[*13·22 . *116·113] &\equiv : (\exists x, x', y, y', z, z') : x, x' \in CʻP . y, y' \in CʻQ . z, z' \in CʻR :\\ +&xPx' . \lor . x = x' . yQy' . \lor . y \downarrow x = y' \downarrow x' . zRz' :\\ +&M = z \downarrow (y \downarrow x) . M' = z' \downarrow (y' \downarrow x') :\\ +[*55·202] &\equiv : (\exists x, x', y, y', z, z') : x, x' \in CʻP . y, y' \in CʻQ . z, z' \in CʻR :\\ +&xPx' . \lor . x = x' . yQy' . \lor . x = x' . y = y' . zRz' :\\ +&M = z \downarrow (y \downarrow x) . M' = z' \downarrow (y' \downarrow x') \colon\ldotp \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*166·401.</b> \[\begin{align}\vdash \colon\ldotp N \{P \times (Q \times R)\} &N'. \equiv : (\exists x, x', y, y', z, z') :\\ +&x, x' \in CʻP . y, y' \in CʻQ . z, z' \in CʻR :\\ +&xPx' . \lor . x = x' . yQy' . \lor . x = x' . y = y' . zRz' :\\ +&N = (z \downarrow y) \downarrow x . N' = (z' \downarrow y') \downarrow xʻ +\quad[\text{Proof as in *166·4}]\end{align}\]</p> + +<p class="nind"><b>*166·41.</b> \[\begin{align}\vdash : T = \hat{M} \hat{N} &\{(\exists x, y, z) . x \in CʻP . y \in CʻQ . z \in CʻR . M = z \downarrow (y \downarrow x)\}.\\ +&N = \{(z \downarrow y) \downarrow x\} . \supset . T \in \{(P \times Q) \times R\} \overline{\text{smor}} \{P \times (Q \times R)\}\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *21·33 . \supset \vdash \colon\colon \text{Hp} . &\supset \colon\ldotp MTN . M'TN . \supset :\\ +&(\exists x, x', y, y', z, z') : x, x' \in CʻP . y, y' \in CʻQ . z, z' \in CʻR :\\ +&M = z \downarrow (y \downarrow x) . M' = z' \downarrow (y'\downarrow x') :\\ +&N = (z \downarrow y) \downarrow x . N = (z' \downarrow y') \downarrow xʻ :\\ +[*55·202] &\supset : (\exists x, x', y, y', z, z') . M = z \downarrow (y \downarrow x) . M' = z' \downarrow (y' \downarrow x') .\\ +&x = x' . y = y' . z = z' :\\ +[*13·22] &\supset : M = M' &\qquad \text{(1)}\\ +\text{Similarly}\quad \vdash \colon\ldotp \text{Hp} . &\supset : MTN . MTNʻ . \supset . N = N' &\qquad \text{(2)}\\ +\vdash . (1) . (2) . &\supset \vdash : \text{Hp} . \supset . T \in 1 \rightarrow 1 &\qquad \text{(3)}\\ +\vdash . *21·33 . *13·19 . &\supset \vdash : \text{Hp} . \supset .\\ +\text{ᗡ}ʻT &= \hat{N} \{(\exists x, y, z) . x \in CʻP . y \in CʻQ . z \in CʻR . N = (z \downarrow y) \downarrow x\}\\ +[*113·101] &= CʻP \times (CʻQ \times CʻR)\\ +[*166·12] &= Cʻ\{P \times (Q \times R)\} &\qquad \text{(4)}\\ +\vdash . *166·401 . &\supset \vdash \colon\colon \text{Hp} . \supset \colon\ldotp M \{T^{;}(P \times (Q \times R))\} M' . \equiv : (\exists x, x', y, y', z, z', N, N') :\\ +&x, x' \in CʻP . y, y' \in CʻQ . z, z' \in CʻR . N = (z \downarrow y) \downarrow x . N' = (z' \downarrow y') \downarrow x' .\\ +&M = z \downarrow (y \downarrow x) . M' = z' \downarrow (y' \downarrow x') :\\ +&xPx'. \lor . x = x' . yQy' . \lor . x = x' . y = y' . zRz' :\\ +[*13·19 . *166·4] &\equiv : M \{(P \times Q) \times R\} M' &\qquad \text{(5)}\\ +\vdash . (3) . (4) . (5) . *151·11 . \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_400">[Pg 400]</span></p> + +<p class="nind"><b><a id="*166·42">*166·42</a>.</b> \(\vdash . (P \times Q) \times R \text{smor} P \times (Q \times R) \quad[*166·41]\)</p> + +<p>This is the associative law for the kind of multiplication concerned in +this number.</p> + +<p class="nind"><b>*166·421.</b> \(P \times Q \times R = (P \times Q) \times R \quad\text{Df}\)</p> + +<p>This definition serves merely for the avoidance of brackets.</p> + +<p>The two following propositions give the distributive law. In +relation-arithmetic, this is in general only true in one of its two +forms, <i>i.e.</i> we have +\[ +\begin{aligned} +(Q \unicode{x2909} R) \times P &= (Q \times P) \unicode{x2909} (R \times P),\\ +\text{but not}\quad P \times (Q \unicode{x2909} R) &= (P \times Q) \unicode{x2909} (P \times R). +\end{aligned} +\] +The latter is true for finite series, but not for infinite series or +(except in exceptional cases) for relations which are not serial.</p> + +<p class="nind"><b>*166·44.</b> \(\vdash . \Sigma ʻ \times P^{;}Q = (\Sigma ʻQ) \times P\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *166·1 . *38·11 . *150·1 . \supset \vdash . \Sigma ʻ \times P^{;}Q &= \Sigma ʻ\Sigma ^{;}(P \downarrow_{.,} ) \dagger ^{;}Q\\ +[*162·34·35] &= \Sigma ʻP \downarrow_{.,} ^{;}\Sigma ʻQ\\ +[*166·1] &= (\Sigma ʻQ) \times P . \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*166·45.</b> \(\vdash . (Q \unicode{x2909} R) \times P = (Q \times P) \unicode{x2909} (R \times P)\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *166·1 . \supset \vdash . (Q \times P) \unicode{x2909} (R \times P) &= \Sigma ʻP \downarrow_{.,} ^{;}Q \unicode{x2909} \Sigma ʻP \downarrow_{.,} + ^{;}R\\ +[*162·31] &= \Sigma ʻ(P \downarrow_{.,} ^{;}Q \unicode{x2909} P \downarrow_{.,} ^{;}R)\\ +[*162·36] &= \Sigma ʻP \downarrow_{.,} ^{;}(Q \unicode{x2909} R)\\ +[*166·1] &= (Q \unicode{x2909} R) \times P . \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p>The following propositions (<a href="#*166·46">*166·46</a>—<a href="#*166·472">·472</a>) exhibit the failure of the +distributive law in the form \(P \times (Q \unicode{x2909} R) = (P\times Q)\unicode{x2909}(P \times R)\), +and give certain results for special cases. They are not referred to +except in this number.</p> + +<p class="nind"><b><a id="*166·46">*166·46</a>.</b> \(\vdash . (P \unicode{x228d} Q) \downarrow_{.,} z = P \downarrow_{.,} z \unicode{x228d} Q \downarrow_{.,} z \quad[*165·01 . *150·3]\)</p> + +<p class="nind"><b>*166·461.</b> \[\begin{align}&\vdash . \dot{s} ʻCʻ(P \unicode{x228d} Q) \downarrow_{.,} ^{;}R = \dot{s} ʻCʻP \downarrow_{.,} ^{;}R + \unicode{x228d} \dot{s} ʻCʻQ \downarrow_{.,} ^{;}R\\ +&[*41·6 . *165·12 . *166·46]\end{align}\]</p> + +<p><span class="pagenum" id="Page_401">[Pg 401]</span></p> + +<p class="nind"><b>*166·462.</b> \[\begin{align}&\vdash .F^{;}(P\unicode{x228d} Q)\downarrow_{.,} ^{;}R=F^{;}P\downarrow_{.,} ^{;}R\unicode{x228d} + F^{;}Q\downarrow_{.,} ^{;}R\unicode{x228d} \hat{M} \hat{N} \{(\exists x,y,z,w):\\ +&zRw:x\in CʻP.y\in CʻQ.\lor.x\in CʻQ.y\in CʻP:M=x\downarrow z.N=y\downarrow w\}\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*165·161.&\supset \vdash .F^{;}(P\unicode{x228d} Q)\downarrow_{.,} ^{;}R\\ +&=\hat{M} \hat{N} \{(\exists x,y,z,w).x,y\in CʻP\cup CʻQ.zRw.M=x\downarrow z.N=y\downarrow w\}\\ +[*22·34] &=\hat{M} \hat{N} \{(\exists x,y,z,w):x,y\in CʻP.\lor.x,y\in CʻQ.\lor.\\ +&x\in CʻP.y\in CʻQ.\lor.x\in CʻQ.y\in CʻP:zRw.M=x\downarrow z.N=y\downarrow w\}\\ +[*11·41.*165·161]&=F^{;}P\downarrow_{.,} ^{;}R\unicode{x228d} F^{;}Q\downarrow_{.,} ^{;}R\unicode{x228d} \hat{M} \hat{N} \{(\exists x,y,w):\\ +&x\in CʻP.y\in CʻQ.\lor.x\in CʻQ.y\in CʻP:\\ +&zRw.M=x\downarrow z.N=y\downarrow w\}.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*166·463.</b> \(\vdash :CʻP\subset CʻQ.\supset .F^{;}P\downarrow_{.,} ^{;}R\,\unicode{x2abd}\, F^{;}Q\downarrow_{.,} ^{;}R \quad[*165·161]\)</p> + +<p class="nind"><b>*166·464.</b> \(\vdash :CʻP\subset CʻQ.\supset .F^{;}(P\unicode{x228d} Q)\downarrow_{.,} ^{;}R=F^{;}Q\downarrow_{.,} + ^{;}R=F^{;}P\downarrow_{.,} ^{;}R\unicode{x228d} F^{;}Q\downarrow_{.,} ^{;}R\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*166·463. \supset \vdash :\text{Hp}.&\supset .F^{;}P\downarrow_{.,} ^{;}R\,\unicode{x2abd}\, F^{;}Q\downarrow_{.,} ^{;}R + &\qquad \text{(1)}\\ +\vdash .*33·262. &\supset \vdash :\text{Hp}.\supset .Cʻ(P\unicode{x228d} Q)=CʻQ.\\ +[*166·463] &\supset .F^{;}(P\unicode{x228d} Q)\downarrow_{.,} ^{;}R=F^{;}Q\downarrow_{.,} ^{;}R &\qquad \text{(2)}\\ +\vdash .(1).(2).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*166·47.</b> \[\begin{align}&\vdash .R\times (P\unicode{x228d} Q)=(R\times P)\unicode{x228d} (R\times Q)\unicode{x228d} \hat{M} \hat{N} \{(\exists x,y,z,w):\\ +&x\in CʻP.y\in CʻQ.\lor.x\in CʻQ.y\in CʻP:zRw.M=x\downarrow z.N=y\downarrow w\}\\ +&[*166·461·462·1.*162·1]\end{align}\]</p> + +<p class="nind"><b>*166·471.</b> \[\begin{align}&\vdash :CʻP\subset CʻQ.\supset .R\times (P\unicode{x228d} Q)=(R\times P)\unicode{x228d} (R\times Q)\\ +&[*166·461·464]\end{align}\]</p> + +<p class="nind"><b><a id="*166·472">*166·472</a>.</b> \(\vdash .R\times (P\unicode{x2909}Q)=(R\times P)\unicode{x228d} (R\times Q)\unicode{x228d} R\times (CʻP\downarrow CʻQ)\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*166·471.*35·85.\supset \\ +\vdash \colon\ldotp \dot{\exists} !Q.&\supset :R\times (P\unicode{x2909}Q)=(R\times P)\unicode{x228d} R\times \{Q\unicode{x228d} (CʻP\uparrow CʻQ)\}:\\ +[*166·471.*35·86] +&\supset :\dot{\exists} !P.\supset .R\times (P\unicode{x2909}Q)=(R\times P)\unicode{x228d} (R\times Q)\unicode{x228d} R\times (CʻP\uparrow CʻQ) &\qquad \text{(1)}\\ +\vdash .*160·21.*166·13.\supset \\ +\vdash :Q=\dot{\Lambda} .&\supset .P\unicode{x2909}Q=P.R\times Q=\dot{\Lambda} .R\times (CʻP\uparrow CʻQ)=\dot{\Lambda} .\\ +[*25·24] &\supset .R\times (P\unicode{x2909}Q)=(R\times P)\unicode{x228d} (R\times Q)\unicode{x228d} R\times (CʻP\downarrow CʻQ) &\qquad \text{(2)}\\ +\text{Similarly}\\ +\vdash :P=\dot{\Lambda} .&\supset .R\times (P\unicode{x2909}Q)=(R\times P)\unicode{x228d} (R\times Q)\unicode{x228d} R\times (CʻP\uparrow CʻQ) &\qquad \text{(3)}\\ +\vdash .(1).(2).(3).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_402">[Pg 402]</span></p> + +<p>The following propositions are concerned with the distributive +law for the addition of a single term to a relation. This law, in +the form in which it holds, is given in <a href="#*166·53">*166·53</a><a href="#*166·531">·531</a> (remembering +\(\text{Nr}ʻP\downarrow_{.,} y=\text{Nr}ʻP\)). <a href="#*166·54">*166·54</a><a href="#*166·541">·541</a> exhibit the +failure of the other form.</p> + +<p class="nind"><b>*166·5.</b> \(\vdash .(Q\unicode{x228d} R)\times P=(Q\times P)\unicode{x228d} (R\times P)\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*166·1.\supset \vdash .(Q\unicode{x228d} R)\times P&=\Sigma ʻP\downarrow_{.,} ^{;}(Q\unicode{x228d} R)\\ +[*162·27] &=\Sigma ʻP\downarrow_{.,} ^{;}Q\unicode{x228d} \Sigma ʻP\downarrow_{.,} ^{;}R\\ +[*166·1] & =(Q\times P)\unicode{x228d} (R\times P).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*166·51.</b> \(\vdash .(Q\unicode{x21f8} y)\times P=(Q\times P)\unicode{x228d} (CʻQ\uparrow {℩}ʻy)\times P \quad[*166·5.*161·1]\)</p> + +<p class="nind"><b>*166·511.</b> \(\vdash .(y\unicode{x21f7}Q)\times P=({℩}ʻy\uparrow CʻP)\times P\unicode{x228d} (Q\times P)\)</p> + +<p class="nind"><b>*166·52.</b> \(\vdash .P\downarrow_{.,} ^{;}(Q\unicode{x21f8} y)=P\downarrow_{.,} ^{;}Q\unicode{x21f8} P\downarrow_{.,} y \quad[*161·4.*165·2]\)</p> + +<p class="nind"><b>*166·521.</b> \(\vdash .P\downarrow_{.,} ^{;}(y\unicode{x21f7}Q)=P\downarrow_{.,} y\unicode{x21f7}P\downarrow_{.,} ^{;}Q\)</p> + +<p class="nind"><b><a id="*166·53">*166·53</a>.</b> \(\vdash :\dot{\exists} !Q.\supset .(Q\unicode{x21f8} y)\times P=(Q\times P)\unicode{x2909}(P\downarrow_{.,} y)\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*162·43.*165·243.\supset \vdash :\text{Hp}.&\supset .\Sigma ʻ(P\downarrow_{.,} ^{;}Q\unicode{x21f8} P\downarrow_{.,} y)=\Sigma ʻP\downarrow_{.,} + ^{;}Q\unicode{x2909}P\downarrow_{.,} y.\\ +[*166·52] &\supset .\Sigma ʻ(P\downarrow_{.,} ^{;}(Q\unicode{x21f8} y)=\Sigma ʻP\downarrow_{.,} ^{;}Q\unicode{x2909}P\downarrow_{.,} + y &\qquad \text{(1)}\\ +\vdash .(1).*166·1.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*166·531">*166·531</a>.</b> \(\vdash :\dot{\exists} !Q.\supset .(y\unicode{x21f7}Q)\times P=(P\downarrow_{.,} y)\unicode{x2909}(Q\times P)\)</p> + +<p class="nind"><b><a id="*166·54">*166·54</a>.</b> \(\vdash .Q\times (P\unicode{x21f8} x)=(Q\times P)\unicode{x228d} Q\times (CʻP\uparrow {℩}ʻx)\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*161·1.\supset \vdash .Q\times (P\unicode{x21f8} x)&=Q\times \{P\unicode{x228d} (CʻP\uparrow {℩}ʻx)\}\\ +[*35·85.*166·471] &=(Q\times P)\unicode{x228d} Q\times (CʻP\uparrow {℩}ʻx).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*166·541">*166·541</a>.</b> \(\vdash .Q\times (x\unicode{x21f7}P)=Q\times ({℩}ʻx\uparrow CʻP)\unicode{x228d} (Q\times P)\)</p> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_403">[Pg 403]</span></p> +<h2 class="nobreak" id="SECTION_C_II">SECTION C.<br> +<span class="allsmcap">THE PRINCIPLE OF FIRST DIFFERENCES, AND THE MULTIPLICATION +AND EXPONENTIATION OF RELATIONS.</span></h2> +</div> + + +<p><i>Summary of Section C.</i></p> + +<p>In the present section, we have to consider various forms of a +principle which is of the utmost utility in relation-arithmetic. This +principle may be called "the principle of first differences." It has +been explained and used by Hausdorff in brilliant articles<a id="FNanchor_15" href="#Footnote_15" class="fnanchor">[15]</a>. The +results there obtained by its use give some measure of its importance +in relation-arithmetic. It has, however, other uses besides those +that are concerned with the multiplication and exponentiation of +relation-numbers, as, for example, in the ordering of segments +and stretches in a series, or of any other set of classes which +are contained in the field of a given relation. In the present +section, after the first two numbers, we shall be concerned with its +arithmetical uses, but other uses will occur later.</p> + +<p>The principle of first differences has various forms which, though +analogous, cannot, in the general case, be reduced to one common +genus. The simplest of these is the relation \(P_{\text{cl}}\), by +which the sub-classes of \(CʻP\) are ordered. This is defined as +follows. If \(\alpha\) and \(\beta\) are both contained in \(CʻP\), we +say that \(\alpha P_{\text{cl}}\beta\) if there are terms belonging +to \(\alpha\) but not to \(\beta\) such that no terms belonging to +\(\beta\) and not to \(\alpha\) precede them; <i>i.e.</i> if, after +taking away the terms (if any) which are common to \(\alpha\) and +\(\beta\), there are terms left in \(\alpha\) which do not come after +any of the terms left in \(\beta\), <i>i.e.</i> if \(\exists !\alpha-\beta -\breve{P} ʻʻ(\beta -\alpha )\). +Thus the definition is +\[ +P_{\text{cl}}=\hat{\alpha} \hat{\beta} \{\alpha ,\beta \in \text{Cl}ʻCʻP.\exists !\alpha -\beta -\breve{P} ʻʻ(\beta -\alpha )\} \quad\text{Df}. +\] +It will be seen that this relation holds if \(\beta \subset \alpha.\beta \neq \alpha\). +Thus it holds between any existent member +of \(\text{Cl}ʻCʻP\) and \(\Lambda\), and between \(CʻP\) and +any member of \(\text{Cl}ʻCʻP\) other than \(CʻP\) itself. When +\(P\) is a serial relation (which is the important case for all +the relations in this section), \(P_{\text{cl}}\) is transitive +\((P^{2}_{\text{cl}}\,\unicode{x2abd}\, P_{\text{cl}})\) and asymmetrical +\((P_{\text{cl}}\dot{\cap} \breve{P} _{\text{cl}}=\dot{\Lambda})\), but +not necessarily <i>connected</i>, <i>i.e.</i> there may<span class="pagenum" id="Page_404">[Pg 404]</span> be two members +of its field of which neither has the relation \(P_{\text{cl}}\) to +the other. This happens whenever \(P\) is not well-ordered; but when +\(P\) is well-ordered, \(P_{\text{cl}}\) is connected, and therefore +generates a series.</p> + +<p>To illustrate the order generated by \(P_{\text{cl}}\) in a simple +case, consider a series of three terms, \(x\), \(y\), \(z\). Let us for +the moment write \((x\downarrow y\downarrow z)\) for the relation +\[ +x\downarrow y\unicode{x228d} x\downarrow z\unicode{x228d} y\downarrow z,\,\, \textit{i.e.}\,\, (x\downarrow y)\unicode{x21f8} z, +\] +and similarly we will write \((x\downarrow y\downarrow z\downarrow w)\) +for \(x\downarrow y\downarrow z\unicode{x21f8} w\), and so on. Then +assuming \(x\neq y.x\neq z.y\neq z\), +\[ +\begin{aligned} +(x\downarrow y\downarrow z)_{\text{cl}}=(\iota ʻx\cup \iota ʻy\cup \iota ʻz)&\downarrow (\iota ʻx\cup \iota ʻy)\\ +&\downarrow (\iota ʻx\cup \iota ʻz)\downarrow \iota ʻx\downarrow (\iota ʻy\cup \iota ʻz)\downarrow \iota ʻy\downarrow \iota ʻz\downarrow \Lambda . +\end{aligned} +\] +In this series, a class containing \(x\) is always earlier than one not +containing \(x\); and of two classes of which both or neither contain +\(x\), one containing \(y\) is earlier than one not containing \(y\); +and of two classes of which both or neither contain \(x\), and both +or neither contain \(y\), one containing \(z\) is earlier than one +not containing \(z\). Thus our relation may be generated as follows: +Begin with \((\iota ʻz)\downarrow \Lambda\), which is \((z\downarrow z)_{\text{cl}}\). +Add before these terms what results from adding \(\iota ʻy\) to each; +then we have \((y\downarrow z)_{\text{cl}}\), which is +\[ +(\iota ʻy\cup \iota ʻz)\downarrow \iota ʻy\downarrow \iota ʻz\downarrow \Lambda . +\] +Now add at the beginning what results from adding \(\iota ʻx\) to each +of the above four classes, and we have \((x\downarrow y\downarrow z)_{\text{cl}}\). +Thus generally, if \(x{\sim}\in CʻP\), +\[ +(x\unicode{x21f7}P)_{\text{cl}}=(\iota ʻx\cup )^{;}P_{\text{cl}}\unicode{x2909}P_{\text{cl}}. +\] +Thus by adding one term to \(P\), we double the number of terms in +\(P_{\text{cl}}\).</p> + +<p>Again, if \(P\) and \(Q\) are two relations which have no common terms +in their fields, we shall have +\[ +\begin{aligned} +\alpha P_{\text{cl}}&\beta .\gamma ,\delta \in \text{Cl}ʻCʻQ.\supset .(\alpha \cup \gamma )(P\unicode{x2909}Q)_{\text{cl}}(\beta \cup \delta )\\ +&\text{and}\quad \alpha \in \text{Cl}ʻCʻP.\gamma Q_{\text{cl}}\delta .\supset .(\alpha \cup \gamma )(P\unicode{x2909}Q)_{\text{cl}}(\alpha \cup \delta ), +\end{aligned} +\] +while conversely +\[ +\begin{aligned} +\alpha ,\beta \in \text{Cl}ʻCʻP.\gamma ,\delta \in \text{Cl}ʻCʻQ.(\alpha \cup \gamma )(P\unicode{x2909}Q)_{\text{cl}}&(\beta \cup \delta ).\supset :\\ +&\alpha P_{\text{cl}}\beta .\lor.\alpha =\beta .\gamma Q_{\text{cl}}\delta . +\end{aligned} +\] +Hence +\[ +\begin{aligned} +(\alpha \cup \gamma )(P\unicode{x2909}Q)_{\text{cl}}(\beta \cup \delta ).&\equiv .(\gamma \downarrow \alpha )(P_{\text{cl}}\times Q_{\text{cl}})(\delta + \downarrow \beta )\\ +&\equiv .(\alpha \cup \gamma ){s^{;}C^{;}(P_{\text{cl}}\times Q_{\text{cl}})}(\beta \cup \delta ), +\end{aligned} +\] +so that \(\text{Nr}ʻ(P\unicode{x2909}Q)_{\text{cl}}=\text{Nr}ʻP_{\text{cl}}\times \text{Nr}ʻQ_{\text{cl}}\).</p> + +<p>These propositions illustrate the connection of \(P_\text{cl}\) with +multiplication.</p> + +<p>Besides \(P_{\text{cl}}\), we often require (though not in this Part) +the relation which is the converse of \((\breve{P})_{\text{cl}}\). +This relation we call \(P_{\text{lc}}\), so that +\[ +P_{\text{lc}}=\text{Cnv}ʻ(\breve{P} )_{\text{cl}} \quad\text{Df}. +\] +This begins with \(\Lambda\), and ends with \(CʻP\).</p> + +<p><span class="pagenum" id="Page_405">[Pg 405]</span></p> + +<p>Thus we shall have, for example, +\[ +\begin{aligned} +(x\downarrow y\downarrow z)_{\text{lc}} = \Lambda \downarrow \iota ʻx\downarrow \iota ʻy&\downarrow (\iota ʻx\cup \iota ʻy)\\ +&\downarrow \iota ʻz\downarrow (\iota ʻx\cup \iota ʻz)\downarrow (\iota ʻy\cup \iota ʻz)\downarrow (\iota ʻx\cup \iota ʻy\cup \iota ʻz). +\end{aligned} +\] +Here, if we start from \(\Lambda \downarrow \iota ʻx\), which is +\((x\downarrow x)_{\text{lc}}\), the series grows by adding terms at +the end: we add \(\iota ʻy\) to each member of \(\Lambda \downarrow\iota ʻx\) +and put the resulting terms \(\iota ʻy, \iota ʻx\cup \iotaʻy\) +after \(\Lambda\) and \(\iota ʻx\); we then add \(\iota ʻz\) to each +of the four terms we already have, and add the resulting terms at the +end; and so we can proceed indefinitely.</p> + +<p>The relation \(P_{\text{lc}}\) with its field limited arranges the +<i>segments</i> of P in ascending order of magnitude; if the class +of segments is \(\sigma\), \(P_{\text{lc}}\unicode{x0294f}\sigma\) +generates what may be called the natural order among the segments (cf. +<a href="#*212">*212</a>).</p> + +<p>A variant of \(P_{\text{cl}}\) is afforded by the relation +\(P_{\text{df}}\) (<a href="#*171">*171</a>), which is to hold between two members +\(\alpha\), \(\beta\) of \(\text{Cl}ʻCʻP\) when the first term of +either which does not belong to both belongs to \(\alpha\), <i>i.e.</i> +the "first difference" belongs to \(\alpha\). This relation implies +\(P_{\text{cl}}\), and coincides with it if \(P\) is well-ordered; but +when \(P\) is not well-ordered, \(P_{\text{cl}}\) may hold between two +classes which have no first point of difference, <i>e.g.</i> (if \(P\) +is "less than" among rationals) if \(\alpha\) consists of rationals +between 0 and 1 (both excluded) and \(\beta\) of rationals between 1 +and 2 (both excluded). The definition of \(P_{\text{df}}\) is +\[ +P_{\text{df}} + = \hat{\alpha} \hat{\beta} \{\alpha ,\beta \in \text{Cl}ʻCʻP:(\exists z).z\in \alpha -\beta .\overrightarrow{P}ʻz\cap \alpha -\iota ʻz = \overrightarrow{P}ʻz\cap \beta\} \quad\text{Df}. +\]</p> + +<p>The relation \(P_{\text{df}}\) has the interesting property that +its relation-number is found by raising \(2_{r}\) to the power +\(\text{Nr}ʻP\) (cf. <a href="#*177">*177</a>). As the field of \(P_{\text{df}}\) +is \(\text{Cl}ʻCʻP\), this theorem is the ordinal analogue of +\(\text{Nc}ʻ\text{Cl}ʻ\alpha = 2^{\text{Nc}ʻ\alpha}\) (<a href="#*116·72">*116·72</a>).</p> + +<p>A somewhat more complicated form of the relation of first differences +arises when we have a series of series. Let us suppose, to begin with, +that \(P\) is a serial relation whose field consists of mutually +exclusive serial relations.</p> + +<figure class="figcenter width500" id="i_405" style="width: 1200px;"> +<img src="images/i_405.jpg" width="1200" height="745" alt="Grid diagram +showing four rows of dots with special markers. Each row has squares +and circles at different positions among dots, with arrows pointing +right to labels Q (top three rows) and R⋎p (bottom). The squares and +circles shift diagonally downward, illustrating relation ordering."> +</figure> + +<p>Thus in the accompanying figure, each row represents a series, the +generating relations of these series being \(Q\), ... \(R\),... But the +series themselves form a series, which may be regarded as generated +by a relation \(P\) whose field consists of the relations \(Q\), ... +\(R\),... (It might be thought more natural to take \(CʻQ\), \(CʻR\), +... as the field of \(P\); but this would lead to confusion in the +case when two or more of the series have the same field.) Suppose +we now wish to find a relation which will order the multiplicative +class of the fields of \(Q\), \(R\), ..., <i>i.e.</i> the class +\(\text{Prod}ʻCʻʻCʻP\). In the case illustrated in the figure, in which +\(P\) generates a well-ordered series, and all the members of \(CʻP\) +are serial, and \(P\in \text{Rel}^{2}\text{excl}\), we might use +\((\Sigma ʻP)_{\text{cl}}\); this relation, with its field limited to +\(\text{Prod}ʻCʻCʻʻP\), will then give us what we want. This relation +will, in the case supposed, put<span class="pagenum" id="Page_406">[Pg 406]</span> a selected class \(\mu\) before +another selected class \(\nu\) if, where they first differ, \(\mu\) +chooses an earlier term than \(\nu\). But if the series \(P\) is not +well-ordered—if it is (say) of the type \(\text{Cnv}ʻʻ\omega\) (cf. +*263)—there may be no <i>first</i> member of the field of \(P\) where +\(\mu\) and \(\nu\) differ. This will happen, for example, if \(\mu\) +consists of all the first terms, and \(\nu\) of all the second terms. +Our ordering relation can be so defined as to put \(\mu\) before +\(\nu\) in this case also, but if it is so defined, the associative +law of multiplication only holds if \(P\) is well-ordered. For this +reason, we define our ordering relation so that, in such a case, +\(\mu\) comes neither before nor after \(\nu\). Again, if \(P\) is +not a \(\text{Rel}^{2}\text{excl}\), a member of a selected class may +occur twice, once as the representative of \(CʻQ\), and once as that +of \(CʻR\), if \(CʻQ\) and \(CʻR\) have terms in common. We wish to +distinguish these two occurrences. Hence we proceed as follows: If +\(\mu\) and \(\nu\) are two selected classes of \(CʻʻCʻP\), let there +be one or more members of \(CʻP\) in which the \(\mu\)-representative +precedes the \(\nu\)-representative, and which are such that, among all +earlier<a id="FNanchor_16" href="#Footnote_16" class="fnanchor">[16]</a> members of \(CʻP\), the \(\mu\)-representative is identical +with the \(\nu\)-representative.</p> + +<p>But a further modification is desirable in order to meet the case +in which two or more of the members of \(CʻP\) have the same field. +Suppose, for example, we had to deal with a series consisting of all +the series that can be formed out of a given set of terms: in this +case, we should have to distinguish occurrences of any given term not +by the field, but by the generating relation. This requires that we +should make an \(F\)-selection from \(CʻP\), not an \({\in}\)-selection +from \(CʻʻCʻP\). Hence we take two members of \(F_{\Delta }ʻCʻP\), say +\(M\) and \(N\), and we arrange them or their domains on the following +principle: We put \(M\) before \(N\) (or \(\text{D}ʻM\) before +\(\text{D}ʻN\)) if there is a relation \(Q\) in the field of \(P\) such +that the \(M\)-representative of \(Q\), <i>i.e.</i> \(MʻQ\), has the +relation \(Q\) to the \(N\)-representative of \(Q\), and such that, if +\(R\) is any earlier member of \(CʻP\), then \(MʻR\) is identical with +\(NʻR\). That is, \(M\) precedes \(N\) if +\[ +(\exists Q):(MʻQ)Q(NʻQ):RPQ.R\neq Q.\supset _{R}.MʻR = NʻR. +\]</p> + +<p>The relation between \(M\) and \(N\) so defined has the properties +required of an arithmetical product; hence we put +\[ +\begin{aligned} +\Pi ʻP=\hat{M} \hat{N} &\{M,N\in F_{\Delta }ʻCʻP\colon\ldotp \\ +&(\exists Q):(MʻQ)Q(NʻQ):RPQ.R\neq Q.\supset _{R}.MʻR=NʻR\} \quad\text{Df}.\\ +\end{aligned} +\]</p> + +<p>This relation is the ordinal analogue of \({\in}_{\Delta}ʻ\kappa\). +The ordinal analogue of \(\text{Prod}ʻ\kappa\) is the corresponding +relation of the domains of \(M\) and \(N\), <i>i.e.</i> \(\text{D}^{;}\PiʻP\); +hence we put +\[ +\text{Prod}ʻP= \text{D}^{;}\Pi ʻP \quad\text{Df}. +\]</p> + +<p>In case \(P\) is a \(\text{Rel}^{2}\text{excl}\), we have +\(\text{Nr}ʻ\text{Prod}ʻP=\text{Nr}ʻ\Pi ʻP\). But when \(P\) is not +a \(\text{Rel}^{2}\text{excl}\), \(\text{Prod}ʻP\) and \(\Pi ʻP\) are +in general not ordinally similar. We can, however, always make a +\(\text{Rel}^{2}\text{excl}\) by replacing the members \(x\), \(y\), etc. +of<span class="pagenum" id="Page_407">[Pg 407]</span> \(CʻQ\) (where \(Q\in CʻP\)) by \(x\downarrow Q,y\downarrow Q\), +etc. In this way, if \(x\) occurs twice in \(Cʻ\Sigma ʻP\), once as a +member of \(CʻQ\), and once as a member of \(CʻR\), the two occurrences +are made to correspond to \(x\downarrow Q\) and \(x\downarrow R\) +respectively, and thus we get a new relation which <i>is</i> a +\(\text{Rel}^{2}\text{excl}\).</p> + +<p>If every member of \(CʻP\) has a first term, \(B\upharpoonright CʻP\) +will be the first term of \(\Pi ʻP\), and \(BʻʻCʻP\) will be the +first term of \(\text{Prod}ʻP\). If further there is a last member of +\(CʻP\), <i>i.e.</i> if \(\text{E}!Bʻ\breve{P}\), and if this last +member has a second term, the second member of \(\Pi ʻP\) is obtained +by taking this second term as the representative of \(Bʻ\breve{P}\), +and leaving all the other representatives unchanged. In any case, +if \(Bʻ\breve{P}\) exists, the earliest successors of any member of +\(\Pi ʻP\) are those obtained by only varying the representative in +\(Bʻ\breve{P}\). Thus, if \(Bʻ\breve{P}\) exists, those members of +\(\Pi ʻP\) which have a given set of representatives in all members +of \(\text{D}ʻP\) form a consecutive stretch of the series, and this +stretch is like \(Bʻ\breve{P}\). If \(Bʻ\breve{P}\) has an immediate +predecessor, the stretches obtained by varying only the representative +in this predecessor are again consecutive, and form a series like the +said predecessor; and so on. This makes it plain why \(\Pi ʻP\) has the +properties of a product.</p> + +<p>As in the case of cardinals, the definition of exponentiation is +derived from that of multiplication. We put +\[ +\begin{aligned} +P\,\,\text{exp}\,\,Q&=\text{Prod}ʻP\downarrow_{.,} ^{;}Q &\quad\text{Df}.\\ +\text{We put also}\quad P^{Q}&=\dot{s} ^{;}(P\,\,\text{exp}\,\,Q) &\quad\text{Df}. +\end{aligned} +\] +This is an important relation, which deserves consideration apart from +the fact that it is useful in connection with exponentiation. It will +be found that +\[ +\begin{aligned} +P^{Q}=\hat{M} \hat{N} &\{M,N\in (CʻP\uparrow CʻQ)_{\Delta }ʻCʻQ\colon\ldotp \\ +&(\exists y):y\in CʻQ.(Mʻy)P(Nʻy):xQy.x\neq y.\supset _{x}.Mʻx=Nʻx\}. +\end{aligned} +\]</p> + +<p>This is a form of the principle of first differences which is +appropriate when <i>two</i> relations are concerned, instead of only +one as in \(P_{\text{cl}}\). The principle, in this case, is as +follows: Let \(M\), \(N\) be any two one-many relations which relate +part (or the whole) of \(CʻP\) to the whole of \(CʻQ\). That is, each +of the two relations assigns a representative in \(CʻP\) to every +term of \(CʻQ\), but different terms of \(CʻQ\) may have the same +representative. Then in travelling along the series \(Q\), there is to +be, sooner or later, a term \(y\) whose \(M\)-representative is earlier +than its \(N\)-representative, and terms which come earlier than \(y\) +in \(Q\) are all to have their \(M\)-representatives identical with +their \(N\)-representatives.</p> + +<p>The relation \(P^{Q}\) may be subjected to various restrictions which +give important results. This subject has been treated by Hausdorff. +For<span class="pagenum" id="Page_408">[Pg 408]</span> example, if \(P=x\downarrow y\) (where \(x\neq y\)), and \(Q\) +is of the ordinal type which Cantor calls \(\omega\), <i>i.e.</i> the +type of progressions (generated by transitive relations), then if \(z\) +is any member of \(CʻQ\), \(Mʻz\) is always either \(x\) or \(y\). +If we impose the condition that \(Mʻz\) is to be \(x\) except for a +finite number of values of \(z\), the resulting series is of the type +of the rationals in order of magnitude, <i>i.e.</i> the type called +\(\eta\). If we impose the condition that there are to be an infinite +number of values of \(z\) for which \(Mʻz=y\), the resulting series is +a continuum, <i>i.e.</i> it is of the ordinal type called \(\theta\); +in this case, the contained "rational" series consists of those \(M\)'s +for which there are only a finite number of \(z\)'s having \(Mʻz=x\). +If we impose no limitation, \(P^{Q}\) is of the type presented by the +real numbers when decimals ending in 9 recurring are counted separately +from the terminating decimals having the same value.</p> + +<p>We may generalize \(P^{Q}\), instead of restricting it. To begin with, we +may allow our \(M\) and \(N\) to have only part of \(CʻQ\) for their +converse domain, and remove the assumption that there is a <i>first</i> +member of \(CʻQ\) for which \(Mʻy\) and \(Nʻy\) differ; this leads to +the relation +\[ +\begin{aligned} +\hat{M} \hat{N} &\{M,N\in (1\rightarrow \text{Cls})\cap \text{Rl}ʻ(CʻP\uparrow CʻQ)\colon\ldotp\\ +&(\exists y):(Mʻy)P(Nʻy):xQy.x\in \text{ᗡ}ʻN.\supset _{x}.(Mʻx)(P\unicode{x228d} I)(Nʻx)\}. +\end{aligned} +\] +Further, we may drop the restriction to one-many relations. It will be +observed that if \((Mʻy)P(Nʻy)\), we have \(y (\breve{M} \mid P\mid N)y\). +Thus we may consider the relation +\[ +\begin{aligned} +MN[\hat{M} ,\hat{N} \in \text{Rl}ʻ&(CʻP\uparrow CʻQ)\colon\ldotp \\ +&(\exists y):y(\breve{M} \mid P\mid N)y:xQy.\supset _{x}.x\{\breve{M} \mid (P\unicode{x228d} I)\mid N\}x]. +\end{aligned} +\] +This relation has for its field all relations contained in +\(CʻP\uparrow CʻQ\). We may, if we like, drop even this restriction, +and consider +\[ +\hat{M} \hat{N} [(\exists y):y\in CʻQ.y(\breve{M} \mid P\mid N)y:xQy.\supset _{x}.x\{\breve{M} \mid (P\unicode{x228d} I\upharpoonright CʻP)\mid N\}x]. +\] +This represents the most general form of the principle of first +differences as applied to a couple of relations \(P\) and \(Q\). In +ordinal arithmetic, however, \(P^{Q}\) is sufficiently general for the +uses we wish to make of it.</p> + +<p>The formal laws, as far as they are true, can be proved without +excessive difficulty. We have +\[ +\vdash :P\neq Q.\supset .\text{Nr}ʻ\Pi ʻ(P\downarrow Q)=\text{Nr}ʻ(P\times Q), +\] +which connects the two kinds of multiplication; +\[ +\begin{aligned} +&\vdash :P \,\text{smor smor}\, Q.\supset .\text{Nr}ʻ\Pi ʻP=\text{Nr}ʻ\Pi ʻQ,\\ +&\vdash :P\in \text{Rel}^{2}\text{excl}.P\,\unicode{x2abd}\, J.\supset .\text{Nr}ʻ\Pi ʻ\Pi ^{;}P=\text{Nr}ʻ\Pi ʻ\Sigma ʻP, +\end{aligned} +\] +which is one form of the associative law, of which another form is +\[ +\vdash :P\neq Q.\supset .\text{Nr}ʻ(\Pi ʻP\times \Pi ʻQ)=\text{Nr}ʻ\Pi ʻ(P\unicode{x2909}Q). +\]<span class="pagenum" id="Page_409">[Pg 409]</span> +Also +\[ +\vdash :P,\Sigma ʻP\in \text{Rel}^{2}\text{excl}.P\,\unicode{x2abd}\, J.\supset .\text{Nr}ʻ\text{Prod}ʻ\text{Prod}^{;}P=\text{Nr}ʻ\text{Prod}ʻ\Sigma ʻP=\text{Nr}ʻ\Pi ʻ\Sigma ʻP, +\] +which is the associative law for "\(\text{Prod}\)." We have +\[ +\begin{aligned} +&\vdash :CʻQ\cap CʻR=\Lambda .\supset .\text{Nr}ʻ(P^{Q}\times P^{R})=\text{Nr}ʻP^{Q\unicode{x2909}R},\\ +&\vdash .\text{Nr}ʻ(P^{Q})^{R}=\text{Nr}ʻ(P^{R\times Q}). +\end{aligned} +\] +But we do not have in general +\[ +\text{Nr}ʻ(P^{R}\times Q^{R})=\text{Nr}ʻ(P\times Q)^{R}, +\] +which obviously would require the commutative law for multiplication, +and therefore does not hold in general in spite of the fact that its +cardinal analogue does always hold.</p> + +<p>As regards the connection with cardinals, we have +\[ +\begin{aligned} +&\vdash :P\in \text{Rel}^{2}\text{excl}.\supset .Cʻ\text{Prod}ʻP=\text{Prod}ʻCʻʻCʻP,\\ +&\vdash :\dot{\exists} !Q.\supset .Cʻ(P\,\,\text{exp}\,\,Q)=(CʻP)\,\,\text{exp}\,\,(CʻQ), +\end{aligned} +\] +and we have already had +\[ +\vdash .Cʻ(P\times Q)=CʻP\times CʻQ. +\] +Moreover the correlators by which similarity is established in +cardinals generally suffice to establish likeness in the analogous +cases in relation-arithmetic. Thus we have +\[ +\begin{aligned} +\vdash :S\in P\,\overline{\text{smor}}\,Q.\supset .S_{\in }\upharpoonright \text{Cl}ʻCʻQ\in P_{\text{cl}}\,\overline{\text{smor}}\,Q_{\text{cl}},\\ +\vdash :P,Q\in \text{Rel}^{2}\text{excl}.S\upharpoonright Cʻ\Sigma ʻQ\in P\overline{\text{smor}} &\overline{\text{smor}}\,Q.\supset .\\ +&S_{\in }\upharpoonright Cʻ\text{Prod}ʻQ\in (\text{Prod}ʻP)\,\overline{\text{smor}}\,(\text{Prod}ʻQ),\\ +\vdash :S\upharpoonright CʻP'\in P\,\overline{\text{smor}}\,P'.&T\upharpoonright CʻQʻ\in Q\,\overline{\text{smor}}\,Q'.\supset .\\ +&(S\parallel \breve{T} )\upharpoonright Cʻ(P'\,\,\text{exp}\,\,Q')\in (P\,\,\text{exp}\,\,Q)\,\overline{\text{smor}}\,(P'\,\,\text{exp}\,\,Q'), +\end{aligned} +\] +which are all closely analogous to propositions which were proved in +cardinals.</p> + +<p>The applications of the propositions of this section are almost wholly +to series, and it is convenient to imagine our relations to be serial. +But the hypothesis that they are serial is not necessary to the +truth of any of the propositions of the present section, and it is a +remarkable fact that so many of the formal laws of ordinal arithmetic +hold for relations in general.</p> + +<p>It should be observed that \(\Pi ʻP\) is not always a series when \(P\) +is a series and all the relations in the field of \(P\) are series. +A series (cf. <a href="#*204">*204</a>) is a relation \(P\) which is (1) contained in +diversity, (2) transitive, (3) connected, <i>i.e.</i> such that every +term of the field of P has the relation P or the relation \(\breve{P}\) +to every other term of the field. It is the third condition which +may fail for \(\Pi ʻP\), and which in fact does fail whenever \(P\) +is not well-ordered. Thus suppose, for the sake of simplicity, that +\(P\) is of the type \(\text{Cnv}ʻʻ\omega\), which we will call a +<i>regression</i>, <i>i.e.</i> the converse of a progression (cf. +*263); and suppose that the field of \(P\) consists entirely of +couples. Take a selection \(M\) which chooses the first term of +every odd couple, and the second term of every even couple ; and take +another selection \(N\) which chooses the second term of every odd +couple, and the first term of every even couple. Neither of these two +selections has the relation \(\Pi ʻP\) to the other, for whatever term +\(Q\) of \(CʻP\) we choose, if \(M\) is the selection which chooses +the first term of \(Q\), there is an earlier term of \(CʻP\) (namely +the immediate predecessor of \(Q\)) in which \(N\) chooses the first +term while \(M\) chooses the second. Hence there is no such \(Q\) as +is required for \(M(\Pi ʻP)N\); and a similar argument holds against +\(N(\Pi ʻP)M\). In such a case, \(\Pi ʻP\) generates a number of +different series, and by suitable restrictions of the field, one of +these series can be extracted. Exactly similar remarks apply to \(P^{Q}\).</p> + + +<div class="footnotes"><h3>FOOTNOTES:</h3> + +<div class="footnote"> + +<p class="nind"> +<a id="Footnote_15" href="#FNanchor_15" class="label">[15]</a> +"Untersuchungen über Ordnungstypen," <i>Berichte der +mathematisch-physischen Klasse der Königlich Sächsischen Gesellschaft +der Wissenschaften zu Leipzig</i>, Feb. 1906 and Feb. 1907. Cf. also +his "Grundzüge einer Theorie der geordneten Mengen," Math. Annalen, 65 +(1908).</p> + +</div> + +<div class="footnote"> + +<p class="nind"> +<a id="Footnote_16" href="#FNanchor_16" class="label">[16]</a> +Here \(Q\) is said to be earlier than \(R\) if \(Q\) has +the relation \(P\) to \(R\) and is not identical with \(R\).</p> + +</div> +</div> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_411">[Pg 411]</span></p> +<h2 class="nobreak" id="*170"> +*170. ON THE RELATION OF FIRST DIFFERENCES AMONG THE +SUB-CLASSES OF A GIVEN CLASS.</h2> +</div> + + +<p class="nind"><i>Summary of</i> *170.</p> + +<p>The definition to be given in this number of the relation of first +differences among the sub-classes of a given class is by no means +the only one possible, in fact a different definition will be +considered in <a href="#*171">*171</a>. In the present number, the definition we choose +is this: \(\alpha\) is said to precede \(\beta\) according to this +definition when \(\alpha\) has at least one member which neither +belongs to \(\beta\) nor follows any term belonging to \(\beta\) and +not to \(\alpha\) (\(\alpha\) and \(\beta\) being both sub-classes +of \(CʻP\)). In other words, if we consider the two classes \(\alpha-\beta\) +and \(\beta -\alpha\), there are members of \(\alpha -\beta\) +which are not preceded by any members of \(\beta -\alpha\). Pictorially, +we may conceive the relation as follows (\(P\) being supposed serial): +\(\alpha\) and \(\beta\) each pick out terms from \(CʻP\), and these +terms have an order conferred by \(P\); we suppose that the earlier +terms selected by \(\alpha\) and \(\beta\) are perhaps the same, but +sooner or later, if \(\alpha \neq \beta\), we must come to terms which +belong to one but not to the other. We assume that the earliest terms +of this sort belong to \(\alpha\), not to \(\beta\); in this case, +\(\alpha\) has to \(\beta\) the relation \(P_{\text{cl}}\). That +is, where \(\alpha\) and \(\beta\) begin to differ, it is terms of +\(\alpha\) that we come to, not terms of \(\beta\). We do not assume +that there is a <i>first</i> term which belongs to \(\alpha\) and not +to \(\beta\), since this would introduce undesirable restrictions in +case \(P\) is not well-ordered.</p> + +<p>A few of the propositions of the present number will be used in the +next number, which deals with a slightly different form of the relation +of first differences, but with this exception the propositions of this +number will not be referred to again until we come to series. Their +chief use occurs in the section on compact series, rational series, and +continuous series (Part V, Section F), especially in *274 and *276, +which respectively establish the existence of rational series (assuming +the axiom of infinity) and the fact that the cardinal number of terms +in a continuous series is the same as the number of classes contained +in the field of a progression, <i>i.e.</i> \(2^{\aleph _{0}}\). The +definitions and a few of the simpler propositions are also used in +connection with the series of segments of a series, since, as explained +above, the segments of a series \(P\) are arranged in the series +<span class="pagenum" id="Page_412">[Pg 412]</span>generated by \(P_{\text{lc}}\).</p> + +<p>The propositions of this number which will be used in dealing with +series are the following:</p> + +<p class="nind"><b>*170·1.</b> \(\vdash :\alpha P_{\text{cl}}\beta .\equiv .\alpha ,\beta \in \text{Cl}ʻCʻP.\exists !\alpha -\beta -\breve{P} ʻʻ(\beta -\alpha )\)</p> + +<p class="nind"><b>*170·101.</b> \(\vdash .P_{\text{lc}}=\text{Cnv}ʻ(\breve{P} )_{\text{cl}}\)</p> + +<p class="nind"><b>*170·102.</b> \(\vdash :\alpha P_{\text{lc}}\beta .\equiv .\alpha ,\beta \in \text{Cl}ʻCʻP.\exists !\beta -\alpha -Pʻʻ(\alpha -\beta )\)</p> + +<p>(These propositions merely embody the definitions.)</p> + +<p class="nind"><b>*170·11.</b> \(\vdash \colon\ldotp \alpha P_{\text{cl}}\beta .\equiv :\alpha ,\beta \in \text{Cl}ʻCʻP:(\exists y).y\in \alpha -\beta .\overrightarrow{P}ʻy\cap \beta \subset \alpha\)</p> + +<p>This form is often more convenient than <a href="#*170·1">*170·1</a>.</p> + +<p class="nind"><b>*170·16.</b> \(\vdash :\alpha \subset CʻP.\beta \subset \alpha .\beta \neq \alpha .\supset .\alpha P_{\text{cl}}\beta\)</p> + +<p><i>I.e.</i> every sub-class of \(CʻP\) has the relation +\(P_{\text{cl}}\) to every proper part of itself.</p> + +<p class="nind"><b>*170·17.</b> \(\vdash .P_{\text{cl}}\,\unicode{x2abd}\, J.P_{\text{lc}}\,\unicode{x2abd}\, J\)</p> + +<p class="nind"><b>*170·2.</b> + \(\vdash \colon\ldotp \alpha ,\beta \in \text{Cl}ʻCʻP:(\exists y).ye\alpha -\beta .\overrightarrow{P}ʻy\cap \alpha =\overrightarrow{P}ʻy\cap \beta :\supset .\alpha P_{\text{cl}}\beta\)</p> + +<p>This proposition deals with the case where there is a definite first +term y which belongs to \(\alpha\) and not to \(\beta\), and whose +predecessors all belong to both or neither.</p> + +<p class="nind"><b>*170·23.</b> \[\begin{align}\vdash \colon\ldotp \alpha \subset CʻP.y \in\alpha -\beta -\breve{P} ʻʻ&(\beta -\alpha).\supset :\\ +&y \text{min}_{P}(\alpha -\beta ).\equiv .\overrightarrow{P}ʻy\cap \alpha =\overrightarrow{P}ʻy\cap \beta\end{align}\]</p> + +<p>This proposition is useful in case \(P\) is well-ordered, since then +\(\alpha -\beta\) must have a minimum if it exists (\(\alpha\) and +\(\beta\) being supposed sub-classes of \(CʻP\)).</p> + +<p class="nind"><b>*170·31.</b> \(\vdash :\beta \subset CʻP.\beta \neq CʻP.\equiv .(CʻP)P_{\text{cl}}\beta\)</p> + +<p>This follows from <a href="#*170·16">*170·16</a>, as does the following proposition:</p> + +<p class="nind"><b>*170·32.</b> \(\vdash :\alpha \subset CʻP.\exists !\alpha .\equiv .\alpha P_{\text{cl}}\Lambda\)</p> + +<p class="nind"><b>*170·35.</b> \(\vdash .\dot{\Lambda} _{\text{cl}}=\dot{\Lambda}\)</p> + +<p class="nind"><b>*170·38.</b> \(\vdash :\dot{\exists} !P.\supset .BʻP_{\text{cl}}=CʻP.Bʻ\text{Cnv}ʻP_{\text{cl}}=\Lambda\)</p> + +<p class="nind"><b>*170·6.</b> \(\vdash :\Lambda P_{\text{lc}}\beta .\equiv .\beta \subset CʻP.\exists !\beta\)</p> + +<p>Besides the above, the following propositions should be noted:</p> + +<p class="nind"><b>*170·36.</b> \(\vdash .\text{D}ʻP_{\text{cl}}=\text{Cl ex}ʻCʻP.\text{ᗡ}ʻP_{\text{cl}}=\text{Cl}ʻCʻP-\iota ʻCʻP\)</p> + +<p class="nind"><b>*170·37.</b> \(\vdash :\dot{\exists} !P.\supset .CʻP_{\text{cl}}=\text{Cl}ʻCʻP\)</p> + +<p class="nind"><b>*170.44.</b> \(\vdash :P\,\text{smor}\,Q.\supset .\text{D}.P_{\text{cl}}\text{smor}Q_{\text{cl}}\)</p> + +<p class="nind"><b>*170·64.</b> \(\vdash :x{\sim}\in CʻP.\supset .(x\unicode{x21f7}P)_{\text{cl}}=(\iota ʻx\cup )^{;}P_{\text{cl}}\unicode{x2909}P_{\text{cl}}\)</p> + +<p>This proposition shows that every term added to \(P\) doubles the +number of terms in \(P_{\text{cl}}\); hence it is not surprising that +\(P_{\text{cl}}\) (when \(P\) is well-ordered) has a power of \(2_{r}\) +for its relation-number (cf. <a href="#*177">*177</a>).</p> + +<p class="nind"><b>*170.67.</b> \(\vdash :\dot{\exists} !P.\dot{\exists} !Q.CʻP\cap CʻQ=\Lambda .\supset .(P\unicode{x2909}Q)_{\text{cl}}=s^{;}C^{;}(P_{\text{cl}}\times + Q_{\text{cl}})\)</p> + +<p class="nind">whence</p> + +<p class="nind"><b>*170·69.</b> \(\vdash :\dot{\exists} !P.\dot{\exists} !Q.CʻP\cap CʻQ=\Lambda .\supset .(P\unicode{x2909}Q)_{\text{cl}}\text{smor}(P_{\text{cl}}xQ_{\text{cl}})\)</p> + +<p><span class="pagenum" id="Page_413">[Pg 413]</span></p> + +<hr class="tb"> + +<p class="nind"><b>*170·01.</b> \(P_{\text{cl}}=\hat{\alpha} \hat{\beta} \{\alpha ,\beta \in \text{Cl}ʻCʻP.\exists !\alpha -\beta -\breve{P} ʻʻ(\beta -\alpha )\} \quad\text{Df}\)</p> + +<p class="nind"><b>*170·02.</b> \(P_{\text{lc}} = \text{Cnv}ʻ(\breve{P} )_{\text{cl}} \quad\text{Df}\)</p> + +<p class="nind"><b><a id="*170·1">*170·1</a>.</b> \(\vdash :\alpha P_{\text{cl}}\beta .\equiv .a,\beta \in \text{Cl}ʻCʻP.\exists !\alpha -\beta -\breve{P} ʻʻ(\beta -\alpha ) \quad[(*170·01)]\)</p> + +<p class="nind"><b>*170·101.</b> \(\vdash .P_{\text{lc}}=\text{Cnv}ʻ(\breve{P} )_{\text{cl}} \quad[(*170·02)]\)</p> + +<p class="nind"><b>*170·102.</b> \(\vdash :\alpha P_{\text{lc}}\beta .\equiv .a,\beta \in \text{Cl}ʻCʻP.\exists !\beta -\alpha -Pʻʻ(\alpha -\beta ) \quad[*170·1·101]\)</p> + +<p>Thus \(\alpha P_{\text{lc}}\beta\) means, roughly speaking, that +\(\beta -\alpha\) goes on longer than \(\alpha -\beta\), just as +\(\alpha P_{\text{cl}}\beta\) means that \(\alpha -\beta\) begins +sooner. Thus if \(P\) is the relation of earlier and later in +time, and \(\alpha\) and \(\beta\) are the times when \(A\) and +\(B\) respectively are out of bed, "\(\alpha P_{\text{cl}}\beta\)" +will mean that \(A\) gets up earlier than \(B\), and +"\(\alpha P_{\text{lc}}\beta\)" will mean that \(B\) goes to bed later +than \(A\).</p> + +<p class="nind"><b>*170·103.</b> \(\vdash :y{\sim}\in \breve{P} ʻʻ(\beta-\alpha ).\equiv .\overrightarrow{P}ʻy\cap \beta \subset \alpha\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*37·105.\supset \vdash \colon\ldotp y{\sim}\in \breve{P} ʻʻ(\beta -\alpha ).&\equiv :{\sim}(\exists x).x\in \beta -\alpha .xPy:\\ +[*10·51] &\equiv:x\in \beta .xPy.\supset _{x}.x\in \alpha :\\ +[*32·18] &\equiv:\overrightarrow{P}ʻy\cap \beta \subset \alpha \colon\ldotp \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*170·11.</b> \[\begin{align}&\vdash \colon\ldotp \alpha P_{\text{cl}}\beta + .\equiv :\alpha ,\beta \in \text{Cl}ʻCʻP:(\exists y).y\in \alpha -\beta .\overrightarrow{P}ʻy\cap \beta \subset \alpha \\ +&[*170·1·103]\end{align}\]</p> + +<p class="nind"><b>*170·12.</b> \[\begin{align}&\vdash :\alpha P_{\text{cl}}\beta + .\equiv .\alpha ,\beta \in \text{Cl}ʻCʻP.\exists !\alpha -(\alpha \cap \beta )-\breve{P} ʻʻ\{\beta -(\alpha \cap \beta )\}\\ +&[*170·1.*22·93]\end{align}\]</p> + +<p class="nind"><b>*170*121.</b> \[\begin{align}&\vdash \colon\ldotp \alpha P_{\text{cl}}\beta + .\equiv .\alpha ,\beta \in \text{Cl}ʻCʻP.\exists !(\alpha \cup \beta )-\beta -\breve{P} ʻʻ\{(\alpha \cup \beta )-\alpha\}\\ +&[*170·1.*22·9]\end{align}\]</p> + +<p class="nind"><b>*170*13.</b> \[\begin{align}&\vdash \colon\ldotp \alpha P_{\text{cl}}\beta .\equiv :(\exists \rho ,\sigma ,\gamma ).\rho ,\sigma ,\gamma \in \text{Cl}ʻCʻP.\\ +&\rho \cap \gamma =\Lambda .\sigma \cap \gamma =\Lambda .\rho \cap \sigma =\Lambda .\alpha =\gamma \cup \rho .\beta =\gamma \cup \sigma .\exists !\rho -\breve{P} ʻʻ\sigma\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*24·24.*22·69 &\supset \vdash :\rho \cap \sigma =\Lambda .\alpha =\gamma \cup \rho .\beta =\gamma \cup \sigma .\supset .\alpha \cap \beta =\gamma &\qquad \text{(1)}\\ +\vdash .*24·4. &\supset \vdash \colon\ldotp \alpha =\gamma \cup \rho .\supset :\rho \cap \gamma =\Lambda .\equiv .\alpha -\gamma =\rho &\qquad \text{(2)}\\ +\vdash .*24·4. &\supset \vdash \colon\ldotp \beta =\gamma \cup \sigma .\supset :\sigma \cap \gamma =\Lambda .\equiv .\beta -\gamma =\sigma &\qquad \text{(3)}\\ +\vdash .(1).(2).(3). &\supset \vdash \colon\ldotp \rho \cap \sigma =\Lambda .\alpha =\gamma \cup \rho .\beta =\gamma \cup \sigma .\supset :\\ +\rho \cap \gamma =\Lambda .\sigma \cap \gamma =\Lambda .&\equiv .\alpha -(\alpha \cap \beta )=\rho .\beta -(\alpha \cap \beta )=\sigma .\\ +[*22·93] &\equiv .\alpha -\beta =\rho .\beta -\alpha =\sigma &\qquad \text{(4)}\\ +\vdash .(1).(4).\supset \vdash :&(\exists \rho ,\sigma ,\gamma ).\rho ,\sigma ,\gamma \in \text{Cl}ʻCʻP.\rho \cap \gamma =\Lambda .\sigma \cap \gamma =\Lambda .\rho \cap \sigma =\Lambda .\\ +&\alpha =\gamma \cup \rho .\beta =\gamma \cup \sigma .\exists !\rho -\breve{P} ʻʻ\sigma .\equiv .\\ +&(\exists \rho ,\sigma ,\gamma ).\rho ,\sigma ,\gamma \in \text{Cl}ʻCʻP.\rho \cap \sigma =\Lambda .\alpha =\gamma \cup \rho .\beta =\gamma \cup \sigma .\\ +&\alpha \cap \beta =\gamma .\alpha -\beta =\rho .\beta -\alpha =\sigma .\exists !\rho -\breve{P} ʻʻ\sigma .\\ +[*13·22] &\equiv .\alpha -\beta ,\beta -\alpha ,\alpha \cap \beta \,\unicode{x2abd}\, \text{Cl}ʻCʻP.\exists !\alpha -\beta -\breve{P} ʻʻ(\beta -\alpha ).\\ +[*60·43.*24·41] &\equiv .\alpha ,\beta \in \text{Cl}ʻCʻP.\exists !\alpha -\beta -\breve{P} ʻʻ(\beta -\alpha ).\\ +[*170·1] &\equiv .\alpha P_{\text{cl}}\beta :\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_414">[Pg 414]</span></p> + +<p class="nind"><b>*170·14.</b> \[\begin{align}&\vdash \colon\ldotp \alpha ,\beta \in \text{Cl}ʻCʻP.\supset :\alpha \dot{-} P_{\text{cl}}\beta + .\equiv .\alpha -\beta \subset \breve{P} ʻʻ(\beta -\alpha )\\ +&[*170·1.*24·55]\end{align}\]</p> + +<p class="nind"><b>*170·141.</b> \[\begin{align}&\vdash \colon\ldotp \alpha ,\beta \in \text{Cl}ʻCʻP.\supset :\alpha \dot{-} P_{\text{lc}}\beta .\equiv .\beta -\alpha \subset Pʻʻ(\alpha - \beta )\\ +&[*170·14·101]\end{align}\]</p> + +<p class="nind"><b>*170·15.</b> \(\vdash :\alpha P_{\text{cl}}\beta .\supset .\beta \cap pʻ\overrightarrow{P}ʻʻ(\alpha -\beta )\subset \alpha\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*40·12.&\supset \vdash :y\in \alpha -\beta .\supset .pʻ\overrightarrow{P}ʻʻ(\alpha -\beta )\subset \overrightarrow{P}ʻy.\\ +[*22·48] &\supset .\beta \cap pʻ\overrightarrow{P}ʻʻ(\alpha -\beta )\subset \beta \cap \overrightarrow{P}ʻy:\\ +[*22·44] &\supset \vdash :y\in \alpha -\beta .\beta \cap \overrightarrow{P}ʻy\subset \alpha .\supset .\beta \cap pʻ\overrightarrow{P}ʻʻ(\alpha -\beta )\subset \alpha :\\ +[*10·11·23] &\supset \vdash :(\exists y).y\in \alpha -\beta .\beta \cap \overrightarrow{P}ʻy\subset \alpha .\supset .\beta \cap pʻ\overrightarrow{P}ʻʻ(\alpha -\beta )\subset \alpha &\qquad \text{(1)}\\ +\vdash .(1).*170·11.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*170·16">*170·16</a>.</b> \(\vdash :\alpha \subset CʻP.\beta \subset \alpha .\beta \neq \alpha .\supset .\alpha P_{\text{cl}}\beta\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*24·6.\supset \vdash :\text{Hp}.&\supset .\exists !\alpha -\beta &\qquad \text{(1)}\\ +\vdash .*24·3.&\supset \vdash :\text{Hp}.\supset .\beta -\alpha =\Lambda .\\ +[*37·29] &\supset .\breve{P} ʻʻ(\beta -\alpha )=\Lambda .\\ +[*24·101·26] &\supset .\alpha -\beta -\breve{P} ʻʻ(\beta -\alpha )=\alpha -\beta &\qquad \text{(2)}\\ +\vdash .(1).(2).*170·1.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*170·161.</b> \(\vdash :\alpha \subset CʻP.\beta \subset \alpha .\beta \neq \alpha .\supset .\beta P_{\text{lc}}\alpha \)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*170·16.\supset \vdash :\text{Hp}.&\supset .\alpha (\breve{P} )_{\text{cl}}\beta .\\ +[*170·101] &\supset .\beta P_{\text{lc}}\alpha :\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*170·17">*170·17</a>.</b> \(\vdash .P_{\text{cl}}\,\unicode{x2abd}\, J.P_{\text{lc}}\,\unicode{x2abd}\, J\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*170·1.\supset \vdash :\alpha P_{\text{cl}}\beta .&\supset .\exists !\alpha -\beta .\\ +[*24·55.*22·42] &\supset .\alpha \neq \beta .\\ +[*50·11] &\supset .\alpha J\beta :\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_415">[Pg 415]</span></p> + +<p>In order that \(P_{\text{cl}}\) should be serial, we need further that +it should be transitive and connected. \(P_{\text{cl}}\) is transitive +if \(P\) is transitive and connected. But \(P_{\text{cl}}\) may still +not be connected: there may be many distinct families in its field, +though all of them must begin with \(CʻP\) and end with \(\Lambda\). +For example, if \(P\) is a regression, the class which takes every +odd member does not have either of the relations \(P_{\text{cl}}\), +\(\breve{P} _{\text{cl}}\) to the class which takes every even member. +In order that \(P_{\text{cl}}\) should be serial, we require that \(P\) +should be not only serial, but well-ordered, <i>i.e.</i> that every +existent sub-class of \(CʻP\) should have a first term. When \(P\) is +serial but not well-ordered, \(P_{\text{cl}}\) will, however, generate +various series contained in it by imposing suitable limitations on the +field.</p> + +<p class="nind"><b>*170·2.</b> + \[\begin{align}&\vdash \colon\ldotp \alpha ,\beta \in \text{Cl}ʻCʻP:(\exists y). y\in \alpha -\beta . \overrightarrow{P}ʻy\cap \alpha = \overrightarrow{P}ʻy\cap \beta :\supset .\alpha P_{\text{cl}}\beta \\ +&[*170·11 .*22·43]\end{align}\]</p> + +<p class="nind"><b>*170·21.</b> \(\vdash \colon\ldotp \alpha \subset CʻP.\supset : y \,\text{min}_{P}(\alpha -\beta ).\equiv . y\in \alpha -\beta . \overrightarrow{P}ʻy\cap \alpha \subset \beta\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *93·11. \supset \vdash \colon\ldotp \text{Hp}.\supset : y \,\text{min}_{P}(\alpha -\beta ).&\equiv . y\in \alpha -\beta -\breve{P} ʻʻ(\alpha -\beta ).\\ +[*170·103] &\equiv .y\in \alpha -\beta .\overrightarrow{P}ʻy\cap \alpha \subset \beta \colon\ldotp \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*170·22.</b> \(\vdash \colon\ldotp \alpha \subset CʻP. y \text{min}_{P} + (\alpha -\beta ).\supset : \overrightarrow{P}ʻy\cap \beta \subset \alpha .\equiv . \overrightarrow{P}ʻy\cap \alpha = \overrightarrow{P}ʻy\cap \beta\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *170·21. *4·73.\supset \vdash \colon\ldotp \text{Hp}.\supset : \overrightarrow{P}ʻy\cap \beta \subset \alpha .&\equiv . \overrightarrow{P}ʻy\cap \alpha \subset \beta . \overrightarrow{P}ʻy\cap \beta \subset \alpha .\\ +[*22·74] &\equiv . \overrightarrow{P}ʻy\cap \alpha = \overrightarrow{P}ʻy\cap \beta \colon\ldotp \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*170·23.</b> \[\begin{align}\vdash \colon\ldotp \alpha \subset CʻP. y\in \alpha -\beta -\breve{P} ʻʻ&(\beta -\alpha ).\supset :\\ +y \,\text{min}_{P} (\alpha-\beta ) .\equiv . \overrightarrow{P}ʻy\cap \alpha =\overrightarrow{P}ʻy\cap \beta \end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*170·103·21. \supset \vdash \colon\ldotp \text{Hp}\supset :\\ +y\,\text{min}_{P} (\alpha -\beta ) .&\equiv . y\in \alpha - \beta .\overrightarrow{P}ʻy\cap \beta \subset \alpha .\overrightarrow{P}ʻy\cap \alpha \subset \beta .\\ +[*22·74.*4·73] &\equiv . y\in \alpha -\beta .\overrightarrow{P}ʻy\cap \beta \subset \alpha .\overrightarrow{P}ʻy\cap \alpha =\overrightarrow{P}ʻy\cap \beta .\\ +[*170·103] &\equiv . y\in \alpha -\beta -\breve{P} ʻʻ(\beta -\alpha ).\overrightarrow{P}ʻy\cap \alpha = \overrightarrow{P}ʻy\cap \beta &\qquad \text{(1)}\\ +\vdash .(1).*5·32.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*170·3.</b> \(\vdash :\alpha \in \text{Cl}ʻCʻP.\beta \subset \alpha .\exists !\alpha -\beta .\supset .\alpha P_{\text{cl}}\beta \quad[*170·16]\)</p> + +<p class="nind"><b>*170·31.</b> \(\vdash :\beta \subset CʻP.\beta \neq CʻP .\equiv . (CʻP) P_{\text{cl}}\beta \quad[*170·16]\)</p> + +<p class="nind"><b>*170·32.</b> \(\vdash :\alpha \subset CʻP.\exists ! \alpha .\equiv . \alpha P_{\text{cl}}\Lambda \quad[*170·3]\)</p> + +<p class="nind"><b>*170·33.</b> \(\vdash :\dot{\exists} !P .\equiv . (CʻP)P_{\text{cl}}\Lambda\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*33·24. *170·32. &\supset \vdash : \dot{\exists} !P.\supset .(CʻP)P_{\text{cl}}\Lambda &\qquad \text{(1)}\\ +\vdash .*170·1. &\supset \vdash : (CʻP)P_{\text{cl}}\Lambda .\supset .\exists !(CʻP)-\Lambda .\\ +[*33·24] &\supset .\dot{\exists} !P &\qquad \text{(2)}\\ +\vdash .(1).(2).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*170·34.</b> \(\vdash :\dot{\exists} !P .\equiv . \dot{\exists} !P_{\text{cl}}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*170·33 .\supset \vdash :\dot{\exists} !P.&\supset .\dot{\exists} !P_{\text{cl}} &\qquad \text{(1)}\\ +\vdash .*170·1. \supset \vdash :\dot{\exists} !P_{\text{cl}}.&\supset .(\exists \alpha ,\beta ).\alpha ,\beta \in \text{Cl}ʻCʻP.\exists !\alpha -\beta .\\ +[*24·561] &\supset .(\exists \alpha ).\alpha \in \text{Cl}ʻCʻP.\exists !\alpha .\\ +[*60·361] &\supset .\exists !CʻP.\\ +[*33·24] &\supset .\dot{\exists} !P &\qquad \text{(2)}\\ +\vdash .(1).(2).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_416">[Pg 416]</span></p> + +<p class="nind"><b><a id="*170·35">*170·35</a>.</b> \(\vdash .\dot{\Lambda} _{\text{cl}}=\dot{\Lambda} \quad[*170·34. \text{Transp}]\)</p> + +<p class="nind"><b><a id="*170·36">*170·36</a>.</b> \(\vdash .\text{D}ʻP_{\text{cl}}=\text{Cl ex}ʻCʻP.\text{ᗡ}ʻP_{\text{cl}}=\text{Cl}ʻCʻP-\iota ʻCʻP\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*170·32. &\supset \vdash .\text{Cl ex}ʻCʻP\subset \text{D}ʻP_{\text{cl}} &\qquad \text{(1)}\\ +\vdash .*170·31. &\supset \vdash .\text{Cl}ʻCʻP-\iota ʻCʻP\subset \text{ᗡ}ʻP_{\text{cl}} &\qquad \text{(2)}\\ +\vdash .*170·1. \supset \vdash :\alpha \in \text{D}ʻP_{\text{cl}}.&\supset .(\exists \beta ).\alpha ,\beta \in \text{Cl}ʻCʻP.\exists !\alpha -\beta .\\ +[*24·561] &\supset . \alpha \in \text{Cl}ʻCʻP.\exists !\alpha &\qquad \text{(3)}\\ +\vdash .*170·1. \supset \vdash :\alpha \in \text{ᗡ}ʻP_{\text{cl}}.&\supset .(\exists \beta ).\alpha ,\beta \in \text{Cl}ʻCʻP.\exists !\beta -\alpha .\\ +[*60·2] &\supset .\alpha \in \text{Cl}ʻCʻP.\exists !CʻP-\alpha .\\ +[*24·6] &\supset .\alpha \in \text{Cl}ʻCʻP-\iota ʻCʻP &\qquad \text{(4)}\\ +\vdash .(1).(2).(3).(4).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*170·37">*170·37</a>.</b> \(\vdash :\dot{\exists} !P.\supset .CʻP_{\text{cl}}=\text{Cl}ʻCʻP \quad[*170·36]\)</p> + +<p class="nind"><b>*170·371.</b> \(\vdash .CʻP_{\text{cl}}\subset \text{Cl}ʻCʻP \quad[*170·37·35.*33·241]\)</p> + +<p class="nind"><b><a id="*170·38">*170·38</a>.</b> \(\vdash :\dot{\exists} !P.\supset .BʻP_{\text{cl}}=CʻP.Bʻ\text{Cnv}ʻP_{\text{cl}}=\Lambda \quad[*170·36]\)</p> + +<p>The following propositions lead up to <a href="#*170·44">*170·44</a>.</p> + +<p class="nind"><b>*170·4.</b> \(\vdash :S\in 1\rightarrow 1.CʻQ=\text{ᗡ}ʻS.\supset .(S^{;}Q)_{\text{cl}}=S_{\in }^{;}Q_{\text{cl}}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*170·1.*150·4.*37·11. &\supset \vdash :\alpha (S_{\in }^{;}Q_{\text{cl}})\beta .\equiv .\\ +&(\exists \gamma ,\delta ).\gamma ,\delta \in \text{Cl}ʻCʻQ.\alpha =Sʻʻ\gamma .\beta =Sʻʻ\delta .\exists !\gamma -\delta -\breve{Q} ʻʻ(\delta -\gamma ) &\qquad \text{(1)}\\ +\vdash .(1).\supset \vdash \colon\ldotp \text{Hp}.\supset :\\ +&\alpha (S_{\in }^{;}Q_{\text{cl}})\beta .\equiv .(\exists \gamma ,\delta ).\gamma ,\delta \in \text{Cl}ʻ\text{ᗡ}ʻS.\alpha =Sʻʻy.\beta =Sʻʻ\delta .\\ +&\exists !\gamma -\delta -\breve{Q} ʻʻ(\delta -\gamma ).\\ +[*37·43] &\equiv .(\exists \gamma ,\delta ).\gamma ,\delta \in \text{Cl}ʻ\text{ᗡ}ʻS.\alpha =Sʻʻ\gamma .\beta =Sʻʻ\delta .\\ +&\exists !Sʻʻ\{\gamma -\delta -\breve{Q} ʻʻ(\delta -\gamma )\}.\\ +[*71·381] &\equiv .(\exists \gamma ,\delta ).\gamma ,\delta \in \text{Cl}ʻ\text{ᗡ}ʻS.\alpha =Sʻʻ\gamma .\beta =Sʻʻ\delta .\\ +&\exists !Sʻʻ\gamma -Sʻʻ\delta -Sʻʻ\breve{Q} ʻʻ(\delta -\gamma ).\\ +[*72·511.*71·38] &\equiv .(\exists \gamma ,\delta ).\gamma ,\delta \in \text{Cl}ʻ\text{ᗡ}ʻS.\alpha =Sʻʻ\gamma .\beta =Sʻʻ\delta .\\ +&\exists !Sʻʻ\gamma -Sʻʻ\delta -Sʻʻ\breve{Q} ʻʻ\breve{S} ʻʻ(\beta -\alpha ).\\ +[*13·193.*37·33] &\equiv .(\exists \gamma ,\delta ).\gamma ,\delta \in \text{Cl}ʻ\text{ᗡ}ʻS.\alpha =Sʻʻ\gamma .\beta =Sʻʻ\delta .\\ +&\exists !\alpha -\beta -(S^{;}\breve{Q} )ʻʻ(\beta -\alpha ).\\ +[*71·48.*37·23] &\equiv .\alpha ,\beta \in \text{Cl}ʻ\text{D}ʻS.\exists !\alpha -\beta -(S^{;}\breve{Q} )ʻʻ(\beta -\alpha ).\\ +[*150·23] &\equiv .\alpha ,\beta \in \text{Cl}ʻCʻ(S^{;}Q).\exists !\alpha -\beta -(S^{;}\breve{Q} )ʻʻ(\beta -\alpha ).\\ +[*150·12.*17·01] &\equiv .\alpha (S^{;}Q)_{\text{cl}}\beta \colon\ldotp \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_417">[Pg 417]</span></p> + +<p class="nind"><b><a id="*170·41">*170·41</a>.</b> \(\vdash .(S\upharpoonright CʻQ)_{\in }^{;}Q_{\text{cl}}=S_{\in }^{;}Q_{\text{cl}} \quad[*150·95.*170·371]\)</p> + +<p class="nind"><b>*170·42.</b> \(\vdash :S\upharpoonright CʻQ\in 1\rightarrow 1.CʻQ\subset \text{ᗡ}ʻS.\supset .(S^{;}Q)_{\text{cl}}=S_{\in }^{;}Q_{\text{cl}}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*150·32. &\supset \vdash .(S^{;}Q)_{\text{cl}}= {(S\upharpoonright CʻQ)^{;}Q}_{\text{cl}} &\qquad \text{(1)}\\ +\vdash .(1).*170·4. \supset \vdash :\text{Hp} .\supset .(S^{;}Q)_{\text{cl}}&=(S\upharpoonright CʻQ)_{\in }^{;}Q_{\text{cl}}\\ +[*170·41] &=S_{\in }^{;}Q_{\text{cl}}:\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*170·43.</b> \(\vdash :S\upharpoonright CʻQ\in P\,\overline{\text{smor}}\,Q.\supset .S_{\in }\upharpoonright CʻQ_{\text{cl}}\in P_{\text{cl}}\,\overline{\text{smor}}\, + Q_{\text{cl}}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*151·22.*170·42. &\supset \vdash :\text{Hp}.\supset .P_{\text{cl}}=S_{\in }^{;}Q_{\text{cl}} &\qquad \text{(1)}\\ +\vdash .*74·131.*170·371. &\supset \vdash :\text{Hp}.\supset .S_{\in }\upharpoonright CʻQ_{\text{cl}}\in 1\rightarrow 1 &\qquad \text{(2)}\\ +\vdash .*37·231. &\supset \vdash .CʻQ_{\text{cl}}\subset \text{ᗡ}ʻS_{\in } &\qquad \text{(3)}\\ +\vdash .(1).(2).(3).*151·22.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*170·44">*170·44</a>.</b> \(\vdash : P\,\text{smor}\,Q.\supset .P_{\text{cl}}\,\text{smor}\,Q_{\text{cl}} \quad[*170·43.*151·23·12]\)</p> + +<p class="nind"><b><a id="*170·5">*170·5</a>.</b> \(\vdash .(x\downarrow x)_{\text{cl}}=(\iota ʻx)\downarrow \Lambda \)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*170·36.*55·15.\supset \vdash .\text{D}ʻ(x\downarrow x)_{\text{cl}}&=\text{Cl ex}ʻ\iota ʻx\\ +[*60·37] &=\iota ʻ\iota ʻx &\qquad \text{(1)}\\ +\vdash .*170·36.*55·15. \supset \vdash .\text{ᗡ}ʻ(x\downarrow x)_{\text{cl}}&=\text{Cl}ʻ\iota ʻx-\iota ʻ\iota ʻx\\ +[*60·362] &=\iota ʻ\Lambda &\qquad \text{(2)}\\ +\vdash .(1).(2).*55·16.\supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*170·51">*170·51</a>.</b> \[\begin{align}\vdash :x\neq y.\supset .(x\downarrow y)_{\text{cl}}=(\iota + ʻx\cup \iota ʻy)&\downarrow \iota ʻx\unicode{x228d} (\iota ʻx\cup \iota ʻy)\downarrow \iota ʻy\unicode{x228d} (\iota ʻx\cup \iota ʻy)\downarrow \Lambda \\ +&\unicode{x228d} \iota ʻx\downarrow \iota ʻy\unicode{x228d} \iota ʻx\downarrow \Lambda \unicode{x228d} \iota ʻy\downarrow \Lambda\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*55·13. \supset \vdash :\text{Hp}.&\supset .\overrightarrow{x\downarrow y}ʻx=\Lambda .\overrightarrow{x\downarrow y}ʻy=\iota ʻx &\qquad \text{(1)}\\ +\vdash .*170·11.*55·15. &\supset \vdash \colon\colon \text{Hp}.\supset \colon\ldotp \alpha (x\downarrow y)_{\text{cl}}\beta .\equiv :\\ +&\alpha ,\beta \in \text{Cl}ʻ(\iota ʻx\cup \iota ʻy):(\exists z).z\in \alpha -\beta .\overrightarrow{x\downarrow y}ʻz\cap \beta \subset \alpha \\ +[*60·39] &\equiv :\alpha =\iota ʻx\cup \iota ʻy.\lor.\alpha =\iota ʻx.\lor.\alpha =\iota ʻy:\beta \subset \iota ʻx\cup \iota ʻy :\\ +&(\exists z).z\in \alpha -\beta .\overrightarrow{x\downarrow y}ʻz\cap \beta \subset \alpha &\qquad \text{(2)}\\ +\vdash .*51·235. &\supset \vdash \colon\colon \alpha =\iota ʻx\cup \iota ʻy.\supset \colon\ldotp (\exists z).z\in \alpha -\beta .\overrightarrow{x\downarrow y}ʻz\cap \beta \subset \alpha .\equiv :\\ +&x\in \alpha -\beta .\overrightarrow{x\downarrow y}ʻx\cap \beta \subset \alpha .\lor.y\in \alpha -\beta .\overrightarrow{x\downarrow y}ʻy\cap \beta \subset \alpha :\\ +[(1)] &\equiv :x\in \alpha -\beta .\lor.y\in \alpha -\beta .\iota ʻx\cap \beta \subset \alpha :\\ +[\text{Hp}.*22·43·58] &\equiv :x\in \alpha -\beta .\lor.y\in \alpha -\beta :\\ +[*51·232.*4·73] &\equiv :x{\sim}\in \beta .\lor.y{\sim}\in \beta &\qquad \text{(3)}\\ +\vdash .*54·4.&\supset \vdash \colon\colon \text{Hp}.\supset \colon\ldotp \beta \subset \iota ʻx\cup \iota ʻy.x{\sim}\in \beta .\equiv :\beta =\iota ʻy.\lor.\beta =\Lambda &\qquad \text{(4)}\\ +\vdash .*54·4.&\supset \vdash \colon\colon \text{Hp}.\supset \colon\ldotp \beta \subset \iota ʻx\cup \iota ʻy.y{\sim}\in \beta .\equiv :\beta =\iota ʻx.\lor.\beta =\Lambda &\qquad \text{(5)}\\ +\vdash .(2).(3).(4).(5).\supset \\ +\vdash \colon\colon \alpha =\iota ʻx\cup \iota ʻy.&\supset \colon\ldotp \alpha (x\downarrow y)_{\text{cl}}\beta + .\equiv :\beta =\iota ʻx.\lor.\beta =\iota ʻy.\lor.\beta =\Lambda &\qquad \text{(6)}\\ +\vdash .(1).(2).&\supset \vdash \colon\colon .\text{Hp}.\supset \colon\colon \alpha =\iota ʻx.\supset \colon\ldotp \alpha (x\downarrow y)_{\text{cl}}\beta + .\equiv :\beta \subset \iota ʻx\cup \iota ʻy.x{\sim}e\beta .\\ +[(4)] &\equiv :\beta =\iota ʻy.\lor.\beta =\Lambda &\qquad \text{(7)}\\ +\vdash .(1).(2).&\supset \vdash \colon\colon \text{Hp}.\supset \colon\ldotp \alpha =\iota ʻy.\supset :\alpha (x\downarrow y)_{\text{cl}}\beta .\equiv .\\ +&\beta \subset \iota ʻx\cup \iota ʻy.y{\sim}\in \beta .\iota ʻx\cap \beta \subset \alpha .\\ +[*51·211] &\equiv .\beta \subset \iota ʻx\cup \iota ʻy.y{\sim}\in \beta .x{\sim}\in \beta .\\ +[*54·4] &\equiv .\beta =\Lambda &\qquad \text{(8)}\\ +\vdash .(2).(6).(7).(8).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_418">[Pg 418]</span></p> + +<p class="nind"><b><a id="*170·52">*170·52</a>.</b> \(\vdash :x\neq y.\supset .(x\downarrow y)_{\text{cl}}=(\iota ʻx\cup \iota ʻy)\downarrow \iota ʻx\unicode{x2909}\iota ʻy\downarrow \Lambda\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*55·15.\supset \vdash .&Cʻ{(\iota ʻx\cup \iota ʻy)\downarrow \iota ʻx}\uparrow Cʻ\{\iota ʻy\downarrow \Lambda \}=\\ +&\{\iota ʻ(\iota ʻx\cup \iota ʻy)\cup \iota ʻ\iota ʻx\}\uparrow \{\iota ʻ\iota ʻy\cup \iota ʻ\Lambda\}\\ +[*55·52] &=(\iota ʻx\cup \iota ʻy)\downarrow \iota ʻy\unicode{x228d} (\iota ʻx\cup \iota ʻy)\downarrow \iota ʻ\Lambda \unicode{x228d} \iota ʻx\downarrow \iota ʻy\unicode{x228d} \iota ʻx\downarrow \Lambda &\qquad \text{(1)}\\ +\vdash .(1).*170·51.*160·1.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*170·6.</b> \(\vdash :\Lambda P_{\text{lc}}\beta .\equiv .\beta \subset CʻP.\exists !\beta \quad[*170·32·101]\)</p> + +<p class="nind"><b>*170·601.</b> \(\vdash :\alpha P_{\text{lc}}(CʻP).\equiv .\alpha \subset CʻP.\alpha \neq CʻP \quad[*170·31·101]\)</p> + +<p class="nind"><b>*170·61.</b> \[\begin{align}\vdash \colon\ldotp x{\sim}&\in CʻP.\dot{\exists} !P.x\in \alpha \cap \beta .\supset :\\ +&\alpha (x\unicode{x21f7}P)_{\text{cl}}\beta .\equiv .\alpha \{(\iota ʻx\cup )^{;}P_{\text{cl}}\}\beta .\equiv .(\alpha -\iota ʻx)P_{\text{cl}}(\beta + -\iota ʻx)\end{align}\]</p> + +<p>This and the following propositions are lemmas for +\[ +x{\sim}\in CʻP.\supset .(x\unicode{x21f7}P)_{\text{cl}}=(\iota ʻx\cup )^{;}P_{\text{cl}}\unicode{x2909}P_{\text{cl}} \quad(*170·64). +\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*161·111.\supset \vdash \colon\colon \text{Hp}.&\supset \colon\ldotp y\in \beta .y(x\unicode{x21f7}P)z.\supset _{y}.y\in \alpha :\equiv :\\ +&y\in \beta .yPz.\supset _{y}.y\in \alpha :y\in \beta .y=x.z\in CʻP.\supset _{y}.y\in \alpha :\\ +[*13·191 . *33·17]&\equiv :y\in \beta -\iota ʻx.yPz.\supset _{y}.y\in \alpha -\iota ʻx:x\in \beta .z\in CʻP.\supset .x\in \alpha :\\ +[\text{Hp}] &\equiv :y\in \beta -\iota ʻx.yPz.\supset _{y}.y\in \alpha -\iota ʻx &\qquad \text{(1)}\\ +\vdash .*51·34.\supset \vdash :\text{Hp}.&\supset .-\beta =-\iota ʻx\cap -\beta .\\ +[*22·481] \supset .\alpha -\beta &=\alpha -\iota ʻx-\beta \\ +[*24·21] &=\alpha -\iota ʻx\cap (\iota ʻx \cup -\beta )\\ +[*22·86] &=\alpha -\iota ʻx-(\beta -\iota ʻx) &\qquad \text{(2)}\\ +\vdash .*170·11.*161·101·14.&\supset \vdash \colon\colon \text{Hp}.\supset \colon\ldotp \alpha (x\unicode{x21f7}P)_{\text{cl}}\beta .\equiv :\\ +&\alpha ,\beta \in \text{Cl}ʻ(CʻP\cup \iota ʻx):(\exists z):z\in \alpha -\beta :y\in \beta .y(x\unicode{x21f7}P)z.\supset _{z}.y\in \alpha :\\ +[(1).(2)] &\equiv :\alpha ,\beta \in \text{Cl}ʻ(CʻP\cup \iota ʻx):(\exists z):z\in \alpha -\iota ʻx-(\beta -\iota ʻx):y\in \beta -\iota ʻx.yPz.\supset _{y}.\\ +&y\in \alpha -\iota ʻx:\\ +[*24·43] &\equiv :\alpha -\iota ʻx,\beta -\iota ʻx\in \text{Cl}ʻCʻP:(\exists z).z\in \alpha -\iota ʻx-(\beta -\iota ʻx).\\ +&\overrightarrow{P}ʻz\cap (\beta -\iota ʻx)\subset \alpha -\iota ʻx:\\ +[*170·11] &\equiv :(\alpha -\iota ʻx)P_{\text{cl}}(\beta -\iota ʻx):\\ +[*51·221] &\equiv :\alpha {(\iota ʻx\cup )^{;}P_{\text{cl}}}\beta \colon\colon \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_419">[Pg 419]</span></p> + +<p class="nind"><b>*170·62.</b> \[\begin{align}\vdash \colon\ldotp x{\sim}\in CʻP.\dot{\exists} !P.x\in &\alpha -\beta .\supset :\\ +&\alpha (x\unicode{x21f7}P)_{\text{cl}}\beta .\equiv .\alpha \subset \iota ʻx\cup CʻP.\beta \subset CʻP\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*161·13. \supset \vdash \colon\ldotp \text{Hp}.&\supset :x{\sim}\in \text{ᗡ}ʻ(x\unicode{x21f7}P):\\ +[*10·53] &\supset :y\in \beta .y(x\unicode{x21f7}P)x.\supset _{y}.y\in \alpha :\\ +[\text{Hp}] &\supset :x\in \alpha -\beta :y\in \beta .y(x\unicode{x21f7}P)x.\supset _y.y\in \alpha :\\ +[*170·11] &\supset :\alpha ,\beta \in \text{Cl}ʻCʻ(x\unicode{x21f7}P).\supset .\alpha (x\unicode{x21f7}P)_{\text{cl}}\beta :\\ +[*161·14.\text{Hp}.*24·49] &\supset :\alpha \subset \iota ʻx\cup CʻP.\beta \subset CʻP.\supset .\alpha (x\unicode{x21f7}P)_{\text{cl}}\beta &\qquad \text{(1)}\\ +\vdash .*170·11.*161·14.\supset\\ +\vdash \colon\ldotp \text{Hp}. \supset :\alpha (x\unicode{x21f7}P)_{\text{cl}}\beta .&\supset .\alpha ,\beta \in \text{Cl}ʻ(\iota ʻx\cup CʻP).\\ +[*24·49.\text{Hp}] &\supset .\alpha \subset \iota ʻx\cup CʻP.\beta \subset CʻP &\qquad \text{(2)}\\ +\vdash .(1).(2).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*170·63.</b> \(\vdash \colon\ldotp x{\sim}\in (\alpha \cup \beta ).\supset :\alpha (x\unicode{x21f7}P)_{\text{cl}}\beta .\equiv .\alpha P_{\text{cl}}\beta\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*24·49.*161·14. &\supset \vdash \colon\ldotp \text{Hp}.\supset :\alpha ,\beta \in \text{Cl}ʻCʻ(x\unicode{x21f7}P).\equiv .\alpha ,\beta \in \text{Cl}ʻCʻP &\qquad \text{(1)}\\ +\vdash .*13·14. &\supset \vdash \colon\colon \text{Hp}.\supset \colon\ldotp y\in \beta .\supset :y\neq x:\\ +[*161·111] &\supset :y(x\unicode{x21f7}P)z .\equiv .yPz &\qquad \text{(2)}\\ +\vdash .*170·11. &\supset \vdash \colon\colon \text{Hp}.\supset \colon\ldotp \alpha (x\unicode{x21f7}P)_{\text{cl}}\beta .\equiv :\\ +&\alpha ,\beta \in \text{Cl}ʻ(x\unicode{x21f7}P):(\exists z):z\in \alpha -\beta :y\in \beta .y(x\unicode{x21f7}P)z.\supset _{y}.y\in \alpha :\\ +[(1).(2)] &\equiv :\alpha ,\beta \in \text{Cl}ʻCʻP:(\exists z):z\in \alpha -\beta :y\in \beta .yPz.\supset _{y}.y\in \alpha :\\ +[*170·11] &\equiv :\alpha P_{\text{cl}}\beta \colon\colon \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*170·64">*170·64</a>.</b> \(\vdash :x{\sim}\in CʻP.\supset .(x\unicode{x21f7}P)_{\text{cl}}=(\iota ʻx\cup )^{;}P_{\text{cl}}\unicode{x2909}P_{\text{cl}}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*170·61·62·63·37.\supset \\ +\vdash \colon\colon \text{Hp}.&\dot{\exists} !P.\supset \colon\ldotp \alpha (x\unicode{x21f7}P)_{\text{cl}}\beta .\equiv :\\ +&x\in \alpha \cap \beta .\alpha {(\iota ʻx\cup )^{;}P_{\text{cl}}}\beta .\lor.x\in \alpha -\beta .\alpha \in Cʻ(\iota ʻx\cup )^{;}P_{\text{cl}}.\beta + \in CʻP_{\text{cl}}.\lor.\\ +&x{\sim}\in (\alpha \cup \beta ).\alpha P_{\text{cl}}\beta &\qquad \text{(1)}\\ +\vdash .*150·4. &\supset \vdash :\alpha {(\iota ʻx\cup )^{;}P}\beta .\supset .x\in \alpha \cap \beta &\qquad \text{(2)}\\ +\vdash .*150·22.*170·37. &\supset \vdash \colon\ldotp \text{Hp}.\supset :\alpha \in Cʻ(\iota ʻx\cup )^{;}P_{\text{cl}}.\beta \in CʻP_{\text{cl}}.\supset + .x\in \alpha -\beta &\qquad \text{(3)}\\ +\vdash .*170·1. &\supset \vdash \colon\ldotp \text{Hp}.\supset :\alpha P_{\text{cl}}\beta .\supset .x{\sim}\in (\alpha \cap \beta ) &\qquad \text{(4)}\\ +\vdash .(1).(2).(3).(4). &\supset \vdash \colon\colon \text{Hp}.\dot{\exists} !P.\supset \colon\ldotp\\ +\alpha (x\unicode{x21f7}P)_{\text{cl}}\beta .&\equiv :\alpha {(\iota ʻx\cup )^{;}P_{\text{cl}}}\beta .\lor.\alpha \in Cʻ(\iota ʻx\cup )^{;}P_{\text{cl}}.\beta + \in CʻP_{\text{cl}}.\lor.\alpha P_{\text{cl}}\beta :\\ +[*160·11] &\equiv :\alpha {(\iota ʻx\cup )^{;}P_{\text{cl}}\unicode{x2909}P_{\text{cl}}}\beta &\qquad \text{(5)}\\ +\vdash .*161·201.&\supset \vdash :P=\dot{\Lambda} .\supset .x\unicode{x21f7}P=\dot{\Lambda} .\\ +[*170·35] &\supset .(x\unicode{x21f7}P)_{\text{cl}}=\dot{\Lambda} &\qquad \text{(6)}\\ +\vdash .*150·42.*160·22.*170·35. &\supset \vdash :P=\dot{\Lambda} .\supset .(\iota ʻx\cup )^{;}P_{\text{cl}}\unicode{x2909}P_{\text{cl}}=\dot{\Lambda} &\qquad \text{(7)}\\ +\vdash .(6).(7). &\supset \vdash :P=\dot{\Lambda} .\supset .(x\unicode{x21f7}P)_{\text{cl}}=(\iota ʻx\cup )^{;}P_{\text{cl}}\unicode{x2909}P_{\text{cl}} + &\qquad \text{(8)}\\ +\vdash .(5).(8).&\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_420">[Pg 420]</span></p> + +<p>The following propositions are lemmas for <a href="#*170·67">*170·67</a>, <i>i.e.</i> +\[ +\dot{\exists} !P.\dot{\exists} !Q.CʻP\cap CʻQ=\Lambda .\supset .(P\unicode{x2909}Q)_{\text{cl}}=s^{;}C^{;}(P_{\text{cl}}\times Q_{\text{cl}}), +\] +which itself leads to <a href="#*170·69">*170·69</a>, <i>i.e.</i> +\[ +\dot{\exists} !P.\dot{\exists} !Q.CʻP\cap CʻQ=\Lambda .\supset .(P\unicode{x2909}Q)_{\text{cl}}\text{smor}(P_{\text{cl}} \times Q_{\text{cl}}). +\]</p> + +<p class="nind"><b>*170·65.</b> \[\begin{align}&\vdash \colon\ldotp \rho (P\unicode{x2909}Q)_{\text{cl}}\sigma + .\equiv :(\exists \alpha ,\beta ,\gamma ,\delta ):\alpha ,\beta \in \text{Cl}ʻCʻP.\gamma ,\delta \in \text{Cl}ʻCʻQ.\\ +&\rho =\alpha \cup \gamma .\sigma =\beta \cup \delta :(\exists y).y\in (\alpha \cup \gamma )-(\beta \cup \delta ).\overrightarrow{P\unicode{x2909}Q} ʻy\cap (\beta \cup \delta )\subset \alpha \cup \gamma \end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*13·193.&\supset \vdash \colon\ldotp (\exists \alpha ,\beta ,\gamma ,\delta ):\alpha ,\beta \in \text{Cl}ʻCʻP.\gamma ,\delta \in \text{Cl}ʻCʻQ.\rho =\alpha \cup \gamma .\sigma =\beta \cup \delta :\\ +&(\exists y).y\in (\alpha \cup \gamma )-(\beta \cup \delta ).\overrightarrow{P\unicode{x2909}Q} ʻy\cap (\beta \cup \delta )\subset \alpha \cup \gamma :\\ +&\equiv :(\exists \alpha ,\beta ,\gamma ,\delta ):\alpha ,\beta \in \text{Cl}ʻCʻP.\gamma ,\delta \in \text{Cl}ʻCʻQ.\rho =\alpha \cup \gamma .\sigma =\beta \cup \delta :\\ +&(\exists y).y\in \rho -\sigma .\overrightarrow{P\unicode{x2909}Q} ʻy\cap \sigma \subset \rho :\\ +[*60·45] &\equiv :\rho ,\sigma \in \text{Cl}ʻ(CʻP\cup CʻQ):(\exists y).y\in \rho -\sigma .\overrightarrow{P\unicode{x2909}Q} ʻy\cap \sigma \subset \rho :\\ +[*160·14] &\equiv :\rho ,\sigma \in \text{Cl}ʻCʻ(P\unicode{x2909}Q):(\exists y).y\in \rho -\sigma .\overrightarrow{P\unicode{x2909}Q} ʻy\cap \sigma \subset \rho :\\ +[*170·11] &\equiv :\rho (P\unicode{x2909}Q)_{\text{cl}}\sigma \colon\ldotp \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*170·651.</b> \[\begin{align}&\vdash \colon\ldotp CʻP\cap CʻQ=\Lambda .\alpha ,\beta \in \text{Cl}ʻCʻP.\gamma ,\delta \in \text{Cl}ʻCʻQ.y\in \alpha .\supset :\\ +&y\in (\alpha \cup \gamma )-(\beta \cup \delta ).\overrightarrow{P\unicode{x2909}Q} ʻy\cap (\beta \cup \delta )\subset \alpha \cup \gamma .\equiv .y\in \alpha -\beta .\overrightarrow{P}ʻy\cap \beta \subset \alpha \end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*24·402·313.&\supset \vdash :\text{Hp}.\supset .(\alpha \cup \gamma )-(\beta \cup \delta )=(\alpha -\beta )\cup (\gamma -\delta ) &\qquad \text{(1)}\\ +\vdash .*160·11. &\supset \vdash :\text{Hp}.\supset .\overrightarrow{P\unicode{x2909}Q} ʻy=\overrightarrow{P}ʻy &\qquad \text{(2)}\\ +\vdash .*24·402. &\supset \vdash \colon\ldotp \text{Hp}.\supset :y{\sim}\in \gamma :\\ +[(1)] &\supset :y\in (\alpha \cup \gamma )-(\beta \cup \delta ).\equiv .y\in \alpha -\beta &\qquad \text{(3)}\\ +\vdash .*33·15·161. &\supset \vdash .\overrightarrow{P}ʻy\subset CʻP.\\ +[*24·402] &\supset \vdash :\text{Hp}.\supset .\overrightarrow{P}ʻy\cap \delta =\Lambda .\\ +[(2)] &\supset .\overrightarrow{P\unicode{x2909}Q} ʻy\cap (\beta \cup \delta )=\overrightarrow{P}ʻy\cap \beta . &\qquad \text{(4)}\\ +[*24·402] &\supset .\overrightarrow{P\unicode{x2909}Q} ʻy\cap (\beta \cup \delta )\cap \gamma =\Lambda &\qquad \text{(5)}\\ +\vdash .(4).(5).*24·49.&\supset \vdash \colon\ldotp \text{Hp}.\supset :\overrightarrow{P\unicode{x2909}Q} ʻy\cap (\beta \cup \delta )\subset \alpha \cup \gamma .\equiv .\\ +&\overrightarrow{P}ʻy\cap \beta \subset \alpha &\qquad \text{(6)}\\ +\vdash .(3).(6).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*170·652.</b> \[\begin{align}\vdash \colon\ldotp CʻP\cap &CʻQ=\Lambda .\alpha ,\beta \in \text{Cl}ʻCʻP.\gamma ,\delta \in \text{Cl}ʻCʻQ.y\in \gamma .\supset :\\ +&y\in (\alpha \cup \gamma )-(\beta \cup \delta ).\overrightarrow{P\unicode{x2909}Q} ʻy\cap (\beta \cup \delta )\subset \alpha \cup \gamma .\equiv .\\ +&\beta \subset \alpha .y\in \gamma -\delta .\overrightarrow{Q}ʻy\cap \delta \subset \gamma\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*24·402·313.&\supset \vdash :\text{Hp}.\supset .(\alpha \cup \gamma )-(\beta \cup \delta )=(\alpha -\beta )\cup (\gamma -\delta ) &\qquad \text{(1)}\\ +\vdash .*24·402. &\supset \vdash :\text{Hp}.\supset .y{\sim}\in \alpha &\qquad \text{(2)}\\ +\vdash .(1).(2).&\supset \vdash \colon\ldotp \text{Hp}.\supset :y\in (\alpha \cup y)-(\beta \cup \delta ).\equiv .y\in \gamma -\delta &\qquad \text{(3)}\\ +\vdash .*160·11.&\supset \vdash :\text{Hp}.\supset .\overrightarrow{P\unicode{x2909}Q} ʻy=CʻP\cup \overrightarrow{Q}ʻy.\\ +[*22·621.*24·402] &\supset .\overrightarrow{P\unicode{x2909}Q} ʻy\cap (\beta \cup \delta )=\beta \cup (\overrightarrow{Q}ʻy\cap \delta ) &\qquad \text{(4)}\\ +\vdash .*24·49. &\supset \vdash \colon\ldotp \text{Hp}.\supset :\beta \subset \alpha \cup \gamma .\equiv .\beta \subset \alpha :\\ +&\overrightarrow{Q}ʻy\cap \delta \subset \alpha \cup \gamma .\equiv .\overrightarrow{Q}ʻy\cap \delta \subset \gamma &\qquad \text{(5)}\\ +\vdash .(4).(5).&\supset \vdash \colon\ldotp \text{Hp}.\supset :\\ +&\overrightarrow{P\unicode{x2909}Q} ʻy\cap (\beta \cup \delta )\subset \alpha \cup \gamma .\equiv .\beta \subset \alpha .\overrightarrow{Q}ʻy\cap \delta \subset \gamma &\qquad \text{(6)}\\ +\vdash .(3).(6).&\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_421">[Pg 421]</span></p> + +<p class="nind"><b>*170·653.</b> \[\begin{align}\vdash \colon\colon CʻP\cap CʻQ&=\Lambda .\alpha ,\beta \in \text{Cl}ʻCʻP.\gamma ,\delta \in \text{Cl}ʻCʻQ.\supset \colon\ldotp \\ +&(\alpha \cup \gamma )(P\unicode{x2909}Q)_{\text{cl}}(\beta \cup \delta ).\equiv :\alpha P_{\text{cl}}\beta + .\lor.\alpha =\beta .\gamma Q_{\text{cl}}\delta\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*170·11. \supset \vdash \colon\colon \text{Hp}.&\supset \colon\ldotp (\alpha \cup \gamma )(P\unicode{x2909}Q)_{\text{cl}}(\beta \cup \delta ).\equiv :\\ +&(\exists y).y\in (\alpha \cup \gamma )-(\beta \cup \delta ).\overrightarrow{P\unicode{x2909}Q} ʻy\cap (\beta \cup \delta )\subset \alpha \cup \gamma :\\ +[*170·651·652]&\equiv :(\exists y).y\in \alpha -\beta .\overrightarrow{P}ʻy\cap \beta \subset \alpha :\lor:\\ +&\beta \subset \alpha :(\exists y).y\in \gamma -\delta .\overrightarrow{Q}ʻy\cap \delta \subset \beta :\\ +[*170·11] &\equiv :\alpha P_{\text{cl}}\beta .\lor.\beta \subset \alpha .\gamma Q_{\text{cl}}\delta :\\ +[*170·16] &\equiv :\alpha P_{\text{cl}}\beta .\lor.\alpha P_{\text{cl}}\beta .\gamma Q_{\text{cl}}\delta .\lor.\alpha =\beta .\gamma Q_{\text{cl}}\delta :\\ +[*4·44] & \equiv :\alpha P_{\text{cl}}\beta .\lor.\alpha =\beta .\gamma Q_{\text{cl}}\delta \colon\colon \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*170·66.</b> \[\begin{align}&\vdash \colon\ldotp \dot{\exists} !P.\dot{\exists} !Q.CʻP\cap CʻQ=\Lambda .\supset :\\ +&\rho (P\unicode{x2909}Q)\sigma .\equiv .(\exists \alpha ,\beta ,\gamma ,\delta ).(\gamma \downarrow \alpha )(P_{\text{cl}}\times Q_{\text{cl}})(\delta + \downarrow \beta ).\rho =\alpha \cup \gamma .\sigma =\beta \cup \delta\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*170·65·11.\supset \\ +\vdash :\rho (P\unicode{x2909}Q)_{\text{cl}}\sigma .&\equiv .(\exists \alpha ,\beta ,\gamma ,\delta ).\alpha ,\beta \in \text{Cl}ʻCʻP.\gamma ,\delta \in \text{Cl}ʻCʻQ.\\ +&\rho =\alpha \cup \gamma .\sigma =\beta \cup \delta .(\alpha \cup \gamma )(P\unicode{x2909}Q)_{\text{cl}}(\beta \cup \delta ) &\qquad \text{(1)}\\ +\vdash .(1).*170·653.&\supset \vdash \colon\colon \text{Hp}.\supset \colon\ldotp \\ +\rho (P\unicode{x2909}Q)_{\text{cl}}\sigma + .&\equiv :(\exists \alpha ,\beta ,\gamma ,\delta ):\alpha ,\beta \in \text{Cl}ʻCʻP.\gamma ,\delta \in \text{Cl}ʻCʻQ.\rho =\alpha \cup \gamma .\sigma =\beta \cup \delta :\\ +&\alpha P_{\text{cl}}\beta .\lor.\alpha =\beta .\gamma Q_{\text{cl}}\delta :\\ +[*170·37] &\equiv :(\exists \alpha ,\beta ,\gamma ,\delta ):\alpha ,\beta \in CʻP_{\text{cl}}.\gamma ,\delta \in CʻQ_{\text{cl}}.\rho + =\alpha \cup \gamma .\sigma =\beta \cup \delta :\\ +&\alpha P_{\text{cl}}\beta .\lor.\alpha =\beta .\gamma Q_{\text{cl}}\delta :\\ +[*166·112] &\equiv :(\exists \alpha ,\beta ,\gamma ,\delta ).(\gamma \downarrow \alpha )(P_{\text{cl}}\times Q_{\text{cl}})(\delta + \downarrow \beta ).\rho =\alpha \cup \gamma .\sigma =\beta \cup \delta \colon\colon\\ +&\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*170·67">*170·67</a>.</b> \(\vdash :\dot{\exists} !P.\dot{\exists} !Q.CʻP\cap CʻQ=\Lambda .\supset .(P\unicode{x2909}Q)_{\text{cl}}=s^{;}C^{;}(P_{\text{cl}}\times + Q_{\text{cl}})\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*170·66.*13·22.&\supset \vdash \colon\colon \text{Hp}.\supset \colon\ldotp \rho (P\unicode{x2909}Q)\sigma .\equiv :\\ +&(\exists \alpha ,\beta ,\gamma ,\delta ,R,S).R=\gamma \downarrow \alpha .S=\delta \downarrow \beta .\rho =\alpha \cup \gamma .\sigma =\beta \cup \delta .\\ +&R(P_{\text{cl}}\times Q_{\text{cl}})S:\\ +[*55·15.*53·11]&\equiv :(\exists \alpha ,\beta ,\gamma ,\delta ,R,S).R=\gamma \downarrow \alpha .S=\delta \downarrow \beta .\\ +&\rho =sʻCʻR.\sigma =sʻCʻS.R(P_{\text{cl}}\times Q_{\text{cl}})S:\\ +[*166·111] &\equiv :(\exists R,S).\rho =sʻCʻR.\sigma =sʻCʻS.R(P_{\text{cl}}\times Q_{\text{cl}})S:\\ +[*150·4] &\equiv :\rho {s^{;}C^{;}(P_{\text{cl}}\times Q_{\text{cl}})}\sigma \colon\colon \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_422">[Pg 422]</span></p> + +<p class="nind"><b><a id="*170·68">*170·68</a>.</b> \[\begin{align}\vdash :\dot{\exists} !P.\dot{\exists} !Q.CʻP&\cap CʻQ=\Lambda .\supset .\\ +&(s\mid C)\upharpoonright Cʻ(P_{\text{cl}}\times Q_{\text{cl}})\in (P\unicode{x2909}Q)_{\text{cl}}\,\overline{\text{smor}}\,(P_{\text{cl}}\times + Q_{\text{cl}})\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*55·15.*53·11.\supset \\ +&\vdash \colon\ldotp R=\gamma \downarrow \alpha .S=\delta \downarrow \beta .sʻCʻR=sʻCʻS.\supset .\alpha \cup \gamma =\beta \cup \delta &\qquad \text{(1)}\\ +\vdash .(1).*24·48.\supset \\ +\vdash \colon\colon \text{Hp}.&\supset \colon\ldotp \alpha ,\beta \in \text{Cl}ʻCʻP.\gamma ,\delta \in \text{Cl}ʻCʻQ.R=\gamma \downarrow \alpha .S=\delta \downarrow \beta .sʻCʻR=sʻCʻS.\supset .\\ +&\alpha =\beta .\gamma =\delta .\\ +[*55·202] &\supset .R=S &\qquad \text{(2)}\\ +&\vdash .(2).*166·12.*170·37.\supset \\ +&\vdash \colon\ldotp \text{Hp}.\supset :R,S\in Cʻ(P_{\text{cl}}\times Q_{\text{cl}}).sʻCʻR=sʻCʻS.\supset .R=S &\qquad \text{(3)}\\ +\vdash .(3).*151·24.*170·67.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*170·69">*170·69</a>.</b> \[\begin{align}&\vdash :\dot{\exists} !P.\dot{\exists} !Q.CʻP\cap CʻQ=\Lambda .\supset .(P\unicode{x2909}Q)_{\text{cl}}\,\text{smor}\,(P_{\text{cl}}\times + Q_{\text{cl}})\\ +&[*170·68]\end{align}\]</p> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_423">[Pg 423]</span></p> +<h2 class="nobreak" id="*171"> +*171. THE PRINCIPLE OF FIRST DIFFERENCES (<i>continued</i>).</h2> +</div> + + +<p class="nind"><i>Summary of</i> *171.</p> + +<p>In this number, we shall consider a more restricted form of the +principle of first differences, which is applicable when there is a +definite first member of one class not belonging to the other class. +In this case, if \(z\) is the first differing member, the part of +\(\alpha\) which precedes \(z\) is to be the same as the part of +\(\beta\) which precedes \(z\). If \(z\) belongs to \(\alpha\) and not +to \(\beta\), we put \(\alpha\) before \(\beta\); in the converse case, +we put \(\beta\) before \(\alpha\). In case \(zPz\), \(z\) itself is +not to be counted among its own predecessors; thus the predecessors +of \(z\) are to be \(\overrightarrow{P}ʻz-\iota ʻz\). Hence the +relation in question will hold between two sub-classes (\(\alpha\) and +\(\beta\)) of \(CʻP\) when there is a \(z\) such that +\[ +z\in \alpha -\beta .\overrightarrow{P}ʻz-\iota ʻz\cap \alpha =\overrightarrow{P}ʻz-\iota ʻz\cap \beta , +\] +or, what comes to the same thing (owing to \(z{\sim}\in \beta\)), +\[ +z\in \alpha -\beta .\overrightarrow{P}ʻz\cap \alpha -\iota ʻz=\overrightarrow{P}ʻz\cap \beta . +\] +This relation between \(\alpha\) and \(\beta\) we denote by +"\(P_{\text{df}}\)," where "\(\text{df}\)" stands for "difference."</p> + +<p>Thus our definition is +\[ +P_{\text{df}}=\hat{\alpha} + \hat{\beta} \{\alpha ,\beta \in \text{Cl}ʻCʻP:(\exists z).z\in \alpha -\beta .\overrightarrow{P}ʻz\cap \alpha -\iota ʻz=\overrightarrow{P}ʻz\cap \beta\} \quad\text{Df}. +\]</p> + +<p>On the analogy of \(P_{\text{lc}}\), we put also +\[ +P_\text{fd}=\text{Cnv}ʻ(\breve{P} )_{\text{df}}. +\]</p> + +<p>When \(P\) is well-ordered, \(P_{\text{df}}\) and \(P_\text{fd}\) coincide +respectively with \(P_{\text{cl}}\) and \(P_{\text{lc}}\). Their +properties are closely analogous to those of \(P_{\text{cl}}\) and +\(P_{\text{lc}}\). Thus <i>e.g.</i> the following propositions remain +true when \(P_{\text{df}}\) is substituted for \(P_{\text{cl}}\): +</p><p class="nindc"> +<a href="#*170·17">*170·17</a><a href="#*170·35">·35</a><a href="#*170·36">·36</a><a href="#*170·37">·37</a><a href="#*170·38">·38</a><a href="#*170·44">·44</a><a href="#*170·5">·5</a><a href="#*170·51">·51</a><a href="#*170·52">·52</a><a href="#*170·64">·64</a><a href="#*170·67">·67</a><a href="#*170·68">·68</a><a href="#*170·69">·69</a>. +</p> + +<p>The only new propositions to be noted in this number are</p> + +<p class="nind"><b>*171·2.</b> \[\begin{align}\vdash :P\,\unicode{x2abd}\, &J.\supset .\\ +&P_{\text{df}}=\hat{\alpha} + \hat{\beta} \{\alpha ,\beta \in \text{Cl}ʻCʻP:(\exists z).z\in \alpha -\beta .\overrightarrow{P}ʻz\cap \alpha =\overrightarrow{P}ʻz\cap \beta\}\end{align}\]</p> + +<p class="nind"><b>*171·21.</b> \(\vdash .P_{\text{df}}\,\unicode{x2abd}\, P_{\text{cl}}\)</p> + +<p><span class="pagenum" id="Page_424">[Pg 424]</span></p> + +<p class="nind"> +and the following formulae suggesting an inductive identification +of \(P_{\text{cl}}\) and \(P_{\text{df}}\) in cases to which such +induction is applicable:</p> + +<p class="nind"><b>*171·7.</b> \(\vdash :P_{\text{df}}=P_{\text{cl}}.x{\sim}\in CʻP.\supset .(x\unicode{x21f7}P)_{\text{df}}=(x\unicode{x21f7}P)_{\text{cl}}\)</p> + +<p class="nind"><b>*171·71.</b> \(\vdash :CʻP\cap CʻQ=\Lambda .P_{\text{df}}=P_{\text{cl}}.Q_{\text{df}}=Q_{\text{cl}}.\supset + .(P\unicode{x2909}Q)_{\text{df}}=(P\unicode{x2909}Q)_{\text{cl}}\)</p> + +<p>These propositions are however superseded (at a later stage) by the +proof that \(P_{\text{cl}}\) and \(P_{\text{df}}\) coincide if \(P\) is +well-ordered (*251·37).</p> + +<p>The chief property of \(P_{\text{df}}\) is that its relation-number is +\(2_{r}\) to the power \(\text{Nr}ʻP\). This will be proved in <a href="#*177">*177</a> and +<a href="#*186·4">*186·4</a></p> + +<hr class="tb"> + +<p class="nind"><b>*171·01.</b> \(P_{\text{df}}=\hat{\alpha} + \hat{\beta} \{\alpha ,\beta \in \text{Cl}ʻCʻP:(\exists z).z\in \alpha -\beta .\overrightarrow{P}ʻz\cap \alpha -\iota ʻz=\overrightarrow{P}ʻz\cap \beta\} \quad\text{Df}\)</p> + +<p class="nind"><b>*171·02.</b> \(P_\text{fd}=\text{Cnv}ʻ(\breve{P} )_{\text{df}} \quad\text{Df}\)</p> + +<p class="nind"><b>*171·1.</b> \[\begin{align}&\vdash \colon\ldotp \alpha P_{\text{df}}\beta + .\equiv :\alpha ,\beta \in \text{Cl}ʻCʻP:(\exists z).z\in \alpha -\beta .\overrightarrow{P}ʻz\cap \alpha -\iota ʻz=\overrightarrow{P}ʻz\cap \beta \\ +&[(*171·01)]\end{align}\]</p> + +<p class="nind"><b>*171·101.</b> \(\vdash .P_\text{fd}=\text{Cnv}ʻ(\breve{P} )_{\text{df}} \quad[(*171·02)]\)</p> + +<p class="nind"><b>*171·102.</b> + \[\begin{align}&\vdash \colon\ldotp \alpha P_\text{fd}\beta .\equiv :\alpha ,\beta \in \text{Cl}ʻCʻP:(\exists z).z\in \beta -\alpha .\overleftarrow{P}ʻz\cap \beta -\iota ʻz=\overleftarrow{P}ʻz\cap \alpha\\ +&[*171·1·101]\end{align}\]</p> + +<p class="nind"><b>*171*11.</b> \[\begin{align}\vdash \colon\colon\ldotp \alpha P_{\text{df}}&\beta .\equiv \colon\colon \alpha ,\beta \in \text{Cl}ʻCʻP\colon\colon \\ +&(\exists z)\colon\ldotp z\in \alpha -\beta \colon\ldotp yPz.y\neq z.\supset _{y}:y\in \alpha .\equiv .y\in \beta \quad [*171·1]\end{align}\]</p> + +<p class="nind"><b>*171·12.</b> \[\begin{align}&\vdash \colon\ldotp \alpha P_{\text{df}}\beta .\equiv :\alpha ,\beta \in \text{Cl}ʻCʻP:\\ +&(\exists z).z\in \alpha -\beta .\overrightarrow{P}ʻz\cap \alpha -\iota ʻz=\overrightarrow{P}ʻz\cap \beta -\iota ʻz \quad[*171·1.*51·222]\end{align}\]</p> + +<p class="nind"><b>*171·13.</b> \(\vdash .CʻP_{\text{df}}\subset \text{Cl}ʻCʻP \quad[*171·1]\)</p> + +<p class="nind"><b>*171·14.</b> \(\vdash :\alpha \subset CʻP.z\in \alpha .\supset .\alpha P_{\text{df}}(\alpha -\iota ʻz)\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*51·21. \supset \vdash :\text{Hp}.&\supset .z\in \alpha -(\alpha -\iota ʻz).\\ +[*13·15] &\supset .z\in \alpha -(\alpha -\iota ʻz).\overrightarrow{P}ʻz\cap \alpha -\iota ʻz=\overrightarrow{P}ʻz\cap (\alpha -\iota ʻz).\\ +[*171·12] &\supset .\alpha P_{\text{df}}(\alpha -\iota ʻz):\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*171·15.</b> \(\vdash :\beta \subset CʻP.z\in CʻP-\beta .(\beta \cup \iota ʻz)P_{\text{df}}\beta\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*51·16. &\supset \vdash :\text{Hp}.\supset .z\in (\beta \cup \iota ʻz)-\beta &\qquad \text{(1)}\\ +\vdash .*51·211·22. \supset \vdash :\text{Hp}.&\supset .(\beta \cup \iota ʻz)-\iota ʻz=\beta .\\ +[*22·481] &\supset .\overrightarrow{P}ʻz\cap (\beta \cup \iota ʻz)-\iota ʻz=\overrightarrow{P}ʻz\cap \beta &\qquad \text{(2)}\\ +\vdash .(1).(2).*171·1.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*171·16.</b> \(\vdash .\text{D}ʻP_{\text{df}}=\text{Cl ex}ʻCʻP.\text{ᗡ}ʻP_{\text{df}}=\text{Cl}ʻCʻP-\iota ʻCʻP\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*171·14. &\supset \vdash :\alpha \in \text{Cl ex}ʻCʻP.\supset .\alpha \in \text{D}ʻP_{\text{df}} &\qquad \text{(1)}\\ +\vdash .*171·1. &\supset \vdash :\alpha \in \text{D}ʻP_{\text{df}}.\supset .\alpha \in \text{Cl ex}ʻCʻP &\qquad \text{(2)}\\ +\vdash .(1).(2). &\supset \vdash .\text{D}ʻP_{\text{df}}=\text{Cl ex}ʻCʻP &\qquad \text{(3)}\\ +\vdash .*171·15. &\supset \vdash :\beta \in \text{Cl}ʻCʻP.\exists !CʻP-\beta .\supset .\beta \in \text{ᗡ}ʻP_{\text{df}}:\\ +[*24·6] &\supset \vdash :\beta \in \text{Cl}ʻCʻP-\iota ʻCʻP.\supset .\beta \in \text{ᗡ}ʻP_{\text{df}} &\qquad \text{(4)}\\ +\vdash .*171·1. \supset \vdash :\beta \in \text{ᗡ}ʻP_{\text{df}}.&\supset .\beta \in \text{Cl}ʻCʻP.\exists !CʻP-\beta .\\ +[*24·6] &\supset .\beta \in \text{Cl}ʻCʻP-\iota ʻCʻP &\qquad \text{(5)}\\ +\vdash .(4).(5).&\supset \vdash .\text{ᗡ}ʻP_{\text{df}}=\text{Cl}ʻCʻP-\iota ʻCʻP &\qquad \text{(6)}\\ +\vdash .(3).(6).&\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_425">[Pg 425]</span></p> + +<p class="nind"><b>*171·17.</b> \(\vdash :\dot{\exists} !P.\supset .CʻP_{\text{df}}=\text{Cl}ʻCʻP\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*171·16. &\supset \vdash :\alpha \in \text{Cl}ʻCʻP.\alpha \neq \Lambda .\supset .\alpha \in \text{D}ʻP_{\text{df}} &\qquad \text{(1)}\\ +\vdash .*171·16. &\supset \vdash :\alpha \in \text{Cl}ʻCʻP.\alpha \neq CʻP.\supset .\alpha \in \text{ᗡ}ʻP_{\text{df}} &\qquad \text{(1)}\\ +\vdash .(1).(2). &\supset \vdash :\alpha \in \text{Cl}ʻCʻP.{\sim}(\alpha =\Lambda .\alpha =CʻP).\supset .\alpha \in CʻP_{\text{df}}:\\ +[*13·171] &\supset \vdash :\alpha \in \text{Cl}ʻCʻP.CʻP\neq \Lambda .\supset .\alpha \in CʻP_{\text{df}} &\qquad \text{(3)}\\ +\vdash .(3).*33·24.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*171·18.</b> \(\vdash :\dot{\exists} !P.\supset .BʻP_{\text{df}}=CʻP.Bʻ\text{Cnv}ʻP_{\text{df}}=\Lambda\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*171·16. \supset \vdash .\overrightarrow{B}ʻP_{\text{df}}&=\text{Cl ex}ʻCʻP-(\text{Cl}ʻCʻP-\iota ʻCʻP)\\ +[*24·3] &=\text{Cl ex}ʻCʻP\cap \iota ʻCʻP &\qquad \text{(1)}\\ +\vdash .(1).*60·35. &\supset \vdash :\dot{\exists} !P.\supset .\overrightarrow{B}ʻP_{\text{df}}=\iota ʻCʻP &\qquad \text{(2)}\\ +\vdash .*171·16. \supset \vdash .\overrightarrow{B}ʻ\text{Cnv}ʻP_{\text{df}}&=\text{Cl}ʻCʻP-\iota ʻCʻP-\text{Cl ex}ʻCʻP\\ +[*60·24] &=\iota ʻ\Lambda -\iota ʻCʻP &\qquad \text{(3)}\\ +\vdash .(3).*33·24.&\supset \vdash :\dot{\exists} !P.\supset .\overrightarrow{B}ʻ\text{Cnv}ʻP_{\text{df}}=\iota ʻ\Lambda &\qquad \text{(4)}\\ +\vdash .(2).(4).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*171·19.</b> \(\vdash :P=\dot{\Lambda} .\supset .P_{\text{df}}=\dot{\Lambda}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*60·33.*171·16. \supset \vdash :\text{Hp}.&\supset .\text{D}ʻP_{\text{df}}=\Lambda .\\ +[*33·241] &\supset .P_{\text{df}}=\dot{\Lambda} :\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*171·2.</b> \(\vdash :P\,\unicode{x2abd}\, J.\supset .P_{\text{df}} + =\hat{\alpha} \hat{\beta} \{\alpha ,\beta \in \text{Cl}ʻCʻP:(\exists z).z\in \alpha -\beta .\overrightarrow{P}ʻz\cap \alpha =\overrightarrow{P}ʻz\cap \beta\}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*50·11.*32·19. \supset \vdash :\text{Hp}.&\supset .\overrightarrow{P}ʻz\subset -\iota ʻz.\\ +[*22·621] &\supset .\overrightarrow{P}ʻz\cap \alpha -\iota ʻz=\overrightarrow{P}ʻz\cap \alpha &\qquad \text{(1)}\\ +\vdash .(1).*171·1.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_426">[Pg 426]</span></p> + +<p class="nind"><b>*171·21.</b> \(\vdash .P_{\text{df}}\,\unicode{x2abd}\, P_{\text{cl}}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*171·1.*22·43.\supset \\ +\vdash \colon\ldotp \alpha P_{\text{df}}\beta .&\supset :\alpha ,\beta \in \text{Cl}ʻCʻP:(\exists z).z\in \alpha -\beta .\overrightarrow{P}ʻz\cap \beta \subset \alpha :\\ +[*170·11] &\supset :\alpha P_{\text{cl}}\beta \colon\ldotp \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*171·22.</b> \(\vdash .P_{\text{df}}\,\unicode{x2abd}\, J \quad[*170·17.*171·21]\)</p> + +<p class="nind"><b>*171·4.</b> \(\vdash :S\in 1\rightarrow 1.CʻQ=\text{ᗡ}ʻS.\supset .(S^{;}Q)_{\text{df}}=S_{\in }^{;}Q_{\text{df}} \quad[\text{Proof as in *170·4}]\)</p> + +<p class="nind"><b>*171·41.</b> \(\vdash :(S\upharpoonright CʻQ)_{\in }^{;}Q_{\text{df}}=S_{\in }^{;}Q_{\text{df}} \quad[\text{Proof as in *170·41}]\)</p> + +<p class="nind"><b>*171·42.</b> \(\vdash :S\upharpoonright CʻQ\in 1\rightarrow 1.CʻQ\subset \text{ᗡ}ʻS.\supset .(S^{;}Q)_{\text{df}}=S_{\in }^{;}Q_{\text{df}} + \quad[*171·4·41]\)</p> + +<p class="nind"><b>*171·43.</b> \[\begin{align}&\vdash :S\upharpoonright CʻQ\in P\,\overline{\text{smor}}\,Q.\supset .S_{\in }\upharpoonright CʻQ_{\text{df}}\in P_{\text{df}}\,\overline{\text{smor}}\, + Q_{\text{df}}\\ +&[\text{Proof as in *170·43}]\end{align}\]</p> + +<p class="nind"><b>*171·44.</b> \(\vdash :P\,\text{smor}\,Q.\supset .P_{\text{df}}\text{smor}Q_{\text{df}} \quad[*171·43]\)</p> + +<p class="nind"><b>*171·5.</b> \(\vdash .(x\downarrow x)_{\text{df}}=(\iota ʻx)\downarrow \Lambda =(x\downarrow x)_{\text{cl}}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*171·1.*55·15.\supset \\ +\vdash \colon\ldotp \alpha (x\downarrow x)_{\text{df}}\beta + .&\equiv :\alpha ,\beta \in \text{Cl}ʻ\iota ʻx:(\exists z).z\in \alpha -\beta.\overrightarrow{x\downarrow x} ʻz\cap \alpha -\iota ʻz=\overrightarrow{x\downarrow x} ʻz\cap \beta :\\ +[*171·16] &\equiv :\alpha \in \text{Cl ex}ʻ\iota ʻx.\beta \in \text{Cl}ʻ\iota ʻx-\iota ʻ\iota ʻx:\\ +&(\exists z).z\in \alpha -\beta .\overrightarrow{x\downarrow x} ʻz\cap \alpha -\iota ʻz=\overrightarrow{x\downarrow x} ʻz\cap \beta :\\ +[*60·362·37] &\equiv :\alpha =\iota ʻx.\beta =\Lambda :(\exists z).z\in \iota ʻx.\overrightarrow{x\downarrow x} ʻz\cap \alpha -\iota ʻz=\overrightarrow{x\downarrow x} ʻz\cap \beta :\\ +[*13·195] &\equiv :\alpha =\iota ʻx.\beta =\Lambda .\iota ʻx\cap \alpha -\iota ʻx=\iota ʻx\cap \beta :\\ +[*24·21·23] &\equiv :\alpha =\iota ʻx.\beta =\Lambda .\Lambda =\Lambda :\\ +[*13·15.*55·13] &\equiv :\alpha \{(\iota ʻx)\downarrow \Lambda\}\beta &\qquad \text{(1)}\\ +\vdash .(1).*170·5.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*171·51.</b> \(\vdash .(x\downarrow y)_{\text{df}}=(x\downarrow y)_{\text{cl}}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*171·1.&\supset \vdash \colon\ldotp \alpha (x\downarrow y)_{\text{df}}\beta .\equiv :\alpha ,\beta \in \text{Cl}ʻ(\iota ʻx\cup \iota ʻy):\\ +&(\exists z).z\in \alpha -\beta .\overrightarrow{x\downarrow y}ʻz\cap \alpha -\iota ʻz=\overrightarrow{x\downarrow y}ʻz\cap \beta :\\ +[*171·16] &\equiv :\alpha \in \text{Cl ex}ʻ(\iota ʻx\cup \iota ʻy).\beta \in \text{Cl}ʻ(\iota ʻx\cup \iota ʻy)-\iota ʻ(\iota ʻx\cup \iota ʻy):\\ +&(\exists z).z\in \alpha -\beta .\overrightarrow{x\downarrow y}ʻz\cap \alpha -\iota ʻz=\overrightarrow{x\downarrow y}ʻz\cap \beta :\\ +[*60·39] &\equiv :\alpha =\iota ʻx\cup \iota ʻy.\lor.\alpha =\iota ʻx.\lor.\alpha =\iota ʻy:\beta =\iota ʻx.\lor.\beta =\iota ʻy.\lor.\beta =\Lambda :\\ +&(\exists z).z\in \alpha -\beta .\overrightarrow{x\downarrow y}ʻz\cap \alpha -\iota ʻz=\overrightarrow{x\downarrow y}ʻz\cap \beta : &\qquad \text{(1)}\\ +\vdash .*55·13.&\supset \vdash \colon\ldotp x\neq y.\supset :\overrightarrow{x\downarrow y}ʻy=\iota ʻx.\overrightarrow{x\downarrow y}ʻx=\Lambda : &\qquad \text{(2)}\\ +[*51·222] &\supset :\alpha =\iota ʻx\cup \iota ʻy.\beta =\iota ʻx.\supset .\\ +&y \in \alpha -\beta .\overrightarrow{x\downarrow y}ʻy\cap \alpha -\iota ʻy=\iota ʻx=\overrightarrow{x\downarrow y}ʻy\cap \beta .\\ +[(1)] &\supset .\alpha P_{\text{df}}\beta &\qquad \text{(3)}\\ +\vdash .(2).&\supset \vdash :x\neq y.\alpha =\iota ʻx\cup \iota ʻy.\beta =\iota ʻy.\supset .\\ +&x\in \alpha -\beta .\overrightarrow{x\downarrow y}ʻx\cap \alpha -\iota ʻx=\Lambda =\overrightarrow{x\downarrow y}ʻx\cap \beta \\ +[(1)] &\supset .\alpha P_{\text{df}}\beta &\qquad \text{(4)}\\ +\vdash .(2).&\supset \vdash :x\neq y.\alpha =\iota ʻx\cup \iota ʻy.\beta =\Lambda .\supset .\\ +&x\in \alpha -\beta .\overrightarrow{x\downarrow y}ʻx\cap \alpha -\iota ʻx=\Lambda =\overrightarrow{x\downarrow y}ʻx\cap \beta .\\ +[(1)] &\supset .\alpha P_{\text{df}}\beta &\qquad \text{(5)}\\ +\vdash .(2).&\supset \vdash \colon\ldotp x\neq y.\alpha =\iota 'x:\beta =\Lambda .\lor.\beta =\iota 'y:\supset .\\ +&x\in \alpha -\beta .\overrightarrow{x\downarrow y}'x\cap \alpha -\iota 'x=\Lambda =\overrightarrow{x\downarrow y}'x\cap \beta .\\ +&\supset . \alpha P_{\text{df}}\beta &\qquad \text{(6)}\\ +\vdash .(2).*24·23.&\supset \vdash :x\neq y.\alpha =\iota 'y.\beta =\Lambda .\supset .\\ +&y\in \alpha -\beta .\overrightarrow{x\downarrow y}'x\cap \alpha -\iota 'y=\Lambda =\overrightarrow{x\downarrow y}'y\cap \beta &\qquad \text{(7)}\\ +\vdash .(3).(4).(5).(6).(7).*170·51.&\supset \vdash :x\neq y.\supset .(x\downarrow y)_{\text{cl}}\,\unicode{x2abd}\, (x\downarrow y)_{\text{df}}.\\ +[*171·21] &\supset .(x\downarrow y)_{\text{df}}=(x\downarrow y)_{\text{cl}} &\qquad \text{(8)}\\ +\vdash .(8).*171·5.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_427">[Pg 427]</span></p> + +<p class="nind"><b>*171·52.</b> \[\begin{align}&\vdash :x\neq y.\supset .(x\downarrow y)_{\text{df}}=(\iota 'x\cup \iota 'y)\downarrow (\iota 'x)\unicode{x2909}(\iota 'y)\downarrow \Lambda \\ +&[*171·51.*170·52]\end{align}\]</p> + +<p class="nind"><b>*171·64.</b> \(\vdash :x{\sim}\in C'P.\supset .(x\unicode{x21f7}P)_{\text{df}}=(\iota 'x\cup )^{;}P_{\text{df}}\unicode{x2909}P_{\text{df}}\)</p> + +<p>The proof proceeds by the same stages as the proof of <a href="#*170·64">*170·64</a>.</p> + +<p class="nind"><b>*171·67.</b> \[\begin{align}&\vdash :\dot{\exists} !P.\dot{\exists} !Q.C'P\cap C'Q=\Lambda .\supset .(P\unicode{x2909}Q)_{\text{df}}=s^{;}C^{;}(P_{\text{df}}\times + Q_{\text{df}})\\ +&[\text{Proof as in *170·67}]\end{align}\]</p> + +<p class="nind"><b>*171·68.</b> \[\begin{align}\vdash : \dot{\exists} ! P . \dot{\exists} ! Q . C'P &\cap C'Q = \Lambda . \supset .\\ +&s\mid C\upharpoonright (P_{\text{df}}\times Q_{\text{df}})\in (P\unicode{x2909}Q)_{\text{df}}\,\overline{\text{smor}}\,(P_{\text{df}}\times Q_{\text{df}})\\ +[\text{Proof as in *170·68}]\end{align}\]</p> + +<p class="nind"><b><a id="*171·69">*171·69</a>.</b> \[\begin{align}&\vdash :\dot{\exists} !P.\dot{\exists} !Q.C'P\cap C'Q=\Lambda .\supset .(P\unicode{x2909}Q)_{\text{df}}\text{smor}(P_{\text{df}}\times + Q_{\text{df}})\\ +&[*171·68]\end{align}\]</p> + +<p class="nind"><b>*171·7.</b> \[\begin{align}&\vdash :P_{\text{df}}=P_{\text{cl}}.x{\sim}\in C'P.\supset .(x\unicode{x21f7}P)_{\text{df}}=(x\unicode{x21f7}P)_{\text{cl}}\\ +&[*171·64.*170·64]\end{align}\]</p> + +<p class="nind"><b>*171·71.</b> \[\begin{align}&\vdash :C'P\cap C'Q=\Lambda .P_{\text{df}}=P_{\text{cl}}.Q_{\text{df}}=Q_{\text{cl}}.\supset + .(P\unicode{x2909}Q)_{\text{df}}=(P\unicode{x2909}Q)_{\text{cl}}\\ +&[*170·67.*171·67.*160·21·22]\end{align}\]</p> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_428">[Pg 428]</span></p> +<h2 class="nobreak" id="*172"> +*172. THE PRODUCT OF THE RELATIONS OF A FIELD.</h2> +</div> + + +<p class="nind"><i>Summary of</i> *172.</p> + +<p>In this number we have to consider the form of product which is +applicable to any relation of relations, whether mutually exclusive +or not. If our relation were a \(\text{Rel}^{2}\text{excl}\), we +could take \(CʻʻCʻP\), and order selected classes from \(CʻʻCʻP\) +by first differences. This would give us a relation whose field +would be \(\text{Prod}ʻCʻʻCʻP\). But if any two fields overlap, +this method fails. We might substitute \({\in}_{\Delta}ʻCʻʻCʻP\) +for \(\text{Prod}ʻCʻʻCʻP\), and order the members of +\({\in}_{\Delta}ʻCʻʻCʻP\) by first differences; but this method will +not give what we want if two or more members of \(CʻP\) have the same +field. In order to avoid any confusion due to repetition, we must, +if \(Q\in CʻP\) and \(x\in CʻQ\), consider \(x\) in connection with +\(Q\), not merely with \(CʻQ\). That is, the relations in the field +of the product of \(P\) must be such as concern themselves with the +ordered couple \(x\downarrow Q\), not merely with \(x\). The simplest +way of effecting this is to consider \(F_{\Delta }ʻCʻP\). A member +of \(F_{\Delta }ʻCʻP\), say \(M\), is a relation which picks out a +representative of \(Q\) from the field of every \(Q\) which is a +member of \(CʻP\); that is, whenever \(Q\in CʻP\), \(MʻQ\in CʻQ\). +Since we have \(MʻQ\), not \(MʻCʻQ\), two relations may have the same +field and yet we can distinguish the occurrence of a given term as the +representative of the one from its occurrence as the representative of +the other. Thus no degree of overlapping will cause confusion.</p> + +<p><span class="pagenum" id="Page_429">[Pg 429]</span></p> + +<p>The relations which compose \(F_{\Delta }ʻCʻP\) are to be ordered by +first differences, but in order to distinguish different occurrences of +a given term, we must give a slightly different form to the principle +of first differences from that employed in <a href="#*170">*170</a> or <a href="#*171">*171</a>. The new form +of the principle is as follows: Consider two relations \(M\) and \(N\) +which are members of \(F_{\Delta }ʻCʻP\). Let \(Q\) be a member of +\(CʻP\) in which \(M\) chooses a representative which precedes that +of \(N\), <i>i.e.</i> in which \((MʻQ)Q(NʻQ)\); and let all earlier +relations than \(Q\), <i>i.e.</i> all relations \(R\) such that \(RPQ\) +and \(R\neq Q\), have \(MʻR=NʻR\). Then we say that \(M\) precedes +\(N\). This principle may also be stated as follows: We may divide +the members of \(CʻP\) into four classes, not in general mutually +exclusive, namely:</p> + +<p>(1) those in which \((MʻQ)Q(NʻQ)\), <i>i.e.</i> in which the +\(M\)-representative precedes the \(N\)-representative;</p> + +<p>(2) those in which \((NʻQ)Q(MʻQ)\),</p> + +<p>(3) those in which \(MʻQ=NʻQ\),</p> + +<p>(4) those in which no one of the above three relations of \(MʻQ\) and +\(NʻQ\) occurs.</p> + +<p>Then we shall say that \(M\) precedes \(N\) if there is a member of +class (1) whose predecessors all belong to class (3).</p> + +<p>In case all the members of \(CʻP\) are serial, the fourth of the above +classes is null, and the other three are mutually exclusive. If, +further,\(P\) is well-ordered, any two different members of \(F_{\Delta}ʻCʻP\) +must be such that one precedes the other in the above-defined +order. Thus in this case the product of a series of series is a series +(cf. *251).</p> + +<p>The definition of the product \(\Pi ʻP\) is +\[ +\begin{aligned} +\Pi ʻP=\hat{M} \hat{N} &\{M,N\in F_{\Delta }ʻCʻP\colon\ldotp \\ +&(\exists Q):(MʻQ)Q(NʻQ):RPQ.R\neq Q.\supset _{R}.MʻR=NʻR\} \quad\text{Df}. +\end{aligned} +\] +Owing to the complication of this definition, the proofs of +propositions of the present number are apt to be long.</p> + +<p>Various other definitions might be adopted for \(\Pi ʻP\), but we have +found the above definition on the whole the best.</p> + +<p>We might, for example, drop the condition \(R\neq Q\) in the +definition; we could then write our definition in the simpler form: +\[ +\Pi ʻP=\hat{M} \hat{N} \{M,N\in F_{\Delta }ʻCʻP:(\exists Q).(MʻQ)Q(NʻQ).M\upharpoonright \overrightarrow{P}ʻQ=N\upharpoonright \overrightarrow{P}ʻQ\}, +\] +which, with our definition, is only available when \(P\,\unicode{x2abd}\, J\). +But if we adopt this simplification, we no longer have +\[ +\Pi ʻ(P\downarrow P)=P\downarrow_{.,} P \quad(*172·2), +\] +which is a very useful proposition, required in the proofs of <a href="#*183·13">*183·13</a>, +<a href="#*185·21">*185·21</a> and other important propositions.</p> + +<p>On the other hand, we might frame our definition on the analogy +of \(P_{\text{cl}}\) rather than, as above, on the analogy of +\(P_{\text{df}}\). The definition would then be: +\[ +\begin{aligned} +\Pi ʻP=\hat{M} \hat{N} &\{M,N\in F_{\Delta }ʻCʻP\colon\ldotp \\ +&(\exists Q):(MʻQ)Q(NʻQ):RPQ.\supset _{R}.(MʻR)(R\unicode{x228d} I)(NʻR)\}. +\end{aligned} +\]</p> + +<p>This definition does not assume that there is a first relation \(Q\) +for which the \(M\)-representative precedes the \(N\)-representative. +Thus it might be thought that it would give better results in cases +where \(P\) is not well-ordered. But in fact this is not the case. If +\(P\) is not well-ordered, it may happen that every \(Q\) for which +\((MʻQ)Q(NʻQ)\) is preceded by one for which \((NʻQ)Q(MʻQ)\), and +<i>vice versa</i>; in this case, we shall have neither \(M(\Pi ʻP)N\) +nor \(N(\Pi ʻP)M\).<span class="pagenum" id="Page_430">[Pg 430]</span> Thus our suggested new definition does not secure +that \(\Pi ʻP\) shall be a series whenever \(P\) and all the members of +\(CʻP\) are series, and therefore has no substantial advantage over the +simpler definition which we have adopted, and has the disadvantage of +greater complication.</p> + +<p>In the present number, we first prove that \(\Pi ʻ\dot{\Lambda}=\dot{\Lambda}\) +(<a href="#*172·13">*172·13</a>) and that \(\dot{\Lambda} \in CʻP.\supset.\Pi ʻP=\dot{\Lambda}\) +(<a href="#*172·14">*172·14</a>), so that a product is null if any one of its factors +is null. We then proceed to propositions about \(Cʻ\Pi ʻP\), +\(\overrightarrow{B}ʻ\Pi ʻP\), etc. We have</p> + +<p class="nind"><b>*172·162.</b> \(\vdash :\dot{\exists} !P.\supset .\overrightarrow{B}ʻ\Pi ʻP=B_{\Delta }ʻCʻP.\overrightarrow{B}ʻ\text{Cnv}ʻ\Pi ʻP=B_{\Delta }ʻ\text{Cnv}ʻʻCʻP\)</p> + +<p class="nind"><b>*172·17.</b> \(\vdash :\dot{\exists} !P.\supset .Cʻ\Pi ʻP=F_{\Delta }ʻCʻP\)</p> + +<p>Hence we derive propositions as to the existence of \(\Pi ʻP\). We have</p> + +<p class="nind"><b>*172·181.</b> \(\vdash \colon\ldotp \text{Mult ax}.\supset :\Lambda {\sim}\in CʻP.\dot{\exists} !P.\equiv .\dot{\exists} !\Pi ʻP\)</p> + +<p>Thus assuming the multiplicative axiom, a product which has factors +none of which are null is not null.</p> + +<p>We then consider \(\Pi ʻ(P\downarrow P)\), and \(\Pi ʻ(P\downarrow Q)\) +where \(P\neq Q\). We have</p> + +<p class="nind"><b>*172·2.</b> \(\vdash .\Pi ʻ(P\downarrow P)=P\downarrow_{.,} P\)</p> + +<p class="nind">which is a useful proposition, and</p> + +<p class="nind"><b>*172·23.</b> \(\vdash :P\neq Q.\supset .\Pi ʻ(P\downarrow Q)\text{smor}P\times Q\)</p> + +<p class="nind"> +which connects the two definitions of multiplication, showing that +they lead to equivalent results for any finite number of factors, +<i>i.e.</i> whenever the definition of <a href="#*166">*166</a> is applicable.</p> + +<p>We next consider \(\Pi ʻ(P\unicode{x21f8} Z)\) and +\(\Piʻ(P\unicode{x2909}Q)\), proving</p> + +<p class="nind"><b>*172·32.</b> \(\vdash :Z{\sim}\in CʻP.\supset .\Pi ʻ(P\unicode{x21f8} Z)\text{smor}\Pi ʻP\times Z\)</p> + +<p class="nind">with a similar proposition for \(Z\unicode{x21f7}P\) (<a href="#*172·321">*172·321</a>), and</p> + +<p class="nind"><b>*172·35.</b> \(\vdash :\dot{\exists} !P.\dot{\exists} !Q.CʻP\cap CʻQ=\Lambda .\supset .\Pi ʻ(P\unicode{x2909}Q)\text{smor}\Pi ʻP\times \Pi ʻQ\)</p> + +<p class="nind">which is a form of the associative law using both kinds of +multiplication. The kind which uses only \(\Pi\) will be proved in <a href="#*174">*174</a>.</p> + +<p>We have next the proof (with its immediate consequences) that if \(P\) +and \(Q\) have double likeness, \(\Pi ʻP\text{smor}\Pi ʻQ\). We prove</p> + +<p class="nind"><b>*172·43.</b> \[\begin{align}\vdash :T\upharpoonright Cʻ\Sigma ʻQ\in P\,\overline{\text{smor}}\, &\overline{\text{smor}}\,Q.\supset .\\ +&(T\mid \text{Cnv}ʻT\dagger )\upharpoonright Cʻ\Pi ʻQ\in (\Pi ʻP)\,\overline{\text{smor}}\,(\Pi ʻQ)\end{align}\]</p> + +<p>This proposition should be compared with <a href="#*114·51">*114·51</a>, which is its cardinal +analogue. It will be seen that the correlator only differs by the +substitution of \(T\dagger\) for \(T_{\in }\). From <a href="#*172·43">*172·43</a> we obtain</p> + +<p class="nind"><b>*172·44.</b> \(\vdash :P\,\text{smor smor}\,Q.\supset .\Pi ʻP\,\text{smor}\,\Pi ʻQ\)</p> + +<p class="nind">whence</p> + +<p><span class="pagenum" id="Page_431">[Pg 431]</span></p> + +<p class="nind"><b>*172·45.</b> \[\begin{align}\vdash \colon\ldotp \text{Mult ax}.\supset :P,Q\in \text{Rel}^{2}\text{excl}.\exists !P\,\overline{\text{smor}}\,Q\cap \text{Rl}ʻ&\text{smor}.\supset .\\ +&\Pi ʻP\,\text{smor}\,\Pi ʻQ\end{align}\]</p> + +<p>Other propositions about \(\Pi ʻP\) will be given in <a href="#*174">*174</a>.</p> + +<hr class="tb"> + +<p class="nind"><b>*172·01.</b> \[\begin{align}\Pi ʻP=\hat{M} \hat{N} &\{M,N\in F_{\Delta }ʻCʻP\colon\ldotp \\ +&(\exists Q):(MʻQ)Q(NʻQ):RPQ.R\neq Q.\supset _{R}.MʻR=NʻR\} \quad\text{Df}\end{align}\]</p> + +<p class="nind"><b>*172·1.</b> \[\begin{align}\vdash \colon\colon M&(\Pi ʻP)N.\equiv \colon\ldotp M,N\in F_{\Delta }ʻCʻP\colon\ldotp \\ +&(\exists Q):(MʻQ)Q(NʻQ):RPQ.R\neq Q.\supset _{R}.MʻR=NʻR\\ +[(*172·01)]\end{align}\]</p> + +<p class="nind"><b>*172·11.</b> \[\begin{align}\vdash \colon\colon &M(\Pi ʻP)N.\equiv \colon\ldotp M,N\in F_{\Delta }ʻCʻP\colon\ldotp \\ +&(\exists Q):Q\in CʻP.(MʻQ)Q(NʻQ):RPQ.R\neq Q.\supset _{R}.MʻR=NʻR\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*14·21. &\supset \vdash :(MʻQ)Q(NʻQ).\supset .\text{E}!MʻQ.\\ +[*33·43] &\supset .Q\in \text{ᗡ}ʻM &\qquad \text{(1)}\\ +\vdash .(1).*80·14.&\supset \vdash \colon\ldotp M\in F_{\Delta }ʻCʻP.\supset :(MʻQ)Q(NʻQ).\supset .Q\in CʻP:\\ +[*4·73] &\supset :(MʻQ)Q(NʻQ).\equiv .Q\in CʻP.(MʻQ)Q(NʻQ) &\qquad \text{(2)}\\ +\vdash .(2).*172·1.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*172·12.</b> \(\vdash .Cʻ\Pi ʻP\subset F_{\Delta }ʻCʻP\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*172·1.\supset \vdash :M(\Pi ʻP)N.\supset .M,N\in F_{\Delta }ʻCʻP &\qquad \text{(1)}\\ +\vdash .(1).*33·352.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*172·13">*172·13</a>.</b> \(\vdash .\Pi ʻ\dot{\Lambda} =\dot{\Lambda}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*172·11. &\supset \vdash :M(\Pi ʻP)N.\supset .\exists !CʻP.\\ +[*33·24] &\supset .\dot{\exists} !P &\qquad \text{(1)}\\ +\vdash .(1).\text{Transp}.&\supset \vdash :P=\dot{\Lambda} .\supset .(M,N).{\sim}{M(\Pi ʻP)N}:\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*172·14">*172·14</a>.</b> \(\vdash :\dot{\Lambda} \in CʻP.\supset .\Pi ʻP=\dot{\Lambda}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*33·24·5.&\supset \vdash .\overrightarrow{F}ʻ\dot{\Lambda} =\Lambda .\\ +[*33·41] &\supset \vdash .\dot{\Lambda} {\sim}\in \text{ᗡ}ʻF.\\ +[*80·21] \supset \vdash :\dot{\Lambda} \in CʻP.&\supset .F_{\Delta }ʻCʻP=\Lambda .\\ +[*172·12.*33·24] &\supset .\Pi ʻP=\dot{\Lambda} :\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*172·141.</b> \(\vdash \colon\ldotp \dot{\exists} !\Pi ʻP.\supset :Q\in CʻP.\supset _{Q}.\dot{\exists} !Q \quad[*172·14.\text{Transp}]\)</p> + +<p><span class="pagenum" id="Page_432">[Pg 432]</span></p> + +<p>The following propositions are concerned with \(Cʻ\Pi ʻP\), +\(\overrightarrow{B}ʻ\Pi ʻP\), etc. <a href="#*172·15">*172·15</a><a href="#*172·151">·151</a><a href="#*172·16">·16</a><a href="#*172·161">·161</a> are lemmas for +<a href="#*172·162">*172·162</a><a href="#*172·17">·17</a>.</p> + +<p class="nind"><b><a id="*172·15">*172·15</a>.</b> \(\vdash :M\in F_{\Delta }ʻCʻP.Q\in CʻP.(MʻQ)Qy.\supset .M(\Pi ʻP)\{M\upharpoonright -\iota ʻQ\unicode{x228d} y\downarrow Q\}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*80·41.&\supset \vdash :\text{Hp}.\supset .M\upharpoonright -\iota ʻQ\unicode{x228d} y\downarrow Q\in F_{\Delta }ʻCʻP &\qquad \text{(1)}\\ +\vdash .*35·101.*55·13.\supset \\ +&\vdash \colon\ldotp z\{M\upharpoonright -\iota ʻQ\unicode{x228d} y\downarrow Q\}R.\equiv :R\neq Q.zMR.\lor.R=Q.z=y &\qquad \text{(2)}\\ +\vdash .(2).*80·3.\supset \\ +\vdash \colon\ldotp \text{Hp}.&\supset :R=Q.\supset .\{M\upharpoonright -\iota ʻQ\unicode{x228d} y\downarrow Q\}ʻR=y:\\ +&R\in CʻP.R\neq Q.\supset .\{M\upharpoonright -\iota ʻQ\unicode{x228d} y\downarrow Q\}ʻR=MʻR:\\ +[\text{Hp}] &\supset :(MʻQ)Q\{M\upharpoonright -\iota ʻQ\unicode{x228d} y\downarrow Q\}ʻQ:\\ +&R\in CʻP.R\neq Q.\supset .\{M\upharpoonright -\iota ʻQ\unicode{x228d} y\downarrow Q\}ʻR=MʻR:\\ +[*33·17]&\supset :(MʻQ)Q{M\upharpoonright -\iota ʻQ\unicode{x228d} y\downarrow Q}ʻQ:\\ +&RPQ.R\neq Q.\supset _{R}.MʻR=\{M\upharpoonright -\iota ʻQ\unicode{x228d} y\downarrow Q\}ʻR:\\ +[*172·1.(1)]&\supset :M(\Pi ʻP)\{M\upharpoonright -\iota ʻQ\unicode{x228d} y\downarrow Q\}\colon\ldotp \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*172·151">*172·151</a>.</b> \[\begin{align}\vdash :N\in F_{\Delta }ʻCʻP.&Q\in CʻP.yQ(NʻQ).\supset .\\ +&\{N\upharpoonright -\iota ʻQ\unicode{x228d} y\downarrow Q\}(\Pi ʻP)N \quad[\text{Proof as in *172·15}]\end{align}\]</p> + +<p class="nind"><b><a id="*172·16">*172·16</a>.</b> \(\vdash :M\in F_{\Delta }ʻCʻP.\dot{\exists} !M\dot{-} B.\supset .M\in \text{ᗡ}ʻ\Pi ʻP\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*72·93.*80·14.\supset \vdash \colon\colon \text{Hp}.&\supset \colon\ldotp M\,\unicode{x2abd}\, B.\equiv :Q\in CʻP.\supset _{Q}.(MʻQ)BQ\colon\ldotp \\ +[\text{Transp}] \supset \colon\ldotp \dot{\exists} !M\dot{-} B.&\equiv :(\exists Q).Q\in CʻP.{\sim}{(MʻQ)BQ}:\\ +[*93·1.*80·3] &\supset :(\exists Q).Q\in CʻP.MʻQ\in \text{ᗡ}ʻQ:\\ +[*33·131] &\supset :(\exists Q,y).Q\in CʻP.yQ(MʻQ):\\ +[*172·151] &\supset :M\in \text{ᗡ}ʻ\Pi ʻP\colon\colon \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*172·161">*172·161</a>.</b> \(\vdash :M\in F_{\Delta }ʻCʻP.\dot{\exists} !M\dot{-} B\mid \text{Cnv}.\supset .M\in \text{D}ʻ\Pi ʻP\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*72·93.*80·14.\supset \vdash \colon\colon &\text{Hp}.\supset \colon\ldotp \\ +M\,\unicode{x2abd}\, B\mid \text{Cnv}.&\equiv :Q\in CʻP.\supset _{Q}.(MʻQ)(B\mid \text{Cnv})Q:\\ +[*71·7] &\equiv :Q\in CʻP.\supset _{Q}.(MʻQ)B\breve{Q} \colon\ldotp \\ +[\text{Transp}] \supset \colon\ldotp \dot{\exists} !M\dot{-} B\mid \text{Cnv}.&\equiv :(\exists Q).Q\in CʻP.{\sim}\{(MʻQ)B\breve{Q}\}:\\ +[*93·1.*80·3] &\supset :(\exists Q).Q\in CʻP.MʻQ\in \text{D}ʻQ:\\ +[*33·13] &\supset :(\exists Q,y).Q\in CʻP.(MʻQ)Qy:\\ +[*172·15] &\supset :M\in \text{D}ʻ\Pi ʻP\colon\colon \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_433">[Pg 433]</span></p> + +<p>The following proposition is important. It shows that, if \(CʻP\) +consists of series, if any member of \(CʻP\) has no first term, \(\PiʻP\) +has no first term, but if every member of \(CʻP\) has a first +term, the selection of all these first terms is the first term of \(\PiʻP\).</p> + +<p class="nind"><b><a id="*172·162">*172·162</a>.</b> \(\vdash :\dot{\exists} !P.\supset .\overrightarrow{B}ʻ\Pi ʻP=B_{\Delta }ʻCʻP.\overrightarrow{B}ʻ\text{Cnv}ʻ\Pi ʻP=B_{\Delta }ʻ\text{Cnv}ʻʻCʻP\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*93·103.*172·12·16.\text{Transp}.&\supset \vdash .\overrightarrow{B}ʻ\Pi ʻP\subset F_{\Delta }ʻCʻP\cap \text{Rl}ʻB &\qquad \text{(1)}\\ +\vdash .*72·93.\supset \\ +\vdash \colon\ldotp M\in F_{\Delta }ʻCʻP.M\,\unicode{x2abd}\, B.Q\in CʻP.&\supset :(MʻQ)BQ:\\ +[*93·1] &\supset :(MʻQ)\in \text{D}ʻQ:\\ +[*33·13] &\supset :(\exists y).(MʻQ)Qy:\\ +[*172·15] &\supset :M\in \text{D}ʻ\Pi ʻP &\qquad \text{(2)}\\ +\vdash .(2).*10·11·23·35.&\supset \vdash \colon\ldotp \dot{\exists} !P.\supset :M\in F_{\Delta }ʻCʻP\cap \text{Rl}ʻB.\supset .M\in \text{D}ʻ\Pi ʻP &\qquad \text{(3)}\\ +\vdash .*172·11. &\supset \vdash :N\in \text{ᗡ}ʻ\Pi ʻP.\supset .(\exists Q,M).Q\in CʻP.(MʻQ)Q(NʻQ).\\ +[*93·1] &\supset .(\exists Q).Q\in CʻP.{\sim}\{(NʻQ)BQ\}.\\ +[*72·93] &\supset .{\sim}(N\,\unicode{x2abd}\, B):\\ +[\text{Transp}. \frac{M}{N}] &\supset \vdash :M\,\unicode{x2abd}\, B.\supset .M{\sim}\in \text{ᗡ}ʻ\Pi ʻP &\qquad \text{(4)}\\ +\vdash .(1).(3).(4). \supset \vdash :\text{Hp}.\supset .\overrightarrow{B}ʻ\Pi ʻP&=F_{\Delta }ʻCʻP\cap \text{Rl}ʻB\\ +[*80·17] &=B_{\Delta }ʻCʻP &\qquad \text{(5)}\\ +\text{Similarly}\quad &\vdash :\text{Hp}.\supset .\overrightarrow{B}ʻ\text{Cnv}ʻ\Pi ʻP=B_{\Delta }ʻ\text{Cnv}ʻʻCʻP &\qquad \text{(6)}\\ +\vdash .(5).(6).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p>The following proposition is much used.</p> + +<p class="nind"><b><a id="*172·17">*172·17</a>.</b> \(\vdash :\dot{\exists} !P.\supset .Cʻ\Pi ʻP=F_{\Delta }ʻCʻP\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*172·16·162.\supset \\ +\vdash \colon\ldotp \text{Hp}.M\in F_{\Delta }ʻCʻP.&\supset :\dot{\exists} !M\dot{-} B.\supset .M\in \text{ᗡ}ʻ\Pi ʻP:M\,\unicode{x2abd}\, B.\supset .M\in \overrightarrow{B}ʻ\Pi ʻP:\\ +[*93·11.*25·55] &\supset :M\in Cʻ\Pi ʻP &\qquad \text{(1)}\\ +\vdash .(1).*172·12.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*172·171.</b> \[\begin{align}\vdash :\dot{\exists} !P.\supset .&\text{D}ʻ\Pi ʻP=F_{\Delta }ʻCʻP-B_{\Delta }ʻ\text{Cnv}ʻʻCʻP.\\ +&\text{ᗡ}ʻ\Pi ʻP=F_{\Delta }ʻCʻP-B_{\Delta }ʻCʻP \quad[*172·162·17]\end{align}\]</p> + +<p class="nind"><b>*172·18.</b> \(\vdash \colon\ldotp \dot{\exists} !P.\supset :\dot{\exists} !\Pi ʻP.\equiv .\exists !F_{\Delta }ʻCʻP \quad[*172·17]\)</p> + +<p class="nind"><b>*172·181.</b> \(\vdash \colon\ldotp \text{Mult ax}.\supset :\dot{\Lambda} {\sim}\in CʻP.\dot{\exists} !P.\equiv .\dot{\exists} !\Pi ʻP\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*88·361.*172·18.\supset \vdash \colon\colon \text{Hp}.\supset \colon\ldotp \dot{\exists} !P.\supset :\dot{\exists} !\Pi ʻP.&\equiv .CʻP\subset \text{ᗡ}ʻF.\\ +[*33·41·5] &\equiv .CʻP\subset \hat{Q} (\exists !CʻQ).\\ +[*33·241] &\equiv .\dot{\Lambda} {\sim}\in CʻP &\qquad \text{(1)}\\ +\vdash .*172·13. &\supset \vdash :\dot{\exists} !\Pi ʻP.\supset .\dot{\exists} !P &\qquad \text{(2)}\\ +\vdash .(1).(2).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_434">[Pg 434]</span></p> + +<p class="nind"><b>*172·182.</b> \[\begin{align}&\vdash \colon\colon \text{Mult ax}.\supset \colon\ldotp \dot{\Lambda} \in CʻP.\lor.P=\dot{\Lambda} :\equiv .\Pi ʻP=\dot{\Lambda}\\ +&[*172·181.\text{Transp}]\end{align}\]</p> + +<p class="nind"><b>*172·19.</b> \(\vdash :\dot{\exists} !\Pi ʻP.\supset .\dot{s} ʻCʻ\Pi ʻP=F\upharpoonright CʻP \quad[*172·17.*80·42]\)</p> + +<p>Note that we cannot proceed to \(\Sigma ʻ\Pi ʻP\), because \(F^{;}\PiʻP\) +is meaningless, owing to the fact that the field of \(\Pi ʻP\) +consists of non-homogeneous relations.</p> + +<p class="nind"><b>*172·191.</b> \(\vdash .\dot{s} ʻCʻ\Pi ʻP\,\unicode{x2abd}\, F\upharpoonright CʻP\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*172·19.*23·42.&\supset \vdash :\dot{\exists} !\Pi ʻP.\supset .\dot{s} ʻCʻ\Pi ʻP\,\unicode{x2abd}\, F\upharpoonright CʻP &\qquad \text{(1)}\\ +\vdash .*41·21. \supset \vdash :\Pi ʻP=\dot{\Lambda} .&\supset .\dot{s} ʻCʻ\Pi ʻP=\dot{\Lambda} .\\ +[*25·12] &\supset .\dot{s} ʻCʻ\Pi ʻP\,\unicode{x2abd}\, F\upharpoonright CʻP &\qquad \text{(2)}\\ +\vdash .(1).(2).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*172·192.</b> \(\vdash .\text{ᗡ}ʻ(F\upharpoonright \beta )=\beta -\iota ʻ\dot{\Lambda}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*35·101.\supset \vdash :Q\in \text{ᗡ}ʻ(F\upharpoonright \beta ).&\equiv .(\exists x).xFQ.Q\in \beta .\\ +[*33·5] &\equiv .\exists !CʻQ.Q\in \beta .\\ +[*33·24] &\equiv .\dot{\exists} !Q.Q\in \beta :\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p>The following proposition is sometimes useful. (It is used in <a href="#*173·22">*173·22</a>. +<a href="#*182·2">*182·2</a>.<a href="#*185·21">*185·21</a>.)</p> + +<p class="nind"><b>*172·2.</b> \(\vdash .\Pi ʻ(P\downarrow P)=P\downarrow_{.,} P\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*172·11.*55·15.\supset \\ +\vdash \colon\colon M\{\Pi ʻ(P\downarrow P)\}N.&\equiv \colon\ldotp M,N\in F_{\Delta }ʻ\iota ʻP\colon\ldotp \\ +(\exists Q):Q\in \iota ʻP.(MʻQ)Q(NʻQ):R&=P.R\neq Q.\supset _{R}.MʻR=NʻR\colon\ldotp \\ +[*13·195·191] &\equiv \colon\ldotp M,N\in F_{\Delta }ʻ\iota ʻP\colon\ldotp (MʻP)P(NʻP)\colon\ldotp \\ +[*85·51.*33·5]&\equiv \colon\ldotp M,N\in \downarrow PʻʻCʻP.(MʻP)P(NʻP)\colon\ldotp \\ +[*38·131] &\equiv \colon\ldotp (\exists x, y).x,y\in CʻP.M=x\downarrow P.N=y\downarrow P.(MʻP)P(NʻP)\colon\ldotp \\ +[*55·13] &\equiv \colon\ldotp (\exists x,y).x,y\in CʻP.M=x\downarrow P.N=y\downarrow P.xPy\colon\ldotp \\ +[*150·11] &\equiv \colon\ldotp M(\downarrow P^{;}P)N\colon\ldotp \\ +[*150·6] &\equiv \colon\ldotp M(P\downarrow_{.,} P)N\colon\colon \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_435">[Pg 435]</span></p> + +<p>The following propositions are concerned with the nature of the +connection between \(\Pi ʻ(P\downarrow Q)\) and \(P\times Q\). The +connection is such as might be desired, except when \(P=Q\), in which +case, as shown above, \(\Pi ʻ(P\downarrow P)\) is like \(P\), and is +therefore not like \(P\times P\).</p> + +<p class="nind"><b>*172·21.</b> \(\vdash :P\neq Q.\supset .P\times Q=\dagger (Q\downarrow P)^{;}\Pi ʻ(P\downarrow Q)\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*172·11.*55·15.\supset \\ +\vdash \colon\colon .M{\Pi ʻ(P\downarrow Q)}N.&\equiv \colon\colon M,N\in F_{\Delta }ʻ(\iota ʻP\cup \iota ʻQ)\colon\ldotp \\ +&(\exists R):R\in \iota ʻP\cup \iota ʻQ.(MʻR)R(NʻR):S(P\downarrow Q)R.S\neq R.\supset _{S}.MʻS=NʻS\colon\colon\\ +[*51·235]&\equiv \colon\colon M,N\in F_{\Delta }ʻ(\iota ʻP\cup \iota ʻQ)\colon\colon\\ +&(MʻP)P(NʻP):S(P\downarrow Q)P.S\neq P.\supset _{S}.MʻS=NʻS\colon\ldotp v\colon\ldotp \\ +&(MʻQ)Q(NʻQ):S(P\downarrow Q)Q.S\neq Q.\supset _{S}.MʻS=NʻS &\qquad \text{(1)}\\ +\vdash .(1).*55·13.\supset \vdash \colon\colon\text{Hp}.\supset \colon\ldotp \\ +M{\Pi ʻ(P\downarrow Q)}N.&\equiv :M,N\in F_{\Delta }ʻ(\iota ʻP\cup \iota ʻQ): +(MʻP)P(NʻP).\lor.(MʻQ)Q(NʻQ).MʻP=NʻP:\\ +[*80·9·91]&\equiv :(\exists x,x',y,y'):x,x'\in CʻP.y,y'\in CʻQ.\\ +&M=x\downarrow P\unicode{x228d} y\downarrow Q.N=x'\downarrow P\unicode{x228d} y'\downarrow Q:\\ +&xPx'.\lor.x=x'.yQy' &\qquad \text{(2)}\\ +\vdash .*150·72.&\supset \vdash :M=x\downarrow P\unicode{x228d} y\downarrow Q.\supset .M^{;}(Q\downarrow P)=y\downarrow x.\\ +[*150·1] &\supset .\dagger (Q\downarrow P)ʻM=y\downarrow x &\qquad \text{(3)}\\ +\vdash .*150·4. &\supset \vdash :R{\dagger (Q\downarrow P)^{;}\Pi ʻ(P\downarrow Q)}S.\equiv .\\ +&(\exists M,N).M{\Pi ʻ(P\downarrow Q)}N.R=\dagger (Q\downarrow P)ʻM.S=\dagger (Q\downarrow P)ʻN &\qquad \text{(4)}\\ +\vdash .(2).(3).(4).&\supset \vdash \colon\colon\text{Hp}.\supset \colon\ldotp \\ +&R\{\dagger (Q\downarrow P)^{;}\Pi ʻ(P\downarrow Q)\}S.\equiv :(\exists M,N,x,x',y,y'):\\ +&x,x'\in CʻP.y,y'\in CʻQ.M=x\downarrow P\unicode{x228d} y\downarrow Q.N=x'\downarrow P\unicode{x228d} y'\downarrow Q:\\ +&R=y\downarrow x.S=y'\downarrow x':xPx'.\lor.x=x'.yQy':\\ +[*13·19] &\equiv :(\exists x,x',y,y'):x,x'\in CʻP.y,y'\in CʻQ.R=y\downarrow x.S=y'\downarrow x':\\ +&xPx'.\lor.x=x'.yQy':\\ +[*166·111] &\equiv :R(P\times Q)S\colon\colon\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*172·22.</b> \(\vdash :P\neq Q.\supset .\{\dagger (Q\downarrow P)\}\upharpoonright F_{\Delta }ʻ(\iota ʻP\cup \iota ʻQ)\in (P\times Q)\,\overline{\text{smor}}\,\Pi ʻ(P\downarrow Q)\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*80·9.*150·71.&\supset \vdash \colon\ldotp \text{Hp}.\supset :\\ +&M\in F_{\Delta }ʻ(\iota ʻP\cup \iota ʻQ).\supset .M^{;}(Q\downarrow P)=(MʻQ)\downarrow (MʻP) &\qquad \text{(1)}\\ +\vdash .(1).*150·1.&\supset \vdash \colon\ldotp \text{Hp}.\supset :\\ +&M,N\in F_{\Delta }ʻ(\iota ʻP\cup \iota ʻQ).\dagger (Q\downarrow P)ʻM=\dagger (Q\downarrow P)ʻN.\supset .\\ +&(MʻQ)\downarrow (MʻP)=(NʻQ)\downarrow (NʻP).\\ +[*55·202] &\supset .MʻP=NʻP.MʻQ=NʻQ.\\ +[*80·91] &\supset .M=N &\qquad \text{(2)}\\ +\vdash .(2).*151·241.*172·21·17.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*172·23">*172·23</a>.</b> \(\vdash :P\neq Q.\supset .\Pi ʻ(P\downarrow Q)\,\text{smor}\,P\times Q \quad[*172·22]\)</p> + +<p>The following propositions are lemmas for <a href="#*172·32">*172·32</a>.</p> + +<p><span class="pagenum" id="Page_436">[Pg 436]</span></p> + +<p class="nind"><b><a id="*172·3">*172·3</a>.</b> \[\begin{align}&\vdash \colon\ldotp \dot{\exists} !P.Z{\sim}\in CʻP.\supset :M{\Pi ʻ(P\unicode{x21f8} Z)}N.\equiv .\\ +&(\exists S,T,u,v).(u\downarrow S)(\Pi ʻP\times Z)(v\downarrow T).M=S\unicode{x228d} u\downarrow Z.N=T\unicode{x228d} v\downarrow Z\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*80·66·44.*161·14. &\supset \vdash \colon\ldotp \text{Hp}.\supset :M\in F_{\Delta }ʻCʻ(P\unicode{x21f8} Z).\equiv .\\ +&(\exists S,u).S\in F_{\Delta }ʻCʻP.u\in CʻZ.M=S\unicode{x228d} u\downarrow Z &\qquad \text{(1)}\\ +\vdash .*55·13.*33·14.*4·73. &\supset \vdash \colon\colon M=S\unicode{x228d} u\downarrow Z.\supset \colon\ldotp \\ +&xMQ.\equiv :xSQ.Q\in \text{ᗡ}ʻS.\lor.x=u.Q=Z &\qquad \text{(2)}\\ +\vdash .(2).*80·14. &\supset \vdash \colon\colon\text{Hp}.\text{Hp}(2).S\in F_{\Delta }ʻCʻP.u\in CʻZ.\supset \colon\ldotp \\ +&xMQ.\equiv :xSQ.Q\in CʻP.\lor.x=u.Q=Z\colon\ldotp \\ +[*24·37] &\supset \colon\ldotp Q\in CʻP.\supset :xMQ.\equiv .xSQ\colon\ldotp Q=Z.\supset :xMQ.\equiv .x=u\colon\ldotp \\ +[*30·341.*80·3.*30·3] &\supset \colon\ldotp Q\in CʻP.\supset .MʻQ=SʻQ:Q=Z.\supset .MʻQ=u &\qquad \text{(3)}\\ +\vdash .(1).*172·11.*161·14.*172·17.\supset \\ +\vdash \colon\colon\ldotp \text{Hp}.&\supset \colon\colon M{\Pi ʻ(P\unicode{x21f8} Z)}N.\equiv \colon\ldotp \\ +&(\exists S,T,u,v)\colon\ldotp S,T\in F_{\Delta }ʻCʻP.u,v\in CʻZ.M=S\unicode{x228d} u\downarrow Z.N=T\unicode{x228d} v\downarrow Z\colon\ldotp \\ +&(\exists Q):Q\in CʻP\cup \iota ʻZ.(MʻQ)Q(NʻQ):R(P\unicode{x21f8} Z)Q.R\neq Q.\supset _{R}.MʻR=NʻP\colon\ldotp \\ +[*51·239.(3).*161·11] +&\equiv \colon\ldotp (\exists S,T,u,v)\colon\ldotp S,T\in F_{\Delta }ʻCʻP.u,v\in CʻZ.M=S\unicode{x228d} u\downarrow Z.N=T\unicode{x228d} v\downarrow Z\colon\ldotp \\ +&(\exists Q):Q\in CʻP.(SʻQ)Q(TʻQ):RPQ.R\neq Q.\supset _{R}.SʻR=TʻR:\lor: +uZv:R\in CʻP.\supset _{R}.SʻR=TʻR\colon\ldotp \\ +[*172·11·17.*71·35.*80·14] +&\equiv \colon\ldotp (\exists S,T,u,v)\colon\ldotp S,T\in Cʻ\Pi ʻP.u,v\in CʻZ.M=S\unicode{x228d} u\downarrow Z.N=T\unicode{x228d} v\downarrow Z:\\ +&S(\Pi ʻP)T.\lor.S=T.uZv\colon\ldotp \\ +[*166·112] &\equiv \colon\ldotp (\exists S,T,u,v):(u\downarrow S)(\Pi ʻP\times Z)(v\downarrow T).\\ +&M=S\unicode{x228d} u\downarrow Z.N=T\unicode{x228d} v\downarrow Z\colon\colon\ldotp \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*172·31.</b> \[\begin{align}&\vdash :\dot{\exists} !P.Z{\sim}\in CʻP.\\ +&W=\hat{M} \hat{R} \{(\dot{\exists} S,u).S\in F_{\Delta }ʻCʻP.u\in CʻZ.R=u\downarrow S.M=S\unicode{x228d} u\downarrow Z\}.\supset .\\ +&W\in \Pi ʻ(P\unicode{x21f8} Z)\,\overline{\text{smor}}\,(\Pi ʻP x Z)\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*172·3. &\supset \vdash :\text{Hp}.\supset .\Pi ʻ(P\unicode{x21f8} Z)=W^{;}(\Pi ʻP\times Z) &\qquad \text{(1)}\\ +\vdash .*21·33. &\supset \vdash \colon\ldotp \text{Hp}.\supset :MWR.MʻWR.\equiv .\\ +&(\exists S,S',u,u').S,S'\in F_{\Delta }ʻCʻP.u,u'\in CʻZ.R=u\downarrow S=u'\downarrow S'.\\ +&M=S\unicode{x228d} u\downarrow Z.Mʻ=S'\unicode{x228d} u'\downarrow Z.\\ +[*55·202]& \supset .(\exists S,Sʻ,u,u').S,S'\in F_{\Delta }ʻCʻP.u,u'\in CʻZ.S=S'.u =u'.\\ +&M=S\unicode{x228d} u\downarrow Z.Mʻ=S'\unicode{x228d} u'\downarrow Z.\\ +[*13·22·172] &\supset .M=M' &\qquad \text{(2)}\\ +\vdash .*21·33. &\supset \vdash \colon\ldotp \text{Hp}.\supset :MWR.MWRʻ.\equiv .\\ +&(\exists S,S',u,u').S,S'\in F_{\Delta }ʻCʻP.u,u'\in CʻZ.R=u\downarrow S.Rʻ=u'\downarrow Sʻ.\\ +&M=S\unicode{x228d} u\downarrow Z.M=Sʻ\unicode{x228d} u'\downarrow Z.\\ +[*80·45·661] &\supset .(\exists S,Sʻ,u,u').R=u\downarrow S.Rʻ=u'\downarrow Sʻ.\\ +&S=M\upharpoonright CʻP.u\downarrow Z=M\upharpoonright \iota ʻZ.Sʻ=M\upharpoonright CʻP.u'\downarrow Z=M\upharpoonright CʻP .\\ +[*13·172] &\supset .(\exists S,S',u,u').R=u\downarrow S.Rʻ=u'\downarrow S'.S=S'.u\downarrow Z=u'\downarrow Z.\\ +[*55·202] &\supset .R=Rʻ &\qquad \text{(3)}\\ +\vdash .(2).(3). &\supset \vdash :\text{Hp}.\supset .W\in 1\rightarrow 1 &\qquad \text{(4)}\\ +\vdash .*166·12.*113·101.*172·17. &\supset \vdash :\text{Hp}.\supset .\text{ᗡ}ʻW=Cʻ(\Pi ʻP\times Z) &\qquad \text{(5)}\\ +\vdash .(1).(4).(5).*151·11.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_437">[Pg 437]</span></p> + +<p class="nind"><b><a id="*172·32">*172·32</a>.</b> \(\vdash :Z{\sim}\in CʻP.\supset .\Pi ʻ(P\unicode{x21f8} Z)\,\text{smor}\,\Pi ʻP\times Z\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*172·31. &\supset \vdash :\text{Hp}.\dot{\exists} !P.\supset .\Pi ʻ(P\unicode{x21f8} Z)\,\text{smor}\,\Pi ʻP\times Z &\qquad \text{(1)}\\ +\vdash .*172·13.*161·2. &\supset \vdash :{\sim}\dot{\exists} !P.\supset .\Pi ʻ(P\unicode{x21f8} Z)=\dot{\Lambda} &\qquad \text{(2)}\\ +\vdash .*172·13.*166·13. &\supset \vdash :{\sim}\dot{\exists} !P.\supset .\Pi ʻP\times Z=\dot{\Lambda} &\qquad \text{(3)}\\ +\vdash .(2).(3).*153·101. &\supset \vdash :{\sim}\dot{\exists} !P.\supset .\Pi ʻ(P\unicode{x21f8} Z)\,\text{smor}\,\Pi ʻP\times Z &\qquad \text{(4)}\\ +\vdash .(1).(4).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*172·321">*172·321</a>.</b> \[\begin{align}&\vdash :Z{\sim}\in CʻP.\supset .\Pi ʻ(Z\unicode{x21f7}CʻP)\,\text{smor}\,Z\times \Pi ʻP\\ +&[\text{Proof by similar stages to those in proof of *172·32}]\end{align}\]</p> + +<p>The following proposition is a lemma for <a href="#*172·34">*172·34</a>, which is required in +proving <a href="#*172·35">*172·35</a> (as well as <a href="#*176·4">*176·4</a>).</p> + +<p class="nind"><b>*172·33.</b> \[\begin{align}&\vdash \colon\colon\dot{\exists} !P.\dot{\exists} !Q.CʻP\cap CʻQ=\Lambda .\supset \colon\ldotp \\ +M\{\Pi ʻ(P\unicode{x2909}Q)\}N.&\equiv :(\exists S,T,S',T'):S,S'\in F_{\Delta }ʻCʻP.T,T'\in F_{\Delta }ʻCʻQ:\\ +&S(\Pi ʻP)S'.\lor.S=S'.T(\Pi ʻQ)Tʻ:M=S\unicode{x228d} T.M'=S'\unicode{x228d} T'\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*80·66. &\supset \vdash \colon\ldotp \text{Hp}.\supset :M\in F_{\Delta }ʻ(CʻP\cup CʻQ).\equiv .\\ +&(\exists S,T).S\in F_{\Delta }ʻCʻP.T\in F_{\Delta }ʻCʻQ.M=S\unicode{x228d} T &\qquad \text{(1)}\\ +\vdash .*80·661.*35·7. &\supset \vdash \colon\ldotp \text{Hp}.S\in F_{\Delta }ʻCʻP.T\in F_{\Delta }ʻCʻQ.M=S\unicode{x228d} T.\supset :\\ +&R\in CʻP.\supset .MʻR=SʻR:R\in CʻQ.\supset .MʻR=TʻR &\qquad \text{(2)}\\ +\vdash .(1).*172·11·17.*160·14. &\supset \vdash \colon\colon\ldotp \text{Hp}.\supset \colon\colon M\{\Pi ʻ(P\unicode{x2909}Q)\}N.\equiv \colon\ldotp\\ +&(\exists S,T,S',T')\colon\ldotp S,S'\in F_{\Delta }ʻCʻP.T,T'\in F_{\Delta }ʻCʻQ.M=S\unicode{x228d} T.N=S'\unicode{x228d} T'\colon\ldotp \\ +&(\exists R):R\in CʻP\cup CʻQ.(MʻR)R(NʻR):Rʻ(P\unicode{x2909}Q)R.R\neq Rʻ.\supset _{Rʻ}.MʻRʻ=NʻRʻ\colon\ldotp \\ +[(2).*160·11] &\equiv \colon\ldotp (\exists S,T,S',T')\colon\ldotp S,S'\in F_{\Delta }ʻCʻP.T,T'\in F_{\Delta }ʻCʻQ.\\ +&M=S\unicode{x228d} T.N=S'\unicode{x228d} T'\colon\ldotp \\ +&(\exists R):R\in CʻP.(SʻR)R(S'ʻR):R'PR.R'\neq R.\supset _{Rʻ}.SʻR'=S'ʻR':\lor:\\ +&(\exists R):R\in CʻQ.(TʻR)R(TʻʻR):R'QR.R'\neq R.\supset _{Rʻ}.TʻR'=TʻʻR': +R'\in CʻP.\supset _{R'}.SʻR'=S'ʻR'\colon\ldotp \\ +[*10·35] &\equiv \colon\ldotp (\exists S,T,S',T')\colon\ldotp S,S'\in F_{\Delta }ʻCʻP.T,T'\in F_{\Delta }ʻCʻQ.M=S\unicode{x228d} T.N=S'\unicode{x228d} T'\colon\ldotp \\ +&(\exists R):R\in CʻP.(SʻR)R(SʻʻR):R'PR.R'\neq R.\supset _{R'}.SʻR'=SʻʻR':\lor:\\ +&R'\in CʻP.\supset _{R'}.SʻR'=S'ʻR':\\ +&(\exists R):R\in CʻQ.(TʻR)R(TʻʻR):RʻQR.R'\neq R.\supset _{R'}.TʻR'=T'ʻR'\colon\ldotp \\ +[*172·11.*71·35.*80·14] +&\equiv \colon\ldotp (\exists S,T,S',T')\colon\ldotp S,S'\in F_{\Delta }ʻCʻP.T,T'\in F_{\Delta }ʻCʻQ.M=S\unicode{x228d} T,Mʻ=S'\unicode{x228d} T'\colon\ldotp \\ +&S(\Pi ʻP)S'.\lor.S=S'.T(\Pi ʻQ)T'\colon\colon\ldotp \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_438">[Pg 438]</span></p> + +<p class="nind"><b><a id="*172·34">*172·34</a>.</b> \[\begin{align}\vdash :\dot{\exists} !P.\dot{\exists} !Q.CʻP\cap CʻQ&=\Lambda .\supset .\\ +&(\dot{s} \mid C)\in \{\Pi ʻ(P\unicode{x2909}Q)\}\,\overline{\text{smor}}\, \{\Pi ʻP\times \Pi ʻQ\}\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*172·33.*55·15.*53·13.\supset \\ +\vdash \colon\colon\text{Hp}.\supset \colon\ldotp M{\Pi ʻ(P\unicode{x2909}Q)}N.&\equiv :(\exists S,T,S',T',R,R'):\\ +&S,S'\in F_{\Delta }ʻCʻP.T,T'\in F_{\Delta }ʻCʻQ.R=T\downarrow S.R'=Tʻ\downarrow Sʻ.M=\dot{s} ʻCʻR.N=\dot{s} ʻCʻR':\\ +&S(\Pi ʻP)S'.\lor.S=S'.T(\Pi ʻQ)Tʻ:\\ +[*166·11.*172·17]&\equiv :(\exists R,R').R\{\Pi ʻP\times \Pi ʻQ\} Rʻ.M=\dot{s} ʻCʻR.N=\dot{s} ʻCʻR':\\ +[*150·4] &\equiv :M\{\dot{s} ^{;}C^{;}(\Pi ʻP\times \Pi ʻQ)\}N &\qquad \text{(1)}\\ +\vdash .*113·153.*172·19.*166·12.&\supset \vdash :\text{Hp}.\supset .(\dot{s} \mid C)\upharpoonright Cʻ(\Pi ʻP\times \Pi ʻQ)\in 1\rightarrow 1 &\qquad \text{(2)}\\ +\vdash .(1).(2).*151·231.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*172·35">*172·35</a>.</b> \[\begin{align}&\vdash :\dot{\exists} !P.\dot{\exists} !Q.CʻP\cap CʻQ=\Lambda .\supset .\Pi ʻ(P\unicode{x2909}Q)\,\text{smor}\,\Pi ʻP\times \Pi ʻQ\\ +&[*172·34]\end{align}\]</p> + +<p>The above proposition is important, being a form of the associative law.</p> + +<p>The following propositions are extensions of <a href="#*172·23">*172·23</a>. It is obvious +that they may be extended to any finite number of factors.</p> + +<p class="nind"><b>*172·36.</b> \(\vdash :X\neq Y.X\neq Z.Y\neq Z.\supset .\Pi ʻ{(X\downarrow Y)\unicode{x21f8} Z}\text{smor}X\times Y\times Z\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*172·32.\supset \vdash :\text{Hp}.\supset .\Pi ʻ{(X\downarrow Y)\unicode{x21f8} Z}\,\text{smor}\,\Pi ʻ(X\downarrow Y)\times Z &\qquad \text{(1)}\\ +\vdash .*172·23.*166·23.\supset \vdash :\text{Hp}.\supset .\Pi ʻ(X\downarrow Y)\times Z\,\text{smor}\,X\times Y\times Z &\qquad \text{(2)}\\ +\vdash .(1).(2).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*172·361.</b> \[\begin{align}&\vdash :X\neq Y.X\neq Z.Y\neq Z.\supset .\Pi ʻ{X\unicode{x21f7}(Y\downarrow Z)}\,\text{smor}\,X\times Y\times Z\\ +&[\text{Proof as in *172·36}]\end{align}\]</p> + +<p class="nind"><b>*172·37.</b> \[\begin{align}\vdash :X\neq Y.X\neq Z.&X\neq W.Y\neq Z.Y\neq W.Z\neq W.\supset .\\ +&\Pi ʻ\{(X\downarrow Y)\unicode{x2909}(Z\downarrow W)\}\,\text{smor}\,X\times Y\times Z\times W\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*172·35.\supset \\ +\vdash :\text{Hp}.\supset .\Pi ʻ{(X\downarrow Y)\unicode{x2909}(Z\downarrow W)}\text{smor}\Pi ʻ(X\downarrow Y)\times \Pi ʻ(Z\downarrow W) &\qquad \text{(1)}\\ +\vdash .*172·23.*166·23.\supset \\ +\vdash :\text{Hp}.\supset .\Pi ʻ(X\downarrow Y)\times \Pi ʻ(Z\downarrow W)\text{smor}(X\times Y)\times (Z\times W) &\qquad \text{(2)}\\ +\vdash .(1).(2).*166·42.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_439">[Pg 439]</span></p> + +<p>The following propositions are concerned with the construction of a +correlator of \(\Pi ʻP\) with \(\Pi ʻQ\) when we are given a double +correlator of \(P\) with \(Q\). If the double correlator is \(T\) or +\(T\upharpoonright Cʻ\Sigma ʻQ\), the correlator of \(\Pi ʻP\) with +\(\Pi ʻQ\) is +\[ +\{T\parallel \text{Cnv}ʻT\dagger\}\upharpoonright Cʻ\Pi ʻQ. +\]</p> + +<p class="nind"><b>*172·4.</b> \(\vdash :T\in P\,\overline{\text{smor smor}}\, Q.\supset .\{T\parallel \text{Cnv}ʻT\dagger \}\upharpoonright Cʻ\Pi ʻQ\in 1\rightarrow 1\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*164·15.\supset \\ +\vdash :\text{Hp}.&\supset .T\upharpoonright Cʻ\Sigma ʻQ,T\dagger \upharpoonright CʻQ\in 1\rightarrow 1.Cʻ\Sigma ʻQ=\text{ᗡ}ʻT.CʻQ\subset \text{ᗡ}ʻT\dagger &\qquad \text{(1)}\\ +\vdash .*41·43. &\supset \vdash .sʻ\text{D}ʻʻCʻ\Pi ʻQ=\text{D}ʻ\dot{s} ʻCʻ\Pi ʻQ.\\ +[*172·191] \supset \vdash .sʻ\text{D}ʻʻCʻ\Pi ʻQ&\subset \text{D}ʻ(F\upharpoonright CʻQ)\\ +[*37·401.*162·23] &\subset Cʻ\Sigma ʻQ &\qquad \text{(2)}\\ +\vdash .*41·44. &\supset \vdash .sʻ\text{ᗡ}ʻʻCʻ\Pi ʻQ=\text{ᗡ}ʻ\dot{s} ʻCʻ\Pi ʻQ.\\ +[*172·191] &\supset \vdash .sʻ\text{ᗡ}ʻʻCʻ\Pi ʻQ\subset \text{ᗡ}ʻ(F\upharpoonright CʻQ)\\ +[*35·64] &\subset CʻQ &\qquad \text{(3)}\\ +\vdash .(1).(2).(3).*74·773 \frac{T,T\dagger ,Cʻ\Sigma ʻQ,CʻQ,Cʻ\Pi ʻQ}{Q,\,R,\,\alpha,\,\beta,\,\lambda}.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*172·401.</b> \[\begin{align}\vdash :T\in P\,\overline{\text{smor smor}}\, Q.N\in F_{\Delta }ʻCʻQ.&S\in CʻQ.\supset .\\ +&\{(T\parallel \text{Cnv}ʻT\dagger )ʻN\}ʻT^{;}S=TʻNʻS\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*43·112.*150·1. \supset \vdash .\{(T\parallel \text{Cnv}ʻT\dagger )ʻN\}ʻT^{;}S&=(T\mid N\mid \text{Cnv}ʻT\dagger )ʻT\dagger ʻS &\qquad \text{(1)}\\ +\vdash .(1).*35·7·48.*80·14.\supset \\ +\vdash :\text{Hp}.\supset .\{(T\parallel \text{Cnv}ʻT\dagger )ʻN\}ʻT^{;}S&=\{T\mid N\mid \text{Cnv}ʻ(T\dagger \upharpoonright CʻQ)\}ʻ(T\dagger \upharpoonright CʻQ)ʻS\\ +[*34·41.*72·601.*164·13] &=(T\mid N)ʻS\\ +[*34·41] & =TʻNʻS:\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*172·402.</b> \[\begin{align}\vdash :T\in P\,\overline{\text{smor smor}}\, Q.&N,N'\in F_{\Delta }ʻCʻQ.S\in CʻQ.M=(T\parallel \text{Cnv}ʻT\dagger )ʻN.\\ +&Mʻ=(T\parallel \text{Cnv}ʻT\dagger )ʻN'.R=T^{;}S.\supset :\\ +&NʻS=N'ʻS.\equiv .MʻR=M'ʻR:(NʻS)S(N'ʻS).\equiv .(MʻR)R(M'ʻR)\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*172·401. &\supset \vdash :\text{Hp}.\supset .MʻR=TʻNʻS.MʻʻR=TʻNʻʻS &\qquad \text{(1)}\\ +\vdash .*162·22.*40·13. \supset \vdash :\text{Hp}.&\supset .CʻS\subset Cʻ\Sigma ʻQ.\\ +[*164·1] &\supset .CʻS\subset \text{ᗡ}ʻT &\qquad \text{(2)}\\ +\vdash .*80·31.*33·5. &\supset \vdash \colon\ldotp \text{Hp}.\supset :NʻS,NʻʻS\in CʻS:\\ +[(2)] &\supset :NʻS,NʻʻS\in \text{ᗡ}ʻT:\\ +[*71·56] \supset :NʻS=NʻʻS.&\equiv .TʻNʻS=TʻNʻʻS.\\ +[(1)] &\equiv .MʻR=MʻʻR &\qquad \text{(3)}\\ +\vdash .(1). \supset \vdash \colon\ldotp \text{Hp}.\supset :(MʻR)R(MʻʻR).&\equiv .(TʻNʻS)(T^{;}S)(TʻNʻʻS).\\ +[*150·41] &\equiv .(NʻS)(\breve{T} ^{;}T^{;}S)(NʻʻS).\\ +[*151·252.(2)] &\equiv .(NʻS)S(NʻʻS) &\qquad \text{(4)}\\ +\vdash .(3).(4).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_440">[Pg 440]</span></p> + +<p class="nind"><b>*172·403.</b> \(\vdash :T\in P\,\overline{\text{smor smor}}\, Q.\supset .(T\parallel \text{Cnv}ʻT\dagger )ʻʻF_{\Delta }ʻCʻQ\subset F_{\Delta }ʻCʻP\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*80·14.*35·48.\supset \\ +\vdash \colon\ldotp \text{Hp}.&\supset :N\in F_{\Delta }ʻCʻQ.\supset .T\mid N\mid \text{Cnv}ʻT\dagger =T\mid N\mid \text{Cnv}ʻ(T\dagger \upharpoonright CʻQ).\\ +[*80·14.*164·13] &\supset .(T\mid N\mid \text{Cnv}ʻT\dagger )\in 1\rightarrow \text{Cls} &\qquad \text{(1)}\\ +\vdash .*37·32.\supset \vdash \colon\ldotp \text{Hp}.\supset :N\in F_{\Delta }ʻCʻQ.\supset .\text{ᗡ}ʻ(T\mid N\mid \text{Cnv}ʻT\dagger )&=T\dagger ʻʻ\breve{N} ʻʻ\text{ᗡ}ʻT\\ +[*37·271.*164·1.*80·33.*162·23] &=T\dagger ʻʻ\text{ᗡ}ʻN\\ +[*80·14] &=T\dagger ʻʻCʻQ\\ +[*164·1.*150·22] &=CʻP &\qquad \text{(2)}\\ +\vdash .*80·14.\supset \vdash \colon\ldotp \text{Hp}.\supset :N\in F_{\Delta }ʻCʻQ.&x(T\mid N\mid \text{Cnv}ʻT\dagger )R.\supset .\\ +&(\exists y,S).xTy.yFS.R=T^{;}S.S\in CʻQ.\\ +[*33·51 . *37·1] &\supset . (\exists S) . x \in TʻʻCʻS . R = T^{;}S . S \in CʻQ .\\ +[*150·22.*164·1] &\supset .x\in CʻR:\\ +[*33·51] &\supset :N\in F_{\Delta }ʻCʻQ.\supset .T\mid N\mid \text{Cnv}ʻT\dagger \,\unicode{x2abd}\, F &\qquad \text{(3)}\\ +\vdash .(1).(2).(3).*80·14.\supset \\ +\vdash \colon\ldotp \text{Hp}.&\supset :N\in F_{\Delta }ʻCʻQ.\supset .(T\mid N\mid \text{Cnv}ʻT\dagger )\in F_{\Delta }ʻCʻP &\qquad \text{(4)}\\ +\vdash .(4).*43·112.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*172·404.</b> \[\begin{align}\vdash \colon\ldotp T\in P\,\overline{\text{smor smor}}\, Q.\supset :&N\in F_{\Delta }ʻCʻQ.M=T\mid N\mid \text{Cnv}ʻT\dagger .\equiv .\\ +&M\in F_{\Delta }ʻCʻP.N=\breve{T} \mid M\mid \text{Cnv}ʻ\breve{T} \dagger\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*164·1.*162·23.*80·33.\supset \vdash \colon\ldotp \text{Hp}.\supset :N\in F_{\Delta }ʻCʻQ.\supset .\text{D}ʻN\subset \text{ᗡ}ʻT.\\ +[*71·191.*50·63] \supset .\breve{T} \mid T\mid N=N:\\ +[*34·28]\supset :N\in F_{\Delta }ʻCʻQ.M=T\mid N\mid \text{Cnv}ʻT\dagger .\supset .\breve{T} \mid M=N\mid \text{Cnv}ʻT\dagger .\\ +[*34·27] \supset .\breve{T} \mid M\mid \text{Cnv}ʻ\breve{T} \dagger =N\mid \text{Cnv}ʻT\dagger \mid \text{Cnv}ʻ\breve{T} \dagger &\qquad \text{(1)}\\ +\vdash .*80·14. \supset \vdash :N\in F_{\Delta }ʻCʻQ.S\in \text{ᗡ}ʻN.\supset .S\in CʻQ.\\ +[*40·13] \supset .CʻS\subset sʻCʻʻCʻQ &\qquad \text{(2)}\\ +\vdash .(2).*164·1.\supset \vdash \colon\ldotp \text{Hp}.\supset :N\in F_{\Delta }ʻCʻQ.S\in \text{ᗡ}ʻN.\supset .CʻS\subset \text{ᗡ}ʻT &\qquad \text{(3)}\\ +\vdash .(1).*150·1.\supset \vdash \colon\ldotp \text{Hp}.N\in F_{\Delta }ʻCʻQ.M=T\mid N\mid \text{Cnv}ʻT\dagger .\supset :\\ +y(\breve{T} \mid M\mid \text{Cnv}ʻ\breve{T} \dagger )Y.\equiv .(\exists S,R).yNS.R=T^{;}S.Y=\breve{T} ^{;}R.\\ +[(3).*151·25]\equiv .(\exists S,R).yNS.R=T^{;}S.Y=S.\\ +[*13·19·195] \equiv .yNS &\qquad \text{(4)}\\ +\vdash .(4).*172·403.\supset \vdash \colon\ldotp \text{Hp}.\supset :N\in F_{\Delta }ʻCʻQ.M=T\mid N\mid \text{Cnv}ʻT\dagger .\supset .\\ +M\in F_{\Delta }ʻCʻP.N=\breve{T} \mid M\mid \text{Cnv}ʻ\breve{T} \dagger &\qquad \text{(5)}\\ +\vdash .(5) \frac{\breve{T},\,Q,\,P}{T,\,P,\,Q}.*164·21.\supset \vdash \colon\ldotp \text{Hp}.\supset :M\in F_{\Delta }ʻCʻP.N=\breve{T} + \mid M\mid \text{Cnv}ʻ\breve{T} \dagger .\supset .\\ +N\in F_{\Delta }ʻCʻQ.M=T\mid N\mid \text{Cnv}ʻT\dagger &\qquad \text{(6)}\\ +\vdash .(5).(6).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_441">[Pg 441]</span></p> + +<p class="nind"><b>*172·41.</b> \(\vdash :T\in P\,\overline{\text{smor smor}}\, Q.\supset .F_{\Delta }ʻCʻP=(T\parallel \text{Cnv}ʻT\dagger )ʻʻF_{\Delta }ʻCʻQ\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*172·404.*43·112. &\supset \vdash \colon\ldotp \text{Hp}.\supset :M\in F_{\Delta }ʻCʻP.\supset .\\ +&\breve{T} \mid M\mid \text{Cnv}ʻ\breve{T} \dagger \in F_{\Delta }ʻCʻQ.M=(T\parallel \text{Cnv}ʻT\dagger )ʻ(\breve{T} \mid M\mid \text{Cnv}ʻ\breve{T} \dagger ).\\ +[*37·6] &\supset .M\in (T\parallel \text{Cnv}ʻT\dagger )ʻʻF_{\Delta }ʻCʻQ &\qquad \text{(1)}\\ +\vdash .(1).*172·403.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p>The following proposition is important, since it gives the required +correlator of \(\Pi ʻP\) with \(\Pi ʻQ\).</p> + +<p class="nind"><b>*172·42.</b> \(\vdash :T\in P\,\overline{\text{smor smor}}\, Q.\supset .(T\mid \text{Cnv}ʻT\dagger )\upharpoonright Cʻ\Pi ʻQ\in (\Pi ʻP)\,\overline{\text{smor}}\,(\Pi ʻQ)\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*164·1.*150·22.\supset \vdash \colon\ldotp \text{Hp}.&\supset :CʻP=T\dagger ʻʻCʻQ:\\ +[*37·6.*150·1] &\supset :R\in CʻP.\equiv .(\exists S).S\in CʻQ.R=T^{;}S &\qquad \text{(1)}\\ +\vdash .*164·1. &\supset \vdash \colon\ldotp \text{Hp}.\supset :RʻPR.\equiv .(\exists Sʻ,Y).Rʻ=T^{;}Sʻ.R=T^{;}Y.SʻQY &\qquad \text{(2)}\\ +\vdash .*151·31.*164·1.&\supset \vdash \colon\colon\text{Hp}.\supset \colon\ldotp S,Y\in CʻQ.R=T^{;}S.R=T^{;}Y.\supset .S=Y\colon\ldotp &\qquad \text{(3)}\\ +[*13·13] &\supset \colon\ldotp S\in CʻQ.R=T^{;}S.\supset :Y\in CʻQ.R=T^{;}Y.\equiv .S=Y &\qquad \text{(4)}\\ +\vdash .(2).(4).*33·17.\supset \\ +\vdash \colon\ldotp \text{Hp}.S\in CʻQ.R=T^{;}S.\supset :RʻPR.&\equiv .(\exists Sʻ,Y).Rʻ=T^{;}Sʻ.S=Y.SʻQY.\\ +[*13·195] &\equiv .(\exists Sʻ).Rʻ=T^{;}Sʻ.SʻQS &\qquad \text{(5)}\\ +\vdash .*150·4.*172·11.&\supset \vdash \colon\colon\ldotp \text{Hp}.\supset \colon\colon M\{(T\parallel \text{Cnv}ʻT\dagger )^{;}\Pi ʻQ\}Mʻ.\equiv \colon\ldotp \\ +&(\exists N,Nʻ)\colon\ldotp M=(T\parallel \text{Cnv}ʻT\dagger )ʻN.Mʻ=(T\parallel \text{Cnv}ʻT\dagger )ʻNʻ.N,Nʻ\in F_{\Delta }ʻCʻQ\colon\ldotp \\ +&(\exists S):S\in CʻQ.(NʻS)S(NʻʻS):SʻQS.Sʻ\neq S.\supset _{Sʻ}.NʻSʻ=NʻʻSʻ\colon\ldotp \\ +[*172·41·402] &\equiv \colon\ldotp M,Mʻ\in F_{\Delta }ʻCʻP\colon\ldotp (\exists S,R):S\in CʻQ.R=T^{;}S.(MʻR)R(MʻʻR):\\ +&SʻQS.Rʻ=T^{;}S.Sʻ\neq S.\supset _{Sʻ,Rʻ}.MʻRʻ= MʻʻRʻ\colon\ldotp \\ +[*10·23.(3).(5)] &\equiv \colon\ldotp M,Mʻ\in F_{\Delta }ʻCʻP\colon\ldotp (\exists S,R):S\in CʻQ.R=T^{;}S.(MʻR)R(MʻʻR):\\ +&Rʻ\neq R.RʻPR.\supset _{Rʻ}.MʻRʻ=MʻʻRʻ\colon\ldotp \\ +[(1).*172·11] &\equiv \colon\ldotp M(\Pi ʻP)Mʻ &\qquad \text{(6)}\\ +\vdash .(6).*43·302.*172·4.*151·22.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p>The following proposition is a lemma for <a href="#*172·43">*172·43</a>.</p> + +<p class="nind"><b>*172·421.</b> \[\begin{align}\vdash :S=T\upharpoonright Cʻ\Sigma ʻQ.S&\in P\,\overline{\text{smor smor}}\, Q.\supset .\\ +&(S\parallel \text{Cnv}ʻS\dagger )\upharpoonright F_{\Delta }ʻCʻQ=(T\parallel \text{Cnv}ʻT\dagger )\upharpoonright F_{\Delta }ʻCʻQ\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*80·33.*164·18.*162·23.&\supset \vdash \colon\ldotp \text{Hp}.N\in F_{\Delta }ʻCʻQ.\supset .\text{D}ʻN\subset Cʻ\Sigma ʻQ.\\ +[*35·481] & \supset .T\mid N=S\mid N &\qquad \text{(1)}\\ +\vdash .*80·14.&\supset \vdash \colon\ldotp \text{Hp}.N\in F_{\Delta }ʻCʻQ.Y\in \text{ᗡ}ʻN.\supset :Y\in CʻQ:\\ +[*150·33.*162·22] &\supset :X=T^{;}Y.\equiv .X=S^{;}Y.\\ +[*150·1] &\supset :Y(\text{Cnv}ʻT\dagger )X.\equiv .Y(\text{Cnv}ʻS\dagger )X &\qquad \text{(2)}\\ +\vdash .(2).*33·14. &\supset \vdash \colon\ldotp \text{Hp}.N\in F_{\Delta }ʻCʻQ.\supset :\\ +&(\exists Y).yNY.Y(\text{Cnv}ʻT\dagger )X.\equiv .(\exists Y).yNY.Y(\text{Cnv}ʻS\dagger )X:\\ +[*34·1] &\supset :N\mid \text{Cnv}ʻT\dagger =N\mid \text{Cnv}ʻS\dagger &\qquad \text{(3)}\\ +\vdash .(1).(3). &\supset \vdash \colon\ldotp \text{Hp}.\supset :N\in F_{\Delta }ʻCʻQ.\supset .T\mid N\mid \text{Cnv}ʻT\dagger =S\mid N\mid \text{Cnv}ʻS\dagger :\\ +[*43·112.*35·71] &\supset :(T\parallel \text{Cnv}ʻT\dagger )\upharpoonright F_{\Delta }ʻCʻQ=(S\parallel \text{Cnv}ʻS\dagger )\upharpoonright F_{\Delta }ʻCʻQ\colon\ldotp + \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_442">[Pg 442]</span></p> + +<p class="nind"><b><a id="*172·43">*172·43</a>.</b> \[\begin{align}\vdash :T\upharpoonright Cʻ\Sigma ʻQ\in &P\,\overline{\text{smor smor}}\,Q.\supset .\\ +&(T\parallel \text{Cnv}ʻT\dagger )\upharpoonright Cʻ\Pi ʻQ\in (\Pi ʻP)\overline{\text{smor}} (\Pi ʻQ)\\ +[*172·42·421]\end{align}\]</p> + +<p class="nind"><b>*172·44.</b> \(\vdash :P\text{ sm }\text{smor}Q.\supset .\Pi ʻP\,\text{smor}\,\Pi ʻQ \quad[*172·42]\)</p> + +<p class="nind"><b>*172·45.</b> \[\begin{align}\vdash \colon\ldotp \text{Mult ax}.\supset :P,Q\in \text{Rel}^{2}\text{excl}.\exists !P\,\overline{\text{smor}}\,Q\cap &\text{Rl}ʻ\text{smor}.\supset .\\ +&\Pi ʻP\text{smor}\Pi ʻQ\\ +[*164·44 . *172·44]\end{align}\]</p> + +<p>The following proposition shows that if two relations have the same +field, and if the parts of them that are contained in diversity +are the same, they have the same product. Thus <i>e.g.</i> +\(\PiʻP_{\text{po}}=\Pi ʻP_{*}\), in virtue of *91·541.</p> + +<p class="nind"><b>*172·5.</b> \(\vdash :CʻP=CʻQ.P\dot{\cap} J=Q\dot{\cap} J.\supset .\Pi ʻP=\Pi ʻQ\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*50·11.\supset \vdash \colon\ldotp \text{Hp}.\supset :RPS.R\neq S.\equiv .RQS.R\neq S &\qquad \text{(1)}\\ +\vdash .(1).*172·11.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p>The following proposition is used in <a href="#*182·42">*182·42</a>.</p> + +<p class="nind"><b><a id="*172·51">*172·51</a>.</b> \(\vdash .\Pi ʻP=\Pi ʻ(P\unicode{x228d} I\upharpoonright CʻP) \quad[*172·5]\)</p> + +<p class="nind"><b>*172·52.</b> \(\vdash \colon\ldotp Q\in \text{ᗡ}ʻP.\supset _{Q}.(\exists R).RPQ.R\neq Q:\supset .\Pi ʻP=\Pi ʻ(P\dot{\cap} J)\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*50·11. &\supset \vdash :\text{Hp}.\supset .\text{ᗡ}ʻP\subset \text{ᗡ}ʻP\subset \text{ᗡ}ʻ(P\dot{\cap} J) &\qquad \text{(1)}\\ +\vdash .*33·14.*93·101.\text{Transp}.&\supset \vdash :QPQ.\supset .Q{\sim}\in \overrightarrow{B}ʻP:\\ +[\text{Transp}.*33·13] &\supset \vdash :Q\in \overrightarrow{B}ʻP.\supset .(\exists R).QPR.R\neq Q.\\ +[*50·11] &\supset .Q\in Cʻ(P\dot{\cap} J) &\qquad \text{(2)}\\ +\vdash .(1).(2).*93·103. &\supset \vdash :\text{Hp}.\supset .CʻP\subset Cʻ(P\dot{\cap} J).\\ +[*33·265] &\supset .CʻP=Cʻ(P\dot{\cap} J) &\qquad \text{(3)}\\ +\vdash .(3).*172·5.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p>Thus we shall always have \(\Pi ʻP=\Pi ʻ(P\dot{\cap} J)\) unless there +are members of \(\text{ᗡ}ʻP\) which have no referent except themselves.</p> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_443">[Pg 443]</span></p> +<h2 class="nobreak" id="*173"> +*173. THE PRODUCT OF THE RELATIONS OF A FIELD (continued).</h2> +</div> + + +<p class="nind"><i>Summary of</i> *173.</p> + +<p>In this number, we shall consider the relation between the +domains of relations related by \(\Pi ʻP\), <i>i.e.</i> we shall +consider \(\text{D}^{;}\Pi ʻP\). This relation bears to \(\Pi ʻP\) a +relation analogous to that which \(\text{Prod}ʻ\kappa\) bears to +\({\in}_{\Delta}ʻ\kappa\). We shall denote it by "\(\text{Prod}ʻP\)." +When \(P\in \text{Rel}^{2}\text{excl}\), \(\text{Prod}ʻP\) is like +\(\Pi ʻP\), and is often more convenient than \(\Pi ʻP\). When +\(P\in \text{Rel}^{2}\text{excl}\), \(\text{Prod}ʻP\) arranges the +multiplicative class of \(CʻʻCʻP\) by first differences, taking first +differences to mean that the earliest member \(Q\) of \(CʻP\) for which +\(\mu \cap CʻQ\neq \nu \cap CʻQ\) has the \(\mu\)-member earlier than +the \(\nu\)-member in the \(Q\)-series.</p> + +<p>The properties of \(\text{Prod}ʻP\) all result immediately from those +of \(\Pi ʻP\), and offer no difficulty of any kind. The most important +of them are:</p> + +<p class="nind"><b>*173·14.</b> \(\vdash :\dot{\exists} !P.C\upharpoonright CʻP\in 1\rightarrow 1.\supset .Cʻ\text{Prod}ʻP=\text{Prod}ʻCʻʻCʻP\)</p> + +<p><i>I.e.</i> if \(P\) is not null, and no two members of \(CʻP\) have +the same field, then the field of \(\text{Prod}ʻP\) is the product +of the fields of \(CʻP\). Observe that \(C\upharpoonright CʻP\in 1\rightarrow 1\) +if \(P\in \text{Rel}^{2}\text{excl}\).</p> + +<p class="nind"><b>*173·16.</b> \[\begin{align}\vdash :P\in &\text{Rel}^{2}\text{excl}.\supset .\\ +&\text{Prod}ʻP\text{smor}\Pi ʻP.\text{D}\upharpoonright Cʻ\Pi ʻP\in (\text{Prod}ʻP)\,\overline{\text{smor}}\,(\Pi ʻP)\end{align}\]</p> + +<p class="nind"><b>*173·2.</b> \(\vdash .\text{Prod}ʻ\dot{\Lambda}=\dot{\Lambda}\)</p> + +<p class="nind"><b>*173·22.</b> \(\vdash .\text{Prod}ʻ(P\downarrow P)=\iota ^{;}P\)</p> + +<p class="nind"><b>*173·23.</b> \(\vdash :P\neq Q.\supset .\text{Prod}ʻ(P\downarrow Q)=C^{;}(P\times Q)\)</p> + +<p class="nind"><b>*173·3.</b> \[\begin{align}\vdash :T\upharpoonright Cʻ\Sigma ʻQ\in &P\,\overline{\text{smor smor}}\,Q.\supset .\\ +&T_{\in }\upharpoonright Cʻ\text{Prod}ʻQ\in (\text{Prod}ʻP)\,\overline{\text{smor}}\,(\text{Prod}ʻQ)\end{align}\]</p> + +<p class="nind"><b>*173·31.</b> \(\vdash :P\,\text{smor smor}\,Q.\supset .\text{Prod}ʻP\,\text{smor}\,\text{Prod}ʻQ\)</p> + +<hr class="tb"> + +<p class="nind"><b>*173·01.</b> \(\text{Prod}ʻP=\text{D}^{;}\Pi ʻP \quad\text{Df}\)</p> + +<p class="nind"><b>*173·1.</b> \(\vdash .\text{Prod}ʻP=\text{D}^{;}\Pi ʻP \quad[(*175·01)]\)</p> + +<p class="nind"><b>*173·11.</b> \[\begin{align}&\vdash :\mu (\text{Prod}ʻP)\nu .\equiv .(\exists M,N).M(\Pi ʻP)N.\mu =\text{D}ʻM.\nu =\text{D}ʻN\\ +&[*173·1.*150·51]\end{align}\]</p> + +<p><span class="pagenum" id="Page_444">[Pg 444]</span></p> + +<p class="nind"><b>*173·12.</b> \(\vdash .Cʻ\text{Prod}ʻP\subset \text{D}ʻʻF_{\Delta }ʻCʻP \quad[*172·12.*150·202]\)</p> + +<p class="nind"><b>*173·121.</b> \(\vdash .Cʻ\text{Prod}ʻP=\text{D}ʻʻCʻ\Pi ʻP \quad[*173·1.*150·22]\)</p> + +<p class="nind"><b>*173·13.</b> \(\vdash :\dot{\exists} !P.\supset .Cʻ\text{Prod}ʻP=\text{D}ʻʻF_{\Delta }ʻCʻP \quad[*172·17.*173·121]\)</p> + +<p class="nind"><b>*173·14.</b> \(\vdash :\dot{\exists} !P.C\upharpoonright CʻP\in 1\rightarrow 1.\supset .Cʻ\text{Prod}ʻP=\text{Prod}ʻCʻʻCʻP\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*85·12.*33·5.\supset \vdash :\text{Hp}.&\supset .\text{D}ʻʻF_{\Delta }ʻCʻP =\text{D}ʻʻ{\in}_{\Delta}ʻCʻʻCʻP.\\ +[*173·13.*115·1] &\supset .Cʻ\text{Prod}ʻP=\text{Prod}ʻCʻʻCʻP:\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*173·15.</b> \[\begin{align}&\vdash :\text{D}\upharpoonright F_{\Delta }ʻCʻP\in + 1\rightarrow 1.\supset .\text{D}\upharpoonright Cʻ\Pi ʻP\in (\text{Prod}ʻP)\,\overline{\text{smor}}\,(\Pi ʻP)\\ +&[*173·1.*172·12.*151·231]\end{align}\]</p> + +<p class="nind"><b>*173·151.</b> \(\vdash :\text{D}\upharpoonright F_{\Delta }ʻCʻP\in 1\rightarrow 1.\supset .\text{Prod}ʻP\,\text{smor}\,\Pi ʻP \quad [*173·15]\)</p> + +<p class="nind"><b>*173·16.</b> \[\begin{align}\vdash :P\in &\text{Rel}^{2}\text{excl}.\supset .\\ +&\text{Prod}ʻP\,\text{smor}\,\Pi ʻP.\text{D}\upharpoonright Cʻ\Pi ʻP\in (\text{Prod}ʻP)\,\overline{\text{smor}}\,(\Pi ʻP)\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*163·12.\supset \vdash :\text{Hp}.&\supset .F\upharpoonright CʻP\in \text{Cls}\rightarrow 1.\\ +[*81·21] &\supset .\text{D}\upharpoonright F_{\Delta }ʻCʻP\in 1\rightarrow 1 &\qquad \text{(1)}\\ +\vdash .(1).*173·151·15.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*173·161.</b> \[\begin{align}&\vdash :P\in \text{Rel}^{2}\text{excl}.\dot{\exists} !P.\supset .Cʻ\text{Prod}ʻP=\text{Prod}ʻCʻʻCʻP\\ +&[*173·14.*163·14]\end{align}\]</p> + +<p class="nind"><b>*173·17.</b> \(\vdash :\dot{\exists} !\text{Prod}ʻP.\supset .sʻCʻ\text{Prod}ʻP=Cʻ\Sigma ʻP\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*173·13. \supset \vdash :\text{Hp}.\supset .sʻCʻ\text{Prod}ʻP&=sʻ\text{D}ʻʻF_{\Delta }ʻCʻP\\ +[*41·43.*80·42] &=\text{D}ʻF\upharpoonright CʻP\\ +[*37·401.*162·23] &=Cʻ\Sigma ʻP:\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*173·2.</b> \(\vdash .\text{Prod}ʻ\dot{\Lambda} =\dot{\Lambda} \quad[*172·13.*150·42]\)</p> + +<p class="nind"><b>*173·21.</b> \(\vdash :\dot{\exists} !\text{Prod}ʻP.\equiv .\dot{\exists} !\Pi ʻP \quad[*173·1.*150·24.*33·12]\)</p> + +<p class="nind"><b><a id="*173·22">*173·22</a>.</b> \(\vdash .\text{Prod}ʻ(P\downarrow P)=\iota ^{;}P\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*172·2. \supset \vdash .\text{Prod}ʻ(P\downarrow P)&=\text{D}^{;}\downarrow P^{;}P\\ +[*150·4] &=\hat{\mu} \hat{\nu} \{(\exists x,y).xPy.\mu =\text{D}ʻ(x\downarrow P).\nu =\text{D}ʻ(y\downarrow P)\}\\ +[*55·16] &=\hat{\mu} \hat{\nu} \{(\exists x,y).xPy.\mu =\iota ʻx.\nu =\iota ʻy\}\\ +[*150·4] &=\iota ^{;}P.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*173·23.</b> \(\vdash :P\neq Q.\supset .\text{Prod}ʻ(P\downarrow Q)=C^{;}(P\times Q)\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*172·21. &\supset \vdash :\text{Hp}.\supset .C^{;}(P\times Q)=C^{;}\dagger (Q\downarrow P)^{;}\Pi ʻ(P\downarrow Q) &\qquad \text{(1)}\\ +\vdash .*80·14.*150·23. &\supset \vdash :M\in F_{\Delta }ʻCʻ(Q\downarrow P).\supset .CʻM^{;}(Q\downarrow P)=\text{D}ʻM:\\ +[*55·15.*150·1] &\supset \vdash :M\in F_{\Delta }ʻCʻ(P\downarrow Q).\supset .Cʻ\dagger (Q\downarrow P)ʻM=\text{D}ʻM:\\ +[*172·12] & \supset \vdash :M\in Cʻ\Pi ʻ(P\downarrow Q).\supset .Cʻ\dagger (Q\downarrow P)ʻM=\text{D}ʻM:\\ +[*150·35] &\supset \vdash .C^{;}\dagger (Q\downarrow P)^{;}\Pi ʻ(P\downarrow Q)=\text{D}^{;}\Pi ʻ(P\downarrow Q) &\qquad \text{(2)}\\ +\vdash .(1).(2).*173·1.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_445">[Pg 445]</span></p> + +<p class="nind"><b>*173·24.</b> \[\begin{align}\vdash :CʻP\cap CʻQ=\Lambda.\supset .C\upharpoonright Cʻ(P\times Q)\in &\{\text{Prod}ʻ(P\downarrow Q)\}\,\overline{\text{smor}}\,(P\times Q).\\ +&\text{Prod}ʻ(P\downarrow Q)\,\text{smor}\,P\times Q\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*166·12. &\supset \vdash .C\upharpoonright Cʻ(P\times Q)=C\upharpoonright (CʻP\times CʻQ) &\qquad \text{(1)}\\ +\vdash .(1).*113·148.&\supset \vdash :\text{Hp}.\supset .C\upharpoonright Cʻ(P\times Q)\in 1\rightarrow 1 &\qquad \text{(2)}\\ +\vdash .(2).*173·23.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*173·25">*173·25</a>.</b> \[\begin{align}&\vdash :P\in \text{Rel}^{2}\text{excl}.Z{\sim}\in CʻP.CʻZ\cap Cʻ\Sigma ʻP=\Lambda.\supset .\\ +&\text{Prod}ʻ(P\unicode{x21f8} Z)\,\text{smor}\,(\text{Prod}ʻP\times Z).\text{Prod}ʻ(Z\unicode{x21f7}P)\,\text{smor}\,(Z\times \text{Prod}ʻP)\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*163·451.\supset \vdash :\text{Hp}.&\supset .P\unicode{x21f8} Z\in \text{Rel}^{2}\text{excl}.\\ +[*173·16] &\supset .\text{Prod}ʻ(P\unicode{x21f8} Z)\,\text{smor}\,\Pi ʻ(P\unicode{x21f8} Z).\\ +[*172·32] &\supset .\text{Prod}ʻ(P\unicode{x21f8} Z)\,\text{smor}\,\Pi ʻPxZ.\\ +[*173·16.*166·23] &\supset .\text{Prod}ʻ(P\unicode{x21f8} Z)\,\text{smor}\,\text{Prod}ʻP\times Z &\qquad \text{(1)}\\ +\text{Similarly}\quad \vdash :\text{Hp}.&\supset .\text{Prod}ʻ(Z\unicode{x21f7}P)\,\text{smor}\,Z\times \text{Prod}ʻP &\qquad \text{(2)}\\ +\vdash .(1).(2).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*173·26">*173·26</a>.</b> \[\begin{align}\vdash :P,Q\in \text{Rel}^{2}\text{excl}.\dot{\exists} !P.\dot{\exists} !Q.&CʻP\cap CʻQ=\Lambda.Cʻ\Sigma ʻP\cap Cʻ\Sigma ʻQ=\Lambda.\supset .\\ +&\text{Prod}ʻ(P\unicode{x2909}Q)\text{smor}\text{Prod}ʻP\times \text{Prod}ʻQ\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*163·441.*173·16.\supset \vdash :\text{Hp}.&\supset .\text{Prod}ʻ(P\unicode{x2909}Q)\text{smor}\Pi ʻ(P\unicode{x2909}Q).\\ +[*172·35] &\supset .\text{Prod}ʻ(P\unicode{x2909}Q)\text{smor}\Pi ʻP\times \Pi ʻQ.\\ +[*173·16.*166·23]&\supset .\text{Prod}ʻ(P\unicode{x2909}Q)\text{smor}\text{Prod}ʻP\times \text{Prod}ʻQ:\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*173·27.</b> \[\begin{align}\vdash :CʻP\cap &CʻQ=\Lambda.CʻP\cap CʻR=\Lambda.CʻQ\cap CʻR=\Lambda.\supset .\\ +&\text{Prod}ʻ\{(P\downarrow Q)\unicode{x21f8} R\}\,\text{smor}\,P\times Q\times R\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*173·25.\supset \vdash :\text{Hp}.R\neq P.R\neq Q.\supset .\\ +&\text{Prod}ʻ\{(P\downarrow Q)\unicode{x21f8} R\}\,\text{smor}\,{\text{Prod}ʻ(P\downarrow Q)}\times R.\\ +[*173·24] &\supset .\text{Prod}ʻ\{(P\downarrow Q)\unicode{x21f8} R\}\,\text{smor}\,P\times Q\times R &\qquad \text{(1)}\\ +\vdash .*33·241.\supset \vdash :\text{Hp}.R=P.&\supset .R=\dot{\Lambda} .P=\dot{\Lambda} .\\ +[*172·14.*166·13] &\supset .\Pi ʻ\{(P\downarrow Q)\unicode{x21f8} R\}=\dot{\Lambda} .P\times Q\times R=\dot{\Lambda} .\\ +[*173·1.*150·42] &\supset .\text{Prod}ʻ\{(P\downarrow Q)\unicode{x21f8} R\}=\dot{\Lambda} .P\times Q\times R=\dot{\Lambda} .\\ +[*153·101] &\supset .\text{Prod}ʻ\{(P\downarrow Q)\unicode{x21f8} R\}\text{smor}(P\times Q\times R) &\qquad \text{(2)}\\ +\text{Similarly}\quad \vdash :\text{Hp}.R=Q.&\supset .\text{Prod}ʻ\{(P\downarrow Q)\unicode{x21f8} R\}\text{smor}(P\times Q\times R) &\qquad \text{(3)}\\ +\vdash .(1).(2).(3).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_446">[Pg 446]</span></p> + +<p>The following proposition gives a correlator of \(\text{Prod}ʻP\) and +\(\text{Prod}ʻQ\) when we are given a double correlator of \(P\) and +\(Q\).</p> + +<p class="nind"><b>*173·3.</b> \[\begin{align}\vdash :T\upharpoonright Cʻ\Sigma ʻQ\in P\,\overline{\text{smor smor}}\, &Q.\supset .\\ +&T\in \upharpoonright Cʻ\text{Prod}ʻQ\in (\text{Prod}ʻP)\,\overline{\text{smor}}\,(\text{Prod}ʻQ)\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*173·11.*172·43.\supset \\ +\vdash \colon\ldotp \text{Hp}.\supset :\mu (\text{Prod}ʻP)\mu'.\equiv .(\in N,N').N(\Pi ʻQ)N'.\mu &=\text{D}ʻ(T\mid N\mid \text{Cnv}ʻT\dagger ).\\ +\mu'&=\text{D}ʻ(T\mid N'\mid \text{Cnv}ʻT\dagger) .\\ +[*37·32·321]&\equiv .(\exists N,N').N(\Pi ʻQ)N'.\mu =Tʻʻ\text{D}ʻN.\mu '=Tʻʻ\text{D}ʻN'.\\ +[*173·11] &\equiv .(\exists \nu ,\nu').\nu (\text{Prod}ʻQ)\nu'.\mu =Tʻʻ\nu .\mu'=Tʻʻ\nu'.\\ +[*37·101] &\equiv .\mu (T_{\in }^{;}\text{Prod}ʻQ)\mu' &\qquad \text{(1)}\\ +\vdash .*173·17.&\supset \vdash .sʻCʻ\text{Prod}ʻQ\subset Cʻ\Sigma ʻQ &\qquad \text{(2)}\\ +[*111·12] &\supset \vdash .(T\upharpoonright Cʻ\Sigma ʻQ)_\in \upharpoonright Cʻ\text{Prod}ʻQ=T_{\in }\upharpoonright Cʻ\text{Prod}ʻQ &\qquad \text{(3)}\\ +\vdash .(2).(3).*72·451.&\supset \vdash :\text{Hp}.\supset .T_{\in }\upharpoonright Cʻ\text{Prod}ʻQ\in 1\rightarrow 1 &\qquad \text{(4)}\\ +\vdash .(1).(4).*151·231.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*173·31.</b> \(\vdash :P\,\text{smor smor}\,Q.\supset .\text{Prod}ʻP\,\text{smor}\,\text{Prod}ʻQ \quad[*173·3]\)</p> + +<p class="nind"><b>*173·32.</b> \(\vdash :R\upharpoonright Cʻ\Sigma ʻQ\in 1\rightarrow 1.Cʻ\Sigma ʻQ\subset \text{ᗡ}ʻR.\supset .\text{Prod}ʻR\dagger ^{;}Q=R_\in ^{;}\text{Prod}ʻQ\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*164·18.\supset \vdash :\text{Hp}.&\supset .R\upharpoonright Cʻ\Sigma ʻQ\in (R\dagger ^{;}Q)\,\overline{\text{smor smor}}\, Q.\\ +[*173·3] &\supset .R_{\in }\upharpoonright Cʻ\text{Prod}ʻQ\in (\text{Prod}ʻR\dagger ^{;}Q)\,\overline{\text{smor}}\,(\text{Prod}ʻQ).\\ +[*151·22] &\supset .\text{Prod}ʻR\dagger ^{;}Q=R_{\in }^{;}\text{Prod}ʻQ:\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*173·33.</b> \(\vdash :\text{D}\upharpoonright Cʻ\Sigma ʻQ\in 1\rightarrow 1.\supset .\text{Prod}ʻ\text{D}\dagger ^{;}Q=\text{D}_{\in }^{;}\text{Prod}ʻQ + \quad\left[*173·32 \frac{\text{D}}{R}\right].\)</p> + +<p>The above proposition is used in proving the associative law for +"\(\text{Prod}\)" (<a href="#*174·401">*174·401</a>).</p> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_447">[Pg 447]</span></p> +<h2 class="nobreak" id="*174"> +*174. THE ASSOCIATIVE LAW OF RELATIONAL MULTIPLICATION.</h2> +</div> + + +<p class="nind"><i>Summary of</i> *174.</p> + +<p>In the present number, we have to prove the associative law for \(\Pi\) +and for \(\text{Prod}\), <i>i.e.</i> we have to prove (with a suitable +hypothesis) +\[ +\Pi ʻ\Pi ^{;}P\text{smor}\Pi ʻ\Sigma ʻP +\] +and \(\text{Prod}ʻ\text{Prod}^{;}P\text{smor}\text{Prod}ʻ\Sigma ʻP\).</p> + +<p>The first of these requires \(P\in \text{Rel}^{2}\text{excl}\) and either +\(P\,\unicode{x2abd}\, J\) or +\[ +QPQ.\supset _{Q}.CʻQ\in 0\cup 1; +\] +the second requires not only this, but also +\(\Sigma ʻP\in\text{Rel}^{2}\text{excl}\). When both \(P\) and +\(\SigmaʻP\) are relations of mutually exclusive relations, we call +\(P\) an <i>arithmetical</i> relation, which we denote by +"\(\text{Rel}^{3}\text{arithm}\)." Arithmetical relations serve exactly +analogous purposes to those served by arithmetical classes in cardinal +arithmetic.</p> + +<p>The proof of the associative law for \(\Pi\) consists in showing that, +under a suitable hypothesis, \(\dot{s} \mid \text{D}\) (with its +converse domain limited) is a correlator of \(\Pi ʻ\Sigma ʻP\) and +\(\Pi ʻ\Pi ^{;}P\) (<a href="#*174·221">*174·221</a><a href="#*174·23">·23</a>). To prove this, we first prove</p> + +<p class="nind"><b>*174·17.</b> \(\vdash :P\in \text{Rel}^{2}\text{excl}.\supset .\dot{s}ʻʻ\text{D}ʻʻCʻ\Pi ʻ\Pi ^{;}P=Cʻ\Pi ʻ\Sigma ʻP\)</p> + +<p class="nind">and</p> + +<p class="nind"><b>*174·19.</b> \(\vdash :P\in \text{Rel}^{2}\text{excl}.\supset .(\dot{s} \mid \text{D})\upharpoonright Cʻ\Pi ʻ\Pi ^{;}P\in 1\rightarrow 1\)</p> + +<p>This gives what we may call the cardinal part of the proof, <i>i.e.</i> +it shows that \((\dot{s} \mid \text{D})\upharpoonright Cʻ\Pi ʻ\Pi ^{;}P\) +is a cardinal correlator of the fields of \(\Pi ʻ\Sigma ʻP\) and \(\Piʻ\Pi ^{;}P\). +We then prove that if \(M\) and \(N\) belong to the field +of \(\Pi ʻ\Pi ^{;}P\), they have the relation \(\Pi ʻ\Pi ^{;}P\) when the +relational sums of their domains have the relation \(\Pi ʻ\Sigma ʻP\). +Here, in addition to the hypothesis \(P\in \text{Rel}^{2}\text{excl}\), +we require that if any relation \(Q\) has the relation \(P\) to itself, +then \(CʻQ\) is not to have more than one term. Thus we have</p> + +<p class="nind"><b>*174·215.</b> \[\begin{align}&\vdash \colon\ldotp P\in \text{Rel}^{2}\text{excl}:QPQ.\supset _{Q}.CʻQ\in 0\cup 1:\supset :\\ +&M(\Pi ʻ\Pi ^{;}P)N.\equiv .M,N\in F_{\Delta }ʻ\Pi ʻʻCʻP.(\dot{s} ʻ\text{D}ʻM)(\Pi ʻ\Sigma ʻP)(\dot{s} ʻ\text{D}ʻN)\end{align}\]</p> + +<p>The hypothesis \(QPQ.\supset _{Q}.CʻQ\in 0\cup 1\) is verified if +\(P\,\unicode{x2abd}\, J\) (<a href="#*174·216">*174·216</a>); thus for most purposes it is more +convenient to substitute the simpler hypothesis \(P\,\unicode{x2abd}\, J\) +for \(QPQ.\supset _{Q}.CʻQ\in 0\cup 1\). We shall, however, have +occasion to use the hypothesis \(QPQ.\supset _{Q}.CʻQ\in 0\cup 1\) in +<a href="#*182·42">*182·42</a><a href="#*182·43">·43</a><a href="#*182·431">·431</a>, where our \(P\) is a relation whose field consists +entirely of relations of the form \(Q\downarrow Q\), whose<span class="pagenum" id="Page_448">[Pg 448]</span> fields are +always unit classes, so that our \(P\) satisfies the above hypothesis +even if \(P\) is not contained in \(J\).</p> + +<p>The proof of <a href="#*174·215">*174·215</a> (above) is effected by first proving</p> + +<p class="nind"><b>*174·2.</b> \(\vdash :P\in \text{Rel}^{2}\text{excl}.Q\in CʻP.M\in Cʻ\Pi ʻ\Pi ^{;}P.\supset .Mʻ\Pi ʻQ=(\dot{s} ʻ\text{D}ʻM)\upharpoonright CʻQ\)</p> + +<p>From <a href="#*174·17">*174·17</a><a href="#*174·19">·19</a><a href="#*174·215">·215</a> we deduce</p> + +<p class="nind"><b>*174·221.</b> \[\begin{align}&\vdash \colon\ldotp P\in \text{Rel}^{2}\text{excl}:QPQ.\supset _{Q}.CʻQ\in 0\cup 1:\supset .\\ +&\Pi ʻ\Sigma ʻP=\dot{s} ^{;}\text{D}^{;}\Pi ʻ\Pi ^{;}P.(\dot{s} \mid \text{D})\upharpoonright Cʻ\Pi ʻ\Pi ^{;}P\in + (\Pi ʻ\Sigma ʻP)\,\overline{\text{smor}}\,(\Pi ʻ\Pi ^{;}P)\end{align}\]</p> + +<p class="nind">whence we obtain the more convenient proposition</p> + +<p class="nind"><b>*174·23.</b> \[\begin{align}\vdash :&P\in \text{Rel}^{2}\text{excl}.P\,\unicode{x2abd}\, J.\supset .\\ +&\Pi ʻ\Sigma ʻP=\dot{s} ^{;}\text{D}^{;}\Pi ʻ\Pi ^{;}P.(\dot{s} \mid \text{D})\upharpoonright Cʻ\Pi ʻ\Pi ^{;}P\in + (\Pi ʻ\Sigma ʻP)\,\overline{\text{smor}}\,(\Pi ʻ\Pi ^{;}P)\end{align}\]</p> + +<p>Thus if the hypothesis of <a href="#*174·221">*174·221</a> or of <a href="#*174·23">*174·23</a> holds, the associative +law holds for \(\Pi\) (<a href="#*174·241">*174·241</a><a href="#*174·25">·25</a>).</p> + +<p>To prove the associative law for \(\text{Prod}\), <i>i.e.</i> +\[ +P\in \text{Rel}^{3}\text{arithm}.P\,\unicode{x2abd}\, J.\supset .\text{Prod}ʻ\Sigma ʻP \,\text{smor} \text{Prod}ʻ\text{Prod}^{;}P, +\] +we observe that, since \(\Pi ʻ\Sigma ʻP=\dot{s} ^{;}\text{D}^{;}\Pi ʻ\Pi^{;}P\) +(<a href="#*174·23">*174·23</a>) +\[ +=\dot{s} ^{;}\text{Prod}ʻ\Pi ^{;}P,\, \text{by the definition of Prod}, +\] +we have (<a href="#*174·41">*174·41</a>) \(\text{Prod}ʻ\Sigma ʻP=\text{D}^{;}\dot{s}^{;}\text{Prod}ʻ\Pi ^{;}P\) +\[ +\begin{aligned} +&=s^{;}\text{D}_{\in }^{;}\text{Prod}ʻ\Pi ^{;}P, &&\quad\text{by *41·33},\\ +&=s^{;}\text{Prod}ʻ\text{D}\dagger ^{;}\Pi ^{;}P, &&\quad\text{by *173·33},\\ +&=s^{;}\text{Prod}ʻ\text{Prod}^{;}P, &&\quad\text{by the definition of Prod}. +\end{aligned} +\] +Also \(s\upharpoonright Cʻ\text{Prod}ʻ\text{Prod}^{;}P\in 1\rightarrow 1\), +by <a href="#*115·46">*115·46</a>. Hence the associative law follows (<a href="#*174·43">*174·43</a>). It will +be observed that in this case the correlator is simply s with its +converse domain limited (<a href="#*174·42">*174·42</a>).</p> + +<p>As in the case of \(\Pi\), "\(P\,\unicode{x2abd}\, J\)" is a stronger +hypothesis than we really need: what we need is \(QPQ.\supset_{Q}.CʻQ\in 0\cup 1\).</p> + +<hr class="tb"> + +<p class="nind"><b>*174·01.</b> \(\text{Rel}^{3}\text{arithm}=\hat{P} (P,\Sigma ʻP\in \text{Rel}^{2}\text{excl}) \quad\text{Df}\)</p> + +<p class="nind"><b>*174·12.</b> \(\vdash :C\upharpoonright CʻP\in 1\rightarrow 1.\supset .\Pi ^{;}P\in \text{Rel}^{2}\text{excl}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*150·202.\supset \\ +\vdash :M,N\in Cʻ\Pi ^{;}P.\exists !CʻM\cap CʻN.&\supset .M,N\in \Pi ʻʻCʻP.\exists !CʻM\cap CʻN.\\ +[*37·6] &\supset .(\exists Q,R).Q,R\in CʻP.M=\Pi ʻQ.N=\Pi ʻR.\exists !CʻM\cap CʻN.\\ +[*172·12] &\supset .(\exists Q,R).Q,R\in CʻP.M=\Pi ʻQ.N=\Pi ʻR.\exists !F_{\Delta }ʻCʻQ\cap F_{\Delta }ʻCʻR.\\ +[*80·82.\text{Transp}] &\supset .(\exists Q,R).Q,R\in CʻP.M=\Pi ʻQ.N=\Pi ʻR.CʻQ=CʻR &\qquad \text{(1)}\\ +\vdash .(1).*71·59.\supset \vdash \colon\ldotp \text{Hp}.&\supset : +M,N\in Cʻ\Pi ^{;}P.\exists !CʻM\cap CʻN.\supset .(\exists Q,R).Q=R.M=\Pi ʻQ.N=\Pi ʻR.\\ +[*13·195·172] &\supset .M=N &\qquad \text{(2)}\\ +\vdash .(2).*163·11.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_449">[Pg 449]</span></p> + +<p class="nind"><b>*174·13.</b> \(\vdash :P\in \text{Rel}^{2}\text{excl}.\supset .\Pi ^{;}P\in \text{Rel}^{2}\text{excl} \quad[*174·12.*163·14]\)</p> + +<p class="nind"><b>*174·16.</b> \(\vdash :\dot{\exists} !P.\supset .Cʻ\Pi ʻ\Pi ^{;}P=F_{\Delta }ʻ\Pi ʻʻCʻP\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*150·25.\supset \vdash :\text{Hp}.&\supset .\dot{\exists} !\Pi ^{;}P.\\ +[*172·17] \supset .Cʻ\Pi ʻ\Pi ^{;}P&=F_{\Delta }ʻCʻ\Pi ^{;}P\\ +[*150·22] &=F_{\Delta }ʻ\Pi ʻʻCʻP:\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*174·161.</b> \[\begin{align}\vdash :\dot{\exists} !P.P\in &\text{Rel}^{2}\text{excl}.\supset .\\ +&Cʻ\text{Prod}ʻ\Pi ^{;}P= \text{D}ʻʻCʻ\Pi ʻ\Pi ^{;}P=\text{Prod}ʻF_{\Delta }ʻʻCʻʻCʻP\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*173·121. &\supset \vdash .Cʻ\text{Prod}ʻ\Pi ^{;}P=\text{D}ʻʻCʻ\Pi ʻ\Pi ^{;}P &\qquad \text{(1)}\\ +\vdash .*173·161. \supset \vdash :\text{Hp}.\supset .Cʻ\text{Prod}ʻ\Pi ^{;}P&=\text{Prod}ʻCʻʻCʻ\Pi ^{;}P\\ +[*150·22] &=\text{Prod}ʻCʻʻ\Pi ʻʻCʻP &\qquad \text{(2)}\\ +\vdash .*172·17. &\supset \vdash :\dot{\Lambda} {\sim}\in CʻP.\supset .Cʻʻ\Pi ʻʻCʻP=F_{\Delta }ʻʻCʻʻCʻP &\qquad \text{(3)}\\ +\vdash .*172·14.*173·21. &\supset \vdash :\dot{\Lambda} \in CʻP.\supset .Cʻ\text{Prod}ʻ\Pi ^{;}P=\Lambda &\qquad \text{(4)}\\ +\vdash .*80·26.*83·11. &\supset \vdash :\dot{\Lambda} \in CʻP.\supset .\text{Prod}ʻF_{\Delta }ʻʻCʻʻCʻP=\Lambda &\qquad \text{(5)}\\ +\vdash .(2).(3). &\supset \vdash :\text{Hp}.\dot{\Lambda} {\sim}\in CʻP.\supset .Cʻ\text{Prod}ʻ\Pi ^{;}P=\text{Prod}ʻF_{\Delta }ʻʻCʻʻCʻP &\qquad \text{(6)}\\ +\vdash .(4).(5). &\supset \vdash :\text{Hp}.\dot{\Lambda} \in CʻP.\supset .Cʻ\text{Prod}ʻ\Pi ^{;}P=\text{Prod}ʻF_{\Delta }ʻʻCʻʻCʻP &\qquad \text{(7)}\\ +\vdash .(1).(6).(7).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*174·162">*174·162</a>.</b> \(\vdash :\dot{\exists} !P.P\in \text{Rel}^{2}\text{excl}.\supset .\dot{s}ʻʻ\text{D}ʻʻCʻ\Pi ʻ\Pi ^{;}P=Cʻ\Pi ʻ\Sigma ʻP=F_{\Delta }ʻCʻ\Sigma ʻP\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*174·161.*115·1. \supset \vdash :\text{Hp}.\supset .\dot{s}ʻʻ\text{D}ʻʻCʻ\Pi ʻ\Pi ^{;}P&=\dot{s}ʻʻ\text{D}ʻʻ{\in}_{\Delta}ʻF_{\Delta }ʻʻCʻʻCʻP\\ +[*85·27.*163·16] &=F_{\Delta }ʻsʻCʻʻCʻP\\ +[*162·22] & =F_{\Delta }ʻCʻ\Sigma ʻP &\qquad \text{(1)}\\ +\vdash .(1).*172·17. &\supset \vdash :\text{Hp}.\dot{\exists} !\Sigma ʻP.\supset .\dot{s}ʻʻ\text{D}ʻʻCʻ\Pi ʻ\Pi ^{;}P=Cʻ\Pi ʻ\Sigma ʻP &\qquad \text{(2)}\\ +\vdash .*162·45. \supset \vdash :\text{Hp}.\Sigma ʻP=\dot{\Lambda} .&\supset .P=\dot{\Lambda} \downarrow \dot{\Lambda} .\\ +[*172·13.*150·71] &\supset .\Pi ^{;}P=\dot{\Lambda} \downarrow \Lambda.\\ +[*172·14] &\supset .\Pi ʻ\Pi ^{;}P=\dot{\Lambda} .\\ +[*33·241] &\supset .\dot{s}ʻʻ\text{D}ʻʻCʻ\Pi ʻ\Pi ^{;}P=\Lambda &\qquad \text{(3)}\\ +\vdash .*172·13.*33·241.\supset \vdash :\Sigma ʻP=\dot{\Lambda} .&\supset .Cʻ\Pi ʻ\Sigma ʻP=\Lambda &\qquad \text{(4)}\\ +\vdash .(3).(4). &\supset \vdash :\text{Hp}.\Sigma ʻP=\dot{\Lambda} .\supset .\dot{s}ʻʻ\text{D}ʻʻCʻ\Pi ʻ\Pi ^{;}P=Cʻ\Pi ʻ\Sigma ʻP &\qquad \text{(5)}\\ +\vdash .(1).(2).(5).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*174·17">*174·17</a>.</b> \(\vdash :P\in \text{Rel}^{2}\text{excl}.\supset .\dot{s}ʻʻ\text{D}ʻʻCʻ\Pi ʻ\Pi ^{;}P=Cʻ\Pi ʻ\Sigma ʻP\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*150·42.*172·13. &\supset \vdash :P=\dot{\Lambda} .\supset .\dot{s}ʻʻ\text{D}ʻʻCʻ\Pi ʻ\Pi ^{;}P=\Lambda &\qquad \text{(1)}\\ +\vdash .*162·4.*172·13. &\supset \vdash :P=\dot{\Lambda} .\supset .Cʻ\Pi ʻ\Sigma ʻP=\Lambda &\qquad \text{(2)}\\ +\vdash .(1).(2).*174·162.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_450">[Pg 450]</span></p> + +<p class="nind"><b>*174·18.</b> \(\vdash :P\in \text{Rel}^{2}\text{excl}.\supset .\text{D}\upharpoonright Cʻ\Pi ʻ\Pi ^{;}P\in 1\rightarrow 1\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*174·12.*163·14·12.\supset \vdash :\text{Hp}.&\supset .F\upharpoonright Cʻ\Pi ^{;}P\in \text{Cls}\rightarrow 1.\\ +[*81·21] &\supset .\text{D}\upharpoonright F_{\Delta }ʻCʻ\Pi ^{;}P\in 1\rightarrow 1.\\ +[*172·12] &\supset .\text{D}\upharpoonright Cʻ\Pi ʻ\Pi ^{;}P\in 1\rightarrow 1:\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*174·19">*174·19</a>.</b> \(\vdash :P\in \text{Rel}^{2}\text{excl}.\supset .(\dot{s} \mid \text{D})\upharpoonright Cʻ\Pi ʻ\Pi ^{;}P\in 1\rightarrow 1\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*163·1.*35·14.&\supset \vdash \colon\ldotp \text{Hp}.\supset :\\ +Q,R\in CʻP.Q\neq R.&\supset _{Q,R}.F\upharpoonright CʻQ\dot{\cap} F\upharpoonright CʻR=\dot{\Lambda} .\\ +[*172·191] &\supset _{Q,R}.\dot{s} ʻCʻ\Pi ʻQ\dot{\cap} \dot{s} ʻCʻ\Pi ʻR=\dot{\Lambda} &\qquad \text{(1)}\\ +\vdash .(1).*33·5.*85·31 \frac{F,\PiʻʻCʻP}{P,\,\alpha}.\supset \\ +\vdash \colon\ldotp \text{Hp}.&\supset :M,N\in F_{\Delta }ʻ\Pi ʻʻCʻP.\dot{s} ʻ\text{D}ʻM=\dot{s} ʻ\text{D}ʻN.\supset .M=N:\\ +[*172·12.*150·22]&\supset :M,N\in Cʻ\Pi ʻ\Pi ^{;}P.\dot{s} ʻ\text{D}ʻM=\dot{s} ʻ\text{D}ʻN.\supset .M=N\colon\ldotp \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*174·191.</b> \(\vdash :P\in \text{Rel}^{2}\text{excl}.\supset .\dot{s} \upharpoonright Cʻ\text{Prod}ʻ\Pi ^{;}P\in 1\rightarrow 1\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*174·19. \supset \vdash \colon\ldotp \text{Hp}.&\supset :M,N\in Cʻ\Pi ʻ\Pi ^{;}P.\dot{s} ʻ\text{D}ʻM=\dot{s} ʻ\text{D}ʻN.\supset .M=N.\\ +[*30·37] &\supset .\text{D}ʻM=\text{D}ʻN:\\ +[*37·63] &\supset :\mu ,\nu \in \text{D}ʻʻCʻ\Pi ʻ\Pi ^{;}P.\dot{s} ʻ\mu =\dot{s} ʻ\nu .\supset .\mu =\nu :\\ +[*173·121] &\supset :\mu ,\nu \in Cʻ\text{Prod}ʻ\Pi ^{;}P.\dot{s} ʻ\mu =\dot{s} ʻ\nu .\supset .\mu =\nu \colon\ldotp \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*174·2.</b> \(\vdash :P\in \text{Rel}^{2}\text{excl}.Q\in CʻP.M\in Cʻ\Pi ʻ\Pi ^{;}P.\supset .Mʻ\Pi ʻQ=(\dot{s} ʻ\text{D}ʻM)\upharpoonright CʻQ\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*172·12.*150·22.\supset \vdash :\text{Hp}.&\supset .M\in F_{\Delta }ʻ\Pi ʻʻCʻP. &\qquad \text{(1)}\\ +[*80·31.*33·5] &\supset .Mʻ\Pi ʻQ\in Cʻʻ\Pi ʻQ.\\ +[*172·12] &\supset .Mʻ\Pi ʻQ\in F_{\Delta }ʻCʻQ.\\ +[*80·14] &\supset .\text{ᗡ}ʻMʻ\Pi ʻQ=CʻQ &\qquad \text{(2)}\\ +\vdash .(1).*80·3.*41·13.\supset \vdash :\text{Hp}.&\supset .Mʻ\Pi ʻQ\,\unicode{x2abd}\, \dot{s} ʻ\text{D}ʻM &\qquad \text{(3)}\\ +\vdash .*174·17. \supset \vdash :\text{Hp}.&\supset .\dot{s} ʻ\text{D}ʻM\in Cʻ\Pi ʻ\Sigma ʻP.\\ +[*172·12.*80·14] &\supset .\dot{s} ʻ\text{D}ʻM\in 1\rightarrow \text{Cls} &\qquad \text{(4)}\\ +\vdash .(3).(4).*72·92. \supset \vdash :\text{Hp}.\supset .Mʻ\Pi ʻQ&=(\dot{s} ʻ\text{D}ʻM)\upharpoonright \text{ᗡ}ʻMʻ\Pi ʻQ\\ +[(2)] &=(\dot{s} ʻ\text{D}ʻM)\upharpoonright CʻQ:\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*174·21.</b> \[\begin{align}\vdash \colon\colon P\in \text{Rel}^{2}\text{excl}.Q\in CʻP.&M,N\in Cʻ\Pi ʻ\Pi ^{;}P.\supset \colon\ldotp \\ +&Mʻ\Pi ʻQ=Nʻ\Pi ʻQ.\equiv :R\in CʻQ.\supset _{R}.(\dot{s} ʻ\text{D}ʻM)ʻR=(\dot{s} ʻ\text{D}ʻN)ʻR\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*71·35.*80·14.*172·12.\supset \vdash \colon\colon\text{Hp}.\supset \colon\ldotp \\ +Mʻ\Pi ʻQ=Nʻ\Pi ʻQ.&\equiv :R\in CʻQ.\supset _{R}.(Mʻ\Pi ʻQ)ʻR=(Nʻ\Pi ʻQ)ʻR:\\ +[*174·2] &\equiv :R\in CʻQ.\supset _{R}.{(\dot{s} ʻ\text{D}ʻM)\upharpoonright CʻQ}ʻR={(\dot{s} ʻ\text{D}ʻN)\upharpoonright CʻQ}ʻR:\\ +[*35·7] &\equiv :R\in CʻQ.\supset _{R}.(\dot{s} ʻ\text{D}ʻM)ʻR=(\dot{s} ʻ\text{D}ʻN)ʻR\colon\colon\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_451">[Pg 451]</span></p> + +<p class="nind"><b>*174·211.</b> \[\begin{align}&\vdash \colon\colon\ldotp P\in \text{Rel}^{2}\text{excl}.\supset \colon\colon M(\Pi ʻ\Pi ^{;}P)N.\equiv \colon\ldotp \\ +&M,N\in F_{\Delta }ʻ\Pi ʻʻCʻP\colon\ldotp (\exists Q,S):Q\in CʻP.S\in CʻQ.{(Mʻ\Pi ʻQ)ʻS}S{(Nʻ\Pi ʻQ)ʻS}:\\ +&TQS.T\neq S.\supset _{T}.(Mʻ\Pi ʻQ)ʻT=(Nʻ\Pi ʻQ)ʻT:\\ +&RPQ.R\neq Q.T\in CʻR.\supset _{R,T}.(Mʻ\Pi ʻR)ʻT=(Nʻ\Pi ʻR)ʻT\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*172·11.*150·22.&\supset \vdash \colon\colon M(\Pi ʻ\Pi ^{;}P)N.\equiv \colon\ldotp \\ +M,N\in F_{\Delta }ʻ\Pi ʻʻCʻP\colon\ldotp (\exists Q):Q&\in CʻP.(Mʻ\Pi ʻQ)(\Pi ʻQ)(Nʻ\Pi ʻQ):\\ +&RPQ.\Pi ʻR\neq \Pi ʻQ.\supset _{R}.Mʻ\Pi ʻR=Nʻ\Pi ʻR &\qquad \text{(1)}\\ +\vdash .*80·31.\supset \vdash \colon\ldotp M\in F_{\Delta }ʻ\Pi ʻʻCʻP.\supset :Q\in CʻP.&\supset _{Q}.Mʻ\Pi ʻQ\in Cʻ\Pi ʻQ. &\qquad \text{(2)}\\ +[*33·24] &\supset _{Q}.\dot{\exists} !\Pi ʻQ. &\qquad \text{(3)}\\ +[*172·19] &\supset _{Q}.\dot{s} ʻCʻ\Pi ʻQ=F\upharpoonright CʻQ &\qquad \text{(4)}\\ +\vdash .(3).(4).\supset \vdash :M\in F_{\Delta }ʻ\Pi ʻʻCʻP.Q,R\in &CʻP.\Pi ʻQ=\Pi ʻR.\supset .\\ +&F\upharpoonright CʻQ=F\upharpoonright CʻR.\dot{\exists} !\Pi ʻQ.\dot{\exists} !\Pi ʻR.\\ +[*172·141·192] &\supset .CʻQ=CʻR &\qquad \text{(5)}\\ +\vdash .(5).*163·14.\supset \vdash \colon\colon\text{Hp}.\supset \colon\ldotp M\in F_{\Delta }ʻ&\Pi ʻʻCʻP.Q,R\in CʻP.\supset :\\ +&\Pi ʻQ=\Pi ʻR.\supset .Q=R:\\ +[*30·37.\text{Transp}] &\supset :\Pi ʻQ\neq \Pi ʻR.\equiv .Q\neq R &\qquad \text{(6)}\\ +\vdash .*71·35.(2).*172·12.*80·14.\supset \\ +\vdash \colon\colon M,N\in &F_{\Delta }ʻ\Pi ʻʻCʻP.R\in CʻP.\supset \colon\ldotp \\ +&Mʻ\Pi ʻR=Nʻ\Pi ʻR.\equiv :T\in CʻR.\supset _{T}.(Mʻ\Pi ʻR)ʻT=(Nʻ\Pi ʻR)ʻT &\qquad \text{(7)}\\ +\vdash .*172·11.&\supset \vdash \colon\colon(Mʻ\Pi ʻQ)(\Pi ʻQ)(Nʻ\Pi ʻQ).\equiv \colon\ldotp Mʻ\Pi ʻQ,Nʻ\Pi ʻQ\in F_{\Delta }ʻCʻQ\colon\ldotp \\ +&(\exists S):S\in CʻQ.{(Mʻ\Pi ʻQ)ʻS}S{(Nʻ\Pi ʻQ)ʻS}:\\ +&TQS.T\neq S.\supset _{T}.(Mʻ\Pi ʻQ)ʻT=(Nʻ\Pi ʻQ)ʻT &\qquad \text{(8)}\\ +\vdash .(1).(2).(6).(7).(8).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*174·212.</b> \[\begin{align}&\vdash \colon\colon\ldotp P\in \text{Rel}^{2}\text{excl}.\supset \colon\colon M(\Pi ʻ\Pi ^{;}P)N.\equiv \colon\ldotp \\ +&M,N\in F_{\Delta }ʻ\Pi ʻʻCʻP\colon\ldotp (\exists Q, S):Q\in CʻP.S\in CʻQ.{(\dot{s} ʻ\text{D}ʻM)ʻS}S\{(\dot{s} ʻ\text{D}ʻN)ʻS\}:\\ +&TQS.T\neq S.\supset _{T}.(\dot{s} ʻ\text{D}ʻM)ʻT=(\dot{s} ʻ\text{D}ʻN)ʻT:\\ +&RPQ.R\neq Q.T\in CʻR.\supset _{R,T}.(\dot{s} ʻ\text{D}ʻM)ʻT=(\dot{s} ʻ\text{D}ʻN)ʻT\\ +&[*174·2·211.*35·7]\end{align}\]</p> + +<p class="nind"><b>*174·213.</b> \[\begin{align}\vdash \colon\ldotp &RPQ.S\in CʻQ.T\in CʻR.S\neq T.\supset _{Q,R,S,T}.R\neq Q:P\in \text{Rel}^{2}\text{excl}:\supset :\\ +&RPQ.S\in CʻQ.T\in CʻR.R\neq Q.\equiv .RPQ.S\in CʻQ.T\in CʻR.S\neq T\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*163·1.&\supset \vdash \colon\ldotp \text{Hp}.\supset :RPQ.R\neq Q.S\in CʻQ.T\in CʻR.\supset .S\neq T &\qquad \text{(1)}\\ +\vdash .*11·1. &\supset \vdash \colon\ldotp \text{Hp}.\supset :RPQ.S\in CʻQ.T\in CʻR.S\neq T.\supset .R\neq Q &\qquad \text{(2)}\\ +\vdash .(1).(2).&\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_452">[Pg 452]</span></p> + +<p class="nind"><b>*174·214.</b> \[\begin{align}&\vdash \colon\colon P\in \text{Rel}^{2}\text{excl}:QPQ.\supset _{Q}.CʻQ\in 0\cup 1:\supset \colon\ldotp \\ +&(\exists Q):Q\in CʻP:TQS.T\neq S.\lor.(\exists R).RPQ.R\neq Q.S\in CʻQ.T\in CʻR:\equiv .\\ +&T(\Sigma ʻP)S.T\neq S\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*52·41.\supset \vdash \colon\ldotp \text{Hp}.&\supset :S,T\in CʻQ.S\neq T.\supset .{\sim}(QPQ):\\ +[*13·12.\text{Transp}] &\supset :RPQ.S\in CʻQ.T\in CʻR.S\neq T.\supset .Q\neq R &\qquad \text{(1)}\\ +\vdash .(1).*174·213.\supset \\ +\vdash \colon\colon\text{Hp}.&\supset \colon\ldotp RPQ.S\in CʻQ.T\in CʻR.R\neq Q.\equiv .RPQ.S\in CʻQ.T\in CʻR.S\neq T\colon\ldotp \\ +[*4·37.*11·341]\supset \colon\ldotp \\ +&(\exists Q).Q\in CʻP.TQS.T\neq S.\lor.(\exists Q,R).RPQ.S\in CʻQ.T\in CʻR.R\neq Q:\equiv :\\ +&(\exists Q).Q\in CʻP.TQS.T\neq S.\lor.(\exists Q,R).RPQ.S\in CʻQ.T\in CʻR.T\neq S:\\ +[*162·13] &\equiv :T(\Sigma ʻP)S.T\neq S &\qquad \text{(2)}\\ +\vdash .(2).*33·17.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*174·215">*174·215</a>.</b> \[\begin{align}\vdash \colon\ldotp &P\in \text{Rel}^{2}\text{excl}:QPQ.\supset _{Q}.CʻQ\in 0\cup 1:\supset :\\ +&M(\Pi ʻ\Pi ^{;}P)N.\equiv .\mu ,N\in F_{\Delta }ʻ\Pi ʻʻCʻP.(\dot{s} ʻ\text{D}ʻM )(\Pi ʻ\Sigma ʻP )(\dot{s} ʻ\text{D}ʻN)\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*174·212·214.&\supset \vdash \colon\colon\ldotp \text{Hp}.\supset \colon\colon M(\Pi ʻ\Pi ^{;}P)N.\equiv \colon\ldotp \\ +&M,N\in F_{\Delta }ʻ\Pi ʻʻCʻP\colon\ldotp (\exists Q,S):Q\in CʻP.S\in CʻQ.\{(\dot{s} ʻ\text{D}ʻM)ʻS\}S\{(\dot{s} ʻ\text{D}ʻN)ʻS\}:\\ +&T(\Sigma ʻP)S.T\neq S.\supset _{T}.(\dot{s} ʻ\text{D}ʻM)ʻT=(\dot{s} ʻ\text{D}ʻN)ʻT &\qquad \text{(1)}\\ +\vdash .*172·13.*152·42.&\supset \vdash :M(\Pi ʻ\Pi ^{;}P)N.\supset .\dot{\exists} !P.\\ +[*172·162] &\supset .\dot{s} ʻ\text{D}ʻM,\dot{s} ʻ\text{D}ʻN\in F_{\Delta }ʻCʻ\Sigma ʻP &\qquad \text{(2)}\\ +\vdash .*162·22.&\supset \vdash :(\exists Q).Q\in CʻP.S\in CʻQ.\equiv .S\in Cʻ\Sigma ʻP &\qquad \text{(3)}\\ +\vdash .(1).(2).(3).*172·11.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*174·216">*174·216</a>.</b> \(\vdash \colon\ldotp P\,\unicode{x2abd}\, J.\supset :QPQ.\supset _{Q}.CʻQ\in 0\cup 1\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*50·24.\supset \vdash \colon\ldotp \text{Hp}.&\supset :(Q).{\sim}(QPQ):\\ +[*10·53] &\supset :QPQ.\supset _{Q}.CʻQ\in 0\cup 1\colon\ldotp \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*174·22.</b> \[\begin{align}\vdash \colon\ldotp &P\in \text{Rel}^{2}\text{excl}.P\,\unicode{x2abd}\, J.\supset :\\ +&M(\Pi ʻ\Pi ^{;}P)N.\equiv .M,N\in F_{\Delta }ʻ\Pi ʻʻCʻP.(\dot{s} ʻ\text{D}ʻM)(\Pi ʻ\Sigma ʻP)(\dot{s} ʻ\text{D}ʻN)\\ +&[*174·215·216]\end{align}\]</p> + +<p class="nind"><b><a id="*174·221">*174·221</a>.</b> \[\begin{align}\vdash \colon\ldotp &P\in \text{Rel}^{2}\text{excl}:QPQ.\supset _{Q}.CʻQ\in 0\cup 1:\supset .\\ +&\Pi ʻ\Sigma ʻP=\dot{s} ^{;}\text{D}^{;}\Pi ʻ\Pi ^{;}P .(\dot{s} \mid \text{D})\upharpoonright Cʻ\Pi ʻ\Pi ^{;}P + \in (\Pi ʻ\Sigma ʻP )\,\overline{\text{smor}}\,(\Pi ʻ\Pi ^{;}P )\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*174·215.*150·41.\supset \\ +\vdash :\text{Hp} .T&=(\dot{s} \mid \text{D})\upharpoonright Cʻ\Pi ʻ\Pi ^{;}P.\supset .\Pi ʻ\Pi ^{;}P =\breve{T} ^{;}\Pi ʻ\Sigma ʻP &\qquad \text{(1)}\\ +\vdash .*174·19. &\supset \vdash :\text{Hp}(1).\supset .T\in 1\rightarrow 1 &\qquad \text{(2)}\\ +\vdash .*174·17. &\supset \vdash :\text{Hp}(1).\supset .\text{D}ʻT=Cʻ\Pi ʻ\Sigma ʻP &\qquad \text{(3)}\\ +\vdash .(1).(2).(3).*151·11.\supset \vdash :\text{Hp}(1).&\supset .\breve{T} \in (\Pi ʻ\Pi ^{;}P)\overline{\text{smor}} (\Pi ʻ\Sigma ʻP).\\ +[*151·131] &\supset .T\in (\Pi ʻ\Sigma ʻP)\overline{\text{smor}} (\Pi ʻ\Pi ^{;}P ) &\qquad \text{(4)}\\ +\vdash .(4).*151.22.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_453">[Pg 453]</span></p> + +<p class="nind"><b><a id="*174·23">*174·23</a>.</b> \[\begin{align}\vdash :P\in &\text{Rel}^{2}\text{excl}.P\,\unicode{x2abd}\, J.\supset .\Pi ʻSʻP=\dot{s} ^{;}\text{D}^{;}\Pi ʻ\Pi ^{;}P.\\ +&(\dot{s} \mid \text{D})\upharpoonright Cʻ\Pi ʻ\Pi ^{;}P\in (\Pi ʻ\Sigma ʻP)\,\overline{\text{smor}}\,(\Pi ʻ\Pi ^{;}P) \quad[*174·221·216]\end{align}\]</p> + +<p class="nind"><b><a id="*174·231">*174·231</a>.</b> \[\begin{align}\vdash :P\in &\text{Rel}^{2}\text{excl}:QPQ.\supset _{Q}.CʻQ\in 0\cup 1:\supset .\\ +&\dot{s} \upharpoonright Cʻ\text{Prod}ʻ\Pi ^{;}P\in (\Pi ʻ\Sigma ʻP)\,\overline{\text{smor}}\,(\text{Prod}ʻ\Pi ^{;}P)\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*174·221.*173·1.\supset \vdash :\text{Hp}.\supset .\Pi ʻ\Sigma ʻP=\dot{s} ^{;}\text{Prod}ʻ\Pi ^{;}P &\qquad \text{(1)}\\ +\vdash .(1).*174·191.*151·231.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*174·24">*174·24</a>.</b> \[\begin{align}\vdash :P\in &\text{Rel}^{2}\text{excl}.P\,\unicode{x2abd}\, J.\supset .\\ +&\dot{s} \upharpoonright Cʻ\text{Prod}ʻ\Pi ^{;}P\in (\Pi ʻ\Sigma ʻP)\,\overline{\text{smor}}\,(\text{Prod}ʻ\Pi ^{;}P) \quad[*174·231·216]\end{align}\]</p> + +<p class="nind"><b><a id="*174·241">*174·241</a>.</b> \[\begin{align}\vdash \colon\ldotp P\in &\text{Rel}^{2}\text{excl}:QPQ.\supset _{Q}.CʻQ\in 0\cup 1:\supset .\\ +&\Pi ʻ\Sigma ʻP \,\text{smor}\,\Pi ʻ\Pi ^{;}P.\Pi ʻ\Sigma ʻP \,\text{smor}\,\text{Prod}ʻ\Pi ^{;}P \quad[*174·221·231]\end{align}\]</p> + +<p class="nind"><b><a id="*174·25">*174·25</a>.</b> \[\begin{align}\vdash :P\in &\text{Rel}^{2}\text{excl}.P\,\unicode{x2abd}\, J.\supset .\\ +&\Pi ʻ\Sigma ʻP \,\text{smor}\,\Pi ʻ\Pi ^{;}P.\Pi ʻ\Sigma ʻP \,\text{smor}\,\text{Prod}ʻ\Pi ^{;}P \quad[*174·23·24]\end{align}\]</p> + +<p>This proposition gives the associative law for \(\Pi\). It remains to +prove the associative law for \(\text{Prod}\).</p> + +<p>The following propositions are concerned with various properties of +"arithmetical" relations, down to <a href="#*174·4">*174·4</a>, where the proof of the +associative law for \(\text{Prod}\) begins.</p> + +<p class="nind"><b>*174·3.</b> \(\vdash :P\in \text{Rel}^{3}\text{arithm}.\equiv .P,\Sigma ʻP\in \text{Rel}^{2}\text{excl} \quad[(*174·01)]\)</p> + +<p class="nind"><b>*174·31.</b> \[\begin{align}\vdash \colon\ldotp P\in &\text{Rel}^{3}\text{arithm}.\equiv :Q,Q'\in CʻP.Q\neq Qʻ.\supset _{Q,Q'}.CʻQ\cap CʻQ'=\Lambda:\\ +&R,R'\in Cʻ\Sigma ʻP.R\neq R'.\supset _{R,R'}.CʻR\cap CʻR'=\Lambda \quad[*174·3.*163·1]\end{align}\]</p> + +<p class="nind"><b>*174·311.</b> \[\begin{align}\vdash \colon\ldotp P\in &\text{Rel}^{3}\text{arithm}.\equiv :Q,Q'\in CʻP.\exists !CʻQ\cap CʻQ'.\supset _{Q,Q'}.Q=Q':\\ +&R,R'\in Cʻ\Sigma ʻP.\exists !CʻR\cap CʻRʻ.\supset _{R,Rʻ}.R=R' \quad[*174·3.*163·11]\end{align}\]</p> + +<p class="nind"><b>*174·32.</b> \[\begin{align}\vdash :P\in \text{Rel}^{3}\text{arithm}.\equiv .F\upharpoonright CʻP,F\upharpoonright Cʻ\Sigma ʻP\in \text{Cls}&\rightarrow 1\\ +&[*174·3.*163·12]\end{align}\]</p> + +<p class="nind"><b>*174·321.</b> \(\vdash :P\in \text{Rel}^{3}\text{arithm}.\supset .C\upharpoonright CʻP,C\upharpoonright Cʻ\Sigma ʻP\in 1\rightarrow 1 \quad[*174·3.*163·14]\)</p> + +<p class="nind"><b>*174·322.</b> \(\vdash :P\in \text{Rel}^{3}\text{arithm}.Q,Q'\in CʻP.\exists !CʻʻCʻQ\cap CʻʻCʻQ'.\supset .Q=Q'\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*37·6.\supset \vdash :\text{Hp}.&\supset .(\exists R,Rʻ).R\in CʻQ.Rʻ\in CʻQ'.CʻR=CʻR'.\\ +[*174·321] &\supset .(\exists R,R').R\in CʻQ.R'\in CʻQ'.R=R'.\\ +[*13·195] &\supset .\exists !CʻQ\cap CʻQ'.\\ +[*174·311] &\supset .Q=Q':\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*174·33.</b> \(\vdash :P\in \text{Rel}^{3}\text{arithm}.\supset .CʻʻʻCʻʻCʻP\in \text{Cls}^{3}\,\text{arithm}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*174·322.\supset \\ +\vdash \colon\ldotp \text{Hp}.&\supset :Q,Qʻ\in CʻP.\exists !CʻʻCʻQ\cap CʻʻCʻQʻ.\supset _{Q,Qʻ}.CʻʻCʻQ=CʻʻCʻQʻ:\\ +[*37·63] &\supset :\gamma ,\delta \in CʻʻʻCʻʻCʻP.\exists !\gamma \cap \delta .\supset _{\gamma ,\delta }.\gamma =\delta :\\ +[*84·11] &\supset :CʻʻʻCʻʻCʻP\in \text{Cls}^{2} \text{excl} &\qquad \text{(1)}\\ +\vdash .*174·3.*163·16.*162·22.\supset \vdash :\text{Hp}.&\supset .CʻʻsʻCʻʻCʻP\in \text{Cls}^{2} \text{excl}.\\ +[*40·38] &\supset .sʻCʻʻʻCʻʻCʻP\in \text{Cls}^{2} \text{excl} &\qquad \text{(2)}\\ +\vdash .(1).(2).*115·2.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_454">[Pg 454]</span></p> + +<p class="nind"><b>*174·34.</b> \[\begin{align}\vdash :P\in &\text{Rel}^{3}\text{arithm}.\equiv .\\ +&CʻʻʻCʻʻCʻP\in \text{Cls}^{3}\,\text{arithm}.C\upharpoonright CʻP,C\upharpoonright Cʻ\Sigma ʻP\in 1\rightarrow 1\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*174·321·33.\supset \\ +\vdash :P\in \text{Rel}^{3}\text{arithm}.\supset .CʻʻʻCʻʻCʻP \in \text{Cls}^{3}\,\text{arithm}.C\upharpoonright + CʻP,C\upharpoonright Cʻ\Sigma ʻP\in 1\rightarrow 1 &\qquad \text{(1)}\\ +\vdash .*115·2.\supset \vdash :CʻʻʻCʻʻCʻP\in \text{Cls}^{3}\,\text{arithm}.C\upharpoonright Cʻ\Sigma ʻP\in 1\rightarrow 1.\supset .\\ +sʻCʻʻʻCʻʻCʻP\in \text{Cls}^{2} \text{excl}.C\upharpoonright Cʻ\Sigma ʻP\in 1\rightarrow 1.\\ +[*40·38.*162·22] \supset .CʻʻCʻ\Sigma ʻP\in \text{Cls}^{2} \text{excl}.C\upharpoonright Cʻ\Sigma ʻP\in 1\rightarrow 1.\\ +[*163·17] \supset .\Sigma ʻP\in \text{Rel}^{2}\text{excl} &\qquad \text{(2)}\\ +\vdash .*37·62.\supset \vdash :Q,Qʻ\in CʻP.R\in CʻQ\cap CʻQʻ.\supset .CʻR\in CʻʻCʻQ\cap CʻʻCʻQʻ &\qquad \text{(3)}\\ +\vdash .(3).*115·2.*84·11.\supset \\ +\vdash :CʻʻʻCʻʻCʻP\in \text{Cls}^{3}\,\text{arithm}.Q,Qʻ\in CʻP.\exists !CʻQ\cap CʻQʻ.\supset .\\ +CʻʻCʻQ=CʻʻCʻQʻ &\qquad \text{(4)}\\ +\vdash .(4).*72·481.*37·421.\supset \\ +\vdash :CʻʻʻCʻʻCʻP\in \text{Cls}^{3}\,\text{arithm}.C\upharpoonright Cʻ\Sigma ʻP\in 1\rightarrow 1.\\ +Q,Qʻ\in CʻP.\exists !CʻQ\cap CʻQʻ.\supset .CʻQ=CʻQʻ &\qquad \text{(5)}\\ +\vdash .(5).*71·59.\supset \\ +\vdash \colon\ldotp CʻʻʻCʻʻCʻP\in \text{Cls}^{3}\,\text{arithm}.C\upharpoonright Cʻ\Sigma ʻP \in 1\rightarrow 1.C\upharpoonright CʻP\in 1\rightarrow 1.\supset :\\ +Q,Qʻ\in CʻP.\exists !CʻQ\cap CʻQʻ.\supset _{Q,Qʻ}.Q=Qʻ:\\ +[*163·11] \supset :P\in \text{Rel}^{2}\text{excl} &\qquad \text{(6)}\\ +\vdash .(2).(6).*174·3.\supset \\ +\vdash :CʻʻʻCʻʻCʻP\in \text{Cls}^{3}\,\text{arithm}.C\upharpoonright Cʻ\Sigma ʻP\in 1\rightarrow 1.C\upharpoonright CʻP\in 1\rightarrow 1.\supset \\. +P\in \text{Rel}^{3}\text{arithm} &\qquad \text{(7)}\\ +\vdash .(1).(7).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*174·35.</b> \(\vdash :P\in \text{Rel}^{3}\text{arithm}.Q,Qʻ\in CʻP.Q\neq Qʻ.\supset .Cʻ\Sigma ʻQ\cap Cʻ\Sigma ʻQʻ=\Lambda\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*174·3.*163·1. &\supset \vdash \colon\ldotp \text{Hp}.\supset :R\in CʻQ.Rʻ\in CʻQʻ.\supset _{R,Rʻ}.R \neq Rʻ &\qquad \text{(1)}\\ +\vdash .*162·22. &\supset \vdash \colon\ldotp \text{Hp}.\supset :R\in CʻQ.Rʻ\in CʻQʻ . \supset _{R,Rʻ}.R,Rʻ\in Cʻ\Sigma ʻP &\qquad \text{(2)}\\ +\vdash .(1).(2).*174·31.\supset \vdash \colon\ldotp \text{Hp}.&\supset :R\in CʻQ.Rʻ\in CʻQʻ.\supset _{R,Rʻ}.CʻR\cap CʻRʻ=\Lambda:\\ +[*40·27] &\supset :sʻCʻʻCʻQ\cap sʻCʻʻCʻQʻ=\Lambda:\\ +[*162·22] &\supset :Cʻ\Sigma ʻQ\cap Cʻ\Sigma ʻQʻ=\Lambda\colon\ldotp \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*174·36.</b> \(\vdash :P\in \text{Rel}^{3}\text{arithm}.\supset .\Sigma ^{;}P\in \text{Rel}^{2}\text{excl}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*174·35.*37·63.*150·22.\supset \\ +\vdash \colon\ldotp \text{Hp}.\supset :R,Rʻ\in Cʻ\Sigma ^{;}P.R\neq Rʻ.\supset .CʻR\cap CʻRʻ=\Lambda &\qquad \text{(1)}\\ +\vdash .(1).*163·1.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_455">[Pg 455]</span></p> + +<p class="nind"><b>*174·361.</b> \(\vdash :P\in \text{Rel}^{3}\text{arithm}.\supset .CʻP\subset \text{Rel}^{2}\text{excl}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*162·1.&\supset \vdash :Q\in CʻP.\supset .Q\,\unicode{x2abd}\, \Sigma ʻP &\qquad \text{(1)}\\ +\vdash .*174·3.&\supset \vdash :\text{Hp}.\supset .\Sigma ʻP\in \text{Rel}^{2}\text{excl} &\qquad \text{(2)}\\ +\vdash .(1).(2).*163·43.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*174·362.</b> \(\vdash :P\in \text{Rel}^{3}\text{arithm} .Q,Qʻ\in CʻP.CʻʻCʻQ=CʻʻCʻQʻ.\supset .Q=Qʻ\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*174·322. &\supset \vdash :\text{Hp}.\exists !CʻʻCʻQ.\supset .Q=Q' &\qquad \text{(1)}\\ +\vdash .*37·45. \supset \vdash :CʻʻCʻQ=\Lambda.&\supset .CʻQ=\Lambda.\\ +[*33·241] &\supset .Q=\dot{\Lambda} &\qquad \text{(2)}\\ +\vdash .(2).*13·172. &\supset \vdash :\text{Hp}.CʻʻCʻQ=\Lambda.\supset .Q=Q' &\qquad \text{(3)}\\ +\vdash .(1).(3).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*174·363.</b> \(\vdash :P\in \text{Rel}^{3}\text{arithm}.\supset .\text{Prod}^{;}P\in \text{Rel}^{2}\text{excl}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*173·161.*174·361.*173·2.\text{Transp}.\supset \\ +\vdash \colon\ldotp \text{Hp}.&\supset :Q,Q'\in CʻP.\exists !Cʻ\text{Prod}ʻQ\cap Cʻ\text{Prod}ʻQ'.\supset .\\ +&\exists !\text{Prod}ʻCʻʻCʻQ\cap \text{Prod}ʻCʻʻCʻQ'.\\ +[*115·23.*174·33] &\supset .CʻʻCʻQ=CʻʻCʻQ'.\\ +[*174·362] &\supset .Q=Q'.\\ +[*30·37] &\supset .\text{Prod}ʻQ=\text{Prod}ʻQ' &\qquad \text{(1)}\\ +\vdash .(1).*163·11.*150·22.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*174·4">*174·4</a>.</b> \[\begin{align}\vdash :P\in \text{Rel}^{2}\text{excl}.&P\,\unicode{x2abd}\, J.\supset .\\ +&\text{Prod}ʻ\Sigma ʻP=\text{D}^{;}\dot{s} ^{;}\text{Prod}ʻ\Pi ^{;}P=s^{;}\text{D}_{\in }^{;}\text{Prod}ʻ\Pi ^{;}P\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*173·1. &\supset \vdash .\text{Prod}ʻ\Sigma ʻP=\text{D}^{;}\Pi ʻ\Sigma ʻP &\qquad \text{(1)}\\ +\vdash .(1).*174·24.&\supset \vdash :P\in \text{Rel}^{2}\text{excl}.P\,\unicode{x2abd}\, J.\supset .\text{Prod}ʻ\Sigma ʻP=\text{D}^{;}\dot{s} ^{;}\text{Prod}ʻ\Pi + ^{;}P\\ +[*41·43] &=s^{;}\text{D}_{\in }^{;}\text{Prod}ʻ\Pi ^{;}P:\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*174·401">*174·401</a>.</b> \(\vdash :P\in \text{Rel}^{3}\text{arithm}.\supset .\text{Prod}ʻ\text{Prod}^{;}P=\text{D}_{\in }^{;}\text{Prod}ʻ\Pi ^{;}P\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*80·33.*162·23.&\supset \vdash :R\in F_{\Delta }ʻCʻQ.\supset .\text{D}ʻR\subset Cʻ\Sigma ʻQ &\qquad \text{(1)}\\ +\vdash .(1).&\supset \vdash :R\in F_{\Delta }ʻCʻQ.R'\in F_{\Delta }ʻCʻQʻ.\exists !\text{D}ʻR\cap \text{D}ʻR'.\supset .\\ +&\exists !Cʻ\Sigma ʻQ \cap Cʻ\Sigma ʻQ' &\qquad \text{(2)}\\ +\vdash .(2).*174·35.&\supset \vdash \colon\ldotp \text{Hp}.\supset :\\ +&Q,Q'\in CʻP.R\in F_{\Delta }ʻCʻQ.R'\in F_{\Delta }ʻCʻQʻ.\text{D}ʻR=\text{D}ʻR'.\exists !\text{D}ʻR.\supset .\\ +&Q=Qʻ.\\ +[*81·21.*174·361.*163·12] &\supset .R=R' &\qquad \text{(3)}\\ +\vdash .(3).*33·241.\supset \\ +\vdash \colon\ldotp \text{Hp}.&\supset :Q,Q'\in CʻP.R\in F_{\Delta }ʻCʻQ.R'\in F_{\Delta }ʻCʻQ'.\text{D}ʻR=\text{D}ʻR'.\supset .R=R':\\ +[*172·12.*150·22] &\supset :\text{D}\upharpoonright sʻCʻʻ\Pi ʻʻCʻP\in 1\rightarrow 1:\\ +[*162·22] &\supset :\text{D}\upharpoonright Cʻ\Sigma ʻ\Pi ^{;}P\in 1\rightarrow 1:\\ +[*173·33] \supset :\text{D}_{\in }^{;}\text{Prod}ʻ\Pi ^{;}P&=\text{Prod}ʻ\text{D}\dagger ^{;}\Pi ^{;}P\\ +[*173·1] &=\text{Prod}ʻ\text{Prod}^{;}P\colon\ldotp \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_456">[Pg 456]</span></p> + +<p class="nind"><b><a id="*174·41">*174·41</a>.</b> \[\begin{align}&\vdash :P\in \text{Rel}^{3}\text{arithm}.P\,\unicode{x2abd}\, J.\supset .\text{Prod}ʻ\Sigma ʻP=s^{;}\text{Prod}ʻ\text{Prod}^{;}P\\ +&[*174·4·401]\end{align}\]</p> + +<p class="nind"><b><a id="*174·42">*174·42</a>.</b> \[\begin{align}\vdash :P\in &\text{Rel}^{3}\text{arithm}.P\,\unicode{x2abd}\, J.\supset .\\ +&s\upharpoonright (Cʻ\text{Prod}ʻ\text{Prod}^{;}P)\in (\text{Prod}ʻ\Sigma ʻP)\,\overline{\text{smor}}\,(\text{Prod}ʻ\text{Prod}^{;}P)\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*173·161·2.*174·363.\supset \\ +\vdash :\text{Hp}.\supset .Cʻ\text{Prod}ʻ\text{Prod}^{;}P&\subset \text{Prod}ʻCʻʻCʻ\text{Prod}^{;}P\\ +[*150·22] &\subset \text{Prod}ʻCʻʻ\text{Prod}ʻʻCʻP\\ +[*173·161.*174·361] &\subset \text{Prod}ʻ\text{Prod}ʻʻCʻʻʻCʻʻCʻP &\qquad \text{(1)}\\ +\vdash .(1).*174·33.*115·46.&\supset \vdash :\text{Hp}.\supset .s\upharpoonright Cʻ\text{Prod}ʻ\text{Prod}^{;}P\in 1\rightarrow 1 &\qquad \text{(2)}\\ +\vdash .(2).*174·41.*151·231.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*174·43">*174·43</a>.</b> \[\begin{align}&\vdash :P\in \text{Rel}^{3}\text{arithm}.P\,\unicode{x2abd}\, J.\supset .\text{Prod}ʻ\Sigma ʻP\,\text{smor}\,\text{Prod}ʻ\text{Prod}^{;}P\\ +&[*174·42]\end{align}\]</p> + +<p>This is the associative law for \(\text{Prod}\).</p> + +<p class="nind"><b>*174·44.</b> \[\begin{align}&\vdash :P\in \text{Rel}^{3}\text{arithm}.\supset .\text{Prod}ʻ\text{Prod}^{;}P=\text{D}_{\in }^{;}\text{D}^{;}\Pi + ʻ\Pi ^{;}P\\ +&[*174·401.*173·1]\end{align}\]</p> + +<p class="nind"><b>*174·45.</b> \[\begin{align}\vdash :P\in &\text{Rel}^{3}\text{arithm}.\supset .\\ +&(\text{D}_{\in }\mid \text{D})\upharpoonright Cʻ\Pi ʻ\Pi ^{;}P\in (\text{Prod}ʻ\text{Prod}^{;}P)\,\overline{\text{smor}}\,(\Pi ʻ\Pi ^{;}P)\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*174·18.\supset \vdash :\text{Hp}.&\supset .\text{D}\upharpoonright Cʻ\Pi ʻ\Pi ^{;}P\in 1\rightarrow 1 &\qquad \text{(1)}\\ +\vdash .*80·33.*162·23.\supset \\ +&\vdash :R\in F_{\Delta }ʻCʻQ.R'\in F_{\Delta }ʻCʻQʻ.\exists !\text{D}ʻR\cap \text{D}ʻR'.\supset .\exists !Cʻ\Sigma ʻQ\cap Cʻ\Sigma ʻQʻ &\qquad \text{(2)}\\ +\vdash .(2).*174·35.\supset \\ +\vdash \colon\ldotp \text{Hp}.&\supset :Q,Q'\in CʻP.R\in F_{\Delta }ʻCʻQ.R'\in F_{\Delta }ʻCʻQʻ.\exists \\!\text{D}ʻR.\text{D}ʻR=\text{D}ʻR'.\supset . +Q=Q'.\\ +[*81·21.*174·361.*163·12] &\supset .R=R' &\qquad \text{(3)}\\ +\vdash .(3).*33·241.\supset \\ +\vdash \colon\ldotp \text{Hp}.&\supset :Q,Q'\in CʻP.R\in F_{\Delta }ʻCʻQ.R'\in F_{\Delta }ʻCʻQʻ.\text{D}ʻR=\text{D}ʻR'.\supset .R=R':\\ +[*172·12]&\supset :Q,Q'\in CʻP.R\in Cʻ\Pi ʻQ.R'\in Cʻ\Pi ʻQʻ.\text{D}ʻR=\text{D}ʻR'.\supset .R=R' &\qquad \text{(4)}\\ +\vdash .*173·161.*37·6.*173·2.\text{Transp}.\supset \\ +&\vdash \colon\colon\text{Hp}.\mu ,\nu \in Cʻ\text{Prod}ʻ\Pi ^{;}P.\text{D}ʻʻ\mu =\text{D}ʻʻ\nu .\supset \colon\ldotp \\ +&R\in \mu .\supset :(\exists Q):Q\in CʻP.R\in Cʻ\Pi ʻQ:\\ +&(\exists Qʻ,R').Qʻ\in CʻP.R'\in Cʻ\Pi ʻQʻ.R'\in \nu .\text{D}ʻR=\text{D}ʻR':\\ +[(4)] &\supset :(\exists R').R'\in \nu .R=R':\\ +[*13·195] &\supset : R\in \nu &\qquad \text{(5)}\\ +\text{Similarly}\quad \vdash \colon\ldotp \text{Hp}(5).&\supset :R\in \nu .\supset .R\in \mu &\qquad \text{(6)}\\ +\vdash .(5).(6).&\supset \vdash \colon\ldotp \text{Hp}.\supset :\mu ,\nu \in Cʻ\text{Prod}ʻ\Pi ^{;}P.\text{D}ʻʻ\mu =\text{D}ʻʻ\nu .\supset .\mu =\nu :\\ +[*71·55] &\supset :\text{D}_{\in }\upharpoonright Cʻ\text{Prod}ʻ\Pi ^{;}P\in 1\rightarrow 1:\\ +[*150·22.*173·1] &\supset :\text{D}_{\in }\upharpoonright \text{D}ʻʻCʻ\Pi ʻ\Pi ^{;}P\in 1\rightarrow 1 &\qquad \text{(7)}\\ +\vdash .(1).(7).*35·481.&\supset \vdash :\text{Hp}.\supset .(\text{D}_{\in }\mid \text{D})\upharpoonright Cʻ\Pi ʻ\Pi ^{;}P\in 1\rightarrow 1 &\qquad \text{(8)}\\ +\vdash .(8).*174·44.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_457">[Pg 457]</span></p> + +<p class="nind"><b>*174·46.</b> \(\vdash :P\in \text{Rel}^{3}\text{arithm}.\supset .\text{Prod}ʻ\text{Prod}^{;}P\,\text{smor}\,\Pi ʻ\Pi ^{;}P \quad[*174·45]\)</p> + +<p class="nind"><b>*174·461.</b> \[\begin{align}&\vdash :P\in \text{Rel}^{3}\text{arithm}.P\,\unicode{x2abd}\, J.\supset .\text{Prod}ʻ\text{Prod}^{;}P\text{smor}\Pi ʻ\Sigma ʻP\\ +&[*174·46·25]\end{align}\]</p> + +<p class="nind"><b>*174·462.</b> \[\begin{align}&\vdash :P\in \text{Rel}^{3}\text{arithm}.\supset .\Pi ʻ\text{Prod}^{;}P\,\text{smor}\,\text{Prod}ʻ\text{Prod}^{;}P\\ +&[*174·363 . *173·16]\end{align}\]</p> + +<p>The two following propositions merely sum up previous results.</p> + +<p class="nind"><b>*174·47.</b> \[\begin{align}&\vdash :P\in \text{Rel}^{3}\text{arithm}.P\,\unicode{x2abd}\, J.\supset .\\ +&\text{Prod}ʻ\Sigma ʻP=s^{;}\text{Prod}ʻ\text{Prod}^{;}P=s^{;}\text{D}_{\in }^{;}\text{D}^{;}\Pi ʻ\Pi ^{;}P=\text{D}^{;}\dot{s} + ^{;}\text{Prod}ʻ\Pi ^{;}P.\\ +&s\upharpoonright Cʻ\text{Prod}ʻ\text{Prod}^{;}P,s\mid \text{D}_{\in }\mid \text{D}\upharpoonright Cʻ\Pi ʻ\Pi ^{;}P,\text{D}\mid + \dot{s} \upharpoonright Cʻ\text{Prod}ʻ\Pi ^{;}P\in 1\rightarrow 1\\ +&[*174·42·45·24.*41·43]\end{align}\]</p> + +<p class="nind"><b>*174·48.</b> \[\begin{align}&\vdash :P\in \text{Rel}^{3}\text{arithm}.P\,\unicode{x2abd}\, J.\supset .\\ +\text{Nr}ʻ\text{Prod}ʻ\text{Prod}^{;}P&=\text{Nr}ʻ\text{Prod}ʻ\Sigma ʻP=\text{Nr}ʻ\Pi ʻ\Sigma ʻP=\text{Nr}ʻ\Pi ʻ\Pi ^{;}P\\ +&=\text{Nr}ʻ\text{Prod}ʻ\Pi ^{;}P=\text{Nr}ʻ\Pi ʻ\text{Prod}^{;}P\\ +&[*174·43·46·25·462.*152·321]\end{align}\]</p> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_458">[Pg 458]</span></p> +<h2 class="nobreak" id="*176">*176. EXPONENTIATION.</h2> +</div> + + +<p class="nind"><i>Summary of</i> *176.</p> + +<p>The definition of exponentiation is framed on the analogy of the +definition in cardinals, <i>i.e.</i> we put +\[ +P\,\,\text{exp}\,\,Q=\text{Prod}ʻP\downarrow_{,.}^{;}Q \quad\text{Df}. +\] +We put also, what is often a more convenient form, +\[ +P^{Q}=\dot{s} ^{;}(P\,\,\text{exp}\,\,Q) \quad\text{Df}. +\]</p> + +<p>The relation \(P^{Q}\) has for its field (unless \(Q=\dot{\Lambda}\)) the +class of Cantor's "Belegungen," <i>i.e.</i> the class \((CʻP\uparrow CʻQ)_{\Delta }ʻCʻQ\). +It arranges these by a form of the principle of +first differences, namely as follows : Suppose \(M\) and \(N\) are two +members of \((CʻP\uparrow CʻQ)_{\Delta }ʻCʻQ\), and suppose there is +in \(CʻQ\) a term \(y\) for which the \(M\)-representative \((Mʻy)\) +precedes the \(N\)-representative \((Nʻy)\), <i>i.e.</i> for which +\((Mʻy)P(Nʻy)\), and suppose further that all terms in \(CʻQ\) which +are earlier than \(y\), <i>i.e.</i> for which \(zQy.z\neq y\), have +their \(M\)-representative and their \(N\)-representative identical; +in this case we say that \(M\) has to \(N\) the relation \(P^{Q}\). This +may be stated as follows, provided we assume that \(P\) and \(Q\) are +series: Let \(M\) and \(N\) be two one-valued functions whose possible +arguments are all the members of \(CʻQ\), while their values are some +or all of the members of \(CʻP\). Then we say that \(M\) has to \(N\) +the relation \(P^{Q}\) if the first argument for which the two functions +do not have the same value gives an earlier value to \(M\) than to +\(N\).</p> + +<figure class="figcenter width500" id="i_458" style="width: 1200px;"> +<img src="images/i_458.jpg" width="1200" height="585" alt="Diagram +showing two ordered sets: P={a₁,a₂,a₃,a₄,a₅} (top row, 5 elements) and +Q={b₁,b₂,b₃,b₄} (bottom row, 4 elements), each represented by dots +with subscripted labels. Arrows indicate P and Q as distinct ordered +sequences, illustrating sets with different cardinalities in formal +logic or set theory."> +</figure> + +<p>Thus for example let \(P\) be the series \(a_{1}\), \(a_{2}\), +\(a_{3}\), \(a_{4}\), \(a_{5}\), and let \(Q\) be the series \(b_{1}\), +\(b_{2}\), \(b_{3}\), \(b_{4}\). Then \(M\) and \(N\) are to be such +that \(Mʻb\) or \(Nʻb\) is defined when, and only when, \(b\) is +\(b_{1}\) or \(b_{2}\) or \(b_{3}\) or \(b_{4}\), and the value of +\(Mʻb\) or \(Nʻb\) is \(a_{1}\) or \(a_{2}\) or \(a_{3}\) or \(a_{4}\) +or \(a_{5}\). Then if \(Mʻb_{1} = a_{1}\) and \(Nʻb_{1} \neq a_{1}\), +\(M\) precedes \(N\); if \(Mʻb_{1} = Nʻb_{1} = a_{1}\), and \(Mʻb_{2} = a_{1} . Nʻb_{2} \neq a_{1}\), +\(M\) precedes \(N\); and so on. Thus in this case the first term of +the series generated by \(P^{Q}\) is the one for which \(Mʻb = a_{1}\) +when \(b\) has any of the values \(b_{1}\), \(b_{2}\), \(b_{3}\), +\(b_{4}\). Thus the first term of the series is \(\iota ʻa_{1}\uparrow CʻQ\), +<i>i.e.</i> \(\iota ʻBʻP\uparrow CʻQ\). The next term will be +\[ +\begin{aligned} +\iota ʻa_{1}\uparrow &(\iota ʻb_{1}\cup \iota ʻb_{2}\cup \iota ʻb_{3})\unicode{x228d} \iota ʻa_{2}\uparrow \iota ʻb_{4},\\ +\textit{i.e.}\quad &\iota ʻBʻP\uparrow \text{D}ʻQ\unicode{x228d} 2_{P}\downarrow BʻQ. +\end{aligned} +\]<span class="pagenum" id="Page_459">[Pg 459]</span> +The next is \(\iota ʻBʻP\uparrow \text{D}ʻQ\unicode{x228d}3_{P}\downarrow Bʻ\breve{Q}\), +and so on. This makes it evident that our series has the structure +required of a series which is to represent the \(Q\)th power of \(P\).</p> + +<p>The two relations \(P\,\,\text{exp}\,\,Q\) and \(P^{Q}\) are ordinally +similar, since \(\dot{s}\) is one-one when its field is limited to +\(Cʻ(P\,\,\text{exp}\,\,Q)\). This follows from <a href="#*116·131">*116·131</a>, together with +\[ +\dot{\exists} !Q.\supset .Cʻ(P\,\,\text{exp}\,\,Q)=(CʻP)\,\,\text{exp}\,\,(CʻQ). +\]</p> + +<p>If \(S\) is a correlator of \(P\) and \(P'\), and \(T\) is a correlator +of \(Q\) and \(Q'\), then \((S \parallel \breve{T} )_{\in }\) and +\((S\parallel \breve{T})\), with their converse domains limited, +are respectively correlators of \((P\,\,\text{exp}\,\,Q)\) with +\((Pʻ\,\,\text{exp}\,\,Qʻ)\) and of \(P^{Q}\) with \(Pʻ^{Qʻ}\). This +shows that the relation-number of \((P\,\,\text{exp}\,\,Q)\) depends +only upon those of \(P\) and \(Q\), which is of course essential if +\((P\,\,\text{exp}\,\,Q)\) is to afford a definition of exponentiation.</p> + +<p>If the multiplicative axiom is assumed, then if \(R\) is a relation +which is like \(Q\), and whose field consists of relations which +are like \(P\), and \(R\in \text{Rel}^{2}\text{excl}\), the product +of \(R\) is like \((P\,\text{exp}\,\,Q)\). That is, if we put +\(\mu =\text{Nr}ʻP.\nu =\text{Nr}ʻQ\), so that \(R\) consists of +\(\nu\) terms each of which has \(\mu\) terms, the product of \(R\) has +\(\mu ^\nu\) terms. This gives the connection of multiplication with +exponentiation.</p> + +<p>There are two formal laws of exponentiation which hold for +relation-numbers, namely +\[ +\begin{aligned} +P^{Q}&\times P^{R}\text{smor}P^{Q\unicode{x2909}R}\\ +\text{and}\quad &(P^{Q})^{R}\text{smor}P^{R\times Q}. +\end{aligned} +\] +They both need a hypothesis: the first needs +\[ +\dot{\exists} !Q.\dot{\exists} !R.CʻQ\cap CʻR=\Lambda, +\] +while the second needs \(R\,\unicode{x2abd}\, J\) because it is proved by +means of the associative law (<a href="#*174·43">*174·43</a>).</p> + +<p>The first of the above formal laws can be generalized, by putting +\(\Sigma ʻS\) in place of \(Q\unicode{x2909}R\), and taking the product +of the various powers +\[ +P\,\text{exp}\,\,Q,\quad P\,\text{exp}\,\,Q', ..., +\] +where \(Q\), \(Q'\), ... \(\in CʻS\), and the products are taken in the +order determined by \(S\). The resulting generalization is +\[ +S\in \text{Rel}^{2}\text{excl}.S\,\unicode{x2abd}\, J.\supset .\{\text{Prod}ʻ(P \,\text{exp}\,\,)^{;}S\}\text{smor}\{P\,\text{exp}\,\,(\Sigma ʻS)\}. +\]</p> + +<p>The proof of this proposition results immediately from <a href="#*174·43">*174·43</a> and +<a href="#*162·35">*162·35</a>.</p> + +<p>The proof of the second of the formal laws is more difficult. We +observe, to begin with, that +\[ +P\,\text{exp}\,\,(R\times Q)=\text{Prod}ʻP\downarrow_{.,} ^{;}\Sigma ʻQ\downarrow_{.,} ^{;}R. +\]</p> + +<p><span class="pagenum" id="Page_460">[Pg 460]</span></p> + +<p>Assuming suitable hypotheses, this, by <a href="#*162·35">*162·35</a>, +\[ +=\text{Prod}ʻ\Sigma ʻ(P\downarrow_{.,} )\dagger ^{;}Q\downarrow_{.,} ^{;}R, +\] +which is like \(\text{Prod}ʻ\text{Prod}^{;}(P\downarrow_{.,})\dagger ^{;}Q\downarrow_{.,} ^{;}R\), +by <a href="#*174·43">*174·43</a>. But +\((P\,\text{exp}\,\,Q)\,\text{exp}\,\,R=\text{Prod}ʻ{\text{Prod}ʻP\downarrow_{.,}^{;}Q}\downarrow_{.,} ^{;}R\). +Thus our result will follow if we can prove +\[ +\{\text{Prod}^{;}(P\downarrow_{.,} )\dagger ^{;}Q\downarrow_{.,} ^{;}R\}\,\text{smor smor}\,\{(\text{Prod}ʻP\downarrow_{.,} + ^{;}Q)\downarrow_{.,} ^{;}R\}. +\] +Now one member of the field of \(\text{Prod}^{;}(P\downarrow_{.,})\dagger ^{;}Q\downarrow_{.,} ^{;}R\) +will be +\[ +\text{Prod}ʻP\downarrow_{.,} ^{;}Q\downarrow_{.,} z,\, \text{where}\, z\in CʻR. +\]</p> + +<p>This is like \(\text{Prod}ʻP\downarrow_{.,} ^{;}Q\), because +\(Q\downarrow_{.,} z\,\text{smor}\,Q\). Hence \(\text{Prod}^{;}(P\downarrow_{.,})\dagger ^{;}Q\downarrow_{.,}^{;}R\) +is a series of terms each of which is like +\(\text{Prod}ʻP\downarrow_{.,} ^{;}Q\), and the whole series of such +terms is like \(R\). If we assumed the multiplicative axiom, this would +suffice to prove the result. But it is possible to obtain our result +without assuming the multiplicative axiom.</p> + +<p>For this purpose, we proceed as follows. The correlator of +\[ +\text{Prod}ʻP\downarrow_{.,} ^{;}Q\downarrow_{.,} z\, \text{and}\, \text{Prod}ʻP\downarrow_{.,} ^{;}Q +\] +is \(\{\mid (\text{Cnv}ʻ\downarrow z)\}_{\in }\), by <a href="#*165·361">*165·361</a> and +<a href="#*172·3">*172·3</a>. Call this \(Mʻz\). Then +\[ +\begin{aligned} +M\in 1\rightarrow 1:&z\in CʻR.\supset _{z}.(Mʻz)\in (\text{Prod}ʻP\downarrow_{.,} ^{;}Q\downarrow_{.,} z)\,\overline{\text{smor}}\,(\text{Prod}ʻP\downarrow_{.,} + ^{;}Q):\\ +&z,w\in CʻR.\exists !\text{D}ʻMʻz\cap \text{D}ʻMʻw.\supset _{z,w}.z=w. +\end{aligned} +\] +This, by the help of two or three lemmas, suffices to prove that +\[ +\{\text{Prod}^{;}(P\downarrow_{.,} )\dagger ^{;}Q\downarrow_{.,} ^{;}R\}\,\text{smor smor}\,\{(\text{Prod}ʻP\downarrow_{.,} + ^{;}Q)\downarrow_{.,} ^{;}R\}, +\] +whence the result follows.</p> + +<p>The principal propositions of the present number are the following:</p> + +<p class="nind"><b>*176·1.</b> \(\vdash .P\,\text{exp}\,\,Q=\text{Prod}ʻP\downarrow_{.,} ^{;}Q=\text{D}^{;}\Pi ʻP\downarrow_{.,} ^{;}Q\)</p> + +<p class="nind"><b>*176·11.</b> \(\vdash .P^{Q}=\dot{s} ^{;}(P\,\text{exp}\,\,Q)=\dot{s} ^{;}\text{Prod}ʻP\downarrow_{.,} ^{;}Q=\dot{s} ^{;}\text{D}^{;}\Pi + ʻP\downarrow_{.,} ^{;}Q\)</p> + +<p>These propositions merely embody the definitions.</p> + +<p class="nind"><b>*176·14.</b> \(\vdash :\dot{\exists} !Q.\supset .Cʻ(P\,\text{exp}\,\,Q)=(CʻP)\,\text{exp}\,\,(CʻQ).CʻP^{Q}=(CʻP\uparrow CʻQ)_{\Delta }ʻCʻQ\)</p> + +<p class="nind"><b>*176·151.</b> \(\vdash \colon\ldotp P=\dot{\Lambda} .\lor.Q=\dot{\Lambda} :\equiv .P\,\text{exp}\,\,Q=\dot{\Lambda} .\equiv .P^{Q}=\dot{\Lambda}\)</p> + +<p>It will be observed that in relation-arithmetic, \(\mu ^{0} = 0\), +whereas in cardinal arithmetic \(\mu ^{0} = 1\). The difference is due to +the fact that there is no ordinal number 1 (cf. <a href="#*153">*153</a>).</p> + +<p><span class="pagenum" id="Page_461">[Pg 461]</span></p> + +<p class="nind"><b>*176·181.</b> \(\vdash .P^{Q}\,\text{smor}\,(P\,\text{exp}\,\,Q)\)</p> + +<p class="nind"><b>*176·182.</b> \(\vdash . (P \,\text{exp}\,\, Q)\,\text{smor}\,(\Pi ʻP \downarrow_{.,} ^{;}Q)\)</p> + +<p class="nind"><b>*176·19.</b> \[\begin{align}\vdash \colon\colon S (P^{Q}) &T . \equiv \colon\ldotp S, T \in (CʻP \uparrow CʻQ)_\Delta ʻCʻQ\colon\ldotp \\ +&(\exists y) : y \in CʻQ . (Sʻy) P (Tʻy) : yʻQy . yʻ \neq y . \supset _y . Sʻyʻ = Tʻyʻ\end{align}\]</p> + +<p class="nind"><b>*176·2.</b> \[\begin{align}\vdash : U \upharpoonright CʻR \in P\, &\overline{\text{smor}}\, R . W \upharpoonright CʻS \in Q \overline{\text{smor}} S . \supset .\\ +&(U \parallel \breve{W} )_\in \upharpoonright Cʻ(R \,\text{exp}\,\, S) \in (P \,\text{exp}\,\, Q) \,\overline{\text{smor}}\,(R \,\text{exp}\,\, S)\end{align}\]</p> + +<p class="nind"><b>*176·21.</b> With the same hypothesis, \((U \parallel \breve{W} ) \upharpoonright Cʻ(R^{S})\) correlates \(P^{Q}\) and \(R^{S}\).</p> + +<p class="nind"><b>*176·22.</b> \(\vdash : P \,\text{smor}\,R . Q\,\text{smor}\,S . \supset . (P \,\text{exp}\,\, Q)\,\text{smor}\,(R \,\text{exp}\,\, S) . P^{Q} \text{smor} R^{S}\)</p> + +<p class="nind"><b>*176·24.</b> \(\vdash \colon\ldotp \text{Mult ax} . \supset : +R \in \text{Rel}^{2}\text{excl} \cap \text{Nr}ʻQ . CʻR \subset \text{Nr}ʻP . \supset . \Pi ʻR\,\text{smor}\,(P \,\text{exp}\,\, Q)\)</p> + +<p>This proposition connects multiplication and exponentiation.</p> + +<p class="nind"><b>*176·31.</b> \(\vdash : \dot{\exists} ! Q . \supset . \overrightarrow{B}ʻ(P \,\text{exp}\,\, Q) = (\overrightarrow{B}ʻP) \,\text{exp}\,\, (CʻQ)\)</p> + +<p class="nind"><b>*176·311·32·321.</b> Similar propositions for \(\overrightarrow{B}ʻ\text{Cnv}ʻ(P \,\text{exp}\,\, Q)\), \(\overrightarrow{B}ʻ(P^{Q}), \overrightarrow{B}ʻ(\text{Cnv}ʻP^{Q})\)</p> + +<p class="nind"><b>*176·34.</b> \[\begin{align}\vdash : \dot{\exists} ! Q . \text{E}! &BʻP . \supset .\\ +&Bʻ(P \,\text{exp}\,\, Q) = (BʻP) \downarrow ʻʻCʻQ . Bʻ(P^{Q}) = ({℩}ʻBʻP) \uparrow CʻQ\end{align}\]</p> + +<p>We come next to the formal laws. We have</p> + +<p class="nind"><b>*176·42.</b> \[\begin{align}\vdash : \dot{\exists} ! Q . \dot{\exists} ! R . CʻQ \cap &CʻR = \Lambda . \supset . P^{Q} \times P^{R}\,\text{smor}\,P^{Q \unicode{x2909} + R} .\\ +&(P \,\text{exp}\,\, Q) \times (P \,\text{exp}\,\, R)\,\text{smor}\,P \,\text{exp}\,\, (Q \unicode{x2909} R)\end{align}\]</p> + +<p class="nind"><b>*176·44.</b> \(\vdash : S \in \text{Rel}^{2}\text{excl} . S \,\unicode{x2abd}\, J . \supset . \{\text{Prod}ʻ(P \,\text{exp}\,\,)^{;}S\} + \, \text{smor}\, \{P \,\text{exp}\, (\Sigma ʻS)\}\)</p> + +<p>This is an extension of <a href="#*176·42">*176·42</a>.</p> + +<p class="nind"><b>*176·57.</b> \[\begin{align}\vdash : R \,\unicode{x2abd}\, J . \supset . \{(P \,\text{exp}\, Q) \,\text{exp}\, R\}\,\text{smor}\,&\{P \,\text{exp}\,\, (R \times Q)\} .\\ +&(P^{Q})^{R}\,\text{smor}\,P^{R \times Q}\end{align}\]</p> + +<hr class="tb"> + +<p class="nind"><b>*176·01.</b> \(P \,\text{exp}\,\, Q = \text{Prod}ʻP\downarrow_{.,} ^{;}Q \quad\text{Df}\)</p> + +<p class="nind"><b>*176·02.</b> \(P^{Q} = \dot{s} ^{;}(P \,\text{exp}\,\, Q) \quad\text{Df}\)</p> + +<p class="nind"><b>*176·1.</b> \(\vdash . P \,\text{exp}\,\, Q = \text{Prod}ʻP\downarrow_{.,} ^{;}Q = \text{D}^{;}\Pi ʻP\downarrow_{.,} ^{;}Q \quad[(*176·01)]\)</p> + +<p class="nind"><b>*176·11.</b> \(\vdash . P^{Q} = \dot{s} ^{;}(P \,\text{exp}\,\, Q) = \dot{s} ^{;}\text{Prod}ʻP\downarrow_{.,} ^{;}Q = \dot{s} ^{;}\text{D}^{;}\Pi + ʻP\downarrow_{.,} ^{;}Q \quad[(*176·02)]\)</p> + +<p class="nind"><b>*176·12.</b> \[\begin{align}&\vdash \colon\colon \mu (P \,\text{exp}\,\, Q) \nu . \equiv \colon\ldotp \mu , \nu \in (CʻP) \,\text{exp}\,\, (CʻQ) \colon\ldotp \\ +&(\exists y, x, x') : x \downarrow y \in \mu . x' \downarrow y \in \nu . xPx' : zQy . z \neq y . w \downarrow z \in \mu . \supset _{w, z} + . w \downarrow z \in \nu \end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *165·21·12 . *163·12 . &\supset \vdash . F \upharpoonright P \downarrow_{.,} ʻʻCʻQ \in \text{Cls} \rightarrow 1 . &\qquad \text{(1)}\\ +[*85·1 . *115·1 . *33·5] \supset \vdash . \text{D}ʻʻF_\Delta ʻP \downarrow_{.,} ʻʻCʻQ &= \text{Prod}ʻCʻʻP \downarrow_{.,} ʻʻCʻQ\\ +[*165·12·14 . (*116·01)] &= (CʻP) \,\text{exp}\,\, (CʻQ) &\qquad \text{(2)}\\ +\vdash . *176·1 . *173·11 . *172·11 . *165·12 . \supset \\ +\vdash \colon\colon\ldotp \mu (P \,\text{exp}\,\, Q) \nu . &\equiv \colon\colon (\exists M, N, y) \colon\ldotp M, N \in F_\Delta ʻP \downarrow_{.,} ʻʻCʻQ :\\ +&y \in CʻQ . (MʻP \downarrow_{.,} y) (P \downarrow_{.,} y) (NʻP \downarrow_{.,} y) :\\ +&zQy . z \neq y . \supset _z . MʻP \downarrow_{.,} z = NʻP \downarrow_{.,} + z : \mu = \text{D}ʻM . \nu = \text{D}ʻN \colon\colon\\ +[*81·5 . (1) . *150·6] &\equiv \colon\colon (\exists M, N, y) \colon\ldotp M, N \in F_{\Delta }ʻP\downarrow_{.,} ʻʻCʻQ . \mu = \text{D}ʻM . \nu = \text{D}ʻN :\\ +&y \in CʻQ . \breve{{℩}} ʻ(\mu \cap \downarrow yʻʻCʻP)(\downarrow y^{;}P)\breve{{℩}} ʻ(\nu \cap \downarrow yʻʻCʻP) :\\ +&zQy . z \neq y . \supset _{z} . \breve{{℩}} ʻ(\mu \cap \downarrow yʻʻCʻP) = \breve{{℩}} ʻ(\nu \cap \downarrow yʻʻCʻP) \colon\colon\\ +[(2) . *150·55] &\equiv \colon\colon (\exists y) \colon\colon \mu , \nu \in (CʻP)\,\text{exp}\,\,(CʻQ) . y \in CʻQ \colon\ldotp (\exists x, xʻ) \colon\ldotp \\ +&x \downarrow y = \breve{{℩}} ʻ(\mu \cap \downarrow yʻʻCʻP) . xʻ \downarrow y = \breve{{℩}} ʻ(\nu \cap \downarrow yʻʻCʻP) . xPxʻ :\\ +&zQy . z \neq y . w \downarrow z = \breve{{℩}} ʻ(\mu \cap \downarrow zʻʻCʻP) .\\ +&wʻ \downarrow z = \breve{{℩}} ʻ(\nu \cap \downarrow zʻʻCʻP) . \supset _{z, w, wʻ} . w = wʻ \colon\colon\\ +[*116·11] &\equiv \colon\colon (\exists y) \colon\colon \mu , \nu \in (CʻP)\,\text{exp}\,\,(CʻQ) \colon\ldotp \\ +&(\exists x, xʻ) : x \downarrow y \in \mu . xʻ \downarrow y \in \nu . xPxʻ :\\ +&zQy . z \neq y . w \downarrow z \in \mu . \supset _{w, z} . w \downarrow z \in \nu \colon\colon\ldotp \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_462">[Pg 462]</span></p> + +<p>The above proposition is used in <a href="#*176·19">*176·19</a>. It has the merit of giving a +direct formula for \(P \,\text{exp}\,\, Q\), instead of one which proceeds +by way of \(\Pi ʻP\downarrow_{.,} ^{;}Q\).</p> + +<p class="nind"><b>*176·13.</b> \(\vdash : \dot{\exists} ! (P \,\text{exp}\,\, Q) . \equiv . \dot{\exists} ! P^{Q} . \equiv . \dot{\exists} ! \Pi ʻP\downarrow_{.,} ^{;}Q + \quad[*150·25 . *176·1·11]\)</p> + +<p class="nind"><b>*176·131.</b> \(\vdash : Q = \dot{\Lambda} . \supset . P \,\text{exp}\,\, Q = \dot{\Lambda} . P^{Q} = \dot{\Lambda} \quad[*165·241 . *173·2 . *150·42]\)</p> + +<p>Owing to this proposition, propositions stating analogies between +ordinal and cardinal powers mostly require the hypothesis +\(\dot{\exists} ! Q\) or its equivalent, because an ordinal power whose +index is zero is itself zero, whereas a cardinal power whose index is +zero is 1.</p> + +<p class="nind"><b>*176·132.</b> \[\begin{align}&\vdash : P = \dot{\Lambda} . \dot{\exists} ! Q . \supset . P \,\text{exp}\,\, Q = \dot{\Lambda} . P^{Q} = \dot{\Lambda} \\ +&[*165·244 . *172·14 . *176·13 . *150·42]\end{align}\]</p> + +<p class="nind"><b>*176·133.</b> \(\vdash . CʻP^{Q} = \dot{s} ʻʻCʻ(P \,\text{exp}\,\, Q) \quad[*176·11 . *150·22]\)</p> + +<p class="nind"><b>*176·14.</b> \(\vdash : \dot{\exists} ! Q . \supset . Cʻ(P \,\text{exp}\,\, Q) = (CʻP) \,\text{exp}\,\, (CʻQ) . CʻP^{Q} = (CʻP \uparrow CʻQ)_{\Delta }ʻCʻQ\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *165·243 . \supset \vdash : \text{Hp} . &\supset . \dot{\exists} ! P \downarrow_{.,} ^{;}Q .\\ +[*173·161 . *165·21] &\supset . Cʻ\text{Prod}ʻP \downarrow_{.,} ^{;}Q = \text{Prod}ʻCʻʻCʻP \downarrow_{.,} ^{;}Q .\\ +[*176·1 . *165·14] \supset . Cʻ(P \,\text{exp}\,\, Q) &= \text{Prod}ʻ(CʻP) \downarrow_{,,} ʻʻ(CʻQ)\\ +[(*116·01)]& = (CʻP) \,\text{exp}\,\, (CʻQ) &\qquad \text{(1)}\\ +\vdash . (1) . *176·133 . \supset \vdash : \text{Hp} . \supset . CʻP^{Q} &= \dot{s} ʻʻ\{(CʻP) \,\text{exp}\,\, (CʻQ)\}\\ +[*116·13] &= (CʻP \uparrow CʻQ)_{\Delta }ʻCʻQ &\qquad \text{(2)}\\ +\vdash . (1) . (2) . \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*176·15.</b> \(\vdash : \dot{\exists} ! P . \dot{\exists} ! Q . \equiv . \dot{\exists} ! (P \,\text{exp}\,\, Q) . \equiv . \dot{\exists} ! P^{Q}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *176·131·132 . &\supset \vdash : \dot{\exists} ! (P \,\text{exp}\,\, Q) . \supset . \dot{\exists} ! P . \dot{\exists} ! Q &\qquad \text{(1)}\\ +\vdash . *116·18 . *176·14 . \supset \vdash : \dot{\exists} ! P . \dot{\exists} ! Q . &\supset . \dot{\exists} ! Cʻ(P \,\text{exp}\,\, Q) .\\ +[*33·24] &\supset . \dot{\exists} ! (P \,\text{exp}\,\, Q) (2) +\vdash . (1) . (2) . *176·13 . \supset \vdash . \text{Prop}\ +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_463">[Pg 463]</span></p> + +<p class="nind"><b>*176·151.</b> \(\vdash \colon\ldotp P = \dot{\Lambda} .\lor.Q = \dot{\Lambda} :\equiv .P \,\text{exp}\, Q = \dot{\Lambda} . \equiv . P^{Q} = \dot{\Lambda} \quad[*176·15]\)</p> + +<p class="nind"><b>*176·16.</b> \[\begin{align}&\vdash . Cʻ(P \,\text{exp}\,\, Q) \subset (CʻP) \,\text{exp}\,\, (CʻQ) . CʻP^{Q} \subset (CʻP \uparrow CʻQ)_{\Delta }ʻCʻQ\\ +&[*176·14·151]\end{align}\]</p> + +<p class="nind"><b>*176·18.</b> \(\vdash . \dot{s} \upharpoonright Cʻ(P \,\text{exp}\,Q) \in (P^{Q})\,\overline{\text{smor}}\,(P \,\text{exp}\,\, Q)\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *116·131 . *176·14 . \supset \\ +\vdash : \dot{\exists} ! Q . \supset . \dot{s} \upharpoonright Cʻ(P \,\text{exp}\,\, Q) \in (CʻP^{Q}) \,\overline{\text{sm}}\, Cʻ(P \,\text{exp}\,\, Q) &\qquad \text{(1)}\\ +\vdash . (1) . *176·11 . *151·191 . \supset \\ +\vdash : \dot{\exists} ! Q . \supset . \dot{s} \upharpoonright Cʻ(P \,\text{exp}\,\, Q) \in (P^{Q})\,\overline{\text{smor}}\,(P \,\text{exp}\,\, Q) &\qquad \text{(2)}\\ +\vdash . *176·151 . *150·42 . *72·1 . \supset \\ +\vdash : Q = \Lambda . \supset . \dot{s} \upharpoonright Cʻ(P \,\text{exp}\,\, Q) \in (P^{Q})\,\overline{\text{smor}}\,(P \,\text{exp}\,\, Q) &\qquad \text{(3)}\\ +\vdash . (2) . (3) . \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*176·181.</b> \(\vdash . P^{Q}\,\text{smor}\,(P \,\text{exp}\,\, Q) \quad[*176·18]\)</p> + +<p class="nind"><b>*176·182.</b> \(\vdash . (P \,\text{exp}\,\, Q)\,\text{smor}\,(\Pi ʻP \downarrow_{.,} ^{;}Q) \quad[*176·1 . *173·16 . *165·21]\)</p> + +<p class="nind"><b><a id="*176·19">*176·19</a>.</b> \[\begin{align}\vdash \colon\colon S(P^{Q})&T . \equiv \colon\ldotp S, T \in (CʻP \uparrow CʻQ)_{\Delta }ʻCʻQ \colon\ldotp \\ +&(\exists y) : y \in CʻQ . (Sʻy) P (Tʻy) : y'Qy . y' \neq y . \supset _{yʻ} . Sʻy' = Tʻy'\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *176·11·12 . \supset \\ +\vdash \colon\colon S(P^{Q})T . &\equiv \colon\ldotp (\exists \mu , \nu ) \colon\ldotp \mu , \nu \in (CʻP) \,\text{exp}\,\, (CʻQ) . S = \dot{s} ʻ\mu . T = \dot{s} ʻ\nu \colon\ldotp \\ +&(\exists y, x, x') : y \in CʻQ . x \downarrow y \in \mu . x' \downarrow y \in \nu . xPx' :\\ +&y'Qy . y' \neq y . w \downarrow y' . w \downarrow y' \in \mu . \supset _{y', w} . w \downarrow y' \in \nu \colon\ldotp \\ +[*56·4] &\equiv \colon\ldotp (\exists \mu , \nu ) \colon\ldotp \mu , \nu \in (CʻP) \,\text{exp}\,\, (CʻQ) . S = \dot{s} ʻ\mu . T = \dot{s} ʻ\nu \colon\ldotp \\ +&(\exists y, x, x') : y \in CʻQ . xSy . x'Ty . xPx' : y'Qy . y' \neq y . wSy' . \supset _{y', w} . wTy' \colon\ldotp \\ +[*116·13 . *80·3] &\equiv \colon\ldotp S, T \in (CʻP \uparrow CʻQ_{\Delta }ʻCʻQ \colon\ldotp (\exists y) : y \in CʻQ . (Sʻy) P (Tʻy) :\\ +&y'Qy . y' \neq y . \supset _{y'} Sʻy' = Tʻy'\colon\colon \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p>The above proposition is often useful, since it gives a direct formula +for \(P^{Q}\), not one which passes by way of \(P \,\text{exp}\,\, Q\) or +\(\Pi ʻP \downarrow_{.,} ^{;}Q\).</p> + +<p class="nind"><b>*176·2.</b> \[\begin{align}\vdash : U \upharpoonright CʻR \in P\,\overline{\text{smor}}\,&R . W \upharpoonright CʻS \in Q \,\overline{\text{smor}}\,S . \supset .\\ +&(U \parallel \breve{W} )_{\in } \upharpoonright Cʻ(R \,\text{exp}\,\, S) \in (P \,\text{exp}\,\, Q)\,\overline{\text{smor}}\,(R \,\text{exp}\,\, S)\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *165·362 . \supset \vdash : \text{Hp} . \supset . (U \parallel \breve{W} ) \upharpoonright Cʻ\Sigma ʻR \downarrow_{.,} ^{;}S \in (P \downarrow_{.,} ^{;}Q) + \,\overline{\text{smor smor}}\, (R \downarrow_{.,} ^{;}S) .\\ +[*173·3] \supset . (U \parallel \breve{W} )_\in \upharpoonright Cʻ\text{Prod}ʻR \downarrow_{.,} ^{;}S \in (\text{Prod}ʻP \downarrow_{.,} ^{;}Q) + \,\overline{\text{smor}}\, (\text{Prod}ʻR \downarrow_{.,} ^{;}S) &\qquad \text{(1)}\\ +\vdash . (1) . *176·1 . \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_464">[Pg 464]</span></p> + +<p class="nind"><b><a id="*176·21">*176·21</a>.</b> \[\begin{align}\vdash : U \upharpoonright CʻR \in P\,\overline{\text{smor}}\, R . W \upharpoonright CʻS \in &Q\,\overline{\text{smor}}\,S . \supset .\\ +&(U \parallel \breve{W} ) \upharpoonright Cʻ(R^{S}) \in (P^{Q})\,\overline{\text{smor}}\,(R^{S})\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *176·2·18 . *151·401 . &\supset \vdash : \text{Hp} . \supset . \dot{s} ^{;}(U \parallel \breve{W} )_{\in } \upharpoonright Cʻ(R \,\text{exp}\,\, S) \in (P^{Q}) + \overline{\text{smor}} (R^{S})\\ +[*150·961] &\supset . (U \parallel \breve{W} ) \upharpoonright \dot{s} ʻʻCʻ(R \,\text{exp}\,\, S) \in (P^{Q}) \overline{\text{smor}} (R^{S}) &\qquad \text{(1)}\\ +\vdash . (1) . *176·11 . *150·22 . \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*176·22.</b> \[\begin{align}&\vdash : P \,\text{smor} R . Q\;\text{smor}\,S . \supset . (P \,\text{exp}\,\, Q)\,\text{smor}\,(P \,\text{exp}\,\, S) . P^{Q}\,\text{smor}\,R^{S}\\ +&[*176·2·21]\end{align}\]</p> + +<p class="nind"><b>*176·23.</b> \(\vdash : R \,\text{smor smor}\, P \downarrow_{.,} ^{;}Q . \supset . \Pi ʻR\,\text{smor}\,(P \,\text{exp}\,\, Q)\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *172·44 . \supset \vdash : \text{Hp} . \supset . \Pi ʻR \,\text{smor}\,\Pi ʻP \downarrow_{.,} ^{;}Q &\qquad \text{(1)}\\ +\vdash . (1) . *176·182 . \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*176·24.</b> \[\begin{align}\vdash \colon\ldotp &\text{Mult ax} . \supset :\\ +&R \in \text{Rel}^{2}\text{excl} \cap \text{Nr}ʻQ . CʻR \subset \text{Nr}ʻP . \supset . \Pi ʻP \,\text{smor}\,(P \,\text{exp}\,\, Q)\\ +&[*165·38 . *176·23]\end{align}\]</p> + +<p class="nind"><b>*176·3.</b> \(\vdash . \text{Cnv}ʻ(P^{Q}) = (\breve{P} )^{Q}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *176·19 . \supset \\ +\vdash \colon\colon T(\breve{P} )^{Q}S . &\equiv \colon\ldotp S, T \in (CʻP \uparrow CʻQ)_{\Delta }ʻCʻQ \colon\ldotp \\ +&(\exists y) : y \in CʻQ . (Tʻy) \breve{P} (Sʻy) : yʻQy . yʻ \neq y . \supset _{yʻ} . Sʻyʻ = Tʻyʻ \colon\ldotp \\ +[*176·19] &\equiv \colon\ldotp S(P^{Q})T \colon\colon \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*176·31.</b> \(\vdash : \dot{\exists} ! Q . \supset . \overrightarrow{B}ʻ(P \,\text{exp}\,\, Q) = (\overrightarrow{B}ʻP) \,\text{exp}\,\, (CʻQ)\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *165·21 . *163·12 . *71·221 . *93·1 . &\supset \vdash . B \upharpoonright CʻP \downarrow_{.,} ^{;}Q \in \text{Cls} \rightarrow 1 &\qquad \text{(1)}\\ +\vdash . *165·12·01 . *37·67 . \supset \vdash . \overrightarrow{B}ʻʻCʻP \downarrow_{.,} ^{;}Q + &= \hat{\alpha} \{(\exists z) . z \in CʻQ . \alpha = \overrightarrow{B}ʻ \downarrow z^{;}P\}\\ +[*165·251 . *151·5 . *38·3] &= (\overrightarrow{B}ʻP) \downarrow_{,,} ʻʻCʻQ &\qquad \text{(2)}\\ +\vdash . *172·162 . *165·243 . &\supset \vdash : \dot{\exists} ! Q . \supset . \overrightarrow{B}ʻ\Pi ʻP \downarrow_{.,} ^{;}Q = B_{\Delta }ʻCʻP \downarrow_{.,} + ^{;}Q .\\ +[*173·16 . *165·21 . *151·5] \supset . \overrightarrow{B}ʻ(P \,\text{exp}\,\, Q) &= \text{D}ʻʻB_{\Delta }ʻCʻP \downarrow_{.,} ^{;}Q\\ +[*85·1 . (1) . *115·1] &= \text{Prod}ʻ\overrightarrow{B}ʻʻCʻP \downarrow_{.,} ^{;}Q\\ +[(2)] &= \text{Prod}ʻ(\overrightarrow{B}ʻP) \downarrow_{,,} ʻʻCʻQ\\ +[(*116·01)] &= (\overrightarrow{B}ʻP) \,\text{exp}\,\, (CʻQ) : \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*176·311.</b> \[\begin{align}&\vdash : \dot{\exists} ! Q . \supset . \overrightarrow{B}ʻ\text{Cnv}ʻ(P \,\text{exp}\,\, Q) = (\overrightarrow{B}ʻP) \,\text{exp}\,\, (CʻQ)\\ +&[\text{Proof as in *176·31}]\end{align} +\]</p> + +<p><span class="pagenum" id="Page_465">[Pg 465]</span></p> + +<p class="nind"><b>*176·32.</b> \(\vdash :\dot{\exists} !Q.\supset .\overrightarrow{B}ʻ(P^{Q})=(\overrightarrow{B}ʻP\uparrow CʻQ)_{\Delta }ʻCʻQ\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*176·31·18.*151·5.\supset \\ +\vdash :\text{Hp}.\supset .\overrightarrow{B}ʻ(P^{Q})&=\dot{s}ʻʻ(\overrightarrow{B}ʻP)\,\text{exp}\,\,(CʻQ)\\ +[*116·13] &=(\overrightarrow{B}ʻP\uparrow CʻQ)_{\Delta }ʻCʻQ:\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*176·321.</b> \(\vdash :\dot{\exists} !Q.\supset .\overrightarrow{B}ʻ\text{Cnv}ʻ(P^{Q})=(\overrightarrow{B}ʻ\breve{P} \uparrow CʻQ)_{\Delta }ʻCʻQ \quad[*176·32·3]\)</p> + +<p class="nind"><b>*176·33.</b> \[\begin{align}\vdash \colon\ldotp \dot{\exists} !Q.\supset :&\exists !\overrightarrow{B}ʻ(P\,\text{exp}\,\,Q).\equiv .\exists !\overrightarrow{B}ʻ(P^{Q}).\equiv + .\exists !\overrightarrow{B}ʻP:\\ +&\exists !\overrightarrow{B}ʻ\text{Cnv}ʻ(P\,\text{exp}\,\,Q).\equiv .\exists !\overrightarrow{B}ʻ\text{Cnv}ʻ(P^{Q}).\equiv .\exists !\overrightarrow{B}ʻ\breve{P}\\ +&[*176·31·311·32·321.*116·18·15]\end{align}\]</p> + +<p class="nind"><b>*176·34.</b> \[\begin{align}\vdash :\dot{\exists} !Q.\text{E}!&BʻP.\supset .\\ +&Bʻ(P\,\text{exp}\,\,Q)=(BʻP)\downarrowʻʻCʻQ.Bʻ(P^{Q})=(\iota ʻBʻP)\uparrow CʻQ\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*176·31.\supset \vdash :\text{Hp}.\supset .\overrightarrow{B}ʻ(P\,\text{exp}\,\,Q)&=(\iota ʻBʻP)\,\text{exp}\,\,(CʻQ)\\ +[(*116·01)] &=\text{Prod}ʻ(\iota ʻBʻP)\downarrow_{,,}ʻʻCʻQ\\ +[*38·3.*53·31] &=\text{Prod}ʻ\iotaʻʻ(BʻP)\downarrowʻʻCʻQ\\ +[*115·143] &=\iota ʻ{(BʻP)\downarrowʻʻCʻQ} &\qquad \text{(1)}\\ +\vdash .*176·32.\supset \vdash :\text{Hp}.\supset .\overrightarrow{B}ʻ(P^{Q})&=(\iota ʻBʻP\uparrow CʻQ)_{\Delta }ʻCʻQ\\ +[*116·12.*51·4] &=\iota ʻ{(\iota ʻBʻP)\uparrow CʻQ} &\qquad \text{(2)}\\ +\vdash .(1).(2).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*176·341.</b> \[\begin{align}\vdash :\dot{\exists} !Q.\text{E}!&Bʻ\breve{P} .\supset .\\ +&Bʻ\text{Cnv}ʻ(P\,\text{exp}\,\,Q)=(Bʻ\breve{P} )\downarrowʻʻCʻQ.Bʻ\text{Cnv}ʻ(P^{Q})=(\iota ʻBʻ\breve{P} )\uparrow CʻQ\\ +&[\text{Proof as in *176·34}]\end{align}\]</p> + +<p class="nind"><b>*176·35.</b> \(\vdash :P\,\unicode{x2abd}\, Q.\supset .P^{R}\,\unicode{x2abd}\, Q^{R}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*116·12.\supset \vdash :\text{Hp}.\supset .(CʻP\uparrow CʻR)_{\Delta }ʻCʻR\subset (CʻQ\uparrow CʻR)_{\Delta }ʻCʻR &\qquad \text{(1)}\\ +\vdash .(1).*176·19.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p>The above proposition is used in the theory of finite ordinals (*261·64).</p> + +<p>The following propositions are concerned in proving (with a suitable +hypothesis) +\[ +P^{Q}\times P^{R}\,\text{smor}\,P^{Q\unicode{x2909}R} +\] +and its extension +\[ +\{\text{Prod}ʻ(P \,\text{exp}\,\,)^{;}S\}\,\text{smor}\,\{P\,\text{exp}\,\,(\Sigma ʻS)\}. +\]</p> + +<p><span class="pagenum" id="Page_466">[Pg 466]</span></p> + +<p class="nind"><b><a id="*176·4">*176·4</a>.</b> \[\begin{align}&\vdash :\dot{\exists} !Q.\dot{\exists} !R.PʻʻCʻQ\cap PʻʻCʻR=\Lambda.CʻQ\cup CʻR\subset \text{ᗡ}ʻP.\supset .\\ +&\dot{s} \mid C\upharpoonright Cʻ\{(\Pi ʻP^{;}Q)\times (\Pi ʻP^{;}R)\}\in \{\Pi ʻP^{;}(Q\unicode{x2909}R)\}\,\overline{\text{smor}}\,\{(\Pi ʻP^{;}Q)\times + (\Pi ʻP^{;}R)\}\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*172·34.*150·22·24. &\supset \vdash :\text{Hp}.\supset .\\ +&\dot{s} \mid C\upharpoonright Cʻ{(\Pi ʻP^{;}Q)\times (\Pi ʻP^{;}R)}\in {\Pi ʻ(P^{;}Q\unicode{x2909}P^{;}R)}\,\overline{\text{smor}}\,\{(\Pi ʻP^{;}Q)\times + (\Pi ʻP^{;}R)\} &\qquad \text{(1)}\\ +\vdash .*162·36. &\supset \vdash :\text{Hp}.\supset .P^{;}Q\unicode{x2909}P^{;}R=P^{;}(Q\unicode{x2909}R) &\qquad \text{(2)}\\ +\vdash .(1).(2).&\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*176·41">*176·41</a>.</b> \[\begin{align}\vdash :\dot{\exists} !Q.\dot{\exists} !R.&PʻʻCʻQ\cap PʻʻCʻR=\Lambda.CʻQ\cup CʻR\subset \text{ᗡ}ʻP.\supset .\\ +&\Pi ʻP^{;}(Q\unicode{x2909}R)\,\text{smor}\,(\Pi ʻP^{;}Q)\times (\Pi ʻP^{;}R) \quad[*176·4]\end{align}\]</p> + +<p class="nind"><b><a id="*176·42">*176·42</a>.</b> \[\begin{align}\vdash :\dot{\exists} !Q.\dot{\exists} !R.CʻQ\cap &CʻR=\Lambda.\supset .P^{Q}\times P^{R} \text{smor} P^{Q\unicode{x2909}R}.\\ +&(P \,\text{exp}\,\, Q)\times (P \,\text{exp}\,\, R)\,\text{smor}\,P \,\text{exp}\,\, (Q\unicode{x2909}R)\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*72·411.*165·22.\supset \\ +\vdash :\dot{\exists} !P.CʻQ\cap CʻR=\Lambda.&\supset .P\downarrow_{.,} ʻʻCʻQ\cap P\downarrow_{.,} ʻʻCʻR=\Lambda &\qquad \text{(1)}\\ +\vdash .(1).*176·41 \frac{P\downarrow_{.,}}{P}.*38·12.*33·431.\supset \\ +\vdash :\text{Hp}.\dot{\exists} !P.&\supset .\Pi ʻP\downarrow_{.,} ^{;}(Q\unicode{x2909}R)\,\text{smor}\,(\Pi ʻP\downarrow_{.,} ^{;}Q)\times (\Pi ʻP\downarrow_{.,} + ^{;}R).\\ +[*176·182.*166·23]&\supset .P \,\text{exp}\,\, (Q\unicode{x2909}R)\,\text{smor}\,(P \,\text{exp}\,\, Q)\times (P \,\text{exp}\,\, R). &\qquad \text{(2)}\\ +[*176·181.*166·23]&\supset .P^{Q\unicode{x2909}R}\text{smor}P^{Q}\times P^{R} &\qquad \text{(3)}\\ +\vdash .*176·151.*166·13.*153·101.\supset \\ +\vdash :P=\dot{\Lambda} .&\supset .P \,\text{exp}\,\, (Q\unicode{x2909}R)\,\text{smor}\,(P \,\text{exp}\,\, Q)\times (P \,\text{exp}\,\, R).\\ +&P^{Q\unicode{x2909}R}\,\text{smor}\,P^{Q}\times P^{R} &\qquad \text{(4)}\\ +\vdash .(2).(3).(4).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*176·43">*176·43</a>.</b> \[\begin{align}\vdash :S\in &\text{Rel}^{2}\text{excl}.S\,\unicode{x2abd}\, J.\supset .\\ +&s\upharpoonright Cʻ\text{Prod}ʻ(P \,\text{exp}\,\,)^{;}S\in {P \,\text{exp}\,\,(\Sigma ʻS)}\,\overline{\text{smor}}\,\text{Prod}ʻ(P \,\text{exp}\,\,)^{;}S\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*165·22.*163·3. &\supset \vdash :\text{Hp}.\dot{\exists} !P.\supset .(P\downarrow_{.,} )\dagger ^{;}S\in \text{Rel}^{2}\text{excl} &\qquad \text{(1)}\\ +\vdash .*162·35.*38·12.*33·431. &\supset \vdash .\Sigma ʻ(P\downarrow_{.,} )\dagger ^{;}S=P\downarrow_{.,} ^{;}\Sigma ʻS &\qquad \text{(2)}\\ +\vdash .*165·21.(2). &\supset \vdash .\Sigma ʻ(P \downarrow_{.,} )\dagger ^{;}S\in \text{Rel}^{2}\text{excl} &\qquad \text{(3)}\\ +\vdash .(1).(3).*174·3. &\supset \vdash :\text{Hp}.\dot{\exists} !P.\supset .(P\downarrow_{.,} )\dagger ^{;}S\in \text{Rel}^{3}\text{arithm} &\qquad \text{(4)}\\ +\vdash .*165·223.\text{Transp}. \supset \vdash \colon\ldotp \text{Hp}.\dot{\exists} !P.&\supset :Q\neq R.\supset .P\downarrow_{.,} ^{;}Q\neq P\downarrow_{.,} + ^{;}R:\\ +[*150·4.*72·14] &\supset :(P\downarrow_{.,} )\dagger ^{;}S\,\unicode{x2abd}\, J &\qquad \text{(5)}\\ +\vdash .*176·1. &\supset \vdash .\text{Prod}ʻ(P\,\text{exp}\,\,)^{;}S=\text{Prod}ʻ\text{Prod}^{;}(P\downarrow_{.,} )\dagger ^{;}S &\qquad \text{(6)}\\ +\vdash .(4).(5).(6).*174·42.\supset\\ +\vdash :\text{Hp}.\dot{\exists} !P.\supset .\\ +&s\upharpoonright Cʻ\text{Prod}ʻ(P\,\text{exp}\,\,)^{;}S\in \{\text{Prod}ʻ\Sigma ʻ(P\downarrow_{.,} )\dagger ^{;}S\}\overline{\text{smor}} + \{\text{Prod}ʻ(P\,\text{exp}\,\,)^{;}S\}.\\ +[(2)] &\supset .s\upharpoonright Cʻ\text{Prod}ʻ(P\,\text{exp}\,\,)^{;}S\in \{\text{Prod}ʻP\downarrow_{.,} ^{;}\Sigma + ʻS\}\overline{\text{smor}} \{\text{Prod}ʻ(P\,\text{exp}\,\,)^{;}S\}.\\ +[*176·1]&\supset .s\upharpoonright Cʻ(P\,\text{exp}\,\,)^{;}S\in \{P\,\text{exp}\,\,(\Sigma ʻS)\}\overline{\text{smor}} \{\text{Prod}ʻ(P\,\text{exp}\,\,)^{;}S\} &\qquad \text{(7)}\\ +\vdash .*176·151.*173·21.*172·13·14.\supset \\ +\vdash :P=\dot{\Lambda} .&\supset .\text{Prod}ʻ(P\,\text{exp}\,\,)^{;}S=\dot{\Lambda} .P\,\text{exp}\,\,(\Sigma ʻS)=\dot{\Lambda} &\qquad \text{(8)}\\ +\vdash .(7).(8).*173·2.*164·32.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_467">[Pg 467]</span></p> + +<p class="nind"><b>*176·44.</b> \[\begin{align}&\vdash :S\in \text{Rel}^{2}\text{excl}.S\,\unicode{x2abd}\, J.\supset .\{\text{Prod}ʻ(P\,\text{exp}\,\,)^{;}S\}\,\text{smor}\,\{P\,\text{exp}\,\,(\Sigma ʻS)\} +&[*176·43]\end{align}\]</p> + +<p>The following propositions are lemmas for +\[ +R\,\unicode{x2abd}\, J.\supset .(P^{Q})^{R}\,\text{smor}\,P^{R\times Q}. +\]</p> + +<p class="nind"><b>*176·5.</b> \[\begin{align}\vdash \colon\ldotp &M\upharpoonright CʻR\in 1\rightarrow 1.CʻR\subset \text{ᗡ}ʻM.CʻQ\subset pʻ\text{ᗡ}ʻʻʻMʻʻCʻR.\\ +&MʻʻCʻR\subset 1\rightarrow 1:z,z'\in CʻR.\exists !\text{D}ʻMʻz\cap \text{D}ʻMʻz'.\supset _{z,z'}.z=z':\\ +&T=\hat{x} \hat{X} \{(\exists u,z).u\in CʻQ.z\in CʻR.x=(Mʻz)ʻu.X=u\downarrow (Mʻz)\}:\\ +&\supset .T\in 1\rightarrow 1\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*21·33.&\supset \vdash \colon\ldotp \text{Hp}.\supset :xTX.x'TX.\supset .\\ +&(\exists u,u',z,z').u,u'\in CʻQ.z,z'\in CʻR.x=(Mʻz)ʻu.x'=(Mʻz')ʻu'.\\ +&X=u\downarrow (Mʻz).X=u'\downarrow (Mʻz').\\ +[*55·202] &\supset .(\exists u,u',z,z').x=(Mʻz)ʻu.x'=(Mʻz')ʻu'.u=u'.Mʻz=Mʻz'.\\ +[*13·22] &\supset .x=x' &\qquad \text{(1)}\\ +\vdash .*21·33.&\supset \vdash \colon\ldotp \text{Hp}.\supset :xTX.xTX'.\supset .\\ +&(\exists u,u',z,z').u,u'\in CʻQ.z,z'\in CʻR.x=(Mʻz)ʻu.x=(Mʻz')ʻu'.\\ +&X=u\downarrow (Mʻz).X'=u'\downarrow (Mʻz').\\ +[*33·43.\text{Hp}] &\supset .(\exists u,u',z,z').u,u'\in CʻQ.z\in CʻR.z=z'.x=(Mʻz)ʻu=(Mʻz')ʻu'.\\ +&X=u\downarrow (Mʻz).X'=u'\downarrow (Mʻz').\\ +[*13·195] &\supset .(\exists u,u',z).u,u'\in CʻQ.z\in CʻR.x=(Mʻz)ʻu=(Mʻz')ʻu'.\\ +&X=u\downarrow (Mʻz).X'=u'\downarrow (Mʻz).\\ +[*71·59.\text{Hp}] &\supset .(\exists u,u',z).u=u'.X=u\downarrow (Mʻz).X'=u'\downarrow (Mʻz).\\ +[*13·195] &\supset .X=X' &\qquad \text{(2)}\\ +\vdash .(1).(2).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*176·501.</b> \(\vdash :\text{Hp}*176·5.\supset .\text{ᗡ}ʻT=Cʻ\Sigma ʻQ\downarrow_{.,} ^{;}M^{;}R\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*71·16.\supset \vdash \colon\ldotp \text{Hp}.&\supset :z\in CʻR.u\in CʻQ.\supset .\text{E}!(Mʻz)ʻu.\\ +[*21·33] &\supset .u\downarrow (Mʻz)\in \text{ᗡ}ʻT &\qquad \text{(1)}\\ +\vdash .*21·33.\supset \\ +\vdash \colon\ldotp \text{Hp}.&\supset :X\in \text{ᗡ}ʻT.\supset .(\exists z,u).z\in CʻR.u\in CʻQ.X=u\downarrow (Mʻz) &\qquad \text{(2)}\\ +\vdash .(1).(2).\supset \vdash \colon\ldotp \text{Hp}.\supset :X\in \text{ᗡ}ʻT.&\equiv .(\exists z,u).z\in CʻR.u\in CʻQ.X=u\downarrow (Mʻz).\\ +[*71·4] &\equiv .(\exists Z,u).Z\in MʻʻCʻR.u\in CʻQ.X=u\downarrow Z.\\ +[*150·22] &\equiv .(\exists Z,u).Z\in CʻM^{;}R.u\in CʻQ.X=u\downarrow Z.\\ +[*165·16.*113·101] &\equiv .X\in Cʻ\Sigma ʻQ\downarrow_{.,} ^{;}M^{;}R\colon\ldotp \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_468">[Pg 468]</span></p> + +<p class="nind"><b>*176·502.</b> \(\vdash :\text{Hp}*176·5.z\in CʻR.\supset .T^{;}Q\downarrow_{.,} ʻMʻz=\dagger QʻMʻz\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*150·4.*165·01.*176·5.\supset \\ +\vdash :\text{Hp}.\supset .T^{;}Q\downarrow_{.,} ʻMʻz=\hat{x} \hat{y} &\{(\exists u,v).uQv.x=Tʻ\downarrow (Mʻz)ʻu.y=Tʻ\downarrow (Mʻz)ʻv\}\\ +[\text{Hp}.*176·5·501] &=\hat{x} \hat{y} \{(\exists u,v).uQv.x=(Mʻz)ʻu.y=(Mʻz)ʻv\}\\ +[*150·4] &=(Mʻz)^{;}Q\\ +[*150·1] &=\dagger QʻMʻz:\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*176·503.</b> \(\vdash :\text{Hp}*176·5.\supset .T\in (\dagger Q^{;}M^{;}R)\,\overline{\text{smor smor}}\, (Q\downarrow_{.,} + ^{;}M^{;}R)\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*176·502.*150·1·35.\supset \vdash :\text{Hp}.\supset .T\dagger ^{;}Q\downarrow_{.,} ^{;}M^{;}R=\dagger Q^{;}M^{;}R &\qquad \text{(1)}\\ +\vdash .(1).*176·5·501.*164·1.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*176·51.</b> \[\begin{align}\vdash \colon\ldotp M\upharpoonright &CʻR\in 1\rightarrow 1.MʻʻCʻR\subset 1\rightarrow 1.\\ +&CʻR\subset \text{ᗡ}ʻM.CʻQ\subset pʻ\text{ᗡ}ʻʻMʻʻCʻR:\\ +&z,z'\in CʻR.\exists !\text{D}ʻMʻz\cap \text{D}ʻMʻz'.\supset _{z,z'}.z=z':\supset .\dagger Q^{;}M^{;}R\,\text{smor}\,\text{smor}Q\downarrow_{.,} + ^{;}R\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*165·361.\supset \vdash :\text{Hp}.\supset .Q\downarrow_{.,} ^{;}M^{;}R\,\text{smor smor}\,Q\downarrow_{.,} ^{;}R &\qquad \text{(1)}\\ +\vdash .(1).*176·503.*164·221.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*176·52.</b> \(\vdash \colon\ldotp z\in CʻR.\supset _{z}.Mʻz\in (Pʻz)\,\overline{\text{smor}}\,Q:\supset .P^{;}R=\dagger Q^{;}M^{;}R\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*151·11.\supset \vdash \colon\ldotp \text{Hp}.\supset :z\in CʻR.\supset _{z}.Pʻz&=(Mʻz)^{;}Q\\ +[*150·1] &=\dagger QʻMʻz &\qquad \text{(1)}\\ +\vdash .(1).*150·35.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*176·53.</b> \[\begin{align}&\vdash \colon\ldotp M\upharpoonright CʻR\in 1\rightarrow 1:z\in CʻR.\supset _{z}.Mʻz\in (Pʻz)\,\overline{\text{smor}}\,Q:\\ +&z,z'\in CʻR.\exists !CʻPʻz\cap CʻPʻz'.\supset _{z,z'}.z=z':\supset .P^{;}R\,\text{smor smor}\,Q\downarrow_{.,} ^{;}R\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*14·21. \supset \vdash \colon\ldotp \text{Hp}.&\supset :z\in CʻR.\supset _{z}.\text{E}!Mʻz: &\qquad \text{(1)}\\ +[*33·43] &\supset :CʻR\subset \text{ᗡ}ʻM &\qquad \text{(2)}\\ +\vdash .*151·11.\supset \vdash \colon\ldotp \text{Hp}.&\supset :z\in CʻR.\supset _{z}.Mʻz\in 1\rightarrow 1:\\ +[*37·61.(1)] &\supset :MʻʻCʻR\subset 1\rightarrow 1 &\qquad \text{(3)}\\ +\vdash .*151·11·131. \supset \vdash \colon\ldotp \text{Hp}.&\supset :z,\in CʻR.\supset _{z}.\text{D}ʻMʻz=CʻPʻz:\\ +[\text{Hp}] &\supset :z,z'\in CʻR.\exists !\text{D}ʻMʻz\cap \text{D}ʻMʻz'.\supset _{z,z'}.z=z' &\qquad \text{(4)}\\ +\vdash .*151·11. &\supset \vdash \colon\ldotp \text{Hp}.\supset :z\in CʻR.\supset .\text{ᗡ}ʻMʻz=CʻQ:\\ +[*37·63] &\supset :Z\in MʻʻCʻR.\supset .\text{ᗡ}ʻZ=CʻQ:\\ +[*40·15] &\supset :CʻQ\subset pʻ\text{ᗡ}ʻʻMʻʻCʻR &\qquad \text{(5)}\\ +\vdash .(2).(3).(4).(5).*176·51.&\supset \vdash :\text{Hp}.\supset .\dagger Q^{;}M^{;}R \text{smor} \text{smor} Q\downarrow_{.,} ^{;}R &\qquad \text{(6)}\\ +\vdash .(6).*176·52.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_469">[Pg 469]</span></p> + +<p class="nind"><b>*176·54.</b> \[\begin{align}&\vdash \colon\ldotp \dot{\exists} !P.\dot{\exists} !Q.M=\hat{Z} \hat{z} [z\in CʻR.Z=\{\mid (\text{Cnv}ʻ\downarrow z)\}_{\in }\upharpoonright + Cʻ(P \,\text{exp}\,\, Q)].\supset :\\ +&M\in 1\rightarrow 1:z\in CʻR.\supset _{z}.Mʻz\in (\text{Prod}ʻP\downarrow_{.,} ^{;}Q\downarrow_{.,} z)\,\overline{\text{smor}}\,(\text{Prod}ʻP\downarrow_{.,} + ^{;}Q)\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*116·606.*176·14.&\supset \vdash :\text{Hp}.\supset .M\in 1\rightarrow 1 &\qquad \text{(1)}\\ +\vdash .*21·33.*30·3.&\supset \vdash :\text{Hp}.z\in CʻR.\supset .Mʻz={\mid (\text{Cnv}ʻ\downarrow z)}_{\in }\upharpoonright Cʻ(P\,\text{exp}\,\,Q) &\qquad \text{(2)}\\ +\vdash .*151·65.*165·361.*166·1.*165·01.\supset \\ +&\vdash .\{\mid (\text{Cnv}ʻ\downarrow z)\}\upharpoonright Cʻ(Q\times P)\in (P\downarrow_{.,} ^{;}Q\downarrow_{.,} + z)\,\overline{\text{smor smor}}\, (P\downarrow_{.,} ^{;}Q).\\ +[(2).*173·3] &\supset \vdash :\text{Hp}.z\in CʻR.\supset .\\ +&Mʻz\in (\text{Prod}ʻP\downarrow_{.,} ^{;}Q\downarrow_{.,} z)\,\overline{\text{smor}}\,(\text{Prod}ʻP\downarrow_{.,} ^{;}Q) &\qquad \text{(3)}\\ +\vdash .(1).(3).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*176·541.</b> \(\vdash .(P\downarrow_{.,} )\dagger ^{;}Q\downarrow_{.,} ^{;}R\in \text{Rel}^{3}\text{arithm}.\Sigma ʻ(P\downarrow_{.,} + )\dagger ^{;}Q\downarrow_{.,} ^{;}R=P\downarrow_{.,} ^{;}\Sigma ʻQ\downarrow_{.,} ^{;}R\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*163·3.*165·21·22.&\supset \vdash :\dot{\exists} !P.\supset .(P\downarrow_{.,} )\dagger ^{;}Q\downarrow_{.,} ^{;}R\in \text{Rel}^{2}\text{excl} + &\qquad \text{(1)}\\ +\vdash .*165·242.&\supset \vdash :P=\dot{\Lambda} .\dot{\exists} !S.\dot{\exists} !S'.\supset .P\downarrow_{.,} ^{;}S=\dot{\Lambda} \downarrow \dot{\Lambda} .P\downarrow_{.,} + ^{;}S'=\dot{\Lambda} \downarrow \dot{\Lambda} :\\ +[\text{Transp}] &\supset \vdash \colon\ldotp P=\dot{\Lambda} .P\downarrow_{.,} ^{;}S\neq P\downarrow_{.,} ^{;}S'.\supset :S=\Lambda.\lor.S'=\Lambda:\\ +[*165·241] &\supset :P\downarrow_{.,} ^{;}S=\dot{\Lambda} .\lor.P\downarrow_{.,} ^{;}S'=\dot{\Lambda} :\\ +[*33·241] &\supset :CʻP\downarrow_{.,} ^{;}S\cap CʻP\downarrow_{.,} ^{;}S'=\Lambda &\qquad \text{(2)}\\ +\vdash .*150·22·1. &\supset \vdash \colon\ldotp P=\dot{\Lambda} .\supset : +T,T'\in Cʻ(P\downarrow_{.,} )\dagger ^{;}Q\downarrow_{.,} ^{;}R.T\neq T'.\supset .\\ +&(\exists z,z').z\neq z'.z,z'\in CʻR.T=(P\downarrow_{.,} )^{;}Q\downarrow_{.,} z.T'=(P\downarrow_{.,} )^{;}Q\downarrow_{.,} z'.\\ +[(2)] &\supset .CʻT\cap CʻT'=\Lambda &\qquad \text{(3)}\\ +\vdash .(1).(3).*163·1. &\supset \vdash .(P\downarrow_{.,} )\dagger ^{;}Q\downarrow_{.,} ^{;}R\in \text{Rel}^{2}\text{excl} &\qquad \text{(4)}\\ +\vdash .*162·35. &\supset \vdash .\Sigma ʻ(P\downarrow_{.,} )\dagger ^{;}Q\downarrow_{.,} ^{;}R=P\downarrow_{.,} ^{;}\Sigma ʻQ\downarrow_{.,} + ^{;}R. &\qquad \text{(5)}\\ +[*165·21] &\supset \vdash .\Sigma ʻ(P\downarrow_{.,} )\dagger ^{;}Q\downarrow_{.,} ^{;}R\in \text{Rel}^{2}\text{excl} &\qquad \text{(6)}\\ +\vdash .(4).(5).(6).*174·3.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_470">[Pg 470]</span></p> + +<p class="nind"><b>*176·55.</b> \(\vdash :\dot{\exists} !P.\dot{\exists} !Q.\supset .\text{Prod}^{;}(P\downarrow_{.,} )\dagger ^{;}Q\downarrow_{.,} ^{;}R\,\text{smor smor}\, +(\text{Prod}ʻP\downarrow_{.,} ^{;}Q)\downarrow_{.,} ^{;}R\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*176·133·15.*37·44·21.\supset \\ +&\vdash :\exists !Cʻ\text{Prod}ʻP\downarrow_{.,} ^{;}Q\downarrow_{.,} z\cap Cʻ\text{Prod}ʻP\downarrow_{.,} ^{;}Q\downarrow_{.,} w.\supset .\\ +&\exists !CʻP^{Q\downarrow_{.,} z}\cap CʻP^{Q\downarrow_{.,} w}.\dot{\exists} !Q\downarrow_{.,} z.\\ +[*176·16.*80·14.*165·212]&\supset .(\exists R).\text{ᗡ}ʻR=CʻQ\downarrow_{.,} z.\text{ᗡ}ʻR=CʻQ\downarrow_{.,} w.\dot{\exists} !Q.\\ +[*13·171.*150·22] &\supset .\downarrow zʻʻCʻQ=\downarrow wʻʻCʻQ.\dot{\exists} !Q.\\ +[*55·232] &\supset .z=w &\qquad \text{(1)}\\ +\vdash .(1).*176·54.*176·53 \frac{\text{Prod}ʻP\downarrow_{.,} ^{;}Q\downarrow_{.,} z,\, \text{Prod}ʻP\downarrow_{.,} ^{;}Q}{Pʻz,\, + Q}.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*176·56.</b> \[\begin{align}\vdash :\dot{\exists} !P.\dot{\exists} !&Q.R\,\unicode{x2abd}\, J.\supset .\\ +&\text{Prod}ʻ\Sigma ʻ(P\downarrow_{.,} )\dagger ^{;}Q\downarrow_{.,} ^{;}R\,\text{smor}\,\text{Prod}ʻ(\text{Prod}ʻP\downarrow_{.,} + ^{;}Q)\downarrow_{.,} ^{;}R\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *165·223.&\supset \vdash \colon\ldotp \dot{\exists} !P.P\downarrow_{.,} ^{;}Q\downarrow_{.,} z=P\downarrow_{.,} ^{;}Q\downarrow_{.,} + z'.\supset :Q\downarrow_{.,} z=Q\downarrow_{.,} z':\\ +[*165·22] &\supset :\dot{\exists} !Q.\supset .z=z' &\qquad \text{(1)}\\ +\vdash .(1).\text{Transp}.&\supset \vdash \colon\ldotp \text{Hp}.zRz'.\supset .P\downarrow_{.,} ^{;}Q\downarrow_{.,} z\neq P\downarrow_{.,} + ^{;}Q\downarrow_{.,} z' &\qquad \text{(2)}\\ +\vdash .(2).*150·4. &\supset \vdash :\text{Hp}.\supset .(P\downarrow_{.,} )\dagger ^{;}Q\downarrow_{.,} ^{;}R\,\unicode{x2abd}\, J &\qquad \text{(3)}\\ +\vdash .(3).*176·541.*174·43.\supset \\ +\vdash :\text{Hp}.&\supset .\text{Prod}ʻ\Sigma ʻ(P\downarrow_{.,} )\dagger ^{;}Q\downarrow_{.,} + ^{;}R\,\text{smor}\,\text{Prod}ʻ\text{Prod}^{;}(P\downarrow_{.,} )\dagger ^{;}Q\downarrow_{.,} ^{;}R &\qquad \text{(4)}\\ +\vdash .*176·55.*173·31.\supset \\ +\vdash :\text{Hp}.&\supset .\text{Prod}ʻ\text{Prod}^{;}(P\downarrow_{.,} )\dagger ^{;}Q\downarrow_{.,} + ^{;}R\,\text{smor}\,\text{Prod}ʻ(\text{Prod}ʻP\downarrow_{.,} ^{;}Q)\downarrow_{.,} ^{;}R &\qquad \text{(5)}\\ +\vdash .(4).(5).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*176·57.</b> \(\vdash :R\,\unicode{x2abd}\, J.\supset .\{(P\,\text{exp}\,\,Q)\,\text{exp}\,\,R\}\,\text{smor}\,\{P\,\text{exp}\,\,(R\times Q)\}.(P^{Q})^{R}\,\text{smor}\,P^{R\times Q}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*176·151. &\supset \vdash \colon\ldotp P=\dot{\Lambda} .\lor.Q=\dot{\Lambda} :\supset .(P\,\text{exp}\,\,Q)\,\text{exp}\,\,R=\dot{\Lambda} &\qquad \text{(1)}\\ +\vdash .*176·151.*166·13. &\supset \vdash \colon\ldotp P=\dot{\Lambda} .\lor.Q=\dot{\Lambda} :\supset .P\,\text{exp}\,\,(P\times Q)=\dot{\Lambda} &\qquad \text{(2)}\\ +\vdash .(1).(2).*153·101. &\supset \vdash \colon\ldotp P=\dot{\Lambda} .\lor.Q=\dot{\Lambda} :\supset .\\ +&\{(P\,\text{exp}\,\,Q)\,\text{exp}\,\,R\}\,\text{smor}\,\{P\,\text{exp}\,\,(R\times Q)\} &\qquad \text{(3)}\\ +\vdash .*176·56·541·1.*166·1.&\supset \vdash :\dot{\exists} !P.\dot{\exists} !Q.R\,\unicode{x2abd}\, J.\supset .\\ +&\{(P\,\text{exp}\,\,Q)\,\text{exp}\,\,R\}\,\text{smor}\,\{P\,\text{exp}\,\,(R\times Q)\} &\qquad \text{(4)}\\ +\vdash .(3).(4).\supset \vdash :\text{Hp}.&\supset .\{(P\,\text{exp}\,\,Q)\,\text{exp}\,\,R\}\,\text{smor}\,\{P\,\text{exp}\,\,(R\times Q)\} &\qquad \text{(5)}\\ +[*176·181·22] &\supset .(P^{Q})^{R}\,\text{smor}\,P^{R\times Q} &\qquad \text{(6)}\\ +\vdash .(5).(6).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p>This completes the proof of the second formal law of exponentiation.</p> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_471">[Pg 471]</span></p> +<h2 class="nobreak" id="*177"> +*177. PROPOSITIONS CONNECTING \(P_{\text{df}}\) WITH PRODUCTS AND POWERS.</h2> +</div> + + +<p class="nind"><i>Summary of</i> *177.</p> + +<p>The principal proposition on this subject is</p> + +<p class="nind"><b>*177*13.</b> \(\vdash :x\neq y.\supset .P_{\text{df}}\,\text{smor}\,\{(x\downarrow y)^{P}\}\)</p> + +<p class="nind">which is the analogue of <a href="#*116·72">*116·72</a>, or rather leads to the analogue of +*116·72 as soon as powers of relation-numbers have been defined; for +then it becomes +\[ +P_{\text{df}}\in 2_{r}^{\text{Nr}ʻP}. +\] +Another proposition is an extension of <a href="#*171·69">*171·69</a>, namely</p> + +<p class="nind"><b>*177·22.</b> \(\vdash :P\in \text{Rel}^{2}\text{excl}.P\,\unicode{x2abd}\, J.\supset .\text{Prod}ʻ\text{df}^{;}P \,\text{smor}\,(\Sigma ʻP)_{\text{df}}\)</p> + +<p>where we put \(\text{df}ʻQ=Q_{\text{df}}\).</p> + +<p>The remaining propositions of this number are lemmas for the above two.</p> + +<p><a href="#*177·13">*177·13</a> shows, for example, that all classes of finite integers can be +arranged in a series of which the relation-number is \(2_{r}^\omega\), +where \(\omega\) is the relation-number of the series of finite +integers. \(2_{r}^\omega\) is not the relation-number of the continuum, +but is closely allied to it.</p> + +<hr class="tb"> + +<p class="nind"><b>*177·1.</b> \[\begin{align}\vdash :x\neq y.T=\hat{\mu} \hat{R} &[R\in \{(\iota ʻx\cup \iota ʻy)\uparrow \alpha\}_{\Delta }ʻ\alpha .\mu =\overleftarrow{R}ʻx].\supset .\\ +&T\in (\text{Cl}ʻ\alpha )\,\overline{\text{ sm }}\, \{(\iota ʻx\cup \iota ʻy)\uparrow \alpha\}_{\Delta }ʻ\alpha \quad[*116·712·713·715]\end{align}\]</p> + +<p>In the propositions of <a href="#*116">*116</a> referred to, \(\Lambda\) and \(\text{V}\) +appear in place of \(x\) and \(y\), but no property of \(\Lambda\) and +\(\text{V}\) is used in the proof except \(\Lambda\neq \text{V}\).</p> + +<p class="nind"><b>*177·11.</b> \(\vdash :\text{Hp}*177·1.\alpha =CʻP.\supset .T^;(x\downarrow y)^P=P_{\text{df}}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*176·19.\supset \\ +\vdash \colon\colon\ldotp \text{Hp}.\supset \colon\colon\mu \{T^{;}(x\downarrow y)^{P}\}\nu + .&\equiv \colon\ldotp (\exists R,S):R,S\in \{(\iota ʻx\cup \iota ʻy)\uparrow CʻP\}_{\Delta }ʻCʻP:\\ +&(\exists z):z\in CʻP.Rʻz(x\downarrow y)Sʻz:\\ +&wPz.w\neq z.\supset _{w}.Rʻw=Sʻw:\mu =\overleftarrow{R}ʻx.\nu =\overleftarrow{S}ʻx\colon\ldotp \\ +[*55·13] &\equiv \colon\ldotp (\exists R,S):R,S\in {(\iota ʻx\cup \iota ʻy)\uparrow CʻP}_{\Delta }ʻCʻP\colon\ldotp \\ +&(\exists z)\colon\ldotp z\in CʻP.Rʻz=x.Sʻz=y\colon\ldotp \mu =\overleftarrow{R}ʻx.\nu =\overleftarrow{S}ʻx\colon\ldotp \\ +&wPz.w\neq z.\supset _{w}:xRw.\equiv .xSw:yRw.\equiv .ySw\colon\ldotp \\ +[*71·36] &\equiv \colon\ldotp (\exists R,S):R,S\in {(\iota ʻx\cup \iota ʻy)\uparrow CʻP}_{\Delta }CʻP\colon\ldotp \\ +&(\exists z)\colon\ldotp z\in CʻP.z\in \mu -\nu .\mu =\overleftarrow{R}ʻx.\nu =\overleftarrow{S}ʻx\colon\ldotp \\ +&wPz.w\neq z.\supset _{w}:w\in \mu .\equiv .w\in \nu \colon\ldotp \\ +[*177·1] &\equiv \colon\ldotp \mu ,\nu \in \text{Cl}ʻCʻP\colon\ldotp (\exists z)\colon\ldotp z\in CʻP.z\in \mu -\nu \colon\ldotp \\ +&wPz.w\neq z.\supset _{w}:w\in \mu .\equiv .w\in \nu \colon\ldotp \\ +[*171·11] &\equiv \colon\ldotp \mu (P_{\text{df}})\nu \colon\colon\ldotp \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_472">[Pg 472]</span></p> + +<p class="nind"><b>*177·12.</b> \(\vdash :\text{Hp}*177·11.\supset .T\in P_{\text{df}}\,\overline{\text{smor}}\,\{(x\downarrow y)^{P}\} \quad[*177·1·11.*151·191]\)</p> + +<p class="nind"><b><a id="*177·13">*177·13</a>.</b> \(\vdash :x\neq y.\supset .P_{\text{df}}\,\overline{\text{smor}}\,\{(x\downarrow y)^{P}\} \quad[*177·12]\)</p> + +<p class="nind"><b>*177·2.</b> \(\text{df}ʻQ=Q_{\text{df}} \quad\text{Dft}[*177]\)</p> + +<p class="nind"><b>*177·21.</b> \(\vdash :P\in \text{Rel}^{2}\text{excl}.P\,\unicode{x2abd}\, J.\supset .s\upharpoonright Cʻ\text{Prod}ʻ\text{df}^{;}P\in (\Sigma ʻP)_{\text{df}}\,\overline{\text{smor}}\, + (\text{Prod}ʻ\text{df}^{;}P)\)</p> + +<p>The proof proceeds as the proof of <a href="#*174·24">*174·24</a> proceeds. If \(Q\in CʻP\), +we shall have, if \(M\in F_{\Delta }ʻ\text{df}ʻʻCʻP\), +\[ +MʻQ_{\text{df}}=(sʻ\text{D}ʻM)\cap CʻQ. +\]</p> + +<p>Hence we easily obtain +\[ +M(\Pi ʻ\text{df}^{;}P)N.\equiv .M,N\in F_{\Delta }ʻdfʻʻCʻP.(sʻ\text{D}ʻM)(\Sigma ʻP)_{\text{df}}(sʻ\text{D}ʻN), +\] +whence +\[ +\mu (\text{Prod}ʻ\text{df}^{;}P)\nu .\equiv .\mu ,\nu \in \text{Prod}ʻ\text{Cl}ʻʻCʻʻCʻP.(sʻ\mu )(\Sigma ʻP)_{\text{df}}(sʻ\nu ), +\] +whence the result follows easily.</p> + +<p class="nind"><b>*177·22.</b> \(\vdash :P\in \text{Rel}^{2}\text{excl}.P\,\unicode{x2abd}\, J.\supset .\text{Prod}ʻ\text{df}^{;}P \,\text{smor}(\Sigma ʻP)_{\text{df}} \quad[*177·21]\)</p> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_473">[Pg 473]</span></p> +<h2 class="nobreak" id="SECTION_D">SECTION D.<br> +<span class="allsmcap">ARITHMETIC OF RELATION-NUMBERS.</span></h2> +</div> + + +<p><i>Summary of Section D.</i></p> + +<p>In the present section, we shall be concerned with the arithmetical +operations on relation-numbers. Their purely logical properties have +been dealt with in Section A; in the present section, it is their +arithmetical properties that are to be established. These properties +result immediately from the arithmetical properties of relations which +have been established in Sections B and C. The subjects treated of +in the present section are analogous to those treated of in Section +B of Part III. with the exception of such as have already had their +analogues discussed in Sections B and C of Part IV. The analogy is +sufficiently close to render it often unnecessary to give proofs, since +these are often step by step analogous to the proofs of corresponding +propositions in Part III, Section B.</p> + +<p>The two chief requisites in defining the arithmetical operations with +relation-numbers are (1) to take due account of types, (2) to construct +what may be called <i>separated</i> relations, <i>i.e.</i> relations +of mutually exclusive relations derived from and ordinally similar +to given relations. Each of these points calls for some preliminary +explanations.</p> + +<p>The sum of two relation-numbers \(\mu\), \(\nu\) will be denoted by +"\(\mu \dot{+} \nu\)," in order to distinguish this kind of addition +from \(\mu + \nu\) (the arithmetical addition of classes) and +\(\mu +_{c}\nu\) (the addition of cardinals). In defining \(\mu \dot{+}\nu\), +we have to take account of the following considerations.</p> + +<p>Suppose \(P\) and \(Q\) are two relations which are of the same type, +and have mutually exclusive fields. Then obviously we shall want +to frame our definition of the sum of two relation-numbers in such +a way that the sum of \(\text{Nr}ʻP\) and \(\text{Nr}ʻQ\) shall be +\(\text{Nr}ʻ(P\unicode{x2909}Q)\). But if \(P\) and \(Q\) are not of +the same type, \(P\unicode{x2909}Q\) is meaningless; and if \(CʻP\) +and \(CʻQ\) overlap, \(P\unicode{x2909}Q\) may be too small to have +as its relation-number the sum of the relation-numbers of \(P\) and +\(Q\). Both these difficulties can be met by observing that, if +\(\text{Nr}ʻP=\text{Nr}ʻR\) and \(\text{Nr}ʻQ\text{Nr}ʻS\), we must +make such definitions as to have +\[ +\text{Nr}ʻP\dot{+} \text{Nr}ʻQ=\text{Nr}ʻR\dot{+} \text{Nr}ʻS. +\]<span class="pagenum" id="Page_474">[Pg 474]</span> +Hence, in defining the sum of the relation-numbers of \(P\) and \(Q\), +we may replace \(P\) and \(Q\) by any two relations \(R\) and \(S\) +which are respectively like \(P\) and \(Q\). Therefore what we require +for our definition is to find two relations \(R\) and \(S\) which +(1) are respectively like \(P\) and \(Q\), (2) are of the same type, +(3) have mutually exclusive fields. All these three requisites are +satisfied if we put +\[ +R=\downarrow (\Lambda\cap CʻQ)^{;}\iota ^{;}P.S=(\Lambda\cap CʻP)\downarrow ^{;}\iota ^{;}Q. +\] +We then define \(P+Q\) as meaning \(R\unicode{x2909}S\), and we +define the sum of the relation-numbers of \(P\) and \(Q\) as the +relation-number of \(P+Q\). This procedure is exactly analogous to that +of <a href="#*110">*110</a>; in fact, we have +\[ +Cʻ(P+Q)=CʻP+CʻQ. +\]</p> + +<p>In defining the sum of the relation-numbers of a field, we do not have +to consider types, because the members of a field are necessarily +all of the same type. But we do have to consider the question of +overlapping. If a term \(x\) occurs both in \(CʻQ\) and in \(CʻR\) +where \(Q,R\in CʻP\), we want a method of counting x twice over in +forming the arithmetical sum. Thus \(\text{Nr}ʻ\Sigma ʻP\) cannot be +taken as the sum of the relation-numbers of members of \(CʻP\), unless +\(P \in \text{Rel}^{2}\text{excl}\). Suppose, for instance, we have three +series +\[ +(a, b, c),\quad ( b, c, a),\quad (c, a, b). +\] +These each have three terms; and we want the sum of their +relation-numbers to be the relation-number of a series of nine terms. +But if we put +\[ +\begin{aligned} +Q &= a\downarrow b\downarrow c \qquad(\text{where}\, a\downarrow b\downarrow c\, \text{is written for}\, a\downarrow b\unicode{x228d} a\downarrow c\unicode{x228d} b\downarrow c),\\ +R &= b\downarrow c\downarrow a,\\ +S &= c\downarrow a\downarrow b,\\ +\end{aligned} +\] +and if we further put +\[ +P = Q\downarrow R\downarrow S, +\] +so that \(P\) places the above three series in the above order, we have +\[ +\Sigma ʻP=(\iota ʻa\cup \iota ʻb\cup \iota ʻc)\uparrow (\iota ʻa\cup \iota ʻb\cup \iota ʻc), +\] +which is not a series, and does not have the relation-number which we +require as the sum of the relation-numbers of \(Q\), \(R\), \(S\).</p> + +<p><span class="pagenum" id="Page_475">[Pg 475]</span></p> + +<p>What is wanted is a method of distinguishing the various occurrences +of \(a\) and \(b\) and \(c\). For this reason, when \(a\) occurs as +a member of the field of \(Q\), we replace it by \(a\downarrow Q\); +when as a member of the field of \(R\), by \(a\downarrow R\); and +when as a member of the field of \(S\), by \(a\downarrow S\). Thus +the series \((a, b, c)\) is replaced by \((a\downarrow Q, b\downarrow Q, c\downarrow Q)\); +\((b, c, a)\) is replaced by \((b\downarrow R, c\downarrow R, a\downarrow R)\); +and \((c, a, b)\) is replaced by \((c\downarrow S, a\downarrow S, b\downarrow S)\). +The sum of these three series then has the relation-number which is +required as the sum of the relation-numbers of \(Q\), \(R\), \(S\).</p> + +<p>The above process is symbolized as follows. The generating relation +of the series \((a\downarrow Q, b\downarrow Q, c\downarrow Q)\) is +\(\downarrow Q^{;}Q\); thus the three relations whose sum is to be +taken are \(\downarrow Q^{;}Q\), \(\downarrow R^{;}R\), \(\downarrow S^{;}S\), +<i>i.e.</i> using the notation of <a href="#*182">*182</a>, according to which +we put \(\unicode{x2640}ʻx=x\unicode{x2640}x\), our three relations +are \(\breve{\downarrow}_{.,}ʻQ\), \(\breve{\downarrow}_{.,}ʻR\), +\(\breve{\downarrow}_{.,}ʻS\). But the generating relation of the +series \((\breve{\downarrow}_{.,}ʻQ,\, \breve{\downarrow}_{.,}ʻR,\,\breve{\downarrow}_{.,}ʻS)\) +is \(\breve{\downarrow}_{.,}^{;}P\), since +\(P=(Q\downarrow R\downarrow S)\). Thus \(\breve{\downarrow}_{.,}^{;}P\) +is the relation required for defining the sum of the relation-numbers +of members of the field of \(P\); <i>i.e.</i> we put +\[ +\Sigma \text{Nr}ʻP=\text{Nr}ʻ\Sigma ʻ\breve{\downarrow}_{.,}^{;}P \quad\text{Df}. +\] +We will call \(\breve{\downarrow}_{.,}^{;}P\) the <i>separated</i> +relation corresponding to \(P\). \(\breve{\downarrow}_{.,}^{;}P\) is +constructed, as above, by replacing every member \(x\) of \(CʻQ\), +where \(Q\in CʻR\), by \(x\downarrow Q\); so that if \(x\) belongs both +to \(CʻQ\) and to \(CʻR\), it is duplicated by being transformed once +into \(x\downarrow Q\), and once again into \(x\downarrow R\).</p> + +<p>For the treatment of products, we do not require +\(\breve{\downarrow}_{.,}^{;}P\), because \(\Pi ʻP\) has been so defined +as to effect the requisite separation. We might, however, by the use +of \(\breve{\downarrow}_{.,}^{;}P\), have dispensed with \(\Pi ʻP\) as a +fundamental notion, and contented ourselves with \(\text{Prod}ʻP\); for +we have +\[ +\Pi ʻP = \dot{s} ^{;}\text{Prod}ʻ\breve{\downarrow}_{.,}^{;}P. +\] +Thus we might have taken \(\text{Prod}\) as the fundamental notion, and +defined \(\Pi\) by means of it.</p> + +<p>The addition of unity to a relation-number has to be treated +separately from the addition of two relation-numbers, for the same +reasons which necessitate the treatment of \(P\unicode{x21f8} x\) and +\(x\unicode{x21f7}P\) separately from \(P\unicode{x2909}Q\). There is no +ordinal number 1, but we can define the <i>addition</i> of one to a +relation-number. If \(\text{Nr}ʻP=\mu\) and \(x{\sim}\in CʻP\), we must +have +\[ +\text{Nr}ʻ(P\unicode{x21f8} x)=\mu \dot{+} \dot{1} , +\] +where we write "\(\dot{1}\)" for unity as an addendum. We do not +write "\(1_r\)," because we shall, at a later stage, give a general +definition of \(\mu _r\), in virtue of which, if \(\mu\) is an +inductive cardinal, \(\mu _r\) will be the corresponding ordinal. This +definition entails \(1_r=\Lambda\), and therefore we use a different +symbol "\(\dot{1}\)" for 1 as addendum. The symbol \(\dot{1}\) is only +defined in its uses, and has no significance except in a use which has +been specially defined.</p> + +<p>We define the product \(\mu \dot{\times} \nu\) as the relation-number +of \(P\times Q\), when \(\mu =\text{N}_0\text{r}ʻP\) and +\(\nu=\text{N}_0\text{r}ʻQ\). The product so defined obeys the +associative law, and obeys the distributive law in the form +\[ +(\nu \dot{+} \varpi )\dot{\times} \mu =(\nu \dot{\times} \mu )\dot{+} (\varpi \dot{\times} \mu ) +\] +but not, in general, in the form +\[ +\mu \dot{\times} (\nu \dot{+} \varpi )=(\mu \dot{\times} \nu )\dot{+} (\mu \dot{\times} \varpi ). +\]<span class="pagenum" id="Page_476">[Pg 476]</span> +The latter form holds when \(\mu\), \(\nu\), \(\varpi\) are finite +ordinals, as we shall prove at a later stage (*262). The commutative +law also does not hold in general for ordinal addition and +multiplication, but holds where finite ordinals are concerned.</p> + +<p>The product of the numbers of the members of \(CʻP\), in the order +generated by \(P\), is defined as being \(\text{Nr}ʻ\Pi ʻP\), and +is denoted by \(\Pi \text{Nr}ʻP\). It will be seen that +\(\Pi\text{Nr}ʻP\) is not a function of \(CʻP\), since the value of a +product depends upon the order of the factors; it is also not a +function of \(\text{Nr}^{;}P\), unless no two members of \(CʻP\) have the +same relation-number. The properties of \(\Pi \text{Nr}ʻP\) result from +<a href="#*172">*172</a> and <a href="#*174">*174</a>.</p> + +<p>"\(\mu\) to the \(\nu\)th power" is denoted by "\(\mu \,\text{exp}\,\,_r\nu\)" +and is defined as the relation-number of \(P \,\text{exp}\,\, Q\), where +\(\mu =\text{N}_0\text{r}ʻP\) and \(\nu = \text{N}_0\text{r}ʻQ\). Its +properties result from the propositions of <a href="#*176">*176</a> and <a href="#*177">*177</a>.</p> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_477">[Pg 477]</span></p> +<h2 class="nobreak" id="*180"> +*180. THE SUM OF TWO RELATION-NUMBERS.</h2> +</div> + + +<p class="nind"><i>Summary of</i> *180.</p> + +<p>In order to define the sum of two relation-numbers, we proceed (as +in <a href="#*110">*110</a>) to construct a relation whose relation-number shall be the +required sum. For this purpose, we put +\[ +P+Q=\{\downarrow (\Lambda\cap CʻQ)^{;}\iota ^{;}P\}\unicode{x2909}\{(\Lambda\cap CʻP)\downarrow ^{;}\iota ^{;}Q\} \quad\text{Df}. +\] +This definition has the following merits: (1) whatever may be the +types of \(P\) and \(Q\), \(\downarrow (\Lambda \cap CʻQ)^{;}\iota ^{;}P\) +is of the same type as \((\Lambda \cap CʻP)\downarrow ^{;}\iota ^{;}Q\); +(2) however the fields of \(P\) and \(Q\) may overlap, and even if +\(P=Q\), the fields of \(\downarrow (\Lambda \cap CʻQ)^{;}\iota ^{;}P\) and +\((\Lambda \cap CʻP)\downarrow ^{;}\iota ^{;}Q\) are mutually exclusive; +(3) these two relations are respectively similar to \(P\) and \(Q\). +Hence it is evident that, without placing any restriction upon \(P\) and \(Q\), +we may take the relation-number of \(P+Q\) as defining the sum of the +relation-numbers of \(P\) and \(Q\). Hence we put +\[ +\mu \dot{+} \nu =\hat{R} \{(\exists P,Q).\mu =\text{N}_{0}\text{r}ʻP.\nu =\text{N}_{0}\text{r}ʻQ.R\,\text{smor}\,(P+Q)\} \quad\text{Df}. +\] +From this definition it follows that \(\mu \dot{+} \nu\) is null unless +\(\mu\) and \(\nu\) are homogeneous relation-numbers, but that if they +are the homogeneous relation-numbers of \(P\) and \(Q\), then \(\mu +\dot{+} \nu\) is the relation-number of \(P+Q\).</p> + +<p>In order to be able to deal with typically ambiguous relation-numbers, +we put, as in <a href="#*110">*110</a>, +\[ +\begin{aligned} +\text{Nr}ʻP\dot{+} \nu &=\text{N}_{0}\text{r}ʻP\dot{+} \nu \quad\text{Df},\\ +\mu \dot{+} \text{Nr}ʻQ&=\mu \dot{+} \text{N}_{0}\text{r}ʻQ \quad\text{Df}. +\end{aligned} +\]</p> + +<p>The principal propositions of the present number are</p> + +<p class="nind"><b>*180·111.</b> \(\vdash .Cʻ(P+Q)=CʻP+CʻQ\)</p> + +<p class="nind"><b>*180·3.</b> \[\begin{align}\vdash .\text{Nr}ʻP\dot{+} \text{Nr}ʻQ&=\text{N}_{0}\text{r}ʻP\dot{+} \text{Nr}ʻQ=\text{Nr}ʻP\dot{+} \text{N}_{0}\text{r}ʻQ\\ +&=\text{N}_{0}\text{r}ʻP\dot{+} \text{N}_{0}\text{r}ʻQ=\text{Nr}ʻ(P+Q)\end{align}\]</p> + +<p class="nind"><b>*180·31.</b> \(\vdash :P\,\text{smor}\,R.Q\text{smor}S.\supset .\text{Nr}ʻP\dot{+} \text{Nr}ʻQ=\text{Nr}ʻR\dot{+} \text{Nr}ʻS\)</p> + +<p>This proposition is essential, since otherwise \(\text{Nr}ʻP\dot{+}\text{Nr}ʻQ\) +would not be a function of \(\text{Nr}ʻP\) and +\(\text{Nr}ʻQ\), but would depend upon the particular \(P\) and \(Q\).</p> + +<p class="nind"><b>*180·32.</b> \(\vdash :CʻP\cap CʻQ=\Lambda .\supset .\text{Nr}ʻP\dot{+} \text{Nr}ʻQ=\text{Nr}ʻ(P\unicode{x2909}Q)\)</p> + +<p><span class="pagenum" id="Page_478">[Pg 478]</span></p> + +<p class="nind"><b>*180·4.</b> \(\vdash :\exists !\mu \dot{+} \nu .\supset .\mu ,\nu \in \text{NR}-\iota ʻ\Lambda .\mu ,\nu \in \text{N}_{0}\text{R}\)</p> + +<p class="nind"><b>*180·42.</b> \(\vdash . \mu \dot{+} \nu \in \text{NR}\)</p> + +<p class="nind"><b>*180·56.</b> \(\vdash . (\mu \dot{+} \nu ) \dot{+} \varpi = \mu \dot{+} (\nu \dot{+} \varpi )\)</p> + +<p class="nind">which is the associative law.</p> + +<p class="nind"><b>*180·61.</b> \(\vdash . \text{Nr}ʻP \dot{+} 0_r = \text{Nr}ʻP = 0_r \dot{+} \text{Nr}ʻP\)</p> + +<p class="nind"><b>*180·71.</b> \(\vdash : \mu , \nu \in \text{NR} . \supset . Cʻʻ(\mu \dot{+} \nu ) = Cʻʻ\mu +_c Cʻʻ\nu \)</p> + +<p>This proposition gives the connection of ordinal and cardinal addition. +It should be observed that, in virtue of <a href="#*154·9">*154·9</a>, \(Cʻʻ\mu\) and +\(Cʻʻ\nu\) are cardinals when \(\mu\) and \(\nu\) are relation-numbers.</p> + +<hr class="tb"> + +<p class="nind"><b>*180·01.</b> \(P + Q = {\downarrow (\Lambda \cap CʻQ)^{;}{℩}^{;}P} \unicode{x2909} \{(\Lambda \cap CʻP) \downarrow ^{;}\iota ^{;}Q\} \quad\text{Df}\)</p> + +<p class="nind"><b>*180·02.</b> \(\mu \dot{+} \nu = \hat{R} \{(\exists P,Q) . \mu = \text{N}_0\text{r}ʻP . \nu = \text{N}_0\text{r}ʻQ . R\,\text{smor}\,(P + Q)\} \quad\text{Df}\)</p> + +<p class="nind"><b><a id="*180·03">*180·03</a>.</b> \(\text{Nr}ʻP \dot{+} \nu = \text{N}_0\text{r}ʻP \dot{+} \nu \quad\text{Df}\)</p> + +<p class="nind"><b><a id="*180·031">*180·031</a>.</b> \(\mu \dot{+} \text{Nr}ʻQ = \mu \dot{+} \text{N}_0\text{r}ʻQ \quad\text{Df}\)</p> + +<p>On the purpose of the definitions <a href="#*180·03">*180·03</a><a href="#*180·031">·031</a>, see the remarks on the +corresponding definitions in <a href="#*110">*110</a> and \(\text{IIT}\) of the Prefatory +Statement.</p> + +<p class="nind"><b>*180·1.</b> \(\vdash . P + Q = {\downarrow (\Lambda \cap CʻQ)^{;}{℩}^{;}P} \unicode{x2909} \{(\Lambda \cap CʻP) \downarrow ^{;}{℩}^{;}Q\} + \quad[(*180·01)]\)</p> + +<p class="nind"><b>*180·101.</b> \[\begin{align}\vdash . Cʻ \downarrow &(\Lambda \cap CʻQ)^{;}{℩}^{;}P = \downarrow (\Lambda \cap CʻQ)ʻʻ{℩}ʻʻCʻP.\\ +&Cʻ(\Lambda \cap CʻP) \downarrow ^{;}\iota {℩}^{;}Q = (\Lambda \cap CʻP) \downarrow ʻʻ{℩}ʻʻCʻQ \quad[*150·22]\end{align}\]</p> + +<p class="nind"><b>*180·11.</b> \(\vdash . Cʻ \downarrow (\Lambda \cap CʻQ)^{;}{℩}^{;}P \cap Cʻ(\Lambda \cap CʻP) \downarrow ^{;}{℩}^{;}Q + = \Lambda \quad[*180·101 . *110·11]\)</p> + +<p class="nind"><b>*180·111.</b> \(\vdash . Cʻ(P + Q) = CʻP + CʻQ\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *180·101 . *160·14 . \supset +\vdash . Cʻ(P + Q) &= \downarrow (\Lambda \cap CʻQ)ʻʻ{℩}ʻʻCʻP \cup (\Lambda \cap CʻP) \downarrow ʻʻ{℩}ʻʻCʻQ\\ +[(*110·01)] &= CʻP + CʻQ . \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*180·12.</b> \(\vdash . \downarrow (\Lambda \cap CʻP)^{;}{℩}^{;}P \,\text{smor} P . (\Lambda \cap CʻP) \downarrow ^{;}{℩}^{;}Q + \,\text{smor}\, Q \quad[*151·61·64·65]\)</p> + +<p class="nind"><b>*180·13.</b> \(\vdash : R \,\text{smor}\, P . S \,\text{smor}\, Q . CʻR \cap CʻS = \Lambda . \supset . R \unicode{x2909} S \,\text{smor}\, P + Q\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *180·12 . \supset \vdash : \text{Hp} . &\supset . R \,\text{smor}\, \downarrow (\Lambda \cap CʻQ)^{;}{℩}^{;}P + . S \,\text{smor}\, (\Lambda \cap CʻP) \downarrow ^{;}{℩}^{;}Q &\qquad \text{(1)}\\ +\vdash . (1) . *180·11 . *160·48 . \supset \\ +\vdash : \text{Hp} . &\supset . R \unicode{x2909} S \,\text{smor}\, \{\downarrow (\Lambda \cap CʻQ)^{;}{℩}^{;}P + \unicode{x2909} (\Lambda \cap CʻP) \downarrow ^{;}{℩}^{;}Q\} .\\ +[(*180·01)] &\supset . R \unicode{x2909} S \,\text{smor}\, (P + Q) : \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*180·14.</b> \(\vdash : CʻP \cap CʻQ = \Lambda . \supset . P \unicode{x2909} Q \,\text{smor}\, P + Q \quad[*180·13 . *151·13]\)</p> + +<p class="nind"><b>*180·15.</b> \(\vdash : R \,\text{smor}\, P . S \,\text{smor}\, Q . \supset . R + S \,\text{smor}\, P + Q\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *180·12 . \supset \vdash : \text{Hp} . &\supset . \downarrow (\Lambda \cap CʻS)^{;}{℩}^{;}R \,\text{smor}\, P . (\Lambda \cap CʻR) \downarrow ^{;}{℩}^{;}S + \,\text{smor}\, Q.\\ +[*180·13] &\supset . \{\downarrow (\Lambda \cap CʻS)^{;}{℩}^{;}R \unicode{x2909} (\Lambda \cap CʻR) \downarrow ^{;}{℩}^{;}S\} \,\text{smor}\, P + Q.\\ +[(*180·01)] &\supset . R + S \,\text{smor}\, P + Q : \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_479">[Pg 479]</span></p> + +<p class="nind"><b>*180·151.</b> \[\begin{align}\vdash \colon\ldotp &CʻP\cap C\supset Q=\Lambda .\supset :Z\,\text{smor}\,(P\unicode{x2909}Q).\equiv .\\ +&(\exists R,S).R\,\,\text{smor}\,\,P.S\,\text{smor}\,Q.CʻR\cap CʻS=\Lambda .Z=R\unicode{x2909}S\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*160·48.&\supset \vdash \colon\ldotp \text{Hp}.\supset :(\exists R,S).R\,\,\text{smor}\,\,P.S\,\text{smor}\,Q.CʻR\cap CʻS=\Lambda .\\ +&Z=R\unicode{x2909}S.\supset .Z\,\text{smor}\,(P\unicode{x2909}Q) &\qquad \text{(1)}\\ +\vdash .*160·44. &\supset \vdash :T\in Z\overline{\,\text{smor}\,} (P\unicode{x2909}Q).\supset .Z=T^{;}P\unicode{x2909}T^{;}Q &\qquad \text{(2)}\\ +\vdash .*160·14.*151·11.&\supset \vdash :T\in Z\overline{\,\text{smor}\,} (P\unicode{x2909}Q).\supset .CʻP\subset \text{ᗡ}ʻT.CʻQ\subset \text{ᗡ}ʻT &\qquad \text{(3)}\\ +\vdash .(3).*151·21.&\supset \vdash :T\in Z\overline{\,\text{smor}\,} (P\unicode{x2909}Q).\supset .T^{;}P\,\,\text{smor}\,\,P.T^{;}Q\,\text{smor}\,Q &\qquad \text{(4)}\\ +\vdash .*72·411.*150·22.&\supset \vdash \colon\ldotp \text{Hp}.\supset :\\ +&T\in Z\,\text{smor}\,(P\unicode{x2909}Q).\supset .CʻT^{;}P\cap CʻT^{;}Q=\Lambda &\qquad \text{(5)}\\ +\vdash .(2).(4).(5). &\supset \vdash \colon\ldotp \text{Hp}.\supset :T\in Z\overline{\,\text{smor}\,} (P\unicode{x2909}Q).\supset .\\ +&T^{;}P\,\,\text{smor}\,\,P.T^{;}Q\,\text{smor}\,Q.CʻT^{;}P\cap CʻT^{;}Q=\Lambda .Z=T^{;}P\unicode{x2909}T^{;}Q:\\ +[*151·12] &\supset :Z\,\text{smor}\,(P\unicode{x2909}Q).\supset .\\ +&(\exists R,S).R\,\,\text{smor}\,\,P.S\,\text{smor}\,Q.CʻR\cap CʻS=\Lambda .Z=R\unicode{x2909}S &\qquad \text{(6)}\\ +\vdash .(1).(6).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*180·152.</b> \[\begin{align}\vdash :Z\,\text{smor}\,&(P+Q).\equiv .\\ +&(\exists R,S).R\,\,\text{smor}\,\,P.S\,\text{smor}\,Q.CʻR\cap CʻS=\Lambda .Z=R\unicode{x2909}S\\ +&[*180·151·11·12]\end{align}\]</p> + +<p class="nind"><b>*180·16.</b> \[\begin{align}\vdash .\text{Nr}ʻ&(P+Q)=\\ +&\hat{Z} \{(\exists R,S).R\in \text{Nr}ʻP.S\in \text{Nr}ʻQ.CʻR\cap CʻS=\Lambda .Z=R\unicode{x2909}S\}\\ +&[*180·152.*152·11]\end{align}\]</p> + +<p class="nind"><b>*180·2.</b> \[\begin{align}&\vdash :Z\in \mu \dot{+} \nu .\equiv .(\exists P,Q).\mu =\text{N}_{0}\text{r}ʻP.\nu =\text{N}_{0}\text{r}ʻQ.Z\,\text{smor}\,(P+Q)\\ +&[(*180·02)]\end{align}\]</p> + +<p class="nind"><b>*180·201.</b> \[\begin{align}&\vdash \colon\ldotp Z\in \mu \dot{+} \nu .\equiv :\mu ,\nu \in \text{N}_{0}\text{R}:(\exists P,Q).P\in \mu .Q\in \nu .Z\,\text{smor}\,(P+Q)\\ +&[*155·27.*180·2]\end{align}\]</p> + +<p class="nind"><b>*180·202.</b> \[\begin{align}\vdash \colon\ldotp Z\in &\mu \dot{+} \nu .\equiv :\\ +&\exists !\mu .\exists !\nu :(\exists P,Q).\mu =\text{Nr}ʻP.\nu =\text{Nr}ʻQ.Z\,\text{smor}\,(P+Q)\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*155·34·22.*180·201.\supset \\ +\vdash \colon\ldotp Z\in \mu \dot{+} \nu .&\equiv :\exists !\mu .\exists !\nu .\mu ,\nu \in \text{NR}:(\exists P,Q).P\in \mu .Q\in \nu .Z\,\text{smor}\,(P+Q):\\ +[*152·44]&\equiv :\exists !\mu .\exists !\nu .\mu ,\nu \in \text{NR}:(\exists P,Q).\mu =\text{Nr}ʻP.\nu =\text{Nr}ʻQ.Z\,\text{smor}\,(P+Q):\\ +[*152·41]&\equiv :\exists !\mu .\exists !\nu :(\exists P,Q).\mu =\text{Nr}ʻP.\nu =\text{Nr}ʻQ.Z\,\text{smor}\,(P+Q)\colon\ldotp \\ +\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p>In the following propositions proofs are omitted, since they are +exactly analogous to proofs of propositions in <a href="#*110">*110</a> whose numbers have +the same decimal part.</p> + +<p class="nind"><b>*180·21.</b> \(\vdash \colon\ldotp \mu ,\nu \in \text{NR}.\supset :Z\in \mu \dot{+} \nu .\equiv .(\exists P,Q).P\in \mu .Q\in \nu .Z\,\text{smor}\,(P\dot{+}Q)\)</p> + +<p><span class="pagenum" id="Page_480">[Pg 480]</span></p> + +<p class="nind"><b>*180·211.</b> \[\begin{align}\vdash \colon\ldotp &\mu , \nu \in \text{NR} . \supset : Z \in \mu \dot{+} \nu . \equiv .\\ +&(\exists R, S) . R \in \,\text{smor}\,ʻʻ\mu . S \in \,\text{smor}\,ʻʻ\nu . CʻR \cap CʻS = \Lambda . Z = R \unicode{x2909} S\end{align}\]</p> + +<p class="nind"><b>*180·212.</b> \[\begin{align}\vdash \colon\ldotp &\mu , \nu \in \text{NR} . \supset : Z \in \mu \dot{+} \nu . \equiv .\\ +&(\exists R) . R \in \,\text{smor}\,ʻʻ\mu . R \,\unicode{x2abd}\, Z . Z \unicode{x0294f} (- CʻR) \in \,\text{smor}\,ʻʻ\nu\end{align}\]</p> + +<p class="nind"><b>*180·22.</b> \(\vdash . \text{N}_{0}\text{r}ʻP \dot{+} \text{N}_{0}\text{r}ʻQ = \text{Nr}ʻ(P \dot{+} Q)\)</p> + +<p class="nind"><b>*180·24.</b> \[\begin{align}&\vdash : R \,\text{smor}\, P . S \,\text{smor}\, Q . \supset . \text{N}_{0}\text{r}ʻR \dot{+} \text{N}_{0}\text{r}ʻS = \text{N}_{0}\text{r}ʻP + \dot{+} \text{N}_{0}\text{r}ʻQ\\ +&[*180·15·22]\end{align}\]</p> + +<p class="nind"><b>*180·3.</b> \[\begin{align}\vdash . \text{Nr}ʻP \dot{+} &\text{Nr}ʻQ = \text{N}_{0}\text{r}ʻP \dot{+} \text{Nr}ʻQ = \text{Nr}ʻP \dot{+} \text{N}_{0}\text{r}ʻQ\\ +&= \text{N}_{0}\text{r}ʻP \dot{+} \text{N}_{0}\text{r}ʻQ = \text{Nr}ʻ(P \dot{+} Q) \quad[*180·22 . (*180·03·031)]\end{align}\]</p> + +<p class="nind"><b>*180·31.</b> \(\vdash : P \,\,\text{smor}\, R . Q \,\text{smor}\, S . \supset . \text{Nr}ʻP \dot{+} \text{Nr}ʻQ = \text{Nr}ʻR \dot{+} \text{Nr}ʻS\)</p> + +<p class="nind"><b>*180·32.</b> \(\vdash : CʻP \cap CʻQ = \Lambda . \supset . \text{Nr}ʻP \dot{+} \text{Nr}ʻQ = \text{Nr}ʻ(P \unicode{x2909} Q) \quad[*180·14·3]\)</p> + +<p class="nind"><b>*180·4.</b> \(\vdash : \exists ! \mu \dot{+} \nu . \supset . \mu , \nu \in \text{NR} - {℩}ʻ\Lambda . \mu , \nu \in \text{N}_{0}\text{R}\)</p> + +<p class="nind"><b>*180·42.</b> \(\vdash . \mu \dot{+} \nu \in \text{NR}\)</p> + +<p class="nind"><b>*180·43.</b> \(\vdash : \mu \dot{+} \nu = \text{N}_{0}\text{r}ʻZ . \equiv . Z \in \mu \dot{+} \nu\)</p> + +<p class="nind"><b>*180·53.</b> \(\vdash . (P + Q) + R \,\text{smor}\, P + (Q + R)\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *160·44 . (*180·01) . \supset \\ +&\vdash : Pʻ = \downarrow (\Lambda \cap CʻR)^{;}{℩}^{;} \downarrow (\Lambda \cap CʻQ)^{;}{℩}^{;}P . Qʻ = \downarrow (\Lambda \cap CʻR)^{;} + \iota ^{;}(\Lambda \cap CʻP) \downarrow ^{;}{℩}^{;}Q .\\ +R' = \{\Lambda \cap Cʻ(P + Q)\} \downarrow ^{;}{℩}^{;}R . &\supset . P' \unicode{x2909} Q' = \downarrow (\Lambda \cap CʻR)^{;}{℩}^{;}(P + + Q) . &\qquad \text{(1)}\\ +[(*180·01)] &\supset . (P' \unicode{x2909} Q') \unicode{x2909} R' = (P + Q) + R .\\ +[*160·31] &\supset . P' \unicode{x2909} (Q' \unicode{x2909} R') = (P + Q) + R &\qquad \text{(2)}\\ +\vdash . (1). *180·11 \frac{P + Q,\, R}{P,\,Q}. *160·14 . \supset \vdash : \text{Hp} (1) . \supset .\\ +&CʻPʻ \cap CʻRʻ = \Lambda . CʻQʻ \cap CʻR' = \Lambda &\qquad \text{(3)}\\ +\vdash . *180·11 . *72·411 . *150·22 .& \supset \vdash : \text{Hp} (1) . \supset . CʻP' \cap CʻQ' = \Lambda &\qquad \text{(4)}\\ +\vdash . (3) . (4) . *160·14 . &\supset \vdash : \text{Hp} (1) . \supset . CʻP' \cap Cʻ(Q' \unicode{x2909} R') = \Lambda &\qquad \text{(5)}\\ +\vdash . *180·12 . &\supset \vdash : \text{Hp} (1) . \supset . P' \,\text{smor}\, P . Q' \,\text{smor}\, Q . R' \,\text{smor}\, R &\qquad \text{(6)}\\ +\vdash . (3) . (6) . *180·13 . \supset \vdash : \text{Hp} (1) . &\supset . Q' \unicode{x2909} R' \,\text{smor}\, Q + R .\\ +[(5) . (6) . *180·13] &\supset . P' \unicode{x2909} (Q' \unicode{x2909} R') \,\text{smor}\, P + (Q + R) .\\ +[(2)] &\supset . (P + Q) + R \,\text{smor}\, P + (Q + R) &\qquad \text{(7)}\\ +\vdash . (7) . *13·19 . \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*180·531.</b> \(P + Q + R = (P + Q) + R \quad\text{Df}\)</p> + +<p class="nind"><b>*180·54.</b> \(\vdash . (\text{Nr}ʻP \dot{+} \text{Nr}ʻQ) \dot{+} \text{Nr}ʻR = \text{Nr}ʻ(P + Q + R)\)</p> + +<p class="nind"><b>*180·541.</b> \(\vdash . \text{Nr}ʻP \dot{+} (\text{Nr}ʻQ \dot{+} \text{Nr}ʻR) = \text{Nr}ʻ(P + Q + R)\)</p> + +<p class="nind"><b>*180·55.</b> \(\vdash . (\text{Nr}ʻP \dot{+} \text{Nr}ʻQ) \dot{+} \text{Nr}ʻR = \text{Nr}ʻP \dot{+} (\text{Nr}ʻQ \dot{+} \text{Nr}ʻR)\)</p> + +<p><span class="pagenum" id="Page_481">[Pg 481]</span></p> + +<p class="nind"><b>*180·551.</b> \(\vdash .(\text{N}_{0}\text{r}ʻP\dot{+} \text{N}_{0}\text{r}ʻQ)\dot{+} \text{N}_{0}\text{r}ʻR = \text{N}_{0}\text{r}ʻP\dot{+} (\text{N}_{0}\text{r}ʻQ\dot{+} + \text{N}_{0}\text{r}ʻR)\)</p> + +<p class="nind"><b>*180·56.</b> \(\vdash .(\mu \dot{+} \nu )\dot{+} \omega = \mu \dot{+} (\nu \dot{+} \omega )\)</p> + +<p class="nind"><b>*180·561.</b> \(\mu \dot{+} \nu \dot{+} \omega = (\mu \dot{+} \nu ])\dot{+} \omega \quad\text{Df}\)</p> + +<p class="nind"><b>*180*57.</b> \(\vdash .(\mu \dot{+} \nu )\dot{+} (\omega \dot{+} \rho ) = \mu \dot{+} \nu \dot{+} \omega \dot{+} \rho\)</p> + +<p class="nind"><b>*180·6.</b> \(\vdash :\mu \in \text{NR}.\supset .\mu \dot{+} 0_{r} = \,\text{smor}\,ʻʻ\mu = 0_{r}\dot{+} \mu\)</p> + +<p>Observe that \(\mu \dot{+} 0_{r} = 0_{r}\dot{+} \mu\) is an equation +depending upon the peculiar properties of \(0_{r}\). We do not in +general have \(\mu \dot{+} \nu = \nu \dot{+} \mu\) unless \(\mu\) and +\(\nu\) are <i>finite</i> ordinals.</p> + +<p class="nind"><b>*180·61.</b> \(\vdash .\text{Nr}ʻP\dot{+} 0_{r} = \text{Nr}ʻP = 0_{r}\dot{+} \text{Nr}ʻP\)</p> + +<p class="nind"><b>*180·62.</b> \(\vdash :\mu \dot{+} \nu = 0_{r}. \equiv .\mu = 0_{r}.\nu = 0_{r}\)</p> + +<p class="nind"><b>*180·64.</b> \(\vdash .0_{r}\dot{+} 0_{r} = 0_{r}\)</p> + +<p class="nind"><b>*180·642.</b> \(\vdash .2_{r}\dot{+} 0_{r} = 0_{r}\dot{+} 2_{r} = 2_{r}\)</p> + +<p>Note that \(\dot{1} \dot{+} 0_{r}\), which will be defined in <a href="#*181">*181</a>, is +\(0_{r}\), not \(\dot{1}\).</p> + +<p>The following propositions, being concerned with the relations of +relation-numbers and cardinal numbers, have no analogues in <a href="#*110">*110</a>.</p> + +<p class="nind"><b>*180·7.</b> \(\vdash .CʻʻN_{r}ʻ(P+Q) = CʻʻN_{r}ʻP+_{c}Cʻʻ\text{Nr}ʻQ = \text{Nc}ʻCʻP+_{c}\text{Nc}ʻCʻQ\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*152·7.\supset \vdash .Cʻʻ\text{Nr}ʻ(P+Q) &= \text{Nc}ʻCʻ(P+Q)\\ +[*180·111] & = \text{Nc}ʻ(CʻP+CʻQ)\\ +[*110·3] & = \text{Nc}ʻCʻP+_{c}\text{Nc}ʻCʻQ &\qquad \text{(1)}\\ +[*152·7] & = Cʻʻ\text{Nr}ʻP+_{c}Cʻʻ\text{Nr}ʻQ &\qquad \text{(2)}\\ +\vdash .(1).(2).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*180·71.</b> \(\vdash :\mu ,\nu \in \text{NR}.\supset .Cʻʻ(\mu \dot{+} \nu ) = Cʻʻ\mu +_{c}Cʻʻ\nu\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*152·4.\supset \vdash :\text{Hp}.&\supset .(\exists P,Q).\mu = \text{Nr}ʻP.\nu = \text{Nr}ʻQ.\\ +[*180·3]&\supset .(\exists P,Q).\mu = \text{Nr}ʻP.\nu = \text{Nr}ʻQ.\mu \dot{+} \nu = \text{Nr}ʻ(P+Q).\\ +[*180·7] \supset .(\exists P,Q).\mu &= \text{Nr}ʻP.\nu = \text{Nr}ʻQ.\\ +&Cʻʻ(\mu \dot{+} \nu = Cʻʻ\text{Nr}ʻP+_{c}Cʻʻ\text{Nr}ʻQ.\\ +[*13·193] &\supset .Cʻʻ(\mu \dot{+} \nu ) = Cʻʻ\mu +_{c}Cʻʻ\nu :\supset \vdash .\text{Prop} +\end{array} +\]</p> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_482">[Pg 482]</span></p> +<h2 class="nobreak" id="*181"> +*181. ON THE ADDITION OF UNITY TO A RELATION-NUMBER.</h2> +</div> + + +<p class="nind"><i>Summary of</i> *181.</p> + +<p>The relation-number \(\dot{1}\) has, according to our definitions, no +meaning in isolation, because our definitions are framed with a view to +series, and a series cannot consist of one term. But we can <i>add</i> +one term to a series; hence \(\dot{1}\) is required as an addendum. +In order to get our definitions in the most manageable form, we first +construct a relation, which we call \(P\dot{\unicode{x21f8}}x\), which +is such that, whenever \(P\) exists, \(P\dot{\unicode{x21f8}}x\) has +one more term in its field than \(P\); the relation-number of this +relation is then defined as \(\text{Nr}ʻP\dot{+} \dot{1}\). We add also +a definition +\[ +\dot{1} \dot{+} \dot{1} =2_r \quad\text{Df} +\] +which is purely formal, and serves to minimize exceptions to the +associative law of addition.</p> + +<p>The definitions are closely analogous to those of <a href="#*180">*180</a>. We put +\[ +P\dot{\unicode{x21f8}} x = \downarrow \Lambda _{x}^{;}{℩}^{;}P\unicode{x21f8} (\Lambda \cap CʻP)\downarrow {℩}ʻx \quad\text{Df} +\] +with a similar definition for \(x\dot{\unicode{x21f7}} P\). \(x\) and +\(P\) may be of any relative types, and we have always +\[ +\downarrow \Lambda _{x}^{;}{℩}^{;}P \,\,\text{smor}\, P.(\Lambda \cap CʻP)\downarrow {℩}ʻx{\sim}\in Cʻ\downarrow \Lambda _{x}^{;}{℩}^{;}P + \quad(*181·11·12). +\] +We put +\[ +\mu \dot{+} \dot{1} = \hat{R} \{(\exists P,x).\text{N}_{0}\text{r}ʻP = \mu .R \,\text{smor}\, (P\dot{\unicode{x21f8}} x)\} \quad\text{Df} +\] +with a similar definition for \(\dot{1} \dot{+} \mu\). We also +introduce definitions analogous to <a href="#*180·03">*180·03</a><a href="#*180·031">·031</a>.</p> + +<p>The principal propositions of this number are</p> + +<p class="nind"><b>*181·3.</b> \(\vdash .\text{Nr}ʻP\dot{+} \dot{1} = \text{N}_{0}\text{r}ʻP\dot{+} \dot{1} = \text{Nr}ʻ(P\dot{\unicode{x21f8}} x)\)</p> + +<p class="nind"><b>*181·31.</b> \(\vdash :P \,\,\text{smor}\, \,Q.\supset .\text{Nr}ʻP\dot{+} \dot{1} = \text{Nr}ʻQ\dot{+} \dot{1}\)</p> + +<p class="nind"><b>*181·32.</b> \(\vdash :x{\sim}\in CʻP.\supset .\text{Nr}ʻP\dot{+} \dot{1} = \text{Nr}ʻ(P\unicode{x21f8} x)\)</p> + +<p class="nind"><b>*181·33.</b> + \(\vdash \colon\ldotp \mu ,\nu \in \text{NR}.\exists !\mu \dot{+} \dot{1} .\supset :\mu = \nu .\equiv .\mu \dot{+} \dot{1} = \nu \dot{+} \dot{1} .\equiv .\dot{1} \dot{+} \mu = \dot{1} \dot{+} \nu\)</p> + +<p class="nind"><b>*181·4.</b> \(\vdash :\exists !\mu \dot{+} \dot{1} .\supset .\mu \in \text{NR}-{℩}ʻ\Lambda .\mu \in \text{N}_{0}\text{R}\)</p> + +<p class="nind"><b>*181·42.</b> \(\vdash .\mu \dot{+} \dot{1} \in \text{NR}\)</p> + +<p><span class="pagenum" id="Page_483">[Pg 483]</span></p> + +<p>The following propositions are formally forms of the associative +law, but they need separate proof on account of the peculiarity of +\(\dot{1}\).</p> + +<p class="nind"><b>*181·54.</b> \(\vdash : \nu \neq 0_{r} . \supset . (\mu \dot{+} \nu) \dot{+} \dot{1} = \mu \dot{+} (\nu \dot{+} \dot{1} )\)</p> + +<p class="nind"><b>*181·56.</b> \(\vdash : \mu \neq 0_{r} . \supset . (\mu \dot{+} \dot{1} ) \dot{+} \dot{1} = \mu \dot{+} (\dot{1} \dot{+} \dot{1} ) = \mu \dot{+} 2_{r}\)</p> + +<p class="nind"><b>*181·58.</b> \(\vdash : \mu \neq 0_{r} . \nu \neq 0_{r} . \supset . (\mu \dot{+} \dot{1} ) \dot{+} \nu = \mu \dot{+} (\dot{1} \dot{+} \nu)\)</p> + +<p class="nind"><b>*181·59.</b> \(\vdash : \mu \neq 0_{r} . \nu \neq 0_{r} . \supset . (\mu \dot{+} \dot{1} ) \dot{+} (\dot{1} \dot{+} \nu) = \mu \dot{+} 2_{r} + \dot{+} \nu\)</p> + +<p>The hypotheses in the above propositions are essential.</p> + +<p class="nind"><b>*181·6.</b> \(\vdash : \dot{\exists} ! P . \supset . Cʻʻ\text{Nr}ʻ(P \dot{\unicode{x21f8}} x) = \text{Nc}ʻCʻP +_{c} 1\)</p> + +<p class="nind"><b>*181·62.</b> \(\vdash : \mu \in \text{NR} - {℩}ʻ0_{r} . \supset . Cʻʻ(\mu \dot{+} \dot{1} ) = Cʻʻ(\dot{1} \dot{+} \mu ) = Cʻʻ\mu +_{c} 1\)</p> + +<p>These propositions give the connection with cardinals.</p> + +<hr class="tb"> + +<p class="nind"><b>*181·01.</b> \(P \dot{\unicode{x21f8}} x = \downarrow \Lambda _{x}^{;}{℩}^{;}P \unicode{x21f8} (\Lambda \cap CʻP) \downarrow {℩}ʻx \quad\text{Df}\)</p> + +<p class="nind"><b>*181·011.</b> \(x \dot{\unicode{x21f7}} P = ({℩}ʻx) \downarrow (\Lambda \cap CʻP) \unicode{x21f7} \Lambda _{x} \downarrow ^{;}{℩}^{;}P \quad\text{Df}\)</p> + +<p class="nind"><b>*181·02.</b> \(\mu \dot{+} \dot{1} = \hat{R} \{(\exists P, x) . \text{N}_{0}\text{r}ʻP = \mu . R \,\text{smor}\, (P \dot{\unicode{x21f8}} x)\} \quad\text{Df}\)</p> + +<p class="nind"><b>*181·021.</b> \(\dot{1} \dot{+} \mu = \hat{R} \{(\exists P, x) . \text{N}_{0}\text{r}ʻP = \mu . R \,\text{smor}\, (x \dot{\unicode{x21f7}} P)\} \quad\text{Df}\)</p> + +<p class="nind"><b>*181·03.</b> \(\text{Nr}ʻP \dot{+} \dot{1} = \text{N}_{0}\text{r}ʻP \dot{+} \dot{1} \quad\text{Df}\)</p> + +<p class="nind"><b>*181·031.</b> \(\dot{1} \dot{+} \text{Nr}ʻP = \dot{1} \dot{+} \text{N}_{0}\text{r}ʻP \quad\text{Df}\)</p> + +<p class="nind"><b>*181·04.</b> \(\dot{1} \dot{+} \dot{1} = 2_{r} \quad\text{Df}\)</p> + +<p>Propositions concerning \(x \dot{\unicode{x21f7}} P\) are omitted +in what follows, since they are proved exactly as the analogous +propositions concerning \(P \dot{\unicode{x21f8}} x\) are proved.</p> + +<p class="nind"><b>*181·1.</b> \[\begin{align}\vdash \colon\ldotp &R(P \dot{\unicode{x21f8}} x)S . \equiv : (\exists y, z) . yPz . R = ({℩}ʻy) \downarrow \Lambda _{x} + . S = ({℩}ʻz) \downarrow \Lambda _{x} . \lor .\\ +&(\exists y) . y \in CʻP . R = ({℩}ʻy) \downarrow \Lambda _{x} . S = (\Lambda \cap CʻP) \downarrow {℩}ʻx \quad[(*181·01)]\end{align}\]</p> + +<p class="nind"><b>*181·11.</b> \(\vdash . (\Lambda \cap CʻP) \downarrow {℩}ʻx {\sim} \in Cʻ \downarrow \Lambda _{x}^{;}{℩}^{;}P\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*150·22. &\supset \vdash . Cʻ \downarrow \Lambda _{x}^{;}{℩}^{;}P = \downarrow \Lambda _{x}ʻʻ{℩}ʻʻCʻP .\\ +[*55·15] &\supset \vdash : Q \in Cʻ \downarrow \Lambda _{x}^{;}{℩}^{;}P . \supset _{Q} . \text{ᗡ}ʻQ = {℩}ʻ\Lambda _{x} &\qquad \text{(1)}\\ +\vdash .*55·15. &\supset \vdash . \text{ᗡ}ʻ(\Lambda \cap CʻP) \downarrow {℩}ʻx = {℩}ʻ{℩}ʻx &\qquad \text{(2)}\\ +\vdash .(1).(2).*51·161. \supset \vdash : Q \in Cʻ \downarrow \Lambda _{x}^{;}{℩}^{;}P . &\supset _{Q} + . \text{ᗡ}ʻQ \neq \text{ᗡ}ʻ(\Lambda \cap CʻP) \downarrow {℩}ʻx .\\ +[*30·37 . \text{Transp}] &\supset _{Q} . Q \neq (\Lambda \cap CʻP) \downarrow {℩}ʻx :\\ +[*13·196] &\supset \vdash . (\Lambda \cap CʻP) \downarrow {℩}ʻx {\sim} \in Cʻ \downarrow \Lambda _{x}^{;}{℩}^{;}P . \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*181·12.</b> \(\vdash . \downarrow \Lambda _{x}^{;}{℩}^{;}P \,\,\text{smor}\, P \quad[*151·61·65]\)</p> + +<p class="nind"><b>*181·13.</b> \(\vdash : Q \,\text{smor}\, P . y {\sim} \in CʻQ . \supset . Q \unicode{x21f8} y \,\text{smor}\, P \dot{\unicode{x21f8}} x\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*181·12. \supset \vdash : \text{Hp} . \supset . Q \,\text{smor}\, \downarrow \Lambda _{x}^{;}{℩}^{;}P &\qquad \text{(1)}\\ +\vdash .(1).*161·31.*181·11. \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_484">[Pg 484]</span></p> + +<p class="nind"><b>*181·2.</b> \(\vdash :Z\in \mu \dot{+} \dot{1} .\equiv .(\exists P,x).\mu =\text{N}_{0}\text{r}ʻP.Z\,\text{smor}\,(P\dot{\unicode{x21f8}} x) \quad[(*181·02)]\)</p> + +<p class="nind"><b>*181·21.</b> \[\begin{align}&\vdash \colon\ldotp \mu \in \text{NR}.\supset :Z\in \mu \dot{+} \dot{1} .\equiv .(\exists P,x).P\in \mu .Z\,\text{smor}\,(P\dot{\unicode{x21f8}} x)\\ +&[*181·2 .*155·26]\end{align}\]</p> + +<p class="nind"><b>*181*22.</b> \(\vdash .\text{N}_{0}\text{r}ʻP\dot{+} \dot{1} =\text{Nr}ʻ(P\dot{\unicode{x21f8}} x)\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *181·21.\supset \vdash :Z\in \text{N}_{0}\text{r}ʻP\dot{+} \dot{1} .\equiv .(\exists Q,y).Q\in \text{N}_{0}\text{r}ʻP.Z\,\text{smor}\,(Q\dot{\unicode{x21f8}} + y) &\qquad \text{(1)}\\ +\vdash .(1).*155·12.*152·11.\supset \vdash .\text{Nr}ʻ(P\dot{\unicode{x21f8}} x)\subset \text{N}_{0}\text{r}ʻP\dot{+} \dot{1} &\qquad \text{(2)}\\ +\vdash . *181·12·11.*161·31.\supset \vdash :Q\in \text{N}_{0}\text{r}ʻP.Z\,\text{smor}\,(Q\dot{\unicode{x21f8}} y).\supset .Z\,\text{smor}\,(P\dot{\unicode{x21f8}} x) &\qquad \text{(3)}\\ +\vdash .(1).(3).*152·11. \supset \vdash :Z\in \text{N}_{0}\text{r}ʻP\dot{+} \dot{1} .\supset .Z\in \text{Nr}ʻ(P\dot{\unicode{x21f8}} x) &\qquad \text{(4)}\\ +\vdash .(2).(4).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*181·24.</b> \(\vdash :P\,\,\text{smor}\,\,Q.\supset .\text{N}_{0}\text{r}ʻP\dot{+} \dot{1} =\text{N}_{0}\text{r}ʻQ\dot{+} 1 \quad[*181·22·12·11.*161·31]\)</p> + +<p class="nind"><b>*181·3.</b> \(\vdash .\text{Nr}ʻP\dot{+} \dot{1} =\text{N}_{0}\text{r}ʻP\dot{+} \dot{1} =\text{Nr}ʻ(P\dot{\unicode{x21f8}} x) \quad[*181·22.(*181·03)]\)</p> + +<p class="nind"><b>*181·31.</b> \(\vdash :P\,\,\text{smor}\,\,Q.\supset .\text{Nr}ʻP\dot{+} \dot{1} =\text{Nr}ʻQ\dot{+} \dot{1} \quad[*181·3·24]\)</p> + +<p class="nind"><b>*181·32.</b> \(\vdash :x{\sim}\in CʻP.\supset .\text{Nr}ʻP\dot{+} \dot{1} =\text{Nr}ʻ(P\unicode{x21f8} x) \quad[*181·3·13]\)</p> + +<p class="nind"><b>*181·33.</b> + \[\begin{align}&\vdash \colon\ldotp \mu ,\nu \in \text{NR}.\exists !\mu \dot{+} \dot{1} .\supset :\mu =\nu .\equiv .\mu \dot{+} \dot{1} =\nu \dot{+} \dot{1} .\equiv .\dot{1} \dot{+} \mu =\dot{1} \dot{+} \nu \\ +&[*161·33.*181·3·11·12]\end{align}\]</p> + +<p>The above proposition is used in *253·23·571.</p> + +<p class="nind"><b>*181·4.</b> \(\vdash :\exists !\mu \dot{+} \dot{1} .\supset .\mu \in \text{NR}-\iota ʻ\Lambda .\mu \in \text{N}_{0}\text{R} \quad[*181·2.*155·22]\)</p> + +<p class="nind"><b>*181·42.</b> \(\vdash .\mu \dot{+} \dot{1} \in \text{NR}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*181·3.\supset \vdash :\mu \in \text{N}_{0}\text{R}.&\supset .(\exists P,x).\mu \dot{+} \dot{1} =\text{Nr}ʻ(P\dot{\unicode{x21f8}} x).\\ +[*152·4] &\supset .\mu \dot{+} 1\in \text{NR} &\qquad \text{(1)}\\ +\vdash .*181·4.\supset \vdash :\mu {\sim}\in \text{N}_{0}\text{R}.&\supset .\mu \dot{+} 1=\Lambda .\\ +[*154·242] &\supset .\mu \dot{+} 1\in \text{NR} &\qquad \text{(2)}\\ +\vdash .(1).(2).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*181·43.</b> \(\vdash :\mu \dot{+} 1=\text{N}_{0}\text{r}ʻZ.\equiv .Z\in \mu \dot{+} 1 \quad[*155·26.*181·42]\)</p> + +<p>The following propositions are concerned with the associative law when +\(\dot{1}\) is one of the addenda.</p> + +<p class="nind"><b>*181·53.</b> \(\vdash :\dot{\exists} !P.x\neq y.\supset .(P\dot{\unicode{x21f8}} x)\dot{\unicode{x21f8}} y\,\text{smor}\,P+(x\downarrow y)\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*13·15.(*181·01).\supset \\ +\vdash .(P\dot{\unicode{x21f8}} x)\dot{\unicode{x21f8}} y=\downarrow \Lambda _{y}^{;}\iota ^{;}\{\downarrow \Lambda _{x}^{;}\iota ^{;}P\unicode{x21f8} + (\Lambda \cap CʻP)\downarrow \iota ʻx\}\unicode{x21f8} \{\Lambda \cap Cʻ(P\dot{\unicode{x21f8}} x)\}\downarrow \iota ʻy\\ +[*161·4]=\downarrow \Lambda _{y}^{;}\iota ^{;}\downarrow \Lambda _{x}^{;}\iota ^{;}P\unicode{x21f8} \downarrow \Lambda _{y}ʻ\iota + ʻ(\Lambda \cap CʻP)\downarrow \iota ʻx\unicode{x21f8} \{\Lambda \cap Cʻ(P\dot{\unicode{x21f8}} x)\}\downarrow \iota ʻy &\qquad \text{(1)}\\ +\vdash .(1).*161·22.\supset \vdash :\text{Hp}.\supset .(P\dot{\unicode{x21f8}} x)\dot{\unicode{x21f8}} y\\ +=\downarrow \Lambda _{y}^{;}\iota ^{;}\downarrow \Lambda _{x}^{;}\iota ^{;}P\unicode{x2909}\{\downarrow \Lambda _{y}ʻ\iota + ʻ(\Lambda \cap CʻP)\downarrow \iota ʻx\}\downarrow [\{\Lambda \cap Cʻ(P\dot{\unicode{x21f8}} x)\}\downarrow \iota ʻy] &\qquad \text{(2)}\\ +\vdash .*180·13.*181·11.\supset \\ +\vdash :\text{Hp}.\supset .\downarrow \Lambda _{y}^{;}\iota ^{;}\downarrow \Lambda _{x}^{;}\iota ^{;}P\unicode{x2909}\{\downarrow \Lambda _{y}ʻ\iota + ʻ(\Lambda \cap CʻP)\downarrow \iota ʻx\}\downarrow [\{\Lambda \cap Cʻ(P\dot{\unicode{x21f8}} x)\}\downarrow \iota ʻy] +\,\text{smor}\,P+(x\downarrow y) &\qquad \text{(3)}\\ +\vdash .(2).(3).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_485">[Pg 485]</span></p> + +<p class="nind"><b><a id="*181·54">*181·54</a>.</b> \(\vdash :\nu \neq 0_{r}.\supset .(\mu \dot{+} \nu )\dot{+} \dot{1} =\mu \dot{+} (\nu \dot{+} \dot{1} )\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*181·2.*180·2.&\supset \vdash :Y\in (\mu \dot{+} \nu )\dot{\unicode{x21f8}} \dot{1} .\equiv .\\ +&(\exists P,Q,R,x).\text{N}_{0}\text{r}ʻP=\mu .\text{N}_{0}\text{r}ʻQ=\nu .R\,\,\text{smor}\,\,P+Q.Y\,\text{smor}\,R\dot{\unicode{x21f8}} x.\\ +[*181·22·31]&\equiv .(\exists P,Q,x).\text{N}_{0}\text{r}ʻP=\mu .\text{N}_{0}\text{r}ʻQ=\nu .Y\,\text{smor}\,(P+Q)\dot{\unicode{x21f8}} x &\qquad \text{(1)}\\ +\vdash .*153·14.&\supset \vdash \colon\ldotp \text{Hp}.\supset :\text{N}_{0}\text{r}ʻQ=\nu .\supset .\dot{\exists} !Q &\qquad \text{(2)}\\ +\vdash .*160·44.(*180·01.*181·01).\supset \\ +&\vdash :P'=\downarrow \Lambda _{x}^{;}\iota ^{;}\downarrow \Lambda _{CʻQ}^{;}\iota ^{;}P.Q'=\downarrow \Lambda _{x}^{;}\iota ^{;}\Lambda + _{CʻP}\downarrow ^{;}\iota ^{;}Q.X=\{\Lambda \cap Cʻ(P+Q)\}\downarrow \iota ʻx.\\ +&\supset .P'\unicode{x2909}Q'=\downarrow \Lambda _{x}^{;}\iota ^{;}(P+Q).(P'\unicode{x2909}Q')\unicode{x21f8} X=(P+Q)\dot{\unicode{x21f8}} x &\qquad \text{(3)}\\ +\vdash .*180·12.*181·12.&\supset \vdash :\text{Hp}(3).\supset .P'\,\text{smor}\,P.Q'\,\text{smor}\,Q &\qquad \text{(4)}\\ +\vdash .*180·11.*72·411.*181·11.(3).\supset \\ +\vdash :\text{Hp}(3).&\supset .CʻP'\cap CʻQ'=\Lambda .X{\sim}\in CʻP'.X{\sim}\in CʻQ' &\qquad \text{(5)}\\ +\vdash .*161·23.(4).&\supset \vdash :\text{Hp}(3).\dot{\exists} !Q.\supset .(Pʻ\unicode{x2909}Qʻ)\unicode{x21f8} X=P'\unicode{x2909}(Qʻ\unicode{x21f8} X) &\qquad \text{(6)}\\ +\vdash .*181·13.(4).(5).&\supset \vdash :\text{Hp}(3).\supset .Q'\unicode{x21f8} X\,\text{smor}\,(Q\dot{\unicode{x21f8}} x).\\ +[*180·13.(4).(5)] &\supset .P'\unicode{x2909}(Q'\unicode{x21f8} X)\,\text{smor}\,P+(Q\dot{\unicode{x21f8}} x) &\qquad \text{(7)}\\ +\vdash .(1).(2).(6).(7).&\supset \vdash \colon\ldotp \text{Hp}.\text{Hp}(3).\supset :Y\in (\mu \dot{+} \nu )\dot{+} \dot{1} .\equiv .\\ +&(\exists P,Q,x).\text{N}_{0}\text{r}ʻP=\mu .\text{N}_{0}\text{r}ʻQ=\nu .Y\,\text{smor}\,P+(Q\dot{\unicode{x21f8}} x).\\ +[*180·3.*181·3]&\equiv .(\exists P,Q,x).\text{N}_{0}\text{r}ʻP=\mu .\text{N}_{0}\text{r}ʻQ=\nu .Y\in \text{N}_{0}\text{r}ʻP\dot{+} (\text{N}_{0}\text{r}ʻQ\dot{+} + \dot{1} ).\\ +[*13·193.*155·2]&\equiv .\mu ,\nu \in \text{N}_{0}\text{R}.Y\in \mu \dot{+} (\nu \dot{+} \dot{1} ).\\ +[*181·4.*180·4] &\equiv .Y\in \mu \dot{+} (\nu \dot{+} \dot{1} ) &\qquad \text{(8)}\\ +\vdash .(8).*13·19.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*181·55.</b> \(\vdash :\mu \neq 0_{r}.\supset .\dot{1} \dot{+} (\mu \dot{+} \nu )=(\dot{1} \dot{+} \mu )\dot{+} \nu \quad[\text{Proof as in *181·54}]\)</p> + +<p class="nind"><b>*181·56.</b> \(\vdash :\mu \neq 0_{r}.\supset .(\mu \dot{+} \dot{1} )\dot{+} \dot{1} =\mu \dot{+} (\dot{1} \dot{+} \dot{1} )=\mu \dot{+} 2_{r}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*153·2.*180·2.\supset \\ +\vdash :Z\in \mu \dot{+} 2_{r}.&\equiv .(\exists P,x,y).\mu =\text{N}_{0}\text{r}ʻP.x\neq y.Z\,\text{smor}\,P+(x\downarrow y) &\qquad \text{(1)}\\ +\vdash .(1).*181·53.\supset \vdash \colon\ldotp \text{Hp}.\supset : +&Z\in \mu \dot{+} 2_{r}.\equiv .(\exists P,x,y).\mu =\text{N}_{0}\text{r}ʻP.x\neq y.Z\,\text{smor}\,(P\dot{\unicode{x21f8}} x)\dot{\unicode{x21f8}} y.\\ +[*181·22] &\equiv .(\exists P,x,y).\mu =\text{N}_{0}\text{r}ʻP.x\neq y.Z\in (\text{N}_{0}\text{r}ʻP\dot{+} \dot{1} )\dot{+} \dot{1} .\\ +[*181·4] &\equiv .(\exists x, y).x\neq y.Z\in (\mu \dot{+} \dot{1} )\dot{+} \dot{1} .\\ +[*24·1] &\equiv .Z\in (\mu \dot{+} \dot{1} )\dot{+} \dot{1} &\qquad \text{(2)}\\ +\vdash .(2).(*181·04).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_486">[Pg 486]</span></p> + +<p>The last line in the above proof, in which *24·1 is used, is legitimate +because \(x\) and \(y\) may be of any type whatever, and therefore +the fact that \(\Lambda \neq \text{V}\) is sufficient to establish +\((\exists x,y).x\neq y\) in the sense wanted.</p> + +<p class="nind"><b>*181·561.</b> \(\mu \dot{+} \dot{1} \dot{+} \dot{1} =\mu \dot{+} (\dot{1} \dot{+} \dot{1} ) \quad\text{Df}\)</p> + +<p>This definition adopts the opposite convention to that usually adopted. +But it is convenient to have \(0_{r}\dot{+} \dot{1} \dot{+} \dot{1}=2_{r}\), +and also to have as much similarity as possible between the +results of adding \(\dot{1}\) at the beginning and end of a relation. +Both reasons lead to the adoption of the above convention. (Cf. +*181·57·571, below.)</p> + +<p class="nind"><b>*181·57.</b> \[\begin{align}&\vdash :\mu \neq 0_{r}.\supset .\dot{1} \dot{+} (\dot{1} \dot{+} \mu )=(\dot{1} \dot{+} \dot{1} )\dot{+} \mu =2_{r}\dot{+} \mu \\ +&[\text{Proof as in *181·56}]\end{align}\]</p> + +<p class="nind"><b>*181·571.</b> \(\dot{1} \dot{+} \dot{1} \dot{+} \mu =(\dot{1} \dot{+} \dot{1} )\dot{+} \mu \quad\text{Df}\)</p> + +<p class="nind"><b>*181·58.</b> \(\vdash :\mu \neq 0_{r}.\nu \neq 0_{r}.\supset .(\mu \dot{+} \dot{1} )\dot{+} \nu =\mu \dot{+} (\dot{1} \dot{+} \nu ) \quad[*161·232]\)</p> + +<p>The proof proceeds in the same way as that of <a href="#*181·54">*181·54</a>.</p> + +<p class="nind"><b>*181·59.</b> \(\vdash :\mu \neq 0_{r}.\nu \neq 0_{r}.\supset .(\mu \dot{+} \dot{1} )\dot{+} (\dot{1} \dot{+} \nu )=\mu \dot{+} 2_{r}\dot{+} \nu \quad[*161·25]\)</p> + +<p>The above propositions show that, except when one of the summands is +zero, the associative law holds for \(\dot{1}\) just as if it were a +relation-number.</p> + +<p>The following propositions are concerned with relations to cardinal +addition.</p> + +<p class="nind"><b>*181·6.</b> \(\vdash :\dot{\exists} !P.\supset .Cʻʻ\text{Nr}ʻ(P\dot{\unicode{x21f8}} x)= \text{Nc}ʻCʻP+_{c}1\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*152·7. &\supset \vdash .Cʻʻ\text{Nr}ʻ(P\dot{\unicode{x21f8}} x)=\text{Nc}ʻCʻ(P\dot{\unicode{x21f8}} x).\\ +\vdash .(1).*161·14.\supset \vdash :\text{Hp}.&\supset .Cʻʻ\text{Nr}ʻ(P\dot{\unicode{x21f8}} x)\\ +&=\text{Nc}ʻ[\downarrow \Lambda _{x}ʻʻ\iotaʻʻCʻP\cup \iota ʻ\{(\Lambda \cap CʻP)\downarrow \iota ʻx\}]\\ +[*110·13·3.*181·11.*110·12]&=\text{Nc}ʻCʻP+_{c}1:\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*181·61.</b> \[\begin{align}&\vdash :\dot{\exists} !P.\supset .Cʻʻ\text{Nr}ʻ(x\dot{\unicode{x21f7}} P)=1+_{c}\text{Nc}ʻCʻP= \text{Nc}ʻCʻP+_{c}1\\ +&[\text{Proof as in *181·6}]\end{align}\]</p> + +<p class="nind"><b>*181·62.</b> \(\vdash :\mu \in \text{NR}-\iota ʻ0_{r}.\supset .Cʻʻ(\mu \dot{+} \dot{1} )=Cʻʻ(\dot{1} \dot{+} \mu )=Cʻ\mu +_{c}1\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*153·16.*152·4. \supset \vdash :\text{Hp}.&\supset .(\exists P).\mu =\text{Nr}ʻP.\dot{\exists} !P.\\ +[*181·3·6] \supset .(\exists P).\mu &=\text{Nr}ʻP.Cʻʻ(\text{Nr}ʻP\dot{+} \dot{1} )= \text{Nc}ʻCʻP+_{c}1\\ +[*152·7] &=Cʻʻ\text{Nr}ʻP+_{c}1.\\ +[*13·193] &\supset .Cʻʻ(\mu \dot{+} \dot{1} )=Cʻʻ\mu +_{c}1 &\qquad \text{(1)}\\ +Similarly \vdash :\text{Hp}.&\supset .Cʻʻ(1\dot{+} \mu )=1+_{c}Cʻʻ\mu &\qquad \text{(2)}\\ +\vdash .(1).(2).*110·51.\supset \vdash .\text{Prop} +\end{array} +\]</p> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_487">[Pg 487]</span></p> +<h2 class="nobreak" id="*182"> +*182 ON SEPARATED RELATIONS.</h2> +</div> + + +<p class="nind"><i>Summary of</i> *182.</p> + +<p>In this number, we have to consider, as a preliminary to the +addition of the relation-numbers of a field, the properties of +the relation \(\breve{\downarrow}_{.,}^{;}P\), which is defined +as follows. If \(x\unicode{x2640}y\) is any function of two +arguments in the sense of *38, we put \(\breve{\unicode{x2640}}ʻx= x\unicode{x2640}x\, \text{Df}\). +Thus \(\breve{\downarrow}_{.,}ʻQ= Q\downarrow_{.,} Q\), +<i>i.e.</i> \(\breve{\downarrow}_{.,}ʻQ= \downarrow Q^{;}Q\). +Hence \(\breve{\downarrow}_{.,}^{;}P\) is +the relation of \(\downarrow Q^{;}Q\) to \(\downarrow R^{;}R\) when +\(QPR\). Thus the symbol \(\breve{\downarrow}_{.,}^{;}P\) is only +significant when \(P\) is a relation of relations; when this is the +case, \(\breve{\downarrow}_{.,}^{;}P\) is the relation which results +when, for every \(Q\) which is a member of \(CʻP\), every member +\(x\) of \(CʻQ\) is replaced by \(x\downarrow Q\). The result is a +\(\text{Rel}^{2}\text{excl}\), whose arithmetical properties serve to +define the arithmetical properties of the sum of the relation-numbers +of members of \(CʻP\) the next number, we shall put +\[ +\Sigma \text{Nr}ʻP = \text{Nr}ʻ\Sigma ʻ\breve{\downarrow}_{.,}^{;}P \quad\text{Df}. +\] +We shall put later +\[ +\Pi \text{Nr}ʻP = \text{Nr}ʻ\Pi ʻP +\] +and we shall find +\[ +\Pi ʻP = \dot{s} ^{;}\text{Prod}ʻ\breve{\downarrow}_{.,}^{;}P.\text{Nr}ʻ\Pi ʻP = \text{Nr}ʻ\text{Prod}ʻ\breve{\downarrow}_{.,}^{;}P. +\] +Thus we might have dispensed with \(\Pi ʻP\) as a fundamental notion, +using \(\text{Prod}\) instead, and putting +\[ +\Pi ʻP = \dot{s} ^{;}\text{Prod}ʻ\breve{\downarrow}_{.,}^{;}P \quad\text{Df}. +\] +But this course is on the whole less convenient than that adopted in +<a href="#*172">*172</a> and <a href="#*173">*173</a>.</p> + +<p>The notation \(\breve{\unicode{x2640}}\) is thus required in +connection with ordinal addition, where it is almost indispensable. +It has besides certain minor uses. The object of the notation is to +enable us to exhibit as a function of \(x\) an expression of the form +\(x\unicode{x2640}x\), where \(\unicode{x2640}\) is any descriptive +double function which exists for all possible pairs of arguments. Thus +for example \(x\downarrow x\) is a function of \(x\), but the notations +hitherto introduced do not enable us to exhibit it in the form \(Rʻx\). +Hence if we wish (say) to deal with the class +\[ +\hat{P} \{(\exists x).x\in \alpha .P = x\downarrow x\} +\]<span class="pagenum" id="Page_488">[Pg 488]</span> +we cannot write it in the form \(Rʻʻ\alpha\) unless we introduce a new +notation. We put +\[ +\breve{{\downarrow}}ʻx=x\downarrow x +\] +whence \(\hat{P} \{(\exists x).x\in \alpha .P=x\downarrow x\}=\breve{{\downarrow}}ʻʻ\alpha\). +We introduce the notation generally for all descriptive double +functions which exist for all possible pairs of arguments. +Thus "\(\unicode{x2640}\)" in this number corresponds to +"\(\unicode{x2640}\)" in *38.</p> + +<p>In the present number, we shall begin by a few propositions +illustrating possible uses of the notation \(\breve{\unicode{x2640}}\). +Thus for example if \(\lambda\) is a class of relations, we have +hitherto had no simple notation for expressing the class of their +squares. But since \(R^{2}=\breve{\mid}ʻR\), the class of the squares +of \(\lambda\)'s is \(\breve{\mid}ʻʻ\lambda\). The notation is, +however, introduced chiefly in order to be applied to \(\downarrow\) +and \(\downarrow_{.,}\). We therefore proceed almost at once to +propositions on \(\breve{\downarrow}_{.,}\), and especially on +\(\breve{\downarrow}_{.,}^{;}P\). We have</p> + +<p class="nind"><b>*182·16·162.</b> \(\vdash .\breve{\downarrow}_{.,}^{;}P\in \text{Rel}^{2}\text{excl}.\breve{\downarrow}_{.,}\in + 1\rightarrow 1.\breve{\downarrow}_{.,}^{;}P\,\,\text{smor}\,\,P\)</p> + +<p class="nind"><b>*182·2.</b> \(\vdash .\breve{\downarrow}_{.,}ʻQ=\Pi ʻ(Q\downarrow Q)=\Pi ʻ\breve{{\downarrow}}ʻQ\)</p> + +<p class="nind"><b>*182·21.</b> \(\vdash .\breve{\downarrow}_{.,}^{;}P=\Pi ^{;}\breve{{\downarrow}}^{;}P\)</p> + +<p>We next prove (<a href="#*182·27">*182·27</a>) that if \(P\in \text{Rel}^{2}\text{excl}\), +then \(P\) has double likeness to \(\breve{\downarrow}_{.,}^{;}P\), +the double correlator being \(\breve{\iota} \mid \text{D}\) with its +converse domain limited to \(Cʻ\Sigma ʻ\breve{\downarrow}_{.,}^{;}P\) +(<a href="#*182·26">*182·26</a>). We then prove (<a href="#*182·33">*182·33</a>) that if \(T\upharpoonright Cʻ\Sigma ʻP\) +is a double correlator of \(P\) with \(Q\), then +\(T\parallel \text{Cnv}ʻT\dagger\) (with its converse domain limited) +is a double correlator of \(\breve{\downarrow}_{.,}^{;}P\) and +\(\breve{\downarrow}_{.,}^{;}Q\), whence we deduce</p> + +<p class="nind"><b>*182·34.</b> \(\vdash :P\,\text{smor}\, \,\text{smor}\,Q.\supset .\breve{\downarrow}_{.,}^{;}P\,\text{smor}\, \,\text{smor}\,\breve{\downarrow}_{.,}^{;}Q\)</p> + +<p>We next proceed to prove</p> + +<p class="nind"><b>*182·42.</b> \(\vdash .\Pi ʻP=\dot{s} ^{;}\text{Prod}ʻ\breve{\downarrow}_{.,}^{;}P=\dot{s} ^{;}\text{D}^{;}\Pi ʻ\breve{\downarrow}_{.,}^{;}P=\Pi + ʻ\Sigma ʻ\breve{{\downarrow}}^{;}P\)</p> + +<p>The proof of this is as follows: In virtue of <a href="#*182·21">*182·21</a> and the +associative law for \(\Pi\), we have +\[ +\dot{s}^{;}\text{D}^{;}\Pi ʻ\breve{\downarrow}_{.,}^{;}P=\Pi ʻ\Sigma ʻ\breve{{\downarrow}}^{;}P. +\] +<span class="pagenum" id="Page_489">[Pg 489]</span> +Now \(\Sigma ʻ\breve{{\downarrow}}^{;}P=P\unicode{x228d} I\upharpoonright CʻP \) +(<a href="#*182·413">*182·413</a>), and \(\Pi ʻ(P\unicode{x228d} I\upharpoonright CʻP)=\Pi ʻP\) (<a href="#*172·51">*172·51</a>).</p> + +<p>Hence our proposition results. Hence we arrive at</p> + +<p class="nind"><b>*182·44.</b> \(\vdash .\text{Nr}ʻ\Pi ʻP=\text{Nr}ʻ\text{Prod}ʻ\breve{\downarrow}_{.,}^{;}P=\text{Nr}ʻ\Pi ʻ\breve{\downarrow}_{.,}^{;}P\)</p> + +<p>Finally we have some propositions showing how the notation +\(\breve{\unicode{x2640}}\) can be applied in cardinals. +It is then applied to \(\downarrow_{,,}\), instead of, as +above, to \(\downarrow_{.,}\). We have (<a href="#*182·5">*182·5</a><a href="#*182·51">·51</a><a href="#*182·52">·52</a>) +\({\in}\unicode{x21A7}\alpha =\breve{\downarrow_{,,}}ʻ\alpha .\in\unicode{x21A7}ʻʻ\kappa =\breve{\downarrow_{,,}}ʻʻ\kappa .\Sigmaʻ\kappa =sʻ\breve{\downarrow_{,,}}ʻʻ\kappa\). +Thus the notation of the present number might have been employed in +dealing with cardinal addition (<a href="#*112">*112</a>) instead of the notation \({\in}\unicode{x21A7}\alpha\). +The general notation \(P\unicode{x21A7}x\) was, however, required for +other purposes (cf. *85) and could not have been dispensed with.</p> + +<p>In <a href="#*183">*183</a> we shall put +\[ +\Sigma \text{Nr}ʻP=\text{Nr}ʻ\Sigma ʻ\breve{\downarrow}_{.,}^{;}P, +\] +and by <a href="#*182·52">*182·52</a> we have +\[ +\Sigma \text{Nc}ʻ\kappa =\text{Nc}ʻsʻ\breve{\downarrow_{,,}}ʻʻ\kappa . +\] +It will be seen that these formulae have the usual kind of analogy.</p> + +<hr class="tb"> + +<p class="nind"><b>*182·01.</b> \(\breve{\unicode{x2640}}=\hat{y} \hat{x} (y=x\unicode{x2640}x) \quad\text{Df}\)</p> + +<p class="nind"><b>*182·02.</b> \(\vdash :y\breve{\unicode{x2640}}x.\equiv .y=x\unicode{x2640}x \quad[(*182·01)]\)</p> + +<p class="nind"><b>*182·021.</b> \(\vdash .\breve{\unicode{x2640}}ʻx=x\unicode{x2640}x \quad[*182·02.*30·3]\)</p> + +<p class="nind"><b>*182·022.</b> \(\vdash .\text{E}!\breve{\unicode{x2640}}ʻx \quad[*182·021.*14·21]\)</p> + +<p class="nind"><b>*182·023.</b> \(\vdash :\breve{\unicode{x2640}}\in 1\rightarrow \text{Cls}:(\alpha ).\alpha \subset \text{ᗡ}ʻ\breve{\unicode{x2640}} \quad[*182·022.*71·166.*33·431]\)</p> + +<p class="nind"><b>*182·03.</b> \(\vdash .\breve{\mid}ʻR=R^{2} \quad[*182·021.(*34·02)]\)</p> + +<p>Thus if \(\lambda\) is a class of relations, the class of their squares +is \(\breve{\mid}ʻʻ\lambda\).</p> + +<p class="nind"><b>*182·031.</b> \(\vdash .\breve{\uparrow}ʻ\alpha =\alpha \uparrow \alpha \quad[*182·021]\)</p> + +<p class="nind"><b>*182·032.</b> \(\vdash .\breve{{\downarrow}}ʻx=x\downarrow x \quad[*182·021]\)</p> + +<p class="nind"><b>*182·033.</b> \(\vdash .\dot{2} -2_{r}=\text{D}ʻ\breve{{\downarrow}}=1_{s} \quad[*56·13.*182·032.*153·3]\)</p> + +<p class="nind"><b>*182·04.</b> \(\vdash .\breve{\downarrow_{,,}}ʻ\alpha =\downarrow \alpha ʻʻ\alpha \quad[*182·021.*38·2]\)</p> + +<p>Observe that in \(\breve{\downarrow_{,,}}\), we first take +\(\downarrow_{,,}\), and then put a circumflex over it. If we first +took \(\breve{{\downarrow}}\), we could not then place two commas +under it, because \(\breve{{\downarrow}}\) is a relation, not a double +descriptive function, and two commas can only significantly be placed +under a double descriptive function.</p> + +<p><span class="pagenum" id="Page_490">[Pg 490]</span></p> + +<p class="nind"><b>*182·05.</b> \(\vdash .\breve{\downarrow}_{.,}ʻQ=\downarrow Q^{;}Q=Q\downarrow_{.,} Q \quad[*182·021.*150·6]\)</p> + +<p>The relation for the sake of which the above notation is chiefly +introduced is \(\breve{\downarrow}_{.,}^{;}P\), where \(P\) is a +relation of relations. If \(P\) relates \(Q\) and \(R\), then +\(\breve{\downarrow}_{.,}^{;}P\) relates \(\downarrow Q^{;}Q\) and +\(\downarrow R^{;}R\). This is stated in the following proposition:</p> + +<p class="nind"><b>*182·1.</b> \[\begin{align}&\vdash .\breve{\downarrow}_{.,}^{;}P=\hat{X} \hat{Y} \{(\exists Q,R).QPR.X=Q\downarrow_{.,} Q.Y=R\downarrow_{.,} R\}\\ +&[*182·023·05.*150·4]\end{align}\]</p> + +<p class="nind"><b>*182·11.</b> \(\vdash .Cʻ\breve{\downarrow}_{.,}^{;}P=\breve{\downarrow}_{.,}ʻʻCʻP \quad[*150·22]\)</p> + +<p class="nind"><b>*182·12.</b> \(\vdash .Cʻ\breve{\downarrow}_{.,}ʻQ=\downarrow QʻʻCʻQ=F\unicode{x21A7}Q \quad[*182·05.*150·22.*33·5.(*85·5)]\)</p> + +<p class="nind"><b>*182·13.</b> \(\vdash .CʻʻCʻ\breve{\downarrow}_{.,}^{;}P=F\unicode{x21A7}ʻʻCʻP \quad[*182·11·12]\)</p> + +<p class="nind"><b>*182·14.</b> \(\vdash .F\unicode{x21A7}\in 1\rightarrow 1\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*182·12. &\supset \vdash :F\unicode{x21A7}Q=F\unicode{x21A7}R.\supset .\downarrow QʻʻCʻQ=\downarrow RʻʻCʻR &\qquad \text{(1)}\\ +\vdash .(1).*55·232.*37·45.&\supset \vdash :F\unicode{x21A7}Q=F\unicode{x21A7}R.\dot{\exists} !Q.\supset .Q=R &\qquad \text{(2)}\\ +\vdash .*37·45. &\supset \vdash :\downarrow QʻʻCʻQ=\downarrow RʻʻCʻR.Q=\dot{\Lambda} .\supset .\downarrow RʻʻCʻR=\dot{\Lambda} .\\ +[*37·45.*33·241] &\supset .R=\dot{\Lambda} &\qquad \text{(3)}\\ +\vdash .(1).(2).(3). &\supset \vdash :F\unicode{x21A7}Q=F\unicode{x21A7}R.\supset .Q=R:\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*182·15.</b> \(\vdash :\exists !F\unicode{x21A7}Q\cap F\unicode{x21A7}R.\supset .Q=R\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*182·12. \supset \vdash :\text{Hp}.&\supset .\exists !\downarrow QʻʻCʻQ\cap \downarrow RʻʻCʻR.\\ +[*55·232] &\supset .Q=R:\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*182·16">*182·16</a>.</b> \(\vdash .\breve{\downarrow}_{.,}^{;}P\in \text{Rel}^{2}\text{excl} \quad[*182·12·15]\)</p> + +<p class="nind"><b>*182·161.</b> \(\vdash :\breve{\downarrow}_{.,}ʻQ=\breve{\downarrow}_{.,}ʻR.\equiv .Q=R\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*182·05.\supset \vdash :\breve{\downarrow}_{.,}ʻQ=\breve{\downarrow}_{.,}ʻR.&\equiv .Q\downarrow_{.,} Q=R\downarrow_{.,} R.\\ +[*165·23] &\supset .Q=R &\qquad \text{(1)}\\ +\vdash .(1).*30·37.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*182·162.</b> \(\vdash .\breve{\downarrow}_{.,}\in 1\rightarrow 1.\breve{\downarrow}_{.,}^{;}P\,\,\text{smor}\,\,P \quad[*182·161.*71·57.*151·243]\)</p> + +<p class="nind"><b>*182·17.</b> \(\vdash .Cʻ\Sigma ʻ\breve{\downarrow}_{.,}^{;}P=\hat{S} \{(\exists Q,x).Q\in CʻP.x\in CʻQ.S=x\downarrow Q\}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*182·12.*162·22.*40·4.\supset \\ +\vdash .Cʻ\Sigma ʻ\breve{\downarrow}_{.,}^{;}P&=\hat{S} \{(\exists Q).Q\in CʻP.S\in \downarrow QʻʻCʻQ\}\\ +[*55·231] &=\hat{S} \{(\exists Q,x).Q\in CʻP.x\in CʻQ.S=x\downarrow Q\}.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_491">[Pg 491]</span></p> + +<p class="nind"><b>*182·18.</b> \(\vdash .\dot{s} ʻCʻ\Sigma ʻ\breve{\downarrow}_{.,}^{;}P = F\upharpoonright CʻP\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *182·17. \supset \\ +\vdash . sʻCʻ\Sigma ʻ \breve{\downarrow}_{.,} ^{;}P &= \hat{y} \hat{R} \{(\exists Q, x).Q\in CʻP .x\in CʻQ.y (x\downarrow Q)R\}\\ +[*55·13] &= \hat{y} \hat{R} \{(\exists Q, x). Q\in CʻP. x\in CʻQ. y = x. R = Q\}\\ +[*13·22] &= \hat{y} \hat{R} \{R\in CʻP. y \in CʻR\}\\ +[*33·51] &= \hat{y} \hat{R} \{R\in CʻP. yFR\}\\ +[*35·101] &= F\upharpoonright CʻP. \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*182·19.</b> \(\vdash . sʻ\text{D}ʻʻCʻ\Sigma ʻ \breve{\downarrow}_{.,}^{;}P = Cʻ\Sigma ʻP . sʻ\text{ᗡ}ʻʻCʻ\Sigma ʻ\breve{\downarrow}_{.,}^{;}P + = CʻP - {℩}ʻ\dot{\Lambda}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *41·43 . *182·18 .\supset \vdash . sʻ\text{D}ʻʻCʻ\Sigma ʻ\breve{\downarrow}_{.,}^{;}P &= \text{D}ʻ(F\upharpoonright CʻP)\\ +[*162·23] &= Cʻ\Sigma ʻP &\qquad \text{(1)}\\ +\vdash . *41·44 . *182·18 .\supset \vdash . sʻ\text{ᗡ}ʻʻCʻ\Sigma ʻ\breve{\downarrow}_{.,}^{;}P &= \text{ᗡ}ʻ(F\upharpoonright CʻP)\\ +[*172·192] &= CʻP - {℩}ʻ\dot{\Lambda} &\qquad \text{(2)}\\ +\vdash . (1). (2) .\supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*182·2">*182·2</a>.</b> \(\vdash .\breve{\downarrow}_{.,}ʻQ=\Pi ʻ(Q\downarrow Q)=\Pi ʻ\breve{{\downarrow}}ʻQ \quad[*182·05·032 . *172·2]\)</p> + +<p class="nind"><b><a id="*182·21">*182·21</a>.</b> \(\vdash .\breve{\downarrow}_{.,}^{;}P=\Pi ^{;}\breve{{\downarrow}}^{;}P \quad[*182·2]\)</p> + +<p>The following propositions lead up to <a href="#*182·26">*182·26</a><a href="#*182·27">·27</a>.</p> + +<p class="nind"><b>*182·22.</b> \(\vdash .\text{D}^{;}\breve{\downarrow}_{.,}ʻQ=\text{Prod}ʻ\breve{{\downarrow}}ʻQ={℩}^{;}Q \quad[*182·2 *173·1·22]\)</p> + +<p class="nind"><b>*182·23.</b> \(\vdash .\breve{{℩}} ^{;}\text{D}^{;}\breve{\downarrow}_{.,}ʻQ=Q \quad[*182·22. *151·252]\)</p> + +<p class="nind"><b>*182·24.</b> \(\vdash .\breve{{℩}} \dagger ^{;}\text{D}\dagger ^{;}\breve{\downarrow}_{.,}^{;}P=P \quad[*182·23]\)</p> + +<p class="nind"><b>*182·25.</b> \(\vdash .\breve{{℩}} ^{;}\text{D}^{;}\Sigma ʻ\breve{\downarrow}_{.,}^{;}P=\Sigma ʻP.Cʻ\Sigma ʻ\breve{\downarrow}_{.,}^{;}P\subset + \text{ᗡ}ʻ(\breve{{℩}} \mid \text{D})\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*55·15.&\supset \vdash .\breve{{℩}} ʻ\text{D}ʻ\downarrow Qʻy=y.\\ +[*33·43] &\supset \vdash .\downarrow Qʻy\in \text{ᗡ}ʻ(\breve{{℩}} \mid \text{D}).\\ +[*182·12] &\supset \vdash .Cʻ\breve{\downarrow}_{.,}ʻQ\subset \text{ᗡ}ʻ(\breve{{℩}} \mid \text{D}).\\ +[*162·22] &\supset \vdash .Cʻ\Sigma ʻ\breve{\downarrow}_{.,}^{;}P\subset \text{ᗡ}ʻ(\breve{{℩}} \mid \text{D}).\\ +[*162·35] \supset \vdash .\breve{{℩}} ^{;}\text{D}^{;}\Sigma ʻ\breve{\downarrow}_{.,}^{;}P&=\Sigma ʻ\breve{{℩}} \dagger ^{;}\text{D}\dagger + ^{;}\breve{\downarrow}_{.,}^{;}P\\ +[*182·24] &= \Sigma ʻP.\supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_492">[Pg 492]</span></p> + +<p class="nind"><b><a id="*182·26">*182·26</a>.</b> \(\vdash :P\in \text{Rel}^{2}\text{excl}.\supset .\breve{\iota} \mid \text{D}\upharpoonright Cʻ\Sigma ʻ\breve{\downarrow}_{.,}^{;}P\in + P\,\overline{\,\text{smor}\,} \,\overline{\,\text{smor}\,}\, \breve{\downarrow}_{.,}^{;}P\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*182·24·25. &\supset \vdash .P=(\breve{\iota} \mid \text{D})\dagger ^{;}\breve{\downarrow}_{.,}^{;}P.Cʻ\Sigma ʻ\breve{\downarrow}_{.,}^{;}P\subset + \text{ᗡ}ʻ(\breve{\iota} \mid \text{D}) &\qquad \text{(1)}\\ +\vdash .*182·17.*55·15.\supset \\ +&\vdash :S,T\in Cʻ\Sigma ʻ\breve{\downarrow}_{.,}^{;}P.\breve{\iota} ʻ\text{D}ʻS=\breve{\iota} ʻ\text{D}ʻT.\supset .\\ +&(\exists Q,R,x,y).Q,R\in CʻP.x\in CʻQ.y\in CʻR.x=y.S=x\downarrow Q.T=y\downarrow R.\\ +[*13·195]&\supset .(\exists Q,R,x).Q,R\in CʻP.x\in CʻQ\cap CʻR.S=x\downarrow Q.T=x\downarrow R &\qquad \text{(2)}\\ +\vdash .(2).*163·11.\supset \\ +\vdash :\text{Hp}.&S,T\in Cʻ\Sigma ʻ\breve{\downarrow}_{.,}^{;}P.\breve{\iota} ʻ\text{D}ʻS=\breve{\iota} ʻ\text{D}ʻT.\supset .\\ +&(\exists Q,R,x).Q=R.S=x\downarrow Q.T=x\downarrow R. +[*13·195·172] \supset .S=T &\qquad \text{(3)}\\ +\vdash .(3).*71·55. &\supset \vdash :\text{Hp}.\supset .\breve{\iota} \mid \text{D}\upharpoonright Cʻ\Sigma ʻ\breve{\downarrow}_{.,}^{;}P\in 1\rightarrow 1 &\qquad \text{(4)}\\ +\vdash .(1).(4).*164·18.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*182·27">*182·27</a>.</b> \(\vdash :P\in \text{Rel}^{2}\text{excl}.\supset .P \,\,\text{smor}\, \,\text{smor}\, \breve{\downarrow}_{.,}^{;}P \quad[*182·26]\)</p> + +<p>The following propositions lead up to <a href="#*182·33">*182·33</a><a href="#*182·34">·34</a>.</p> + +<p class="nind"><b>*182·3.</b> + \(\vdash :T\upharpoonright Cʻ\Sigma ʻQ\in P\,\overline{\,\text{smor}\,} \,\overline{\,\text{smor}\,}\, Q.\supset .(T\parallel \text{Cnv}ʻT\dagger )\upharpoonright Cʻ\Sigma ʻ\breve{\downarrow}_{.,}^{;}Q\in 1\rightarrow 1\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*74·775 \frac{T,\,T\dagger,\,Cʻ\Sigma ʻ\breve{\downarrow}_{.,}^{;}Q}{Q,\,R,\,\lambda}.\supset \\ +&\vdash :T\upharpoonright sʻ\text{D}ʻʻCʻ\Sigma ʻ\breve{\downarrow}_{.,}^{;}Q,T\dagger \upharpoonright sʻ\text{ᗡ}ʻʻCʻ\Sigma ʻ\breve{\downarrow}_{.,}^{;}Q\in + \text{Cls}\rightarrow 1.\\ +&sʻ\text{D}ʻʻCʻ\Sigma ʻ\breve{\downarrow}_{.,}^{;}Q\subset \text{ᗡ}ʻT.sʻ\text{ᗡ}ʻʻCʻ\Sigma ʻ\breve{\downarrow}_{.,}^{;}Q\subset \text{ᗡ}ʻT\dagger .\supset .\\ +&(T\parallel \text{Cnv}ʻ\breve{T} )\upharpoonright Cʻ\Sigma ʻ\breve{\downarrow}_{.,}^{;}Q\in 1\rightarrow 1 &\qquad \text{(1)}\\ +\vdash .*182·19.*164·18.&\supset \vdash :\text{Hp}.\supset .T\upharpoonright sʻ\text{D}ʻʻCʻ\Sigma ʻ\breve{\downarrow}_{.,}^{;}Q\in 1\rightarrow 1 &\qquad \text{(2)}\\ +\vdash .*164·18·13. \supset \vdash :\text{Hp}.&\supset .T\dagger \upharpoonright CʻQ\in 1\rightarrow 1.\\ +[*182·19] &\supset .T\dagger \upharpoonright sʻ\text{ᗡ}ʻʻCʻ\Sigma ʻ\breve{\downarrow}_{.,}^{;}Q\in 1\rightarrow 1 &\qquad \text{(3)}\\ +\vdash .*164·18.*182·19.&\supset \vdash :\text{Hp}.\supset .sʻ\text{D}ʻʻCʻ\Sigma ʻ\breve{\downarrow}_{.,}^{;}Q\subset \text{ᗡ}ʻT &\qquad \text{(4)}\\ +\vdash .*150·1.*33·431. &\supset \vdash .sʻ\text{ᗡ}ʻʻCʻ\Sigma ʻ\breve{\downarrow}_{.,}^{;}Q\subset \text{ᗡ}ʻT\dagger &\qquad \text{(5)}\\ +\vdash .(1).(2).(3).(4).(5).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_493">[Pg 493]</span></p> + +<p class="nind"><b>*182·31.</b> \(\vdash :\text{E}!!TʻʻCʻS.\supset .\breve{\downarrow}_{.,}ʻT^{;}S=(T\parallel \text{Cnv}ʻT\dagger )^{;}\breve{\downarrow}_{.,}ʻS\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*182·05.\supset \vdash .\breve{\downarrow}_{.,}ʻT^{;}S&=(T^{;}S)\downarrow_{.,} (T^{;}S) &\qquad \text{(1)}\\ +\vdash .(1).*165·31.\supset \\ +\vdash :\text{Hp}.\supset .\breve{\downarrow}_{.,}ʻT^{;}S&=T\mid ^{;}S\downarrow_{.,} (T^{;}S)\\ +[*150·1.*165·321] &=T\mid ^{;}(\mid \text{Cnv}ʻT\dagger )^{;}S\downarrow_{.,} S\\ +[*150·13.*182·05] &=(T\parallel \text{Cnv}ʻT\dagger )^{;}\breve{\downarrow}_{.,}ʻS:\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*182·32.</b> \(\vdash :\text{E}!!TʻʻCʻ\Sigma ʻQ.\supset .\breve{\downarrow}_{.,}^{;}T\dagger ^{;}Q=(T\parallel + \text{Cnv}ʻT\dagger )\dagger ^{;}\breve{\downarrow}_{.,}^{;}Q\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*162·22.\supset \vdash \colon\ldotp \text{Hp}.\supset :S\in CʻQ.&\supset _{S}.\text{E}!!TʻʻCʻS.\\ +[*182·31] &\supset _{S}.\breve{\downarrow}_{.,}ʻT^{;}S=(T\parallel \text{Cnv}ʻT\dagger )^{;}\breve{\downarrow}_{.,}ʻS:\\ +[*150·35·1] &\supset :\breve{\downarrow}_{.,}^{;}T\dagger ^{;}S=(T\parallel \text{Cnv}ʻT\dagger )\dagger ^{;}\breve{\downarrow}_{.,}^{;}Q\colon\ldotp + \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*182·33">*182·33</a>.</b> \[\begin{align}\vdash :T\upharpoonright Cʻ\Sigma ʻP\in P&\,\overline{\,\text{smor}\,} \,\overline{\,\text{smor}\,}\, Q.\supset .\\ +&(T\parallel \text{Cnv}ʻT\dagger )\upharpoonright Cʻ\Sigma ʻ\breve{\downarrow}_{.,}^{;}Q\in (\breve{\downarrow}_{.,}^{;}P)\overline{\,\text{smor}\,} + \overline{\,\text{smor}\,} (\breve{\downarrow}_{.,}^{;}Q)\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*164·18.\supset \vdash .\text{Hp}.&\supset .\text{ᗡ}ʻT\subset Cʻ\Sigma ʻQ.T\upharpoonright Cʻ\Sigma ʻP\in 1\rightarrow 1.P=T\dagger ^{;}Q.\\ +[*74·11] &\supset .\text{E}!!TʻʻCʻ\Sigma ʻQ.P=T\dagger ^{;}Q.\\ +[*182·32] &\supset .\breve{\downarrow}_{.,}^{;}P=(T\parallel \text{Cnv}ʻT\dagger )\dagger ^{;}\breve{\downarrow}_{.,}^{;}Q &\qquad \text{(1)}\\ +\vdash .(1).*182·3.*164·18.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*182·34">*182·34</a>.</b> \(\vdash :P\,\text{smor}\,\text{smor}Q.\supset .\breve{\downarrow}_{.,}^{;}P\,\text{smor}\,\text{smor}\breve{\downarrow}_{.,}^{;}Q \quad[*182·33]\)</p> + +<p>The converse of the above proposition is false. For +example, if \(Q=\breve{\downarrow}_{.,}^{;}P\), we shall have +\(\breve{\downarrow}_{.,}^{;}P\,\text{smor}\,\text{smor}\breve{\downarrow}_{.,}^{;}Q\), +by <a href="#*182·16">*182·16</a><a href="#*182·27">·27</a>, but we shall not have \(P\,\text{smor}\,\text{smor}Q\) +unless \(P \in \text{Rel}^{2}\text{excl}\), as appears from <a href="#*182·16">*182·16</a> and +<a href="#*164·23">*164·23</a>.</p> + +<p><a href="#*182·411">*182·411</a><a href="#*182·412">·412</a> are lemmas for <a href="#*182·413">*182·413</a>. All the following propositions +lead up to <a href="#*182·42">*182·42</a>, which leads to <a href="#*182·44">*182·44</a>.</p> + +<p class="nind"><b><a id="*182·411">*182·411</a>.</b> \(\vdash .\dot{s} ʻCʻ\breve{{\downarrow}}^{;}P=I\upharpoonright CʻP\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*150·22.\supset \vdash .\dot{s} ʻCʻ\breve{{\downarrow}}^{;}P&=\dot{s} ʻ\breve{{\downarrow}}ʻʻCʻP\\ +[*182·032] &=\dot{s} ʻ\hat{R} \{(\exists y).y\in CʻP.R=y\downarrow y\}\\ +[*41·11] &=\hat{x} \hat{z} \{(\exists y).y\in CʻP.x(y\downarrow y)z\}\\ +[*55·13.*13·195] &=\hat{x} \hat{z} \{z\in CʻP.x=z\}\\ +[*50·1.*35·101] &=I\upharpoonright CʻP.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_494">[Pg 494]</span></p> + +<p class="nind"><b><a id="*182·412">*182·412</a>.</b> \(\vdash .F^{;}\breve{{\downarrow}}^{;}P=P\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*150·11.*182·032. \supset \vdash .F^{;}\breve{{\downarrow}}^{;}P&=\hat{x} \hat{y} \{(\exists z,w).zPw.xF(z\downarrow z).yF(w\downarrow w)\}\\ +[*33·51.*55·15] &=\hat{x} \hat{y} \{(\exists z,w).zPw.x=z.y=w\}\\ +[*13·22] &=P.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*182·413">*182·413</a>.</b> \(\vdash .\Sigma ʻ\breve{{\downarrow}}^{;}P=P\unicode{x228d} I\upharpoonright CʻP \quad[*182·411·412.*162·1]\)</p> + +<p class="nind"><b>*182·414.</b> \(\vdash .\breve{{\downarrow}}^{;}P\in \text{Rel}^{2}\text{excl}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*150·22.\supset \vdash .Cʻ\breve{{\downarrow}}^{;}P&=\breve{{\downarrow}}ʻʻCʻP\\ +[*182·032] & =\hat{Q} \{(\exists x).x\in CʻP.Q=x\downarrow x\} &\qquad \text{(1)}\\ +\vdash .(1).*55·15. &\supset \vdash :Q,R\in Cʻ\breve{{\downarrow}}^{;}P.\exists !CʻQ\cap CʻR.\supset .\\ +&(\exists x,y).x,y\in CʻP.Q=x\downarrow x.R=y\downarrow y.\exists !\iota ʻx\cap \iota ʻy.\\ +[*51·231.\text{Transp}] &\supset .(\exists x,y).Q=x\downarrow x.R=y\downarrow y.x=y.\\ +[*13·195·172] &\supset .Q=R:\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*182·415.</b> \(\vdash :Q\in Cʻ\breve{{\downarrow}}^{;}P.\supset .CʻQ\in 1\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*150·22.\supset \vdash :\text{Hp}.&\supset .(\exists x).x\in CʻP.Q=x\downarrow x.\\ +[*55·15] &\supset .(\exists x).x\in CʻP.CʻQ=\iota ʻx.\\ +[*52·1] &\supset .CʻQ\in 1:\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p>The purpose of the above proposition is to enable us to apply +<a href="#*174·221">*174·221</a><a href="#*174·231">·231</a> to \(\Pi ʻ\Pi ^{;}\breve{{\downarrow}}^{;}P\), as is done in +<a href="#*182·42">*182·42</a><a href="#*182·43">·43</a><a href="#*182·431">·431</a> below.</p> + +<p class="nind"><b><a id="*182·42">*182·42</a>.</b> \(\vdash .\Pi ʻP=\dot{s} ^{;}\text{Prod}ʻ\breve{\downarrow}_{.,}^{;}P=\dot{s} ^{;}\text{D}^{;}\Pi ʻ\breve{\downarrow}_{.,}^{;}P=\Pi + ʻ\Sigma ʻ\breve{{\downarrow}}^{;}P\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*182·21.\supset \vdash .\dot{s} ^{;}\text{D}^{;}\Pi ʻ\breve{\downarrow}_{.,}^{;}P&=\dot{s} ^{;}\text{D}^{;}\Pi + ʻ\Pi ^{;}\breve{{\downarrow}}^{;}P\\ +[*174·221.*182·414·415] & =\Pi ʻ\Sigma ʻ\breve{{\downarrow}}^{;}P &\qquad \text{(1)}\\ +[*182·413] &=\Pi ʻ(P\unicode{x228d} I\upharpoonright CʻP)\\ +[*172·51] & =\Pi ʻP &\qquad \text{(2)}\\ +\vdash .(1).(2).*173·1.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*182·43">*182·43</a>.</b> \(\vdash .\dot{s} \upharpoonright (Cʻ\text{Prod}ʻ\breve{\downarrow}_{.,}^{;}P)\in + (\Pi ʻP)\overline{\,\text{smor}\,} (\text{Prod}ʻ\breve{\downarrow}_{.,}^{;}P)\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*174·231.*182·414·415.\supset \\ +\vdash .\dot{s} \upharpoonright (Cʻ\text{Prod}ʻ\Pi ^{;}\breve{{\downarrow}}^{;}P)\in (\Pi ʻ\Sigma ʻ\breve{{\downarrow}}^{;}P)\overline{\,\text{smor}\,} + (\text{Prod}ʻ\Pi ^{;}\breve{{\downarrow}}^{;}P) &\qquad \text{(1)}\\ +\vdash .(1).*182·21·42.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_495">[Pg 495]</span></p> + +<p class="nind"><b><a id="*182·431">*182·431</a>.</b> \[\begin{align}&\vdash .\dot{s} \mid \text{D} \upharpoonright (Cʻ\Pi ʻ\breve{\downarrow}_{.,}^{;}P) + \in (\Pi ʻP) \overline{\,\text{smor}\,} (\Pi ʻ\breve{\downarrow}_{.,}^{;}P)\\ +&[*174·221. *182·414·415·21·42]\end{align}\]</p> + +<p class="nind"><b><a id="*182·44">*182·44</a>.</b> \(\vdash . \text{Nr}ʻ\Pi ʻP = \text{Nr}ʻ\text{Prod}ʻ\breve{\downarrow}_{.,}^{;}P = \text{Nr}ʻ\Pi ʻ\breve{\downarrow}_{.,}^{;}P + \quad[*182·43·431 . *152·321]\)</p> + +<p class="nind"><b>*182·45.</b> \(\vdash : P \in \text{Rel}^{2}\text{excl} . \supset . \text{Nr}ʻ\text{Prod}ʻP = \text{Nr}ʻ\text{Prod}ʻ\breve{\downarrow}_{.,}^{;}P + \quad[*182·44 . *173·16]\)</p> + +<p>The following propositions are concerned with cardinals. They show how +to express the propositions and definitions of <a href="#*112">*112</a> in the notation of +this number, and they thereby illustrate the analogy of cardinal and +ordinal addition.</p> + +<p class="nind"><b><a id="*182·5">*182·5</a>.</b> \(\vdash . \in \unicode{x21A7} \alpha = \breve{\downarrow_{,,}} ʻ\alpha \quad[*182·04 . *85·601]\)</p> + +<p class="nind"><b><a id="*182·51">*182·51</a>.</b> \(\vdash . \in \unicode{x21A7}ʻʻ\kappa = \breve{\downarrow_{,,}} ʻʻ\kappa \quad[*182·5]\)</p> + +<p class="nind"><b><a id="*182·52">*182·52</a>.</b> \(\vdash . \Sigma ʻ\kappa = sʻ\breve{\downarrow_{,,}}ʻʻ\kappa . \Sigma \text{Nc}ʻ\kappa = \text{Nc}ʻsʻ\breve{\downarrow_{,,}}ʻʻ\kappa + \quad[*182·51 . *112·1·101]\)</p> + +<p class="nind"><b>*182·53.</b> \[\begin{align}\vdash : C \upharpoonright CʻP \in 1 &\rightarrow 1 . \supset .\\ +&(\mid \breve{C} ) \upharpoonright (Cʻ\Sigma ʻ\breve{\downarrow}_{.,}^{;}P) \in (\breve{\downarrow_{,,}}ʻʻCʻʻCʻP) + \,\overline{\text{ sm }}\, \,\overline{\text{ sm }}\, (CʻʻCʻ\breve{\downarrow}_{.,}^{;}P)\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *182·18 . *41·44 . &\supset \vdash . sʻ\text{ᗡ}ʻʻCʻ\Sigma ʻ\breve{\downarrow}_{.,}^{;}P = \text{ᗡ}ʻ(F \upharpoonright CʻP)\\ +[*35·64] &\subset CʻP &\qquad \text{(1)}\\ +\vdash . (1) . *74·75 . &\supset \vdash : \text{Hp} . \supset . (\mid \breve{C} ) \upharpoonright Cʻ\Sigma ʻ\breve{\downarrow}_{.,}^{;}P + \in 1 \rightarrow 1 &\qquad \text{(2)}\\ +\vdash . *55·581 \frac{\breve{C}}{S}. &\supset \vdash . (x \downarrow Q) \mid \breve{C} = x \downarrow CʻQ .\\ +[*38·11] &\supset \vdash . \mid \breve{C} ʻ \downarrow Qʻx = \downarrow (CʻQ)ʻx .\\ +[*182·12·04] &\supset \vdash . \mid \breve{C} ʻʻCʻ\breve{\downarrow}_{.,}ʻQ = \breve{\downarrow_{,,}}ʻCʻQ .\\ +[*150·22] &\supset \vdash . \mid \breve{C} ʻʻʻCʻʻCʻ\breve{\downarrow}_{.,}^{;}P = \breve{\downarrow_{,,}}ʻʻCʻʻCʻP &\qquad \text{(3)}\\ +\vdash . (2) . (3) . *111·14 . *162·22 . \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*182·54.</b> \[\begin{align}&\vdash : C \upharpoonright CʻP \in 1 \rightarrow 1 . \supset . \text{Nc}ʻCʻ\Sigma ʻ\breve{\downarrow}_{.,}^{;}P = \Sigma \text{Nc}ʻCʻʻCʻP\\ +&[*182·52·53 . *111·44]\end{align}\]</p> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_496">[Pg 496]</span></p> +<h2 class="nobreak" id="*183"> +*183. THE SUM OF THE RELATION-NUMBERS OF A FIELD.</h2> +</div> + + +<p class="nind"><i>Summary of</i> *183.</p> + +<p>In this number we have to define and consider the sum of the +relation-numbers of the members of \(CʻP\), where \(P\) is a relation +of relations. Since relational sums are not commutative, we cannot +define the sum of the relation-numbers of members of a class of +relations \(\lambda\): it is necessary that \(\lambda\) should be given +as the field of a relation \(P\), where \(P\) determines the order in +which the summation is to be effected.</p> + +<p>In order to avoid repetition, we replace \(P\) by +\(\breve{\downarrow}_{.,}^{;}P\), so that if \(Q\) is a member of +\(CʻP\), \(Q\) is replaced by \(\breve{\downarrow}_{.,}ʻQ\), +<i>i.e.</i> by \(\downarrow Q^{;}Q\). This relation is like \(Q\), and +its field has no members in common with the field of \(\downarrow R^{;}R\), +unless \(Q=R\). Hence we are led to the following definition:</p> + +<p class="nind"><b>*183·01.</b> \(\Sigma \text{Nr}ʻP=\text{Nr}ʻ\Sigma ʻ\breve{\downarrow}_{.,}^{;}P \quad\text{Df}\)</p> + +<p>This definition is analogous to <a href="#*112·01">*112·01</a>, as appears from <a href="#*182·52">*182·52</a>, and +the propositions of the present number are analogous to some of the +propositions of <a href="#*112">*112</a>.</p> + +<p>We have not merely</p> + +<p class="nind"><b>*183·11.</b> \(\vdash :P \,\,\text{smor}\, \,\text{smor}\, Q.\supset .\Sigma \text{Nr}ʻP=\Sigma \text{Nr}ʻQ\)</p> + +<p class="nind">but also</p> + +<p class="nind"><b>*183·15.</b> \(\vdash :\breve{\downarrow}_{.,}^{;}P \,\,\text{smor}\, \,\text{smor}\, \breve{\downarrow}_{.,}^{;}Q.\supset .\Sigma \text{Nr}ʻP=\Sigma \text{Nr}ʻQ\)</p> + +<p class="nind">which is a proposition with a weaker hypothesis than that of <a href="#*183·11">*183·11</a> +(cf. note to <a href="#*182·34">*182·34</a>).</p> + +<p>Important propositions in this number are</p> + +<p class="nind"><b>*183·13.</b> \(\vdash :P\in \text{Rel}^{2}\text{excl}.\supset . \text{Nr}ʻ\Sigma ʻP=\Sigma \text{Nr}ʻP\)</p> + +<p class="nind"><b>*183·2.</b> \(\vdash :\Sigma \text{Nr}ʻP=0_{r}.\equiv .\Sigma ʻP=\dot{\Lambda}\)</p> + +<p><i>I.e.</i> a sum is only zero when there is no summand except (at +most) zero. (Cf. <a href="#*162·4">*162·4</a><a href="#*162·45">·45</a>.)</p> + +<p><span class="pagenum" id="Page_497">[Pg 497]</span></p> + +<p class="nind"><b>*183·25.</b> \(\vdash .\Sigma \text{Nr}ʻP\downarrow_{.,} ^{;}Q=\text{Nr}ʻ(Q\times P)\)</p> + +<p class="nind"><b>*183-26.</b> \(\vdash \colon\ldotp \text{Mult ax}.\supset :P\in \text{Nr}ʻR.CʻP\subset \text{Nr}ʻS.\supset .\Sigma \text{Nr}ʻP= \text{Nr}ʻ(R\times S)\)</p> + +<p>This proposition connects addition and multiplication.</p> + +<p class="nind"><b>*183·31.</b> \(\vdash :P\neq Q.\supset .\Sigma \text{Nr}ʻ(P\downarrow Q)=\text{Nr}ʻP\dot{+} \text{Nr}ʻQ\)</p> + +<p>This proposition connects the two kinds of addition. We have also</p> + +<p class="nind"><b>*183·33.</b> \(\vdash :\dot{\exists} !P.Z{\sim}\in CʻP.\supset .\Sigma \text{Nr}ʻ(P\unicode{x21f8} Z)=\Sigma \text{Nr}ʻP\dot{+} \text{Nr}ʻZ\)</p> + +<p>The associative law of addition in a very general form is</p> + +<p class="nind"><b>*183·43.</b> \(\vdash :P\in \text{Rel}^{2}\text{excl}.\supset .\Sigma \text{Nr}ʻ\Sigma ^{;}\breve{\downarrow}_{.,}\dagger ^{;}P=\Sigma \text{Nr}ʻ\Sigma ʻP\)</p> + +<p>Finally the connection of ordinal and cardinal addition is given by</p> + +<p class="nind"><b>*183·5.</b> \(\vdash :C\upharpoonright CʻP\in 1\rightarrow 1.\supset .Cʻʻ\Sigma \text{Nr}ʻP=\Sigma \text{Nc}ʻCʻʻCʻP\)</p> + +<hr class="tb"> + +<p class="nind"><b>*183·01.</b> \(\Sigma \text{Nr}ʻP=\text{Nr}ʻ\Sigma ʻ\breve{\downarrow}_{.,}^{;}P \quad\text{Df}\)</p> + +<p class="nind"><b>*183·1.</b> \(\vdash .\Sigma \text{Nr}ʻP=\text{Nr}ʻ\Sigma ʻ\breve{\downarrow}_{.,}^{;}P \quad[(*183·01)]\)</p> + +<p class="nind"><b><a id="*183·11">*183·11</a>.</b> \(\vdash :P \,\,\text{smor}\, \,\text{smor}\, Q.\supset .\Sigma \text{Nr}ʻP=\Sigma \text{Nr}ʻQ\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*182·34.\supset \vdash :\text{Hp}.&\supset .(\breve{\downarrow}_{.,}^{;}P)\,\text{smor}\, \,\text{smor}\,(\breve{\downarrow}_{.,}^{;}Q).\\ +[*164·151] &\supset .(\Sigma ʻ\breve{\downarrow}_{.,}^{;}P)\,\text{smor}\,(\Sigma ʻ\breve{\downarrow}_{.,}^{;}Q).\\ +[*183·1.*152·321] &\supset .\Sigma \text{Nr}ʻP=\Sigma \text{Nr}ʻQ:\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*183·12.</b> \(\vdash :P \,\,\text{smor}\, \,\text{smor}\,\breve{\downarrow}_{.,}^{;}Q.\supset .\text{Nr}ʻ\Sigma ʻP=\Sigma \text{Nr}ʻQ \quad[*164·151.*183·1]\)</p> + +<p class="nind"><b><a id="*183·13">*183·13</a>.</b> \(\vdash :P\in \text{Rel}^{2}\text{excl}.\supset .\text{Nr}ʻ\Sigma ʻP=\Sigma \text{Nr}ʻP \quad[*182·27.*183·12]\)</p> + +<p class="nind"><b>*183·14.</b> \(\vdash .\Sigma \text{Nr}ʻP=\Sigma \text{Nr}ʻ\breve{\downarrow}_{.,}^{;}P\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*182·16.*183·13.\supset \vdash .\text{Nr}ʻ\Sigma ʻ\breve{\downarrow}_{.,}^{;}P=\Sigma \text{Nr}ʻ\breve{\downarrow}_{.,}^{;}P &\qquad \text{(1)}\\ +\vdash .(1).*183·1.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*183·15.</b> \(\vdash :\breve{\downarrow}_{.,}^{;}P\,\text{smor}\, \,\text{smor}\,\breve{\downarrow}_{.,}^{;}Q.\supset + .\Sigma \text{Nr}ʻP=\Sigma \text{Nr}ʻQ \quad[*183·11·14]\)</p> + +<p class="nind"><b>*183·2.</b> \(\vdash :\Sigma \text{Nr}ʻP=0_{r}.\equiv .\Sigma ʻP=\dot{\Lambda}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*183·1.*153·17.\supset\\ +\vdash \colon\ldotp \Sigma \text{Nr}ʻP=0_{r}.&\equiv :\Sigma ʻ\breve{\downarrow}_{.,}^{;}P=\dot{\Lambda} :\\ +[*162·42] & \equiv :Cʻ\breve{\downarrow}_{.,}^{;}P\subset \iota ʻ\dot{\Lambda} :\\ +[*182·05] &\equiv :Q\in CʻP.\supset _{Q}.\downarrow Q^{;}Q=\dot{\Lambda} :\\ +[*151·65.*153·101]&\equiv :Q\in CʻP.\supset _{Q}.Q=\dot{\Lambda} :\\ +[*162·42] &\equiv :\Sigma ʻP=\dot{\Lambda} \colon\ldotp \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_498">[Pg 498]</span></p> + +<p class="nind"><b>*183·22.</b> \(\vdash \colon\ldotp \text{Mult ax} . \supset : \exists ! (\breve{\downarrow}_{.,}^{;}P) \overline{\,\text{smor}\,} (\breve{\downarrow}_{.,}^{;}Q) + \cap \text{Rl}ʻ\,\text{smor}\, . \supset . \Sigma \text{Nr}ʻP = \Sigma \text{Nr}ʻQ\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *164·46 . *182·16 . \supset \\ +\vdash \colon\ldotp \text{Mult ax} . &\supset : \exists ! (\breve{\downarrow}_{.,}^{;}P) \overline{\,\text{smor}\,} (\breve{\downarrow}_{.,}^{;}Q) + \cap \text{Rl}ʻ\,\text{smor}\, . \supset . \Sigma ʻ\breve{\downarrow}_{.,}^{;}P \,\,\text{smor}\, \Sigma ʻ\breve{\downarrow}_{.,}^{;}Q .\\ +[*183·1 . *152·321] &\supset . \Sigma \text{Nr}ʻP = \Sigma \text{Nr}ʻQ : \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*183·23.</b> \(\vdash \colon\ldotp \text{Mult ax} . \supset : P, Q \in \text{Rel}^{2}\text{excl} . \exists ! P \overline{\,\text{smor}\,} Q \cap \text{Rl}ʻ\,\text{smor}\, . \supset . +\Sigma \text{Nr}ʻP = \Sigma \text{Nr}ʻQ \quad[*164·46 . *183·13]\)</p> + +<p class="nind"><b>*183·231.</b> \(\vdash : P \in \text{Nr}ʻR . CʻP \subset \text{Nr}ʻS . \equiv . \breve{\downarrow}_{.,}^{;}P \in \text{Nr}ʻR . Cʻ\breve{\downarrow}_{.,}^{;}P + \subset \text{Nr}ʻS\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *182·162 . *152·31·321 . \supset \vdash : P \in \text{Nr}ʻR . &\equiv . \breve{\downarrow}_{.,}^{;}P \in \text{Nr}ʻR &\qquad \text{(1)}\\ +\vdash . *182·05·11 . \supset \vdash : Cʻ\breve{\downarrow}_{.,}^{;}P \subset \text{Nr}ʻS . &\equiv : Q \in CʻP . \supset _{Q} . \downarrow Q^{;}Q + \in \text{Nr}ʻS :\\ +[*151·65] &\equiv : Q \in CʻP . \supset _{Q} . Q \in \text{Nr}ʻS :\\ +[*22·1] &\equiv : CʻP \subset \text{Nr}ʻS &\qquad \text{(2)}\\ +\vdash . (1) . (2) . \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*183·24.</b> \(\vdash \colon\ldotp \text{Mult ax} . \supset : P, Q \in \text{Nr}ʻR . CʻP, CʻQ \in \text{Cl}ʻ\text{Nr}ʻS . \supset . \Sigma \text{Nr}ʻP = \Sigma \text{Nr}ʻQ\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *183·231 . &\supset \vdash \colon\ldotp P, Q \in \text{Nr}ʻR . CʻP, CʻQ \in \text{Cl}ʻ\text{Nr}ʻS . \supset :\\ +&\breve{\downarrow}_{.,}^{;}P, \breve{\downarrow}_{.,}^{;}Q \in \text{Nr}ʻR . Cʻ\breve{\downarrow}_{.,}^{;}P, + Cʻ\breve{\downarrow}_{.,}^{;}Q \in \text{Cl}ʻ\text{Nr}ʻS :\\ +[*164·48 . *182·16] &\supset : \text{Mult ax} . \supset . \breve{\downarrow}_{.,}^{;}P \,\,\text{smor}\, \,\text{smor}\, \breve{\downarrow}_{.,}^{;}Q .\\ +[*183·15] &\supset . \Sigma \text{Nr}ʻP = \Sigma \text{Nr}ʻQ &\qquad \text{(1)}\\ +\vdash . (1) . \text{Comm} . \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*183·25.</b> \(\vdash . \Sigma \text{Nr}ʻP \downarrow_{.,} ^{;}Q = \text{Nr}ʻ(Q \times P)\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *165·21 . *183·13 . \supset \vdash . \Sigma \text{Nr}ʻP \downarrow_{.,} ^{;} Q &= \text{Nr}ʻ\Sigma ʻP \downarrow_{.,} ^{;}Q\\ +[*166·1] &= \text{Nr}ʻ(Q \times P) . \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*183·26.</b> \(\vdash \colon\ldotp \text{Mult ax} . \supset : P \in \text{Nr}ʻR . CʻP \subset \text{Nr}ʻS . \supset . \Sigma \text{Nr}ʻP = \text{Nr}ʻ(R \times S)\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *165·27 . *183·24 . \supset \\ +\vdash \colon\ldotp \text{Mult ax} . &\supset : \dot{\exists} ! S . P \in \text{Nr}ʻR . CʻP \subset \text{Nr}ʻS . \supset . \Sigma \text{Nr}ʻP = \Sigma \text{Nr}ʻS \downarrow_{.,} + ^{;}R\\ +[*183·13 . *166·1] &= \text{Nr}ʻ(R \times S) &\qquad \text{(1)}\\ +\vdash . *153·11·101 . \supset \\ +\vdash : S = \dot{\Lambda} . P \in \text{Nr}ʻR . CʻP \subset \text{Nr}ʻS . &\supset . CʻP \subset {℩}ʻ\dot{\Lambda} .\\ +[*162·42] &\supset . \Sigma ʻP = \dot{\Lambda} .\\ +[*183·2] &\supset . \Sigma \text{Nr}ʻP = 0_{r} &\qquad \text{(2)}\\ +\vdash . *166·13 . &\supset \vdash : S = \dot{\Lambda} . \supset . R \times S = \dot{\Lambda} &\qquad \text{(3)}\\ +\vdash . (2) . (3) . *153·17 . \supset \\ +\vdash : S = \dot{\Lambda} . P \in \text{Nr}ʻR . CʻP \subset \text{Nr}ʻS . &\supset . \Sigma \text{Nr}ʻP = \text{Nr}ʻ(R \times S) &\qquad \text{(4)}\\ +\vdash . (1) . (4) . \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_499">[Pg 499]</span></p> + +<p class="nind"><b>*183·3.</b> \(\vdash .\Sigma \text{Nr}ʻ\dot{\Lambda} =0_{r} \quad[*183·2.*162·4]\)</p> + +<p class="nind"><b>*183·301.</b> \(\vdash .\Sigma \text{Nr}ʻ(\dot{\Lambda} \downarrow \dot{\Lambda} )=0_{r} \quad[*183·2.*162·41]\)</p> + +<p class="nind"><b>*183·302.</b> \(\vdash .\Sigma \text{Nr}ʻ(P\downarrow P)=\text{Nr}ʻ(CʻP\uparrow CʻP)\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*183·13.*163·41.\supset \vdash .\Sigma \text{Nr}ʻ(P\downarrow P)&=\text{Nr}ʻ\Sigma ʻ(P\downarrow P)\\ +[*162·3.*160·1] &=\text{Nr}ʻ(CʻP\uparrow CʻP).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*183·31.</b> \(\vdash :P\neq Q.\supset .\Sigma \text{Nr}ʻ(P\downarrow Q)=\text{Nr}ʻP\dot{+} \text{Nr}ʻQ\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*183·1.\supset \vdash .\Sigma \text{Nr}ʻ(P\downarrow Q)&=\text{Nr}ʻ\Sigma ʻ\breve{\downarrow}_{.,}^{;}(P\downarrow Q)\\ +[*150·71] & =\text{Nr}ʻ\Sigma ʻ\{(\breve{\downarrow}_{.,}ʻP)\downarrow (\breve{\downarrow}_{.,}ʻQ)\}\\ +[*162·3] & =\text{Nr}ʻ(\breve{\downarrow}_{.,}ʻP\unicode{x2909}\breve{\downarrow}_{.,}ʻQ) &\qquad \text{(1)}\\ +\vdash .(1).*180·32.*182·12·15.\supset \vdash :\text{Hp}.\supset .\\ +\Sigma \text{Nr}ʻ(P\downarrow Q)&=\text{Nr}ʻ\breve{\downarrow}_{.,}ʻP\dot{+} \text{Nr}ʻ\breve{\downarrow}_{.,}ʻQ\\ +[*182·05.*151·65.*180·31] & =\text{Nr}ʻP\dot{+} \text{Nr}ʻQ:\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*183·32.</b> \(\vdash :CʻP\cap CʻQ=\Lambda .\supset .\Sigma \text{Nr}ʻ(P\unicode{x2909}Q)=\Sigma \text{Nr}ʻP\dot{+} \Sigma \text{Nr}ʻQ\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*183·1.\supset \vdash .\Sigma \text{Nr}ʻ(P\unicode{x2909}Q)&=\text{Nr}ʻ\Sigma ʻ\breve{\downarrow}_{.,}^{;}(P\unicode{x2909}Q)\\ +[*162·31.*160·44] &=\text{Nr}ʻ(\Sigma ʻ\breve{\downarrow}_{.,}^{;}P\unicode{x2909}\Sigma ʻ\breve{\downarrow}_{.,}^{;}Q) &\qquad \text{(1)}\\ +\vdash .*182·17.*55·202.\supset \\ +\vdash :\exists !Cʻ\Sigma ʻ\breve{\downarrow}_{.,}^{;}P\cap Cʻ\Sigma ʻ\breve{\downarrow}_{.,}^{;}Q.&\equiv + .(\exists S,R,x).R\in CʻP\cap CʻQ.x\in CʻR.S=x\downarrow R.\\ +[*10·5] &\supset .\exists !CʻP\cap CʻQ &\qquad \text{(2)}\\ +\vdash .(2).\text{Transp}.&\supset \vdash :\text{Hp}.\supset .Cʻ\Sigma ʻ\breve{\downarrow}_{.,}^{;}P\cap Cʻ\Sigma ʻ\breve{\downarrow}_{.,}^{;}Q=\Lambda .\\ +[*180·32] &\supset .\text{Nr}ʻ(\Sigma ʻ\breve{\downarrow}_{.,}^{;}P\unicode{x2909}\Sigma ʻ\breve{\downarrow}_{.,}^{;}Q)=\text{Nr}ʻ\Sigma + ʻ\breve{\downarrow}_{.,}^{;}P\dot{+} \text{Nr}ʻ\Sigma ʻ\breve{\downarrow}_{.,}^{;}Q\\ +[*183·1] & =\Sigma \text{Nr}ʻP\dot{+} \Sigma \text{Nr}ʻQ &\qquad \text{(3)}\\ +\vdash .(1).(3).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*183·33.</b> \(\vdash :\dot{\exists} !P.Z{\sim}\in CʻP.\supset .\Sigma \text{Nr}ʻ(P\unicode{x21f8} Z)=\Sigma \text{Nr}ʻP\dot{+} \text{Nr}ʻZ\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*183·1.\supset \vdash .\Sigma \text{Nr}ʻ(P\unicode{x21f8} Z)&=\text{Nr}ʻ\Sigma ʻ\breve{\downarrow}_{.,}^{;}(P\unicode{x21f8} Z)\\ +[*161·4] &=\text{Nr}ʻ\Sigma ʻ(\breve{\downarrow}_{.,}^{;}P\unicode{x21f8} \breve{\downarrow}_{.,}ʻZ) &\qquad \text{(1)}\\ +\vdash .(1).*162·43.\supset \vdash :\text{Hp}.\supset .\Sigma \text{Nr}ʻ(P\unicode{x21f8} Z)&=\text{Nr}ʻ(\Sigma ʻ\breve{\downarrow}_{.,}^{;}P\unicode{x2909}\breve{\downarrow}_{.,}ʻZ)\\ +[*182·12·15.*180·32] &=\text{Nr}ʻ\Sigma ʻ\breve{\downarrow}_{.,}^{;}P\dot{+} \text{Nr}ʻ\breve{\downarrow}_{.,}ʻZ\\ +[*183·1.*182·05.*151·65] & =\Sigma \text{Nr}ʻP\dot{+} \text{Nr}ʻZ:\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_500">[Pg 500]</span></p> + +<p class="nind"><b>*183·331.</b> \[\begin{align}&\vdash :\dot{\exists} !P.Z{\sim}\in CʻP.\supset .\Sigma \text{Nr}ʻ(Z\unicode{x21f7}P) = \text{Nr}ʻZ\dot{+} \Sigma \text{Nr}ʻP\\ +&[\text{Proof as in *183·33}]\end{align}\]</p> + +<p class="nind"><b>*183·42.</b> \(\vdash :P\in \text{Rel}^{2}\text{excl}.\supset .\breve{\downarrow_{,,}}\dagger^{{} +;}P\in \text{Rel}^{3}\text{arithm}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*163·3.*182·162.&\supset \vdash :\text{Hp}.\supset .\breve{\downarrow}_{.,}\dagger ^{;}P\in \text{Rel}^{2}\text{excl}&\qquad \text{(1)}\\ +\vdash .*162·35. &\supset \vdash .\Sigma ʻ\breve{\downarrow_{,,}}\dagger^{{} +;}P = \breve{\downarrow}_{.,}^{;}\Sigma ʻP.\\ +[*182·16] &\supset \vdash .\Sigma ʻ\breve{\downarrow_{,,}}\dagger^{{} +;}P\in \text{Rel}^{2}\text{excl} &\qquad \text{(2)}\\ +\vdash .(1).(2).*174·3.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*183·43.</b> \(\vdash :P\in \text{Rel}^{2}\text{excl}.\supset .\Sigma \text{Nr}ʻ\Sigma ^{;}\breve{\downarrow_{,,}}\dagger^{{} +;}P = \Sigma \text{Nr}ʻ\Sigma ʻP\)</p> + +<p>This is a form of the associative law of addition.</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*183·42.*174·36. \supset \vdash :\text{Hp}.&\supset .\Sigma ^{;}\breve{\downarrow}_{.,}\dagger ^{;}P\in \text{Rel}^{2}\text{excl}.\\ +[*183·13] \supset .\Sigma \text{Nr}ʻ\Sigma ^{;}\breve{\downarrow_{,,}}\dagger^{{} +;}P &= \text{Nr}ʻ\Sigma ʻ\Sigma ^{;}\breve{\downarrow}_{.,}\dagger ^{;}P\\ +[*162·34] &= \text{Nr}ʻ\Sigma ʻ\Sigma ʻ\breve{\downarrow_{,,}}\dagger^{{} +;}P\\ +[*162·35] &= \text{Nr}ʻ\Sigma ʻ\breve{\downarrow_{,,}}\dagger^{{} +;}\Sigma ʻP\\ +[*183·1] & = \Sigma \text{Nr}ʻ\Sigma ʻP:\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*183·5.</b> \(\vdash :C\upharpoonright CʻP\in 1\rightarrow 1.\supset .Cʻʻ\Sigma \text{Nr}ʻP = \Sigma \text{Nc}ʻCʻʻCʻP\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*152·7.*183·1.\supset \vdash :\text{Hp}.\supset .Cʻʻ\Sigma \text{Nr}ʻP &= \text{Nc}ʻCʻ\Sigma ʻ\breve{\downarrow_{,,}}\dagger^{{} +;}P\\ +[*182·54] &= \Sigma \text{Nc}ʻCʻʻCʻP.\supset \vdash .\text{Prop} +\end{array} +\]</p> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_501">[Pg 501]</span></p> +<h2 class="nobreak" id="184_THE_PRODUCT_OF_TWO_RELATION-NUMBERS"> +*184. THE PRODUCT OF TWO RELATION-NUMBERS.</h2> +</div> + + +<p class="nind"><i>Summary of</i> *184.</p> + +<p>The propositions of this number are for the most part analogous to +those of the propositions of <a href="#*113">*113</a> which are concerned with \(\mu \times _{c} \nu\). +Those of *113 which are concerned with \(\alpha \times \beta\) +have their analogues in <a href="#*166">*166</a>. We put</p> + +<p class="nind"><b>*184·01.</b> \(\mu \dot{\times} \nu =\hat{R} \{(\exists P,Q).\mu =\text{N}_{0}\text{r}ʻP.\nu =\text{N}_{0}\text{r}ʻQ.R\,\text{smor}\,(P\times Q)\} \quad\text{Df}\)</p> + +<p class="nind"><b>*184·02.</b> \(\text{Nr}ʻP\dot{\times} \nu =\text{N}_{0}\text{r}ʻP\dot{\times} \nu \quad\text{Df}\)</p> + +<p class="nind"><b>*184·03.</b> \(\mu \dot{\times} \text{Nr}ʻQ=\mu \dot{\times} \text{N}_{0}\text{r}ʻQ \quad\text{Df}\)</p> + +<p>We prove that \(\mu \dot{\times} \nu\) is only zero when one of its +factors is zero (<a href="#*184·16">*184·16</a>); we prove the associative law (<a href="#*184·31">*184·31</a>), and +the distributive law in the forms</p> + +<p class="nind"><b>*184·33.</b> \(\vdash :P\in \text{Rel}^{2}\text{excl}.\supset .\Sigma \text{Nr}ʻP\dot{\times} \text{Nr}ʻR=\Sigma \text{Nr}ʻ(\times R)^{;}P\)</p> + +<p class="nind"><b>*184·35.</b> \(\vdash .(\nu \dot{+} \varpi )\dot{\times} \mu =(\nu \dot{\times} \mu )\dot{+} (\varpi \dot{\times} \mu )\)</p> + +<p class="nind">and we prove \(2_{r}\dot{\times} \mu =\mu \dot{+} \mu\) (<a href="#*184·4">*184·4</a>). Also +we extend the distributive law to the case where one of the summands is +\(\dot{1}\), <i>i.e.</i> we prove</p> + +<p class="nind"><b>*184·41.</b> \(\vdash :\nu \neq 0_{r}.\supset .(\nu \dot{+} \dot{1} )\dot{\times} \mu =(\nu \dot{\times} \mu )\dot{+} \mu\)</p> + +<p class="nind"><b>*184·42.</b> \(\vdash :\nu \neq 0_{r}.\supset .(\dot{1} \dot{+} \nu) \dot{\times} \mu =\mu \dot{+} (\nu \dot{\times} \mu )\)</p> + +<p class="nind">and the connection of cardinal and ordinal multiplication is given by</p> + +<p class="nind"><b>*184·5.</b> \(\vdash :\mu ,\nu \in \text{NR}.\supset .Cʻʻ(\mu \dot{\times} \nu )=Cʻʻ\mu \times _{c}Cʻʻ\nu\)</p> + +<hr class="tb"> + +<p class="nind"><b>*184·01.</b> \(\mu \dot{\times} \nu =\hat{R} \{(\exists P,Q).\mu =\text{N}_{0}\text{r}ʻP.\nu =\text{N}_{0}\text{r}ʻQ.R\,\text{smor}\,(P\times Q)\} \quad\text{Df}\)</p> + +<p class="nind"><b>*184·02.</b> \(\text{Nr}ʻP\dot{\times} \nu =\text{N}_{0}\text{r}ʻP\dot{\times} \nu \quad\text{Df}\)</p> + +<p class="nind"><b>*184·03.</b> \(\mu \dot{\times} \text{Nr}ʻQ=\mu \dot{\times} \text{N}_{0}\text{r}ʻQ \quad\text{Df}\)</p> + +<p class="nind"><b>*184·1.</b> \[\begin{align}&\vdash :R\in \mu \dot{\times} \nu .\equiv .(\exists P,Q).\mu =\text{N}_{0}\text{r}ʻP.\nu =\text{N}_{0}\text{r}ʻQ.R\,\text{smor}\,(P\times Q)\\ +&[(*184·01)]\end{align}\]</p> + +<p><span class="pagenum" id="Page_502">[Pg 502]</span></p> + +<p>The proofs of the following propositions are omitted, since they are +analogous to those of the corresponding propositions of <a href="#*113">*113</a>.</p> + +<p class="nind"><b>*184·11.</b> \(\vdash :\exists !\mu \dot{\times} \nu .\supset .\mu ,\nu \in \text{N}_{0}\text{R}.\exists !\mu .\exists !\nu\)</p> + +<p class="nind"><b>*184·111.</b> \(\vdash :{\sim}(\mu ,\nu \in \text{N}_{0}\text{R}).\supset .\mu \dot{\times} \nu =\Lambda\)</p> + +<p class="nind"><b>*184·12.</b> \(\vdash \colon\ldotp \mu ,\nu \in \text{NR}.\supset :R\in \mu \dot{\times} \nu .\equiv .(\exists P,Q).P\in \mu .Q\in \nu .R\,\text{smor}\,(P\times Q)\)</p> + +<p class="nind"><b>*184·13.</b> \[\begin{align}\vdash .\text{Nr}ʻP\dot{\times} \text{Nr}ʻQ=\text{N}_{0}\text{r}ʻP\dot{\times} \text{Nr}ʻQ&=\text{Nr}ʻP\dot{\times} \text{N}_{0}\text{r}ʻQ\\ +&=\text{N}_{0}\text{r}ʻP\dot{\times} \text{N}_{0}\text{r}ʻQ=\text{Nr}ʻ(P\times Q)\end{align}\]</p> + +<p class="nind"><b>*184·14.</b> \(\vdash :P\,\,\text{smor}\,\,R.Q\,\text{smor}\,S.\supset .\text{Nr}ʻP\dot{\times} \text{Nr}ʻQ=\text{Nr}ʻR\dot{\times} \text{Nr}ʻS\)</p> + +<p class="nind"><b>*184·15.</b> \(\vdash .\mu \dot{\times} \nu \in \text{NR}\)</p> + +<p class="nind"><b><a id="*184·16">*184·16</a>.</b> \(\vdash \colon\ldotp \mu \dot{\times} \nu =0_{r}.\equiv :\mu ,\nu \in \text{NR}-\iota ʻ\Lambda :\mu =0_{r}.\lor.\nu =0_{r}\)</p> + +<p class="nind"><b>*184·2.</b> \[\begin{align}&\vdash \colon\ldotp \text{Mult ax}.\supset :P\in \text{Nr}ʻR.CʻP\subset \text{Nr}ʻS.\supset .\Sigma \text{Nr}ʻP =\text{Nr}ʻR\dot{\times} \text{Nr}ʻS\\ +&[*183·26.*184·13]\end{align}\]</p> + +<p class="nind"><b>*184·21.</b> \(\vdash \colon\ldotp \text{Mult ax}.\supset :\mu ,\nu \in \text{NR}.\nu \neq \Lambda .P \in \mu .CʻP\subset \nu .\supset .\Sigma \text{Nr}ʻP =\mu \dot{\times} \nu\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*152·45.&\supset \vdash :\mu \in \text{NR}.P\in \mu .\supset .\mu =\text{Nr}ʻP &\qquad \text{(1)}\\ +\vdash .*152·45.&\supset \vdash :\nu \in \text{NR}.CʻP\subset \nu .S\in CʻP.\nu =\text{Nr}ʻS.CʻP\subset \text{Nr}ʻS &\qquad \text{(2)}\\ +\vdash .(1).(2).*184·2.\supset \\ +\vdash \colon\ldotp \text{Mult ax}.&\supset :\mu ,\nu \in \text{NR}.P\in \mu .CʻP\subset \nu .S\in CʻP.\supset .\\ +&\Sigma \text{Nr}ʻP=\text{Nr}ʻP\dot{\times} \text{Nr}ʻS.\mu =\text{Nr}ʻP.\nu =\text{Nr}ʻS.\\ +[*13·13] &\supset .\Sigma \text{Nr}ʻP=\mu \dot{\times} \nu &\qquad \text{(3)}\\ +\vdash .(3).*10·11·21·23.\supset \\ +\vdash \colon\ldotp \text{Mult ax}.&\supset :\mu ,\nu \in \text{NR}.P\in \mu .CʻP\subset \nu .\exists !CʻP.\supset .\Sigma ΝrʻP=\mu \dot{\times} \nu &\qquad \text{(4)}\\ +\vdash .*183·2.*162·4. &\supset \vdash :P=\dot{\Lambda} .\supset .\Sigma \text{Nr}ʻP=0_{r} &\qquad \text{(5)}\\ +\vdash .*153·16.\text{Transp}. &\supset \vdash \colon\ldotp \mu \in \text{NR}.P\in \mu .P=\dot{\Lambda} .\supset :\mu =0_{r}:\\ +[*184·16] &\supset :\nu \in \text{NR}-\iota ʻ\Lambda .\supset .\mu \dot{\times} \nu =0_{r} &\qquad \text{(6)}\\ +\vdash .(5).(6).&\supset \vdash :\mu ,\nu \in \text{NR}.\nu \neq \Lambda .P\in \mu .CʻP\subset \nu .P=\dot{\Lambda} .\supset .\\ +&\Sigma \text{Nr}ʻP=\mu \dot{\times} \nu &\qquad \text{(7)}\\ +\vdash .(4).(7).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*184·3.</b> \(\vdash .(\text{Nr}ʻP\dot{\times} \text{Nr}ʻQ)\dot{\times} \text{Nr}ʻR=\text{Nr}ʻP\dot{\times} (\text{Nr}ʻQ\dot{\times} \text{Nr}ʻR)=\text{Nr}ʻ(P\times Q\times R)\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*184·13. \supset \vdash .(\text{Nr}ʻP\dot{\times} \text{Nr}ʻQ)\dot{\times} \text{Nr}ʻR&=\text{Nr}ʻ(P\times Q)\dot{\times} \text{Nr}ʻR\\ +[*184·13] &=\text{Nr}ʻ(P\times Q\times R)\\ +[*166·42] &=\text{Nr}ʻ\{P\times (Q\times R)\}\\ +[*184·13] &=\text{Nr}ʻP\dot{\times} (\text{Nr}ʻQ\dot{\times} \text{Nr}ʻR).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*184·31">*184·31</a>.</b> \(\vdash .(\mu \dot{\times} \nu )\dot{\times} \varpi =\mu \dot{\times} (\nu \dot{\times} \varpi )\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*184·111.&\supset \vdash :{\sim}(\mu ,\nu ,\varpi \in \text{N}_{0}\text{R}).\supset + .(\mu \dot{\times} \nu )\dot{\times} \varpi =\Lambda .\mu \dot{\times} (\nu \dot{\times} \varpi )=\Lambda &\qquad \text{(1)}\\ +\vdash .*155·2. &\supset \vdash :\mu ,\nu ,\varpi \in \text{N}_{0}\text{R}.\supset .\\ +&(\exists P,Q,R).\mu =\text{N}_{0}\text{r}ʻP.\nu =Ν_{0}rʻQ.\varpi =\text{N}_{0}\text{r}ʻR &\qquad \text{(2)}\\ +\vdash .*184·13.\supset \\ +\vdash :\mu =\text{N}_{0}\text{r}ʻP.\nu =\text{N}_{0}\text{r}ʻQ.\varpi =\text{N}_{0}\text{r}ʻR.\supset + .(\mu \dot{\times} \nu )\dot{\times} \varpi &=\text{Nr}ʻ\{(P \times Q) \times R\}\\ +[*184·3] &=\text{Nr}ʻ\{P\times (Q\times R)\}\\ +[*184·13] &=\mu \dot{\times} (\nu \dot{\times} \varpi ) &\qquad \text{(3)}\\ +\vdash .(2).(3).&\supset \vdash :\mu ,\nu ,\varpi \in \text{N}_{0}\text{R}.\supset + .(\mu \dot{\times} \nu )\dot{\times} \varpi =\mu \dot{\times} (\nu \dot{\times} \varpi ) &\qquad \text{(4)}\\ +\vdash .(1).(4).&\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_503">[Pg 503]</span></p> + +<p class="nind"><b>*184·32.</b> \(\mu \dot{\times} \nu \dot{\times} \varpi =(\mu \dot{\times} \nu )\dot{\times} \varpi \quad\text{Df}\)</p> + +<p class="nind"><b>*184·33.</b> \(\vdash :P\in \text{Rel}^{2}\text{excl}.\supset .\Sigma \text{Nr}ʻP\dot{\times} \text{Nr}ʻR=\Sigma \text{Nr}ʻ(\times R)^{;}P\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*183·13.\supset \vdash :\text{Hp}.\supset .\Sigma \text{Nr}ʻP\dot{\times} \text{Nr}ʻR&=\text{Nr}ʻ\Sigma ʻP\dot{\times} \text{Nr}ʻR\\ +[*184·13] &=\text{Nr}ʻ(\Sigma ʻP\times R)\\ +[*166·44] &=\text{Nr}ʻ\Sigma ʻ(\times R)^{;}P &\qquad \text{(1)}\\ +\vdash .*166·3. \supset \vdash :\text{Hp}.&\supset .(\times R)^{;}P\in \text{Rel}^{2}\text{excl}.\\ +[*183·13] &\supset .\text{Nr}ʻ\Sigma ʻ(\times R)^{;}P=\Sigma \text{Nr}ʻ(\times R)^{;}P &\qquad \text{(2)}\\ +\vdash .(1).(2).&\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*184·34.</b> \(\vdash .(\text{Nr}ʻP\dot{+} \text{Nr}ʻQ)\dot{\times} \text{Nr}ʻR=(\text{Nr}ʻP\dot{\times} \text{Nr}ʻR)\dot{+} (\text{Nr}ʻQ\dot{\times} \text{Nr}ʻR)\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*180·3.*184·13.\supset \\ +\vdash .(\text{Nr}ʻP\dot{+} \text{Nr}ʻQ)\dot{\times} \text{Nr}ʻR&=\text{Nr}ʻ\{(P+Q)\times R\}\\ +[*166·45.*180·1] &=\text{Nr}ʻ[\{\downarrow (\Lambda \cap CʻQ)^{;}\iota ^{;}P\}\times R\unicode{x2909}\{(\Lambda \cap CʻP)\downarrow ^{;}\iota ^{;}Q\}\times R]\\ +[*166·3.*180·11·32] &=\text{Nr}ʻ[\{\downarrow (\Lambda \cap CʻQ)^{;}\iota ^{;}P\}\times R]\dot{+} \text{Nr}ʻ[\{(\Lambda \cap CʻP)\downarrow ^{;}\iota ^{;}Q\}\times + R]\\ +[*184·14.*180·12] &=\text{Nr}ʻ(P\times R)\dot{+} \text{Nr}ʻ(Q\times R)\\ +[*184·13] &=(\text{Nr}ʻP\dot{\times} \text{Nr}ʻQ)\dot{+} (\text{Nr}ʻQ\dot{\times} \text{Nr}ʻR).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*184·35.</b> \(\vdash .(\nu \dot{+} \varpi )\dot{\times} \mu =(\nu \dot{\times} \mu )\dot{+} (\varpi \dot{\times} \mu ) \quad[*184·34]\)</p> + +<p>The proof proceeds as in <a href="#*184·31">*184·31</a>.</p> + +<p class="nind"><b><a id="*184·4">*184·4</a>.</b> \(\vdash .2_{r}\dot{\times} \mu =\mu \dot{+} \mu \)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*184·111.*180·4.&\supset \vdash :\mu {\sim}\in \text{N}_{0}\text{R}.\supset .2_{r}\dot{\times} \mu =\Lambda .\mu \dot{+} \mu =\Lambda &\qquad \text{(1)}\\ +\vdash .*153·24.*184·13.\supset \\ +\vdash :\mu& =\text{N}_{0}\text{r}ʻP.\supset .2_{r}\dot{\times} \mu =\text{Nr}ʻ\{\Lambda \downarrow (\iota ʻx)\times P\} &\qquad \text{(2)}\\ +\vdash .*166·1.\supset \vdash .\Lambda \downarrow (\iota ʻx)\times P&=\Sigma ʻP\downarrow_{.,} ^{;}(\Lambda \downarrow \iota ʻx)\\ +[*150·71] &=\Sigma ʻ\{(P\downarrow_{.,} \Lambda )\downarrow (P\downarrow_{.,} \iota ʻx)\}\\ +[*162·3] & =(P\downarrow_{.,} \Lambda )\unicode{x2909}(P\downarrow_{.,} \iota ʻx) &\qquad \text{(3)}\\ +\vdash .*180·31·32.*165·251·211.\text{Transp}.\supset \\ +\vdash .\text{Nr}ʻ\{(P\downarrow_{.,} \Lambda )\unicode{x2909}(P\downarrow_{.,} \iota ʻx)\}&=\text{Nr}ʻP\dot{+} \text{Nr}ʻP &\qquad \text{(4)}\\ +\vdash .(2).(3).(4).\supset \vdash :\mu =\text{N}_{0}\text{r}ʻP.\supset .2_{r}\dot{\times} \mu &=\text{Nr}ʻP\dot{+} \text{Nr}ʻP\\ +[*180·3] &=\mu \dot{+} \mu &\qquad \text{(5)}\\ +\vdash .(1).(5).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_504">[Pg 504]</span></p> + +<p class="nind"><b>*184·41.</b> \(\vdash :\nu \neq 0_{r}.\supset .(\nu \dot{+} \dot{1} )\dot{\times} \mu = (\nu \dot{\times} \mu) \dot{+} \mu\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*166·53.*180·32.*165·251.\supset \\ +\vdash :\dot{\exists} !Q.y{\sim}\in CʻQ.\supset .\text{Nr}ʻ\{(Q\unicode{x21f8} y)xP\} = \text{Nr}ʻ(QxP)\dot{+} \text{Nr}ʻP &\qquad \text{(1)}\\ +\vdash .(1).*181·32.*184·13.\supset \\ +\vdash :\mu = \text{Nr}ʻP.\nu = \text{Nr}ʻQ.\nu \neq 0_{r}.y{\sim}\in CʻQ.\supset .(\nu \dot{+} \dot{1} )\dot{\times} \mu = (\nu \dot{\times} \mu )\dot{+} \mu &\qquad \text{(2)}\\ +\vdash .*181·11·12.\supset \vdash :\nu \in \text{NR}-\iota ʻ\Lambda .\supset .(\exists Q,y).\nu = \text{Nr}ʻQ.y{\sim}\in CʻQ &\qquad \text{(3)}\\ +\vdash .(2).(3).\supset \\ +\vdash :\mu \in \text{NR}.\nu \in \text{NR}-\iota ʻ\Lambda .\nu \neq 0_{r}.\supset .(\nu \dot{+} \dot{1} )\dot{\times} \mu = (\nu \dot{\times} \mu )\dot{+} \mu &\qquad \text{(4)}\\ +\vdash .*184·111.*181·4.\supset \\ +\vdash :{\sim}(\mu \in \text{NR}.\nu \in \text{NR}-\iota ʻ\Lambda ).\supset .(\nu \dot{+} 1)\dot{\times} \mu = \Lambda .(\nu \dot{\times} \mu )\dot{+} \mu = \Lambda &\qquad \text{(5)}\\ +\vdash .(4).(5).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*184·42.</b> \(\vdash :\nu \neq 0_{r}.\supset .(\dot{1} \dot{+} \nu )\dot{\times} \mu = \mu \dot{+} (\nu \dot{\times} \mu ) \quad[\text{Proof as in *184·41}]\)</p> + +<p class="nind"><b>*184·5.</b> \(\vdash :\mu ,\nu \in \text{NR}.\supset .Cʻʻ(\mu \dot{\times} \nu ) = Cʻʻ\mu \times _{c}Cʻʻ\nu \)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*184·13.\supset \vdash :\text{Hp}.P\in \mu .Q\in \nu .\supset .Cʻʻ(\mu \dot{\times} \nu ) &= Cʻʻ\text{Nr}ʻ(P\times Q)\\ +[*152·7.*166·12] & = \text{Nc}ʻ(CʻP\times CʻQ)\\ +[*152·7.*113·25] &= Cʻʻ\text{Nr}ʻP\times _{c}Cʻʻ\text{Nr}ʻQ\\ +[*152·45] & = Cʻʻ\mu x_{c}Cʻʻ\nu &\qquad \text{(1)}\\ +\vdash .*184·11.*113·204.\supset \\ +\vdash :{\sim}(\exists !\mu .\exists !\nu ).\supset .Cʻʻ(\mu \dot{\times} \nu )& = \Lambda .Cʻʻ\mu \times _{c}Cʻʻ\nu = \Lambda &\qquad \text{(2)}\\ +\vdash .(1).(2).\supset \vdash .\text{Prop} +\end{array} +\]</p> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_505">[Pg 505]</span></p> +<h2 class="nobreak" id="185_THE_PRODUCT_OF_THE_RELATION-NUMBERS_OF_A_FIELD"> +*185. THE PRODUCT OF THE RELATION-NUMBERS OF A FIELD.</h2> +</div> + + +<p><i>Summary of</i> *185.</p> + +<p>The subject of this number is analogous to part of the subject of <a href="#*114">*114</a>. +The propositions concerned are immediate consequences of previously +proved properties of \(\Pi ʻP\), and offer no difficulty of any kind.</p> + +<hr class="tb"> + +<p class="nind"><b>*185·01.</b> \(\Pi \text{Nr}ʻP=\text{Nr}ʻ\Pi ʻP \quad\text{Df}\)</p> + +<p class="nind"><b>*185·1.</b> \(\vdash .\Pi \text{Nr}ʻP=\text{Nr}ʻ\Pi ʻP \quad[(*185·01)]\)</p> + +<p class="nind"><b>*185·11.</b> \(\vdash :P \,\,\text{smor}\, \,\text{smor}\, Q.\supset .\Pi \text{Nr}ʻP=\Pi \text{Nr}ʻQ \quad[*172·44]\)</p> + +<p class="nind"><b>*185·12.</b> \(\vdash .\Pi \text{Nr}ʻP=\text{Nr}ʻ\text{Prod}ʻ\breve{\downarrow}_{.,}^{;}P=\text{Nr}ʻ\Pi ʻ\breve{\downarrow}_{.,}^{;}P \quad[*182·44]\)</p> + +<p class="nind"><b>*185·2.</b> \(\vdash .\Pi \text{Nr}ʻ\dot{\Lambda} =0_{r} \quad[*172·13]\)</p> + +<p class="nind"><b><a id="*185·21">*185·21</a>.</b> \(\vdash .\Pi \text{Nr}ʻ(P\downarrow P)=\text{Nr}ʻP \quad[*172·2.*165·251]\)</p> + +<p class="nind"><b>*185·22.</b> \(\vdash .\Pi \text{Nr}ʻ(\dot{\Lambda} \downarrow \dot{\Lambda} )=0_{r} \quad[*185·21]\)</p> + +<p class="nind"><b>*185·23.</b> \(\vdash :\dot{\Lambda} \in CʻP.\supset .\Pi \text{Nr}ʻP=0_{r} \quad [*172·14]\)</p> + +<p class="nind"><b>*185·25.</b> \(\vdash \colon\colon\text{Mult ax}.\supset \colon\ldotp \Pi \text{Nr}ʻP=0_{r}.\equiv :\dot{\Lambda} \in CʻP.\lor.P=\dot{\Lambda} \quad[*172·182]\)</p> + +<p class="nind"><b>*185·27.</b> \[\begin{align}\vdash \colon\ldotp \text{Mult ax}.\supset :P,Q\in &\text{Rel}^{2}\text{excl}.\exists !P\,\overline{\,\text{smor}\,}\,Q\cap \text{Rl}ʻ\,\text{smor}\,.\supset .\\ +&\Pi \text{Nr}ʻP=\Pi \text{Nr}ʻQ \quad[*172·45]\end{align}\]</p> + +<p class="nind"><b>*185·28.</b> \[\begin{align}\vdash \colon\ldotp \text{Mult ax}.\supset :P,Q\in &\text{Rel}^{2}\text{excl}.P,Q\in \text{Nr}ʻR.CʻP,CʻQ\in \text{Cl}ʻ\text{Nr}ʻS.\supset .\\ +&\Pi \text{Nr}ʻP=\Pi \text{Nr}ʻQ \quad[*164·48.*185·11]\end{align}\]</p> + +<p class="nind"><b>*185·29.</b> \[\begin{align}\vdash \colon\ldotp \text{Mult ax}.\supset :P\in &\text{Rel}^{2}\text{excl}.P\in \text{Nr}ʻR.CʻP\subset \text{Nr}ʻS.\supset .\\ +&\Pi \text{Nr}ʻP=\text{Nr}ʻ(S\,\text{exp}\,\,R) \quad[*176·24]\end{align}\]</p> + +<p class="nind"><b>*185·31.</b> \[\begin{align}&\vdash :\dot{\exists} !P.\dot{\exists} !Q.CʻP\cap CʻQ=\Lambda .\supset .\Pi \text{Nr}ʻ(P\unicode{x2909}Q)=\Pi \text{Nr}ʻP\dot{\times} \Pi \text{Nr}ʻQ\\ +&[*172·35]\end{align}\]</p> + +<p class="nind"><b>*185·32.</b> \(\vdash :Z{\sim}\in CʻP.\supset .\Pi \text{Nr}ʻ(P\unicode{x21f8} Z)=\Pi \text{Nr}ʻP\dot{\times} \text{Nr}ʻZ \quad[*172·32]\)</p> + +<p class="nind"><b>*185·321.</b> \(\vdash :Z{\sim}\in CʻP.\supset .\Pi \text{Nr}ʻ(Z\unicode{x21f7}P)=\text{Nr}ʻZ\dot{\times} \Pi \text{Nr}ʻP \quad[*172·321]\)</p> + +<p class="nind"><b>*185·35.</b> \(\vdash :P\neq Q.\supset .\Pi \text{Nr}ʻ(P\downarrow Q)=\text{Nr}ʻP\dot{\times} \text{Nr}ʻQ \quad [*172·23]\)</p> + +<p><span class="pagenum" id="Page_506">[Pg 506]</span></p> + +<p class="nind"><b>*185·4.</b> \[\begin{align}&\vdash \colon\ldotp P\in \text{Rel}^{2}\text{excl}:QPQ.\supset _{Q}.CʻQ\in 0\cup 1:\supset .\Pi \text{Nr}ʻ\Pi ^{;}P=\Pi + \text{Nr}ʻ\Sigma ʻP\\ +&[*174·241]\end{align}\]</p> + +<p class="nind"><b>*185·41.</b> \(\vdash :P\in \text{Rel}^{2}\text{excl}.P\,\unicode{x2abd}\, J.\supset .\Pi \text{Nr}ʻ\Pi ^{;}P=\Pi \text{Nr}ʻ\Sigma ʻP \quad[*174·25]\)</p> + +<p>The following proposition gives the connection between ordinal and +cardinal multiplication.</p> + +<p class="nind"><b>*185·5.</b> \(\vdash : P \in \text{Rel}^{2}\text{excl} . \dot{\exists} ! P . \supset . Cʻʻ\Pi \text{Nr}ʻP = \Pi \text{Nc}ʻCʻʻCʻP\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*173·16.\supset \vdash :\text{Hp}.\supset .Cʻʻ\Pi \text{Nr}ʻP&=Cʻʻ\text{Nr}ʻ\text{Prod}ʻP\\ +[*152·7] &= \text{Nc}ʻCʻ\text{Prod}ʻP\\ +[*173·161] &= \text{Nc}ʻ\text{Prod}ʻCʻʻCʻP\\ +[*163·16.*115·12] &=\Pi \text{Nc}ʻCʻʻCʻP:\supset \vdash .\text{Prop} +\end{array} +\]</p> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_507">[Pg 507]</span></p> +<h2 class="nobreak" id="186_POWERS_OF_RELATION-NUMBERS"> +*186. POWERS OF RELATION-NUMBERS.</h2> +</div> + + +<p class="nind"><i>Summary of</i> *186.</p> + +<p>For "\(\mu\) to the \(\nu\)th power," where ordinal powers are +concerned, we use the notation "\(\mu \,\text{exp}\,\,_{r}\nu\)." We cannot +use "\(\mu ^\nu\)" or "\(\mu \,\text{exp}\,\,\nu\)" because these have +been already used for cardinals and classes (<a href="#*116">*116</a>). We therefore put a +suffix \(r\) to "\(\,\text{exp}\,\)" to show that it is <i>relational</i> +powers that we are dealing with. We put +\[ +\mu \,\text{exp}\,\,_{r}\nu =\hat{R} \{(\exists P,Q).\mu =\text{N}_{0}\text{r}ʻP.\nu =\text{N}_{0}\text{r}ʻQ.R\,\text{smor}\,(P\,\text{exp}\,\,Q)\} \quad\text{Df}. +\]</p> + +<p>The following are the principal propositions of this number:</p> + +<p class="nind"><b>*186·2.</b> \(\vdash :\mu \in \text{N}_{0}\text{R}.\supset .0_{r}\,\text{exp}\,\,_{r}\mu =0_{r}.\mu \,\text{exp}\,\,_{r}0_{r}=0_{r}\)</p> + +<p>We do not have \(\mu \,\text{exp}\,\,_{r}0_{r}=1\), because there is no +ordinal 1.</p> + +<p class="nind"><b>*186·21.</b> \(\vdash .\mu \,\text{exp}\,\,_{r}2_{r}=\mu \dot{\times} \mu\)</p> + +<p class="nind"><b>*186·22.</b> \(\vdash .\alpha \,\text{exp}\,\,_{r}(\beta \dot{+} \dot{1} )=(\alpha \,\text{exp}\,\,_{r}\beta )\dot{\times} \alpha\)</p> + +<p class="nind"><b>*186·23.</b> \(\vdash .\alpha \,\text{exp}\,\,_{r}(\dot{1} \dot{+} \beta )=\alpha \dot{\times} (\alpha \,\text{exp}\,\,_{r}\beta )\)</p> + +<p class="nind"><b>*186·14.</b> \(\vdash :\nu \neq 0_{r}.\varpi \neq 0_{r}.\supset .\mu \,\text{exp}\,\,_{r}(\nu \dot{+} \varpi )=(\mu \,\text{exp}\,\,_{r}\nu + )\dot{\times} (\mu \,\text{exp}\,\,_{r}\varpi )\)</p> + +<p class="nind"><b>*186·15.</b> \(\vdash :\varpi \subset \text{Rl}ʻJ.\supset .\mu \,\text{exp}\,\,_{r}(\varpi \dot{\times} \nu )=(\mu \,\text{exp}\,\,_{r}\nu )\,\text{exp}\,\,_{r}\varpi\)</p> + +<p class="nind"><b>*186·31.</b> \[\begin{align}\vdash \colon\ldotp \text{Mult ax}.\supset :\mu \nu \in \text{NR}-\iota ʻ\Lambda .P\in \text{Rel}^{2}\text{excl}\cap &\mu .CʻP\subset \nu .\supset .\\ +&\Pi \text{Nr}ʻP=\mu \,\text{exp}\,\,_{r}\nu\end{align}\]</p> + +<p class="nind">which connects exponentiation with multiplication.</p> + +<p class="nind"><b>*186·4.</b> \(\vdash .\text{Nr}ʻP_{\text{df}}=2_{r}\,\text{exp}\,\,_{r}(\text{Nr}ʻP)\) (cf.*177)</p> + +<p class="nind"><b>*186·5.</b> \(\vdash :\mu ,\nu \in \text{N}_{0}\text{R}.\nu \neq 0_{r}.\supset .Cʻʻ(\mu \,\text{exp}\,\,_{r}\nu )=(Cʻʻ\mu )^{Cʻʻ\nu }\)</p> + +<p class="nind">which connects ordinal and cardinal exponentiation.</p> + +<hr class="tb"> + +<p class="nind"><b>*186·01.</b> \(\mu \,\text{exp}\,\,_{r}\nu =\hat{R} \{(\exists P,Q).\mu =\text{N}_{0}\text{r}ʻP.\nu =\text{N}_{0}\text{r}ʻQ.R\,\text{smor}\,(P\,\text{exp}\,\,Q)\} \quad\text{Df}\)</p> + +<p class="nind"><b>*186·02.</b> \((\text{Nr}ʻP)\,\text{exp}\,\,_{r}\nu =(\text{N}_{0}\text{r}ʻP)\,\text{exp}\,\,_{r}\nu \quad\text{Df}\)</p> + +<p class="nind"><b>*186·03.</b> \(\mu \,\text{exp}\,\,_{r}(\text{Nr}ʻQ)=\mu \,\text{exp}\,\,_{r}(\text{N}_{0}\text{r}ʻQ) \quad\text{Df}\)</p> + +<p class="nind"><b>*186·1.</b> \[\begin{align}&\vdash :R\in \mu \,\text{exp}\,\,_{r}\nu .\equiv .(\exists P,Q).\mu =\text{N}_{0}\text{r}ʻP.\nu + =\text{N}_{0}\text{r}ʻQ.R\,\text{smor}\,(P\,\text{exp}\,\,Q)\\ +&[(*186·01)]\end{align}\]</p> + +<p><span class="pagenum" id="Page_508">[Pg 508]</span></p> + +<p class="nind"><b>*186·11.</b> \(\vdash .\exists !\mu \,\text{exp}\,\,_{r}\nu .\supset .\mu ,\nu \in \text{N}_{0}\text{R}.\mu ,\nu \in \text{NR}-\iota ʻ\Lambda\)</p> + +<p class="nind"><b>*186·111.</b> \(\vdash :{\sim}(\mu \nu \in \text{N}_{0}\text{R}).\supset .\mu \,\text{exp}\,\,_{r}\nu =\Lambda\)</p> + +<p class="nind"><b>*186·12.</b> \[\begin{align}&\vdash :R\in \mu \,\text{exp}\,\,_{r}\nu .\equiv .(\exists P,Q).\mu =\text{N}_{0}\text{r}ʻP.\nu + =\text{N}_{0}\text{r}ʻQ.R\,\,\text{smor}\,\,P^{Q}\\ +&[*176·181.*186·1]\end{align}\]</p> + +<p class="nind"><b>*186·13.</b> + \[\begin{align}\vdash .(\text{Nr}ʻP)\,\text{exp}\,\,_{r}(\text{Nr}ʻQ)&=(\text{N}_{0}\text{r}ʻP)\,\text{exp}\,\,_{r}(\text{Nr}ʻQ)=(\text{Nr}ʻP)\,\text{exp}\,\,_{r}(\text{N}_{0}\text{r}ʻQ)\\ +&=(\text{N}_{0}\text{r}ʻP)\,\text{exp}\,\,_{r}(\text{N}_{0}\text{r}ʻQ)=\text{Nr}ʻ(P\,\text{exp}\,\,Q)=\text{Nr}ʻ(P^{Q})\\ +[\text{Proof as in *180·3}]\end{align}\]</p> + +<p class="nind"><b>*186·14.</b> \(\vdash :\nu \neq 0_{r}.\varpi \neq 0_{r}.\supset .\mu \,\text{exp}\,\,_{r}(\nu \dot{+} \varpi )=(\mu \,\text{exp}\,\,_{r}\nu + )\dot{\times} (\mu \,\text{exp}\,\,_{r}\varpi )\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*180·4.*186·111.\supset \\ +\vdash :{\sim}(\mu ,\nu ,\varpi \in \text{N}_{0}\text{R}).&\supset .\mu \,\text{exp}\,\,_{r}(\nu \dot{+} \varpi )=\Lambda .(\mu \,\text{exp}\,\,_{r}\nu + )\dot{\times} (\mu \,\text{exp}\,\,_{r}\varpi )=\Lambda &\qquad \text{(1)}\\ +\vdash .*186·13.*180·3.\supset \\ +\vdash :\mu =\text{N}_{0}\text{r}ʻP.\nu &=\text{N}_{0}\text{r}ʻQ.\varpi =\text{N}_{0}\text{r}ʻR.\supset .\mu \,\text{exp}\,\,_{r}(\nu \dot{+} \varpi )=\text{Nr}ʻP^{Q+R} + &\qquad \text{(2)}\\ +\vdash .176·42.*180·11.\supset \\ +\vdash :\text{Hp}.\text{Hp}(2).\supset .\text{Nr}ʻP^{Q+R}&=\text{Nr}ʻ(P^{\downarrow(\Lambda \cap CʻP){;}\iota{;}Q}\times + P^{(\Lambda \cap CʻP)\downarrow{;}\iota{;}R})\\ +[*180·12.*176·22.*166·23] &=\text{Nr}ʻ(P^{Q}\times P^{R})\\ +[*186·13.*184·13] &=(\mu \,\text{exp}\,_{r}\nu )\dot{\times} (\mu \,\text{exp}\,_{r}\varpi ) &\qquad \text{(3)}\\ +\vdash .(2).(3).*155·2.&\supset \vdash :\mu ,\nu ,\varpi \in \text{N}_{0}\text{R}.\nu \neq 0_{r}.\varpi \neq 0_{r}.\supset .\\ +&\mu \,\text{exp}\,_{r}(\nu \dot{+} \varpi )=(\mu \,\text{exp}\,_{r}\nu )\dot{\times} (\mu \,\text{exp}\,_{r}\varpi ) &\qquad \text{(4)}\\ +\vdash .(1).(4).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*186·15.</b> \(\vdash :\varpi \subset \text{Rl}ʻJ.\supset .\mu \,\text{exp}\,\,_{r}(\varpi \dot{\times} \nu )=(\mu \,\text{exp}\,\,_{r}\nu )\,\text{exp}\,\,_{r}\varpi )\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*186·111.*184·111.\supset \\ +\vdash :{\sim}(\mu ,\nu ,\varpi \in \text{N}_{0}\text{R}).&\supset .\mu \,\text{exp}\,\,_{r}(\varpi \dot{\times} \nu )=\Lambda .(\mu \,\text{exp}\,\,_{r}\nu )\,\text{exp}\,\,_{r}\varpi + =\Lambda &\qquad \text{(1)}\\ +\vdash .*186·13.*184·13.\supset \\ +\vdash :\mu =\text{N}_{0}\text{r}ʻP.\nu& =\text{N}_{0}\text{r}ʻQ.\varpi =\text{N}_{0}\text{r}ʻR.\supset .\mu \,\text{exp}\,\,_{r}(\varpi + \dot{\times} \nu )=\text{Nr}ʻ(P^{R\times Q}) &\qquad \text{(2)}\\ +\vdash .*176·57.\supset \vdash :\text{Hp}.\text{Hp}(2).\supset .\text{Nr}ʻ(P^{P\times Q})&=\text{Nr}ʻ(P^{Q})^{R}\\ +[*186·13] & =\{(\text{N}_{0}\text{r}ʻP)\,\text{exp}\,\,_{r}(\text{N}_{0}\text{r}ʻQ)\}\,\text{exp}\,\,_{r}(\text{N}_{0}\text{r}ʻR)\\ +[\text{Hp}] &=(\mu \,\text{exp}\,\,_{r}\nu )\,\text{exp}\,\,_{r}\varpi &\qquad \text{(3)}\\ +\vdash .(2).(3).*155·2.\supset \\ +\vdash :\mu ,\nu ,\varpi \in &\text{N}_{0}\text{R}.\varpi \subset \text{Rl}ʻJ.\supset .\mu \,\text{exp}\,\,_{r}(\varpi \dot{\times} \nu )=(\mu \,\text{exp}\,\,_{r}\nu + )\,\text{exp}\,\,_{r}\varpi &\qquad \text{(4)}\\ +\vdash .(1).(4).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*186·2.</b> \(\vdash :\mu \in \text{N}_{0}\text{R}\supset .0_{r}\,\text{exp}\,\,_{r}\mu =0_{r}.\mu \,\text{exp}\,\,_{r}0_{r}=0_{r} + \quad[*176·151]\)</p> + +<p class="nind"><b>*186·21.</b> \(\vdash .\mu \,\text{exp}\,\,_{r}2_{r}=\mu \dot{\times} \mu\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*186·111.*184·111.&\supset \vdash :\mu {\sim}\in \text{N}_{0}\text{R}.\supset .\mu \,\text{exp}\,\,_{r}2_{r}=\Lambda + .\mu \dot{\times} \mu =\Lambda &\qquad \text{(1)}\\ +\vdash .*186·13.*176·1. &\supset \vdash :\mu =N_{O}rʻP.x\neq y.\supset .\\ +\mu \,\text{exp}\,\,_{r}2_{r}&=\text{Nr}ʻ\text{Prod}ʻP\downarrow_{.,} ^{;}(x\downarrow y)\\ +[*150·71] &=\text{Nr}ʻ\text{Prod}ʻ\{(P\downarrow_{.,} x)\downarrow (P\downarrow_{.,} y)\}\\ +[*173·24.*165·211.\text{Transp}] &=\text{Nr}ʻ\{(P\downarrow_{.,} x)\times (P\downarrow_{.,} y)\}\\ +[*165·251.*166·23] &=\text{Nr}ʻ(P\times P)\\ +[*184·13] & =\mu \dot{\times} \mu &\qquad \text{(2)}\\ +\vdash .(2).*155·2.*24·1. &\supset \vdash :\mu \in N_{O}R.\supset .\mu \,\text{exp}\,\,_{r}2_{r}=\mu \dot{\times} \mu &\qquad \text{(3)}\\ +\vdash .(1).(3).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_509">[Pg 509]</span></p> + +<p class="nind"><b>*186·22.</b> \(\vdash .\alpha \,\text{exp}\,\,_{r}(\beta \dot{+} \dot{1} )=(\alpha \,\text{exp}\,\,_{r}\beta )\dot{\times} \alpha\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*186·111.*181·4.\supset\\ +\vdash :{\sim}(\alpha ,\beta \in N_{O}R).&\supset .\alpha \,\text{exp}\,\,_{r}(\beta \dot{+} \dot{1} )=\Lambda .(\alpha \,\text{exp}\,\,_{r}\beta + )\dot{\times} \alpha =\Lambda &\qquad \text{(1)}\\ +\vdash .*186·13.*181·22.\supset \\ +\vdash :\alpha =N_{O}rʻP.\beta =N_{O}rʻQ.&\supset .\alpha \,\text{exp}\,\,_{r}(\beta \dot{+} \dot{1} )=\text{Nr}ʻ\{P\,\text{exp}\,\,(Q\dot{\unicode{x21f8}} z)\}.\\ +&(\alpha \,\text{exp}\,\,_{r}\beta )\dot{\times} \alpha =\text{Nr}ʻ(P\,\text{exp}\,\,Q)\dot{\times} \text{Nr}ʻP &\qquad \text{(2)}\\ +\vdash .(2).*176·151.*166·13.\supset \\ +\vdash :\text{Hp}(2).P=\dot{\Lambda} .&\supset .\alpha \,\text{exp}\,\,_{r}(\beta \dot{+} \dot{1} )=0_{r}.(\alpha \,\text{exp}\,\,_{r}\beta )\dot{\times} \alpha =0_{r}&\qquad + \text{(3)}\\ +\vdash .*165·2.*161·4.*176·1.(*181·01).\supset \\ +\vdash .\text{Nr}ʻ\{P\,\text{exp}\,\,(Q\dot{\unicode{x21f8}} z)\}&=\text{Nr}ʻ\text{Prod}ʻ[P\downarrow_{.,} ^{;}\downarrow \Lambda _{x}^{;}\iota ^{;}Q\unicode{x21f8} + P\downarrow_{.,} \{(\Lambda \cap CʻP)\downarrow \iota ʻx\}] &\qquad \text{(4)}\\ +\vdash .*165·221·222.*181·11.*162·22.\supset \\ +\vdash :\dot{\exists} !P.\supset .P\downarrow_{.,} &\{(\Lambda \cap CʻP)\downarrow \iota ʻx\}{\sim}\in CʻP\downarrow_{.,} ^{;}\downarrow \Lambda _{x}^{;}\iota + ^{;}Q.\\ +&CʻP\downarrow_{.,} \{(\Lambda \cap CʻP)\downarrow \iota ʻx\}\cap Cʻ\Sigma ʻP\downarrow_{.,} ^{;}\downarrow \Lambda _{x}^{;}\iota ^{;}Q=\Lambda + &\qquad \text{(5)}\\ +\vdash .(4).(5).*165·21.*173·25.\supset \vdash :\dot{\exists} !P.\supset .\\ +&\text{Nr}ʻ\{P\,\text{exp}\,\,(Q\dot{\unicode{x21f8}} z)\}=\text{Nr}ʻ[(\text{Prod}ʻP\downarrow_{.,} ^{;}\downarrow \Lambda _{x}^{;}\iota ^{;}Q)\times + P\downarrow_{.,} \{(\Lambda \cap CʻP)\downarrow \iota ʻx\}]\\ +[*181·12.*165·251.*176·1·22.*184·13]&=\text{Nr}ʻ(P\,\text{exp}\,\,Q)\dot{\times} \text{Nr}ʻP &\qquad \text{(6)}\\ +\vdash .(2).(6).\supset \vdash :\text{Hp}(2).\dot{\exists} !P.&\supset .\alpha \,\text{exp}\,\,_{r}(\beta \dot{+} \dot{1} )=(\alpha \,\text{exp}\,\,_{r}\beta + )\dot{\times} \alpha &\qquad \text{(7)}\\ +\vdash .(1).(3).(7).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*186·23.</b> \(\vdash .\alpha \,\text{exp}\,\,_{r}(\dot{1} \dot{+} \beta )=\alpha \dot{\times} (\alpha \,\text{exp}\,\,_{r}\beta ) \quad[\text{Proof as in *186·22}]\)</p> + +<p class="nind"><b>*186·3.</b> \[\begin{align}\vdash \colon\ldotp \text{Mult ax}.\supset :P\in &\text{Rel}^{2}\text{excl}\cap \text{Nr}ʻR.CʻP\subset \text{Nr}ʻS.\supset .\\ +&\Pi \text{Nr}ʻP=(\text{Nr}ʻP)\,\text{exp}\,\,_{r}(\text{Nr}ʻS) \quad[*185·29]\end{align}\]</p> + +<p class="nind"><b>*186·31.</b> \[\begin{align}\vdash \colon\ldotp \text{Mult ax}.\supset :\mu ,\nu \in \text{NR}-\iota ʻ\Lambda .P\in &\text{Rel}^{2}\text{excl}\cap \mu .CʻP\subset \nu .\supset .\\ +&\Pi \text{Nr}ʻP=\mu \,\text{exp}\,\,_{r}\nu \quad[*186·3]\end{align}\]</p> +<p><span class="pagenum" id="Page_510">[Pg 510]</span></p> + +<p class="nind"><b><a id="*186·4">*186·4</a>.</b> \(\vdash .\text{Nr}ʻP_{\text{df}}=2_{r}\,\text{exp}\,\,_{r}(\text{Nr}ʻP) \quad[*177·13]\)</p> + +<p class="nind"><b>*186·5.</b> \(\vdash :\mu ,\nu \in \text{N}_{0}\text{R}.\nu \neq 0_{r}.\supset .Cʻʻ(\mu \,\text{exp}\,\,_{r}\nu )=(Cʻʻ\mu )^{Cʻʻ\nu }\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*152·7.*186·13.\supset \vdash :\mu =\text{N}_{0}\text{r}ʻP.\nu &=\text{N}_{0}\text{r}ʻQ.\supset .\\ +Cʻʻ(\mu \,\text{exp}\,\,_{r}\nu )&=\text{Nc}ʻCʻ(P\,\text{exp}\,\,Q) &\qquad \text{(1)}\\ +\vdash .(1).*176·14.\supset \vdash :\text{Hp}(1).\nu \neq 0_{r}.\supset .Cʻʻ(\mu \,\text{exp}\,\,_{r}\nu )&=\text{Nc}ʻ\{(CʻP)\,\text{exp}\,\,(CʻQ)\}\\ +[*116·222] &=(\text{N}_{0}\text{c}ʻCʻP)^{\text{N}_{0}\text{c}ʻCʻQ}\\ +[*155·6] &=(Cʻʻ\text{N}_{0}\text{r}ʻP)^{Cʻʻ\text{N}_{0}\text{r}ʻQ}\\ +[\text{Hp}] & =(Cʻʻ\mu )^{cʻʻ\nu }:\supset \vdash .\text{Prop} +\end{array} +\]</p> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_511">[Pg 511]</span></p> +<h2 class="nobreak" id="PART_V">PART V.<br> +<br> +SERIES.</h2> +</div> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_513">[Pg 513]</span></p> +<h2 class="nobreak" id="SUMMARY_OF_PART_V">SUMMARY OF PART V.</h2> +</div> + + +<p><span class="allsmcap">A RELATION</span> is said to be <i>serial</i>, or to generate a +series, when it possesses three different properties, namely (1) being +contained in diversity, (2) transitiveness, (3) connexity, <i>i.e.</i> +the property that the relation or its converse holds between any two +different members of its field. Thus \(P\) is a serial relation if +(1) \(P\,\unicode{x2abd}\, J\), (2) \(P^{2}\,\unicode{x2abd}\, P\), (3)\(x,y\in CʻP.x\neq y.\supset _{x,y}:xPy.\lor.yPx\). +The third characteristic, that of connexity, may be written more shortly +\[ +x\in CʻP.\supset _{x}.\overrightarrow{P}ʻx\cup \iota ʻx\cup \overleftarrow{P}ʻx=CʻP, +\] +<i>i.e.</i> \(x\in CʻP.\supset _{x}.\overleftrightarrow{P}ʻx=CʻP\), +using the notation of *97; and this, in virtue of *97·23, is equivalent +to +\[ +\overleftrightarrow{P}ʻʻCʻP\in 0\cup 1. +\]</p> + +<p>In virtue of *50·47, the first two characteristics are equivalent to +\[ +P\dot{\cap} \breve{P} =\dot{\Lambda} .P^{2}\,\unicode{x2abd}\, P. +\] +When \(P\dot{\cap} \breve{P} =\dot{\Lambda}\), we say that \(P\) is +"asymmetrical." Thus serial relations are such as are asymmetrical, +transitive, and connected.</p> + +<p>It might be thought that a serial relation need not be contained +in diversity, since we commonly speak of series in which there are +repetitions, <i>i.e.</i> in which an earlier term is identical with a +later term. Thus, <i>e.g.</i> +\[ +a,\, b,\, c,\, a,\, e,\, f,\, b,\, g,\, h +\] +would be called a series of letters, although the letters \(a\) and +\(b\) recur. But in all such cases, there is some means (in the above +case, position in space) by which one <i>occurrence</i> of a given +term is distinguished from another occurrence, and this will be found +to mean that there is some other series (in the above case, the +series of positions in a line) free from repetitions, with which our +pseudo-series has a one-many correlation. Thus, in the above instance, +we have a series of nine positions, which we may call +\[ +1,\, 2,\, 3,\, 4,\, 5,\, 6,\, 7,\, 8,\, 9, +\] +which form a true series without repetitions; we have a one-many +relation, that of <i>occupying</i> these positions, by means of which +we distinguish occurrences of \(a\), the first occurrence being a as +the correlate of 1, the second being<span class="pagenum" id="Page_514">[Pg 514]</span> \(a\) as the correlate of 4. All +series in which there are repetitions (which we may call pseudo-series) +are thus obtained by correlation with true series, i.e, with series +in which there is no repetition. That is to say, a pseudo-series has +as its generating relation a relation of the form \(S^{;}P\), where +\(P\) is a serial relation, and \(S\) is a one-many relation whose +converse domain contains the field of \(P\). Thus what we may call +self-subsistent series must be series without repetitions, <i>i.e.</i> +series whose generating relations are contained in diversity.</p> + +<p>For our purposes, there is no use in distinguishing a series from +its generating relation. A series is not a class, since it has a +definite order, while a class has no order, but is capable of many +orders (unless it contains only one term or none). The generating +relation determines the order, and also the class of terms ordered, +since this class is the field of the generating relation. Hence the +generating relation completely determines the series, and may, for all +mathematical purposes, be taken to be the series.</p> + +<p>When \(P\) is transitive, we have +\[ +P_{\text{po}}=P.P_{*}=P\unicode{x228d} I\upharpoonright CʻP. +\] +Hence all the propositions of Part II, Section E become greatly +simplified when applied to series.</p> + +<p>Also, since the field of a connected relation consists of a single +family, a series has one first term or none, and one last term or none.</p> + +<p>In the case of a serial relation \(P\), the relation \(P_{1}\) (defined +in <a href="#*121·02">*121·02</a>) becomes \(P\dot{-} P^{2}\), <i>i.e.</i> the relation +"immediately preceding." In a <i>discrete</i> series, the terms in +general immediately precede other terms. A <i>compact</i> series, on +the contrary, is defined as one in which there are terms between any +two: in such a series, \(P_{1}=\dot{\Lambda}\).</p> + +<p>It very frequently occurs that we wish to consider the relations of +various series which are all contained in some one series; for example, +we may wish to consider various series of real numbers, all arranged +in order of magnitude. In such a case, if \(P\) is the series in which +all the others are contained, and \(\alpha\), \(\beta\), \(\gamma\), +... are the fields of the contained series, the contained series +themselves are \(P\unicode{x0294f}\alpha\), \(P\unicode{x0294f}\beta\), +\(P\unicode{x0294f}\gamma\), .... Thus when series are given as +contained in a given series, they are completely determined by their +fields.</p> + +<p>In what follows, Section A deals with the elementary properties of +series, including maximum and minimum points, sequent points and limits.</p> + +<p><span class="pagenum" id="Page_515">[Pg 515]</span></p> + +<p>Section B will deal with the theory of segments and kindred topics; +in this section we shall define "Dedekindian" series, and shall prove +the important proposition that the series of segments of a series is +always Dedekindian, <i>i.e.</i> that every class of segments has either +a maximum or a limit.</p> + +<p>Section C, which stands outside the main developments of the book, +is concerned with convergence and the limits of functions and the +definition of a continuous function. Its purpose is to show how these +notions can be expressed, and many of their properties established, in +a much more general way than is usually done, and without assuming that +the arguments or values of the functions concerned are either numerical +or numerically measurable.</p> + +<p>Section D will deal with "well-ordered" series, <i>i.e.</i> series in +which every class containing members of the field has a first term. The +properties of well-ordered series are many and important; most of them +depend upon the fact that an extended variety of mathematical induction +is possible in dealing with well-ordered series. The term "ordinal +number" is confined by usage to the relation-number of a well-ordered +series; ordinal numbers will also be considered in our fourth section.</p> + +<p>Section E will deal with finite and infinite. We shall show that the +distinction between "inductive" and "non-reflexive" does not arise in +well-ordered series.</p> + +<p>Section F will deal with "compact" series, <i>i.e.</i> series in which +there is a term between any two, <i>i.e.</i> in which \(P^{2}=P\). In +particular we shall consider "rational" series (<i>i.e.</i> series +like the series of rationals in order of magnitude) and continuous +series (<i>i.e.</i> series like the series of real numbers in order of +magnitude). Our treatment of this subject will follow Cantor closely.</p> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_516">[Pg 516]</span></p> +<h2 class="nobreak" id="SECTION_A_III">SECTION A.<br> +<span class="allsmcap">GENERAL THEORY OF SERIES.</span></h2> +</div> + + +<p class="nind"><i>Summary of Section A.</i></p> + +<p>In the present section, we shall be concerned with the properties +common to all series. Such properties, for the most part, are very +simple, and present no difficulties of any kind. Many of the properties +of series do not require all the three characteristics by which serial +relations are defined, but only one or two of these properties: we +therefore begin with numbers in which, though the properties proved +derive their chief importance from their applicability to series, the +hypotheses are only that the relations in question have one or two +of the properties of serial relations. Thence we proceed to the most +elementary properties peculiar to series, and thence to the theory +of minimum and maximum members of classes contained in a series, +and of the successors and limits of classes. We then proceed to the +correlation of a series with part of itself. The ground covered is +familiar, and the difficulties encountered are less than in most +previous sections.</p> + +<p>It will be observed that where series are concerned, if +\(\alpha\) is an existent class contained in \(CʻP\), +\(pʻ\overleftarrow{P}ʻʻ\alpha\) is correlative to \(Pʻʻ\alpha\) (which +is \(sʻ\overrightarrow{P}ʻʻ\alpha\)): \(Pʻʻ\alpha\) is "predecessors of +some \(\alpha\)," and \(pʻ\overleftarrow{P}ʻʻ\alpha\) is "successors +of all \(\alpha\)'s." If \(\alpha\) is an existent class contained in +\(CʻP\), the whole of \(CʻP\), with the exception of the last term +of \(\alpha\) (if there is such a term), belongs to one or other of +the classes \(Pʻʻ\alpha\), \(pʻ\overleftarrow{P}ʻʻ\alpha\), of which +the first wholly precedes the second. The division of \(CʻP\) into +these two classes is the Dedekind "cut" defined by \(\alpha\). But +when only part of \(\alpha\) is contained in \(CʻP\), we must replace +\(pʻ\overleftarrow{P}ʻʻ\alpha\) by \(pʻ\overleftarrow{P}ʻʻ(\alpha\cap CʻP)\), +since \(pʻ\overleftarrow{P}ʻʻ\alpha =\Lambda\) if +\(\alpha\) has any member not belonging to \(CʻP\). Again, if +\(\alpha \cap CʻP=\Lambda\), we have \(pʻ\overleftarrow{P}ʻʻ(\alpha\cap CʻP)=\text{V}\). +But what we want is the complement to +\(Pʻʻ\alpha\), which in this case is null. Hence we must +replace \(pʻ\overleftarrow{P}ʻʻ(\alpha \cap CʻP)\) by +\(CʻP\cap pʻ\overleftarrow{P}ʻʻ(\alpha \cap CʻP)\): this is \(CʻP\) when +\(Pʻʻ\alpha =\Lambda\), <i>i.e.</i> when \(\alpha \cap CʻP=\Lambda\). +In any other event it is equal to \(pʻ\overleftarrow{P}ʻʻ(\alpha\cap CʻP)\). +If \(\alpha\)<span class="pagenum" id="Page_517">[Pg 517]</span> is contained in \(CʻP\) and is not null, +\(CʻP\cap pʻ\overleftarrow{P}ʻʻ(\alpha \cap CʻP)=pʻ\overleftarrow{P}ʻʻ\alpha\). +Thus the Dedekind "cut" defined by a class \(\alpha\), whether or not +this class is contained in whole or part in \(CʻP\), is always the two +classes +\[ +Pʻʻ\alpha , CʻP\cap pʻ\overleftarrow{P}ʻʻ(\alpha \cap CʻP). +\]</p> + +<p>Throughout the elementary propositions of this section, we have +been careful to avoid stronger hypotheses than are required: we +have not assumed \(P\) to be serial, if our conclusion would follow +(<i>e.g.</i>) from the hypothesis that \(P\) is transitive and +connected. It will be found that many properties of series depend +upon the fact that, if \(x\), \(y\) are two different terms of a +series \(P\), then \(xPy.\equiv .{\sim}(yPx)\) (<a href="#*204·3">*204·3</a>). Here the +implication \(xPy.\supset .{\sim}(yPx)\) requires that \(P\) should be +asymmetrical, <i>i.e.</i> that we should have \(P\dot{\cap} \breve{P}=\dot{\Lambda}\), +or \(P^{2}\,\unicode{x2abd}\, J\). The implication +\({\sim}(yPx).\supset .xPy\) requires that \(P\) should be connected. +Thus the hypothesis required is not that \(P\) should be serial, but +that \(P\) should be connected and asymmetrical (<a href="#*202·5">*202·5</a>).</p> + +<p>Again, consider the proposition that if \(P\) is a series, +\(P_{1}=P\dot{-} P^{2}\). This relation \(P_{1}\) is the very useful +relation "immediately preceding"; thus the above proposition is +important, as is the further proposition that if \(P\) is a series, +\(P_{1}\) is a one-one relation. It will be remembered that (by +<a href="#*121">*121</a>) "\(xP_{1}y\)" means that \(P(x\vdash\dashv y)\) consists of +two terms. It was shown in <a href="#*121·304">*121·304</a><a href="#*121·305">·305</a> that if \(P_{\text{po}}\) +is contained in diversity, "\(xP_{1}y\)" implies "\(xPy\)" and is +equivalent to the statement that \(x\) and \(y\) constitute the +whole interval \(P(x\vdash\dashv y)\) and are not identical. Also +by <a href="#*121·254">*121·254</a>, \(P_{1}=(P_{\text{po}})_{1}\). It is evident that, if +\(P_{\text{po}}\) is contained in diversity, and \(xP_{1}y\), we +cannot have \(xP^{2}y\), because there is no term other than \(x\) +and \(y\) in the interval \(P(x\vdash\dashv y)\), and we cannot have +\(xPx\) or \(yPy\). Hence if \(P_{\text{po}}\,\unicode{x2abd}\, J\), we +have \(P_{1}\,\unicode{x2abd}\, \dot{-} P^{2}\). Hence by what was said +above (<a href="#*121·305">*121·305</a>), if \(P_{\text{po}}\,\unicode{x2abd}\, J\), we shall have +\(P_{1}\,\unicode{x2abd}\, P\dot{-} P^{2}\). On the other hand, if \(P\) is +transitive, we have \(P\dot{-} P^{2}\,\unicode{x2abd}\, P_{1}\) (<a href="#*201·61">*201·61</a>). +Combining these two facts, and remembering that if \(P\) is transitive, +\(P=P_{\text{po}}\) (<a href="#*201·18">*201·18</a>), we find that \(P_{1}=P\dot{-} P^{2}\) +if \(P\) is transitive and contained in diversity. We find further +(<a href="#*202·7">*202·7</a>) that if \(P\) is connected, \(P\dot{-} P^{2}\) is one-one. Hence +we need the full hypothesis that \(P\) is a series in order to prove +that \(P_{1}\) is a one-one (<a href="#*204·7">*204·7</a>). This is a good example of the +way in which the various separate characteristics that make up the +definition of series are relevant in proving the properties of series.</p> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_518">[Pg 518]</span></p> +<h2 class="nobreak" id="*200"> +*200. RELATIONS CONTAINED IN DIVERSITY.</h2> +</div> + + +<p class="nind"><i>Summary of</i> *200.</p> + +<p>Some of the propositions of this number are repetitions or +immediate consequences of previous propositions, especially those +of the propositions of *50 which deal with diversity. But we are +chiefly concerned here with propositions which will be useful in +the theory of series; this leads us to introduce propositions +on \(pʻ\overleftarrow{P}ʻʻ\alpha\) and on matters connected +with relation-arithmetic and other topics. It will be seen that +"\(P^{2}\,\unicode{x2abd}\, J\)" (<i>i.e.</i> "\(P\) is asymmetrical") is an +important hypothesis, as is also \(P_{\text{po}}\,\unicode{x2abd}\, J\), of +the use of which we have already had examples in *96 and <a href="#*121">*121</a>.</p> + +<p>The following are among the most useful propositions in this number:</p> + +<p class="nind"><b>*200·12.</b> \(\vdash :P\in \text{Rl}ʻJ.\supset .CʻP{\sim}\in 1\)</p> + +<p>This is the proposition which makes it impossible to define an ordinal +number 1 which shall take its place among relation-numbers applicable +to series.</p> + +<p class="nind"><b>*200·35.</b> \(\vdash :P\,\unicode{x2abd}\, J.\alpha \in 1.\supset .P\unicode{x0294f}\alpha =\dot{\Lambda}\)</p> + +<p>This is a consequence of <a href="#*200·12">*200·12</a>.</p> + +<p class="nind"><b>*200·36.</b> \(\vdash :P^{2}\,\unicode{x2abd}\, J.\supset .P\,\unicode{x2abd}\, J\)</p> + +<p class="nind"><b>*200·361.</b> \(\vdash :P^{2}\,\unicode{x2abd}\, + J.\supset .\overrightarrow{P}ʻx\cap ({℩}ʻx\cup \overleftarrow{P}ʻx)=\Lambda .\overleftarrow{P}ʻx\cap (\overrightarrow{P}ʻx\cup {℩}ʻx)=\Lambda\)</p> + +<p><i>I.e.</i> if \(P^{2}\,\unicode{x2abd}\, J\), no term precedes itself or +any of its predecessors, and no term succeeds itself or any of its +successors.</p> + +<p class="nind"><b>*200·38.</b> \(\vdash :P_{\text{po}}\,\unicode{x2abd}\, J.\supset .P_{\text{po}}=P_{*}\dot{\cap} J\)</p> + +<p class="nind"><b>*200·39.</b> \(\vdash :P_{\text{po}}\,\unicode{x2abd}\, J.x\in CʻP.\supset .\overrightarrow{P}_{*}ʻx\cap \overleftarrow{P}_{*}ʻx={℩}ʻx\)</p> + +<p>We then have a collection of propositions concerned with +relation-arithmetic.</p> + +<p class="nind"><b>*200·211.</b> \(\vdash :P\,\unicode{x2abd}\, J.P\,\,\text{smor}\,\,Q.\supset .Q\,\unicode{x2abd}\, J\)</p> + +<p><i>I.e.</i> the property of being contained in diversity is invariant +for likeness-transformations;</p> + +<p class="nind"><b>*200·4.</b> \(\vdash :P\unicode{x2909}Q\in \text{Rl}ʻJ.\equiv .P,Q\in \text{Rl}ʻJ.CʻP\cap CʻQ=\Lambda\)</p> + +<p><span class="pagenum" id="Page_519">[Pg 519]</span></p> + +<p class="nind"><b>*200·41.</b> \(\vdash : P \unicode{x21f8} x \,\unicode{x2abd}\, J . \equiv . x \unicode{x21f7} P \,\unicode{x2abd}\, J . \equiv . P \,\unicode{x2abd}\, J . x {\sim} \in CʻP\)</p> + +<p class="nind">and other such propositions.</p> + +<p>We then have a set of propositions concerned with +\(pʻ\overrightarrow{P}ʻʻ\alpha\) and \(pʻ\overleftarrow{P}ʻʻ\alpha\). +The most important are</p> + +<p class="nind"><b>*200·5.</b> \(\vdash : P \,\unicode{x2abd}\, J . \supset . \alpha \cap pʻ\overrightarrow{P}ʻʻ\alpha = \Lambda . \alpha \cap pʻ\overleftarrow{P}ʻʻ\alpha = \Lambda\)</p> + +<p class="nind"><b>*200·52.</b> \(\vdash : P \,\unicode{x2abd}\, J . \supset . CʻP {\sim} \in \overrightarrow{P}ʻʻCʻP\)</p> + +<p class="nind"><b>*200·53.</b> \(\vdash : P^{2} + \,\unicode{x2abd}\, J . \supset . Pʻʻ\alpha \cap Pʻ\overleftarrow{P}ʻʻ\alpha = \Lambda . \breve{P} ʻʻ\alpha \cap pʻ\overrightarrow{P}ʻʻ\alpha = \Lambda\)</p> + +<p><i>I.e.</i> if \(P\) is asymmetrical, the terms which precede part of +\(\alpha\) do not succeed the whole of \(\alpha\), and vice versa.</p> + +<hr class="tb"> + +<p class="nind"><b>*200·11.</b> \(\vdash : P \in \text{Rl}ʻJ . \equiv . \breve{P} \in \text{Rl}ʻJ \quad[*50·23]\)</p> + +<p class="nind"><b><a id="*200·12">*200·12</a>.</b> \(\vdash : P \in \text{Rl}ʻJ . \supset . CʻP {\sim} \in 1\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*50·11.*33·17. &\supset \vdash \colon\ldotp \text{Hp} : xPy . \lor . yPx : \supset . y \neq x . y \in CʻP &\qquad \text{(1)}\\ +\vdash .(1).*33·132. \supset \vdash \colon\ldotp \text{Hp} . &\supset : x \in CʻP . \supset . (\exists y) . y \neq x . y \in CʻP :\\ +[*52·181] &\supset : CʻP {\sim} \in 1 \colon\ldotp \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*200·2.</b> \(\vdash : T \in 1 \rightarrow 1 . \supset . T^{;}(P \dot{\cap} J) = T^{;}P \dot{\cap} J\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*150·4. &\supset \vdash \colon\ldotp \text{Hp} . \supset :\\ +x \{T^{;}(P \dot{\cap} J)\} y . &\equiv . (\exists z, w) . x = Tʻz . y = Tʻw . zPw . z \neq w .\\ +[*71·56] &\equiv . (\exists z, w) . x \neq y . x = Tʻz . y = Tʻw . zPw .\\ +[*150·4] &\equiv . x \{T^{;}P \dot{\cap} J\} y \colon\ldotp \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*200·21.</b> \(\vdash : T \in \text{Cls} \rightarrow 1. P \,\unicode{x2abd}\,J. \supset . T^{;}P \,\unicode{x2abd}\, J\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*150·1.*50·24. \supset \vdash \colon\ldotp \text{Hp} . &\supset : x (T^{;}P) y . \supset . (\exists z, w) . xTz . yTw . z \neq w .\\ +[*71·171 . \text{Transp}] &\supset . x \neq y \colon\ldotp \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*200·211">*200·211</a>.</b> \(\vdash : P \,\unicode{x2abd}\, J . P \,\,\text{smor}\, \,Q . \supset . Q \,\unicode{x2abd}\, J \quad[*200·21 . *151·1]\)</p> + +<p>The properties of relations are very frequently common to all +relations which are like a given relation, and this applies specially +to the kinds of properties with which we are most concerned. The +above proposition is an illustration of this fact: it shows that +the property of being contained in diversity is invariant for +likeness-transformations.</p> + +<p class="nind"><b>*200·22.</b> \(\vdash : P \,\unicode{x2abd}\, J . \equiv . \text{N}_{0}\text{r}ʻP \subset \text{Rl}ʻJ . \equiv . \exists ! \text{N}_{0}\text{r}ʻP \cap \text{Rl}ʻJ\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*155·11.*200·211. &\supset \vdash : P \,\unicode{x2abd}\, J . \supset . \text{N}_{0}\text{r}ʻP \subset \text{Rl}ʻJ &\qquad \text{(1)}\\ +\vdash .*155·12. &\supset \vdash : \text{N}_{0}\text{r}ʻP \subset \text{Rl}ʻJ . \supset . P \,\unicode{x2abd}\, J &\qquad \text{(2)}\\ +\vdash .*155·12. &\supset \vdash : P \,\unicode{x2abd}\, J . \supset . \exists ! \text{N}_{0}\text{r}ʻP \cap \text{Rl}ʻJ &\qquad \text{(3)}\\ +\vdash .*155·11.*200·211. &\supset \vdash : \exists ! \text{N}_{0}\text{r}ʻP \cap \text{Rl}ʻJ . \supset . P \,\unicode{x2abd}\, J &\qquad \text{(4)}\\ +\vdash .(1).(2).(3).(4).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_520">[Pg 520]</span></p> + +<p>We have, without the need of typical definiteness, +\[ +\begin{aligned} +&\vdash :P\,\unicode{x2abd}\, J.\supset .\text{Nr}ʻP\subset \text{Rl}ʻJ\\ +\text{and}\quad &\vdash :\exists !\text{Nr}ʻP\cap \text{Rl}ʻJ.\supset .P\,\unicode{x2abd}\, J, +\end{aligned} +\] +both of which are immediate consequences of <a href="#*200·211">*200·211</a>. The converse +implications, however, fail if \(\text{Nr}ʻP\) is taken in a type in +which \(\text{Nr}ʻP=\Lambda\).</p> + +<p class="nind"><b>*200·3.</b> \(\vdash .\dot{\Lambda} \in \text{Rl}ʻJ \quad[*25·12]\)</p> + +<p class="nind"><b>*200·31.</b> \(\vdash :x\neq y.\equiv .x\downarrow y\in \text{Rl}ʻJ \quad[*55·3]\)</p> + +<p class="nind"><b>*200·32.</b> \(\vdash :\alpha \uparrow \beta \,\unicode{x2abd}\, J.\equiv .\alpha \cap \beta =\Lambda \quad[*50·55]\)</p> + +<p class="nind"><b>*200·33.</b> \(\vdash :P\,\unicode{x2abd}\, J.\supset .P\unicode{x0294f}\alpha \,\unicode{x2abd}\, J \quad[*35·442]\)</p> + +<p class="nind"><b>*200·34.</b> \(\vdash :P\unicode{x0294f}\alpha \,\unicode{x2abd}\, J.\equiv .P\upharpoonright \alpha \,\unicode{x2abd}\, J.\equiv .\alpha \upharpoonleft P\,\unicode{x2abd}\, J \quad[*50·58]\)</p> + +<p class="nind"><b>*200·35.</b> \(\vdash :P\,\unicode{x2abd}\, J.\alpha \in 1.\supset .P\unicode{x0294f}\alpha =\dot{\Lambda}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*52·16.\supset \vdash \colon\ldotp \text{Hp}.\supset :x,y\in \alpha .&\supset _{x,y}.{\sim}(xJy).\\ +[*23·81] &\supset _{x,y}.{\sim}(xPy):\\ +[*11·521] &\supset :(x,y).{\sim}\{x,y\in \alpha .xPy\}\colon\ldotp \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*200·36.</b> \(\vdash :P^{2}\,\unicode{x2abd}\, J.\supset .P\,\unicode{x2abd}\, J \quad[*50·45]\)</p> + +<p class="nind"><b>*200·361.</b> \(\vdash :P^{2}\,\unicode{x2abd}\, + J.\supset .\overrightarrow{P}ʻx\cap ({℩}ʻx\cup \overleftarrow{P}ʻx)=\Lambda .\overleftarrow{P}ʻx\cap (\overrightarrow{P}ʻx\cup {℩}ʻx)=\Lambda\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*51·15. &\supset \vdash :y\in \overrightarrow{P}ʻx\cap {℩}ʻx.\supset .xPx &\qquad \text{(1)}\\ +\vdash .*200·36. \supset \vdash :\text{Hp}.&\supset .{\sim}(xPx).\\ +[(1).\text{Transp}] &\supset .\overrightarrow{P}ʻx\cap {℩}ʻx=\Lambda &\qquad \text{(2)}\\ +\vdash .*34·11. &\supset \vdash :\exists !\overrightarrow{P}ʻx\cap \overleftarrow{P}ʻx.\equiv .xP^{2}x &\qquad \text{(3)}\\ +\vdash .(3).\text{Transp}. &\supset \vdash :\text{Hp}.\supset .\overrightarrow{P}ʻx\cap \overleftarrow{P}ʻx=\Lambda &\qquad \text{(4)}\\ +\vdash .(2).(4). &\supset \vdash :\text{Hp}.\supset .\overrightarrow{P}ʻx\cap ({℩}ʻx\cup \overleftarrow{P}ʻx)=\Lambda &\qquad \text{(5)}\\ +\text{Similarly}\quad &\vdash :\text{Hp}.\supset .\overleftarrow{P}ʻx\cap (\overrightarrow{P}ʻx\cup {℩}ʻx)=\Lambda &\qquad \text{(6)}\\ +\vdash .(5).(6).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*200·37">*200·37</a>.</b> \(\vdash :\exists !\text{Pot}ʻP\cap \text{Rl}ʻJ.\supset .P\,\unicode{x2abd}\, J\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*91·373 \frac{xSx}{\phi S}.\supset \\ +\vdash \colon\colon xPx:S\in \text{Pot}ʻP.xSx.&\supset _{s}.x(S\mid P)x:\supset :Q\in \text{Pot}ʻP.\supset _{Q}.xQx &\qquad \text{(1)}\\ +\vdash .*3·2. \supset \vdash \colon\ldotp xPx.&\supset :xSx.\supset .xSx.xPx.\\ +[*34·1] & \supset .x(S\mid P)x &\qquad \text{(2)}\\ +\vdash .(1).(2). \supset \vdash \colon\ldotp xPx.&\supset :Q\in \text{Pot}ʻP.\supset _{Q}.xQx.\\ +[*50·24] &\supset _{Q}.{\sim}(Q\,\unicode{x2abd}\, J) &\qquad \text{(3)}\\ +\vdash .(3).\text{Transp}.\supset \vdash :(\exists Q).Q\in \text{Pot}ʻP.Q\,\unicode{x2abd}\, J.&\supset .{\sim}(xPx).\\ +[*50·24] & \supset .P\,\unicode{x2abd}\, J:\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_521">[Pg 521]</span></p> + +<p class="nind"><b>*200·38.</b> \(\vdash :P_{\text{po}}\,\unicode{x2abd}\, J.\supset .P_{\text{po}}=P_{*}\dot{\cap} J \quad[*91·541]\)</p> + +<p class="nind"><b>*200·381.</b> \(\vdash :P_{\text{po}}\,\unicode{x2abd}\, J.\supset .\overrightarrow{P}_{\text{po}}ʻx\cap \overleftarrow{P}_{*}ʻx=\Lambda .\overleftarrow{P}_{\text{po}}ʻx\cap + \overrightarrow{P}_{*}=\Lambda\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*91·56.\supset \vdash :\text{Hp}.&\supset .P_{\text{po}}^{2}\,\unicode{x2abd}\, J.\\ +[*200·361] & \supset .\overrightarrow{P}_{\text{po}}ʻx\cap (\iota ʻx\cup \overleftarrow{P}_{\text{po}}ʻx)=\Lambda .\overleftarrow{P}_{\text{po}}ʻx\cap + (\overrightarrow{P}_{\text{po}}ʻx\cup \iota ʻx)=\Lambda .\\ +[*91·54] &\supset .\overrightarrow{P}_{\text{po}}ʻx\cap \overleftarrow{P}_{*}ʻx=\Lambda .\overleftarrow{P}_{\text{po}}ʻx\cap \overrightarrow{P}_{*}ʻx=\Lambda + :\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*200·39.</b> \(\vdash :P_{\text{po}}\,\unicode{x2abd}\, J.x\in CʻP.\supset .\overrightarrow{P}_{*}ʻx\cap \overleftarrow{P}_{*}ʻx=\iota ʻx\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *91·54. \supset \vdash :\text{Hp}.\supset .\overrightarrow{P}_{*}ʻx\cap \overleftarrow{P}_{*}ʻx&=(\overrightarrow{P}_{\text{po}}ʻx\cup + \iota ʻx)\cap (\overleftarrow{P}_{\text{po}}ʻx\cup \iota ʻx)\\ +[*22·69] & =(\overrightarrow{P}_{\text{po}}ʻx\cap \overleftarrow{P}_{\text{po}}ʻx)\cup \iota ʻx &\qquad \text{(1)}\\ +\vdash .*91·56. & \supset \vdash :y\in \overrightarrow{P}_{\text{po}}ʻx\cap \overleftarrow{P}_{\text{po}}ʻx.\supset .yP_{\text{po}}y &\qquad \text{(2)}\\ +\vdash .(2).\text{Transp}.&\supset \vdash :\text{Hp}.\supset .\overrightarrow{P}_{\text{po}}ʻx\cap \overleftarrow{P}_{\text{po}}ʻx=\Lambda &\qquad \text{(3)}\\ +\vdash .(1).(3).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*200·391.</b> \(\vdash :P_{\text{po}}\,\unicode{x2abd}\, J.\supset .\overrightarrow{P}_{*}^{;}P\,\,\text{smor}\,\,P.\overrightarrow{P}_{*}\upharpoonright + CʻP\in (\overrightarrow{P}_{*}^{;}P)\,\overline{\,\text{smor}\,}\,P\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*90·12.\supset \vdash :\text{Hp}.x,y\in CʻP.\overrightarrow{P}_[*]ʻx=\overrightarrow{P}_{*}ʻy.&\supset .xP_{*}y.yP_{*}x.\\ +[*200·39] &\supset .x=y &\qquad \text{(1)}\\ +\vdash .(1).*151·24.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p>The above proposition is useful in the theory of segments.</p> + +<p>The following propositions are concerned with the ideas of +relation-arithmetic. Analogous propositions will be proved for +transitiveness and connection in <a href="#*201">*201</a> and <a href="#*202">*202</a>, whence analogous +propositions concerning series will be deduced in <a href="#*204">*204</a>.</p> + +<p class="nind"><b><a id="*200·4">*200·4</a>.</b> \(\vdash :P\unicode{x2909}Q\in \text{Rl}ʻJ.\equiv .P,Q\in \text{Rl}ʻJ.CʻP\cap CʻQ=\Lambda\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*23·59.*160·1.\supset \\ +\vdash :P\unicode{x2909}Q\in \text{Rl}ʻJ.&\equiv .P,Q\in \text{Rl}ʻJ.CʻP\uparrow CʻQ\,\unicode{x2abd}\, J.\\ +[*200·32] &\equiv .P,Q\in \text{Rl}ʻJ.CʻP\cap CʻQ=\Lambda :\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p>This proposition is part of the proof that the sum of two mutually +exclusive series is a series.</p> + +<p class="nind"><b>*200·41.</b> \(\vdash :P\unicode{x21f8} x\,\unicode{x2abd}\, J.\equiv .x\unicode{x21f7}P\,\unicode{x2abd}\, J.\equiv .P\,\unicode{x2abd}\, J.x{\sim}\in CʻP \quad[*23·59.*200·32]\)</p> + +<p class="nind"><b><a id="*200·42">*200·42</a>.</b> \(\vdash :\Sigma ʻP\,\unicode{x2abd}\, J.\equiv .CʻP\subset \text{Rl}ʻJ.F^{;}P\,\unicode{x2abd}\, J\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*23·59.*162·1.\supset \vdash :\Sigma ʻP\,\unicode{x2abd}\, J.&\equiv .\dot{s} ʻCʻP\,\unicode{x2abd}\, J.F^{;}P\,\unicode{x2abd}\, J.\\ +[*61·52] &\equiv .CʻP\subset \text{Rl}ʻJ.F^{;}P\,\unicode{x2abd}\, J:\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_522">[Pg 522]</span></p> + +<p>The following propositions (<a href="#*200·421">*200·421</a><a href="#*200·422">·422</a><a href="#*200·423">·423</a>) are lemmas for <a href="#*204·53">*204·53</a>.</p> + +<p class="nind"><b><a id="*200·421">*200·421</a>.</b> \(\vdash :P\in \text{Rel}^{2}\text{excl}.P\,\unicode{x2abd}\, J.Q\in CʻP.\supset .Q=(\Sigma ʻP)\unicode{x0294f}CʻQ\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*163·11.*162·13.\supset \\ +\vdash \colon\colon\text{Hp}.\supset \colon\ldotp &x\{(\Sigma ʻP)\unicode{x0294f}CʻQ\}y.\equiv :\\ +&(\exists R).R\in CʻP.x,y\in CʻQ.xRy.R=Q.\lor.\\ +&(\exists R,S).RPS.x,y\in CʻQ.x\in CʻR.y\in CʻS.R=Q.S=Q:\\ +[*13·195·22] &\equiv :xQy.\lor.QPQ.x,y\in CʻQ:\\ +[*50·24.\text{Hp}] &\equiv :xQy\colon\colon\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*200·422">*200·422</a>.</b> \(\vdash :\Sigma ʻP\,\unicode{x2abd}\, J.\supset .P\unicode{x0294f}(-\iota ʻ\dot{\Lambda} )\,\unicode{x2abd}\, J\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*162·13.*50·24.\supset \vdash \colon\colon\text{Hp}.&\supset \colon\ldotp QPR.\supset :x\in CʻQ.y\in CʻR.\supset .x\neq y:\\ +[*24·37] & \supset :CʻQ\cap CʻR=\Lambda :\\ +[*24·57] &\supset :\dot{\exists} !Q.\supset .CʻQ\neq CʻR.\\ +[*30·37] &\supset .Q\neq R\colon\colon\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*200·423">*200·423</a>.</b> \(\vdash \colon\ldotp P\in \text{Rel}^{2}\text{excl}.\Lambda {\sim}\in CʻP.\supset :\Sigma ʻP\,\unicode{x2abd}\, J.\equiv .P\,\unicode{x2abd}\, J.CʻP\subset \text{Rl}ʻJ\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*200·422·42. &\supset \vdash :\text{Hp}.\Sigma ʻP\,\unicode{x2abd}\, J.\supset .P\,\unicode{x2abd}\, J.CʻP\subset \text{Rl}ʻJ &\qquad \text{(1)}\\ +\vdash .*61·52. &\supset \vdash :CʻP\subset \text{Rl}ʻJ.\supset .\dot{s} ʻCʻP\,\unicode{x2abd}\, J &\qquad \text{(2)}\\ +\vdash .*163·12.*200·21. &\supset \vdash :\text{Hp}.P\,\unicode{x2abd}\, J.\supset .F^{;}P\,\unicode{x2abd}\, J &\qquad \text{(3)}\\ +\vdash .(2).(3).*162·1. &\supset \vdash :\text{Hp}.P\,\unicode{x2abd}\, J.CʻP\subset \text{Rl}ʻJ.\supset .\Sigma ʻP\,\unicode{x2abd}\, J &\qquad \text{(4)}\\ +\vdash .(1).(4).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*200·43.</b>\[\begin{align}\vdash :&P\,\unicode{x2abd}\, J.\supset .\Pi ʻP=\\ +&\hat{M} \hat{N} \{M,N\in F_{\Delta }ʻCʻP:(\exists Q).(MʻQ)Q(NʻQ).M\upharpoonright \overrightarrow{P}ʻQ=N\upharpoonright \overrightarrow{P}ʻQ\}\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*4·71.*172·1.\supset \vdash :\text{Hp}.\supset .\\ +\Pi ʻP=\hat{M} \hat{N} \{M,N\in F_{\Delta }ʻCʻP\colon\ldotp (\exists Q):(MʻQ)Q(NʻQ):RPQ.\supset _{R}.MʻR=NʻR\}\\ +[*35·71.*71·35] +=\hat{M} \hat{N} \{M,N\in F_{\Delta }ʻCʻP:(\exists Q).(MʻQ)Q(NʻQ).M\upharpoonright \overrightarrow{P}ʻQ=N\upharpoonright \overrightarrow{P}ʻQ\}.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_523">[Pg 523]</span></p> + +<p>The following propositions, with the exception of <a href="#*200·52">*200·52</a>, +are concerned with \(pʻ\overrightarrow{P}ʻʻ\alpha\) and +\(pʻ\overleftarrow{P}ʻʻ\alpha\), <i>i.e.</i> the class of terms +preceding (or succeeding) the whole of \(\alpha\).</p> + +<p class="nind"><b>*200·5.</b> \(\vdash :P\,\unicode{x2abd}\, J.\supset .\alpha \cap pʻ\overrightarrow{P}ʻʻ\alpha =\Lambda .\alpha \cap pʻ\overleftarrow{P}ʻʻ\alpha =\Lambda\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*40·51.\supset \vdash \colon\ldotp x\in \alpha \cap pʻ\overrightarrow{P}ʻʻ\alpha .&\supset :x\in \alpha :y\in \alpha .\supset _{y}.xPy:\\ +[*10·26] &\supset :xPx:\\ +[*50·24] &\supset :{\sim}(P\,\unicode{x2abd}\, J) &\qquad \text{(1)}\\ +\vdash .(1).\text{Transp}.&\supset \vdash :\text{Hp}.\supset .\alpha \cap pʻ\overrightarrow{P}ʻʻ\alpha =\Lambda &\qquad \text{(2)}\\ +\text{Similarly}\quad &\vdash :\text{Hp}.\supset .\alpha \cap pʻ\overleftarrow{P}ʻʻ\alpha =\Lambda &\qquad \text{(3)}\\ +\vdash .(2).(3).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*200·51.</b> \(\vdash :P\,\unicode{x2abd}\, J.\dot{\exists} !P.\supset .pʻ\overrightarrow{P}ʻʻCʻP=\Lambda .pʻ\overleftarrow{P}ʻʻCʻP=\Lambda\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*40·62.\supset \vdash :\text{Hp}.&\supset .pʻ\overrightarrow{P}ʻʻCʻP\subset CʻP.\\ +[*22·621] \supset .pʻ\overrightarrow{P}ʻʻCʻP&=CʻP\cap pʻ\overrightarrow{P}ʻʻCʻP\\ +[*200·5] & =\Lambda &\qquad \text{(1)}\\ +\text{Similarly}\quad &\vdash :\text{Hp}.\supset .pʻ\overleftarrow{P}ʻʻCʻP=\Lambda &\qquad \text{(2)}\\ +\vdash .(1).(2).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*200·52">*200·52</a>.</b> \(\vdash :P\,\unicode{x2abd}\, J.\supset .CʻP{\sim}\in \overrightarrow{P}ʻʻCʻP\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*50·24.\supset \vdash \colon\ldotp \text{Hp}.\supset :x\in CʻP.&\supset _{x}.x{\sim}\in \overrightarrow{P}ʻx.\\ +[*13·14] &\supset _{x}.CʻP\neq \overrightarrow{P}ʻx:\\ +[*37·7.\text{Transp}] &\supset :CʻP{\sim}\in \overrightarrow{P}ʻʻCʻP\colon\ldotp \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p>This proposition is often used in the theory of well-ordered series.</p> + +<p class="nind"><b>*200·53.</b> \(\vdash :P^{2}\,\unicode{x2abd}\, J.\supset .Pʻʻ\alpha \cap pʻ\overleftarrow{P}ʻʻ\alpha =\Lambda .\breve{P} ʻʻ\alpha \cap pʻ\overrightarrow{P}ʻʻ\alpha =\Lambda\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*37·1.*40·53.\supset \vdash \colon\ldotp x\in Pʻʻ\alpha \cap pʻ\overleftarrow{P}ʻʻ\alpha .&\supset :(\exists y).y\in \alpha .xPy:y\in \alpha .\supset _{y}.yPx:\\ +[*10·56] &\supset :(\exists y).xPy.yPx:\\ +[*34·5] &\supset :xP^{2}x:\\ +[*50·24] &\supset :{\sim}(P^{2}\,\unicode{x2abd}\, J) &\qquad \text{(1)}\\ +\vdash .(1).\text{Transp}.&\supset \vdash :\text{Hp}.\supset .(x).x{\sim}\in Pʻʻ\alpha \cap pʻ\overleftarrow{P}ʻʻ\alpha &\qquad \text{(2)}\\ +\text{Similarly}\quad &\vdash :\text{Hp}.\supset .(x).x{\sim}\in \breve{P} ʻʻ\alpha \cap pʻ\overrightarrow{P}ʻʻ\alpha &\qquad \text{(3)}\\ +\vdash .(2).(3).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_524">[Pg 524]</span></p> + +<p>The above proposition is frequently used. If \(\alpha\) is +an existent class contained in \(CʻP\), \(Pʻʻ\alpha\) and +\(pʻ\overleftarrow{P}ʻʻ\alpha\) are the two parts of the Dedekind "cut" +determined by \(\alpha\) (excluding the maximum of \(\alpha\), if any). +The above proposition shows that these two parts are mutually exclusive.</p> + +<p class="nind"><b>*200·54.</b> \(\vdash :P\,\unicode{x2abd}\, J.\dot{\exists} !P.\supset .pʻ\overrightarrow{P}ʻʻ{CʻP\cap pʻ\overleftarrow{P}ʻʻ\alpha }=pʻ\overrightarrow{P}ʻʻpʻ\overleftarrow{P}ʻʻ\alpha\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*40·62. &\supset \vdash :\exists !\alpha.\supset .CʻP\cap pʻ\overleftarrow{P}ʻʻ\alpha =pʻ\overleftarrow{P}ʻʻ\alpha &\qquad \text{(1)}\\ +\vdash .*40·2. \supset \vdash :\alpha=\Lambda .&\supset .pʻ\overleftarrow{P}ʻʻ\alpha=\text{V}. &\qquad \text{(2)}\\ +[*40·16] &\supset .pʻ\overrightarrow{P}ʻʻpʻ\overleftarrow{P}ʻʻ\alpha \subset pʻ\overrightarrow{P}ʻʻCʻP &\qquad \text{(3)}\\ +\vdash .(3).*200·51.&\supset \vdash :\text{Hp}.\alpha =\Lambda .\supset .pʻ\overrightarrow{P}ʻʻpʻ\overleftarrow{P}ʻʻ\alpha =\Lambda &\qquad \text{(4)}\\ +\vdash .(2).*24·26. &\supset \vdash :\alpha =\Lambda .\supset .CʻP\cap pʻ\overleftarrow{P}ʻʻ\alpha =CʻP &\qquad \text{(5)}\\ +\vdash .(5).*200·51.&\supset \vdash :\text{Hp}.\alpha =\Lambda .\supset .pʻ\overrightarrow{P}ʻʻ(CʻP\cap pʻ\overleftarrow{P}ʻʻ\alpha )=\Lambda &\qquad \text{(6)}\\ +\vdash .(1).(4).(6).&\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p>This proposition is a lemma whose purpose is to avoid the necessity of +introducing the hypothesis \(\exists !\alpha\) or in proofs in which it +is not really necessary. The first use of this proposition occurs in +<a href="#*206·551">*206·551</a>.</p> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_525">[Pg 525]</span></p> +<h2 class="nobreak" id="*201"> +*201. TRANSITIVE RELATIONS.</h2> +</div> + + +<p class="nind"><i>Summary of</i> *201.</p> + +<p>There are two main varieties of transitive relations, namely +those that are symmetrical \((P=\breve{P})\), and those that are +asymmetrical \((P\dot{\cap} \breve{P} =\dot{\Lambda})\). Transitive +<i>symmetrical</i> relations have the formal properties of equality: +examples of such relations have occurred above, <i>e.g.</i> identity, +similarity, and likeness. The propositions of the present number, +however, are rather such as will be useful in connection with +transitive <i>asymmetrical</i> relations, since they are intended to be +applied to series.</p> + +<p>We denote the class of transitive relations by "\(\text{trans}\)"; thus +\[ +\text{trans}=\hat{P} (P^{2}\,\unicode{x2abd}\, P) \quad\text{Df}. +\] +Many propositions of this number are analogous to propositions whose +numbers have the same decimal part in <a href="#*200">*200</a>. Such are: If \(P\) is +transitive, so is its converse (<a href="#*201·11">*201·11</a>), and so is any relation +which is like P (<a href="#*201·211">*201·211</a>); \(\dot{\Lambda}\) and \(x\downarrow y\) +are transitive (<a href="#*201·3">*201·3</a><a href="#*201·31">·31</a>); if \(P\) is transitive, so is +\(P\unicode{x0294f}\alpha\) (<a href="#*201·33">*201·33</a>). The propositions <a href="#*201·4">*201·4</a>—<a href="#*201·42">·42</a>, +which deal with the ideas of relation-arithmetic, are also analogous to +<a href="#*200·4">*200·4</a>—<a href="#*200·42">·42</a>.</p> + +<p>Most of the other propositions of this number, however, have no +analogues in <a href="#*200">*200</a>. Among the most important of these are the following :</p> + +<p class="nind"><b>*201·14.</b> \(\vdash :P\in \text{trans}.xPy.\supset .\overrightarrow{P}ʻx\subset \overrightarrow{P}ʻy\)</p> + +<p class="nind"><b>*201·15.</b> \(\vdash .R_{*}\in \text{trans}\)</p> + +<p class="nind"><b>*201·18.</b> \(\vdash :P^{2}\,\unicode{x2abd}\, P.\supset .P_{\text{po}}=P.P_{*}=P\unicode{x228d} I\upharpoonright CʻP\)</p> + +<p>This proposition is very important, since it effects an immense +simplification in the use of all propositions involving +\(P_{\text{po}}\) or \(P_{*}\), when these propositions are to be +applied to transitive relations. Owing to the above proposition, +\(P_{\text{po}}\) drops out where transitive relations are concerned. +\(P_{*}\), on the other hand, remains useful: If \(y\in CʻP\), +"\(xP_{*}y\)" will mean "\(x\) precedes or is \(y\)," which, if \(P\) +generates a series of which \(x\) and \(y\) are members, is equivalent +to "\(x\) does not follow \(y\)."</p> + +<p><span class="pagenum" id="Page_526">[Pg 526]</span></p> + +<p>We have a series of propositions (<a href="#*201·5">*201·5</a>—<a href="#*201·5">·56</a>) on \(Pʻʻ\alpha\) and +\(pʻ\overrightarrow{P}ʻʻ\alpha\). The chief of these are</p> + +<p class="nind"><b>*201·5.</b> \(\vdash :P\in \text{trans}.\supset .PʻʻPʻʻ\alpha \subset Pʻʻ\alpha \)</p> + +<p class="nind"><b>*201·501.</b> \(\vdash :P\in \text{trans}.\supset .Pʻʻ\overrightarrow{P}ʻx\subset \overrightarrow{P}ʻx\)</p> + +<p>These two propositions express the fact that a predecessor of a +predecessor is a predecessor.</p> + +<p class="nind"><b>*201·52.</b> \(\vdash :P\in \text{trans}.\supset .P_{*}ʻʻ\alpha =Pʻʻ\alpha \cup (\alpha \cap CʻP)\)</p> + +<p>Thus if \(\alpha \subset CʻP,P_{*}ʻʻ\alpha\) consists of \(\alpha\) +together with the predecessors of its members.</p> + +<p class="nind"><b>*201·521.</b> \(\vdash :P\in \text{trans}.x\in CʻP.\supset .\overrightarrow{P}_{*}ʻx=\overrightarrow{P}ʻx\cup {℩}ʻx\)</p> + +<p class="nind"><b>*201·55.</b> \(\vdash :P\in \text{trans}.\supset .Pʻʻ(\alpha \cup Pʻʻ\alpha )=Pʻʻ\alpha\)</p> + +<p>We have next a set of important propositions on \(P\dot{-} P^{2}\) and +\(P_{1}\). The chief are</p> + +<p class="nind"><b>*201·63.</b> \(\vdash :P\in \text{trans}\cap \text{Rl}ʻJ.\supset .P_{1}=P\dot{-} P^{2}\)</p> + +<p class="nind"><b>*201·65.</b> \(\vdash \colon\ldotp P\in \text{trans}\cap \text{Rl}ʻJ.\supset :P_{1}=\dot{\Lambda} .\equiv .P^{2}=P\)</p> + +<p>On these two propositions, see the notes appended to them below.</p> + +<hr class="tb"> + +<p class="nind"><b>*201·01.</b> \(\text{trans}=\hat{P} (P^{2}\,\unicode{x2abd}\, P) \quad\text{Df}\)</p> + +<p class="nind"><b>*201·1.</b> \(\vdash :P\in \text{trans}.\equiv .P^{2}\,\unicode{x2abd}\, P \quad[(*201·01)]\)</p> + +<p class="nind"><b><a id="*201·11">*201·11</a>.</b> \(\vdash :P\in \text{trans}.\equiv .\breve{P} \in \text{trans}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*201·1.*31·4.\supset \vdash :P\in \text{trans}.&\equiv .\text{Cnv}ʻP^2\,\unicode{x2abd}\, \breve{P} .\\ +[*34·63.*201·1] &\equiv .\breve{P} \in \text{trans}:\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*201·12.</b> \(\vdash \colon\ldotp P\in \text{trans}.\supset :P\,\unicode{x2abd}\, J.\equiv .P^{2}\,\unicode{x2abd}\, J.\equiv .P\dot{\cap} \breve{P} =\dot{\Lambda} \quad[*50·47]\)</p> + +<p>In virtue of this proposition, being contained in diversity is +equivalent (where transitive relations are concerned) to asymmetry. +This is not in general the case with relations which are not +transitive; thus <i>e.g.</i> diversity itself is contained in +diversity, but is symmetrical.</p> + +<p class="nind"><b>*201·13.</b> \(\vdash .\text{Rl}ʻI\subset \text{trans}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*34·34.\supset \vdash :R\,\unicode{x2abd}\, I.&\supset .R^{2}\,\unicode{x2abd}\, R\mid I.\\ +[*50·4] &\supset .R^{2}\,\unicode{x2abd}\, R:\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*201·14.</b> \(\vdash :P\in \text{trans}.xPy.\supset .\overrightarrow{P}ʻx\subset \overrightarrow{P}ʻy\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*201·1.\supset \vdash :\text{Hp}.zPx.\supset .zPy &\qquad \text{(1)}\\ +\vdash .(1).*32·18.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_527">[Pg 527]</span></p> + +<p>The following propositions (<a href="#*201·15">*201·15</a>—<a href="#*201·19">·19</a>) are concerned with \(R_{*}\) +and \(R_{\text{po}}\).</p> + +<p class="nind"><b><a id="*201·15">*201·15</a>.</b> \(\vdash .R_{*}\in \text{trans} \quad[*90·17]\)</p> + +<p class="nind"><b>*201·16.</b> \(\vdash .R_{\text{po}}\in \text{trans} \quad[*91·56]\)</p> + +<p>This proposition is important, since it often happens that a series is +given as defined by a one-one relation \(R\), as in <a href="#*122">*122</a> for example, +and in such cases \(R_{\text{po}}\) is a serial relation in our +present sense. By the above proposition, \(R_{\text{po}}\) is always +transitive; by *96·421, \(R_{\text{po}}\) is connected when confined to +the posterity of a given term, provided \(R\in \text{Cls}\rightarrow 1\); +by *96·23, if \(R\in 1\rightarrow \text{Cls}\) and \(xBR\), +\(R_{\text{po}}\) is contained in diversity throughout the posterity of +\(x\). Thus if \(R\) is a one-one, \(R_{\text{po}}\) confined to any +family which has a beginning will be a serial relation.</p> + +<p class="nind"><b>*201·17.</b> \(\vdash :P^{2}\,\unicode{x2abd}\, P.Q\in \text{Pot}ʻP.\supset .Q\,\unicode{x2abd}\, P\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*34·34.\supset \vdash \colon\ldotp \text{Hp}.\supset :S\,\unicode{x2abd}\, P.\supset _{S}.S\mid P\,\unicode{x2abd}\, P &\qquad \text{(1)}\\ +\vdash .*91·171 \frac{S\,\unicode{x2abd}\, P}{\phi S}. \supset \\ +\vdash \colon\ldotp Q\in \text{Pot}ʻP:S\,\unicode{x2abd}\, P.\supset _{S}.S\mid P\,\unicode{x2abd}\, P:P\,\unicode{x2abd}\, P:\supset .Q\,\unicode{x2abd}\, P &\qquad \text{(2)}\\ +\vdash .(1).(2).*23·42.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*201·18">*201·18</a>.</b> \(\vdash :P^{2}\,\unicode{x2abd}\, P.\supset .P_{\text{po}}=P.P_{*}=P\unicode{x228d} I\upharpoonright CʻP\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*201·17.*41·151.(*91·05).\supset \vdash :\text{Hp}.&\supset .P_{\text{po}}\,\unicode{x2abd}\, P &\qquad \text{(1)}\\ +\vdash .(1).*91·502. & \supset \vdash :\text{Hp}.\supset .P_{\text{po}}=P &\qquad \text{(2)}\\ +\vdash .(2).*91·54.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p>This proposition is important, since it simplifies all propositions +concerning \(P_{\text{po}}\) and \(P_{*}\) in case \(P\) is transitive. +The following proposition is an instance of this simplification.</p> + +<p class="nind"><b><a id="*201·19">*201·19</a>.</b> \(\vdash :P\in \text{trans}.\supset .P(x-y)=\overleftarrow{P}ʻx\cap \overrightarrow{P}ʻy \quad[*201·18.(*121·01)]\)</p> + +<p>The following propositions (<a href="#*201·2">*201·2</a>—<a href="#*201·2">·22</a>) are concerned in proving that +transitiveness is unaffected by likeness-transformations, and therefore +belongs to every member of a relation-number or to none.</p> + +<p class="nind"><b><a id="*201·2">*201·2</a>.</b> \(\vdash :S\in \text{Cls}\rightarrow 1.\text{ᗡ}ʻQ\subset \text{ᗡ}ʻS.\supset .(S^{;}Q)^{2}=S^{;}Q^{2}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .150·1. &\supset \vdash .(S^{;}Q)^{2}=S\mid Q\mid \breve{S} \mid S\mid Q\mid \breve{S} &\qquad \text{(1)}\\ +\vdash .*72·601.&\supset \vdash :\text{Hp}.\supset .Q\mid \breve{S} \mid S=Q &\qquad \text{(2)}\\ +\vdash .(1).(2).&\supset \vdash :\text{Hp}.\supset .(S^{;}Q)^{2}=S\mid Q^{2}\mid \breve{S} :\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*201·201.</b>\[\begin{align}&\vdash :S\in \text{Cls}\rightarrow 1.\text{D}ʻQ\subset \text{ᗡ}ʻS.\supset .(S^{;}Q)^{2}=S^{;}Q^{2}\\ +&[\text{Proof as in *201·2}]\end{align}\]</p> + +<p><span class="pagenum" id="Page_528">[Pg 528]</span></p> + +<p class="nind"><b>*201·21.</b> \(\vdash :S\in \text{Cls}\rightarrow 1.Q\in \text{trans}.\supset .S^{;}Q\in \text{trans}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*150·36.*35·452.&\supset \vdash .S^{;}Q=S^{;}Q\unicode{x0294f}\text{ᗡ}ʻS &\qquad \text{(1)}\\ +\vdash .(1).*201·2. \supset \vdash :\text{Hp}.&\supset .(S^{;}Q)^{2}=S^{;}(Q\unicode{x0294f}\text{ᗡ}ʻS)^{2}.\\ +[*150·31.*201·1] &\supset .(S^{;}Q)^{2}\,\unicode{x2abd}\, S^{;}Q:\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*201·211">*201·211</a>.</b> \(\vdash :P\in \text{trans}.Q\,\,\text{smor}\,\,P.\supset .Q\in \text{trans} \quad[*201·21.*151·1]\)</p> + +<p>This shows that transitiveness is a property which is unchanged by +likeness-transformations. Hence</p> + +<p class="nind"><b>*201·212.</b> \(\vdash :P\in \text{trans}.\supset .\text{Nr}ʻP\subset \text{trans} \quad[*201·211]\)</p> + +<p class="nind"><b>*201·22.</b>\[\begin{align}&\vdash :P\in \text{trans}.\equiv .\text{N}_{0}\text{r}ʻP\subset \text{trans}.\equiv .\exists !\text{N}_{0}\text{r}ʻP\cap \text{trans}\\ +&[\text{Proof as in *200*22}]\end{align}\]</p> + +<p class="nind"><b><a id="*201·3">*201·3</a>.</b> \(\vdash .\dot{\Lambda} \in \text{trans}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*34·32. &\supset \vdash .\dot{\Lambda} ^{2}=\dot{\Lambda} &\qquad \text{(1)}\\ +\vdash .(1).*23·42. &\supset \vdash .\dot{\Lambda} ^{2}\,\unicode{x2abd}\, \dot{\Lambda} .\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*201·31">*201·31</a>.</b> \(\vdash .x\downarrow y\in \text{trans}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*55·13.\supset \vdash :z(x\downarrow y)^{2} w.&\equiv .(\exists u).z=x.u=y.u=x.w=y.\\ +[*10·35] &\supset .z=x.w=y.\\ +[*55·13] &\supset .z(x\downarrow y)w:\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p>Unless \(x=y\), \((x\downarrow y)^{2}=\dot{\Lambda}\). A relation whose +square is \(\dot{\Lambda}\) is transitive, because \(\dot{\Lambda}\) is +contained in every relation.</p> + +<p class="nind"><b>*201·32.</b> \(\vdash .\alpha \uparrow \beta \in \text{trans}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*35·103.\supset \vdash :x(\alpha \uparrow \beta )^{2}z.&\equiv .(\exists y).x\in \alpha .y\in \beta .y\in \alpha .z\in \beta .\\ +[*10·35] &\supset .x\in \alpha .z\in \beta .\\ +[*35·103] &\supset .x(\alpha \uparrow \beta )z:\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*201·33">*201·33</a>.</b> \(\vdash :P\in \text{trans}.\supset .P\unicode{x0294f}\alpha \in \text{trans}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*36·13.&\supset \vdash :x(P\unicode{x0294f}\alpha )^{2}z.\equiv .(\exists y).x,y,z\in \alpha .xPy.yPz &\qquad \text{(1)}\\ +\vdash .(1). \supset \vdash \colon\ldotp \text{Hp}.\supset :x(P\unicode{x0294f}\alpha )^{2}z.&\supset .(\exists y).x,y,z\in \alpha .xPz.\\ +[*10·35.*36·13]& \supset .x(P\unicode{x0294f}\alpha )z\colon\ldotp \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p>The following propositions (<a href="#*201·4">*201·4</a>—<a href="#*201·4">·42</a>) are concerned with the ideas +of relation-arithmetic.</p> + +<p class="nind"><b><a id="*201·4">*201·4</a>.</b> \(\vdash :P,Q\in \text{trans}.CʻP\cap CʻQ=\Lambda .\supset .P\unicode{x2909}Q\in \text{trans}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*160·51. &\supset \vdash :\text{Hp}.\supset .(P\unicode{x2909}Q)^{2}=P^{2}\unicode{x228d} Q^{2}\unicode{x228d} + \text{D}ʻP\uparrow CʻQ\unicode{x228d} CʻP\uparrow \text{ᗡ}ʻQ &\qquad \text{(1)}\\ +\vdash .*201·1. &\supset \vdash :\text{Hp}.\supset .P^{2}\,\unicode{x2abd}\, P.Q^{2}\,\unicode{x2abd}\, Q &\qquad \text{(2)}\\ +\vdash .*35·432·82. &\supset \vdash .\text{D}ʻP\uparrow CʻQ\,\unicode{x2abd}\, CʻP\uparrow CʻQ.CʻP\uparrow \text{ᗡ}ʻQ\,\unicode{x2abd}\, CʻP\uparrow CʻQ &\qquad \text{(3)}\\ +\vdash .(1).(2).(3).&\supset \vdash :\text{Hp}.\supset .(P\unicode{x2909}Q)^{2}\,\unicode{x2abd}\, P\unicode{x228d} Q\unicode{x228d} CʻP\uparrow CʻQ:\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_529">[Pg 529]</span></p> + +<p class="nind"><b>*201·401.</b> \(\vdash \colon\ldotp CʻP\cap CʻQ=\Lambda .\supset :P\unicode{x2909}Q\in \text{trans}.\equiv .P,Q\in \text{trans}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*160·51.\supset \\ +\vdash \colon\ldotp \text{Hp}.\supset :P\unicode{x2909}Q\in \text{trans}.&\equiv .P^{2}\unicode{x228d} Q^{2}\unicode{x228d} + \text{D}ʻP\uparrow CʻQ\unicode{x228d} CʻP\uparrow \text{ᗡ}ʻQ\,\unicode{x2abd}\, P\unicode{x2909}Q.\\ +[*160·1] &\equiv .P^{2}\unicode{x228d} Q^{2}\,\unicode{x2abd}\, P\unicode{x2909}Q.\\ +[*160·5] &\supset .(P^{2}\unicode{x228d} Q^{2})\unicode{x0294f}CʻP\,\unicode{x2abd}\, P.(P^{2}\unicode{x228d} Q^{2})\unicode{x0294f}CʻQ\,\unicode{x2abd}\, Q.\\ +[*36·4.*34·56] &\supset .P^{2}\,\unicode{x2abd}\, P.Q^{2}\,\unicode{x2abd}\, Q &\qquad \text{(1)}\\ +\vdash .(1).*201·4.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*201·41.</b> \(\vdash :.x{\sim}\in CʻP.\supset :P\in \text{trans}.\equiv .P\unicode{x21f8} x\in \text{trans}.\equiv .x\unicode{x21f7}P\in \text{trans}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*34·301.\supset \vdash :\text{Hp}.&\supset .(CʻP\uparrow {℩}ʻx)\mid P=\dot{\Lambda} .\\ +[*161·1] \supset .(P\unicode{x21f8} x)^{2}&=P^{2}\unicode{x228d} (CʻP\uparrow {℩}ʻx)^{2}\unicode{x228d} P\mid (CʻP\uparrow {℩}ʻx)\\ +[*35·881] & =P^{2}\unicode{x228d} (CʻP\uparrow {℩}ʻx)^{2}\unicode{x228d} (\text{D}ʻP\uparrow {℩}ʻx)\\ +[*35·895] & =P^{2}\unicode{x228d} (\text{D}ʻP\uparrow {℩}ʻx) &\qquad \text{(1)}\\ +\vdash .(1).*201·1.\supset \\ +\vdash \colon\ldotp \text{Hp}.\supset :(P\unicode{x21f8} x)\in \text{trans}.&\equiv .P^{2}\unicode{x228d} (\text{D}ʻP\uparrow {℩}ʻx)\,\unicode{x2abd}\, P\unicode{x228d} (CʻP\uparrow {℩}ʻx).\\ +[*35·432·82] & \equiv .P^{2}\,\unicode{x2abd}\, P\unicode{x228d} (CʻP\uparrow {℩}ʻx) &\qquad \text{(2)}\\ +\vdash .*33·33.*34·56.*35·86.&\supset \vdash :\text{Hp}.\supset .P^{2}\dot{\cap} (CʻP\uparrow {℩}ʻx)=\dot{\Lambda} &\qquad \text{(3)}\\ +\vdash .(2).(3)*25·49.\supset \vdash \colon\ldotp \text{Hp}.\supset :P\unicode{x21f8} x\in \text{trans}.&\equiv .P^{2}\,\unicode{x2abd}\, P.\\ +[*201·1] &\equiv .P\in \text{trans} &\qquad \text{(4)}\\ +\text{Similarly}\quad \vdash \colon\ldotp \text{Hp}.\supset :x\unicode{x21f7}P\in \text{trans}.&\equiv .P\in \text{trans} &\qquad \text{(5)}\\ +\vdash .(4).(5).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*201·411.</b> \(\vdash :z\neq x.z\neq y.\supset .x\downarrow y\unicode{x21f8} z\in \text{trans} \quad[*201·41·31]\)</p> + +<p class="nind"><b><a id="*201·42">*201·42</a>.</b> \(\vdash :P\in \text{trans}\cap \text{Rel}^{2}\text{excl}.CʻP\subset \text{trans}.\supset .\Sigma ʻP\in \text{trans}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*162·1.\supset \\ +\vdash .(\Sigma ʻP)^{2}&=(\dot{s} ʻCʻP)^{2}\unicode{x228d} (F^{;}P)^{2}\unicode{x228d} (\dot{s} ʻCʻP)\mid (F^{;}P)\unicode{x228d} (F^{;}P)\mid + (\dot{s} ʻCʻP) &\qquad \text{(1)}\\ +\vdash .*41·11.\supset \vdash :x(\dot{s} ʻCʻP)^{2}z.&\equiv .(\exists Q,R,y).Q,R\in CʻP.xQy.yRz.\\ +[*33·17] &\equiv .(\exists Q,R,y).Q,R\in CʻP.xQy.yRz.\exists !CʻQ\cap CʻR &\qquad \text{(2)}\\ +\vdash .(2).*163·11.\supset \\ +\vdash \colon\ldotp \text{Hp}.&\supset :x(\dot{s} ʻCʻP)^{2}z.\supset .(\exists Q,R,y).Q,R\in CʻP.xQy.yRz.Q=R.\\ +[*13·195] &\supset .(\exists Q).Q\in CʻP.xQ^{2}z.\\ +[*201·1.\text{Hp}] &\supset .(\exists Q).Q\in CʻP.xQz.\\ +[*41·11] &\supset .x(\dot{s} ʻCʻP)z &\qquad \text{(3)}\\ +\vdash .*201·21.*163·12.\supset \vdash :\text{Hp}.&\supset .(F^{;}P)^{2}\,\unicode{x2abd}\, F^{;}P &\qquad \text{(4)}\\ +\vdash .*34·1.*41·11.*150·52.\supset\\ +\vdash :x(\dot{s} ʻCʻP)\mid (F^{;}P)z.&\supset .(\exists Q,R,S,y).Q\in CʻP.xQy.RPS.y\in CʻR.z\in CʻS &\qquad \text{(5)}\\ +\vdash .(5).*163·11.*13·195.\supset \\ +\vdash \colon\ldotp \text{Hp}.&\supset :x(\dot{s} ʻCʻP)\mid (F^{;}P)z.\supset .(\exists Q,S,y).Q\in CʻP.xQy.QPS.z\in CʻS.\\ +[*33·17.*150·52] & \supset .x(F^{;}P)z &\qquad \text{(6)}\\ +\text{Similarly}\quad \vdash \colon\ldotp \text{Hp}.&\supset :x(F^{;}P)\mid (\dot{s} ʻCʻP)z.\supset .x(F^{;}P)z &\qquad \text{(7)}\\ +\vdash .(1).(3).(4).(6).(7).\supset \\ +\vdash :\text{Hp}.&\supset .(\Sigma ʻP)^{2}\,\unicode{x2abd}\, \dot{s} ʻCʻP\unicode{x228d} F^{;}P:\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_530">[Pg 530]</span></p> + +<p>The following propositions (<a href="#*201·5">*201·5</a>—<a href="#*201·56">·56</a>) are concerned with +\(Pʻʻ\alpha\) and \(pʻ\overrightarrow{P}ʻʻ\alpha\), <i>i.e.</i> with +the predecessors of some part of a class and the predecessors of the +whole of a class.</p> + +<p class="nind"><b><a id="*201·5">*201·5</a>.</b> \(\vdash :P\in \text{trans}.\supset .PʻʻPʻʻ\alpha \subset Pʻʻ\alpha \quad[*37·33·201]\)</p> + +<p class="nind"><b>*201·501.</b> \(\vdash :P\in \text{trans}.\supset .Pʻʻ\overrightarrow{P}ʻx\subset \overrightarrow{P}ʻx \quad[*53·301.*201·5]\)</p> + +<p class="nind"><b>*201·51.</b> \(\vdash :P\in \text{trans}.\supset .Pʻʻpʻ\overrightarrow{P}ʻʻ\alpha \subset pʻ\overrightarrow{P}ʻʻ\alpha\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*37·1.*40·51.\supset \vdash \colon\ldotp x\in Pʻʻpʻ\overrightarrow{P}ʻʻ\alpha . &\equiv :(\exists y):z\in \alpha .\supset _{z}.yPz:xPy:\\ +[*5·31] &\supset :z\in \alpha .\supset _{z}.xP^{2}z &\qquad \text{(1)}\\ +\vdash .(1).*201·1. \supset \vdash \colon\ldotp \text{Hp}.&\supset :x\in Pʻʻpʻ\overrightarrow{P}ʻʻ\alpha .\supset :z\in \alpha .\supset _{z}.xPz:\\ +[*40·51] & \supset :x\in pʻ\overrightarrow{P}ʻʻ\alpha \colon\ldotp \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*201·52.</b> \(\vdash :P\in \text{trans}.\supset .P_{*}ʻʻ\alpha = Pʻʻ\alpha \cup (\alpha \cap CʻP) \quad[*91·543.*201·18]\)</p> + +<p class="nind"><b>*201·521.</b> \(\vdash :P\in \text{trans}.x\in CʻP.\supset .\overrightarrow{P}_{*}ʻx = \overrightarrow{P}ʻx\cup \iota ʻx \quad[*201·52.*53·301]\)</p> + +<p class="nind"><b>*201·53.</b> \(\vdash :P\in \text{trans}.\supset .P_{*}ʻʻPʻʻ\alpha = Pʻʻ\alpha \quad[*201·5·52.*37·265]\)</p> + +<p class="nind"><b>*201·54.</b> \(\vdash :P\in \text{trans}.\supset .P_{*}ʻʻpʻ\overrightarrow{P}ʻʻ\alpha \subset pʻ\overrightarrow{P}ʻʻ\alpha \quad[*201·51·52]\)</p> + +<p class="nind"><b>*201·55.</b> \(\vdash :P\in \text{trans}.\supset .Pʻʻ(\alpha \cup Pʻʻ\alpha ) = Pʻʻ\alpha\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*201·5.\supset \vdash :\text{Hp}.\supset .Pʻʻ\alpha &= Pʻʻ\alpha \cup PʻʻPʻʻ\alpha \\ +[*37·22] &= Pʻʻ(\alpha \cup Pʻʻ\alpha ):\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p>The following proposition is a lemma which is used in <a href="#*205·192">*205·192</a> and +<a href="#*206·24">*206·24</a>.</p> + +<p class="nind"><b><a id="*201·56">*201·56</a>.</b>\[\begin{align}&\vdash :P\in \text{trans}.\beta \subset Pʻʻ\alpha .\supset .\\ +&Pʻʻ(\alpha \cup \beta ) = Pʻʻ\alpha .pʻ\overleftarrow{P}ʻʻ{\alpha \cup \beta )\cap CʻP} = pʻ\overleftarrow{P}ʻʻ(\alpha \cap CʻP)\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*37·22. &\supset \vdash .Pʻʻ(\alpha \cup \beta ) = Pʻʻ\alpha \cup Pʻʻ\beta &\qquad \text{(1)}\\ +\vdash .*37·2. \supset \vdash :\text{Hp}.&\supset .Pʻʻ\beta \subset PʻʻPʻʻ\alpha .\\ +[*201·5] &\supset .Pʻʻ\beta \subset Pʻʻ\alpha &\qquad \text{(2)}\\ +\vdash .(1).(2).\supset \vdash :\text{Hp}.&\supset .Pʻʻ(\alpha \cup \beta ) = Pʻʻ\alpha &\qquad \text{(3)}\\ +\vdash .*40·51.*37·265.\supset \\ +\vdash \colon\colon\text{Hp}.&\supset :z\in pʻ\overleftarrow{P}ʻʻ(\alpha \cap CʻP).x\in \beta \cap CʻP.\supset :\\ +&y\in \alpha \cap CʻP.\supset _{y}.yPz:(\exists y).y\in \alpha \cap CʻP.xPy:\\ +[*10·56] &\supset :(\exists y).xPy.yPz:\\ +[*34·5.\text{Hp}] &\supset :xPz &\qquad \text{(4)}\\ +\vdash .(4).*40·51.\supset \vdash :\text{Hp}.&\supset .pʻ\overleftarrow{P}ʻʻ(\alpha \cap CʻP)\subset pʻ\overleftarrow{P}ʻʻ(\beta \cap CʻP).\\ +[*22·621] \supset .pʻ\overleftarrow{P}ʻʻ(\alpha \cap CʻʻP)&=pʻ\overleftarrow{P}ʻʻ(\alpha \cap CʻP)\cap pʻ\overleftarrow{P}ʻʻ(\beta \cap CʻP)\\ +[*40·18.*37·22] & =pʻ\overleftarrow{P}ʻʻ\{(\alpha \cup \beta )\cap CʻP\} &\qquad \text{(5)}\\ +\vdash .(3).(5).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_531">[Pg 531]</span></p> + +<p>The following propositions, to the end of the number, are concerned +with the relation \(P_{1}\) defined in <a href="#*121">*121</a>. We may regard \(P_{1}\) as +meaning "immediately precedes." <a href="#*201·6">*201·6</a><a href="#*201·61">·61</a><a href="#*201·62">·62</a> are lemmas for <a href="#*201·63">*201·63</a>.</p> + +<p class="nind"><b><a id="*201·6">*201·6</a>.</b> \(\vdash :P\in \text{trans}.{\sim}(xPx).{\sim}(yPy).xP_{1}y.\supset .x(P\dot{-} P^{2})y\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*121·32·242.\supset \vdash :\text{Hp}.\supset .P(x\vdash\dashv y)&=\iota ʻx\cup \iota ʻy\cup P(x-y)\\ +[*201·19] &=\iota ʻx\cup \iota ʻy\cup \overleftarrow{P}ʻx\cap \overrightarrow{P}ʻy &\qquad \text{(1)}\\ +\vdash .*121·321.*201·18. & \supset :\text{Hp}.\supset .xPy &\qquad \text{(2)}\\ +\vdash .(2).*13·14.& \supset :\text{Hp}.\supset .x\neq y &\qquad \text{(3)}\\ +\vdash .(1).(3).*54·53.*121·11. &\supset \vdash :\text{Hp}.\supset .\overleftarrow{P}ʻx\cap \overrightarrow{P}ʻy\subset \iota ʻx\cup \iota ʻy &\qquad \text{(4)}\\ +\vdash .*32·18·181. &\supset \vdash :\text{Hp}.\supset .x{\sim}\in \overleftarrow{P}ʻx.y{\sim}\in \overrightarrow{P}ʻy &\qquad \text{(5)}\\ +\vdash .(4).(5). \supset \vdash :\text{Hp}.&\supset .\overleftarrow{P}ʻx\cap \overrightarrow{P}ʻy=\Lambda .\\ +[*34·11] &\supset .{\sim}(xP^{2}y) &\qquad \text{(6)}\\ +\vdash .(2).(6).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*201·61">*201·61</a>.</b> \(\vdash :P\in \text{trans}.\supset .P\dot{-} P^{2}\,\unicode{x2abd}\, P_{1}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*121·242.*90·151. &\supset \vdash :xPy.\supset .P(x\vdash\dashv y)=\iota ʻx\cup \iota ʻy\cup P(x-y) &\qquad \text{(1)}\\ +\vdash .(1).*201·19. &\supset \vdash \colon\ldotp \text{Hp}.\supset :xPy.\supset .P(x\vdash\dashv y)=\iota ʻx\cup \iota ʻy\cup (\overleftarrow{P}ʻx\cap \overrightarrow{P}ʻy) &\qquad \text{(2)}\\ +\vdash .*34·11. &\supset \vdash :{\sim}(xP^{2}y).\supset .\overleftarrow{P}ʻx\cap \overrightarrow{P}ʻy=\Lambda &\qquad \text{(3)}\\ +\vdash .*34·54. &\supset \vdash :xPy.{\sim}(xP^{2}y).\supset .x\neq y &\qquad \text{(4)}\\ +\vdash .(2).(3).(4). \supset \vdash \colon\ldotp \text{Hp}.&\supset :xPy.{\sim}(xP^{2}y).\supset .P(x\vdash\dashv y)=\iota ʻx\cup \iota ʻy.x\neq y.\\ +[*54·101] &\supset .P(x\vdash\dashv y)\in 2.\\ +[*121·11] &\supset .xP_{1}y\colon\ldotp \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*201·62">*201·26</a>.</b> \[\begin{align}&\vdash \colon\ldotp P\in \text{trans}.{\sim}(xPx).{\sim}(yPy).\supset :xP_{1}y.\equiv .x(P\dot{-} P^{2})y\\ +&[*201·6·61]\end{align}\]</p> + +<p class="nind"><b><a id="*201·63">*201·63</a>.</b> \(\vdash :P\in \text{trans}\cap \text{Rl}ʻJ.\supset .P_{1}=P\dot{-} P^{2} \quad[*201·62]\)</p> + +<p>The above proposition is of fundamental importance. The relation +\(P_{1}\) (defined in <a href="#*121">*121</a>) plays a great part in the theory of series. +It is the relation<span class="pagenum" id="Page_532">[Pg 532]</span> "immediately preceding." Its domain consists of +those terms which have immediate successors; its converse domain, +of those that have immediate predecessors. In well-ordered series, +\(\text{D}ʻP_{1}=\text{D}ʻP\), while \(\text{ᗡ}ʻP_{1}\) consists +of all terms (except the first) which do not belong to the first +derivative (cf. <a href="#*216">*216</a>). In any series, \(\text{ᗡ}ʻP-\text{ᗡ}ʻP_{1}\) +consists of all the terms which are limits of ascending series, and +\(\text{D}ʻP-\text{D}ʻP_{1}\). consists of all the terms which are +limits of descending series.</p> + +<p class="nind"><b>*201·64.</b> \(\vdash \colon\ldotp P\in \text{trans}.\supset :P\dot{-} P^{2}=\dot{\Lambda} .\equiv .P^{2}=P\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*23·41.\supset \vdash \colon\ldotp \text{Hp}.\supset :P^{2}=P.&\equiv .P\,\unicode{x2abd}\, P^{2}.\\ +[*25·3] &\equiv .P\dot{-} P^{2}=\dot{\Lambda} \colon\ldotp \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*201·65">*201·65</a>.</b> \(\vdash \colon\ldotp P\in \text{trans}\cap \text{Rl}ʻJ.\supset :P_{1}=\dot{\Lambda} .\equiv .P^{2}=P \quad [*201·64·63]\)</p> + +<p>When \(P\) is a series, \(P^{2}=P\) is the condition for its being +a <i>compact</i> series, <i>i.e.</i> one in which there are terms +between any two. In virtue of <a href="#*201·65">*201·65</a>, this condition is equivalent to +\(P_{1}=\dot{\Lambda}\), which states that no term has an immediate +predecessor.</p> + +<p>The following proposition is first used in *253·521.</p> + +<p class="nind"><b>*201·66.</b> \(\vdash :P\in \text{trans}.\text{E}!Pʻx.Pʻx\neq x.\supset .(Pʻx)P_{1}x\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*201·521.*121·11.\supset \\ +\vdash \colon\ldotp \text{Hp}.&\supset :(Pʻx)P_{1}x.\equiv .({℩}ʻPʻx\cup \overleftarrow{P}ʻPʻx)\cap ({℩}ʻx\cup \overrightarrow{P}ʻx)\in 2 &\qquad \text{(1)}\\ +\vdash .*53·31.\supset \vdash :\text{Hp}.\supset .\\ +({℩}ʻPʻx\cup \overleftarrow{P}ʻPʻx)\cap ({℩}ʻx\cup \overrightarrow{P}ʻx)&=({℩}ʻPʻx\cup \overleftarrow{P}ʻPʻx)\cap ({℩}ʻx\cup {℩}ʻPʻx)\\ +[*30·32.*22·68] & ={℩}ʻx\cup {℩}ʻPʻx &\qquad \text{(2)}\\ +\vdash .*54·26.&\supset \vdash :\text{Hp}.\supset .({℩}ʻx\cup {℩}ʻPʻx)\in 2 &\qquad \text{(3)}\\ +\vdash .(1).(2).(3).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*201·661.</b> \(\vdash :P\in \text{trans}.\text{ᗡ}ʻP\in 1.\exists !\text{D}ʻP-\text{ᗡ}ʻP.\supset .\text{ᗡ}ʻP\subset \text{ᗡ}ʻP_{1}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*33·151·4.*60·38.\supset \\ +\vdash :\text{Hp}.y\in \text{D}ʻP-\text{ᗡ}ʻP.&\supset .\overleftarrow{P}ʻy\in 1.y{\sim}\in \overleftarrow{P}ʻy.\overleftarrow{P}ʻy=\text{ᗡ}ʻP.\\ +[*53·3] &\supset .\text{E}!\breve{P} ʻy.y\neq \breve{P} ʻy.{℩}ʻ\breve{P} ʻy=\text{ᗡ}ʻP.\\ +[*201·66·11.*121·26] &\supset .yP_{1}(\breve{P} ʻy).{℩}ʻ\breve{P} ʻy=\text{ᗡ}ʻP:\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p>The above proposition is a lemma for the following.</p> + +<p class="nind"><b>*201·662.</b>\[\begin{align}&\vdash :P\in \text{trans}.\exists !\overrightarrow{B}ʻP.\exists !\text{ᗡ}ʻP-\text{ᗡ}ʻP_{1}.\supset .\text{ᗡ}ʻP{\sim}\in 1\\ +&[*201·661.\text{Transp}]\end{align}\]</p> + +<p>This proposition is first used in *253·521.</p> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_533">[Pg 533]</span></p> +<h2 class="nobreak" id="*202">*202. CONNECTED RELATIONS.</h2> +</div> + + +<p class="nind"><i>Summary of</i> *202.</p> + +<p>A relation is said to be <i>connected</i> when either it or its +converse holds between any two different members of its field, +<i>i.e.</i> when, if \(x\), \(y \in CʻP . x \neq y\), we have \(xPy. \lor . yPx\). +Thus the field of a connected relation consists of +a single family, unless the relation is null, in which case it has +no families. Conversely, a relation which has one family or none is +connected. Connection is necessary, in addition to transitiveness and +asymmetry, in order that a relation may generate a single series. +If \(\lambda\) is a class of transitive or asymmetrical relations, +\(\dot{s} ʻ\lambda\) is transitive or asymmetrical; but if \(\lambda\) +is a class of connected relations, \(\dot{s} ʻ\lambda\) is not in +general connected. Hence if \(\lambda\) is a class of series, \(\dot{s}ʻ\lambda\) +is not one series, but many detached series. This is one +reason why the arithmetical sum of a relation of relations is not +defined as \(\dot{s} ʻCʻP\), but as \(\dot{s} ʻCʻP \unicode{x228d}F^{;}P\) +(cf. <a href="#*162">*162</a>), because the latter, but not in general the former, +is connected when \(P\) and all the members of \(CʻP\) are connected +(<a href="#*202·42">*202·42</a>).</p> + +<p>When \(P\) is connected, if \(\alpha\) is any class contained in +\(CʻP\), we have +\[ +CʻP = Pʻʻ\alpha \cup \alpha \cup (CʻP \cap pʻ\overleftarrow{P}ʻʻ\alpha ), +\] +and there is at most one member of \(\alpha\) belonging neither +to \(Pʻʻ\alpha\) nor to \(CʻP \cap pʻ\overleftarrow{P}ʻʻ\alpha\). +This member of \(\alpha\), if it exists, is the maximum of \(\alpha\). +If, further, \(P^{2} \,\unicode{x2abd}\, J\) (<i>i.e.</i> if \(P\) +is asymmetrical), \((Pʻʻ\alpha \cup \alpha ) \cap (CʻP \cap pʻ\overleftarrow{P}ʻʻ\alpha) = \Lambda\). +Thus when \(P\) is both connected and asymmetrical, \(Pʻʻ\alpha\cup \alpha\) +and \(CʻP \cap pʻ\overleftarrow{P}ʻʻ\alpha\) are each +other's complements, and the two together constitute the Dedekind +cut defined by \(\alpha , Pʻʻ\alpha \cup \alpha\) being all the +terms that do not follow the whole of \(\alpha\), and \(CʻP \cap pʻ\overleftarrow{P}ʻʻ\alpha\) +being all the terms that do follow the whole of \(\alpha\).</p> + +<p>More generally, if \(\alpha\) is any class, not necessarily contained +in \(CʻP\), then when \(P\) is connected, we have +\[ +CʻP - pʻ\overleftarrow{P}ʻʻ(\alpha \cap CʻP) \subset Pʻʻ\alpha \cup (\alpha \cap CʻP), +\] +and when \(P\) is asymmetrical, we have +\[ +Pʻʻ\alpha \cup (\alpha \cap CʻP) \subset CʻP - pʻ\overleftarrow{P}ʻʻ\alpha . +\\] +Thus when both conditions are fulfilled, we have (<a href="#*202·503">*202·503</a>) +\[ +CʻP - pʻ\overleftarrow{P}ʻʻ(\alpha \cap CʻP) = Pʻʻ\alpha \cup (\alpha \cap CʻP). +\]</p> + +<p><span class="pagenum" id="Page_534">[Pg 534]</span></p> + +<p>The above inclusions and the consequent equality will be constantly +required throughout what follows. The division of \(CʻP\) into the two +mutually exclusive parts +\[ +Pʻʻ\alpha \cup (\alpha \cap CʻP) \,\text{and}\, CʻP\cap pʻ\overleftarrow{P}ʻʻ(\alpha \cap CʻP) +\] +is the Dedekind "cut" defined by the class \(\alpha\). If \(\alpha\subset CʻP\), +the two parts become, as above mentioned, +\[ +Pʻʻ\alpha \cup \alpha \,\text{and}\, CʻP\cap pʻ\overleftarrow{P}ʻʻ\alpha . +\] +If, further, \(\alpha\) is not null, they become +\[ +Pʻʻ\alpha \cup \alpha \,\text{and}\, pʻ\overleftarrow{P}ʻʻ\alpha . +\] +If \(\alpha\) is contained in \(CʻP\) and contains all its own +predecessors, they become +\[ +\alpha\, \text{and}\, CʻP\cap pʻ\overleftarrow{P}ʻʻ\alpha. +\] +In this simplified form, Dedekind "cuts" will be considered later +(<a href="#*211">*211</a>).</p> + +<p>We take as our definition +\[ +\text{connex} = \hat{P} \{x\in CʻP.\supset _{x}.\overleftrightarrow{P}ʻx = CʻP\} \quad\text{Df}. +\]</p> + +<p>Some of the propositions of the present number are analogues of +propositions in <a href="#*200">*200</a> and <a href="#*201">*201</a>. Such are: If \(P\) is connected, +so is \(\breve{P}\) (<a href="#*202·11">*202·11</a>); if \(P\) is connected, so is any +similar relation (<a href="#*202·211">*202·211</a>); \(\dot{\Lambda}\) and \(x\downarrow y\) +are connected (<a href="#*202·3">*202·3</a><a href="#*202·31">·31</a>); if \(P\) is connected, so is +\(P\unicode{x0294f}\alpha\) (<a href="#*202·33">*202·33</a>); and various propositions +connected with relation-arithmetic (<a href="#*202·4">*202·4</a>—<a href="#*202·42">·42</a>). The majority of the +propositions of this number, however, deal with properties peculiar to +connexity. Among the most important of these are:</p> + +<p class="nind"><b>*202·101.</b> \(\vdash \colon\ldotp P\in \text{connex} . \equiv :x\in CʻP.\supset _{x}.\overrightarrow{P}ʻx\cup \iota ʻx\cup \overleftarrow{P}ʻx = CʻP\)</p> + +<p class="nind"><b>*202·103.</b> \(\vdash \colon\colon P\in \text{connex} . \equiv \colon\ldotp x,y\in CʻP.\supset _{x,y}:xPy.\lor.x = y.\lor.yPx\)</p> + +<p>These are merely alternative forms of the definition.</p> + +<p class="nind"><b>*202·13.</b> \(\vdash :R_{*}\in \text{connex} . \equiv .R_{\text{po}}\in \text{connex}\)</p> + +<p class="nind"><b>*202·5.</b> \(\vdash \colon\ldotp P\in \text{connex} .P^{2}\,\unicode{x2abd}\, J.x,y\in CʻP.\supset :x \neq y.{\sim}(xPy). \equiv .yPx\)</p> + +<p class="nind"><b>*202·501.</b> \(\vdash :P\in \text{connex} .\supset .CʻP-\alpha -Pʻʻ\alpha \subset pʻ\overleftarrow{P}ʻʻ(\alpha \cap CʻP)\)</p> + +<p class="nind"><b>*202·503.</b> \(\vdash :P\in \text{connex} .P^{2}\,\unicode{x2abd}\, J.\supset .CʻP-pʻ\overleftarrow{P}ʻʻ(\alpha \cap CʻP) = (\alpha \cap CʻP)\cup Pʻʻ\alpha\)</p> + +<p class="nind"><b>*202·505.</b> \(\vdash :P\in \text{connex} .\supset .CʻP = Pʻʻ\alpha \cup (\alpha \cap CʻP)\cup \{CʻP\cap pʻ\overleftarrow{P}ʻʻ(\alpha \cap CʻP)\}\)</p> + +<p class="nind"><b>*202·52.</b> \(\vdash :P\in \text{connex} .\supset .\overrightarrow{B}ʻP,\overrightarrow{B}ʻ\breve{P} \in 0\cup 1\)</p> + +<p class="nind"><b>*202·524.</b> \(\vdash :P\in \text{connex} .\exists !\overrightarrow{B}ʻP.\supset .\text{ᗡ}ʻP = \overleftarrow{P}ʻBʻP\)</p> + +<p class="nind"><b>*202·55.</b> \(\vdash :P\unicode{x0294f}\alpha \in \text{connex} .\alpha \subset CʻP.\alpha {\sim}\in 1.\supset .CʻP\unicode{x0294f}\alpha = \alpha\)</p> + +<p><span class="pagenum" id="Page_535">[Pg 535]</span></p> + +<p>In virtue of this proposition (and others) if \(P\) is a series +and \(\alpha\) is a class (not a unit class) contained in \(CʻP\), +\(P\unicode{x0294f}\alpha\) is the generating relation of the series +consisting of the class \(\alpha\) in the order which it has in the +series \(P\).</p> + +<p class="nind"><b>*202·7.</b> \(\vdash :P\in \text{connex} .\supset .P\dot{-} P^{2}\in 1\rightarrow 1\)</p> + +<p>This proposition is to be taken in connection with <a href="#*201·63">*201·63</a>. The two +together show that when \(P\) is a series, \(P_{1}\) is one-one.</p> + +<hr class="tb"> + +<p class="nind"><b>*202·01.</b> \(\text{connex} =\hat{P} \{x\in CʻP.\supset _{x}.\overleftrightarrow{P}ʻx=CʻP\} \quad\text{Df}\)</p> + +<p>For the definition of \(\overleftrightarrow{P}ʻx\), see *97·01.</p> + +<p class="nind"><b>*202·1.</b> \(\vdash \colon\ldotp P\in \text{connex} .\equiv :x\in CʻP.\supset _{x}.\overleftrightarrow{P}ʻx=CʻP \quad[(*202·01)]\)</p> + +<p class="nind"><b>*202·101.</b>\[\begin{align}& \vdash \colon\ldotp P\in \text{connex} .\equiv .x\in CʻP.\supset _{x}.\overrightarrow{P}ʻx\cup \iota ʻx\cup \overleftarrow{P}ʻx=CʻP\\ +&[*202·1.*97·1]\end{align}\]</p> + +<p class="nind"><b>*202·102.</b> \(\vdash :P\in \text{connex} .\equiv .\overleftrightarrow{P}ʻʻCʻP\in 0\cup 1 \quad[*97·231.*202·101]\)</p> + +<p class="nind"><b>*202·103.</b>\[\begin{align}& \vdash \colon\colon P\in \text{connex} .\equiv \colon\ldotp x,y\in CʻP.\supset _{x,y}:xPy.\lor.x=y.\lor.yPx\\ +&[*97·23.*202·102]\end{align}\]</p> + +<p class="nind"><b>*202·104.</b>\[\begin{align}& \vdash \colon\colon P\in \text{connex} .\equiv \colon\ldotp x,y\in CʻP.x\neq y.\supset _{x,y}:xPy.\lor.yPx\\ +&[*202·103.*5·6]\end{align}\]</p> + +<p class="nind"><b><a id="*202·11">*202·11</a>.</b> \(\vdash :P\in \text{connex} .\equiv .\breve{P} \in \text{connex} \quad[*202·104.*33·22]\)</p> + +<p class="nind"><b>*202·12.</b> \(\vdash \colon\ldotp \dot{\exists} !P.\supset :P\in \text{connex} .\equiv .\overleftrightarrow{P}ʻʻCʻP\in 1.\equiv .\overleftrightarrow{P}ʻʻCʻP=\iota ʻCʻP\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*202·1. &\supset \vdash :P\in \text{connex} .\equiv .\overleftrightarrow{P}ʻʻCʻP\subset \iota ʻCʻP &\qquad \text{(1)}\\ +\vdash .*37·45. \supset \vdash \colon\ldotp \text{Hp}.&\supset :\exists !\overleftrightarrow{P}ʻʻCʻP:\\ +[*54·102] &\supset :\overleftrightarrow{P}ʻʻCʻP{\sim}\in 0:\\ +[*202·102] &\supset :P\in \text{connex} .\supset .\overleftrightarrow{P}ʻʻCʻP\in 1 &\qquad \text{(2)}\\ +\vdash .*202·102. &\supset \vdash :\overleftrightarrow{P}ʻʻCʻP\in 1.\supset .P\in \text{connex} &\qquad \text{(3)}\\ +\vdash .(2).(3).& \supset \vdash \colon\ldotp \text{Hp}.\supset :P\in \text{connex} .\equiv .\overleftrightarrow{P}ʻʻCʻP\in 1 &\qquad \text{(4)}\\ +\vdash .(1).(4).*52·46.&\supset \vdash \colon\ldotp \text{Hp}.\supset :P\in \text{connex} .\supset .\overleftrightarrow{P}ʻʻCʻP=\iota ʻCʻP &\qquad \text{(5)}\\ +\vdash .(1).*22·42. &\supset \vdash :\overleftrightarrow{P}ʻʻCʻP=\iota ʻCʻP.\supset .P\in \text{connex} &\qquad \text{(6)}\\ +\vdash .(4).(5).(6).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_536">[Pg 536]</span></p> + +<p>The following propositions, down to <a href="#*202·181">*202·181</a> inclusive (excepting +<a href="#*202·16">*202·16</a><a href="#*202·161">·161</a>) are concerned with \(R_{*}\) and \(R_{\text{po}}\). +It often happens that these are connected when \(R\) is not so, +<i>e.g.</i> if \(R\) is the relation \(+_{c}1\) among inductive +cardinals.</p> + +<p class="nind"><b>*202·13.</b> \(\vdash : R_{*} \in \text{connex} . \equiv . R_{\text{po}} \in \text{connex}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *202·104 . \supset \\ +\vdash \colon\colon R_{*} \in \text{connex} . &\equiv \colon\ldotp x, y \in CʻR_{*} . x \neq y . \supset _{x, y} : xR_{*}y . \lor . yR_{*}x + \colon\ldotp \\ +[*91·542] &\equiv \colon\ldotp x, y \in CʻR_{*} . x \neq y . \supset _{x, y} : xR_{\text{po}}y . \lor . yR_{\text{po}}x \colon\ldotp \\ +[*90·14 . *91·504] &\equiv \colon\ldotp x, y \in CʻR_{\text{po}} . x \neq y . \supset _{x, y} : xR_{\text{po}}y . \lor . yR_{\text{po}}x \colon\ldotp \\ +[*202·104] &\equiv \colon\ldotp R_{\text{po}} \in \text{connex} \colon\colon \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*202·131.</b> \(\vdash : P \in \text{connex} . CʻP = CʻQ . P \,\unicode{x2abd}\, Q . \supset . Q \in \text{connex} \quad[*202·103]\)</p> + +<p class="nind"><b>*202·132.</b>\[\begin{align}&\vdash : P \in \text{connex} . \supset . P_{\text{po}}, P_{*} \in \text{connex} \\ +&[*202·131 . *90·14·151 . *91·502·504]\end{align}\]</p> + +<p class="nind"><b>*202·133.</b> \(\vdash \colon\colon I \upharpoonright CʻP \,\unicode{x2abd}\, P . \supset \colon\ldotp P \in \text{connex} . \equiv : x \in CʻP . \supset _{x} + . CʻP = \overrightarrow{P}ʻx \cup \overleftarrow{P}ʻx\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *35·101 . \supset \vdash \colon\ldotp \text{Hp} . \supset : x \in CʻP . \supset . \iota ʻx \subset \overrightarrow{P}ʻx &\qquad \text{(1)}\\ +\vdash . (1) . *202·101 . \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*202·134.</b>\[\begin{align}&\vdash + \colon\colon\ldotp I \upharpoonright CʻP \,\unicode{x2abd}\, P . \supset \colon\colon P \in \text{connex} . \equiv \colon\ldotp x, y \in CʻP . \supset _{x, y} : xPy . \lor . yPx\\ +&[*202·103]\end{align}\]</p> + +<p class="nind"><b>*202·135.</b> \(\vdash : P \in \text{connex} . \equiv . P \unicode{x228d} I \upharpoonright CʻP \in \text{connex}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *202·134 . \supset \vdash \colon\colon &P \unicode{x228d} I \upharpoonright CʻP \in \text{connex} . \equiv \colon\ldotp \\ +&x, y \in CʻP . \supset _{x, y} : x (P \unicode{x228d} I \upharpoonright CʻP) y . \lor . y (P \unicode{x228d} I \upharpoonright CʻP) x \colon\ldotp \\ +[*202·103] &\equiv \colon\ldotp P \in \text{connex} \colon\colon \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*202·136.</b>\[\begin{align}&\vdash \colon\ldotp P_{*} \in \text{connex} . \equiv : x \in CʻP . \supset _{x} . CʻP = \overrightarrow{P}_{*}ʻx + \cup \overleftarrow{P}_{*}ʻx\\ +&[*202·133 . *90·14·15]\end{align}\]</p> + +<p class="nind"><b>*202·137.</b>\[\begin{align}&\vdash \colon\colon P_{*} \in \text{connex} . \equiv \colon\ldotp x, y \in CʻP . \supset _{x, y} : xP_{*}y + . \lor . yP_{*}x\\ +&[*202·134 . *90·15]\end{align}\]</p> + +<p class="nind"><b>*202·138.</b> \(\vdash \colon\ldotp P \in \text{trans} . \supset : P \in \text{connex} . \equiv . P_{*} \in \text{connex} \quad[*202·13 . *201·18]\)</p> + +<p class="nind"><b>*202·14.</b> \(\vdash : R \in \text{Cls} \rightarrow 1 . \supset . R_{\text{po}} \unicode{x0294f} \overleftarrow{R}_{*}ʻx \in \text{connex} \quad[*96·303 . *202·104]\)</p> + +<p class="nind"><b>*202·141.</b> \(\vdash : R \in 1 \rightarrow \text{Cls} . \supset . R_{\text{po}} \unicode{x0294f} \overrightarrow{R}_{*}ʻx + \in \text{connex} \quad\left[*202·14 \frac{\breve{R}}{R}. *202·11\right]\)</p> + +<p class="nind"><b>*202·15.</b> \(\vdash : R \in 1 \rightarrow 1 . \supset . R_{\text{po}} \unicode{x0294f} \overleftrightarrow{R}_{*}ʻx \in \text{connex}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *97·13 . \supset \vdash \colon\ldotp y, z \in \overleftrightarrow{R}_{*}ʻx . \supset _{y, z} : &y, z \in \overrightarrow{R}_{*}ʻx + . \lor . y, z \in \overleftarrow{R}_{*}ʻx . \lor .\\ +&y \in \overrightarrow{R}_{*}ʻx . z \in \overleftarrow{R}_{*}ʻx . \lor . y \in \overleftarrow{R}_{*}ʻx + . z \in \overrightarrow{R}_{*}ʻx &\qquad \text{(1)}\\ +\vdash .*202·141·104. &\supset \vdash \colon\colon\text{Hp}.\supset \colon\ldotp y,z\in \overrightarrow{R}_{*}ʻx.y \neq z.\supset :yR_{\text{po}}x.\lor.xR_{\text{po}}y + &\qquad \text{(2)}\\ +\vdash .*202·14·104. &\supset \vdash \colon\colon\text{Hp}.\supset \colon\ldotp y,z\in \overleftarrow{R}_{*}ʻx.y \neq z.\supset :yR_{\text{po}}x.\lor.xR_{\text{po}}y + &\qquad \text{(3)}\\ +\vdash .*90·17. \supset \vdash :y\in \overrightarrow{R}_{*}ʻx.z\in \overleftarrow{R}_{*}ʻx.y \neq z.&\supset .yR_{*}z.y \neq z.\\ +[*91·542] &\supset .yR_{\text{po}}z &\qquad \text{(4)}\\ +\text{Similarly}\quad &\vdash :y\in \overleftarrow{R}_{*}ʻx.z\in \overrightarrow{R}_{*}ʻx.y \neq z.\supset .zR_{\text{po}}y &\qquad \text{(5)}\\ +\vdash .(1).(2).(3).(4).(5).\supset \\ +\vdash \colon\colon\text{Hp}.&\supset \colon\ldotp y,z\in \overleftrightarrow{R}_{*}ʻx.y \neq z.\supset _{y,z}:yR_{\text{po}}z.\lor.zR_{\text{po}}y + &\qquad \text{(6)}\\ +\vdash .(6).*202·104.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_537">[Pg 537]</span></p> + +<p>The above proposition is used in the ordinal theory of finite and +infinite (*260·4).</p> + +<p class="nind"><b><a id="*202·16">*202·16</a>.</b> \(\vdash :P\in \text{connex} .x,y\in CʻP.{\sim}(xPx).{\sim}(yPy).\overrightarrow{P}ʻx = \overrightarrow{P}ʻy.\supset .x = y\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*32·18·181.\supset \vdash :\text{Hp}.&\supset .{\sim}(xPy).{\sim}(yPx).\\ +[*202·103] &\supset .x = y:\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*202·161">*202·161</a>.</b> \(\vdash :P\in \text{connex} \cap R1ʻJ.\supset .\overrightarrow{P}\upharpoonright CʻP\in 1\rightarrow 1.\overrightarrow{P}\upharpoonright CʻP\in (\overrightarrow{P}^{;}P) + \overline{\,\text{smor}\,} P\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*202·16.\supset \vdash \colon\ldotp \text{Hp}.\supset :x,y\in CʻP.\overrightarrow{P}ʻx = \overrightarrow{P}ʻy.\supset .x = y &\qquad \text{(1)}\\ +\vdash .(1).*71·55.*151·24.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*202·162.</b> \(\vdash :P\in \text{connex} .P_{\text{po}}\,\unicode{x2abd}\, J.\supset .P\unicode{x0294f}^{;}\overrightarrow{P}_{*}^{;}P + \,\text{smor}\, P.P\unicode{x0294f}\mid \overrightarrow{P}_{*}\upharpoonright CʻP\in 1\rightarrow 1\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*36·13.\supset \vdash \colon\ldotp P\unicode{x0294f}\overrightarrow{P}_{*}ʻx = &P\unicode{x0294f}\overrightarrow{P}_{*}ʻy. \equiv :\\ +&uPv.u,v\in \overrightarrow{P}_{*}ʻx. \equiv _{u,v}.uPv.u,v\in \overrightarrow{P}_{*}ʻy &\qquad \text{(1)}\\ +\vdash .(1).*11·1.*90·12.\supset \\ +\vdash \colon\ldotp x,y\in CʻP.P\unicode{x0294f}\overrightarrow{P}_{*}ʻx = P\unicode{x0294f}\overrightarrow{P}_{*}ʻy.\supset :&xPy.yP_{*}x. \equiv .xPy.xP_{*}y:\\ +&yPx.yP_{*}x. \equiv .yPx.xP_{*}y:\\ +[*90·151.*91·52]&\supset :xPy.\supset .xP_{\text{po}}x:yPx.\supset .yP_{\text{po}}y &\qquad \text{(2)}\\ +\vdash .(2).&\supset \vdash :\text{Hp}.x,y\in CʻP.P\unicode{x0294f}\overrightarrow{P}_{*}ʻx = P\unicode{x0294f}\overrightarrow{P}_{*}ʻy.\supset .{\sim}(xPy).{\sim}(yPx).\\ +[*202·103] &\supset .x = y:\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*202·17.</b> \(\vdash :P_{\text{po}}\in \text{connex} .y\in P(x\vdash\dashv z).\supset .P(x\vdash\dashv y)\cup P(y\vdash\dashv z) = P(x\vdash\dashv z)\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*201·14·15.*121·103.\supset \\ +\vdash :\text{Hp}.&\supset .P(x\vdash\dashv y)\subset P(x\vdash\dashv z).P(y\vdash\dashv z)\subset P(x\vdash\dashv z) &\qquad \text{(1)}\\ +\vdash .*202·13·137.*121·103.\supset \\ +\vdash \colon\ldotp \text{Hp}.\omega \in P(x\vdash\dashv z).&\supset :\omega P_{*}y.\lor.yP_{*}\omega :xP_{*}\omega .\omega P_{*}z:\\ +[*121·103] &\supset :\omega \in P(x\vdash\dashv y)\cup P(y\vdash\dashv z) &\qquad \text{(2)}\\ +\vdash .(1).(2).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_538">[Pg 538]</span></p> + +<p class="nind"><b>*202·171.</b>\[\begin{align}&\vdash :P_{\text{po}}\in \text{connex} .y\in P(x\vdash\dashv z).\supset .\\ +&P(x\dashv z)=P(x\dashv y)\cup P(y\dashv z).P(x\unicode{x27dd} z)=P(x\vdash y)\cup P(y\unicode{x27dd} z)\\ +&[\text{Proof as in *202·17}]\end{align}\]</p> + +<p class="nind"><b>*202·172.</b>\[\begin{align}\vdash :P_{\text{po}}\in &\text{connex} .y\in P(x-z).\supset .\\ +&P(x-y)=P(x\dashv y)\cup P(y-z)=P(x-y)\cup P(y \unicode{x27dd} z)\\ +&[\text{Proof as in *202·17}]\end{align}\]</p> + +<p class="nind"><b>*202·18.</b> \(\vdash :P_{\text{po}}\in \text{connex} .\text{E}!BʻP.\supset .CʻP=\overleftarrow{P}_{*}ʻBʻP\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*202·1.\supset \vdash :\text{Hp}.\supset .CʻP&=\overleftrightarrow{P}_{\text{po}}ʻBʻP\\ +[*97·2.*91·504] &=\overleftarrow{P}_{*}ʻBʻP:\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*202·181">*202·181</a>.</b> \(\vdash :P_{\text{po}}\in \text{connex} .\text{E}!BʻP.\text{E}!Bʻ\breve{P} .\supset .CʻP=P(BʻP\vdash\dashv Bʻ\breve{P})\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*202·18.\supset \vdash :\text{Hp}.\supset .CʻP&=\overleftarrow{P}_{*}ʻBʻP\cap \overrightarrow{P}_{*}ʻBʻ\breve{P} \\ +[*121·103] &=P(BʻP\vdash\dashv Bʻ\breve{P} ):\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p>The above proposition is used in the ordinal theory of finite and +infinite (*261·2).</p> + +<p>The following proposition is a lemma for <a href="#*202·211">*202·211</a>, which shows that if +a relation is connected, so are all similar relations.</p> + +<p class="nind"><b>*202·21.</b> \(\vdash :P\in \text{connex} .S\in 1\rightarrow \text{Cls}.\supset .S^{;}P\in \text{connex}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*150·202.&\supset \vdash \colon\colon\text{Hp}.\supset \colon\ldotp x,y\in CʻS^{;}P.x\neq y.\supset :x,y\in SʻʻCʻP.x\neq y:\\ +[*71·4.*30·37]&\supset :(\exists z,w).z,w\in CʻP.x=Sʻz.y=Sʻw.z\neq w:\\ +[*202·104] &\supset :(\exists z,w):x=Sʻz.y=Sʻw:zPw.\lor.wPz:\\ +[*150·4] &\supset :x(S^{;}P)y.\lor.y(S^{;}P)x &\qquad \text{(1)}\\ +\vdash .(1).*202·104.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p>The proofs of the three following propositions proceed like the proofs +of the analogous propositions in <a href="#*200">*200</a> and <a href="#*201">*201</a>.</p> + +<p class="nind"><b><a id="*202·211">*202·211</a>.</b> \(\vdash :P\in \text{connex} .Q\,\,\text{smor}\,\,P.\supset .Q\in \text{connex}\)</p> + +<p class="nind"><b>*202·212.</b> \(\vdash :P\in \text{connex} .\supset .\text{Nr}ʻP\subset \text{connex}\)</p> + +<p class="nind"><b>*202·22.</b> \(\vdash :P\in \text{connex} .\equiv .\text{N}_{0}\text{r}ʻP\subset \text{connex} .\equiv .\exists !\text{N}_{0}\text{r}ʻP\cap \text{connex}\)</p> + +<p class="nind"><b><a id="*202·3">*202·3</a>.</b> \(\vdash .\dot{\Lambda} \in \text{connex}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*37·29.\supset \vdash :P=\dot{\Lambda} .&\supset .\overleftrightarrow{P}ʻʻCʻP=\Lambda .\\ +[*202·102] &\supset .P\in \text{connex} :\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_539">[Pg 539]</span></p> + +<p class="nind"><b><a id="*202·31">*202·31</a>.</b> \(\vdash .x\downarrow y\in \text{connex}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*55·15.\supset \vdash \colon\ldotp &z,w\in Cʻ(x\downarrow y).\supset :\\ +&z,w\in \iota ʻx.\lor.z,w\in \iota ʻy.\lor.z\in \iota ʻx.w\in \iota ʻy.\lor.z\in \iota ʻy.w\in \iota ʻx:\\ +[*51·15.*13·172] &\supset :z=w.\lor.z=x.w=y.\lor.z=y.w=x:\\ +[*55·15] &\supset :z=w.\lor.z(x\downarrow y)w.\lor.w(x\downarrow y)z &\qquad \text{(1)}\\ +\vdash .(1).*202·103.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*202·33">*202·33</a>.</b> \(\vdash :P\in \text{connex} .\supset .P\unicode{x0294f}\alpha \in \text{connex} \)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*37·41.&\supset \vdash :x,y\in CʻP\unicode{x0294f}\alpha .\supset .x,y\in \alpha .x,y\in CʻP &\qquad \text{(1)}\\ +\vdash .(1).*202·103.\supset \\ +\vdash \colon\colon\text{Hp}.\supset \colon\ldotp x,y\in CʻP\unicode{x0294f}\alpha .&\supset :x,y\in \alpha :xPy.\lor.x=y.\lor.yPx:\\ +[*36·13] &\supset :x(P\unicode{x0294f}\alpha )y.\lor.x=y.\lor.y(P\unicode{x0294f}\alpha )x &\qquad \text{(2)}\\ +\vdash .(2).*202·103.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p>The following propositions (<a href="#*202·4">*202·4</a>—<a href="#*202·42">·42</a>) are concerned with +applications of relation-arithmetic.</p> + +<p class="nind"><b><a id="*202·4">*202·4</a>.</b> \(\vdash :P,Q\in \text{connex} .\supset .P\unicode{x2909}Q\in \text{connex}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*160·14.&\supset \vdash \colon\ldotp x,y\in Cʻ(P\unicode{x2909}Q).\equiv :\\ +&x,y\in CʻP.\lor.x,y\in CʻQ.\lor.x\in CʻP.y\in CʻQ.\lor.x\in CʻQ.y\in CʻP &\qquad \text{(1)}\\ +\vdash .*202·103.\supset \vdash \colon\colon\text{Hp}.\supset \colon\ldotp x,y\in &CʻP.\supset :xPy.\lor.x=y.\lor.yPx:\\ +[*160·1] &\supset :x(P\unicode{x2909}Q)y.\lor.x=y.\lor.y(P\unicode{x2909}Q)x &\qquad \text{(2)}\\ +\text{Similarly}\quad \vdash \colon\colon\text{Hp}.&\supset \colon\ldotp x,y\in CʻQ.\supset :x(P\unicode{x2909}Q)y.\lor.x=y.\lor.y(P\unicode{x2909}Q)x &\qquad \text{(3)}\\ +\vdash .*160·1.*35·103.&\supset \vdash :x\in CʻP.y\in CʻQ.\supset .x(P\unicode{x2909}Q)y &\qquad \text{(4)}\\ +\vdash .*160·1.*35·103.&\supset \vdash :x\in CʻQ.y\in CʻP.\supset .y(P\unicode{x2909}Q)x &\qquad \text{(5)}\\ +\vdash .(1).(2).(3).(4).(5).\supset \\ +\vdash \colon\colon\text{Hp}.&\supset \colon\ldotp x,y\in Cʻ(P\unicode{x2909}Q).\supset :x(P\unicode{x2909}Q)y.\lor.x=y.\lor.y(P\unicode{x2909}Q)x &\qquad \text{(6)}\\ +\vdash .(6).*202·103.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p>The above proposition illustrates the reasons for defining +\(P\unicode{x2909}Q\) as was done in <a href="#*160">*160</a>. When \(P\) and \(Q\) are +connected, \(P\unicode{x228d} Q\) is in general not connected: it is +the additional term \(CʻP\uparrow CʻQ\) which insures connection.</p> + +<p class="nind"><b>*202·401.</b> \(\vdash \colon\ldotp CʻP\cap CʻQ=\Lambda .\supset :P\unicode{x2909}Q\in \text{connex} .\equiv .P,Q\in \text{connex}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*202·33.&\supset \vdash :P\unicode{x2909}Q\in \text{connex} .\supset .(P\unicode{x2909}Q)\unicode{x0294f}CʻP,(P\unicode{x2909}Q)\unicode{x0294f}CʻQ\in \text{connex} &\qquad \text{(1)}\\ +\vdash .(1).*160·5.&\supset \vdash \colon\ldotp \text{Hp}.\supset :P\unicode{x2909}Q\in \text{connex} .\supset .P,Q\in \text{connex} &\qquad \text{(2)}\\ +\vdash .(2).*202·4.&\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_540">[Pg 540]</span></p> + +<p class="nind"><b>*202·41.</b> \(\vdash : P \in \text{connex} . \supset . P \unicode{x21f8} z \in \text{connex} . z \unicode{x21f7} P \in \text{connex}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *161·14·2 . &\supset \vdash \colon\ldotp x, y \in Cʻ(P \unicode{x21f8} z) . x \neq y . \supset : x, y \in (CʻP \cup {℩}ʻz) . x \neq y :\\ +[*51·236] &\supset : x, y \in CʻP . x\neq y . \lor . x \in CʻP . y = z . \lor . y \in CʻP . x = z &\qquad \text{(1)}\\ +\vdash . (1) . *202·104 . \supset \\ +\vdash \colon\colon P \in \text{connex} . \supset \colon\ldotp &x, y \in Cʻ(P \unicode{x21f8} z) . x \neq y . \supset :\\ +&xPy . \lor . yPx . \lor . x \in CʻP . y = z . \lor . y \in CʻP . x = z :\\ +[*161·11] &\supset : x (P \unicode{x21f8} z) y . \lor . y (P \unicode{x21f8} z) x \colon\ldotp \\ +[*202·104] &\supset \colon\ldotp P \unicode{x21f8} z \in \text{connex} &\qquad \text{(2)}\\ +\text{Similarly}\quad \vdash : P \in \text{connex} . &\supset . z \unicode{x21f7} P \in \text{connex} &\qquad \text{(3)}\\ +\vdash . (2) . (3) . \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*202·411.</b> \(\vdash . x \downarrow y \unicode{x21f8} z \in \text{connex} \quad[*202·41·31]\)</p> + +<p class="nind"><b><a id="*202·412">*202·412</a>.</b> + \(\vdash \colon\ldotp z {\sim} \in CʻP . \supset : P \in \text{connex} . \equiv . P \unicode{x21f8} z \in \text{connex} . \equiv . z \unicode{x21f7} P \in \text{connex}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *161·16 . &\supset \vdash \colon\ldotp \text{Hp} . \supset : P = (P \unicode{x21f8} z) \unicode{x0294f} CʻP :\\ +[*202·33] &\supset : P \unicode{x21f8} z \in \text{connex} . \supset . P \in \text{connex} &\qquad \text{(1)}\\ +\text{Similarly}\quad &\vdash \colon\ldotp \text{Hp} . \supset . z \unicode{x21f7} P \in \text{connex} . \supset . P \in \text{connex} &\qquad \text{(2)}\\ +\vdash . (1) . (2) . *202·41 . \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*202·42">*202·42</a>.</b> \(\vdash : P \in \text{connex} . CʻP \subset \text{connex} . \supset . \Sigma ʻP \in \text{connex}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *162·22. &\supset \vdash : x, y \in Cʻ\Sigma ʻP . \equiv . (\exists Q, R) . Q, R \in CʻP . x \in CʻQ . y \in CʻR &\qquad \text{(1)}\\ +\vdash . (1) . *202·103 . \supset \\ +\vdash \colon\colon P \in \text{connex} . \supset \colon\ldotp x, y \in Cʻ\Sigma ʻP . \supset :\\ +&(\exists Q, R) : QPR . \lor . Q = R . Q, R \in CʻP . \lor . RPQ : x \in CʻQ . y \in CʻR &\qquad \text{(2)}\\ +\vdash . *162·13 . &\supset \vdash \colon\ldotp QPR . \lor . RPQ : x \in CʻQ . y \in CʻR : \supset :\\ +&x (\Sigma ʻP) y . \lor . y (\Sigma ʻP) x &\qquad \text{(3)}\\ +\vdash . *13·195 . &\supset \vdash : (\exists Q, R) . Q = R . Q, R \in CʻP . x \in CʻQ . y \in CʻR . \supset .\\ +&(\exists Q) . Q \in CʻP . x, y \in CʻQ &\qquad \text{(4)}\\ +\vdash . *202·103 . &\supset \vdash \colon\colon CʻP \subset \text{connex} . \supset \colon\ldotp (\exists Q) . Q \in CʻP . x, y \in CʻQ . \supset :\\ +&(\exists Q) : Q \in CʻP : xQy . \lor . x = y . \lor . yQx :\\ +[*162·13] &\supset : x (\Sigma ʻP) y . \lor . x = y . \lor . y (\Sigma ʻP) x &\qquad \text{(5)}\\ +\vdash . (4) . (5) . \supset \\ +\vdash \colon\colon CʻP \subset \text{connex} . &\supset \colon\ldotp (\exists Q, R) . Q = R . Q, R \in CʻP . x \in CʻQ . y \in CʻR . \supset :\\ +&x (\Sigma ʻP) y . \lor . x = y . \lor . y (\Sigma ʻP) x &\qquad \text{(6)}\\ +\vdash . (2) . (3) . (6) . &\supset \vdash \colon\colon \text{Hp} . \supset \colon\ldotp \\ +&x, y \in Cʻ\Sigma ʻP . \supset : x (\Sigma ʻP) y . \lor . x = y . \lor . y (\Sigma ʻP) x &\qquad \text{(7)}\\ +\vdash . (7) . *202·103 . \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_541">[Pg 541]</span></p> + +<p class="nind"><b><a id="*202·5">*202·5</a>.</b> \(\vdash \colon\ldotp P\in \text{connex} .P^{2}\,\unicode{x2abd}\, J.x,y\in CʻP.\supset :x\neq y.{\sim}(xPy).\equiv .yPx\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*50·43. &\supset \vdash \colon\ldotp P^{2}\,\unicode{x2abd}\, J.\supset :yPx.\supset .{\sim}(xPy) &\qquad \text{(1)}\\ +\vdash .*200·36. &\supset \vdash \colon\ldotp P^{2}\,\unicode{x2abd}\, J.\supset :yPx.\supset .x\neq y &\qquad \text{(2)}\\ +\vdash .*202·104. &\supset \vdash \colon\ldotp P\in \text{connex} .x,y\in CʻP.\supset :x\neq y.{\sim}(xPy).\supset .yPx &\qquad \text{(3)}\\ +\vdash .(1).(2).(3).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p>The following propositions (<a href="#*202·501">*202·501</a>-<a href="#*202·51">·51</a>) are concerned with the +relations of \(Pʻʻ\alpha\) and \(pʻ\overleftarrow{P}ʻʻ(\alpha \cap CʻP)\). +They are important, and <a href="#*202·501">*202·501</a><a href="#*202·503">·503</a><a href="#*202·505">·505</a> will be often used.</p> + +<p class="nind"><b><a id="*202·501">*202·501</a>.</b> \(\vdash :P\in \text{connex} .\supset .CʻP-\alpha -Pʻʻ\alpha \subset pʻ\overleftarrow{P}ʻʻ(\alpha \cap CʻP)\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*13·14.*37·1.\supset \vdash \colon\ldotp y\in CʻP-\alpha -Pʻʻ\alpha .x\in \alpha .\supset .x\neq y.{\sim}(yPx) &\qquad \text{(1)}\\ +\vdash .(1).*202·103.\supset \vdash \colon\ldotp \text{Hp}.\supset :y\in CʻP-\alpha -Pʻʻ\alpha .x\in \alpha \cap CʻP.\supset .xPy:\\ +[*40·53]\supset :y\in CʻP-\alpha -Pʻʻ\alpha .\supset .y\in pʻ\overleftarrow{P}ʻʻ(a\cap CʻP)\colon\ldotp \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*202·502.</b> \(\vdash :P\in \text{connex} .P^{2}\,\unicode{x2abd}\, J.\exists !\alpha \cap CʻP.\supset .CʻP-\alpha -Pʻʻ\alpha =pʻ\overleftarrow{P}ʻʻ(\alpha \cap CʻP)\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*40·62. &\supset \vdash :\text{Hp}.\supset .pʻ\overleftarrow{P}ʻʻ(\alpha \cap CʻP)\subset CʻP &\qquad \text{(1)}\\ +\vdash .*200·5. &\supset \vdash :\text{Hp}.\supset .pʻ\overleftarrow{P}ʻʻ(\alpha \cap CʻP)\subset -\alpha &\qquad \text{(2)}\\ +\vdash .*200·53. &\supset \vdash :\text{Hp}.\supset .pʻ\overleftarrow{P}ʻʻ(\alpha \cap CʻP)\subset -Pʻʻ\alpha &\qquad \text{(3)}\\ +\vdash .(1).(2).(3).*202·501.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*202·503">*202·503</a>.</b> \(\vdash :P\in \text{connex} .P^{2}\,\unicode{x2abd}\, J.\supset .CʻP-pʻ\overleftarrow{P}ʻʻ(\alpha \cap CʻP)=(\alpha \cap CʻP)\cup Pʻʻ\alpha\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*202·501.*24·43.&\supset \vdash :\text{Hp}.\supset .CʻP-pʻ\overleftarrow{P}ʻʻ(\alpha \cap CʻP)\subset \alpha \cup Pʻʻ\alpha &\qquad \text{(1)}\\ +\vdash .(1).*22·43. &\supset \vdash :\text{Hp}.\supset .CʻP-pʻ\overleftarrow{P}ʻʻ(\alpha \cap CʻP)\subset (\alpha \cup Pʻʻ\alpha )\cap CʻP\\ +[*22·68.*37·15] & \subset (\alpha \cap CʻP)\cup Pʻʻ\alpha &\qquad \text{(2)}\\ +\vdash .*200·5·36. &\supset \vdash :\text{Hp}.\supset .\alpha \cap CʻP\subset -pʻ\overleftarrow{P}ʻʻ(\alpha \cap CʻP) &\qquad \text{(3)}\\ +\vdash .*200·53.&\supset \vdash :\text{Hp}.\supset .Pʻʻ\alpha \subset -pʻ\overleftarrow{P}ʻʻ(\alpha \cap CʻP) &\qquad \text{(4)}\\ +\vdash .*22·43.*37·15. &\supset \vdash .\alpha \cap CʻP\subset CʻP.Pʻʻ\alpha \subset CʻP &\qquad \text{(5)}\\ +\vdash .(3).(4).(5).&\supset \vdash :\text{Hp}.\supset .(\alpha \cap CʻP)\cup Pʻʻ\alpha \subset CʻP-pʻ\overleftarrow{P}ʻʻ(\alpha \cap CʻP) &\qquad \text{(6)}\\ +\vdash .(2).(6).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*202·504</b> \(\vdash :P\in \text{connex} .P^{2}\,\unicode{x2abd}\, J.\supset .CʻP\cap pʻ\overleftarrow{P}ʻʻ(\alpha \cap CʻP)=CʻP-\alpha -Pʻʻ\alpha\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*200·5·36. &\supset \vdash :\text{Hp}.\supset .pʻ\overleftarrow{P}ʻʻ(\alpha \cap CʻP)\subset -\alpha &\qquad \text{(1)}\\ +\vdash .*200·53. &\supset \vdash :\text{Hp}.\supset .pʻ\overleftarrow{P}ʻʻ(\alpha \cap CʻP)\subset -Pʻʻ\alpha &\qquad \text{(2)}\\ +\vdash .(1).(2).*22·48.&\supset \vdash :\text{Hp}.\supset .CʻP\cap pʻ\overleftarrow{P}ʻʻ(\alpha \cap CʻP)\subset CʻP-\alpha -Pʻʻ\alpha &\qquad \text{(3)}\\ +\vdash .(3).*202·501.&\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_542">[Pg 542]</span></p> + +<p class="nind"><b><a id="*202·505">*202·505</a>.</b> \(\vdash :P\in \text{connex} .\supset .CʻP=Pʻʻ\alpha \cup (\alpha \cap CʻP)\cup \{CʻP\cap pʻ\overleftarrow{P}ʻʻ(\alpha \cap CʻP)\}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*202·501. &\supset \vdash :\text{Hp}.\supset .CʻP-\alpha -Pʻʻ\alpha \subset pʻ\overleftarrow{P}ʻʻ(\alpha \cap CʻP).\\ +[*24·43] &\supset .CʻP\subset \alpha \cup Pʻʻ\alpha \cup \{pʻ\overleftarrow{P}ʻʻ(\alpha \cap CʻP)\}.\\ +[*22·621.*37·15] &\supset .CʻP=(\alpha \cap CʻP)\cup Pʻʻ\alpha \cup \{CʻP\cap pʻ\overleftarrow{P}ʻʻ(\alpha \cap CʻP)\}:\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*202·51">*202·51</a>.</b>\[\begin{align}\vdash :P\in \text{connex} .\alpha \subset CʻP.&\exists !\alpha .\supset .\\ +&CʻP=Pʻʻ\alpha \cup \alpha \cup pʻ\overleftarrow{P}ʻʻ\alpha =\breve{P} ʻʻ\alpha \cup \alpha \cup pʻ\overrightarrow{P}ʻʻ\alpha\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*40·62. &\supset \vdash :\text{Hp}.\supset .pʻ\overleftarrow{P}ʻʻ\alpha \subset CʻP &\qquad \text{(1)}\\ +\vdash .*22·621. &\supset \vdash :\text{Hp}.\supset .\alpha =\alpha \cap CʻP &\qquad \text{(2)}\\ +\vdash .(1).(2).*202·505.&\supset \vdash :\text{Hp}.\supset .CʻP=Pʻʻ\alpha \cup \alpha \cup pʻ\overleftarrow{P}ʻʻ\alpha &\qquad \text{(3)}\\ +\vdash .(3)\frac{\breve{P}}{P}.*202·11.&\supset \vdash :\text{Hp}.\supset .CʻP=\breve{P} ʻʻ\alpha \cup \alpha \cup pʻ\overrightarrow{P}ʻʻ\alpha &\qquad \text{(4)}\\ +\vdash .(3).(4).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p>The following propositions (<a href="#*202·511">*202·511</a>—<a href="#*202·524">·524</a>) are concerned with +\(\overrightarrow{B}ʻP\). <a href="#*202·52">*202·52</a> shows that if \(P\in \text{connex}\), +\(P\) cannot have more than one first term or more than one last +term, and <a href="#*202·523">*202·523</a> shows that this still holds if only \(P_{*}\) +is connected. *202·511 shows that if \(P\) is a connected relation +which has a first term, then if \(\alpha\) is any class, there are +predecessors of the whole of \(\alpha \cap CʻP\) when and only when +\(BʻP\) is such a predecessor, and when and only when \(BʻP{\sim}\in \alpha\). +<a href="#*202·524">*202·524</a> shows that if \(P\) is connected and has a first +term, \(\text{ᗡ}ʻP\) consists of the successors of the first term. +These propositions are much used.</p> + +<p class="nind"><b><a id="*202·511">*202·511</a>.</b>\[\begin{align}\vdash \colon\ldotp P\in &\text{connex} .\text{E}!BʻP.\supset :\\ +&\exists !pʻ\overrightarrow{P}ʻʻ(\alpha \cap CʻP).\equiv .BʻP{\sim}\in \alpha .\equiv .BʻP\in pʻ\overrightarrow{P}ʻʻ(\alpha \cap CʻP)\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*202·104.*93·1.\supset \vdash \colon\ldotp \text{Hp}.BʻP{\sim}\in \alpha .&\supset :x\in (\alpha \cap CʻP).\supset _{x}.(BʻP)Px:\\ +[*40·51] & \supset :BʻP\in pʻ\overrightarrow{P}ʻʻ(\alpha \cap CʻP): &\qquad \text{(1)}\\ +[*10·24] &\supset :\exists !pʻ\overrightarrow{P}ʻʻ(\alpha \cap CʻP) &\qquad \text{(2)}\\ +\vdash .*93·1. \supset \vdash :\text{Hp}.BʻP\in \alpha .&\supset .(x).{\sim}\{xP(BʻP)\}.BʻP\in \alpha \cap CʻP.\\ +[*40·51] &\supset .pʻ\overrightarrow{P}ʻʻ(\alpha \cap CʻP)=\Lambda . &\qquad \text{(3)}\\ +[*24·105] &\supset .BʻP{\sim}\in pʻ\overrightarrow{P}ʻʻ(\alpha \cap CʻP) &\qquad \text{(4)}\\ +\vdash .(2).(3).&\supset \vdash \colon\ldotp \text{Hp}.\supset :BʻP{\sim}\in \alpha .\equiv .\exists !pʻ\overrightarrow{P}ʻʻ(\alpha \cap CʻP) &\qquad \text{(5)}\\ +\vdash .(1).(4).&\supset \vdash \colon\ldotp \text{Hp}.\supset :BʻP{\sim}\in \alpha .\equiv .BʻP\in pʻ\overrightarrow{P}ʻʻ(\alpha \cap CʻP) &\qquad \text{(6)}\\ +\vdash .(5).(6).&\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_543">[Pg 543]</span></p> + +<p class="nind"><b><a id="*202·52">*202·52</a>.</b> \(\vdash :P\in \text{connex} .\supset .\overrightarrow{B}ʻP,\overrightarrow{B}ʻ\breve{P} \in 0\cup 1\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*93·103. \supset \vdash :x,y\in \overrightarrow{B}ʻP.&\supset .x,y\in CʻP.x{\sim}\in \text{ᗡ}ʻP.y{\sim}\in \text{ᗡ}ʻP.\\ +[*33·14] &\supset .x,y\in CʻP.{\sim}(xPy).{\sim}(yPx) &\qquad \text{(1)}\\ +\vdash .(1).*202·103. \supset \vdash \colon\ldotp \text{Hp}.&\supset :x,y\in \overrightarrow{B}ʻP.\supset .x=y:\\ +[*52·4] &\supset :\overrightarrow{B}ʻP\in 0\cup 1 &\qquad \text{(2)}\\ +\vdash .(2).*202·11. &\supset \vdash :\text{Hp}.\supset .\overrightarrow{B}ʻ\breve{P} \in 0\cup 1 &\qquad \text{(3)}\\ +\vdash .(2).(3).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*202·521.</b> \(\vdash :P_{*}\in \text{connex} .\supset .\overrightarrow{B}ʻP\subset pʻ\overrightarrow{P}_{*}ʻʻCʻP\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*202·13·103.\supset \\ +\vdash \colon\colon\text{Hp}.&\supset \colon\ldotp x\in \overrightarrow{B}ʻP.y\in CʻP.\supset :xP_{\text{po}}y.\lor.x=y.\lor.yP_{\text{po}}x &\qquad \text{(1)}\\ +\vdash .*91·504.&\supset \vdash :x\in \overrightarrow{B}ʻP.\supset .{\sim}(yP_{\text{po}}x) &\qquad \text{(2)}\\ +\vdash .(1).(2).&\supset \vdash \colon\colon\text{Hp}.\supset \colon\ldotp x\in \overrightarrow{B}ʻP.y\in CʻP.\supset :xP_{\text{po}}y.\lor.x=y:\\ +[*91·54] &\supset :xP_{*}y\colon\colon\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*202·522.</b> \(\vdash .\overrightarrow{B}ʻP=\overrightarrow{B}ʻP_{\text{po}} \quad[*91·504]\)</p> + +<p class="nind"><b><a id="*202·523">*202·523</a>.</b> \(\vdash :P_{*}\in \text{connex} .\supset .\overrightarrow{B}ʻP\in 0\cup 1 \quad[*202·13·52·522]\)</p> + +<p class="nind"><b><a id="*202·524">*202·524</a>.</b> \(\vdash :P\in \text{connex} .\exists !\overrightarrow{B}ʻP.\supset .\text{ᗡ}ʻP=\overleftarrow{P}ʻBʻP\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*202·52.\supset \vdash \colon\ldotp \text{Hp}.&\supset :\text{E}!BʻP:\\ +[*202·104.*93·103] &\supset :x\in \text{ᗡ}ʻP.\supset .(BʻP)Px &\qquad \text{(1)}\\ +\vdash .(1).*33·151.\supset \vdash \text{Prop} +\end{array} +\]</p> + +<p>The following propositions (<a href="#*202·53">*202·53</a>—<a href="#*202·55">·55</a>) are concerned with relations +with limited fields. Such relations are constantly used in the theory +of series.</p> + +<p class="nind"><b><a id="*202·53">*202·53</a>.</b> \(\vdash :Q\in \text{connex} .P^{2}\,\unicode{x2abd}\, J.Q\,\unicode{x2abd}\, P.\supset .Q=P\unicode{x0294f}CʻQ\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*33·17.*36·13. &\supset \vdash \colon\ldotp \text{Hp}.\supset :xQy.\supset .x(P\unicode{x0294f}CʻQ)y &\qquad \text{(1)}\\ +\vdash .*50·43. \supset \vdash \colon\ldotp \text{Hp}.&\supset :xPy.\supset .{\sim}(yPx).\\ +[*23·81] &\supset .{\sim}(yQx). &\qquad \text{(2)}\\ +\vdash .*200·36.&\supset \vdash \colon\ldotp \text{Hp}.\supset :xPy.\supset .x\neq y &\qquad \text{(3)}\\ +\vdash .(2).(3).*202·104.\supset \vdash \colon\ldotp \text{Hp}.&\supset :x,y\in CʻQ.xPy.\supset .xQy:\\ +[*36·13]&\supset :x(P\unicode{x0294f}CʻQ)y.\supset .xQy &\qquad \text{(4)}\\ +\vdash .(1).(4).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_544">[Pg 544]</span></p> + +<p>This proposition is important in series. If \(P\) and \(Q\) are +serial relations, and \(Q\,\unicode{x2abd}\, P\), they verify the above +hypothesis; hence if \(Q\) is a series contained in a given series +\(P\), \(Q\) is simply \(P\) with its field limited. Thus series +contained in a given series are completely determined by their fields.</p> + +<p class="nind"><b><a id="*202·54">*202·54</a>.</b> \(\vdash :P\unicode{x0294f}\alpha \in \text{connex} .\alpha \cap CʻP{\sim}\in 1.\supset .CʻP\unicode{x0294f}\alpha = \alpha \cap CʻP\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*52·181.\supset \\ +\vdash \colon\colon\text{Hp}.\supset \colon\ldotp x\in \alpha \cap CʻP.&\supset _{x}:(\exists y).y\in \alpha \cap CʻP.y \neq x:\\ +[*202·104] &\supset _{x}:(\exists y).y\in \alpha \cap CʻP:xPy.\lor.yPx:\\ +[*36·13] &\supset _{x}:(\exists y):x(P\unicode{x0294f}\alpha )y.\lor.y(P\unicode{x0294f}\alpha )x:\\ +[*33·132] &\supset _{x}:x\in CʻP\unicode{x0294f}\alpha &\qquad \text{(1)}\\ +\vdash .*37·41·15·16.&\supset \vdash .CʻP\unicode{x0294f}\alpha \subset \alpha \cap CʻP &\qquad \text{(2)}\\ +\vdash .(1).(2).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p>The above proposition is frequently used. <a href="#*202·55">*202·55</a>, which is an +immediate consequence of <a href="#*202·54">*202·54</a>, is used incessantly.</p> + +<p>The following proposition is used in <a href="#*232·14">*232·14</a>.</p> + +<p class="nind"><b>*202·541.</b> \(\vdash :P\in \text{trans} \cap \text{connex} .\alpha \cap CʻP{\sim}\in 1.\supset .(P\unicode{x0294f}\alpha )_{*} = P_{*}\unicode{x0294f}\alpha\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*201·18·33 .\supset \vdash :\text{Hp}.\supset .(P\unicode{x0294f}\alpha )_{*} &= P\unicode{x0294f}\alpha \unicode{x228d} I\upharpoonright (CʻP\unicode{x0294f}\alpha )\\ +[*202·54] &= P\unicode{x0294f}\alpha \unicode{x228d} I\upharpoonright (CʻP\cap \alpha )\\ +[*201·18.*36·23.*50·5] &=P_{*}\unicode{x0294f}\alpha +\end{array} +\]</p> + +<p class="nind"><b><a id="*202·55">*202·55</a>.</b> \(\vdash :P\unicode{x0294f}\alpha \in \text{connex} .\alpha \subset CʻP.\alpha {\sim}\in 1.\supset .CʻP\unicode{x0294f}\alpha = \alpha \quad[*202·54]\)</p> + +<p class="nind"><b>*202·56.</b> \(\vdash .P\in \text{connex} .P\,\unicode{x2abd}\, J.x\in CʻP.\beta \subset CʻP.Pʻʻ\beta \subset \overrightarrow{P}ʻx.\supset .\beta \subset \overrightarrow{P}ʻx\cup \iota ʻx\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*37·1. &\supset \vdash :Pʻʻ\beta \subset \overrightarrow{P}ʻx.y\in \beta .xPy.\supset .xPx &\qquad \text{(1)}\\ +\vdash .(1).\text{Transp}. &\supset \vdash :\text{Hp}.y\in \beta .\supset .{\sim}(xPy) &\qquad \text{(2)}\\ +\vdash .(2).*32·18. &\supset \vdash :\text{Hp}.y\in \beta - \overrightarrow{P}ʻx.\supset .{\sim}(xPy).{\sim}(yPx).\\ +[*202·103] &\supset .y = x:\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p>The above proposition is used in <a href="#*212·652">*212·652</a>.</p> + +<p class="nind"><b>*202·6.</b> \(\vdash \colon\colon P\in \text{connex} .P\,\unicode{x2abd}\, J.\supset \colon\ldotp x,y\in CʻP.x \neq y. \equiv :xPy.\lor.yPx\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*202·104.& \supset \vdash \colon\colon\text{Hp}.\supset \colon\ldotp x,y\in CʻP.x \neq y.\supset :xPy.\lor.yPx &\qquad \text{(1)}\\ +\vdash .*50·11.*33·17. &\supset \vdash \colon\colon\text{Hp}.\supset \colon\ldotp xPy.\lor.yPx:\supset .x,y\in CʻP.x \neq y &\qquad \text{(2)}\\ +\vdash .(1).(2).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p>The following proposition is a lemma for <a href="#*202·62">*202·62</a>, which is itself a +lemma for <a href="#*204·52">*204·52</a>.</p> + +<p class="nind"><b>*202·61.</b>\[\begin{align}\vdash \colon\colon P\in &\text{connex} .P\,\unicode{x2abd}\, J:\phi (x,y). \equiv _{x,y}.\phi (y,x):\supset \colon\ldotp \\ +&xPy.\supset _{x,y}.\phi (x,y): \equiv :x,y\in CʻP.x \neq y.\supset _{x,y}.\phi (x,y)\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*202·6.\supset \vdash \colon\colon\ldotp &\text{Hp}.\supset \colon\colon x,y\in CʻP.x \neq y.\supset _{x,y}.\phi (x,y): \equiv \colon\ldotp \\ +&xPy.\lor.yPx:\supset _{x,y}.\phi (x,y)\colon\ldotp \\ +[*4·77] &\equiv \colon\ldotp xPy.\supset _{x,y}.\phi (x,y):yPx.\supset _{x,y}.\phi (x,y)\colon\ldotp \\ +[*4·85.\text{Hp}] &\equiv \colon\ldotp xPy.\supset _{x,y}.\phi (x,y):yPx.\supset _{x,y}.\phi (y,x)\colon\ldotp \\ +[*4·24] &\equiv \colon\ldotp xPy.\supset _{x,y}.\phi (x,y)\colon\colon\ldotp \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_545">[Pg 545]</span></p> + +<p class="nind"><b>*202·611.</b>\[\begin{align}&\vdash + \colon\ldotp P \in \text{connex} . P \,\unicode{x2abd}\, J . R = \breve{R} . \supset : P \,\unicode{x2abd}\, R . \equiv . J \unicode{x0294f} CʻP \,\unicode{x2abd}\, R\\ +&\left[*202·61 \frac{xRy}{\phi(x, y)}\right]\end{align}\]</p> + +<p class="nind"><b><a id="*202·62">*202·62</a>.</b> \(\vdash \colon\ldotp P \in \text{connex} . P \,\unicode{x2abd}\, J . \supset : P \in \text{Rel}^{2}\text{excl} . \equiv . F^{;}P \,\unicode{x2abd}\, J\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *202·61 . *163·1 . &\supset \vdash \colon\colon \text{Hp} . \supset \colon\ldotp \\ +P \in \text{Rel}^{2}\text{excl} . &\equiv : QPR . \supset _{Q, R} . CʻQ \cap CʻR = \Lambda :\\ +[*24·37] &\equiv : QPR . x \in CʻQ . y \in CʻR . \supset _{Q, R, x, y} . x \neq y :\\ +[*150·52] &\equiv . x (F^{;}P) y . \supset _{x, y} . x \neq y \colon\colon \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p>The three following propositions (<a href="#*202·7">*202·7</a>—<a href="#*202·72">·72</a>) are concerned with \(P\dot{-} P^{2}\). +Of these, *202·7 is important: it shows that if \(P\) +is connected, no term can have more than one immediate predecessor +or successor. *202·72 is used in <a href="#*204·71">*204·71</a>, which is an important +proposition.</p> + +<p class="nind"><b><a id="*202·7">*202·7</a>.</b> \(\vdash : P \in \text{connex} . \supset . P \dot{-} P^{2} \in 1 \rightarrow 1\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *34·5 . \text{Transp} . &\supset \vdash : zPx . {\sim} (yP^{2}x) . \supset . {\sim} (yPz) &\qquad \text{(1)}\\ +\text{Similarly}\quad &\vdash : yPx . {\sim} (zP^{2}x) . \supset . {\sim} (zPy) &\qquad \text{(2)}\\ +\vdash . (1) . (2) . &\supset \vdash : y (P \dot{-} P^{2}) x . z (P \dot{-} P^{2}) x . \supset . {\sim} (yPz) . {\sim} (zPy) &\qquad \text{(3)}\\ +\vdash . (3) . *202·103 . &\supset \vdash \colon\ldotp \text{Hp} . \supset : y (P \dot{-} P^{2}) x . z (P \dot{-} P^{2}) x . \supset . y = z &\qquad \text{(4)}\\ +\text{Similarly}\quad &\vdash \colon\ldotp \text{Hp} . \supset : x (P \dot{-} P^{2}) y . x (P \dot{-} P^{2}) z . \supset . y = z &\qquad \text{(5)}\\ +\vdash . (4) . (5) . \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*202·71.</b> \(\vdash : P \in \text{connex} . x (P \dot{-} P^{2}) y . \supset . \overrightarrow{P}_{\text{po}}ʻy = \overrightarrow{P}_{*}ʻx\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *91·52 . \supset \vdash : \text{Hp} . &\supset . \overrightarrow{P}_{*}ʻx \subset \overrightarrow{P}_{\text{po}}ʻy &\qquad \text{(1)}\\ +\vdash . *91·57 . \supset \vdash \colon\ldotp zP_{\text{po}}y . &\supset : zPy . \lor . zP_{\text{po}} \mid Py :\\ +[*25·41] &\supset : z (P \dot{-} P^{2}) y . \lor . z (P \dot{\cap} P^{2}) y . \lor . z (P_{\text{po}} \mid P) y :\\ +[*91·502] &\supset : z (P \dot{-} P^{2}) y . \lor . z (P_{\text{po}} \mid P) y &\qquad \text{(2)}\\ +\vdash . *202·7. &\supset \vdash : \text{Hp} . z (P \dot{-} P^{2}) y . \supset . z = x &\qquad \text{(3)}\\ +\vdash . (2) . (3) . &\supset \vdash : \text{Hp} . zP_{\text{po}}y . z \neq x . \supset . z (P_{\text{po}} \mid P) y .\\ +[*34·1] &\supset . (\exists w) . zP_{\text{po}} w . wPy &\qquad \text{(4)}\\ +\vdash . *34·5 . &\supset \vdash : wPy . xPw . \supset . xP^{2}y &\qquad \text{(5)}\\ +\vdash . (5) . &\text{Transp} . \supset \vdash \colon\ldotp \text{Hp} . wPy . \supset : {\sim} (xPw) :\\ +[*202·103] &\supset : wPx . \lor . w = x &\qquad \text{(6)}\\ +\vdash . (4) . (6) . &\supset \vdash \colon\ldotp \text{Hp} . zP_{\text{po}}y . z \neq x . \supset : zP_{\text{po}}x . \lor . (\exists w) . zP_{\text{po}}w . wPx :\\ +[*91·511] &\supset : zP_{\text{po}}x &\qquad \text{(7)}\\ +\vdash . (7) . *91·54 . &\supset \vdash : \text{Hp} . \supset . \overrightarrow{P}_{\text{po}}ʻy \subset \overrightarrow{P}_{*}ʻx &\qquad \text{(8)}\\ +\vdash . (1) . (8) . \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_546">[Pg 546]</span></p> + +<p class="nind"><b><a id="*202·72">*202·72</a>.</b>\[\begin{align}&\vdash :P\in \text{trans}\cap \text{connex} .x(P\dot{-} P^{2})y.\supset .\overrightarrow{P}ʻy=\overrightarrow{P}ʻx\cup \iota ʻx\\ +&[*202·71.*201·18·521]\end{align}\]</p> + +<p class="nind"><b>*202·8.</b>\[\begin{align}\vdash :Q\in \text{connex} .S\in P\,\overline{\,\text{smor}\,}\,Q.CʻQ\cap &\beta {\sim}\in 1.\supset .\\ +&S\upharpoonright \beta \in (P\unicode{x0294f}Sʻʻ\beta )\overline{\,\text{smor}\,} Q\unicode{x0294f}\beta\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*71·29. &\supset \vdash :\text{Hp}.\supset .S\upharpoonright \beta \in 1\rightarrow 1 &\qquad \text{(1)}\\ +\vdash .*35·64.*151·11. \supset \vdash :\text{Hp}.\supset .\text{ᗡ}ʻ(S\upharpoonright \beta )&=CʻQ\cap \beta \\ +[*202·54] & =Cʻ(Q\unicode{x0294f}\beta ) &\qquad \text{(2)}\\ +\vdash .*150·37. & \supset \vdash :\text{Hp}.\supset .(S\upharpoonright \beta )^{;}Q=P\unicode{x0294f}Sʻʻ\beta &\qquad \text{(3)}\\ +\vdash .(1).(2).(3).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*202·81.</b> \(\vdash :Q\in \text{connex} .S\in P\,\overline{\,\text{smor}\,}\,Q.\supset .(P\unicode{x0294f}Sʻʻ\beta )\text{ sm }Q\unicode{x0294f}\beta\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*202·8. & \supset \vdash :\text{Hp}.CʻQ\cap \beta {\sim}\in 1.\supset .(P\unicode{x0294f}Sʻʻ\beta )\,\text{smor}\,Q\unicode{x0294f}\beta &\qquad \text{(1)}\\ +\vdash .*36·13.*33·17. & \supset \vdash :CʻQ\cap \beta =\iota ʻy.\supset .Q\unicode{x0294f}\beta \,\unicode{x2abd}\, y\downarrow y &\qquad \text{(2)}\\ +\vdash .*36·13. & \supset \vdash \colon\ldotp \text{Hp}(2).\supset :y(Q\unicode{x0294f}\beta )y.\equiv .yQy &\qquad \text{(3)}\\ +\vdash .(2).(3).*55·341.& \supset \vdash :\text{Hp}(2).yQy.\supset .Q\unicode{x0294f}\beta =y\downarrow y &\qquad \text{(4)}\\ +\vdash .*35·64.*151·11. &\supset \vdash :\text{Hp}.\supset .\text{ᗡ}ʻ(S\upharpoonright \beta )=CʻQ\cap \beta &\qquad \text{(5)}\\ +\vdash .(4).(5).& \supset \vdash :\text{Hp}(4).\supset .\text{ᗡ}ʻ(S\upharpoonright \beta )=Cʻ(Q\unicode{x0294f}\beta ) &\qquad \text{(6)}\\ +\vdash .*71·29.*150·37. &\supset \vdash :\text{Hp}.\supset .S\upharpoonright \beta \in 1\rightarrow 1.P\unicode{x0294f}Sʻʻ\beta =S^{;}(Q\unicode{x0294f}\beta ) &\qquad \text{(7)}\\ +\vdash .(6).(7).*151·1. & \supset \vdash :\text{Hp}(4).\supset .(P\unicode{x0294f}Sʻʻ\beta )\,\text{smor}\,Q\unicode{x0294f}\beta &\qquad \text{(8)}\\ +\vdash .(2).(3).*55·341. &\supset \vdash :\text{Hp}(2).{\sim}(yQy).\supset .Q\unicode{x0294f}\beta =\dot{\Lambda} . &\qquad \text{(9)}\\ +[(7).*150·42] &\supset .P\unicode{x0294f}Sʻʻ\beta =\dot{\Lambda} &\qquad \text{(10)}\\ +\vdash .(9).(10).*153·101. &\supset \vdash :\text{Hp}(9).\supset .(P\unicode{x0294f}Sʻʻ\beta )\,\text{smor}\,Q\unicode{x0294f}\beta &\qquad \text{(11)}\\ +\vdash .(8).(11).*52·1.&\supset \vdash :\text{Hp}.CʻQ\cap \beta \in 1.\supset .(P\unicode{x0294f}Sʻʻ\beta )\,\text{smor}\,Q\unicode{x0294f}\beta &\qquad \text{(12)}\\ +\vdash .(1).(12).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p>The above proposition shows that if \(Q\) is connected, and any class +\(\beta\) is picked out of \(CʻQ\), then \(Q\) arranges \(\beta\) in an +order which is similar to that in which \(P\) arranges the correlates +of \(\beta\).</p> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_547">[Pg 547]</span></p> +<h2 class="nobreak" id="*204"> +*204. ELEMENTARY PROPERTIES OF SERIES.</h2> +</div> + + +<p class="nind"><i>Summary of</i> *204.</p> + +<p>In this number we give the definition and a few of the simpler +properties of series. Most of the propositions of this number result +immediately from those of <a href="#*200">*200</a>, <a href="#*201">*201</a>, and <a href="#*202">*202</a>. Our definition is +\[ +\text{Ser} =\text{Rl}ʻJ\cap \text{trans}\cap \text{connex} \quad\text{Df}. +\]</p> + +<p>We have</p> + +<p class="nind"><b>*204·16.</b> \(\vdash :P\in \text{Ser}.\equiv .P\in \text{connex} .P^{2}\,\unicode{x2abd}\, J.P^{3}\,\unicode{x2abd}\, J.\equiv .P\in \text{connex} .P_{\text{po}}\,\unicode{x2abd}\, + J\)</p> + +<p class="nind">either of which might have been taken as the definition.</p> + +<p>After a few propositions giving other possible forms of the definition +of series, we proceed to a set of propositions which follow immediately +from those of *200, *201, and *202. Such are</p> + +<p class="nind"><b>*204·2.</b> \(\vdash :P\in \text{Ser}.\equiv .\breve{P} \in \text{Ser}\)</p> + +<p class="nind"><b>*204·21.</b> \(\vdash :P\in \text{Ser}.P\,\,\text{smor}\,\,Q.\supset .Q\in \text{Ser}\)</p> + +<p class="nind"><b>*204·24.</b> \(\vdash .\dot{\Lambda} \in \text{Ser}\)</p> + +<p class="nind"><b>*204·25.</b> \(\vdash :x\neq y.\equiv .x\downarrow y\in \text{Ser}\)</p> + +<p>Another important proposition on couples is</p> + +<p class="nind"><b>*204·272.</b> \(\vdash \colon\ldotp P\in \text{Ser}.\supset :\text{D}ʻP\in 1.\equiv .P\in 2_{r}.\equiv .\text{ᗡ}ʻP\in 1\)</p> + +<p class="nind">so that couples are the only series having unit classes for their +domains or converse domains.</p> + +<p>We then proceed to a set of propositions on \(\overrightarrow{P}ʻx\). +We have</p> + +<p class="nind"><b>*204·33.</b> \(\vdash \colon\ldotp P\in \text{Ser}.x,y\in CʻP.\supset :x\neq y.\overrightarrow{P}ʻyC\overrightarrow{P}ʻx.\equiv .yPx\)</p> + +<p>Also, if \(P\in \text{Ser},\overrightarrow{P}\upharpoonright CʻP\) is a +one-one and \(\overrightarrow{P}^{;}P\,\,\text{smor}\,\,P\) (*204·34·35).</p> + +<p>We then have some propositions (<a href="#*204·4">*204·4</a>—<a href="#*204·44">·44</a>) on relations with limited +fields. The most important of these are</p> + +<p class="nind"><b>*204·4.</b> \(\vdash :P\in \text{Ser}.\supset .P\unicode{x0294f}\alpha \in \text{Ser}\)</p> + +<p class="nind"><b>*204·41.</b> \(\vdash :P,Q\in \text{Ser}.Q\,\unicode{x2abd}\, P.\supset .Q=P\unicode{x0294f}CʻQ\)</p> + +<p><span class="pagenum" id="Page_548">[Pg 548]</span></p> + +<p>This proposition is important, since it shows that any series contained +in a given series is wholly determined when its field is given.</p> + +<p>We have next a number of propositions (<a href="#*204·45">*204·45</a>—<a href="#*204·59">·59</a>) applying +relation-arithmetic to series. The first set of these (<a href="#*204·45">*204·45</a>—<a href="#*204·483">·483</a>) +are concerned with the proof that if a "cut" is made in a series, the +series is the sum of the two parts into which the cut divides it, where +the sum is taken in the sense of <a href="#*160">*160</a> or <a href="#*161">*161</a>, according as one part +of the cut does not or does consist of a single term. Most of these +propositions do not require the full hypothesis that \(P\) is a series, but +only some part of it. Thus we have for instance</p> + +<p class="nind"><b>*204·46.</b> \[\begin{align}\vdash : &P \in \text{connex} . \text{E}! BʻP . \text{ᗡ}ʻP {\sim} \in 1 . \supset .\\ +&P = BʻP \unicode{x21f7} P \unicode{x0294f} \text{ᗡ}ʻP . \text{Nr}ʻP = \dot{1} \dot{+} \text{Nr}ʻ(P \unicode{x0294f} \text{ᗡ}ʻP)\end{align}\]</p> + +<p class="nind">with a similar proposition for \(Bʻ\breve{P}\) and \(\text{D}ʻP\) +(<a href="#*204·461">*204·461</a>).</p> + +<p>We next prove that if \(P\), \(Q\) are mutually exclusive series, +their sum \((P \unicode{x2909} Q)\) is a series, and vice versa +(<a href="#*204·5">*204·5</a>); that if \(P\) is a series to which \(x\) does not belong, +\(P \unicode{x21f8} x\) and \(x \unicode{x21f7} P\) are series, and +vice versa (<a href="#*204·51">*204·51</a>); that if \(P\) is a series of mutually exclusive +series, its sum \(\Sigma ʻP\) is a series (<a href="#*204·52">*204·52</a>); that if \(P\), +\(Q\) are series, so is \(P \times Q\) (<a href="#*204·55">*204·55</a>); that if \(P\) is +a series of series, \(\Pi ʻP\) is contained in diversity and is +transitive (<a href="#*204·561">*204·561</a>), while if \(P\) is also well-ordered, <i>i.e.</i> +such that every existent sub-class of \(CʻP\) has a first term, then +\(\Pi ʻP\) is a series (<a href="#*204·57">*204·57</a>); and that if \(P\) and \(Q\) are +series, and \(Q\) is well-ordered, then \(P^{Q}\) and \(P \,\text{exp}\,\, +Q\) are series (<a href="#*204·59">*204·59</a>). These propositions are essential to ordinal +arithmetic, but they will not be referred to again until we reach that +stage (Sections D and E of this Part).</p> + +<p>We have next a collection of propositions (<a href="#*204·6">*204·6</a>—<a href="#*204·65">·65</a>) on +\(pʻ\overrightarrow{P}ʻʻ\alpha\) for various values of \(\alpha\), and +finally three propositions on \(P_{1}\). Two of these are much used, +namely</p> + +<p class="nind"><b>*204·7.</b> \(\vdash : P \in \text{Ser} . \supset . P_{1} \in 1 \rightarrow 1\)</p> + +<p class="nind"><b>*204·71.</b> \(\vdash : P \in \text{Ser} . xP_{1}y . \supset . \overrightarrow{P}ʻy = \overrightarrow{P}ʻx \cup {℩}ʻx\)</p> + +<hr class="tb"> + +<p class="nind"><b>*204·01.</b> \(\text{Ser} = \text{Rl}ʻJ \cap \text{trans} \cap \text{connex} \quad\text{Df}\)</p> + +<p class="nind"><b>*204·1.</b> \[\begin{align}\vdash : P \in \text{Ser} . &\equiv . P \,\unicode{x2abd}\, J . P^{2} \,\unicode{x2abd}\, P . P \in \text{connex}.\\ +&\equiv . P \in \text{Rl}ʻJ . P \in \text{trans} . P \in \text{connex} \quad[(*204·01)]\end{align}\]</p> + +<p class="nind"><b>*204·11.</b> \[\begin{align}&\vdash \colon\ldotp P \in \text{Ser} . \equiv : P \,\unicode{x2abd}\, J . P^{2} \,\unicode{x2abd}\, P : x \in CʻP . \supset _{x} + . \overrightarrow{P}ʻx \cup {℩}ʻx \cup \overleftarrow{P}ʻx = CʻP\\ +&[*204·1 . *202·101]\end{align}\]</p> + +<p class="nind"><b>*204·12.</b> \[\begin{align}\vdash \colon\colon P \in \text{Ser} . \equiv \colon\ldotp P \,\unicode{x2abd}\, &J . P^{2} + \,\unicode{x2abd}\, P \colon\ldotp x, y \in CʻP . \supset _{x, y} :\\ +&xPy . \lor . x = y . \lor . yPx \quad[*204·1 . *202·103]\end{align}\]</p> + +<p class="nind"><b>*204·121.</b> \[\begin{align}\vdash \colon\colon P \in \text{Ser} . \equiv \colon\ldotp P \,\unicode{x2abd}\, J . P^{2} + \,\unicode{x2abd}\, &P \colon\ldotp x, y \in CʻP . x \neq y . \supset _{x, y} :\\ +&xPy . \lor . yPx \quad[*204·1 . *202·104]\end{align}\]</p> + +<p><span class="pagenum" id="Page_549">[Pg 549]</span></p> + +<p class="nind"><b>*204·13.</b> \(\vdash :P\in \text{Ser}.\supset .P^{2}\,\unicode{x2abd}\, J.P\dot{\cap} \breve{P} = \dot{\Lambda} \)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*204·1.*23·44.\supset \vdash :P\in \text{Ser}.\supset .P^{2}\,\unicode{x2abd}\, J.P\in \text{trans} &\qquad \text{(1)}\\ +\vdash .(1).*201·12.\supset \vdash .\text{Prop} +\end{array}\]</p> + +<p class="nind"><b>*204·14.</b> \[\begin{align}&\vdash :P\in \text{Ser}. \equiv .P\dot{\cap} \breve{P} = \dot{\Lambda} .P^{2}\,\unicode{x2abd}\, P.P\in \text{connex} \\ +&[*204·1.*50·47]\end{align}\]</p> + +<p class="nind"><b>*204·15.</b> \(\vdash :P\in \text{connex} .P^{2}\,\unicode{x2abd}\, J.P^{3}\,\unicode{x2abd}\, J.\supset .P\in \text{trans}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*34·5. &\supset \vdash \colon\ldotp P^{2}\,\unicode{x2abd}\, J.\supset :xPy.yPz.\supset .x \neq z &\qquad \text{(1)}\\ +\vdash .*50·41. &\supset \vdash \colon\ldotp P^{3}\,\unicode{x2abd}\, J.\supset :xPy.yPz.\supset .{\sim}(zPx) &\qquad \text{(2)}\\ +\vdash .(1).(2).\supset \vdash \colon\ldotp \text{Hp}.&\supset :xPy.yPz.\supset .x \neq z.{\sim}(zPx).\\ +[*202·103] &\supset .xPz\colon\ldotp \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*204·151.</b> \[\begin{align}&\vdash :P\in \text{connex} .P_{\text{po}}\,\unicode{x2abd}\, J.\supset .P\in \text{trans}\\ +&[*204·15.*91·502·503·511]\end{align}\]</p> + +<p class="nind"><b>*204·16.</b> \[\begin{align}&\vdash :P\in \text{Ser}. \equiv .P\in \text{connex} .P^{2}\,\unicode{x2abd}\, J.P^{3}\,\unicode{x2abd}\, + J. \equiv .P\in \text{connex} .P_{\text{po}}\,\unicode{x2abd}\, J\\ +&[*204·15·151.*200·36.*201·18]\end{align}\]</p> + +<p>We have also +\[ +\vdash :P\in \text{Ser}. \equiv .P\in \text{connex} .P^{6}\,\unicode{x2abd}\, J. +\] +For, by <a href="#*200·37">*200·37</a>, since \(P^{6} = (P^{2})^{3} = (P^{3})^{2}\), it +follows that +\[ +P^{6}\,\unicode{x2abd}\, J.\supset .P^{2}\,\unicode{x2abd}\, J.P^{3}\,\unicode{x2abd}\, J. +\]</p> + +<p>A relation such as \(x\downarrow y\unicode{x228d} y\downarrow z\unicode{x228d} z\downarrow x\), +where \(x \neq y.y \neq z.z \neq x\), satisfies +\(P\in \text{connex} .P^{2}\,\unicode{x2abd}\, J\), but not +\(P^{3}\,\unicode{x2abd}\, J\). On the other hand, +\[ +x\downarrow y\unicode{x228d} y\downarrow z\unicode{x228d} z\downarrow w\unicode{x228d} w\downarrow x +\] +satisfies \(P^{2}\,\unicode{x2abd}\, J.P^{3}\,\unicode{x2abd}\, J\), but not +\(P\in \text{connex}\).</p> + +<p class="nind"><b>*204·2.</b> \(\vdash :P\in \text{Ser}. \equiv .\breve{P} \in \text{Ser} \quad[*200·11.*201·11.*202·11]\)</p> + +<p class="nind"><b>*204·21.</b> \[\begin{align}&\vdash :P\in \text{Ser}.P \,\,\text{smor}\, \,Q.\supset .Q\in \text{Ser}\\ +&[*200·211.*201·211.*202·211]\end{align}\]</p> + +<p class="nind"><b>*204·22.</b> \(\vdash :P\in \text{Ser}.\supset .\text{Nr}ʻP\subset \text{Ser} \quad[*204·21]\)</p> + +<p class="nind"><b>*204·23.</b> \[\begin{align}&\vdash :P\in \text{Ser}. \equiv .\text{N}_{0}\text{r}ʻP\subset \text{Ser}. \equiv .\exists !\text{N}_{0}\text{r}ʻP\cap \text{Ser}\\ +&[*200·22.*201·22.*202·22]\end{align}\]</p> + +<p class="nind"><b>*204·24.</b> \(\vdash .\dot{\Lambda} \in \text{Ser} \quad[*200·3.*201·3.*202·3]\)</p> + +<p class="nind"><b>*204·25.</b> \(\vdash :x \neq y. \equiv . x\downarrow y \in \text{Ser} \quad[*200·31.*201·31.*202·31]\)</p> + +<p class="nind"><b>*204·26.</b> \[\begin{align}&\vdash :x \neq y.x \neq z.y \neq z.\supset .x\downarrow y\unicode{x21f8} z\in \text{Ser}\\ +&[*200·31·41.*201·411.*202·411]\end{align}\]</p> + +<p>The three following propositions deal with couples. Couples +often require special treatment, owing to the fact that, +if \(P\) is a couple, \(P\unicode{x0294f}\text{D}ʻP =\dot{\Lambda}\), so that +\(Cʻ(P\unicode{x0294f}\text{D}ʻP) \neq\text{D}ʻP\), whereas in any other case, +if \(P\) is<span class="pagenum" id="Page_550">[Pg 550]</span> a series, \(Cʻ(P\unicode{x0294f}\text{D}ʻP)=\text{D}ʻP\). +Hence the following propositions are often required.</p> + +<p class="nind"><b>*204·27.</b> \(\vdash :P\in \text{Ser}.xPy.\text{D}ʻP={℩}ʻx.\supset .P=x\downarrow y\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*33·14. \supset \vdash :\text{Hp}.zPw.&\supset .z=x &\qquad \text{(1)}\\ +\vdash .(1).*50·24. \supset \vdash :\text{Hp}.zPw.&\supset .w\neq x.\\ +\left[(1).\text{Transp} \frac{w,\,y}{z,\,w}\right] &\supset .{\sim}(wPy) &\qquad \text{(2)}\\ +\vdash .(1).\text{Transp}.*50·24.&\supset \vdash :\text{Hp}.\supset .{\sim}(yPw) &\qquad \text{(3)}\\ +\vdash .(2).(3).*204·12. &\supset \vdash :\text{Hp}.zPw.\supset .y=w &\qquad \text{(4)}\\ +\vdash .(1).(4).&\supset \vdash \colon\ldotp \text{Hp}.\supset :zPw.\supset .z=x.y=w &\qquad \text{(5)}\\ +\vdash .(5).*55·34.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*204·271.</b> \(\vdash :P\in \text{Ser}.\text{D}ʻP\in 1.\supset .P\in 2_r\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*204·27.\supset \vdash :\text{Hp}.&\supset .(\exists x,y).P=x\downarrow y.\\ +[*204·25] &\supset .(\exists x,y).x\neq y.P=x\downarrow y.\\ +[*56·11] &\supset .P\in 2_r:\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*204·272.</b> \[\begin{align}&\vdash \colon\ldotp P\in \text{Ser}.\supset :\text{D}ʻP\in 1.\equiv .P\in 2_r.\equiv .\text{ᗡ}ʻP\in 1\\ +&[*204·271·2.*56·111]\end{align}\]</p> + +<p class="nind"><b><a id="*204·3">*204·3</a>.</b> \[\begin{align}&\vdash \colon\ldotp P\in \text{Ser}.x,y\in CʻP.\supset :x\neq y.{\sim}(yPx).\equiv .xPy\\ +&[*202·5.*204·13]\end{align}\]</p> + +<p class="nind"><b>*204·32.</b> \(\vdash \colon\ldotp P\in \text{Ser}.x,y\in CʻP.\supset :\overrightarrow{P}ʻy\subset \overrightarrow{P}ʻx.\equiv .y\in \overrightarrow{P}ʻx\cup {℩}ʻx\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*204·1. \supset \vdash \colon\ldotp \text{Hp}.&\supset :yPx.zPy.\supset .zPx:\\ +[*32·18] &\supset :y\in \overrightarrow{P}ʻx.\supset .\overrightarrow{P}ʻy\subset \overrightarrow{P}ʻx &\qquad \text{(1)}\\ +\vdash .*22·42. &\supset \vdash :y=x.\supset .\overrightarrow{P}ʻy\subset \overrightarrow{P}ʻx &\qquad \text{(2)}\\ +\vdash .(1).(2). &\supset \vdash \colon\ldotp \text{Hp}.\supset :y\in \overrightarrow{P}ʻx\cup {℩}ʻx.\supset .\overrightarrow{P}ʻy\subset \overrightarrow{P}ʻx &\qquad \text{(3)}\\ +\vdash .*204·11. &\supset \vdash \colon\ldotp \text{Hp}.\supset :y{\sim}\in \overrightarrow{P}ʻx\cup {℩}ʻx.\supset .y\in \overleftarrow{P}ʻx.\\ +[*32·18·181] & \supset .x\in \overrightarrow{P}ʻy &\qquad \text{(4)}\\ +\vdash .*50·24. &\supset \vdash :\text{Hp}.\supset .x{\sim}\in \overrightarrow{P}ʻx &\qquad \text{(5)}\\ +\vdash .(4).(5). &\supset \vdash \colon\ldotp \text{Hp}.\supset :y{\sim}\in \overrightarrow{P}ʻx\cup {℩}ʻx.\supset .{\sim}(\overrightarrow{P}ʻy\subset \overrightarrow{P}ʻx) &\qquad \text{(6)}\\ +\vdash .(3).(6). &\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*204·33.</b> \(\vdash \colon\ldotp P\in \text{Ser}.x,y\in CʻP.\supset :x\neq y.\overrightarrow{P}ʻy\subset \overrightarrow{P}ʻx.\equiv .yPx\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*204·32.\supset \vdash \colon\ldotp \text{Hp}.\supset :x\neq y.\overrightarrow{P}ʻy\subset \overrightarrow{P}ʻx.&\equiv .x\neq y.y\in \overrightarrow{P}ʻx\cup {℩}ʻx.\\ +[*51·15] &\equiv .x\neq y.y\in \overrightarrow{P}ʻx.\\ +[\text{Hp}.*4·71] &\equiv .yPx\colon\ldotp \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_551">[Pg 551]</span></p> + +<p>The three following propositions only require \(P\in \text{Rl}ʻJ\cap\text{connex}\), +but are required for application to series, and are +therefore convenient in the form here given.</p> + +<p class="nind"><b>*204·331.</b> \[\begin{align}&\vdash \colon\ldotp P\in \text{Ser}.x,y\in CʻP.\supset :\overrightarrow{P}ʻx=\overrightarrow{P}ʻy.\equiv .x=y\\ +&[*202·161.*71·55]\end{align}\]</p> + +<p class="nind"><b>*204·34.</b> \(\vdash :P\in \text{Ser}.\supset .\overrightarrow{P}\upharpoonright CʻP\in 1\rightarrow 1.\overrightarrow{P}\upharpoonright CʻP\in (\overrightarrow{P}^{;}P)\overline{\,\text{smor}\,} + P \quad[*202·161]\)</p> + +<p class="nind"><b>*204·35.</b> \(\vdash :P\in \text{Ser}.\supset .\overrightarrow{P}^{;}P\,\,\text{smor}\,\,P \quad[*204·34]\)</p> + +<p>This proposition shows that the series of segments which have upper +limits is like the original series, for a segment whose upper limit is +\(x\) is \(\overrightarrow{P}ʻx\), and the series of such segments is +\(\overrightarrow{P}^{;}P\).</p> + +<p>The following propositions (<a href="#*204·4">*204·4</a>—<a href="#*204·44">·44</a>) are concerned with relations +with limited fields.</p> + +<p class="nind"><b><a id="*204·4">*204·4</a>.</b> \(\vdash :P\in \text{Ser}.\supset .P\unicode{x0294f}\alpha \in \text{Ser} \quad[*200·33.*201·33.*202·33]\)</p> + +<p class="nind"><b>*204·41.</b> \(\vdash :P,Q\in \text{Ser}.Q\,\unicode{x2abd}\, P.\supset .Q=P\unicode{x0294f}CʻQ \quad[*202·53.*204·13]\)</p> + +<p>In virtue of the above two propositions, the series contained in a +given series are the relations resulting from limitations of the field; +the process of limiting the field is merely the process of selecting a +part of the original series without changing the order.</p> + +<p class="nind"><b>*204·42.</b> + \[\begin{align}&\vdash \colon\ldotp P\in \text{Ser}.\supset :Q\in \text{Ser}.Q\,\unicode{x2abd}\, P.\equiv .(\exists \alpha ).Q=P\unicode{x0294f}\alpha .\equiv .Q\in \text{D}ʻP\unicode{x0294f}\\ +&[*204·4·41]\end{align}\]</p> + +<p class="nind"><b>*204·421.</b> \(\vdash :P\in \text{Ser}.\supset .\text{Ser}\cap \text{Rl}ʻP=\text{D}ʻP\unicode{x0294f} \quad[*204·42]\)</p> + +<p class="nind"><b>*204·43.</b> \(\vdash :P^{2}\,\unicode{x2abd}\, P.P\,\unicode{x2abd}\, J.Q\,\unicode{x2abd}\, P.Q\in \text{connex} .\supset .Q\in \text{Ser}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*23·1.*34·55.\supset \vdash \colon\ldotp \text{Hp}.&\supset :xQy.yQz.\supset .xPz.\\ +[*50·43.\text{Hp}] &\supset .{\sim}(zPx).x\neq z.\\ +[*23·81.\text{Hp}] &\supset .{\sim}(zQx).x\neq z.\\ +[*202·103] &\supset .xQz:\\ +[*34·55] &\supset :Q^{2}\,\unicode{x2abd}\, Q &\qquad \text{(1)}\\ +\vdash .*23·44.&\supset \vdash :\text{Hp}.\supset .Q\,\unicode{x2abd}\, J &\qquad \text{(2)}\\ +\vdash .(1).(2).*204·1.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*204·44">*204·44</a>.</b> \(\vdash :P\in \text{Rl}ʻJ\cap \text{trans}.\supset .\text{Rl}ʻP\cap \text{connex} \subset \text{Ser} \quad[*204·43]\)</p> + +<p><span class="pagenum" id="Page_552">[Pg 552]</span></p> + +<p>The following propositions (<a href="#*204·45">*204·45</a>—<a href="#*204·483">*204·483</a>) are concerned with the +division of a series into two parts, one of which wholly precedes +the other. The case where one of the parts consists of a single term +requires special treatment, and so does the case where both parts +consist of single terms, <i>i.e.</i> where the series is a couple.</p> + +<p class="nind"><b><a id="*204·45">*204·45</a>.</b> \[\begin{align}\vdash :P\in \text{connex} .&\alpha \in \text{Cl}ʻCʻP-1.Pʻʻ\alpha \subset \alpha .\beta =CʻP-\alpha .\beta {\sim}\in 1.\supset .\\ +&P=P\unicode{x0294f}\alpha \unicode{x2909}P\unicode{x0294f}\beta .\text{Nr}ʻP=\text{Nr}ʻP\unicode{x0294f}\alpha \dot{+} \text{Nr}ʻP\unicode{x0294f}\beta\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*24·411.*33·17.\supset \vdash \colon\colon\text{Hp}.&\supset \colon\ldotp \\ +&xPy.\equiv :y\in \alpha .xPy.\lor.x\in \alpha .y\in \beta .xPy.\lor.x,y\in \beta .xPy &\qquad \text{(1)}\\ +\vdash .*37·17. &\supset \vdash \colon\ldotp \text{Hp}.\supset :y\in \alpha .xPy.\supset .x\in \alpha &\qquad \text{(2)}\\ +\vdash .(2).\text{Transp}.*202·103. &\supset \vdash \colon\ldotp \text{Hp}.\supset :y\in \alpha .x\in \beta .\supset .yPx &\qquad \text{(3)}\\ +\vdash .*202·55. & \supset \vdash :\text{Hp}.\supset .\alpha =CʻP\unicode{x0294f}\alpha .\beta =CʻP\unicode{x0294f}\beta &\qquad \text{(4)}\\ +\vdash .(1).(2).(3).(4).&\supset \vdash \colon\colon\text{Hp}.\supset \colon\ldotp \\ +xPy.&\equiv :x(P\unicode{x0294f}\alpha )y.\lor.x\in CʻP\unicode{x0294f}\alpha .y\in CʻP\unicode{x0294f}\beta .\lor.x(P\unicode{x0294f}\beta )y:\\ +[*160·1] &\equiv :x{P\unicode{x0294f}\alpha \unicode{x2909}P\unicode{x0294f}\beta }y &\qquad \text{(5)}\\ +\vdash .(5).*180·32.&\supset \vdash :\text{Hp}.\supset .\text{Nr}ʻP=\text{Nr}ʻP\unicode{x0294f}\alpha \dot{+} \text{Nr}ʻP\unicode{x0294f}\beta &\qquad \text{(6)}\\ +\vdash .(5).(6).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*204·46.</b> \[\begin{align}\vdash :P\in \text{connex} .&\text{E}!BʻP.\text{ᗡ}ʻP{\sim}\in 1.\supset .\\ +&P=BʻP\unicode{x21f7}P\unicode{x0294f}\text{ᗡ}ʻP.\text{Nr}ʻP=\dot{1} \dot{+} \text{Nr}ʻ(P\unicode{x0294f}\text{ᗡ}ʻP)\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*202·524. \supset \vdash \colon\ldotp \text{Hp}.&\supset :x=BʻP.y\in \text{ᗡ}ʻP.\supset .xPy &\qquad \text{(1)}\\ +\vdash .(1).*161·111.\supset \vdash \colon\colon\text{Hp}.&\supset \colon\ldotp x(BʻP\unicode{x21f7}P\unicode{x0294f}\text{ᗡ}ʻP)y.\equiv :\\ +&x=BʻP.y\in \text{ᗡ}ʻP.xPy.\lor.x,y\in \text{ᗡ}ʻP.xPy:\\ +[*93·103] &\equiv :x\in CʻP.y\in \text{ᗡ}ʻP.xPy:\\ +[*33·14·17] &\equiv :xPy &\qquad \text{(2)}\\ +\vdash .(2).*181·32. &\supset \vdash :\text{Hp}.\supset .\text{Nr}ʻP=\dot{1} \dot{+} \text{Nr}ʻ(P\unicode{x0294f}\text{ᗡ}ʻP) &\qquad \text{(3)}\\ +\vdash .(2).(3).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*204·461">*204·461</a>.</b> \[\begin{align}\vdash :P\in \text{connex} .&\text{E}!Bʻ\breve{P} .\text{D}ʻP{\sim}\in 1.\supset .\\ +&P=P\unicode{x0294f}\text{D}ʻP\unicode{x21f8} Bʻ\breve{P} .\text{Nr}ʻP=\text{Nr}ʻ(P\unicode{x0294f}\text{D}ʻP)\dot{+} \dot{1}\\ +&[\text{Proof as in *204·46}]\end{align}\]</p> + +<p class="nind"><b>*204·462.</b> \[\begin{align}\vdash \colon\ldotp P,Q\in &\text{connex} .\text{E}!BʻP.\text{ᗡ}ʻP{\sim}\in 1.\text{E}!BʻQ.\text{ᗡ}ʻQ{\sim}\in 1.\supset :\\ +&P\,\,\text{smor}\,\,Q.\equiv .P\unicode{x0294f}\text{ᗡ}ʻP\,\,\text{smor}\,\,Q\unicode{x0294f}\text{ᗡ}ʻQ \quad[*161·33.*204·46]\end{align}\]</p> + +<p class="nind"><b>*204·463.</b> \[\begin{align}\vdash :P,Q\in &\text{Rl}ʻJ.\text{E}!BʻP.\text{ᗡ}ʻP\in 1.\text{E}!.BʻQ.\text{ᗡ}ʻQ\in 1.\supset .\\ +&P\,\,\text{smor}\,\,Q.P\unicode{x0294f}\text{ᗡ}ʻP\,\,\text{smor}\,\,Q\unicode{x0294f}\text{ᗡ}ʻQ\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*56·37. &\supset \vdash :\text{Hp}.\supset .P,Q\in 2_{r} &\qquad \text{(1)}\\ +\vdash .*200·35.&\supset \vdash :\text{Hp}.\supset .P\unicode{x0294f}\text{ᗡ}ʻP=\dot{\Lambda} .Q\unicode{x0294f}\text{ᗡ}ʻQ=\dot{\Lambda} &\qquad \text{(2)}\\ +\vdash .(1).(2).*153·202·101.\supset \\ +\vdash :\text{Hp}.&\supset .P\,\,\text{smor}\,\,Q.P\unicode{x0294f}\text{ᗡ}ʻP\,\,\text{smor}\,\,Q\unicode{x0294f}\text{ᗡ}ʻQ:\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*204·47.</b> \[\begin{align}\vdash \colon\ldotp P,Q\in \text{connex} \cap \text{Rl}ʻJ.\text{E}!BʻP.&\text{E}!BʻQ.\supset :\\ +P\,\,\text{smor}\,\,Q.\equiv .P\unicode{x0294f}\text{ᗡ}ʻrP\,\,\text{smor}\,\,Q\unicode{x0294f}\text{ᗡ}ʻQ\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*151·18.*200·35.*202·55.*153·102.\supset \\ +\vdash :\text{Hp}.\text{ᗡ}ʻP\in 1.\text{ᗡ}ʻQ{\sim}\in 1.\supset .{\sim}(P\,\,\text{smor}\,\,Q).{\sim}(P\unicode{x0294f}\text{ᗡ}ʻP\,\,\text{smor}\,\,Q\unicode{x0294f}\text{ᗡ}ʻQ) &\qquad \text{(1)}\\ +\vdash .(1).*204·462·463.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_553">[Pg 553]</span></p> + +<p class="nind"><b>*204·48.</b> \[\begin{align}\vdash \colon\colon &P\in \text{Ser}.\supset \colon\ldotp \\ +&\text{E}!BʻP.\equiv :(\exists Q).\dot{\exists} !Q.\text{Nr}ʻP=\dot{1} \dot{+} \text{Nr}ʻQ.\lor.\text{Nr}ʻP=2_{r}\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*204·46. &\supset \vdash :\text{Hp}.\text{E}!BʻP.\text{ᗡ}ʻP{\sim}\in 1.\supset .(\exists Q).\text{Nr}ʻP=\dot{1} \dot{+} \text{Nr}ʻQ &\qquad \text{(1)}\\ +\vdash .*161·2. &\supset \vdash :\dot{\exists} !P.\text{Nr}ʻP=\dot{1} \dot{+} \text{Nr}ʻQ.\supset .\dot{\exists} !Q &\qquad \text{(2)}\\ +\vdash .(1).(2). &\supset \vdash :\text{Hp}.\text{E}!BʻP.\text{ᗡ}ʻP{\sim}\in 1.\supset .\\ +&(\exists Q).\dot{\exists} !Q.\text{Nr}ʻP=\dot{1} \dot{+} \text{Nr}ʻQ &\qquad \text{(3)}\\ +\vdash .*204·272.&\supset \vdash :\text{Hp}.\text{ᗡ}ʻP\in 1.\supset .P\in 2_{r} &\qquad \text{(4)}\\ +\vdash .(3).(4). &\supset \vdash :\text{Hp}.\text{E}!BʻP.\supset :\\ +&(\exists Q).\dot{\exists} !Q.\text{Nr}ʻP=\dot{1} +\text{Nr}ʻQ.\lor.\text{Nr}ʻP=2_{r} &\qquad \text{(5)}\\ +\vdash .*181·11·12·32.&\supset \vdash :\text{Nr}ʻP=\dot{1} \dot{+} \text{Nr}ʻQ.\supset .\\ +&(\exists R,z).R\,\,\text{smor}\,\,Q.z{\sim}\in CʻR.\text{Nr}ʻP=\text{Nr}ʻ(z\unicode{x21f8} R) &\qquad \text{(6)}\\ +\vdash .*161·15·12.\supset \vdash \colon\ldotp \dot{\exists} !R.z{\sim}\in CʻR.&\supset :\text{E}!Bʻ(z\unicode{x21f8} R):\\ +[*151·5] &\supset :\text{Nr}ʻP=\text{Nr}ʻ(z\unicode{x21f8} R).\supset .\text{E}!BʻP &\qquad \text{(7)}\\ +\vdash .(6).(7). &\supset \vdash :\text{Nr}ʻP=\dot{1} \dot{+} \text{Nr}ʻQ.\dot{\exists} !Q.\supset .\text{E}!BʻP &\qquad \text{(8)}\\ +\vdash .*153·281.&\supset \vdash :P\in 2_{r}.\supset .\text{E}!BʻP &\qquad \text{(9)}\\ +\vdash .(5).(8).(9).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*204·481.</b> \[\begin{align}\vdash \colon\colon P\in &\text{Ser}.\supset \colon\ldotp \\ +&\text{E}!Bʻ\breve{P} .\equiv :(\exists Q).\dot{\exists} !Q.\text{Nr}ʻP=\text{Nr}ʻQ\dot{+} \dot{1} .\lor.\text{Nr}ʻP=2_{r}\\ +&[\text{Proof as in *204·48}]\end{align}\]</p> + +<p class="nind"><b>*204·482.</b> \[\begin{align}\vdash \colon\colon\alpha \in \text{N}_{0}\text{r}ʻʻ\text{Ser}.\supset + \colon\ldotp \alpha &\subset \text{ᗡ}ʻB:\equiv :\exists !\alpha \cap \text{ᗡ}ʻB:\\ +&\equiv :(\exists \beta ).\beta \in \text{NR}-{℩}ʻ0_{r}.\alpha =\dot{1} \dot{+} \beta .\lor.\alpha =2_{r}\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*151·5.*155·13.&\supset \vdash \colon\ldotp \text{Hp}.\supset :\alpha \subset \text{ᗡ}ʻB.\equiv .\exists !\alpha \cap \text{ᗡ}ʻB &\qquad \text{(1)}\\ +\vdash .*204·23·48. &\supset \vdash \colon\colon \text{Hp}.P\in \alpha .\supset \colon\ldotp \\ +&\text{E}!BʻP.\equiv :(\exists \beta ).\beta \in \text{NR}-{℩}ʻ0_{r}.\alpha =\dot{1} \dot{+} \beta .\lor.\alpha =2_{r} &\qquad \text{(2)}\\ +\vdash .(1).(2).*202·52.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*204·483">*204·483</a>.</b> \[\begin{align}\vdash \colon\colon\alpha \in \text{N}_{0}\text{r}ʻʻ\text{Ser}.\supset + \colon\ldotp \alpha &\subset \text{ᗡ}ʻ(B\mid \text{Cnv}):\equiv :\exists !\alpha \cap \text{ᗡ}ʻ(B\mid \text{Cnv}):\\ +&\equiv :(\exists \beta ).\beta \in \text{NR}-{℩}ʻ0_{r}.\alpha =\beta \dot{+} \dot{1} .\lor.\alpha =2_{r}\\ +[\text{Proof as in *204·482}]\end{align}\]</p> + +<p>The following propositions are concerned with the application of +relation-arithmetic to series.</p> + +<p class="nind"><b><a id="*204·5">*204·5</a>.</b> \[\begin{align}&\vdash :P,Q\in \text{Ser}.CʻP\cap CʻQ=\Lambda .\equiv .P\unicode{x2909}Q\in \text{Ser}\\ +&[*200·4.*201·401.*202·401]\end{align}\]</p> + +<p class="nind"><b><a id="*204·51">*204·51</a>.</b> \[\begin{align}&\vdash :P\in \text{Ser}.x{\sim}\in CʻP.\equiv .P\unicode{x21f8} x\in \text{Ser}.\equiv .\chi\unicode{x21f7}P\in \text{Ser}\\ +&[*200·41.*201·41.*202·412]\end{align}\]</p> + +<p><span class="pagenum" id="Page_554">[Pg 554]</span></p> + +<p class="nind"><b><a id="*204·52">*204·52</a>.</b> \(\vdash :P\in \text{Rel}^{2}\text{excl}\cap \text{Ser}.CʻP\subset \text{Ser}.\supset .\Sigma ʻP \in \text{Ser}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*200·42.*202·62.\supset \vdash :\eta P .\supset .\Sigma ʻP\,\unicode{x2abd}\, J &\qquad \text{(1)}\\ +\vdash .(1).*201·42.*202·42.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*204·53">*204·53</a>.</b> \(\vdash \colon\ldotp P\in \text{Rel}^{2}\text{excl}.\dot{\Lambda} {\sim}\in CʻP.\supset :\Sigma ʻP\in \text{Ser}.\equiv .P\in \text{Ser}.CʻP\subset \text{Ser}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*200·423. \supset \vdash \colon\ldotp \text{Hp}.\Sigma ʻP\in \text{Ser}.&\supset :P\,\unicode{x2abd}\, J: &\qquad \text{(1)}\\ +[*200·421] &\supset :Q\in CʻP.\supset .Q=(\Sigma ʻP)\unicode{x0294f}CʻQ.\\ +[*204·4] &\supset .Q\in \text{Ser} &\qquad \text{(2)}\\ +\vdash .*162·13.\supset \\ +\vdash \colon\ldotp \text{Hp}.\Sigma ʻP\in \text{Ser}.QPR.RPS.x\in CʻQ.y\in CʻR.z\in CʻS.&\supset :x(\Sigma ʻP)z: &\qquad \text{(3)}\\ +[*162·13.*163·11] &\supset :(\exists M,N).MPN.x\in CʻM.z\in CʻN.M=Q.N=S.\lor.\\ +&(\exists M).M\in CʻP.xMz.M=Q.M=S:\\ +[*13·22·195] &\supset :QPS.\lor.Q=S &\qquad \text{(4)}\\ +\vdash .(3).*50·24.*24·37. \supset \vdash :\text{Hp}(3).&\supset .CʻQ\cap CʻS=\Lambda .\\ +[*24·57.*30·37] &\supset .Q\neq S &\qquad \text{(5)}\\ +\vdash .(4).(5). \supset \vdash \colon\ldotp \text{Hp}.\Sigma ʻP\in \text{Ser}.&\supset :QPR.RPS.\supset .QPS &\qquad \text{(6)}\\ +\vdash .*162·1. \supset \vdash \colon\ldotp \text{Hp}.\Sigma ʻP\in \text{Ser}.&Q,R\in CʻP.x\in CʻP.y\in CʻQ.Q\neq R.\supset :x\neq y:\\ +[*202·104] &\supset :x(\Sigma ʻP)y.\lor.y(\Sigma ʻP)x:\\ +[*162·13.*163·11] &\supset :QPR.\lor.RPQ &\qquad \text{(7)}\\ +\vdash .(6).(7).& \supset \vdash :\text{Hp}.\Sigma ʻP\in \text{Ser}.\supset . P\in \text{trans}\cap \text{connex} &\qquad \text{(8)}\\ +\vdash .(1).(2).(8).*204·52.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*204·54.</b> \(\vdash :P\in \text{Rel}^{3}\text{arithm}\cap \text{Ser}.CʻP\subset \text{Ser}.Cʻ\Sigma ʻP\subset \text{Ser}.\supset .\Sigma ʻ\Sigma ʻP\in \text{Ser}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*204·52.&\supset \vdash :\text{Hp}.\supset .\Sigma ʻP\in \text{Ser} &\qquad \text{(1)}\\ +\vdash .*174·3. &\supset \vdash :\text{Hp}.\supset .\Sigma ʻP\in \text{Rel}^{2}\text{excl} &\qquad \text{(2)}\\ +\vdash .(1).(2).*204·52.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*204·55">*204·55</a>.</b> \(\vdash :P,Q\in \text{Ser}.\supset .Q\times P\in \text{Ser}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*165·27.*204·22. &\supset \vdash \colon\ldotp \text{Hp}.\supset :\dot{\exists} !P.\supset .P\downarrow_{.,} ^{;}Q\in \text{Ser} &\qquad \text{(1)}\\ +\vdash .*165·26.*204·22. &\supset \vdash :\text{Hp}.\supset .CʻP\downarrow_{.,} ^{;}Q\subset \text{Ser}. &\qquad \text{(2)}\\ +\vdash .(1).(2).*165·21.*204·52.\supset \vdash :\text{Hp}.\dot{\exists} !P.&\supset .\Sigma ʻP\downarrow_{.,} ^{;}Q\in \text{Ser}.\\ +[*166·1] &\supset .Q\times P\in \text{Ser} &\qquad \text{(3)}\\ +\vdash .*166·13.*204·24. &\supset \vdash :P=\dot{\Lambda} .\supset .Q\times P\in \text{Ser} &\qquad \text{(4)}\\ +\vdash .(3).(4).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_555">[Pg 555]</span></p> + +<p class="nind"><b>*204·551.</b> \(\vdash \colon\ldotp \dot{\exists} !P.\dot{\exists} !Q.\supset :P\times Q\in \text{Ser}.\equiv .P,Q\in \text{Ser}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*165·21·212. \supset \vdash \colon\ldotp \text{Hp}.&\supset :P\downarrow_{.,} ^{;}Q\in \text{Rel}^{2}\text{excl}.\dot{\Lambda} {\sim}\in CʻP\downarrow_{.,} + ^{;}Q:\\ +[*204·53.*166·1] \supset :P\times Q\in \text{Ser}.&\equiv .P\downarrow_{.,} ^{;}Q\in \text{Ser}.CʻP\downarrow_{.,} ^{;}Q\subset \text{Ser}.\\ +[*165·27.*204·22] &\equiv .P,Q\in \text{Ser}\colon\ldotp \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*204·56.</b> \(\vdash :CʻP\subset \text{Rl}ʻJ.\supset .\Pi ʻP\,\unicode{x2abd}\, J\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*172·11. &\supset \vdash :M(\Pi ʻP)N.\supset .(\exists Q).Q\in CʻP.(MʻQ)Q(NʻQ) &\qquad \text{(1)}\\ +\vdash .(1). \supset \vdash \colon\ldotp \text{Hp}.\supset :M(\Pi ʻP)N.&\supset .(\exists Q).MʻQ\neq NʻQ.\\ +[*30·37.\text{Transp}] &\supset .M\neq N\colon\ldotp \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*204·561">*204·561</a>.</b> \(\vdash :P\in \text{Ser}.CʻP\subset \text{Ser}.\supset .\Pi ʻP\in \text{Rl}ʻJ\cap \text{trans}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*200·43. &\supset \vdash \colon\colon\text{Hp}.\supset \colon\ldotp L(\Pi ʻP)M.M(\Pi ʻP)N.\supset :\\ +&(\exists Q,R).Q,R\in CʻP.(LʻQ)Q(MʻQ).(MʻR)R(NʻR).L\upharpoonright \overrightarrow{P}ʻQ=M\upharpoonright \overrightarrow{P}ʻQ.\\ +&M\upharpoonright \overrightarrow{P}ʻR=N\upharpoonright \overrightarrow{P}ʻR:\\ +[*204·12] +&\supset :(\exists Q,R):Q=R.\lor.QPR.\lor.RPQ:(LʻQ)Q(MʻQ).(MʻR)R(NʻR).\\ +&L\upharpoonright \overrightarrow{P}ʻQ=M\upharpoonright \overrightarrow{P}ʻQ.M\upharpoonright \overrightarrow{P}ʻR=N\upharpoonright \overrightarrow{P}ʻR &\qquad \text{(1)}\\ +\vdash .*204·1. &\supset \vdash :\text{Hp}.L(\Pi ʻP)M.M(\Pi ʻP)N.\\ +&Q=R.(LʻQ)Q(MʻQ).(MʻR)R(NʻR).L\upharpoonright \overrightarrow{P}ʻQ=M\upharpoonright \overrightarrow{P}ʻQ.\\ +&M\upharpoonright \overrightarrow{P}ʻR=N\upharpoonright \overrightarrow{P}ʻR.\supset .(LʻQ)Q(NʻQ).L\upharpoonright \overrightarrow{P}ʻQ=N\upharpoonright \overrightarrow{P}ʻQ &\qquad \text{(2)}\\ +\vdash .*204·33.\supset \\ +&\vdash :\text{Hp}.L(\Pi ʻP)M.M(\Pi ʻP)N.QPR.(LʻQ)Q(MʻQ).(MʻR)R(NʻR).\\ +&L\upharpoonright \overrightarrow{P}ʻQ=M\upharpoonright \overrightarrow{P}ʻQ.M\upharpoonright \overrightarrow{P}ʻR=N\upharpoonright \overrightarrow{P}ʻR.\supset .\\ +&L\upharpoonright \overrightarrow{P}ʻQ=N\upharpoonright \overrightarrow{P}ʻQ.MʻQ=NʻQ.\\ +[*13·12] &\supset .L\upharpoonright \overrightarrow{P}ʻQ=N\upharpoonright \overrightarrow{P}ʻQ.(LʻQ)Q(NʻQ) &\qquad \text{(3)}\\ +\vdash .*204·33.\supset\\ +&\vdash :\text{Hp}.L(\Pi ʻP)M.M(\Pi ʻP)N.RPQ.(LʻQ)Q(MʻQ).(MʻR)R(NʻR).\\ +&L\upharpoonright \overrightarrow{P}ʻQ=M\upharpoonright \overrightarrow{P}ʻQ.M\upharpoonright \overrightarrow{P}ʻR=N\upharpoonright \overrightarrow{P}ʻR.\supset .\\ +&L\upharpoonright \overrightarrow{P}ʻR=N\upharpoonright \overrightarrow{P}ʻR.LʻR=MʻR.\\ +[*13·12] &\supset .L\upharpoonright \overrightarrow{P}ʻR=N\upharpoonright \overrightarrow{P}ʻR.(LʻR)R(NʻR) &\qquad \text{(4)}\\ +\vdash .(1).(2).(3).(4).*200·43.\supset \\ +\vdash \colon\ldotp \text{Hp}.&\supset :L(\Pi ʻP)M.M(\Pi ʻP)N.\supset .L(\Pi ʻP)N &\qquad \text{(5)}\\ +\vdash .(5).*204·56.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_556">[Pg 556]</span></p> + +<p>In order to prove that \(\Pi ʻP\) is connected, we require a further +hypothesis, namely that \(P\) is <i>well-ordered</i>, <i>i.e.</i> that +every class contained in \(CʻP\) and not null has a first term.</p> + +<p class="nind"><b>*204·562.</b> \(\vdash \colon\ldotp CʻP\subset ʻ\text{Ser}:\alpha \subset CʻP.\exists !\alpha .\supset _{\alpha }.\exists + !\alpha -\breve{P} ʻʻ\alpha :\supset .\Pi ʻP\in \text{connex}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*172·1.*33·45.\text{Transp}.\supset \\ +\vdash \colon\colon\text{Hp}.\supset \colon\ldotp M,N\in Cʻ&\Pi ʻP.M\neq N.\supset :(\exists Q).Q\in CʻP.MʻQ\neq NʻQ:\\ +[\text{Hp}] &\supset :(\exists Q):Q\in CʻP.MʻQ\neq NʻQ:RPQ.\supset _{R}.MʻR=NʻR:\\ +[*204·121.*172·12] &\supset :(\exists Q):Q\in CʻP:(MʻQ)Q(NʻQ).\lor.(NʻQ)Q(MʻQ):\\ +&RPQ.\supset _{R}.MʻR=NʻR:\\ +[*172·11] &\supset :M(\Pi ʻP)N.\lor.N(\Pi ʻP )\mu &\qquad \text{(1)}\\ +\vdash .(1).*202·104.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*204·57">*204·57</a>.</b> \[\begin{align}&\vdash \colon\ldotp P\in \text{Ser}.CʻP\subset \text{Ser}:\alpha \subset CʻP.\exists !\alpha .\supset _{\alpha }.\exists + !\alpha -\breve{P} ʻʻ\alpha :\supset .\Pi ʻP\in \text{Ser}\\ +&[*204·561·562]\end{align}\]</p> + +<p class="nind"><b>*204·58.</b> \[\begin{align}&\vdash \colon\ldotp P\in \text{Ser}.CʻP\subset \text{Ser}.Cʻ\Sigma ʻP\subset \text{Ser}.P\in \text{Rel}^{2}\text{excl}:\\ +&\alpha \subset Cʻ\Sigma ʻP .\exists !\alpha .\supset _{\alpha }.\exists !\alpha -(\text{Cnv}ʻ\Sigma ʻP)ʻʻ\alpha :\supset .\Pi ʻ\Sigma ʻP,\Pi ʻ\Pi ^{;}P\in + \text{Ser}\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*204·52. &\supset \vdash :\text{Hp}.\supset .\Sigma ʻP\in \text{Ser} &\qquad \text{(1)}\\ +\vdash .(1).*204·57. &\supset \vdash :\text{Hp}.\supset .\Pi ʻ\Sigma ʻP \in \text{Ser} &\qquad \text{(2)}\\ +\vdash .*174·25. &\supset \vdash :\text{Hp}.\supset .\Pi ʻ\Sigma ʻP\,\text{smor}\,\Pi ʻ\Pi ^{;}P &\qquad \text{(3)}\\ +\vdash .(2).(3).*204·21. &\supset \vdash :\text{Hp}.\supset .\Pi ʻ\Pi ^{;}P\in \text{Ser} &\qquad \text{(4)}\\ +\vdash .(2).(4).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*204·581.</b> \[\begin{align}&\vdash :\text{Hp}*204·58.\Sigma ʻP\in \text{Rel}^{2}\text{excl}.\supset .\text{Prod}ʻ\text{Prod}^{;}P,\text{Prod}ʻ\Sigma ʻP\in \text{Ser}\\ +&[*174·461·43.*204·58·21]\end{align}\]</p> + +<p class="nind"><b><a id="*204·59">*204·59</a>.</b> \[\begin{align}\vdash \colon\ldotp P,Q\in \text{Ser}:\alpha \subset CʻQ.\exists !\alpha .\supset _{\alpha }.\exists !\alpha -&\breve{Q} ʻʻ\alpha :\supset .\\ +&P^{Q}\in \text{Ser}.(P\,\text{exp}\,\,Q)\in \text{Ser}\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*165·27·241.*204·22·24.&\supset \vdash :\text{Hp}.\supset .P\downarrow_{.,} ^{;}Q\in \text{Ser} &\qquad \text{(1)}\\ +\vdash .*165·26.*204·22.&\supset \vdash :\text{Hp}.\supset .CʻP\downarrow_{.,} ^{;}Q\subset \text{Ser} &\qquad \text{(2)}\\ +\vdash .*150·22.*71·47.&\supset \vdash :\beta \subset CʻP\downarrow_{.,} ^{;}Q.\exists !\beta .\supset .(\exists \alpha ).\alpha \subset CʻQ.\exists !\alpha .\beta =P\downarrow_{.,} + ʻʻ\alpha :\\ +[\text{Hp}] &\supset \vdash :\text{Hp}.\beta \subset CʻP\downarrow_{.,} ^{;}Q.\exists + !\beta .\supset .(\exists \alpha ).\exists !\alpha -\breve{Q} ʻʻ\alpha .\beta =P\downarrow_{.,} ʻʻ\alpha &\qquad \text{(3)}\\ +\vdash .*37·45.&\supset \vdash :\exists !\alpha -\breve{Q} ʻʻ\alpha .\equiv .\exists !P\downarrow_{.,} ʻʻ(\alpha -\breve{Q} ʻʻ\alpha ) &\qquad \text{(4)}\\ +\vdash .(4).*71·381.*165·22.&\supset \vdash :\dot{\exists} !P.\exists !\alpha -\breve{Q} ʻʻ\alpha .\supset .\exists !P\downarrow_{.,} ʻʻ\alpha -P\downarrow_{.,} + ʻʻ\breve{Q} ʻʻ\alpha &\qquad \text{(5)}\\ +\vdash .*72·503.*165·22. &\supset \vdash :\dot{\exists} !P.\supset .\alpha =(\text{Cnv}ʻP\downarrow_{.,} )ʻʻP\downarrow_{.,} ʻʻ\alpha &\qquad \text{(6)}\\ +\vdash .(5).(6).\supset \vdash :\dot{\exists} !P.\exists !\alpha -\breve{Q} ʻʻ\alpha .&\supset .\exists !P\downarrow_{.,} ʻʻ\alpha -P\downarrow_{.,} + ʻʻ\breve{Q} ʻʻ(\text{Cnv}ʻP\downarrow_{.,} )ʻʻP\downarrow_{.,} ʻʻ\alpha .\\ +[*165·18] &\supset .\exists !P\downarrow_{.,} ʻʻ\alpha -(\text{Cnv}ʻP\downarrow_{.,} ^{;}Q)ʻʻP\downarrow_{.,} + ʻʻ\alpha &\qquad \text{(7)}\\ +\vdash .(3).(7).\supset \vdash \colon\ldotp \text{Hp}.&\dot{\exists} !P.\supset :\\ +&\beta \subset CʻP\downarrow_{.,} ^{;}Q.\exists !.\supset .\exists !\beta -(\text{Cnv}ʻP\downarrow_{.,} ^{;}Q)ʻʻ\beta &\qquad \text{(8)}\\ +\vdash .(1).(2).(8).*204·57. &\supset \vdash :\text{Hp}.\dot{\exists} !P.\supset .\Pi ʻP\downarrow_{.,} ^{;}Q\in \text{Ser} &\qquad \text{(9)}\\ +\vdash .(9).*176·182.*204·21. & \supset \vdash :\text{Hp}.\dot{\exists} !P.\supset .(P \,\text{exp}\,\, Q)\in \text{Ser} &\qquad \text{(10)}\\ +\vdash .*176·151.*204·24. &\supset \vdash :P = \dot{\Lambda} .\supset .(P \,\text{exp}\,\, Q)\in \text{Ser} &\qquad \text{(11)}\\ +\vdash .(10).(11). &\supset \vdash :\text{Hp}.\supset .(P \,\text{exp}\,\, Q)\in \text{Ser} &\qquad \text{(12)}\\ +\vdash .(12).*176·181.*204·21. &\supset \vdash :\text{Hp}.\supset .P^{Q}\in \text{Ser} &\qquad \text{(13)}\\ +\vdash .(12).(13).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_557">[Pg 557]</span></p> + +<p>The two following propositions are lemmas for <a href="#*204·62">*204·62</a>.</p> + +<p class="nind"><b><a id="*204·6">*204·6</a>.</b> \(\vdash :P\in \text{trans} .\supset .\alpha \cup \breve{P} ʻʻ\alpha \subset pʻ\overleftarrow{P}ʻʻpʻ\overrightarrow{P}ʻʻ\alpha \)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*40·53.\supset \vdash \colon\colon x\in pʻ\overleftarrow{P}ʻʻpʻ\overrightarrow{P}ʻʻ\alpha . &\equiv \colon\ldotp y\in pʻ\overrightarrow{P}ʻʻ\alpha .\supset _{y}.yPx\colon\ldotp + \\ +[*40·51] &\equiv \colon\ldotp z\in \alpha .\supset _{z}.yPz:\supset _{y}.yPx &\qquad \text{(1)}\\ +\vdash .*10·26.&\supset \vdash \colon\ldotp x\in \alpha :z\in \alpha .\supset _{z}.yPz:\supset .yPx:\\ +[\text{Exp.(1)}] &\supset \vdash :x\in \alpha .\supset .x\in pʻ\overleftarrow{P}ʻʻpʻ\overrightarrow{P}ʻʻ\alpha &\qquad \text{(2)}\\ +\vdash .*10·1. &\supset \vdash \colon\ldotp u\in \alpha .uPx:z\in \alpha .\supset _{z}.yPz:\supset .yPu.uPx &\qquad \text{(3)}\\ +\vdash .(3).*201·1.\supset \vdash \colon\colon\text{Hp}.&\supset \colon\ldotp u\in \alpha .uPx:z\in \alpha .\supset _{z}.yPz:\supset .yPx\colon\ldotp \\ +[*37·105] &\supset \colon\ldotp x\in \breve{P} ʻʻ\alpha :z\in \alpha .\supset _{z}.yPz:\supset .yPx\colon\ldotp \\ +[Exp.(1)] &\supset \colon\ldotp x\in \breve{P} ʻʻ\alpha .\supset .x\in pʻ\overleftarrow{P}ʻʻpʻ\overrightarrow{P}ʻʻ\alpha &\qquad \text{(4)}\\ +\vdash .(2).(4).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*204·61.</b> \(\vdash :P\in \text{Rl}ʻJ\cap \text{connex} .\supset .CʻP\cap pʻ\overleftarrow{P}ʻʻpʻ\overrightarrow{P}ʻʻ(\alpha \cap CʻP)\subset \alpha \cup \breve{P} ʻʻ\alpha\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*200·5.\supset \vdash :\text{Hp}.&\supset .pʻ\overrightarrow{P}ʻʻ(\alpha \cap CʻP)\cap pʻ\overleftarrow{P}ʻʻpʻ\overrightarrow{P}ʻʻ(\alpha \cap CʻP) = \Lambda .\\ +[*24·311] &\supset .pʻ\overleftarrow{P}ʻʻpʻ\overrightarrow{P}ʻʻ(\alpha \cap CʻP)\subset -pʻ\overrightarrow{P}ʻʻ(\alpha \cap CʻP).\\ +[*22·48] &\supset .CʻP\cap pʻ\overleftarrow{P}ʻʻpʻ\overrightarrow{P}ʻʻ(\alpha \cap CʻP)\subset CʻP-pʻ\overrightarrow{P}ʻʻ(\alpha \cap CʻP)\\ +[*24·43.*202·505] &\subset \alpha \cup \breve{P} ʻʻ\alpha :\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*204·62">*204·62</a>.</b> \(\vdash :P\in \text{Ser}.\supset .CʻP\cap pʻ\overleftarrow{P}ʻʻpʻ\overrightarrow{P}ʻʻ(\alpha \cap CʻP) = (\alpha \cap CʻP)\cup \breve{P} ʻʻ\alpha\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*204·6.*37·265.&\supset \vdash :\text{Hp}.\supset .(\alpha \cap CʻP)\cup \breve{P} ʻʻ\alpha \subset pʻ\overleftarrow{P}ʻʻpʻ\overrightarrow{P}ʻʻ(\alpha \cap CʻP) &\qquad \text{(1)}\\ +\vdash .*37·16.*22·43. &\supset \vdash .(\alpha \cap CʻP)\cup \breve{P} ʻʻ\alpha \subset CʻP &\qquad \text{(2)}\\ +\vdash .*204·61.*22·43.&\supset \vdash :\text{Hp}.\supset .CʻP\cap pʻ\overleftarrow{P}ʻʻpʻ\overrightarrow{P}ʻʻ(\alpha \cap CʻP)\subset (\alpha \cup \breve{P} ʻʻ\alpha )\cap CʻP.\\ +[*37·16] &\supset .CʻP\cap pʻ\overleftarrow{P}ʻʻpʻ\overrightarrow{P}ʻʻ(\alpha \cap CʻP)\subset (\alpha \cap CʻP)\cup \breve{P} ʻʻ\alpha &\qquad \text{(3)}\\ +\vdash .(1).(2).(3).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_558">[Pg 558]</span></p> + +<p class="nind"><b>*204·63.</b> \(\vdash :P\in \text{Ser}.\exists !pʻ\overrightarrow{P}ʻʻ\alpha .\supset .pʻ\overleftarrow{P}ʻʻpʻ\overrightarrow{P}ʻʻ\alpha =\alpha \cup \breve{P} ʻʻ\alpha\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*40·65.\text{Transp}.&\supset \vdash :\text{Hp}.\supset .\alpha \subset CʻP &\qquad \text{(1)}\\ +\vdash .*40·62. &\supset \vdash :\text{Hp}.\supset .pʻ\overleftarrow{P}ʻʻpʻ\overrightarrow{P}ʻʻ\alpha \subset CʻP &\qquad \text{(2)}\\ +\vdash .(1).(2).*204·62.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*204·64.</b> \(\vdash :P\in \text{Ser}.x\in \text{D}ʻP.\supset .pʻ\overrightarrow{P}ʻʻ\overleftarrow{P}ʻx=\overrightarrow{P}_{*}ʻx\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*40·62. &\supset \vdash :\text{Hp}.\supset .pʻ\overrightarrow{P}ʻʻ\overleftarrow{P}ʻx\subset CʻP &\qquad \text{(1)}\\ +\vdash .*40·51.&\supset \vdash \colon\ldotp z\in pʻ\overrightarrow{P}ʻʻ\overleftarrow{P}ʻx.\equiv :xPy.\supset _{y}.zPy &\qquad \text{(2)}\\ +\vdash .(2).*50·11. \supset \vdash \colon\colon\text{Hp}.\supset \colon\ldotp z\in pʻ\overrightarrow{P}ʻʻ\overleftarrow{P}ʻx.&\supset :xPy.\supset _{y}.z\neq y:\\ + [(1).*202·103] &\supset :zPx.\lor.z=x &\qquad \text{(3)}\\<br> +\vdash .*201·521. \supset \vdash \colon\colon\text{Hp}.\supset \colon\ldotp z\in \overrightarrow{P}_{*}ʻx.&\equiv :zPx.\lor.z=x: &\qquad \text{(4)}\\ +[*201·1.*13·12] & \supset :xPy.\supset .zPy:\\ +[(2)] &\supset :z\in pʻ\overrightarrow{P}ʻʻ\overleftarrow{P}ʻx &\qquad \text{(5)}\\ +\vdash .(3).(4).(5).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p>The following proposition is used in <a href="#*234·101">*234·101</a>.</p> + +<p class="nind"><b><a id="*204·65">*204·65</a>.</b> \(\vdash :P\in \text{Ser}.x\in CʻP.\supset .pʻ\overrightarrow{P}ʻʻ\overleftarrow{P}ʻx\cap CʻP=\overrightarrow{P}_{*}ʻx\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*40·2.\supset \vdash :\text{Hp}.x{\sim}\in \text{D}ʻP.\supset .pʻ\overrightarrow{P}ʻʻ\overleftarrow{P}ʻx\cap CʻP&=CʻP\\ +[*204·11] &=\overrightarrow{P}ʻx\cup \iota ʻx\\ +[*201·521] &=\overrightarrow{P}_{*}ʻx &\qquad \text{(1)}\\ +\vdash .*40·62.*204·64.\supset \vdash :\text{Hp}.x\in \text{D}ʻP.\supset .pʻ\overrightarrow{P}ʻʻ\overleftarrow{P}ʻx\cap CʻP&=\overrightarrow{P}_{*}ʻx &\qquad \text{(2)}\\ +\vdash .(1).(2).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*204·7">*204·7</a>.</b> \(\vdash :P\in \text{Ser}.\supset .P_{1}\in 1\rightarrow 1 \quad[*201·63.*202·7]\)</p> + +<p>On this proposition, compare the remarks preceding <a href="#*201·6">*201·6</a>.</p> + +<p class="nind"><b><a id="*204·71">*204·71</a>.</b> \(\vdash :P\in \text{Ser}.xP_{1}y.\supset .\overrightarrow{P}ʻy=\overrightarrow{P}ʻx\cup \iota ʻx \quad[*202·72.*201·63]\)</p> + +<p class="nind"><b>*204·72.</b> \(\vdash \colon\colon P\in \text{Ser}.\supset \colon\ldotp xP_{1}y.\equiv :xPy:xPz.z\neq y.\supset _{z}.yPz\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*201·63. &\supset \vdash \colon\ldotp \text{Hp}.\supset :xP_{1}y.\supset .xPy &\qquad \text{(1)}\\ +\vdash .*204·71.*121·26.\supset \vdash \colon\ldotp \text{Hp}.xP_{1}y.&\supset :\overleftarrow{P}ʻx=\overleftarrow{P}ʻy\cup \iota ʻy:\\ +[*24·43.*32·181] &\supset :xPz.z\neq y.\supset _{z}.yPz &\qquad \text{(2)}\\ +\vdash .*24·43.*32·181. &\supset \vdash \colon\ldotp xPz.z\neq y.\supset _{z}.yPz:\supset .\overleftarrow{P}ʻx\subset \iota ʻy\cup \overleftarrow{P}ʻy &\qquad \text{(3)}\\ +\vdash .(3).*200·361. \supset \vdash \colon\ldotp \text{Hp}:xPz.z\neq y.\supset _{z}.yPz:&\supset .\overleftarrow{P}ʻx\cap \overrightarrow{P}ʻy=\Lambda .\\ +[*34·11] &\supset .{\sim}(xP^{2}y) &\qquad \text{(4)}\\ +\vdash .(4).\text{Fact}.\supset \vdash \colon\ldotp \text{Hp}:xPy:xPz.z\neq y.&\supset _{z}.yPz:\supset .x(P\dot{-} P^{2})y.\\ +[*201·63] &\supset .xP_{1}y &\qquad \text{(5)}\\ +\vdash .(1).(2).(5).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p>The above proposition is used in *274·23.</p> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_559">[Pg 559]</span></p> +<h2 class="nobreak" id="205_MAXIMUM_AND_MINIMUM_POINTS"> +*205. MAXIMUM AND MINIMUM POINTS.</h2> +</div> + + +<p class="nind"><i>Summary of</i> *205.</p> + +<p>The minimum points of a class \(\alpha\) with respect to a relation +\(P\) are those members of \(\alpha\) which belong to the field of +\(P\) but to which no members of \(\alpha\) have the relation \(P\); +that is, they are those members of \(\alpha\) which belong to \(CʻP\) +but have no predecessors in \(\alpha\). Similarly the maximum points +of \(\alpha\) are those members of \(\alpha\) which belong to \(CʻP\) +but have no successors in \(\alpha\). Both these notions have been +already defined in *93, but they were there only used for the special +purpose of studying generations. Their chief utility is in connection +with <i>series</i>, and it is in this connection that we shall now +consider them. Many of the properties of maxima and minima in series do +not demand the whole hypothesis "\(P\in \text{Ser}\)," but only "\(P\in\text{connex}\)." +This is the case, in particular, with the fundamental +property of maxima and minima in series, namely that each class has at +most one maximum and one minimum. The minimum of a class, if it exists, +is the first term of the class, and the maximum, if it exists, is the +last term. The maxima with respect to \(P\) are the minima with respect +to \(\breve{P}\); hence properties of maxima result immediately from +the corresponding properties of minima, and will be set down without +proof in what follows.</p> + +<p>It will be seen that the maxima and minima of \(\alpha\) depend only +upon \(\alpha \cap CʻP\): the part of \(\alpha\) (if any) which is not +contained in \(CʻP\) is irrelevant.</p> + +<p>In accordance with the definitions of *93, the class of minima of +\(\alpha\) is denoted by \(\overrightarrow{\text{min}}_{p}ʻ\alpha\), +where +\[ +\overrightarrow{\text{min}}_{p}ʻ\alpha =(\alpha \cap CʻP)-\breve{P} ʻʻ\alpha , +\] +the definition being +\[ +\text{min}_{P}=\hat{x} \hat{\alpha} \{x\in (\alpha \cap CʻP)-\breve{P} ʻʻ\alpha\}. +\] +Thus \(\text{min}_{P}\) is a relation contained in \({\in}\). When +\(P\) is connected, we have \(\overrightarrow{\text{min}}_{P}ʻ\alpha\in 0\cup 1\), +<i>i.e.</i> (by *71·12) +\[ +\text{min}_{P}\in 1\rightarrow \text{Cls}. +\] +It follows that, if \(\kappa\) is a set of classes which all have +minima, \(\text{min}_{P}\upharpoonright \kappa\) is a selective +relation for \(\kappa\), <i>i.e.</i> +\[ +\text{min}_{P}\upharpoonright \kappa \in {\in}_{\Delta}ʻ\kappa . +\]<span class="pagenum" id="Page_560">[Pg 560]</span> +Owing to this fact, the existence of selections can sometimes be proved +in dealing with series (especially with well-ordered series), in cases +where such proof would be impossible if no serial arrangement were +given.</p> + +<p>The definition of \(\text{min}_{P}\) is so chosen as to exclude +from \(\overrightarrow{\text{min}}_{P}ʻ\alpha\) whatever +part of \(\alpha\) is not contained in \(CʻP\), and to make +\(\overrightarrow{\text{min}}_{P}ʻ{℩}ʻx={℩}ʻx\), <i>i.e.</i> +\(\text{min}_{P}ʻ{℩}ʻx=x\), provided \(x\in CʻP.{\sim}(xPx)\). For +these two reasons we have to reject two simpler definitions which might +otherwise be thought preferable. One of these would give +\[ +\overrightarrow{\text{min}}_{P}ʻ\alpha =\alpha -\breve{P} ʻʻ\alpha , +\] +which might be obtained by putting +\[ +\text{min}_{P}=\in \dot{-} \in \mid \breve{P} \quad\text{Df}. +\] +This agrees with our definition whenever \(\alpha\subset CʻP\), +but not otherwise, since it includes in +\(\overrightarrow{\text{min}}_{P}ʻ\alpha\) any part of \(\alpha\) not +contained in \(CʻP\). Hence it necessitates the hypothesis \(\alpha\) +\(\subset CʻP\) in many propositions which, with our definition, do not +require this hypothesis, and in particular in the proposition +\[ +P\in \text{connex} .\supset .\overrightarrow{\text{min}}_{P}ʻ\alpha \in 0\cup 1, +\] +so that instead of having (as with our definition) +\[ +P\in \text{connex} .\supset .\overrightarrow{\text{min}}_{P}\in 1\rightarrow \text{Cls} +\] +we should only have +\[ +P\in \text{connex} .\supset .\overrightarrow{\text{min}}_{P}\upharpoonright \text{Cl}ʻCʻP\in 1\rightarrow \text{Cls}. +\] +For these reasons, this definition is less convenient than the one we +have adopted.</p> + +<p>The other definition which suggests itself is one which will give +\[ +\overrightarrow{\text{min}}_{P}ʻ\alpha =\overrightarrow{B}ʻP\unicode{x0294f}\alpha . +\] +If this definition were adopted, we might dispense +with a special notation altogether, using +\(\overrightarrow{B}ʻP\unicode{x0294f}\alpha\), +\(BʻP\unicode{x0294f}\alpha\) in place of +\(\overrightarrow{\text{min}}_{P}ʻ\alpha\), \(\text{min}_{P}ʻ\alpha\). +This definition, however, has the drawback that, if \(\alpha \in 1\) +and \(P\,\unicode{x2abd}\, J\), +\[ +P\unicode{x0294f}\alpha =\dot{\Lambda} , +\] +so that we have +\[ +\overrightarrow{\text{min}}_{P}ʻ\alpha =\dot{\Lambda}\,\text{when}\, \alpha \in 1.\alpha \subset CʻP. +\] +This necessitates the addition of the hypothesis \(\alpha{\sim}\in 1\) +(as in <a href="#*204·45">*204·45</a> above, for example) in cases where, +with our definition, no such hypothesis is required. If we take +\(\overrightarrow{B}ʻ\alpha \upharpoonleft P\), instead of +\(\overrightarrow{B}ʻP\unicode{x0294f}\alpha\), as the class +of minimum points, we secure \(\text{min}_{P}ʻ{℩}ʻx=x\) when +\(P\,\unicode{x2abd}\, J\) and \(x\in \text{D}ʻP\), but not when +\(x\in\overrightarrow{B}ʻ\breve{P}\). Thus we still have exceptions to +provide against which do not arise with the definition we have adopted.</p> + +<p><span class="pagenum" id="Page_561">[Pg 561]</span></p> + +<p>The first few propositions of this number have already been proved in +*93, but are repeated here for convenience of reference.</p> + +<p>The propositions of this number are numerous and much used. Among the +elementary properties of \(\text{max}_{P}\) and \(\text{min}_{P}\) with +which the number begins, the following should be noted:</p> + +<p class="nind"><b>*205·12.</b> \(\vdash .\overrightarrow{B}ʻP=\overrightarrow{\text{min}}_{P}ʻ\text{D}ʻP=\overrightarrow{\text{min}}_{P}ʻCʻP\)</p> + +<p class="nind"><b>*205·123.</b> \(\vdash :\overrightarrow{\text{max}}_{P}ʻ\alpha =\Lambda .\equiv .\alpha \cap CʻP\subset Pʻʻ\alpha\)</p> + +<p class="nind"><b>*205·14.</b> \(\vdash .\overrightarrow{\text{min}}_{P}ʻ\alpha =\hat{x} \{x\in \alpha \cap CʻP.\alpha \cap \overrightarrow{P}ʻx= \Lambda \}\)</p> + +<p class="nind"><b>*205·15.</b> \(\vdash .\overrightarrow{\text{min}}_{P}ʻ(\alpha \cap CʻP)=\overrightarrow{\text{min}}_{P}ʻ\alpha\)</p> + +<p class="nind"><b>*205·16.</b> \(\vdash .\overrightarrow{\text{min}}_{P}ʻ\Lambda =\Lambda\)</p> + +<p class="nind"><b>*205·18.</b> \(\vdash :{\sim}(xPx).x\in CʻP.\supset .\text{min}_{P}ʻ{℩}ʻx=\text{max}_{P}ʻ{℩}ʻx=x\)</p> + +<p class="nind"><b>*205·19.</b> \(\vdash :P\in \text{trans}.\supset .\overrightarrow{\text{min}}_{P}ʻ\alpha =\overrightarrow{\text{min}}_{P}ʻ(\alpha + \cup \breve{P} ʻʻ\alpha )=\overrightarrow{\text{min}}_{P}ʻ\breve{P} _{*}ʻʻ\alpha\)</p> + +<p class="nind"><b>*205·194.</b> \(\vdash :x\text{min}_{P}\alpha .\supset .{\sim}(xPx)\)</p> + +<p>Owing to this proposition, we can sometimes dispense with the +hypothesis \(P\,\unicode{x2abd}\, J\) in propositions about minima which +would otherwise require this hypothesis.</p> + +<p class="nind"><b>*205·197.</b> \(\vdash \colon\ldotp P\in \text{Rl}ʻJ\cap \text{trans}.\supset :x\in CʻP.\equiv .x=\text{max}_{P}ʻ(\overrightarrow{P}ʻx\cup {℩}ʻx)\)</p> + +<p>Our next set of propositions (<a href="#*205·2">*205·2</a>—<a href="#*205·27">·27</a>) introduces the hypothesis +that \(P\) is connected, or transitive and connected. The chief of them +are</p> + +<p class="nind"><b>*205·21.</b> \(\vdash :P\in \text{connex} .\text{E}!\text{min}_{P}ʻ\alpha .y\in \alpha \cap CʻP-{℩}ʻ\text{min}_{P}ʻ\alpha .\supset .\text{min}_{P}ʻ\alpha Py\)</p> + +<p><i>I.e.</i> if the minimum of \(\alpha\) exists, it precedes every other member of \(\alpha \cap CʻP\).</p> + +<p class="nind"><b>*205·22.</b> \(\vdash :P\in \text{trans}\cap \text{connex} .\text{E}!\text{min}_{P}ʻ\alpha .\supset .\breve{P} ʻʻ\alpha =\overleftarrow{P}ʻ\text{min}_{P}ʻ\alpha\)</p> + +<p><i>I.e.</i> the terms which come after some part of \(\alpha\) are +those that come after its minimum (when the minimum exists).</p> + +<p class="nind"><b>*205·25.</b> \(\vdash .\overrightarrow{\text{min}}_{P}ʻ\overleftarrow{P}ʻx=(\overleftarrow{P}\dot{-} P^{2})ʻx\)</p> + +<p>We have next the fundamental proposition:</p> + +<p class="nind"><b>*205·3.</b> \(\vdash :P\in \text{connex} .\supset .\overrightarrow{\text{min}}_{P}ʻ\alpha \in 0\cup 1.\overrightarrow{\text{max}}_{P}ʻ\alpha \in 0\cup 1\)</p> + +<p class="nind">whence</p> + +<p class="nind"><b>*205·31.</b> \(\vdash :P\in \text{connex} .\supset .\text{min}_{P},\text{max}_{P}\in 1\rightarrow \text{Cls}\)</p> + +<p class="nind">which leads to</p> + +<p class="nind"><b>*205·33.</b> \(\vdash :P\in \text{connex} .\kappa \subset \text{ᗡ}ʻ\text{min}_{P}.\supset .\text{min}_{P}\upharpoonright \kappa \in {\in}_{\Delta}ʻ\kappa\)</p> + +<p>This proposition is useful in the theory of well-ordered series. +Observe that "\(\kappa \subset \text{ᗡ}ʻ\text{min}_{P}\)" means that +\(\kappa\) consists of classes which have minima.</p> + +<p><span class="pagenum" id="Page_562">[Pg 562]</span></p> + +<p>We have next a set of propositions (<a href="#*205·4">*205·4</a>—<a href="#*205·44">·44</a>) dealing with the +relations of \(\text{min}_{P}ʻ\alpha\) to \(BʻP\unicode{x0294f}\alpha\) +and \(Bʻ\alpha \upharpoonleft P\); next we have propositions on the +relations of the minima of two different classes, of which the most +useful is</p> + +<p class="nind"><b>*205·55.</b> \(\vdash :P\in \text{connex} .BʻP\in \alpha .\supset .BʻP=\text{min}_{P}ʻ\alpha\)</p> + +<p>We have next various propositions on \(pʻ\overrightarrow{P}ʻʻ(\alpha\cap CʻP)\), +of which the chief is</p> + +<p class="nind"><b>*205·65.</b> \(\vdash :P\in \text{trans}\cap \text{connex} .\text{E}!\text{min}_{P}ʻ\alpha + .\supset .pʻ\overrightarrow{P}ʻʻ(\alpha \cap CʻP)=\overrightarrow{P}ʻ\text{min}_{P}ʻ\alpha\)</p> + +<p><i>I.e.</i> the predecessors of the whole of a class contained in +\(CʻP\) are the predecessors of its minimum (if it has one).</p> + +<p>A useful proposition is</p> + +<p class="nind"><b>*205·68.</b> \(\vdash :\breve{P} ʻʻ\alpha \subset \alpha .\supset .\overrightarrow{\text{min}}_{P}ʻ\alpha =\overrightarrow{\text{min}}(P_{\text{po}})ʻ\alpha\)</p> + +<p><i>I.e.</i> if \(\alpha\) is a hereditary class, its minima with +respect to \(P\) are the same as its minima with respect to +\(P_{\text{po}}\).</p> + +<p>We prove next that if \(Pʻʻ\alpha\) has a maximum, so has \(\alpha\) +(<a href="#*205·7">*205·7</a>), and that if \(P\in \text{connex}\), only a unit class can +have its maximum identical with its minimum (<a href="#*205·73">*205·73</a>).</p> + +<p><a href="#*205·8">*205·8</a>—<a href="#*205·85">·85</a> are concerned with relation-arithmetic. The chief +proposition here is</p> + +<p class="nind"><b>*205·8.</b> \(\vdash :S\in P\,\overline{\,\text{smor}\,}\,Q.\supset .\overrightarrow{\text{min}}_{P}ʻ\alpha =Sʻʻ\overrightarrow{\text{min}}_{Q}ʻ\breve{S} ʻʻ\alpha\)</p> + +<p><i>I.e.</i> in any correlation, the minima of the correlates of a class +are the correlates of the minima.</p> + +<p>We end with two propositions on relations with limited fields. The more +useful of these is</p> + +<p class="nind"><b>*205·9.</b> \(\vdash :P\in \text{connex} ·\kappa \subset CʻP.\kappa {\sim}\in 1.\supset .\overrightarrow{\text{min}}(P\unicode{x0294f}\kappa )ʻ\alpha =\overrightarrow{\text{min}}_{P}ʻ(\alpha + \cap \kappa )\)</p> + +<hr class="tb"> + +<p class="nind"><b>*205·1.</b> \(\vdash :x\text{min}_{P}\alpha .\equiv .x\in \alpha \cap CʻP-\breve{P} ʻʻ\alpha \quad[*93·11]\)</p> + +<p class="nind"><b>*205·101.</b> \(\vdash :x\text{max}_{P}\alpha .\equiv .x\in \alpha \cap CʻP-Pʻʻ\alpha .\equiv .x \text{min}(\breve{P} )\alpha \quad[*93·115]\)</p> + +<p class="nind"><b>*205·102.</b> \(\vdash .\text{max}_{P}=\text{min}(\breve{P} ) \quad[*93·114]\)</p> + +<p class="nind"><b>*205·11.</b> \(\vdash .\overrightarrow{\text{min}}_{P}ʻ\alpha =\alpha \cap CʻP-\breve{P} ʻʻ\alpha \quad [*93·111]\)</p> + +<p class="nind"><b>*205·111.</b> \(\vdash .\overrightarrow{\text{max}}_{P}ʻ\alpha =\alpha \cap CʻP-Pʻʻ\alpha \quad [*93·116]\)</p> + +<p class="nind"><b>*205·12.</b> \(\vdash .\overrightarrow{B}ʻP=\overrightarrow{\text{min}}_{P}ʻ\text{D}ʻP=\overrightarrow{\text{min}}_{P}ʻCʻP \quad[*93·112]\)</p> + +<p class="nind"><b>*205·121.</b> \(\vdash .\overrightarrow{B}ʻ\breve{P} =\overrightarrow{\text{max}}_{P}ʻ\text{ᗡ}ʻP=\overrightarrow{\text{max}}_{P}ʻCʻP \quad[*93·117]\)</p> + +<p class="nind"><b>*205·122.</b> \(\vdash :\overrightarrow{\text{min}}_{P}ʻ\alpha =\Lambda .\equiv .\alpha \cap CʻP\subset \breve{P} ʻʻ\alpha \quad[*205·11.*24·3]\)</p> + +<p><span class="pagenum" id="Page_563">[Pg 563]</span></p> + +<p class="nind"><b>*205·123.</b> \(\vdash :\overrightarrow{\text{max}}_{P}ʻ\alpha =\Lambda .\equiv .\alpha \cap CʻP\subset Pʻʻ\alpha\)</p> + +<p class="nind"><b>*205·13.</b> \(\vdash .\overrightarrow{\text{min}}_{P}ʻ\alpha \cup \breve{P} ʻʻ\alpha =(\alpha \cap CʻP)\cup \breve{P} ʻʻ\alpha \quad[*22·91.*205·11]\)</p> + +<p class="nind"><b>*205·131.</b> \(\vdash .\overrightarrow{\text{max}}_{P}ʻ\alpha \cup Pʻʻ\alpha =(\alpha \cap CʻP)\cup Pʻʻ\alpha\)</p> + +<p class="nind"><b>*205·14.</b> \(\vdash .\overrightarrow{\text{min}}_{P}ʻ\alpha =\hat{x} \{x\in \alpha \cap CʻP.\alpha \cap \overrightarrow{P}ʻx=\Lambda\} \quad[*37·462.*205·11]\)</p> + +<p class="nind"><b>*205·141.</b> \(\vdash .\overrightarrow{\text{max}}_{P}ʻ\alpha =\hat{x} \{x\in \alpha \cap CʻP.\alpha \cap \overleftarrow{P}ʻx=\Lambda\}\)</p> + +<p class="nind"><b>*205·15.</b> \(\vdash .\overrightarrow{\text{min}}_{P}ʻ(\alpha \cap CʻP)=\overrightarrow{\text{min}}_{P}ʻ\alpha \quad[*37·265.*205·11]\)</p> + +<p class="nind"><b>*205·151.</b> \(\vdash .\overrightarrow{\text{max}}_{P}ʻ(\alpha \cap CʻP)=\overrightarrow{\text{max}}_{P}ʻ\alpha\)</p> + +<p class="nind"><b>*205·16.</b> \(\vdash .\overrightarrow{\text{min}}_{P}ʻ\Lambda =\Lambda \quad[*205·11.*24·23]\)</p> + +<p class="nind"><b>*205·161.</b> \(\vdash .\overrightarrow{\text{max}}_{P}ʻ\Lambda =\Lambda\)</p> + +<p class="nind"><b>*205·17.</b> \[\begin{align}\vdash \colon\ldotp x\in (\alpha \cap CʻP).\supset _{x}.{\sim}(xPx):\alpha &\cap CʻP\in 1:\supset .\\ +&\overrightarrow{\text{min}}_{P}ʻ\alpha =\overrightarrow{\text{max}}_{P}ʻ\alpha =\alpha \cap CʻP\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*13·14. &\supset \vdash \colon\ldotp \text{Hp}.\supset :x\in \alpha .xPy.\supset _{x,y}.x\neq y &\qquad \text{(1)}\\ +\vdash .*52·16. &\supset \vdash \colon\ldotp \text{Hp}.\supset :x,y\in \alpha \cap CʻP.\supset _{x,y}.x=y &\qquad \text{(2)}\\ +\vdash .(1).(2).*33·17.\supset \vdash \colon\ldotp \text{Hp}.&\supset :x\in \alpha .xPy.\supset _{x,y}.y{\sim}\in \alpha :\\ +[*37·1] & \supset :\breve{P} ʻʻ\alpha \subset -\alpha :\\ +[*22·811] &\supset :\alpha \subset -\breve{P} ʻʻ\alpha &\qquad \text{(3)}\\ +\vdash .(3).*205·11.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*205·18.</b> \(\vdash :{\sim}(xPx).x\in CʻP.\supset .\text{min}_{P}ʻ\iota ʻx=\text{max}_{P}ʻ\iota ʻx=x\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*205·17.\supset \vdash :\text{Hp}.\supset .\overrightarrow{\text{min}}_{P}ʻ\iota ʻx=\overrightarrow{\text{max}}_{P}ʻ\iota ʻx=\iota ʻx &\qquad \text{(1)}\\ +\vdash .(1).*53·4.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*205·181.</b> \(\vdash :xPy.{\sim}(xPx).{\sim}(yPx).\supset .\text{min}_{P}ʻ(\iota ʻx\cup \iota ʻy)=x\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*37·105.&\supset \vdash :\text{Hp}.\supset .x{\sim}\in )\breve{P} ʻʻ(\iota ʻx\cup \iota ʻy).y\in \breve{P} ʻʻ(\iota ʻx\cup \iota ʻy) &\qquad \text{(1)}\\ +\vdash .*33·17. &\supset \vdash :\text{Hp}.\supset .\iota ʻx\cup \iota ʻy\subset CʻP &\qquad \text{(2)}\\ +\vdash .(1).(2).*205·11.&\supset \vdash :\text{Hp}.\supset .\overrightarrow{\text{min}}_{P}ʻ(\iota ʻx\cup \iota ʻy)=\iota ʻx:\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*205·182.</b> \(\vdash :P^{2}\,\unicode{x2abd}\, J.xPy.\supset .\text{min}_{P}ʻ(\iota ʻx\cup \iota ʻy)=x\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*200·36.*50·43.\supset \vdash :\text{Hp}.\supset .{\sim}(xPx).{\sim}(yPx) &\qquad \text{(1)}\\ +\vdash .(1).*205·181.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_564">[Pg 564]</span></p> + +<p class="nind"><b>*205·183.</b> \[\begin{align}\vdash \colon\ldotp P^{2} \,\unicode{x2abd}\, J . P \in \text{connex} . &x, y \in CʻP . \supset :\\ +&\text{min}_{P}ʻ({℩}ʻx \cup {℩}ʻy) = x . \lor . \text{min}_{P}ʻ({℩}ʻx \cup {℩}ʻy) = y\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *202·103 . &\supset \vdash \colon\ldotp \text{Hp} . \supset : x = y . \lor . xPy . \lor . yPx &\qquad \text{(1)}\\ +\vdash . *205·18 . &\supset \vdash : \text{Hp} . x = y . \supset . \text{min}_{P}ʻ({℩}ʻx \cup {℩}ʻy) = x &\qquad \text{(2)}\\ +\vdash . *205·182 . &\supset \vdash : \text{Hp} . xPy . \supset . \text{min}_{P}ʻ({℩}ʻx \cup {℩}ʻy) = x &\qquad \text{(3)}\\ +\vdash . *205·182 . &\supset \vdash : \text{Hp} . yPx . \supset . \text{min}_{P}({℩}ʻx \cup {℩}ʻy) = y &\qquad \text{(4)}\\ +\vdash . (1) . (2) . (3) . (4) . \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*205·19.</b> \(\vdash : P \in \text{trans} . \supset . \overrightarrow{\text{min}}_{P}ʻ\alpha = \overrightarrow{\text{min}}_{P}ʻ(\alpha + \cup \breve{P} ʻʻ\alpha ) = \overrightarrow{\text{min}}_{P}ʻ\breve{P} _{*}ʻʻ\alpha\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *205·11 . & \supset \vdash . \overrightarrow{\text{min}}_{P}ʻ(\alpha + \cup \breve{P} ʻʻ\alpha ) = (\alpha \cup \breve{P} ʻʻ\alpha ) \cap CʻP - \breve{P} ʻʻ(\alpha \cup \breve{P} ʻʻ\alpha ) &\qquad \text{(1)}\\ +\vdash . (1) . *201·55 . \supset \vdash : \text{Hp} . \supset . \overrightarrow{\text{min}}_{P}ʻ(\alpha + \cup \breve{P} ʻʻ\alpha ) &= (\alpha \cup \breve{P} ʻʻ\alpha ) \cap CʻP - \breve{P} ʻʻ\alpha \\ +[*22·9] &= \alpha \cap CʻP - \breve{P} ʻʻ\alpha\\ +[*205·11] & = \overrightarrow{\text{min}}_{P}ʻ\alpha &\qquad \text{(2)}\\ +\vdash . *201·52 . *37·265 . \supset \vdash : \text{Hp} . &\supset . \breve{P} _{*}ʻʻ\alpha = (\alpha \cap CʻP) \cup \breve{P} ʻʻ(\alpha \cap CʻP) .\\ +[(2)] \supset . \overrightarrow{\text{min}}_{P}ʻ\breve{P} _{*}ʻʻ\alpha &= \overrightarrow{\text{min}}_{P}ʻ(\alpha \cap CʻP)\\ +[*205·15] &= \overrightarrow{\text{min}}_{P}ʻ\alpha &\qquad \text{(3)}\\ +\vdash . (2) . (3) . \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*205·191.</b> \(\vdash : P \in \text{trans} . \supset . \overrightarrow{\text{max}}_{P}ʻ\alpha = \overrightarrow{\text{max}}_{P}ʻ(\alpha + \cup Pʻʻ\alpha) = \overrightarrow{\text{max}}_{P}ʻP_{*}ʻʻ\alpha\)</p> + +<p class="nind"><b><a id="*205·192">*205·192</a>.</b> \(\vdash : P \in \text{trans} . \beta \subset \breve{P} ʻʻ\alpha . \supset . \overrightarrow{\text{min}}_{P}ʻ(\alpha + \cup \beta ) = \overrightarrow{\text{min}}_{P}ʻ\alpha\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *205·11 . *201·56 . \supset \\ +\vdash : \text{Hp} . \supset . \overrightarrow{\text{min}}_{P}ʻ(\alpha \cup \beta ) &= (\alpha \cup \beta ) \cap CʻP - \breve{P} ʻʻ\alpha \\ +[*22·68] & = (\alpha \cap CʻP - \breve{P} ʻʻ\alpha ) \cup (\beta \cap CʻP - \breve{P} ʻʻ\alpha )\\ +[*24·3] & = \alpha \cap CʻP - \breve{P} ʻʻ\alpha \\ +[*205·11] & = \overrightarrow{\text{min}}_{P}ʻ\alpha : \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*205·193.</b> \(\vdash : P \in \text{trans} . \beta \subset Pʻʻ\alpha . \supset . \overrightarrow{\text{max}}_{P}ʻ(\alpha + \cup \beta ) = \overrightarrow{\text{max}}_{P}ʻ\alpha\)</p> + +<p class="nind"><b>*205·194.</b> \(\vdash : x \text{min}_{P} \alpha . \supset . {\sim} (xPx)\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *37·105 . &\supset \vdash . x \in \alpha . xPx . \supset . x \in \breve{P} ʻʻ\alpha &\qquad \text{(1)}\\) +\vdash . (1) . \text{Transp} . &\supset \vdash : x \in \alpha - \breve{P} ʻʻ\alpha . \supset . {\sim} (xPx) &\qquad \text{(2)}\\ +\vdash . (2) . *205·1 . &\supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_565">[Pg 565]</span></p> + +<p class="nind"><b>*205·195.</b> \(\vdash : x\, \text{max}_{P}\, \alpha . \supset . {\sim} (xPx)\)</p> + +<p class="nind"><b>*205·196.</b> \(\vdash \colon\ldotp P\in \text{Rl}ʻJ\cap \text{trans}.\supset :x\in CʻP.\equiv .x=\text{min}_{P}ʻ({℩}ʻx\cup \overleftarrow{P}ʻx)\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*205·19.\supset \vdash \colon\ldotp \text{Hp}.&\supset :\overrightarrow{\text{min}}_{P}ʻ({℩}ʻx\cup \overleftarrow{P}ʻx)=\overrightarrow{\text{min}}_{P}ʻ{℩}ʻx:\\ +[*205·18] &\supset :x\in CʻP.\supset .\text{min}_{P}ʻ({℩}ʻx\cup \overleftarrow{P}ʻx)=x &\qquad \text{(1)}\\ +\vdash .*205·11.&\supset \vdash :\text{min}_{P}ʻ({℩}ʻx\cup \overleftarrow{P}ʻx)=x.\supset .x\in CʻP &\qquad \text{(2)}\\ +\vdash .(1).(2).&\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*205·197.</b> \(\vdash \colon\ldotp P\in \text{Rl}ʻJ\cap \text{trans}.\supset :x\in CʻP.\equiv .x=\text{max}_{P}ʻ(\overrightarrow{P}ʻx\cup {℩}ʻx)\)</p> + +<p class="nind"><b><a id="*205·2">*205·2</a>.</b> \(\vdash \colon\ldotp P\in \text{connex} .\text{E}!\text{min}_{P}ʻ\alpha .y\in \alpha \cap CʻP.\supset :\text{min}_{P}ʻ\alpha =y.\lor.\text{min}_{P}ʻ\alpha + Py\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*202·103. &\supset \vdash \colon\ldotp \text{Hp}.\supset :yP\,\text{min}_{P}\,ʻ\alpha .\lor.\text{min}_{P}ʻ\alpha =y.\lor.\text{min}_{P}ʻ\alpha + Py &\qquad \text{(1)}\\ +\vdash .*205·14. &\supset \vdash :\text{Hp}.\supset .{\sim}(yP\,\text{min}_{P}\,ʻ\alpha ) &\qquad \text{(2)}\\ +\vdash .(1).(2).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p>In the remainder of the present number, when a proposition has been +proved for \(\text{min}_{P}\), we shall not state the corresponding +proposition for \(\text{max}_{P}\), unless it is specially important. +When propositions concerning \(\text{max}_{P}\) are required for +reference in the sequel, we shall refer to the corresponding +propositions for \(\text{min}_{P}\), in case no reference exists for +\(\text{max}_{P}\).</p> + +<p class="nind"><b>*205·*21.</b> \[\begin{align}&\vdash :P\in \text{connex} .\text{E}!\text{min}_{P}ʻ\alpha .y\in \alpha \cap CʻP-{℩}ʻ\text{min}_{P}ʻ\alpha.\supset .\text{min}_{P}ʻ\alpha + Py\\ +&[*205·2]\end{align}\]</p> + +<p class="nind"><b>*205·211.</b> \(\vdash :P\in \text{trans}\cap \text{connex} .\text{E}!\text{min}_{P}ʻ\alpha .y\in \breve{P} ʻʻ\alpha .\supset .\text{min}_{P}ʻ\alpha Py\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*37·105. &\supset \vdash :\text{Hp}.\supset .(\exists x).x\in \alpha .xPy &\qquad \text{(1)}\\ +\vdash .*13·13. &\supset \vdash :x\in \alpha .xPy.x=\text{min}_{P}ʻ\alpha .\supset .\text{min}_{P}ʻ\alpha Py &\qquad \text{(2)}\\ +\vdash .*205·21. \supset \vdash :\text{Hp}.x\in \alpha .xPy.x\neq \text{min}_{P}ʻ\alpha .&\supset .\text{min}_{P}ʻ\alpha Px.xPy.\\ +[\text{Hp}.*201·1] &\supset .\text{min}_{P}ʻ\alpha Py &\qquad \text{(3)}\\ +\vdash .(1).(2).(3).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*205·22.</b> \[\begin{align}&\vdash :P\in \text{trans}\cap \text{connex} .\text{E}!\text{min}_{P}ʻ\alpha .\supset .\breve{P} ʻʻ\alpha =\overleftarrow{P}ʻ\text{min}_{P}ʻ\alpha + \\ +&[*205·211.*37·181]\end{align}\]</p> + +<p class="nind"><b>*205·23.</b> \(\vdash :P\in \text{connex} .x\in \text{D}ʻP.y\in \overrightarrow{B}ʻ\breve{P} .\supset .xPy\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*93·101.\supset \vdash :\text{Hp}.&\supset .x\neq y.{\sim}(yPx).\\ +[*202·103] &\supset .xPy:\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*205·24.</b> \(\vdash :P\in \text{connex} .\supset .\overrightarrow{B}ʻ\breve{P} \subset pʻ\overleftarrow{P}ʻʻ\text{D}ʻP \quad[*205·23]\)</p> + +<p class="nind"><b>*205·241.</b> \(\vdash :P\in \text{connex} .\supset .\overrightarrow{B}ʻP\subset pʻ\overrightarrow{P}ʻʻ\text{ᗡ}ʻP \quad[\text{Proof as in *205·24}]\)</p> + +<p><span class="pagenum" id="Page_566">[Pg 566]</span></p> + +<p class="nind"><b>*205·25.</b> \(\vdash .\overrightarrow{\text{min}}_{P}ʻ\overleftarrow{P}ʻx=(\overleftarrow{P\dot{-} P^{2}})ʻx\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*205·11.\supset \vdash .\overrightarrow{\text{min}}_{P}ʻ\overleftarrow{P}ʻx&=\overleftarrow{P}ʻx-\breve{P} ʻʻ\overleftarrow{P}ʻx\\ +[*37·301] &=\overleftarrow{P}ʻx-\overleftarrow{P}^{2}ʻx\\ +[*32·31·35] &=(\overleftarrow{P\dot{-} P^{2}})ʻx.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p>The following proposition is used in the theory of well-ordered series +(*250·2).</p> + +<p class="nind"><b>*205·251.</b> \(\vdash :\exists !\overrightarrow{\text{min}}_{P}ʻ\overleftarrow{P}ʻx.\equiv .x\in \text{D}ʻ(P\dot{-} P^{2}) \quad[*205·25]\)</p> + +<p class="nind"><b>*205·252.</b> \(\vdash :\exists !\overrightarrow{\text{max}}_{P}ʻ\overrightarrow{P}ʻx.\equiv .x\in \text{ᗡ}ʻ(P\dot{-} P^{2})\)</p> + +<p class="nind"><b>*205·253.</b> \(\vdash :P\in \text{connex} .\text{E}!BʻP.\supset .\text{ᗡ}ʻP=\overleftarrow{P}ʻBʻP \quad[*202·524]\)</p> + +<p class="nind"><b>*205·254.</b> \(\vdash :P\in \text{connex} .\text{E}!BʻP.\supset .\overleftarrow{\text{min}}_{P}ʻ\text{ᗡ}ʻP=\overleftarrow{P\dot{-} P^{2}}ʻBʻP \quad[*205·253·25]\)</p> + +<p class="nind"><b>*205·255.</b> \(\vdash :\exists !\overrightarrow{\text{min}}_{P}ʻ\text{ᗡ}ʻP.\supset .\exists !\overrightarrow{B}ʻP\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*93·101.\supset \vdash :\overrightarrow{B}ʻP=\Lambda .&\supset .\text{D}ʻP\subset \text{ᗡ}ʻP.\\ +[*37·271] &\supset .\text{ᗡ}ʻP=\breve{P} ʻʻ\text{ᗡ}ʻP.\\ +[*205·122] &\supset .\overrightarrow{\text{min}}_{P}ʻ\text{ᗡ}ʻP=\Lambda &\qquad \text{(1)}\\ +\vdash .(1).\text{Transp}.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*205·256.</b> \[\begin{align}\vdash \colon\ldotp P\in \text{Ser}.&\supset :\\ +&\text{E}!\text{min}_{P}ʻ\text{ᗡ}ʻP.\equiv .\text{E}!\breve{P} _{1}ʻBʻP.\equiv .\text{min}_{P}ʻ\text{ᗡ}ʻP=\breve{P} _{1}ʻBʻP\\ +&[*205·254·255.*201·63.*202·52·7]\end{align}\]</p> + +<p class="nind"><b>*205·26.</b> \(\vdash :Q\,\unicode{x2abd}\, P.\supset .\text{min}_{P}\upharpoonright \text{Cl}ʻCʻQ\,\unicode{x2abd}\, \text{min}_{Q}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*37·201.\supset \vdash \colon\ldotp \text{Hp}.\alpha \subset CʻQ.&\supset :\breve{Q} ʻʻ\alpha \subset \breve{P} ʻʻ\alpha .\alpha \subset CʻQ.\alpha \subset CʻP:\\ +[\text{Transp}.*22·621] &\supset :\alpha - \breve{P} ʻʻ\alpha \subset \alpha -\breve{Q} ʻʻ\alpha .\alpha =\alpha \cap CʻQ=\alpha \cap CʻP:\\ +[*205·11] &\supset :\overrightarrow{\text{min}}_{P}ʻ\alpha \subset \overrightarrow{\text{min}}_{Q}ʻ\alpha :\\ +[*32·18] &\supset :x\text{min}_{P}\alpha .\supset .x\text{min}_{Q}\alpha \colon\ldotp \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*205·261.</b> + \(\vdash :P\unicode{x0294f}\beta \in \text{connex} .\beta \cap CʻP{\sim}\in 1.\supset .\overrightarrow{\text{min}}(P\unicode{x0294f}\beta )ʻ\alpha =\overrightarrow{\text{min}}_{P}ʻ(\alpha + \cap \beta )\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*205·11.*202·54.*37·413.*36·34.\supset \\ +\vdash :\text{Hp}.\supset .\overrightarrow{\text{min}}(P\unicode{x0294f}\beta )ʻ\alpha &=\alpha \cap \beta \cap CʻP-\{\beta \cap \breve{P} ʻʻ(\alpha \cap \beta )\}\\ +[*22·93.*205·11] & =\overrightarrow{\text{min}}_{P}ʻ(\alpha \cap \beta ):\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_567">[Pg 567]</span></p> + +<p class="nind"><b>*205·262.</b> \[\begin{align}\vdash :P\in \text{trans}\cap \text{connex} .x\in \alpha \cap CʻP.\beta &=\overrightarrow{P}ʻx\cup {℩}ʻx.\supset .\\ +&\overrightarrow{\text{min}}_{P}ʻ\alpha =\overrightarrow{\text{min}}_{P}ʻ(\alpha \cap \beta )\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*32·18.\supset \vdash \colon\ldotp \text{Hp}.y\in \alpha ·yPx.&\supset :y\in \alpha \cap \beta :\\ +[*37·105] &\supset :yPz.\supset .z\in \breve{P} ʻʻ(\alpha \cap \beta ) &\qquad \text{(1)}\\ +\vdash .*51·15.\supset \vdash \colon\ldotp \text{Hp}.y\in \alpha .y=x.&\supset :y\in \alpha \cap \beta :\\ +[*37·105] &\supset :yPz.\supset .z\in \breve{P} ʻʻ(\alpha \cap \beta ) &\qquad \text{(2)}\\ +\vdash .*51·15.*201·1.\supset \vdash :\text{Hp}.y\in \alpha .xPy.yPz.&\supset .x\in \alpha .xPz.\\ +[*37·105] &\supset .z\in \breve{P} ʻʻ(\alpha \cap \beta ) &\qquad \text{(3)}\\ +\vdash .(1).(2).(3).*202·103.\supset \vdash \colon\ldotp \text{Hp}.&\supset :y\in \alpha .yPz.\supset .z\in \breve{P} ʻʻ(\alpha \cap \beta ):\\ +[*37·105·2] &\supset :\breve{P} ʻʻ\alpha =\breve{P} ʻʻ(\alpha \cap \beta ) &\qquad \text{(4)}\\ +\vdash .*37·181.*202·101.\supset \vdash :\text{Hp}.&\supset .\overleftarrow{P}ʻx\subset \overleftarrow{P}ʻʻ\alpha .\overleftarrow{P}ʻx=CʻP-\beta .\\ +[*22·82] &\supset .CʻP-\breve{P} ʻʻ\alpha \subset \beta .\\ +[*22·621.(4)] & \supset .CʻP\cap \alpha -\breve{P} ʻʻ\alpha=CʻP\cap \alpha \cap \beta -\breve{P} ʻʻ(\alpha \cap \beta ).\\ +[*205·11] & \supset .\overrightarrow{\text{min}}_{P}ʻ\alpha =\overrightarrow{\text{min}}_{P}ʻ(\alpha \cap \beta ):\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*205·27">*205·27</a>.</b> \[\begin{align}\vdash :P\in \text{trans}\cap \text{connex} .x\in &\alpha \cap \text{ᗡ}ʻP.\beta =\overrightarrow{P}ʻx\cup \iota ʻx.\supset .\\ +&\overrightarrow{\text{min}}_{P}ʻ\alpha =\overrightarrow{\text{min}}(P\unicode{x0294f}\beta )ʻ\alpha =\overrightarrow{\text{min}}_{P}ʻ(\alpha \cap \beta )\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*52·41. \supset \vdash :\text{Hp}.\overrightarrow{P}ʻx\neq \iota ʻx.&\supset .\beta {\sim}\in 1.\\ +[*205·261] &\supset .\overrightarrow{\text{min}}(P\unicode{x0294f}\beta )ʻ\alpha =\overrightarrow{\text{min}}_{P}ʻ(\alpha \cap \beta ) &\qquad \text{(1)}\\ +\vdash .*202·101. \supset \vdash :\text{Hp}.\overrightarrow{P}ʻx=\iota ʻx.&\supset .CʻP-\iota ʻx=\overleftarrow{P}ʻx.xPx.\\ +[*37·105] &\supset .CʻP\subset \breve{P} ʻʻ(\alpha \cap \beta ).\\ +[*205·122.*37·413] &\supset .\overrightarrow{\text{min}}_{P}ʻ(\alpha + \cap \beta )=\Lambda .\overrightarrow{\text{min}}(P\unicode{x0294f}\beta )ʻ\alpha =\Lambda &\qquad \text{(2)}\\ +\vdash .(1).(2).&\supset \vdash :\text{Hp}.\supset .\overrightarrow{\text{min}}(P\unicode{x0294f}\beta )ʻa=\overrightarrow{\text{min}}_{P}ʻ(\alpha \cap \beta ) &\qquad \text{(3)}\\ +\vdash .(3).*205·262.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p>The above proposition is used in *250·7.</p> + +<p class="nind"><b><a id="*205·3">*205·3</a>.</b> \(\vdash :P\in \text{connex} .\supset .\overrightarrow{\text{min}}_{P}ʻ\alpha \in 0\cup 1.\overrightarrow{\text{max}}_{P}ʻ\alpha \in 0\cup 1\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*205·11.\supset \vdash \colon\ldotp x,y\in \overrightarrow{\text{min}}_{P}ʻ\alpha .&\supset :x,y\in \alpha \cap CʻP:z\in \alpha .\supset _{z}.{\sim}(zPx).{\sim}(zPy):\\ +[*10·1] &\supset :x,y\in \alpha \cap CʻP.{\sim}(yPx).{\sim}(xPy) &\qquad \text{(1)}\\ +\vdash .(1).*202·103.\supset \vdash \colon\ldotp \text{Hp}.&\supset :x,y\in \overrightarrow{\text{min}}_{P}ʻ\alpha .\supset .x=y:\\ +[*52·4] &\supset :\overrightarrow{\text{min}}_{P}ʻ\alpha \in 0\cup 1 &\qquad \text{(2)}\\ +\text{Similarly}\quad & \vdash \colon\ldotp \text{Hp}.\supset :\overrightarrow{\text{max}}_{P}ʻ\alpha \in 0\cup 1 &\qquad \text{(3)}\\) +\vdash .(2).(3).\supset \vdash .\text{Prop}\ +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_568">[Pg 568]</span></p> + +<p>The above proposition is of great importance in the theory of maxima +and minima.</p> + +<p class="nind"><b>*205·31.</b> \(\vdash : P \in \text{connex} . \supset . \text{min}_{P}, \text{max}_{P} \in 1 \rightarrow \text{Cls} \quad[*205·3.*71·12]\)</p> + +<p class="nind"><b>*205·32.</b> \[\begin{align}&\vdash \colon\ldotp P \in \text{connex} . \supset : \exists ! \overrightarrow{\text{min}}_{P}ʻ\alpha . \equiv . \text{E}! \text{min}_{P}ʻ\alpha + . \equiv . \alpha \in \text{ᗡ}ʻ\text{min}_{P}\\ +&[*205·31 . *71·163 . *33·41]\end{align}\]</p> + +<p class="nind"><b>*205·33.</b> \(\vdash : P \in \text{connex} . \kappa \subset \text{ᗡ}ʻ\text{min}_{P} . \supset . \text{min}_{P} + \upharpoonright \kappa \in {\in}_{\Delta}ʻ\kappa\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *205·31 . &\supset \vdash : \text{Hp} . \supset . \text{min}_{P} \upharpoonright \kappa \in 1 \rightarrow \text{Cls} &\qquad \text{(1)}\\ +\vdash . *205·1 . &\supset \vdash : \text{Hp} . \supset . \text{min}_{P} \upharpoonright \kappa \,\unicode{x2abd}\, \in &\qquad \text{(2)}\\ +\vdash . *35·65 . &\supset \vdash : \text{Hp} . \supset . \text{ᗡ}ʻ\text{min}_{P} \upharpoonright \kappa = \kappa &\qquad \text{(3)}\\ +\vdash . (1) . (2) . (3) . *80·14 . \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*205·34.</b> \(\vdash : P \in \text{connex} . \kappa \subset \text{ᗡ}ʻ\text{min}_{P} . \supset . \kappa \in \text{Cls}^{2}\, \text{mult} \quad[*205·33 . *88·2]\)</p> + +<p>The following proposition is used in *260·17.</p> + +<p class="nind"><b>*205·35.</b> \[\begin{align}\vdash \colon\colon P^{2} \,\unicode{x2abd}\, J . &P \in \text{connex} . \supset \colon\ldotp \\ +&x = \text{min}_{P}ʻ\alpha . \equiv : x \in \alpha \cap CʻP : y \in \alpha \cap CʻP - \iota ʻx . \supset _{y} . xPy\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *205·31 . *71·36 . &\supset \vdash \colon\colon \text{Hp} . \supset \colon\ldotp x = \text{min}_{P}ʻ\alpha . \equiv : x \text{min}_{P}\alpha :\\ +[*205·1 . *37·265] &\equiv : x \in \alpha \cap CʻP - \breve{P} ʻʻ(\alpha \cap CʻP) :\\ +[*37·105] &\equiv : x \in \alpha \cap CʻP : y \in \alpha \cap CʻP . \supset _{y} . {\sim} (yPx) :\\ +[*51·221] &\equiv : x \in \alpha \cap CʻP : y = x . \supset _{y} . {\sim} (yPx) : y \in \alpha \cap CʻP - \iota ʻx . \supset _{x} . {\sim} (yPx) &\qquad \text{(1)}\\ +\vdash . *200·36 . \supset \vdash \colon\ldotp \text{Hp} . &\supset : yPz . \supset _{z, y} .y \neq z :\\ +[\text{Transp} . *10·1] &\supset : y = x . \supset _{y} . {\sim} (yPx) &\qquad \text{(2)}\\ +\vdash . (1) . (2) . &\supset \vdash \colon\colon \text{Hp}. \supset \colon\ldotp \\ +x = \text{min}_{P}ʻ\alpha . &\equiv : x \in \alpha \cap CʻP : y \in \alpha \cap CʻP - \iota ʻx . \supset _{y} . {\sim} (yPx) :\\ +[*202·5] &\equiv : x \in \alpha \cap CʻP : y \in \alpha \cap CʻP - \iota ʻx . \supset _{y} . xPy \colon\colon \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*205·36.</b> \(\vdash : P \in \text{trans} \cap \text{connex} . \supset . \overrightarrow{\text{min}}_{P}ʻ\alpha \subset pʻP_{*}ʻʻ(\alpha \cap CʻP)\)</p> + +<p><i>Dem.</i> +\[ +\vdash . *205·2 . *201·18 . \supset \vdash \colon\ldotp \text{Hp} . x = \text{min}_{P}ʻ\alpha . \supset : y \in (\alpha \cap CʻP) . \supset _{y} . xP_{*}y + \colon\ldotp \supset \vdash . \text{Prop} +\]</p> + +<p>The above proposition is used in <a href="#*230·53">*230·53</a>.</p> + +<p class="nind"><b>*205·37.</b> \(\vdash : P \in \text{trans} . \overrightarrow{\text{max}}_{P}ʻ\alpha = \Lambda . \supset . P_{*}ʻʻ\alpha = Pʻʻ\alpha \quad[*201·52.*205·123]\)</p> + +<p>The following proposition is used in *257·21.</p> + +<p class="nind"><b>*205·38.</b> \(\vdash : P_{\text{po}} \,\unicode{x2abd}\, J . \supset . \mu \cap pʻ\overrightarrow{P}_{*}ʻʻ\mu \subset \overrightarrow{\text{min}} (P_{\text{po}})ʻ\mu + \)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *200·381 . \supset \vdash \colon\colon \text{Hp} . &\supset \colon\ldotp x \in \mu . \supset _{x} . yP_{*}x : \supset : x \in \mu . \supset _{x}\\ + . {\sim} (xP_{\text{po}}y) \colon\ldotp \\ +[*40·51 . *37·105] &\supset \colon\ldotp pʻ\overrightarrow{P}_{*}ʻʻ\mu \subset - \breve{P} _{\text{po}}ʻʻ\mu &\qquad \text{(1)}\\ +\vdash . *40·62 . &\supset \vdash : \exists ! \mu . \supset . pʻ\overrightarrow{P}_{*}ʻʻ\mu \subset CʻP &\qquad \text{(2)}\\ +\vdash . *24·12 . &\supset \vdash : {\sim} \exists ! \mu . \supset . \mu \subset CʻP &\qquad \text{(3)}\\ +\vdash . (1) . (2) . (3) . \supset \vdash : \text{Hp} . \supset . \mu \cap pʻ\overrightarrow{P}_{*}ʻʻ\mu &\subset \mu \cap CʻP - \breve{P} _{\text{po}}ʻʻ\mu \\ +[*205·11 . *91·504] &\subset \overrightarrow{\text{min}} (P_{\text{po}})ʻ\mu : \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_569">[Pg 569]</span></p> + +<p class="nind"><b>*205·381.</b> \(\vdash :P_{\text{po}}\,\unicode{x2abd}\, J.\overrightarrow{\text{max}}_{P}ʻ\mu = \Lambda .\supset .pʻ\overleftarrow{P}_{*}ʻʻ\mu + = pʻ\overleftarrow{P}_{\text{po}}ʻʻ\mu\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*205·38 \frac{\breve{P}}{P}.\supset \vdash :\text{Hp}.&\supset .\mu \cap pʻ\overleftarrow{P}_{*}ʻʻ\mu = \Lambda &\qquad \text{(1)}\\ +\vdash .(1).*40·53.*24·37.\supset \\ +\vdash \colon\colon\text{Hp}.\supset \colon\ldotp x\in pʻ\overleftarrow{P}_{*}ʻʻ\mu . &\equiv :y\in \mu .\supset _{y}.yP_{*}x.y \neq x:\\ +[*200·38] &\equiv :y\in \mu .\supset _{y}.yP_{\text{po}}x:\\ +[*40·53] &\equiv :x\in pʻ\overleftarrow{P}_{\text{po}}ʻʻ\mu \colon\colon\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p>The three following propositions lead up to <a href="#*205·42">*205·42</a>, which is used in +*261·26.</p> + +<p class="nind"><b><a id="*205·4">*205·4</a>.</b> \(\vdash :CʻP\in 1.\supset .\overrightarrow{B}ʻP = \Lambda .\overrightarrow{B}ʻ\breve{P} = \Lambda\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*56·381.*55·15.&\supset \vdash :\text{Hp}.\supset .(\exists x).\text{D}ʻP = \iota ʻx.\text{ᗡ}ʻP = \iota ʻx.\\ +[*93·101] & \supset .\overrightarrow{B}ʻP = \Lambda .\overrightarrow{B}ʻ\breve{P} = \Lambda :\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*205·401.</b> \(\vdash :\exists !\overrightarrow{B}ʻP\unicode{x0294f}\alpha .\supset .\alpha \cap CʻP{\sim}\in 0\cup 1.CʻP\unicode{x0294f}\alpha {\sim}\in 0\cup 1\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*205·4.\text{Transp}.&\supset \vdash :\text{Hp}.\supset .CʻP\unicode{x0294f}\alpha {\sim}\in 1 &\qquad \text{(1)}\\ +\vdash .*93·103. &\supset \vdash :\text{Hp}.\supset .\exists !CʻP\unicode{x0294f}\alpha &\qquad \text{(2)}\\ +\vdash .(1).(2). &\supset \vdash :\text{Hp}.\supset .CʻP\unicode{x0294f}\alpha {\sim}\in 0\cup 1 &\qquad \text{(3)}\\ +\vdash .*37·41. &\supset \vdash .CʻP\unicode{x0294f}\alpha \subset \alpha \cap CʻP &\qquad \text{(4)}\\ +\vdash .(4).*60·32·371.\text{Transp}.&\supset \vdash :CʻP\unicode{x0294f}\alpha {\sim}\in 0\cup 1.\supset .\alpha \cap CʻP{\sim}\in 0\cup 1 &\qquad \text{(5)}\\ +\vdash .(3).(5).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p>The following proposition, besides being required for <a href="#*205·41">*205·41</a>, is used +in *250·151.</p> + +<p class="nind"><b><a id="*205·41">*205·41</a>.</b> \(\vdash :P\in \text{connex} .\alpha \cap CʻP{\sim}\in 1.\supset .\overrightarrow{\text{min}}_{P}ʻ\alpha = \overrightarrow{B}ʻP\unicode{x0294f}\alpha\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*202·54. &\supset \vdash :\text{Hp}.\supset .CʻP\unicode{x0294f}\alpha = \alpha \cap CʻP &\qquad \text{(1)}\\ +\vdash .*37·41. & \supset \vdash .\text{ᗡ}ʻP\unicode{x0294f}\alpha = \alpha \cap \breve{P} ʻʻ\alpha &\qquad \text{(2)}\\ +\vdash .(1).(2).*93·103. \supset \vdash :\text{Hp}.\supset .\overrightarrow{B}ʻP\unicode{x0294f}\alpha & = \alpha \cap CʻP-(\alpha \cap \breve{P} ʻʻ\alpha )\\ +[*22·93.*205·11] &= \overrightarrow{\text{min}}_{P}ʻ\alpha :\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*205·42">*205·42</a>.</b> \(\vdash :P\in \text{connex} .\text{E}!BʻP\unicode{x0294f}\alpha .\supset .BʻP\unicode{x0294f}\alpha = \text{min}_{P}ʻ\alpha\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*205·401.\supset \vdash :\text{Hp}.&\supset .\alpha \cap CʻP{\sim}\in 1.\\ +[*205·41] &\supset .\overrightarrow{\text{min}}_{P}ʻ\alpha = \overrightarrow{B}ʻP\unicode{x0294f}\alpha &\qquad \text{(1)}\\ +\vdash .(1).*32·41.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_570">[Pg 570]</span></p> + +<p>The following proposition leads up to <a href="#*205·44">*205·44</a>.</p> + +<p class="nind"><b>*205·43.</b> \(\vdash :P\in \text{connex} .\exists !\alpha \cap \text{D}ʻP.\supset .\overrightarrow{\text{min}}_{P}ʻ\alpha =\overrightarrow{B}ʻ\alpha \upharpoonleft P\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*205·11. \supset \vdash .\overrightarrow{\text{min}}_{P}ʻ\alpha + &=(\alpha \cap \text{D}ʻP-\breve{P} ʻʻ\alpha )\cup (\alpha \cap \overrightarrow{B}ʻ\breve{P} -\breve{P} ʻʻ\alpha )\\ +[*35·61.*37·4] &=\overrightarrow{B}ʻ\alpha \upharpoonleft P\cup (\alpha \cap \overrightarrow{B}ʻ\breve{P} -\breve{P} ʻʻ\alpha )&\qquad \text{(1)}\\ +\vdash .*205·23. &\supset \vdash :P\in \text{connex} .x\in \alpha \cap \text{D}ʻP.y\in \overrightarrow{B}ʻ\breve{P} .\supset .x\in \alpha .xPy.\\ +[*37·1] &\supset .y\in \breve{P} ʻʻ\alpha &\qquad \text{(2)}\\ +\vdash .(2).*10·23. \supset \vdash :\text{Hp}.&\supset .\overrightarrow{B}ʻ\breve{P} \subset \breve{P} ʻʻ\alpha .\\ +[*24·3] &\supset .\alpha \cap \overrightarrow{B}ʻ\breve{P} -\breve{P} ʻʻ\alpha =\Lambda &\qquad \text{(3)}\\ +\vdash .(1).(3).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*205·44">*205·44</a>.</b> \(\vdash :P\in \text{connex} .\text{E}!Bʻ\alpha \upharpoonleft P.\supset .\text{min}_{P}ʻ\alpha =Bʻ\alpha \upharpoonleft P \quad[*205·43.*32·41]\)</p> + +<p>The following propositions deal with the circumstances under which +the minimum of one class is identical with, or earlier than, that of +another.</p> + +<p class="nind"><b>*205·5.</b> \(\vdash :P\in \text{connex} .\alpha \subset \beta .\text{min}_{P}ʻ\beta \in \alpha .\supset .\text{E}!\text{min}_{P}ʻ\alpha .\text{min}_{P}ʻ\alpha + =\text{min}_{P}ʻ\beta\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*37·2. \text{Transp}.\supset \vdash :\text{Hp}.&\supset .-Pʻʻ\beta \subset -Pʻʻ\alpha .\\ +[*205·11.\text{Hp}] &\supset .\text{min}_{P}ʻ\beta \in \alpha -Pʻʻ\alpha .\\ +[*205·1] &\supset .\text{min}_{P}ʻ\beta \in \overrightarrow{\text{min}}_{P}ʻ\alpha &\qquad \text{(1)}\\ +\vdash .(1).*205·3.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*205·501.</b> \(\vdash :P\in \text{connex} .\text{min}_{P}ʻ\alpha =\text{min}_{P}ʻ\beta .\supset .\beta \subset -pʻ\overrightarrow{P}ʻʻ\alpha\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*205·11. \supset \vdash \colon\ldotp \text{Hp}.&\supset :\text{min}_{P}ʻ\alpha {\sim}\in \breve{P} ʻʻ\beta :\\ +[*37·105] &\supset :y\in \beta .\supset _{y}.{\sim}(yP\text{min}_{P}ʻ\alpha ):\\ +[*205·11] &\supset :y\in \beta .\supset _{y}.(\exists x).x\in \alpha .{\sim}(yPx):\\ +[*40·51] &\supset :\beta \subset -pʻ\overrightarrow{P}ʻʻ\alpha \colon\ldotp \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*205·51.</b> \[\begin{align}\vdash \colon\ldotp P\in \text{connex} .\alpha \subset \beta .\text{E}!&\text{min}_{P}ʻ\alpha .\text{E}!\text{min}_{P}ʻ\beta .\supset :\\ +&\text{min}_{P}ʻ\alpha =\text{min}_{P}ʻ\beta .\lor.\text{min}_{P}ʻ\beta P\text{min}_{P}ʻ\alpha\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*22·1.*205·1.\supset \vdash :\text{Hp}.\supset .\text{min}_{P}ʻ\alpha \in \beta \cap CʻP &\qquad \text{(1)}\\ +\vdash .(1).*205·2.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*205·52.</b> \[\begin{align}\vdash :P\in \text{trans} \cap \text{connex} .&\exists !\alpha \cap pʻ\overrightarrow{P}ʻʻ\beta .\\ +&\text{E}!\text{min}_{P}ʻ\alpha .\text{E}!\text{min}_{P}ʻ\beta .\supset .\text{min}_{P}ʻ\alpha P\text{min}_{P}ʻ\beta\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*40·51.\supset \vdash \colon\ldotp \text{Hp}.&\supset :(\exists x):x\in \alpha :y\in \beta .\supset _{y}.xPy &\qquad \text{(1)}\\ +\vdash .*205·2.\supset \vdash \colon\colon\text{Hp}.&\supset \colon\ldotp x\in \alpha \cap CʻP.\supset _{x}:\text{min}_{P}ʻ\alpha + =x.\lor.\text{min}_{P}ʻ\alpha Px &\qquad \text{(2)}\\ +\vdash .(1). *205·1 . \supset \vdash \colon\ldotp \text{Hp}. &\supset : (\exists x) . x \in \alpha . xP \text{min}_{P}ʻ\beta :\\ +[*33·17] &\supset : (\exists x) . x \in \alpha \cap CʻP . xP \text{min}_{P}ʻ\beta &\qquad \text{(3)}\\ +\vdash .(2).(3). \supset \vdash \colon\ldotp \text{Hp}. &\supset : (\exists x) : xP \text{min}_{P}ʻ\beta : \text{min}_{P}ʻ\alpha = x . \lor . \text{min}_{P}ʻ\alpha + Px :\\ +[*201·1 . *13·195] &\supset : \text{min}_{P}ʻ\alpha P \text{min}_{P}ʻ\beta \colon\ldotp \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_571">[Pg 571]</span></p> + +<p class="nind"><b>*205·53.</b> \(\vdash : P \in \text{connex} \cap \text{Rl}ʻJ . x \in \alpha \cap CʻP . \overrightarrow{P}ʻx = Pʻʻ\alpha . \supset . x = \text{max}_{P}ʻ\alpha\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *50·24 . \supset \vdash : \text{Hp} . &\supset . x \in \alpha \cap CʻP - \overrightarrow{P}ʻx .\\ +[\text{Hp}] &\supset . x \in \alpha \cap CʻP - Pʻʻ\alpha .\\ +[*205·111] &\supset . x \in \overrightarrow{\text{max}}_{P}ʻ\alpha &\qquad \text{(1)}\\ +\vdash . (1) . *205·3 . \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*205·54">*205·54</a>.</b> \[\begin{align}&\vdash \colon\ldotp P \in \text{Ser} . \supset : x \in \alpha \cap CʻP . \overrightarrow{P}ʻx = Pʻʻ\alpha . \equiv . x = \text{max}_{P}ʻ\alpha + \\ +&[*205·53·22]\end{align}\]</p> + +<p class="nind"><b>*205·55.</b> \(\vdash : P \in \text{connex} . BʻP \in \alpha . \supset . BʻP = \text{min}_{P}ʻ\alpha\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *93·101 . *37·16 . &\supset \vdash : \text{E}! BʻP . \supset . BʻP \in CʻP - \breve{P} ʻʻ\alpha &\qquad \text{(1)}\\ +\vdash . (1) . *205·1 . &\supset \vdash : BʻP \in \alpha . \supset . BʻP \text{min}_{P}\alpha &\qquad \text{(2)}\\ +\vdash . (2) . *205·31 . \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*205·56.</b> \(\vdash . \overrightarrow{\text{max}}_{P}ʻsʻ\kappa \subset \text{max}_{P}ʻʻ\kappa\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *205·111 . *40·38 . &\supset \vdash . \overrightarrow{\text{max}}_{P}ʻsʻ\kappa \subset sʻ\kappa \cap CʻP - sʻPʻʻʻ\kappa \\ +[*40·11] &\subset \hat{y} \{(\exists \alpha ) . \alpha \in \kappa . y \in \alpha \cap CʻP : {\sim} (\exists \alpha ) . \alpha \in \kappa . y \in Pʻʻ\alpha\}\\ +[*10·56] &\subset \hat{y} \{(\exists \alpha ) . \alpha \in \kappa . y \in \alpha \cap CʻP - Pʻʻ\alpha\}\\ +[*205·111] &\subset \hat{y} \{(\exists \alpha ) . \alpha \in \kappa . y \in \overrightarrow{\text{max}}_{P}ʻ\alpha\}\\ +[*40·5] &\subset \text{max}_{P}ʻʻ\kappa . \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*205·561.</b> \(\vdash : \kappa \subset - \text{ᗡ}ʻ\text{max}_{P} . \supset . sʻ\kappa {\sim} \in \text{ᗡ}ʻ\text{max}_{P} \quad[*205·56 . *37·26·29]\)</p> + +<p class="nind"><b>*205·6.</b> \(\vdash \colon\ldotp P \in \text{connex} . \supset : {\sim} \text{E}! \text{min}_{P}ʻ\alpha + . \equiv . \alpha \cap CʻP \subset \breve{P} ʻʻ\alpha \quad[*205·32·122]\)</p> + +<p class="nind"><b>*205·601.</b> \(\vdash : P \in \text{connex} . \alpha \subset CʻP . \supset : {\sim} \text{E}! \text{min}_{P}ʻ\alpha + . \equiv . \alpha \subset \breve{P} ʻʻ\alpha \quad[*205·6]\)</p> + +<p class="nind"><b>*205·61.</b> \[\begin{align}&\vdash : P \in \text{connex} . \supset . CʻP = {CʻP \cap pʻ\overrightarrow{P}ʻʻ(\alpha \cap CʻP)} \cup \overrightarrow{\text{min}}_{P}ʻ\alpha + \cup \breve{P} ʻʻ\alpha\\ +&[*202·505 . *205·13]\end{align}\]</p> + +<p class="nind"><b>*205·62.</b> + \[\begin{align}&\vdash : P \in \text{connex} . \exists ! \alpha \cap CʻP . \supset . CʻP = pʻ\overrightarrow{P}ʻʻ(\alpha \cap CʻP) \cup \overrightarrow{\text{min}}_{P}ʻ\alpha + \cup \breve{P} ʻʻ\alpha \\ +&[*40·62 . *205·61]\end{align}\]</p> + +<p class="nind"><b>*205·63.</b> \[\begin{align}\vdash : P \in \text{connex} . P^{2} \,\unicode{x2abd}\, J . &\exists ! (\alpha \cap CʻP) . \supset .\\ +&pʻ\overrightarrow{P}ʻʻ(\alpha \cap CʻP) = CʻP - \breve{P} ʻʻ\alpha - \overrightarrow{\text{min}}_{P}ʻ\alpha\\ +[*202·502 . *205·13]\end{align}\]</p> + +<p><span class="pagenum" id="Page_572">[Pg 572]</span></p> + +<p class="nind"><b>*205·64.</b> \[\begin{align}\vdash :P\in \text{connex} .\exists !&(\alpha \cap CʻP).\supset .\\ +&\overrightarrow{\text{min}}_{P}ʻ\alpha =CʻP-\breve{P} ʻʻ\alpha -pʻ\overrightarrow{P}ʻʻ(\alpha \cap CʻP)\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*205·62.&\supset \vdash :\text{Hp}.\supset .\\ +&CʻP-\breve{P} ʻʻ\alpha -pʻ\overrightarrow{P}ʻʻ(\alpha \cap CʻP)=\overrightarrow{\text{min}}_{P}ʻ\alpha + -\breve{P} ʻʻ\alpha -pʻ\overrightarrow{P}ʻʻ(\alpha \cap CʻP) &\qquad \text{(1)}\\ +\vdash .*205·11.&\supset \vdash .\overrightarrow{\text{min}}_{P}ʻ\alpha -\breve{P} ʻʻ\alpha =\overrightarrow{\text{min}}_{P}ʻ\alpha &\qquad \text{(2)}\\ +\vdash .*205·14.\supset \vdash \colon\ldotp x\in \overrightarrow{\text{min}}_{P}ʻ\alpha .&\supset :y\in \alpha .\supset _{y}.{\sim}(yPx):\\ +[*205·11.*10·1] &\supset :{\sim}(xPx).x\in \alpha \cap CʻP:\\ +[*40·51] &\supset :x{\sim}\in pʻ\overrightarrow{P}ʻʻ(\alpha \cap CʻP) &\qquad \text{(3)}\\ +\vdash .(3).&\supset \vdash .\overrightarrow{\text{min}}_{P}ʻ\alpha -pʻ\overrightarrow{P}ʻʻ(\alpha \cap CʻP)=\overrightarrow{\text{min}}_{P}ʻ\alpha &\qquad \text{(4)}\\ +\vdash .(1).(2).(4).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*205·65.</b> \(\vdash :P\in \text{trans}\alpha \cap \text{connex} .\text{E}!\text{min}_{P}ʻ\alpha + .\supset .pʻ\overrightarrow{P}ʻʻ(\alpha \cap CʻP)=\overrightarrow{P}ʻ\text{min}_{P}ʻ\alpha\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*205·2.\supset \vdash \colon\colon\text{Hp}.\supset \colon\ldotp xP\text{min}_{P}ʻ\alpha .&\supset :y\in \alpha \cap CʻP.\supset _{y}.xPy:\\ +[*40·51] &\supset :x\in pʻ\overrightarrow{P}ʻʻ(\alpha \cap CʻP) &\qquad \text{(1)}\\ +\vdash .*205·1.*40·12.&\supset \vdash :\text{Hp}.\supset .pʻ\overrightarrow{P}ʻʻ(\alpha \cap CʻP)\subset \overrightarrow{P}ʻ\text{min}_{P}ʻ\alpha &\qquad \text{(2)}\\ +\vdash .(1).(2).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*205·66.</b> \[\begin{align}\vdash :P\in \text{trans}\alpha &\cap \text{connex} .\text{E}!\text{min}_{P}ʻ\alpha .\supset .\\ +&pʻ\overrightarrow{P}ʻʻ(\alpha \cap CʻP)=\overrightarrow{P}ʻ\text{min}_{P}ʻ\alpha .\breve{P} ʻʻ\alpha =\overleftarrow{P}ʻ\text{min}_{P}ʻ\alpha .\\ +&CʻP=pʻ\overrightarrow{P}ʻʻ(\alpha \cap CʻP)\cup \iota ʻ\text{min}_{P}ʻ\alpha \cup \breve{P} ʻʻ\alpha\\ +&[*205·65·22.*202·101]\end{align}\]</p> + +<p class="nind"><b>*205·67.</b> \(\vdash \colon\ldotp P\in \text{Ser}.\supset :x=\text{min}_{P}ʻ\alpha .\equiv .\overrightarrow{P}ʻx=pʻ\overrightarrow{P}ʻʻ(\alpha \cap CʻP).x\in CʻP\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*205·65·11.\supset \\ +\vdash \colon\ldotp \text{Hp}.&\supset :x=\text{min}_{P}ʻ\alpha .\supset .\overrightarrow{P}ʻx=pʻ\overrightarrow{P}ʻʻ(\alpha \cap CʻP).x\in CʻP &\qquad \text{(1)}\\ +\vdash .*50·24.&\supset \vdash :\text{Hp}.\overrightarrow{P}ʻx=pʻ\overleftarrow{P}ʻʻ(\alpha \cap CʻP).\supset .x{\sim}\in pʻ\overrightarrow{P}ʻʻ(\alpha \cap CʻP) &\qquad \text{(2)}\\ +\vdash .*200·5.\supset \vdash :\text{Hp}(2).&\supset .\alpha \cap \overrightarrow{P}ʻx=\Lambda .\\ +[*37·462] &\supset .x{\sim}\in \breve{P} ʻʻ\alpha &\qquad \text{(3)}\\ +\vdash .(2).(3).*202·505.\supset \vdash :\text{Hp}(2).x\in CʻP.&\supset .x\in \alpha \cap CʻP-\breve{P} ʻʻ\alpha .\\ +[*205·3·11] &\supset .x=\text{min}_{P}ʻ\alpha &\qquad \text{(4)}\\ +\vdash .(1).(4).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_573">[Pg 573]</span></p> + +<p class="nind"><b>*205·68.</b> \(\vdash :\breve{P} ʻʻ\alpha \subset \alpha .\supset .\overrightarrow{\text{min}}_{P}ʻ\alpha =\overrightarrow{\text{min}}(P_{\text{po}})ʻ\alpha\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*91·711.&\supset \vdash :\text{Hp}.\supset .\breve{P} _{\text{po}}ʻʻ\alpha =\breve{P} ʻʻ\alpha .\\ +[*205·11] &\supset .\overrightarrow{\text{min}}(P_{\text{po}})ʻ\alpha =\overrightarrow{\text{min}}_{P}ʻ\alpha :\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*205·681.</b> \(\vdash :P_{\text{po}}\in \text{connex} .\breve{P} ʻʻ\alpha \subset \alpha .\supset .\overrightarrow{\text{min}}_{P}ʻ\alpha \in 0\cup 1 \quad[*205·68·3]\)</p> + +<p class="nind"><b><a id="*205·7">*205·7</a>.</b> \(\vdash :\exists !\overrightarrow{\text{max}}_{P}ʻPʻʻ\alpha .\supset .\exists !\overrightarrow{\text{max}}_{P}ʻ\alpha\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*37·2·265. &\supset \vdash :\alpha \cap CʻP\subset Pʻʻ\alpha .\supset .Pʻʻ\alpha \subset PʻʻPʻʻ\alpha &\qquad \text{(1)}\\ +\vdash .(1).\text{Transp}.&\supset \vdash :\exists !Pʻʻ\alpha -PʻʻPʻʻ\alpha .\supset .\exists !\alpha \cap CʻP-Pʻʻ\alpha &\qquad \text{(2)}\\ +\vdash .(2).*205·111.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*205·71.</b> \(\vdash :P\in \text{connex} .\exists !\overrightarrow{\text{max}}_{P}ʻPʻʻ\alpha .\supset .\text{max}_{P}ʻPʻʻ\alpha + (P\dot{-} P^{2})\text{max}_{P}ʻ\alpha\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*205·7·3.\supset \vdash :\text{Hp}.&\supset .\text{E}!\text{max}_{P}ʻPʻʻ\alpha .\text{E}!\text{max}_{P}ʻ\alpha . &\qquad \text{(1)}\\ +[*205·101] &\supset .\text{max}_{P}ʻPʻʻ\alpha \in Pʻʻ\alpha &\qquad \text{(2)}\\ +\vdash .(1).*205·101.\supset \vdash \colon\ldotp \text{Hp}.&\supset :\text{max}_{P}ʻPʻʻ\alpha {\sim}\in PʻʻPʻʻ\alpha :\\ +[*37·39] &\supset :y\in \alpha .\supset _{y}.{\sim}(\text{max}_{P}ʻPʻʻ\alpha P^{2}y):\\ +[(1)] &\supset :{\sim}(\text{max}_P{}ʻPʻʻ\alpha P^{2}\text{max}_{P}ʻ\alpha ): &\qquad \text{(3)}\\ +[*34·5.\text{Transp}] & \supset :zP\text{max}_{P}ʻ\alpha .\supset .{\sim}(\text{max}_{P}ʻPʻʻ\alpha Pz):\\ +[*205·21] &\supset :z\in \alpha -\iota ʻ\text{max}_{P}ʻ\alpha .\supset .{\sim}(\text{max}_{P}ʻPʻʻ\alpha Pz) &\qquad \text{(4)}\\ +\vdash .(2).*37·1.&\supset \vdash :\text{Hp}.\supset .(\exists z).z\in \alpha .\text{max}_{P}ʻPʻʻ\alpha Pz &\qquad \text{(5)}\\ +\vdash .(4).(5). &\supset \vdash :\text{Hp}.\supset .\text{max}_{P}ʻPʻʻ\alpha P\text{max}_{P}ʻ\alpha &\qquad \text{(6)}\\ +\vdash .(3).(6).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*205·72.</b> \(\vdash :P\in \text{connex} .P\,\unicode{x2abd}\, P^{2}.\supset .{\sim}\exists !\overrightarrow{\text{max}}_{P}ʻPʻʻ\alpha \quad[*205·71.\text{Transp}]\)</p> + +<p class="nind"><b><a id="*205·73">*205·73</a>.</b> \(\vdash :P\in \text{connex} .\text{min}_{P}ʻ\gamma =\text{max}_{P}ʻ\gamma .\supset .\gamma \cap CʻP\in 1.\gamma \cap CʻP=\iota ʻ\text{min}_{P}ʻ\gamma\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *205·21. \supset \vdash :.\text{Hp}.\supset :x\in \gamma \cap CʻP-\iota ʻ\text{min}_{P}ʻ\gamma .&\supset .\text{max}_{P}ʻ\gamma Px.\\ +[*37·1] &\supset .\text{max}_{P}ʻ\gamma \in Pʻʻ\gamma &\qquad \text{(1)}\\ +\vdash .*205·111.&\supset \vdash :\text{Hp}.\supset .\text{max}_{P}ʻ\gamma {\sim}\in Pʻʻ\gamma &\qquad \text{(2)}\\ +\vdash .(2).(1).\text{Transp}.\supset \vdash :\text{Hp}.&\supset .\gamma \cap CʻP-\iota ʻ\text{min}_{P}ʻ\gamma =\Lambda .\\ +[*205·11] &\supset .\gamma \cap CʻP=\iota ʻ\text{min}_{P}ʻ\gamma :\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*205·731.</b> \[\begin{align}&\vdash \colon\ldotp P\in \text{connex} \cap \text{Rl}ʻJ.\supset :\text{min}_{P}ʻ\gamma =\text{max}_{P}ʻ\gamma .\equiv .\gamma \cap CʻP\in 1\\ +&[*205·17·73]\end{align}\]</p> + +<p class="nind"><b>*205·732.</b> \[\begin{align}\vdash :P\in \text{connex} .\gamma \cap CʻP{\sim}\in 1.\text{E}!&\text{min}_{P}ʻ\gamma .\text{E}!\text{max}_{P}ʻ\gamma .\supset .\\ +&\text{min}_{P}ʻ\gamma P\text{max}_{P}ʻ\gamma\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*205·73.\text{Transp}.\supset \vdash :\text{Hp}.&\supset .\text{max}_{P}ʻ\alpha \neq \text{min}_{P}ʻ\alpha .\\ +[*205·21] &\supset .\text{min}_{P}ʻ\alpha P\text{max}_{P}ʻ\alpha :\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_574">[Pg 574]</span></p> + +<p>The following propositions lead up to <a href="#*205·75">*205·75</a>, which shows that the +minimum of a class belongs to \(\text{D}ʻP\) unless the part of the +class contained in \(CʻP\) is \(\iota ʻBʻ\breve{P}\).</p> + +<p class="nind"><b>*205·74.</b> \(\vdash : \alpha \cap CʻP \subset \overrightarrow{B}ʻ\breve{P} . \supset . \overrightarrow{\text{min}}_{P}ʻ\alpha = \alpha \cap CʻP\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *93·101 . \supset \vdash : \text{Hp} . &\supset . \alpha \cap \text{D}ʻP = \Lambda .\\ +[*37·261·29] &\supset . \breve{P} ʻʻ\alpha = \Lambda .\\ +[*205·11] &\supset · \overrightarrow{\text{min}}_{P}ʻ\alpha = \alpha \cap CʻP : \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*205·741.</b> \(\vdash : P \in \text{connex} . \alpha \cap CʻP {\sim} \in 1 . \supset . \overrightarrow{\text{min}}_{P}ʻ\alpha \subset \text{D}ʻP\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *205·21 . &\supset \vdash : P \in \text{connex} . y = \text{min}_{P}ʻ\alpha . z \in \alpha \cap CʻP - {℩}ʻy . \supset . yPz :\\ +[*205·3] &\supset \vdash : P \in \text{connex} . y \in \overrightarrow{\text{min}}_{P}ʻ\alpha . z \in \alpha \cap CʻP - {℩}ʻy . \supset . yPz :\\ +[*33·13] &\supset \vdash : P \in \text{connex} . y \in \overrightarrow{\text{min}}_{P}ʻ\alpha . \exists ! \alpha \cap CʻP - {℩}ʻy . \supset . y \in \text{D}ʻP :\\ +[*52·181] &\supset \vdash : P \in \text{connex} . \alpha \cap CʻP {\sim} \in 1 . \supset . \overrightarrow{\text{min}}_{P}ʻ\alpha + \subset \text{D}ʻP : \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*205·742.</b> \(\vdash \colon\ldotp P \in \text{connex} . \supset : \exists ! \overrightarrow{\text{min}}_{P}ʻ\alpha + - \text{D}ʻP . \equiv . \alpha \cap CʻP = {℩}ʻBʻ\breve{P}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *205·74 . \supset \vdash : \alpha \cap CʻP = {℩}ʻBʻ\breve{P} . &\supset . \overrightarrow{\text{min}}_{P}ʻ\alpha = {℩}ʻBʻ\breve{P} .\\ +[*93·101] &\supset . \exists ! \text{min}_{P}ʻ\alpha - \text{D}ʻP &\qquad \text{(1)}\\ +\vdash . *205·741 . &\supset \vdash : \text{Hp} . \exists ! \overrightarrow{\text{min}}_{P}ʻ\alpha - \text{D}ʻP . \supset . \alpha \cap CʻP \in 1 &\qquad \text{(2)}\\ +\vdash . *205·11 . \supset \vdash : \exists ! \overrightarrow{\text{min}}_{P}ʻ\alpha - \text{D}ʻP . &\supset . \exists ! \alpha \cap CʻP - \text{D}ʻP .\\ +[*93·103] &\supset . \exists ! \alpha \cap \overrightarrow{B}ʻ\breve{P} &\qquad \text{(3)}\\ +\vdash . (2) . (3) . *202·52 . &\supset \vdash : \text{Hp} . \exists ! \overrightarrow{\text{min}}_{P}ʻ\alpha + - \text{D}ʻP . \supset . \alpha \cap CʻP = {℩}ʻBʻ\breve{P} &\qquad \text{(4)}\\ +\vdash . (1) . (4). \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*205·75">*205·75</a>.</b> \[\begin{align}&\vdash \colon\ldotp P \in \text{connex} . \supset : {\sim} (\alpha \cap CʻP = {℩}ʻBʻ\breve{P} ) . \equiv . \overrightarrow{\text{min}}_{P}ʻ\alpha + \subset \text{D}ʻP\\ +&[*205·742]\end{align}\]</p> + +<p>Observe that \({\sim} (\alpha \cap CʻP = {℩}ʻBʻ\breve{P})\) is not in +general equivalent to \(\alpha \cap CʻP \neq {℩}ʻBʻ\breve{P}\), since +the latter implies \(\text{E}! Bʻ\breve{P}\), while the former does not.</p> + +<p>The following proposition is important.</p> + +<p class="nind"><b><a id="*205·8">*205·8</a>.</b> \(\vdash : S \in P \overline{\,\text{smor}\,} Q . \supset . \overrightarrow{\text{min}}_{P}ʻ\alpha = Sʻʻ\overrightarrow{\text{min}}_{Q}ʻ\breve{S} ʻʻ\alpha\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *205·11 . &\supset \vdash . Sʻʻ\overrightarrow{\text{min}}_{Q}ʻ\breve{S} + ʻʻ\alpha = Sʻʻ\{\breve{S} ʻʻ\alpha \cap CʻQ - \breve{Q} ʻʻ\breve{S} ʻʻ\alpha\} &\qquad \text{(1)}\\ +\vdash . *151·11 . &\supset \vdash : \text{Hp} . \supset . \breve{S} ʻʻ\alpha \subset CʻQ &\qquad \text{(2)}\\ +\vdash . (1) . (2) . \supset \vdash : \text{Hp} . \supset . Sʻʻ\overrightarrow{\text{min}}_{Q}ʻ\breve{S} ʻʻ\alpha &= Sʻʻ\{\breve{S} ʻʻ\alpha - \breve{Q} ʻʻ\breve{S} ʻʻ\alpha\}\\ +[*71·381] &= Sʻʻ\breve{S} ʻʻ\alpha - Sʻʻ\breve{Q} ʻʻ\breve{S} ʻʻ\alpha \\ +[*72·5 . *150·23] &= \alpha \cap CʻP - \breve{P} ʻʻ\alpha \\ +[*205·11] &= \overrightarrow{\text{min}}_{P}ʻ\alpha : \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_575">[Pg 575]</span></p> + +<p class="nind"><b>*205·81.</b> \(\vdash \colon\ldotp S\in P\,\overline{\,\text{smor}\,}\,Q.\supset :\text{E}!\text{min}_{P}ʻ\alpha .\equiv .\text{E}!\text{min}_{Q}ʻ\breve{S}ʻʻ\alpha\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*205·8.*73·22.\supset \vdash \colon\ldotp \text{Hp}.&\supset :\overrightarrow{\text{min}}_{P}ʻ\alpha \text{ sm }\overrightarrow{\text{min}}_{Q}ʻ\breve{S} ʻʻ\alpha :\\ +[*73·44] &\supset :\overrightarrow{\text{min}}_{P}ʻ\alpha \in 1.\equiv .\overrightarrow{\text{min}}_{Q}ʻ\breve{S} ʻʻ\alpha \in 1:\\ +[*53·3] &\supset :\text{E}!\text{min}_{P}ʻ\alpha .\equiv .\text{E}!\text{min}_{Q}ʻ\breve{S} ʻʻ\alpha \colon\ldotp \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*205·82.</b> \[\begin{align}&\vdash :S\in P\,\overline{\,\text{smor}\,}\,Q.\text{E}!\text{min}_{P}ʻ\alpha .\supset .\text{min}_{P}ʻ\alpha =Sʻ\text{min}_{Q}ʻ\breve{S} + ʻʻ\alpha\\ +&[*53·31.*205·8·81]\end{align}\]</p> + +<p>The two following propositions are used in *251·13.</p> + +<p class="nind"><b>*205·83.</b> \(\vdash :z{\sim}\in CʻP.\exists !CʻP\cap \alpha .\supset .\overrightarrow{\text{min}}_{P}ʻ\alpha =\overrightarrow{\text{min}}(P\unicode{x21f8} z)ʻ\alpha\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*161·1. \supset \vdash :\text{Hp}.&\supset .\{\text{Cnv}ʻ(P\unicode{x21f8} z)\}ʻʻ\alpha =\breve{P} ʻʻ\alpha \cup \iota ʻz.\\ +[*161·14·2.*24·495] &\supset .\alpha \cap Cʻ(P\unicode{x21f8} z)-\{\text{Cnv}ʻ(P\unicode{x21f8} z)\}ʻʻ\alpha =\alpha \cap CʻP-\breve{P} ʻʻ\alpha .\\ +[*205·11] &\supset .\overrightarrow{\text{min}}(P\unicode{x21f8} z)ʻ\alpha =\overrightarrow{\text{min}}_{P}ʻ\alpha :\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*205·831.</b> \(\vdash :z{\sim}\in CʻP.Cʻ(P\unicode{x21f8} z)\cap \alpha =\iota ʻz.\supset .\overrightarrow{\text{min}}(P\unicode{x21f8} z)ʻ\alpha =\iota ʻz\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*161·11. \supset \vdash \colon\ldotp \text{Hp}.&\supset :x\in \alpha .\supset _{x}.{\sim}\{x(P\unicode{x21f8} z)z\}:\\ +[*37·1.\text{Transp}] &\supset :z{\sim}\in \{\text{Cnv}ʻ(P\unicode{x21f8} z)\}ʻʻ\alpha &\qquad \text{(1)}\\ +\vdash .(1).*22·621.\supset \vdash :\text{Hp}.\supset .\iota ʻz&=Cʻ(P\unicode{x21f8} z)\cap \alpha -\{\text{Cnv}ʻ(P\unicode{x21f8} z)\}ʻʻ\alpha \\ +[*205·11] &=\overrightarrow{\text{min}}(P\unicode{x21f8} z)ʻʻ\alpha :\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p>The two following propositions are used in *251·14.</p> + +<p class="nind"><b>*205·832.</b> \(\vdash :z{\sim}\in CʻP.z{\sim}\in \alpha .\supset .\overrightarrow{\text{max}}_{P}ʻ\alpha =\overrightarrow{\text{max}}(P\unicode{x21f8} z)ʻ\alpha\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*205·111.*161·2.&\supset \vdash :P=\dot{\Lambda} .\supset .\overrightarrow{\text{max}}_{P}ʻ\alpha + =\Lambda .\overrightarrow{\text{max}}(P\unicode{x21f8} z)ʻ\alpha =\Lambda &\qquad \text{(1)}\\ +\vdash .*205·111.*161·11·14.\supset \\ +\vdash :\text{Hp}.\exists !CʻP\cap \alpha .\supset .\overrightarrow{\text{max}}(P\unicode{x21f8} z)ʻ\alpha &=\alpha \cap (CʻP\cup \iota ʻz)-(\breve{P} ʻʻ\alpha \cup \iota ʻz)\\ +[*24·495.*205·111] &=\overrightarrow{\text{max}}_{P}ʻ\alpha &\qquad \text{(2)}\\ +\vdash .*161·14.*205·151·161.\supset\\ +\vdash :\text{Hp}.\dot{\exists} !P.CʻP\cap \alpha &=\Lambda .\supset .\overrightarrow{\text{max}}_{P}ʻ\alpha + =\Lambda .\overrightarrow{\text{max}}(P\unicode{x21f8} z)ʻ\alpha =\Lambda &\qquad \text{(3)}\\ +\vdash .(1).(2).(3).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*205·833.</b> \(\vdash :\dot{\exists} !P.z{\sim}\in CʻP.z\in \alpha .\supset .\overrightarrow{\text{max}}(P\unicode{x21f8} z)ʻ\alpha =\iota ʻz\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*161·11.\supset \vdash :\text{Hp}.&\supset .(P\unicode{x21f8} z)ʻʻ\alpha =CʻP.\\ +[*161·14.*205·111] \supset .\overrightarrow{\text{max}}(P\unicode{x21f8} z)ʻ\alpha &=\alpha \cap (CʻP\cup \iota ʻz)-CʻP\\ +[*22·621.\text{Hp}] &=\iota ʻz:\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_576">[Pg 576]</span></p> + +<p>The following proposition is used in *251·25.</p> + +<p class="nind"><b>*205·84.</b> \(\vdash :CʻP\cap CʻQ=\Lambda .\exists !CʻP\cap \alpha .\supset .\overrightarrow{\text{min}}(P\unicode{x2909}Q)ʻ\alpha =\overrightarrow{\text{min}}_{P}ʻ\alpha\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*160·11.\supset \vdash :\text{Hp}.&\supset .\{\text{Cnv}ʻ(P\unicode{x2909}Q)\}ʻʻ\alpha =\breve{P} ʻʻ\alpha \cup CʻQ.\\ +[*205·11.*160·14] \supset .\overrightarrow{\text{min}}(P\unicode{x2909}Q)ʻ\alpha &=\alpha \cap (CʻP\cup CʻQ)-(\breve{P} ʻʻ\alpha \cup CʻQ)\\ +[*24·495] &=\alpha \cap CʻP-\breve{P} ʻʻ\alpha \\ +[*205·11] &=\overrightarrow{\text{min}}_{P}ʻ\alpha :\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*205·841.</b> \(\vdash :CʻP\cap \alpha =\Lambda .\supset .\overrightarrow{\text{min}}(P\unicode{x2909}Q)ʻ\alpha =\overrightarrow{\text{min}}_{Q}ʻ\alpha\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*160·11.\supset \vdash :\text{Hp}.&\supset .\{\text{Cnv}ʻ(P\unicode{x2909}Q)\}ʻʻ\alpha =\breve{Q} ʻʻ\alpha .\\ +[*205·11.*160·14] \supset .\overrightarrow{\text{min}}(P\unicode{x2909}Q)ʻ\alpha &=\alpha \cap (CʻP\cup CʻQ)-\breve{Q} ʻʻ\alpha \\ +[\text{Hp}] &=\alpha \cap CʻQ-\breve{Q} ʻʻ\alpha \\ +[*205·11] & =\overrightarrow{\text{min}}_{Q}ʻ\alpha :\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p>The following proposition is used in *251·2.</p> + +<p class="nind"><b><a id="*205·85">*205·85</a>.</b> \(\vdash \colon\ldotp P\in \text{Rel}^{2}\text{excl}.\supset :x\{\text{min}(\Sigma ʻP)\}\alpha .\equiv .(\exists Q).Q\text{min}_{P}(\breve{F} + ʻʻ\alpha ).x\text{min}_{Q}\alpha\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*162·12·23.*205·1.&\supset \vdash \colon\ldotp x{\text{min}(\Sigma ʻP)}\alpha .\equiv :\\ +&x\in \alpha :(\exists Q).Q\in CʻP.xFQ:{\sim}(\exists Q,y).Q\in CʻP.y\in \alpha .yQx:\\ +&{\sim}(\exists Q,R,y).xFQ.RPQ.yFR.y\in \alpha :\\ +[*37·105]&\equiv :x\in \alpha :(\exists Q).Q\in CʻP.xFQ:xFQ.Q\in CʻP.\supset _{Q}.x{\sim}\in \breve{Q} ʻʻ\alpha :\\ +&xFQ.Q\in CʻP.\supset _{Q}.Q{\sim}\in \breve{P} ʻʻ\breve{F} ʻʻ\alpha &\qquad \text{(1)}\\ +\vdash .(1).*163·12.*14·26.&\supset \vdash \colon\colon\text{Hp}.\supset \colon\ldotp x{\text{min}(\Sigma ʻP)}\alpha .\equiv :\\ +&(\exists Q).xFQ.Q\in CʻP.x\in \alpha -\breve{Q} ʻʻ\alpha :xFQ.Q\in CʻP.\supset _{Q}.Q\in \breve{F} ʻʻ\alpha -\breve{P} ʻʻ\breve{F} ʻʻ\alpha :\\ +[*163·12.*14·26]&\equiv :(\exists Q).xFQ.Q\in CʻP.x\in \alpha -\breve{Q} ʻʻ\alpha .Q\in \breve{F} ʻʻ\alpha -\breve{P} ʻʻ\breve{F} ʻʻ\alpha :\\ +[*205·1] &\equiv :(\exists Q).Q\text{min}_{P}(\breve{F} ʻʻ\alpha ).x\text{min}_{Q}a\colon\colon\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*205·9.</b> + \[\begin{align}&\vdash :P\in \text{connex} .\kappa \subset CʻP.\kappa {\sim}\in 1.\supset .\overrightarrow{\text{min}}(P\unicode{x0294f}\kappa )ʻ\alpha =\overrightarrow{\text{min}}_{P}ʻ(a\cap + \kappa )\\ +&[*205·261]\end{align}\]</p> + +<p class="nind"><b>*205·91.</b> \(\vdash :\breve{P} ʻʻ\alpha \subset \alpha .P_{\text{po}}\unicode{x0294f}\alpha \in \text{connex} .\supset .\overrightarrow{\text{min}}_{P}ʻ\alpha + \in 0\cup 1\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*205·261.\supset \vdash :\text{Hp}.\alpha \cap CʻP{\sim}\in 1.\supset .\overrightarrow{\text{min}}(P_{\text{po}}\unicode{x0294f}\alpha + )ʻ\alpha &=\overrightarrow{\text{min}}(P_{\text{po}})ʻ\alpha \\ +[*205·68] &=\overrightarrow{\text{min}}_{P}ʻ\alpha .\\ +[*205·3] &\supset .\overrightarrow{\text{min}}_{P}ʻ\alpha \in 0\cup 1 &\qquad \text{(1)}\\ +\vdash .*93·113.*60·371.&\supset \vdash :\alpha \cap CʻP\in 1.\supset .\overrightarrow{\text{min}}_{P}ʻ\alpha \in 0\cup 1 &\qquad \text{(2)}\\ +\vdash .(1).(2).\supset \vdash .\text{Prop} +\end{array} +\]</p> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_577">[Pg 577]</span></p> +<h2 class="nobreak" id="*206">*206. SEQUENT POINTS.</h2> +</div> + + +<p class="nind"><i>Summary of</i> *206.</p> + +<p>A "sequent" of a class \(\alpha\) is a minimum of the terms that come +after the whole of \(\alpha \cap CʻP\); that is, we put +\[ +\overrightarrow{\text{seq}} _{P}ʻ\alpha =\overrightarrow{\text{min}}_{P}ʻpʻ\overleftarrow{P}ʻʻ(\alpha \cap CʻP). +\] +Thus the sequents of \(\alpha\) are its immediate successors. If +\(\alpha\) has a maximum, the sequents are the immediate successors +of the maximum; but if \(\alpha\) has no maximum, there will be no +one term of \(\alpha\) which is immediately succeeded by a sequent +of \(\alpha\); in this case, if \(\alpha\) has a single sequent, +the sequent is the "upper limit" of \(\alpha\). Whenever \(P\) is +connected, and therefore whenever \(P\) is serial, every class has one +sequent or none with respect to \(P\), by <a href="#*205·3">*205·3</a>.</p> + +<p>It will be seen that the sequents of \(\alpha\) are the same +as the sequents of \(\alpha \cap CʻP\), and therefore that +\(\overrightarrow{\text{seq}}_{P}ʻ\alpha\) depends only upon \(\alpha\cap CʻP\): +if \(\alpha\) has terms not belonging to \(CʻP\), they are +irrelevant.</p> + +<p>For the immediate predecessors of a class \(\alpha\), we put +\[ +\overrightarrow{\text{prec}} _{P}ʻ\alpha =\overrightarrow{\text{max}}_{P}ʻpʻ\overrightarrow{P}ʻʻ(\alpha \cap CʻP). +\] +We have \(\text{prec}_{P}=\text{seq}(\breve{P})\), so that propositions +about \(\text{prec}_{P}\) result from those about \(\text{seq}_{P}\) by +merely writing \(\breve{P}\) in place of \(P\); they will therefore not +be given in what follows.</p> + +<p>Among the elementary properties of \(\text{seq}_{P}\) with which this +number begins, the following are the most important:</p> + +<p class="nind"><b>*206·13.</b> \(\vdash .\overrightarrow{\text{seq}} _{P}ʻ\alpha =\overrightarrow{\text{min}}_{P}ʻpʻ\overleftarrow{P}ʻʻ(\alpha \cap CʻP)\)</p> + +<p>This merely embodies the definition.</p> + +<p class="nind"><b>*206·131.</b> \(\vdash .\overrightarrow{\text{seq}} _{P}ʻ\alpha =\overrightarrow{\text{seq}} _{P}ʻ(\alpha \cap CʻP)\)</p> + +<p class="nind"><b>*206·134.</b> \(\vdash .\overrightarrow{\text{seq}} _{P}ʻ\alpha + =CʻP\cap \hat{x} \{\alpha \cap CʻP\subset \overrightarrow{P}ʻx.\overrightarrow{P}ʻx\subset -pʻ\overleftarrow{P}ʻʻ(\alpha \cap CʻP)\}\)</p> + +<p><span class="pagenum" id="Page_578">[Pg 578]</span></p> + +<p class="nind"><b>*206·14.</b> \(\vdash :\alpha \cap CʻP=\Lambda .\supset .\overrightarrow{\text{seq}} _{P}ʻ\alpha =\overrightarrow{B}ʻP\)</p> + +<p>Thus if \(P\) has a first term, this is the sequent of the null class, +or of any other class which has no members in common with \(CʻP\).</p> + +<p class="nind"><b>*206·16.</b> \(\vdash :P\in \text{connex} .\supset .\overrightarrow{\text{seq}} _{P}ʻ\alpha \in 0\cup 1\)</p> + +<p>This follows at once from <a href="#*205·3">*205·3</a>. It leads to</p> + +<p class="nind"><b>*206·161.</b> \(\vdash :P\in \text{connex} .\supset .\text{seq}_{P}\in 1\rightarrow \text{Cls}\)</p> + +<p>Thus if \(P\) is a connected relation, no class has more than one +sequent. This is not in general the case with relations which are +not connected, even where the idea of sequents is quite naturally +applicable. Take, <i>e.g.</i>, the relation of descendent to ancestor, +and let \(\alpha\) be the class of monarchs of England. Then +\(\overrightarrow{\text{seq}} _{P}ʻ\alpha\) will be such parents of +monarchs as were not themselves monarchs.</p> + +<p class="nind"><b>*206·171.</b> \[\begin{align}\vdash :P&\in \text{connex} .P^{2}\,\unicode{x2abd}\, J.\supset .\\ +&\overrightarrow{\text{seq}} _{P}ʻ\alpha =CʻP\cap \hat{x} \{\alpha \cap CʻP\subset \overrightarrow{P}ʻx.\overrightarrow{P}ʻx\subset (\alpha \cap CʻP)\cup Pʻʻ\alpha\}\end{align}\]</p> + +<p>This proposition states that \(x\) is a sequent of \(\alpha\) if the +whole of \(\alpha \cap CʻP\) precedes \(x\), but every term that +precedes \(x\) either belongs to \(\alpha\) or precedes some term of +\(\alpha\). When \(P\) is a series and \(\alpha\) has no maximum, we +have +\[ +\overrightarrow{\text{seq}} _{P}ʻ\alpha =CʻP\cap \hat{x} (\overrightarrow{P}ʻx=Pʻʻ\alpha ) \quad(*206·174), +\] +<i>i.e.</i> the sequent of \(\alpha\), if any, is a term whose +predecessors are identical with the predecessors of members of +\(\alpha\). This is the case of a limit (cf. <a href="#*207">*207</a>).</p> + +<p>We have next a set of propositions (<a href="#*206·211">*206·211</a><a href="#*206·28">·28</a>) concerned +with \(\overrightarrow{P}ʻ\text{seq}_{P}ʻ\alpha\) and +\(\overleftarrow{P}ʻ\text{seq}_{P}ʻ\alpha\). When \(P\) is transitive +and connected, and \(\alpha\) is an existent class contained in \(CʻP\) +and having a sequent, we shall have +\[ +\overrightarrow{P}ʻ\text{seq}_{P}ʻ\alpha =\alpha \cup Pʻʻ\alpha .{℩}ʻ\text{seq}_{P}ʻ\alpha \cup \overleftarrow{P}ʻ\text{seq}_{P}ʻ\alpha =pʻ\overleftarrow{P}ʻʻ\alpha . +\] +That is, the predecessors of the sequent are the members of \(\alpha\) +and the predecessors of members, while the sequent and its successors +are the successors of the whole of \(\alpha\). The various parts of +this statement require various parts of the hypothesis. Thus we have</p> + +<p class="nind"><b>*206·211.</b> \(\vdash :\text{E}!\text{seq}_{P}ʻ\alpha .\supset .\alpha \cap CʻP\subset \overrightarrow{P}ʻ\text{seq}_{P}ʻ\alpha\)</p> + +<p class="nind"><b>*206·213.</b> \(\vdash :P\in \text{connex} .\text{E}!\text{seq}_{P}ʻ\alpha .\supset .\overrightarrow{P}ʻ\text{seq}_{P}ʻ\alpha \subset (\alpha \cap CʻP)\cup Pʻʻ\alpha\)</p> + +<p class="nind"><b>*206·22.</b> \[\begin{align}\vdash :P\in \text{trans}\cap \text{connex} .&\text{E}!\text{seq}_{P}ʻ\alpha .\supset .\\ +&\overrightarrow{P}ʻ\text{seq}_{P}ʻ\alpha =(\alpha \cap CʻP)\cup Pʻʻ\alpha =\overrightarrow{\text{max}}_{P}ʻ\alpha \cup Pʻʻ\alpha\end{align}\]</p> + +<p class="nind"><b>*206·23.</b> \[\begin{align}\vdash :P\in \text{trans}\cap \text{connex} .&\text{E}!\text{seq}_{P}ʻ\alpha .\supset .\\ +&{℩}ʻ\text{seq}_{P}ʻ\alpha \cup \overleftarrow{P}ʻ\text{seq}_{P}ʻ\alpha + =pʻ\overleftarrow{P}ʻʻ(\alpha \cap CʻP)\cap CʻP\end{align}\]</p> + +<p><span class="pagenum" id="Page_579">[Pg 579]</span></p> + +<p>If \(P\) is transitive, the value of \(\overrightarrow{\text{seq}}_{P}ʻ\alpha\) +is unchanged if we add to \(\alpha\) any set of +terms contained in \(Pʻʻ\alpha\) (<a href="#*206·24">*206·24</a>); thus in particular, +\(\overrightarrow{\text{seq}} _{P}ʻ(\alpha \cup Pʻʻ\alpha ) =\overrightarrow{\text{seq}} _{P}ʻ\alpha\) +(<a href="#*206·25">*206·25</a>). Thus we can fill up any gaps in \(\alpha\), and take the +whole series up to the end of \(\alpha\), without altering the sequent.</p> + +<p>We have next a set of propositions (<a href="#*206·3">*206·3</a>—<a href="#*206·38">*206·38</a>) on the sequent of +\(Pʻʻ\alpha\), <i>i.e.</i> of the segment defined by \(\alpha\). If +\(P\) is a series, \(\text{seq}_{P}ʻPʻʻ\alpha\) is the maximum of +\(\alpha\) if \(\alpha\) has a maximum, the sequent of \(\alpha\) if +\(\alpha\) has a sequent but no maximum, and non-existent if \(\alpha\) has +neither a maximum nor a sequent (<a href="#*206·35">*206·35</a><a href="#*206·331">·331</a><a href="#*206·36">·36</a>).</p> + +<p>Our next set of propositions (<a href="#*206·4">*206·4</a>—<a href="#*206·52">·52</a>) concerns the sequents of +unit classes, especially of \(\iota ʻ\text{max}_{P}ʻ\alpha\), and of +classes of the form \(\overrightarrow{P}ʻx\). We have</p> + +<p class="nind"><b>*206·4.</b> \(\vdash :P\,\unicode{x2abd}\, J.x\in CʻP.\supset .x \text{seq}_{P}\overrightarrow{P}ʻx\)</p> + +<p class="nind"><b>*206·42.</b> \(\vdash :x\in CʻP.\supset .\overrightarrow{\text{seq}} _{P}ʻ\iota ʻx = \overleftarrow{P\dot{-} P^{2}}ʻx + = \overrightarrow{\text{min}}_{P}ʻ\overleftarrow{P}ʻx\)</p> + +<p class="nind">whence the three following propositions:</p> + +<p class="nind"><b>*206·43.</b> \(\vdash :P\in \text{trans} \cap \text{Rl}ʻJ.x\in CʻP.\supset .\overrightarrow{\text{seq}} _{P}ʻ\iota ʻx = \overleftarrow{P}_{1}ʻx\)</p> + +<p class="nind"><b>*206·45.</b> \(\vdash \colon\ldotp P\in \text{Ser}.x\in CʻP.\supset :\text{E}!\text{seq}_{P}ʻ\iota ʻx. \equiv .x\in \text{D}ʻP_{1}\)</p> + +<p class="nind"><b>*206·46.</b> \(\vdash :P\in \text{trans} \cap \text{connex} .\text{E}! \text{max}_{P}ʻ\alpha .\supset .\overrightarrow{\text{seq}} _{P}ʻ\alpha + = \overrightarrow{\text{seq}} _{P}ʻ\overrightarrow{\text{max}}_{P}ʻ\alpha\)</p> + +<p>From the above propositions it results that, when \(P\) is a series, +any member of \(CʻP\) is the sequent of the class of its predecessors, +\(\breve{P} _{1}ʻx\) is the sequent of \(\iota ʻx\) if either exists, +and the sequent of a class which has a maximum is the immediate +successor (if any) of the maximum, <i>i.e.</i></p> + +<p class="nind"><b>*206·5.</b> \(\vdash :P\in \text{trans} \cap \text{connex} .\text{E}!\text{max}_{P}ʻ\alpha .\text{E}!\text{seq}_{P}ʻ\alpha .\supset .\text{max}_{P}ʻ\alpha + (P\dot{-} P^{2})\text{seq}_{P}ʻ\alpha\)</p> + +<p>We then have a set of propositions (<a href="#*206·53">*206·53</a>—<a href="#*206·57">·57</a>) on the sequent +of \(pʻ\overrightarrow{P}ʻʻ(\alpha \cap CʻP)\), <i>i.e.</i> +the sequent of the predecessors of the whole of \(\alpha \cap CʻP\). +These propositions are specially useful in connection with +"Dedekindian" series, <i>i.e.</i> series in which every class +has either a maximum or a sequent (<a href="#*214">*214</a>). These propositions all +require the full hypothesis that \(P\) is a series. In this case, +\(\overrightarrow{\text{seq}} _{P}ʻpʻ\overrightarrow{P}ʻʻ(\alpha \cap CʻP) = \overrightarrow{\text{min}}_{P}ʻ\alpha\), +<i>i.e.</i> the sequent (if any) of the predecessors of the whole of +\(\alpha \cap CʻP\) is the minimum (if any) of \(\alpha\). Moreover +by definition the maximum of \(pʻ\overrightarrow{P}ʻʻ(\alpha \cap CʻP)\), +if any, is the precedent of \(\alpha\). Hence \(\alpha\) has +either a minimum or a precedent if \(pʻ\overrightarrow{P}ʻʻ(\alpha\cap CʻP)\) +has either a sequent or a maximum (<a href="#*206·54">*206·54</a>). Moreover the +sequent and maximum of \(\alpha\) are respectively (if they exist) +the sequent and maximum of the predecessors of all the successors +of the whole of \(\alpha \cap CʻP\) (<a href="#*206·551">*206·551</a>). Hence we arrive at +the conclusion that the assumption that every class of the form +\(pʻ\overrightarrow{P}ʻʻ(\alpha \cap CʻP)\) has either a maximum or +a sequent is equivalent both to the<span class="pagenum" id="Page_580">[Pg 580]</span> assumption that every class has +either a maximum or a sequent (<a href="#*206·56">*206·56</a>) and to the assumption that +every class has either a minimum or a precedent (<a href="#*206·55">*206·55</a>). It follows +that these two latter assumptions are equivalent (<a href="#*206·57">*206·57</a>), <i>i.e.</i> +that a series is Dedekindian when, and only when, its converse is +Dedekindian (<a href="#*214·14">*214·14</a>).</p> + +<p>We deal next (<a href="#*206·6">*206·6</a>—<a href="#*206·63">*206·63</a>) with correlations, showing that if two +relations are correlated, the sequents of the correlates of any class are the +correlates of the sequents, <i>i.e.</i></p> + +<p class="nind"><b>*206·61.</b> \(\vdash :S\in P\overline{\,\text{smor}\,}Q.\supset .\overrightarrow{\text{seq}} _{P}ʻ\alpha =Sʻʻ\overrightarrow{\text{seq}} _{Q}ʻ\breve{S} ʻʻ\alpha\)</p> + +<p>We end with a set of propositions (<a href="#*206·7">*206·7</a>—<a href="#*206·732">·732</a>) showing that the +sequent of a class is unchanged if we remove from the class any +term other than its maximum (<a href="#*206·72">*206·72</a>); that if a class has terms in +\(CʻP\), and has both a precedent and a sequent, the precedent has the +relation \(P^{2}\) to the sequent (<a href="#*206·73">*206·73</a>), and that the precedent is +not identical with the sequent (<a href="#*206·732">*206·732</a>). These propositions are in +the nature of lemmas, whose use is chiefly in the theory of stretches +(<a href="#*215">*215</a>).</p> + +<hr class="tb"> + +<p class="nind"><b>*206·01.</b> \(\text{seq}_{P}=\hat{x} \hat{\alpha} \{x\text{min}_{P}pʻ\overleftarrow{P}ʻʻ(\alpha \cap CʻP)\} \quad\text{Df}\)</p> + +<p class="nind"><b>*206·02.</b> \(\text{prec}_{P}=\hat{x} \hat{\alpha} \{x\text{max}_{P}pʻ\overrightarrow{P}ʻʻ(\alpha \cap CʻP)\} \quad\text{Df}\)</p> + +<p class="nind"><b>*206·1.</b> \(\vdash :x\text{seq}_{P}\alpha .\equiv .x\text{min}_{P}pʻ\overleftarrow{P}ʻʻ(\alpha \cap CʻP) \quad[(*206·01)]\)</p> + +<p class="nind"><b>*206·101.</b> \(\vdash .\text{prec}_{P}=\text{seq}(\breve{P} ) \quad[*32·241.*33·22.*205·102]\)</p> + +<p>We shall not enunciate any other propositions on \(\text{prec}_{P}\) +(unless for some special reason), since the above proposition enables +them to be immediately deduced from the corresponding propositions on +\(\text{seq}_{P}\).</p> + +<p class="nind"><b>*206·11.</b> \[\begin{align}&\vdash :x\text{seq}_{P}\alpha .\equiv .x\in pʻ\overleftarrow{P}ʻʻ(\alpha \cap CʻP)\cap CʻP-\breve{P} ʻʻpʻ\overleftarrow{P}ʻʻ(\alpha \cap CʻP)\\ +&[*206·1.*205·1]\end{align}\]</p> + +<p>Observe that when \(\alpha \cap CʻP\) is not null, +\(pʻ\overleftarrow{P}ʻʻ(\alpha \cap CʻP)\subset CʻP\), so that +the factor \(CʻP\) on the right is unnecessary; but when \(\alpha\cap CʻP=\Lambda\), +we have \(pʻ\overleftarrow{P}ʻʻ(\alpha \cap CʻP)=\text{V}\), so that +the factor \(CʻP\) becomes relevant. Owing to this factor, the sequents +of \(\Lambda\) are \(\overrightarrow{B}ʻP\), so that if \(BʻP\) exists, +is the sequent of \(\Lambda\).</p> + +<p class="nind"><b>*206·12.</b> \[\begin{align}\vdash \colon\colon &x\text{seq}_{P}\alpha .\equiv \colon\ldotp y\in \alpha \cap CʻP.\supset _{y}.yPx:x\in CʻP\colon\ldotp \\ +&y\in \alpha \cap CʻP.\supset _{y}.yPz:\supset _{z}.{\sim}(zPx) \quad[*206·11.*40·53.*37·105]\end{align}\]</p> + +<p class="nind"><b><a id="*206·13">*206·13</a>.</b> \(\vdash .\overrightarrow{\text{seq}} _{P}ʻ\alpha =\overrightarrow{\text{min}}_{P}pʻ\overleftarrow{P}ʻʻ(\alpha \cap CʻP) \quad[*206·1]\)</p> + +<p><span class="pagenum" id="Page_581">[Pg 581]</span></p> + +<p class="nind"><b>*206·131.</b> \(\vdash .\overrightarrow{\text{seq}} _{P}ʻ\alpha =\overrightarrow{\text{seq}} _{P}ʻ(\alpha \cap CʻP) \quad[*206·13.*22·43·621]\)</p> + +<p class="nind"><b><a id="*206·132">*206·132</a>.</b> \(\vdash . \overrightarrow{\text{seq}} _{P}ʻ\alpha + = pʻ\overleftarrow{P}ʻʻ(\alpha \cap CʻP) \cap CʻP - \breve{P} ʻʻpʻ\overleftarrow{P}ʻʻ(\alpha \cap CʻP) \quad[*206·11]\)</p> + +<p class="nind"><b>*206·133.</b> \(\vdash : x \text{seq}_{P}\alpha . \supset . {\sim} (xPx) \quad[*205·194 . *206·13]\)</p> + +<p class="nind"><b>*206·134.</b> \(\vdash . \overrightarrow{\text{seq}} _{P}ʻ\alpha + = CʻP \cap \hat{x} \{\alpha \cap CʻP \subset \overrightarrow{P}ʻx . \overrightarrow{P}ʻx \subset - pʻ\overleftarrow{P}ʻʻ(\alpha \cap CʻP)\}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *206·12 . *32·18 . \supset \\ +\vdash . \overrightarrow{\text{seq}} _{P}ʻ\alpha &= CʻP \cap \hat{x} (\alpha \cap CʻP \subset \overrightarrow{P}ʻx) \cap \hat{x} \{y \in \alpha \cap CʻP . \supset _{y} + . yPz : \supset _{z} . {\sim} (zPx)\}\\ +[*40·53] &= CʻP \cap \hat{x} (\alpha \cap CʻP \subset \overrightarrow{P}ʻx) \cap \hat{x} \{z \in pʻ\overleftarrow{P}ʻʻ(\alpha \cap CʻP) . \supset _{z} . {\sim} (zPx)\}\\ +[\text{Transp} . *32·18]\\ +&= CʻP \cap \hat{x} (\alpha \cap CʻP \subset \overrightarrow{P}ʻx) \cap \hat{x}\{\overrightarrow{P}ʻx \subset - pʻ\overleftarrow{P}ʻʻ(\alpha \cap CʻP)\} . \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p>This formula for \(\overrightarrow{\text{seq}} _{P}ʻ\alpha\) is usually +more convenient than <a href="#*206·13">*206·13</a><a href="#*206·132">·132</a>.</p> + +<p class="nind"><b>*206·14.</b> \(\vdash : \alpha \cap CʻP = \Lambda . \supset . \overrightarrow{\text{seq}} _{P}ʻ\alpha = \overrightarrow{B}ʻP\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *206·13 . *40·2 . \supset \vdash : \text{Hp} . \supset . \overrightarrow{\text{seq}} _{P}ʻ\alpha &= \overrightarrow{\text{min}}_{P}ʻ\text{V}\\ +[*205·15 . *24·26] &= \overrightarrow{\text{min}}_{P}ʻCʻP\\ +[*205·12] &= \overrightarrow{B}ʻP : \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*206·141.</b> \(\vdash : \exists ! \alpha \cap CʻP . \supset . \overrightarrow{\text{seq}} _{P}ʻ\alpha + = pʻ\overleftarrow{P}ʻʻ(\alpha \cap CʻP) - \breve{P} ʻʻpʻ\overleftarrow{P}ʻʻ(\alpha \cap CʻP)\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *40·62 . \supset \vdash : \text{Hp} . \supset . pʻ\overleftarrow{P}ʻʻ(\alpha \cap CʻP) \subset CʻP &\qquad \text{(1)}\\ +\vdash . (1) . *206·132 . \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*206·142.</b> \(\vdash : \exists ! \alpha \cap CʻP . \supset . \overrightarrow{\text{seq}} _{P}ʻ\alpha \subset \breve{P} ʻʻ\alpha \quad[*40·61 . *206·141]\)</p> + +<p class="nind"><b>*206·143.</b> \[\begin{align}&\vdash : \alpha \subset CʻP . \supset . \overrightarrow{\text{seq}} _{P}ʻ\alpha + = pʻ\overleftarrow{P}ʻʻ\alpha \cap CʻP - \breve{P} ʻʻpʻ\overleftarrow{P}ʻʻ\alpha \\ +&[*206·132 . *22·621]\end{align}\]</p> + +<p class="nind"><b>*206·144.</b> \(\vdash : \exists ! \overrightarrow{\text{seq}} _{P}ʻ\alpha . \supset . \exists ! pʻ\overleftarrow{P}ʻʻ(\alpha \cap CʻP) \quad[*206·132]\)</p> + +<p class="nind"><b>*206·15.</b> \[\begin{align}&\vdash : \alpha \subset CʻP . \exists ! \alpha . \supset . \overrightarrow{\text{seq}} _{P}ʻ\alpha + = pʻ\overleftarrow{P}ʻʻ\alpha - \breve{P} ʻʻpʻ\overleftarrow{P}ʻʻ\alpha\\ +&[*206·141 . *22·621]\end{align}\]</p> + +<p class="nind"><b>*206·16.</b> \(\vdash : P \in \text{connex} . \supset . \overrightarrow{\text{seq}} _{P}ʻ\alpha \in 0 \cup 1 \quad[*205·3 . *206·13]\)</p> + +<p class="nind"><b>*206·161.</b> \(\vdash : P \in \text{connex} . \supset . \text{seq}_{P} \in 1 \rightarrow \text{Cls} \quad[*206·16 . *71·12]\)</p> + +<p>Thus in a series, or in any connected relation, no class has more than one +sequent.</p> + +<p><span class="pagenum" id="Page_582">[Pg 582]</span></p> + +<p class="nind"><b>*206·17.</b> \[\begin{align}\vdash \colon\ldotp x \text{seq}_{P}\alpha .\equiv :y\in \alpha \cap CʻP.\supset _{y}.&yPx:x\in CʻP:\\ +&yPx.\supset _{y}.(\exists z).z\in \alpha \cap CʻP.{\sim}(zPy)\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*37·462.*206·11.\supset \\ +\vdash \colon\ldotp x \text{seq}_{P}\alpha .&\equiv :x\in pʻ\overleftarrow{P}ʻʻ(\alpha \cap CʻP)\cap CʻP.\overrightarrow{P}ʻx\subset -pʻ\overleftarrow{P}ʻʻ(\alpha \cap CʻP):\\ +[*40·53] &\equiv :y\in \alpha \cap CʻP.\supset _{y}.yPx:x\in CʻP:\\ +&yPx.\supset _{y}.(\exists z).z\in \alpha \cap CʻP.{\sim}(zPy)\colon\ldotp \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p>The following propositions give simplified formulae for +\(\overrightarrow{\text{seq}} _{P}ʻ\alpha\) in various special cases.</p> + +<p class="nind"><b>*206·171.</b> \[\begin{align}\vdash :P\in &\text{connex} .P^{2}\,\unicode{x2abd}\, J.\supset .\\ +&\overrightarrow{\text{seq}} _{P}ʻ\alpha =CʻP\cap \hat{x} \{\alpha \cap CʻP\subset \overrightarrow{P}ʻx.\overrightarrow{P}ʻx\subset (\alpha \cap CʻP)\cup Pʻʻ\alpha\}\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*206·134.*33·152.\supset \\ +\vdash .\overrightarrow{\text{seq}} _{P}ʻ\alpha + =CʻP\cap \hat{x}\{\alpha \cap CʻP\subset \overrightarrow{P}ʻx.\overrightarrow{P}ʻx\subset CʻP-pʻ\overleftarrow{P}ʻʻ(\alpha \cap CʻP)\} &\qquad \text{(1)}\\ +\vdash .(1).*202·503.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*206·172.</b> \[\begin{align}\vdash :P\in &\text{connex} .P^{2}\,\unicode{x2abd}\, J.Pʻʻ\alpha \subset \alpha .\supset .\\ +&\overrightarrow{\text{seq}} _{P}ʻ\alpha =CʻP\cap \hat{x} (\alpha \cap CʻP=\overrightarrow{P}ʻx) \quad[*206·171.*22·62]\end{align}\]</p> + +<p class="nind"><b><a id="*206·173">*206·173</a>.</b> \[\begin{align}\vdash :P\in &\text{connex} .P^{2}\,\unicode{x2abd}\, J.\alpha \cap CʻP\subset Pʻʻ\alpha .\supset .\\ +&\overrightarrow{\text{seq}} _{P}ʻ\alpha =CʻP\cap \hat{x} \{\alpha \cap CʻP\subset \overrightarrow{P}ʻx.\overrightarrow{P}ʻx\subset Pʻʻ\alpha\} +&[*206·171.*22·62]\end{align}\]</p> + +<p class="nind"><b><a id="*206·174">*206·174</a>.</b> \(\vdash :P\in \text{Ser}.\alpha \cap CʻP\subset Pʻʻ\alpha .\supset .\overrightarrow{\text{seq}} _{P}ʻ\alpha =CʻP\cap \hat{x} (\overrightarrow{P}ʻx=Pʻʻ\alpha )\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*13·12.*22·42.&\supset \vdash \colon\ldotp \text{Hp}.\supset :\\ +&\overrightarrow{P}ʻx=Pʻʻ\alpha .\supset .\alpha \cap CʻP\subset \overrightarrow{P}ʻx.\overrightarrow{P}ʻx\subset Pʻʻ\alpha &\qquad \text{(1)}\\ +\vdash .*37·265.&\supset \vdash :\alpha \cap CʻP\subset \overrightarrow{P}ʻx.\supset .Pʻʻ\alpha \subset Pʻʻ\overrightarrow{P}ʻx:\\ +[*201·501] \supset \vdash \colon\ldotp \text{Hp}.&\supset :\alpha \cap CʻP\subset \overrightarrow{P}ʻx.\supset .Pʻʻ\alpha \subset \overrightarrow{P}ʻx:\\ +[\text{Fact}] &\supset :\alpha \cap CʻP\subset \overrightarrow{P}ʻx.\overrightarrow{P}ʻx\subset Pʻʻ\alpha .\supset .Pʻʻ\alpha =\overrightarrow{P}ʻx &\qquad \text{(2)}\\ +\vdash .(1).(2).*206·173.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_583">[Pg 583]</span></p> + +<p>The propositions <a href="#*206·173">*206·173</a><a href="#*206·174">·174</a> deal with limits. When a class +\(\alpha\) has no maximum, <i>i.e.</i> when \(\alpha \cap CʻP\subset Pʻʻ\alpha\), +its sequent (if any) is called its <i>limit</i>. By the +above propositions, the limit is a term \(x\) such that \(\alpha \cap CʻP\) +precedes \(x\), but every predecessor of \(x\) precedes some +member of \(\alpha \cap CʻP\) (*206·173); it is also a term \(x\) +whose predecessors are identical with the predecessors of \(\alpha\) +(*206·174). The subject of limits will be explicitly treated in <a href="#*207">*207</a>.</p> + +<p class="nind"><b>*206·18.</b> \(\vdash .\overrightarrow{\text{seq}} _{P}ʻ\alpha \subset CʻP \quad[*206·132]\)</p> + +<p class="nind"><b>*206·181.</b> \(\vdash :\exists !\alpha \cap CʻP.\supset .\overrightarrow{\text{seq}} _{P}ʻ\alpha \subset \text{ᗡ}ʻP \quad[*206·142.*37·16]\)</p> + +<p class="nind"><b>*206·2.</b> \(\vdash .\overrightarrow{\text{seq}} _{P}ʻ\alpha \subset -\alpha\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*40·68.\text{Transp}.\supset \vdash .pʻ\overleftarrow{P}ʻʻ(\alpha \cap CʻP)-\breve{P} ʻʻpʻ\overleftarrow{P}ʻʻ(\alpha \cap CʻP)\subset -(\alpha \cap CʻP) &\qquad \text{(1)}\\ +\vdash .(1).*206·132.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*206·21.</b> \(\vdash :P^{2}\,\unicode{x2abd}\, J.\supset .\overrightarrow{\text{seq}} _{P}ʻ\alpha \subset -Pʻʻ\alpha \quad[*200·53.*206·132]\)</p> + +<p class="nind"><b><a id="*206·211">*206·211</a>.</b> \(\vdash :\text{E}!\text{seq}_{P}ʻ\alpha .\supset .\alpha \cap CʻP\subset \overrightarrow{P}ʻ\text{seq}_{P}ʻ\alpha\)</p> + +<p><i>Dem.</i> +\[ +\vdash .*206·17.\supset \vdash \colon\ldotp \text{Hp}.\supset :y\in \alpha \cap CʻP.\supset _{y}.yP\text{seq}_{P}ʻ\alpha \colon\ldotp \supset \vdash .\text{Prop} +\]</p> + +<p class="nind"><b>*206·212.</b> \(\vdash :P\in \text{trans}.\text{E}!\text{seq}_{P}ʻ\alpha .\supset .Pʻʻ\alpha \subset \overrightarrow{P}ʻ\text{seq}_{P}ʻ\alpha\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *206·211.\supset \vdash :\text{Hp}.\supset .Pʻʻ\alpha &\subset Pʻʻ\overrightarrow{P}ʻ\text{seq}_{P}ʻ\alpha \\ +[*201·501] &\subset \overrightarrow{P}ʻ\text{seq}_{P}ʻ\alpha :\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*206·213.</b> \(\vdash :P\in \text{connex} .\text{E}!\text{seq}_{P}ʻ\alpha .\supset .\overrightarrow{P}ʻ\text{seq}_{P}ʻ\alpha \subset (\alpha \cap CʻP)\cup Pʻʻ\alpha\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*206·17.\supset \vdash \colon\colon\text{Hp}.\supset \colon\ldotp yP\text{seq}_{P}ʻ\alpha .&\supset _{y}:(\exists z).z\in (\alpha \cap CʻP).{\sim}(zPy):\\ +[*202·103] &\supset _{y}:(\exists z):z\in \alpha \cap CʻP:y=z.\lor.yPz:\\ +[*13·195.*37·1] &\supset _{y}:y\in \alpha \cap CʻP.\lor.y\in Pʻʻ(\alpha \cap CʻP):\\ +[*37·265] & \supset _{y}:y\in (\alpha \cap CʻP)\cup Pʻʻ\alpha \colon\colon\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*206·22.</b> \[\begin{align}\vdash :P\in \text{trans} \cap \text{connex} .&\text{E}!\text{seq}_{P}ʻ\alpha .\supset .\\ +&\overrightarrow{P}ʻ\text{seq}_{P}ʻ\alpha =(\alpha \cap CʻP)\cup Pʻʻ\alpha =\overrightarrow{\text{max}}_{P}ʻ\alpha \cup Pʻʻ\alpha \\ +&[*206·211·212·213.*205·131]\end{align}\]</p> + +<p class="nind"><b>*206·23.</b> \[\begin{align}\vdash :P\in \text{trans} \cap \text{connex} .&\text{E}!\text{seq}_{P}ʻ\alpha .\supset .\\ +&\iota ʻ\text{seq}_{P}ʻ\alpha \cup \overleftarrow{P}ʻ\text{seq}_{P}ʻ\alpha =pʻ\overleftarrow{P}ʻʻ(\alpha \cap CʻP)\cap CʻP\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*205·22.*206·13.\supset \\ +\vdash :\text{Hp}.\supset .\iota ʻ\text{seq}_{P}ʻ\alpha \cup \overleftarrow{P}ʻ\text{seq}_{P}ʻ\alpha &=\iota ʻ\text{seq}_{P}ʻ\alpha + \cup \breve{P} ʻʻpʻ\overleftarrow{P}ʻʻ(\alpha \cap CʻP)\\ +[*206·13.*53·31]&=\overrightarrow{\text{min}}_{P}ʻpʻ\overleftarrow{P}ʻʻ(\alpha \cap CʻP)\cup \breve{P} ʻʻpʻ\overleftarrow{P}ʻʻ(\alpha \cap CʻP)\\ +[*205·13] &=pʻ\overleftarrow{P}ʻʻ(\alpha \cap CʻP)\cap CʻP\cup \breve{P} ʻʻpʻ\overleftarrow{P}ʻʻ(\alpha \cap CʻP)\\ +[*201·51.*37·16] &=pʻ\overleftarrow{P}ʻʻ(\alpha \cap CʻP)\cap CʻP:\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_584">[Pg 584]</span></p> + +<p class="nind"><b><a id="*206·24">*206·24</a>.</b> \(\vdash : P \in \text{trans} . \beta \subset Pʻʻ\alpha . \supset . \overrightarrow{\text{seq}} _{P}ʻ(\alpha + \cup \beta ) = \overrightarrow{\text{seq}} _{P}ʻ\alpha\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *201·56 . \supset \vdash : \text{Hp} . \supset . pʻ\overleftarrow{P}ʻʻ\{(\alpha \cup \beta ) \cap CʻP\} = pʻ\overleftarrow{P}ʻʻ(\alpha \cap CʻP) &\qquad \text{(1)}\\ +\vdash . (1) . *206·13 . \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*206·25">*206·25</a>.</b> \(\vdash : P \in \text{trans} . \supset . \overrightarrow{\text{seq}} _{P}ʻ(\alpha \cup Pʻʻ\alpha ) = \overrightarrow{\text{seq}} _{P}ʻ\alpha + \quad[*206·24]\)</p> + +<p class="nind"><b>*206·26.</b> \[\begin{align}\vdash : P \in \text{trans} \cap \text{connex} . \exists ! \alpha \cap &CʻP . \text{E}! \text{seq}_{P}ʻ\alpha . \supset .\\ +&pʻ\overleftarrow{P}ʻʻ(\alpha \cap CʻP) = {℩}ʻ\text{seq}_{P}ʻ\alpha \cup \overleftarrow{P}ʻ\text{seq}_{P}ʻ\alpha\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *40·62 . \supset \vdash : \text{Hp} . \supset . pʻ\overleftarrow{P}ʻʻ(\alpha \cap CʻP) \subset CʻP &\qquad \text{(1)}\\ +\vdash . (1) . *206·23 . \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*206·27.</b> \[\begin{align}\vdash : P \in \text{trans} \cap \text{connex} . \text{E}! \text{seq}_{P}ʻ&\alpha . \text{E}! \text{max}_{P}ʻ\alpha . \supset .\\ +&\overrightarrow{P}ʻ\text{seq}_{P}ʻ\alpha = \overrightarrow{P}ʻ\text{max}_{P}ʻ\alpha \cup {℩}ʻ\text{max}_{P}ʻ\alpha.\\ +&\overleftarrow{P}ʻ\text{max}_{P}ʻ\alpha = \overleftarrow{P}ʻ\text{seq}_{P}ʻ\alpha \cup {℩}ʻ\text{seq}_{P}ʻ\alpha\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *206·22 . \supset \vdash : \text{Hp} . \supset . \overrightarrow{P}ʻ\text{seq}_{P}ʻ\alpha &= \overrightarrow{\text{max}}_{P}ʻ\alpha \cup Pʻʻ\alpha \\ +[*205·22] &= {℩}ʻ\text{max}_{P}ʻ\alpha \cup \overrightarrow{P}ʻ\text{max}_{P}ʻ\alpha &\qquad \text{(1)}\\ +\vdash . *205·65 . &\supset \vdash : \text{Hp} . \supset . \overleftarrow{P}ʻ\text{max}_{P}ʻ\alpha = pʻ\overleftarrow{P}ʻʻ(\alpha \cap CʻP) &\qquad \text{(2)}\\ +\vdash . *205·151·161 . &\supset \vdash : \text{Hp} . \supset . \exists ! (\alpha \cap CʻP) &\qquad \text{(3)}\\ +\vdash . (3) . *206·26 . &\supset \vdash : \text{Hp} . \supset . pʻ\overleftarrow{P}ʻʻ(\alpha \cap CʻP) = {℩}ʻ\text{seq}_{P}ʻ\alpha \cup \overleftarrow{P}ʻ\text{seq}_{P}ʻ\alpha + &\qquad \text{(4)}\\ +\vdash . (2) . (4) . &\supset \vdash : \text{Hp} . \supset . \overleftarrow{P}ʻ\text{max}_{P}ʻ\alpha = {℩}ʻ\text{seq}_{P}ʻ\alpha \cup \overleftarrow{P}ʻ\text{seq}_{P}ʻ\alpha + &\qquad \text{(5)}\\ +\vdash . (1) . (5) . \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*206·28">*206·28</a>.</b> \[\begin{align}\vdash \colon\ldotp P \in \text{Ser} . &\supset :\\ +&x \in CʻP - \alpha . \overrightarrow{P}ʻx = Pʻʻ\alpha . \equiv . x = \text{seq}_{P}ʻ\alpha . {\sim} \text{E}! \text{max}_{P}ʻ\alpha \end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *206·174 . *205·6 . \supset \\ +\vdash \colon\ldotp \text{Hp} . \supset : x = \text{seq}_{P}ʻ\alpha . {\sim} \text{E}! \text{max}_{P}ʻ\alpha . &\supset . x \in CʻP . \overrightarrow{P}ʻx = Pʻʻ\alpha .\\ +[*206·2] &\supset . x \in CʻP - \alpha . \overrightarrow{P}ʻx = Pʻʻ\alpha &\qquad \text{(1)}\\ +\vdash . *37·1 . &\supset \vdash : xPy . y \in \alpha . \overrightarrow{P}ʻx = Pʻʻ\alpha . \supset . x \in \overrightarrow{P}ʻx &\qquad \text{(2)}\\ +\vdash . (2) . \text{Transp} . &\supset \vdash : P \,\unicode{x2abd}\, J . y \in \alpha . \overrightarrow{P}ʻx = Pʻʻ\alpha . \supset . {\sim} (xPy) &\qquad \text{(3)}\\ +\vdash . *13·14 . &\supset \vdash : x \in CʻP - \alpha . y \in \alpha . \supset . x \neq y &\qquad \text{(4)}\\ +\vdash . (3) . (4) . *202·103 . &\supset \vdash \colon\ldotp \text{Hp} . \supset :\\ +&x \in CʻP - \alpha . \overrightarrow{P}ʻx = Pʻʻ\alpha . y \in \alpha \cap CʻP . \supset . yPx :\\ +[*32·18] &\supset : x \in CʻP - \alpha . \overrightarrow{P}ʻx = Pʻʻ\alpha . \supset . \alpha \cap CʻP \subset \overrightarrow{P}ʻx &\qquad \text{(5)}\\ +\vdash .(5).*206·171·16.&\supset \vdash \colon\ldotp \text{Hp}.\supset :\\ +&x\in CʻP-\alpha .\overrightarrow{P}ʻx = Pʻʻ\alpha .\supset .x = \text{seq}_{P}ʻ\alpha &\qquad \text{(6)}\\ +\vdash .(5).*205·123. &\supset \vdash \colon\ldotp \text{Hp}.\supset :\\ +&x\in CʻP-\alpha .\overrightarrow{P}ʻx = Pʻʻ\alpha .\supset .{\sim}\text{E}!\text{max}_{P}ʻ\alpha &\qquad \text{(7)}\\ +\vdash .(1).(6).(7).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_585">[Pg 585]</span></p> + +<p class="nind"><b><a id="*206·3">*206·3</a>.</b> \[\begin{align}\vdash :P\in \text{trans} \cap \text{connex} .\alpha \subset CʻP.Pʻʻ\alpha \subset &\alpha .\text{E}!\text{seq}_{P}ʻ\alpha .\supset .\\ +&\overrightarrow{P}ʻ\text{seq}_{P}ʻ\alpha = \alpha \quad[*206·22]\end{align}\]</p> + +<p class="nind"><b>*206·31.</b> \[\begin{align}&\vdash :P\in \text{trans} \cap \text{connex} .\text{E}!\text{seq}_{P}ʻPʻʻ\alpha .\supset .\overrightarrow{P}ʻ\text{seq}_{P}ʻPʻʻ\alpha + = Pʻʻ\alpha\\ +&[*206·3.*201·5]\end{align}\]</p> + +<p class="nind"><b><a id="*206·32">*206·32</a>.</b> \[\begin{align}\vdash :P\in \text{trans} \cap \text{connex} .\text{E}!\text{max}_{P}ʻ\alpha .\text{E}!\text{seq}_{P}ʻ&Pʻʻ\alpha .\supset .\\ +&\text{max}_{P}ʻ\alpha = \text{seq}_{P}ʻPʻʻ\alpha\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*206·31.*205·22.&\supset \vdash \colon\ldotp \text{Hp}.\supset :\overrightarrow{P}ʻ\text{max}_{P}ʻ\alpha = \overrightarrow{P}ʻ\text{seq}_{P}ʻPʻʻ\alpha :\\ +[*205·194.*206·133]&\supset :{\sim}(\text{seq}_{P}ʻPʻʻ\alpha P \text{max}_{P}ʻ\alpha ).{\sim}(\text{max}_{P}ʻ\alpha P \text{seq}_{P}ʻʻ\alpha ):\\ +[*202·103] &\supset :\text{max}_{P}ʻ\alpha = \text{seq}_{P}ʻPʻʻ\alpha \colon\ldotp \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p>In the hypothesis of <a href="#*206·32">*206·32</a>, we have both +\(\text{E}!\text{max}_{P}ʻ\alpha\) and +\(\text{E}!\text{seq}_{P}ʻPʻʻ\alpha\). So long as \(P\) is not +contained in diversity, these are both necessary. For example, suppose +we take +\[ +P = \alpha \uparrow (\alpha \cup \iota ʻx),\,\text{where}\, x{\sim}\in \alpha .\exists !\alpha . +\] +Then \(P\) is transitive and connected, but not contained in diversity. +We have +\[ +\alpha \cup \iota ʻx = CʻP.Pʻʻ(\alpha \cup \iota ʻx) = \alpha = \text{D}ʻP. +\] +Also +\[ +\begin{aligned} +\text{max}_{P}ʻ(\alpha \cup \iota ʻx) &= x\,\\ +\overrightarrow{\text{seq}} _{P}ʻPʻʻ(\alpha \cup \iota ʻx) &= \overrightarrow{\text{min}}_{P}ʻpʻ\overleftarrow{P}ʻʻ\alpha = \overrightarrow{\text{min}}_{P}ʻ(\alpha + \cup \iota ʻx) = \Lambda . +\end{aligned} +\] +Thus in this case \(\text{max}_{P}ʻ(\alpha \cup \iota ʻx)\) +exists, but \(\text{seq}_{P}ʻPʻʻ(\alpha \cup \iota ʻx)\) does not +exist. When \(P\) is serial, <i>i.e.</i> when \(P\) is contained +in diversity, in addition to being transitive and connected, +the existence of \(\text{max}_{P}ʻ\alpha\) involves that of +\(\text{seq}_{P}ʻPʻʻ\alpha\), and therefore the hypothesis +\(\text{E}!\text{seq}_{P}ʻPʻʻ\alpha\), which appears in <a href="#*206·32">*206·32</a>, +becomes unnecessary.</p> + +<p class="nind"><b>*206·33.</b> \(\vdash :P\in \text{trans} \cap \text{connex} .{\sim}\text{E}!\text{max}_{P}ʻ\alpha .\supset .\overrightarrow{\text{seq}} _{P}ʻPʻʻ\alpha + = \overrightarrow{\text{seq}} _{P}ʻ\alpha\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*205·6. \supset \vdash :\text{Hp}.&\supset .\alpha \cap CʻP\subset Pʻʻ\alpha .\\ +[*22·62.*37·15] &\supset .(\alpha \cup Pʻʻ\alpha )\cap CʻP = Pʻʻ\alpha &\qquad \text{(1)}\\ +\vdash .*206·25 .\supset \vdash :\text{Hp}.\supset .\overrightarrow{\text{seq}} _{P}ʻ\alpha &= \overrightarrow{\text{seq}} _{P}ʻ(\alpha \cup Pʻʻ\alpha )\\ +[*206·131] &= \overrightarrow{\text{seq}} _{P}ʻ\{(\alpha \cup Pʻʻ\alpha )\cap CʻP\}\\ +[(1)] &= \overrightarrow{\text{seq}} _{P}ʻPʻʻ\alpha :\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_586">[Pg 586]</span></p> + +<p class="nind"><b><a id="*206·331">*206·331</a>.</b> \[\begin{align}&\vdash :P\in \text{trans}\cap \text{connex} .{\sim}\text{E}!\text{max}_{P}ʻ\alpha .\text{E}!\text{seq}_{P}ʻ\alpha .\supset .\text{seq}_{P}ʻPʻʻ\alpha + =\text{seq}_{P}ʻ\alpha \\ +&[*206·33]\end{align}\]</p> + +<p class="nind"><b>*206·34.</b> \(\vdash :P\in \text{Ser}.\supset .\overrightarrow{\text{max}}_{P}ʻ\alpha \subset \overrightarrow{\text{seq}} _{P}ʻPʻʻ\alpha\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash *205·101.*37·265.\supset \\ +\vdash \colon\ldotp y\in \overrightarrow{\text{max}}_{P}ʻ\alpha .\equiv :&y\in \alpha \cap CʻP:z\in \alpha \cap CʻP.\supset _{z}.{\sim}(yPz) &\qquad \text{(1)}\\ +\vdash .(1).*202·103.&\supset \vdash \colon\colon\ldotp \text{Hp}.\supset \colon\colon\\ +&y\in \overrightarrow{\text{max}}_{P}ʻ\alpha .\supset \colon\ldotp y\in \alpha \cap CʻP\colon\ldotp z\in \alpha \cap CʻP.\supset _{z}:z=y.\lor.zPy &\qquad \text{(2)}\\ +\vdash .(2).*13·195.*201·1.\supset \vdash \colon\colon\ldotp \text{Hp}.\supset \colon\colon\\ +&y\in \overrightarrow{\text{max}}_{P}ʻ\alpha .\supset \colon\ldotp y\in \alpha \cap CʻP\colon\ldotp z\in \alpha \cap CʻP.uPz.\supset _{u,z}.uPy\colon\ldotp \\ +[*37·1·265] &\supset \colon\ldotp u\in Pʻʻ\alpha .\supset _{u}.uPy\colon\ldotp \\ +[*40·53] &\supset \colon\ldotp y\in pʻ\overleftarrow{P}ʻʻPʻʻ\alpha &\qquad \text{(3)}\\ +\vdash .(1).*37·1. &\supset \vdash :y\in \overrightarrow{\text{max}}_{P}ʻ\alpha .vPy.\supset .v\in Pʻʻ\alpha &\qquad \text{(4)}\\ +\vdash .*50·24. & \supset \vdash :\text{Hp}.\supset .{\sim}(vPv) &\qquad \text{(5)}\\ +\vdash .(4).(5). & \supset \vdash \colon\ldotp \text{Hp}.\supset :y\in \overrightarrow{\text{max}}_{P}ʻ\alpha .vPy.\supset .(\exists w).w\in Pʻʻ\alpha .{\sim}(wPv).\\ +[*40·53] & \supset .v{\sim}\in pʻ\overleftarrow{P}ʻʻPʻʻ\alpha :\\ +[*10·51] \supset :y\in \overrightarrow{\text{max}}_{P}ʻ\alpha .&\supset .{\sim}(\exists v)·v\in pʻ\overleftarrow{P}ʻʻPʻʻ\alpha .vPy.\\ +[*37·105] &\supset .y{\sim}\in \breve{P} ʻʻpʻ\overleftarrow{P}ʻʻPʻʻ\alpha &\qquad \text{(6)}\\ +\vdash .(3).(6).(1).&\supset \vdash \colon\ldotp \text{Hp}.\supset :\\ +&y\in \overrightarrow{\text{max}}_{P}ʻ\alpha .\supset .y\in pʻ\overleftarrow{P}ʻʻPʻʻ\alpha \cap CʻP-\breve{P} ʻʻpʻ\overleftarrow{P}ʻʻPʻʻ\alpha .\\ +[*206·143] &\supset .y\in \overrightarrow{\text{seq}} _{P}ʻPʻʻ\alpha \colon\ldotp \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*206·35">*206·35</a>.</b> \(\vdash :P\in \text{Ser}.\text{E}!\text{max}_{P}ʻ\alpha .\supset .\text{max}_{P}ʻ\alpha =\text{seq}_{P}ʻPʻʻ\alpha + .\text{E}!\text{seq}_{P}ʻPʻʻ\alpha\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*206·34. &\supset \vdash :\text{Hp}.\supset .\text{max}_{P}ʻ\alpha \in \overrightarrow{\text{seq}} _{P}ʻPʻʻ\alpha &\qquad \text{(1)}\\ +\vdash .(1).*206·16.&\supset \vdash :\text{Hp}.\supset .\text{max}_{P}ʻ\alpha =\text{seq}_{P}ʻPʻʻ\alpha &\qquad \text{(2)}\\ +\vdash .(2).*14·21. &\supset \vdash :\text{Hp}.\supset .\text{E}!\text{seq}_{P}ʻPʻʻ\alpha &\qquad \text{(3)}\\ +\vdash .(2).(3).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*206·36">*206·36</a>.</b> \(\vdash \colon\colon P\in \text{Ser}.\supset \colon\ldotp \text{E}!\text{seq}_{P}ʻPʻʻ\alpha .\equiv :\text{E}!\text{max}_{P}ʻ\alpha + .\lor.\text{E}!\text{seq}_{P}ʻ\alpha\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*206·35·331.&\supset \vdash \colon\ldotp \text{Hp}:\text{E}!\text{max}_{P}ʻ\alpha .\lor.\text{E}!\text{seq}_{P}ʻ\alpha :\supset .\text{E}!\text{seq}_{P}ʻPʻʻ\alpha + &\qquad \text{(1)}\\ +\vdash .*206·34. \supset \vdash \colon\ldotp \text{Hp}.&\supset :\text{E}!\text{seq}_{P}ʻPʻʻ\alpha .\supset .{\sim}\text{E}!\text{max}_{P}ʻ\alpha . &\qquad \text{(2)}\\ +[*206·33] &\supset .{\sim}\text{E}!\text{seq}_{P}ʻ\alpha &\qquad \text{(3)}\\ +\vdash .(1).(2).(3).&\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_587">[Pg 587]</span></p> + +<p>The condition \((\alpha ):\text{E}!\text{max}_{P}ʻ\alpha.\lor.\text{E}!\text{seq}_{P}ʻ\alpha\) +is the definition of what may be called "Dedekindian" series, +<i>i.e.</i> series in which, when any division of the field into two +parts is made in such a way that the first part wholly precedes the +second, then either the first part has a last term or the second part +has a first term. (When these alternatives are also mutually exclusive, +the series has "Dedekindian continuity.") If \(\alpha\) is any class, +\(Pʻʻ\alpha\) is the segment of \(CʻP\) defined by \(\alpha\). In +virtue of the above proposition, every segment of a Dedekindian series +has a sequent. The sequent of a class having no maximum is what is +commonly called a <i>limit</i>. Thus in a series having Dedekindian +continuity (in which segments never have maxima), every segment has a +limit.</p> + +<p class="nind"><b>*206·37.</b> \(\vdash :P\in \text{Ser}.\supset .\overrightarrow{\text{seq}} _{P}ʻPʻʻ\alpha =\overrightarrow{\text{min}}_{P}ʻ(\overrightarrow{\text{max}}_{P}ʻ\alpha + \cup \overrightarrow{\text{seq}} _{P}ʻ\alpha )\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*205·16.\supset \vdash :&\overrightarrow{\text{max}}_{P}ʻ\alpha =\Lambda .\overrightarrow{\text{seq}} _{P}ʻ\alpha =\Lambda .\supset .\\ +&\overrightarrow{\text{min}}_{P}ʻ(\overrightarrow{\text{max}}_{P}ʻ\alpha \cup \overrightarrow{\text{seq}} _{P}ʻ\alpha )=\Lambda &\qquad \text{(1)}\\ +\vdash .*206·36.\supset \vdash :\text{Hp}.\text{Hp}(1).&\supset .{\sim}\text{E}!\text{seq}_{P}ʻPʻʻ\alpha .\\ +[*206·16] &\supset .\overrightarrow{\text{seq}} _{P}ʻPʻʻ\alpha =\Lambda &\qquad \text{(2)}\\ +\vdash .*24·24. \supset \vdash :\text{Hp}.&\overrightarrow{\text{max}}_{P}ʻ\alpha =\Lambda .\exists !\overrightarrow{\text{seq}} _{P}ʻ\alpha .\supset .\\ +&\overrightarrow{\text{min}}_{P}ʻ(\overrightarrow{\text{max}}_{P}ʻ\alpha \cup \overrightarrow{\text{seq}} _{P}ʻ\alpha )=\overrightarrow{\text{min}}_{P}ʻ\overrightarrow{\text{seq}} + _{P}ʻ\alpha \\ +[*205·17.*206·16] &=\overrightarrow{\text{seq}} _{P}ʻ\alpha \\ +[*206·33] &=\overrightarrow{\text{seq}} _{P}ʻPʻʻ\alpha &\qquad \text{(3)}\\ +\vdash .*205·17·3.&\supset \vdash :\text{Hp}.\exists !\overrightarrow{\text{max}}_{P}ʻ\alpha .\overrightarrow{\text{seq}} _{P}ʻ\alpha =\Lambda .\supset .\\ +&\overrightarrow{\text{min}}_{P}ʻ(\overrightarrow{\text{max}}_{P}ʻ\alpha \cup \overrightarrow{\text{seq}} _{P}ʻ\alpha )=\overrightarrow{\text{max}}_{P}ʻ\alpha \\ +[*206·35] & =\overrightarrow{\text{seq}} _{P}ʻPʻʻ\alpha &\qquad \text{(4)}\\ +\vdash .*206·16.*205·3.\supset \\ +\vdash :\text{Hp}.\exists !&\overrightarrow{\text{max}}_{P}ʻ\alpha .\exists !\overrightarrow{\text{seq}} _{P}ʻ\alpha .\supset .\\ +&\overrightarrow{\text{min}}_{P}ʻ(\overrightarrow{\text{max}}_{P}ʻ\alpha \cup \overrightarrow{\text{seq}} _{P}ʻ\alpha + )=\overrightarrow{\text{min}}_{P}ʻ({℩}ʻ\text{max}_{P}ʻ\alpha \cup {℩}ʻ\text{seq}_{P}ʻ\alpha )\\ +[*206·27.*205·182] & ={℩}ʻ\text{max}_{P}ʻ\alpha \\ +[*206·35] &=\overrightarrow{\text{seq}} _{P}ʻPʻʻ\alpha &\qquad \text{(5)}\\ +\vdash .(1).(2).(3).(4).(5).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*206·38">*206·38</a>.</b> \(\vdash :P\in \text{Ser}.\supset .\overrightarrow{\text{max}}_{P}ʻ\alpha =\alpha \cap \overrightarrow{\text{seq}} _{P}ʻPʻʻ\alpha\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*206·35.*205·111.\supset\\ +\vdash :\text{Hp}.\text{E}!\text{max}_{P}ʻ\alpha .&\supset .\overrightarrow{\text{max}}_{P}ʻ\alpha =\overrightarrow{\text{seq}} _{P}ʻPʻʻ\alpha + .\overrightarrow{\text{max}}_{P}ʻ\alpha \subset \alpha .\\ +[*22·621] &\supset .\overrightarrow{\text{max}}_{P}ʻ\alpha =\alpha \cap \overrightarrow{\text{seq}} _{P}ʻPʻʻ\alpha + &\qquad \text{(1)}\\ +\vdash .*205·3. \supset \vdash :\text{Hp}.{\sim}\text{E}!\text{max}_{P}ʻ\alpha .&\supset .\overrightarrow{\text{max}}_{P}ʻ\alpha =\Lambda &\qquad \text{(2)}\\ +\vdash .*206·33. \supset \vdash :\text{Hp}.{\sim}\text{E}!\text{max}_{P}ʻ\alpha .&\supset .\overrightarrow{\text{seq}} _{P}ʻPʻʻ\alpha =\overrightarrow{\text{seq}} _{P}ʻ\alpha + .\\ +[*206·2] &\supset .\alpha \cap \overrightarrow{\text{seq}} _{P}ʻPʻʻ\alpha =\Lambda .\\ +[(2)] &\supset .\overrightarrow{\text{max}}_{P}ʻ\alpha =\alpha \cap \overrightarrow{\text{seq}} _{P}ʻPʻʻ\alpha &\qquad \text{(3)}\\ +\vdash .(1).(3).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_588">[Pg 588]</span></p> + +<p class="nind"><b><a id="*206·4">*206·4</a>.</b> \(\vdash :P\,\unicode{x2abd}\, J.x\in CʻP.\supset .x\text{seq}_{P}\overrightarrow{P}ʻx\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*206·134.*22·43.\supset \\ +\vdash :x\text{seq}_{P}\overrightarrow{P}ʻx.\equiv .x\in .CʻP.\overrightarrow{P}ʻx\subset -pʻ\overleftarrow{P}ʻʻ\overrightarrow{P}ʻx &\qquad \text{(1)}\\ +\vdash .*200·5.\supset \vdash :P\,\unicode{x2abd}\, J.\supset .\overrightarrow{P}ʻx\subset -pʻ\overleftarrow{P}ʻʻ\overrightarrow{P}ʻx &\qquad \text{(2)}\\ +\vdash .(1).(2).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*206·401.</b> \(\vdash :P\in \text{connex} \cap \text{Rl}ʻJ.x\in CʻP.\supset .x=\text{seq}_{P}ʻ\overrightarrow{P}ʻx \quad[*206·4·161]\)</p> + +<p class="nind"><b>*206·41.</b> \(\vdash .\overrightarrow{\text{min}}_{P}ʻ\overleftarrow{P}ʻx=\overleftarrow{P\dot{-} P^{2}}ʻx \quad[*205·25]\)</p> + +<p class="nind"><b>*206·42.</b> \(\vdash :x\in CʻP.\supset .\overrightarrow{\text{seq}} _{P}ʻ{℩}ʻx=\overleftarrow{P\dot{-} + P^{2}}ʻx=\overrightarrow{\text{min}}_{P}ʻ\overleftarrow{P}ʻx\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*53·01·31.\supset \vdash .pʻ\overleftarrow{P}ʻʻ{℩}ʻx=\overleftarrow{P}ʻx &\qquad \text{(1)}\\ +\vdash .(1).*206·41·143.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*206·43.</b> \[\begin{align}&\vdash :P\in \text{trans}\cap \text{Rl}ʻJ.x\in CʻP.\supset .\overrightarrow{\text{seq}} _{P}ʻ{℩}ʻx=\overleftarrow{P}_{1}ʻx\\ +&[*206·42.*201·63]\end{align}\]</p> + +<p class="nind"><b>*206·44.</b> \[\begin{align}\vdash \colon\ldotp P\in &\text{trans}\cap \text{Rl}ʻJ.x\in CʻP.\supset :\\ +&\text{E}!\text{seq}_{P}ʻ{℩}ʻx.\equiv .\text{E}!\breve{P} _{1}ʻx:\text{E}!\text{seq}_{P}ʻ{℩}ʻx.\supset .\text{seq}_{P}ʻ{℩}ʻx=\breve{P} _{1}ʻx\\ +[*206·43]\end{align}\]</p> + +<p class="nind"><b>*206·45.</b> \[\begin{align}&\vdash \colon\ldotp P\in \text{Ser}.x\in CʻP.\supset :\text{E}!\text{seq}_{P}ʻ{℩}ʻx.\equiv .x\in \text{D}ʻP_{1}\\ +&[*206·44.*204·7.*71·165]\end{align}\]</p> + +<p class="nind"><b>*206·451.</b> \(\vdash :P\in \text{Ser}.\text{E}!\text{seq}_{P}ʻ\alpha .\supset .\overrightarrow{\text{max}}_{P}ʻ\alpha + =\overrightarrow{P}_{1}ʻ\text{seq}_{P}ʻ\alpha\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*206·41.\supset \vdash :\text{Hp}.\supset .\overrightarrow{P}_{1}ʻ\text{seq}_{P}ʻ\alpha &=\overrightarrow{\text{max}}pʻ\overrightarrow{P}ʻ\text{seq}_{P}ʻ\alpha \\ +[*206·22] &=\overrightarrow{\text{max}}pʻ{(\alpha \cap CʻP)\cup Pʻʻ\alpha } +[*205·191] &=\overrightarrow{\text{max}}_{P}ʻ\alpha :\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_589">[Pg 589]</span></p> + +<p class="nind"><b>*206·46.</b> \(\vdash : P \in \text{trans} \cap \text{connex} . \text{E}! \text{max}_{P}ʻ\alpha . \supset . \overrightarrow{\text{seq}} _{P}ʻ\alpha + = \overrightarrow{\text{seq}} _{P}ʻ\overrightarrow{\text{max}}_{P}ʻ\alpha\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *206·42 . \supset \vdash : \text{Hp} . \supset . &\overrightarrow{\text{seq}} _{P}ʻ\overrightarrow{\text{max}}_{P}ʻ\alpha\\ + &= \overrightarrow{\text{min}}_{P}ʻ\overleftarrow{P}ʻ\text{max}_{P}ʻ\alpha\\ +[*205·65] &= \overrightarrow{\text{min}}_{P}ʻpʻ\overleftarrow{P}ʻʻ(\alpha \cap CʻP)\\ +[*206·13] &= \overrightarrow{\text{seq}} _{P}ʻ\alpha : \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*206·47.</b> \(\vdash : P \in \text{trans} . \text{E}! \text{seq}_{P}ʻ\alpha . \supset . \text{seq}_{P}ʻ\alpha = \text{max}_{P}ʻ(\alpha + \cup \overrightarrow{\text{seq}} _{P}ʻ\alpha )\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *206·134 . \supset \vdash : \text{Hp} . &\supset . \alpha \cap CʻP \subset \overrightarrow{P}ʻ\text{seq}_{P}ʻ\alpha .\\ +[*205·193·151] \supset . \overrightarrow{\text{max}}_{P}ʻ(\alpha \cup \overrightarrow{\text{seq}} _{P}ʻ\alpha ) &=\overrightarrow{\text{max}}_{P}ʻ\overrightarrow{\text{seq}} + _{P}ʻ\alpha \\ +[*206·133 . *205·18] &= {℩}ʻ\text{seq}_{P}ʻ\alpha : \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*206·48.</b> \(\vdash : P \in \text{trans} \cap \text{connex} . \text{E}! \text{seq}_{P}ʻ\alpha . \supset . \overrightarrow{\text{seq}} _{P}ʻ\overrightarrow{\text{seq}} + _{P}ʻ\alpha = \overrightarrow{\text{seq}} _{P}ʻ(\alpha \cup \overrightarrow{\text{seq}} _{P}ʻ\alpha )\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *206·47 . &\supset \vdash : \text{Hp} . \supset .\\ +&\overrightarrow{\text{seq}} _{P}ʻ\overrightarrow{\text{seq}} _{P}ʻ\alpha = \overrightarrow{\text{seq}} _{P}ʻ\overrightarrow{\text{max}}_{P}ʻ(\alpha + \cup \overrightarrow{\text{seq}} _{P}ʻ\alpha ) . \text{E}! \text{max}_{P}ʻ(\alpha \cup \text{seq}_{P}ʻ\alpha ) .\\ +[*206·46] &\supset . \overrightarrow{\text{seq}} _{P}ʻ\overrightarrow{\text{seq}} _{P}ʻ\alpha = \overrightarrow{\text{seq}} _{P}ʻ(\alpha + \cup \overrightarrow{\text{seq}} _{P}ʻ\alpha ) : \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*206·5.</b> \[\begin{align}\vdash : P \in \text{trans} \cap \text{connex} . \text{E}! \text{max}_{P}ʻ\alpha . \text{E}! &\text{seq}_{P}ʻ\alpha . \supset .\\ +&\text{max}_Pʻ\alpha (P \dot{-} P^{2}) \text{seq}_{P}ʻ\alpha\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *206·46 . \supset \vdash : \text{Hp} . \supset . \overrightarrow{\text{seq}} _{P}ʻ\alpha &= \overrightarrow{\text{seq}} _{P}ʻ{℩}ʻ\text{max}_{P}ʻ\alpha\\ +[*206·42] &= \overleftarrow{P\dot{-} P^{2}}ʻ\text{max}_{P}ʻ\alpha : \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*206·51.</b> \(\vdash : \exists ! \overrightarrow{\text{max}}_{P}ʻ\overrightarrow{P}ʻx . \supset . x \text{seq}_{P} \overrightarrow{P}ʻx\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *205·161 . \supset \vdash : \text{Hp} . &\supset . \exists ! \overrightarrow{P}ʻx .\\ +[*33·42] &\supset . x \in CʻP &\qquad \text{(1)}\\ +\vdash . (1) . *206·134 . \supset \vdash \colon\ldotp \text{Hp} . \supset : x \text{seq}_{P}\overrightarrow{P}ʻx + . &\equiv . \overrightarrow{P}ʻx \subset \overrightarrow{P}ʻx . \overrightarrow{P}ʻx \subset - pʻ\overleftarrow{P}ʻʻ\overrightarrow{P}ʻx .\\ +[*22·42] &\equiv . \overrightarrow{P}ʻx \subset - pʻ\overleftarrow{P}ʻʻ\overrightarrow{P}ʻx &\qquad \text{(2)}\\ +\vdash . *205·101 . \supset \vdash \colon\ldotp y \in \overrightarrow{\text{max}}_{P}ʻ\overrightarrow{P}ʻx . &\supset : yPx . y {\sim} \in Pʻʻ\overrightarrow{P}ʻx :\\ +[*37·1] &\supset : yPx : zPx . \supset _{z} . {\sim} (yPz) :\\ +[*32·18 . *5·31] &\supset : zPx . \supset _{z} . y \in \overrightarrow{P}ʻx . {\sim} (yPz) .\\ +[*40·53] &\supset _{z} . z {\sim} \in pʻ\overleftarrow{P}ʻʻ\overrightarrow{P}ʻx :\\ +[*32·18] &\supset : \overrightarrow{P}ʻx \subset - pʻ\overleftarrow{P}ʻʻ\overrightarrow{P}ʻx &\qquad \text{(3)}\\ +\vdash . (2) . (3) . &\supset \vdash : \text{Hp} . y \in \overrightarrow{\text{max}}_{P}ʻ\overrightarrow{P}ʻx . \supset . x \text{seq}_{P}\overrightarrow{P}ʻx + : \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_590">[Pg 590]</span></p> + +<p class="nind"><b><a id="*206·52">*206·52</a>.</b> \[\begin{align}\vdash :P\in \text{trans} \cap \text{connex} .\text{E}!\text{max}_{P}ʻ&Pʻʻ\alpha .\supset .\\ +&\text{E}!\text{seq}_{P}ʻPʻʻ\alpha .\text{seq}_{P}ʻPʻʻ\alpha =\text{max}_{P}ʻ\alpha\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*205·7. \supset \vdash :\text{Hp}.&\supset .\text{E}!\text{max}_{P}ʻ\alpha . &\qquad \text{(1)}\\ +[*205·22] &\supset .Pʻʻ\alpha =\overrightarrow{P}ʻ\text{max}_{P}ʻ\alpha &\qquad \text{(2)}\\ +\vdash .(2).*206·51. \supset \vdash :\text{Hp}.&\supset .\text{max}_{P}ʻ\alpha \text{seq}_{P}Pʻʻ\alpha .\\ +[*206·161] &\supset .\text{max}_{P}ʻ\alpha =\text{seq}_{P}ʻPʻʻ\alpha &\qquad \text{(3)}\\ +\vdash .(1).(3).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*206·53">*206·53</a>.</b> \(\vdash :P\in \text{Ser}.\supset .\overrightarrow{\text{seq}} _{P}ʻpʻ\overrightarrow{P}ʻʻ(\alpha \cap CʻP)=\overrightarrow{\text{min}}_{P}ʻ\alpha\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*206·13.\supset \vdash .\overrightarrow{\text{seq}} _{P}ʻpʻ\overrightarrow{P}ʻʻ(\alpha \cap CʻP)&=\overrightarrow{\text{min}}_{P}ʻpʻ\overleftarrow{P}ʻʻ\{pʻ\overrightarrow{P}ʻʻ(\alpha + \cap CʻP)\cap CʻP\}\\ +[*205·15·16.*206·18.*200·54] &=\overrightarrow{\text{min}}_{P}ʻ\{CʻP\cap pʻ\overleftarrow{P}ʻʻpʻ\overrightarrow{P} ʻʻ(\alpha \cap CʻP)\} &\qquad \text{(1)}\\ +\vdash .(1).*204·62. \supset \vdash :\text{Hp}.\supset .\overrightarrow{\text{seq}} _{P}ʻpʻ\overrightarrow{P}ʻʻ(\alpha \cap CʻP )&=\overrightarrow{\text{min}}_{P}ʻ\{(\alpha + \cap CʻP)\cup \breve{P} ʻʻ\alpha\}\\ +[*205·19.*201·52] &=\overrightarrow{\text{min}}_{P}ʻ\alpha :\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*206·531.</b> \[\begin{align}\vdash :P&\in \text{Ser}.\supset .\\ +&CʻP\cap \hat{x} \{pʻ\overrightarrow{P}ʻʻ(\alpha \cap CʻP)=\overrightarrow{P}ʻx\}=\overrightarrow{\text{seq}} _{P}ʻpʻ\overrightarrow{P}ʻʻ(\alpha + \cap CʻP)=\overrightarrow{\text{min}}_{P}ʻ\alpha \end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*206·172.*201·51.\supset \\ +\vdash :\text{Hp}.\supset .&\overrightarrow{\text{seq}} _{P}ʻpʻ\overleftarrow{P}ʻʻ(\alpha + \cap CʻP)=CʻP\cap \hat{x} \{pʻ\overrightarrow{P}ʻʻ(\alpha \cap CʻP)\cap CʻP=\overrightarrow{P}ʻx\} &\qquad \text{(1)}\\ +\vdash .(1).*40·62.&\supset \vdash :\text{Hp}.\exists !(\alpha \cap CʻP).\supset .\\ +&\overrightarrow{\text{seq}} _{P}ʻpʻ\overleftarrow{P}ʻʻ(\alpha \cap CʻP)=CʻP\cap \hat{x} \{pʻ\overrightarrow{P}ʻʻ(\alpha \cap CʻP)=\overrightarrow{P}ʻx\} &\qquad \text{(2)}\\ +\vdash .*205·16.*206·53. &\supset \vdash :\text{Hp}.\alpha \cap CʻP=\Lambda .\supset .\overrightarrow{\text{seq}} _{P}ʻpʻ\overleftarrow{P}ʻʻ(\alpha + \cap CʻP)=\Lambda &\qquad \text{(3)}\\ +\vdash .*40·2.\supset \vdash :\alpha \cap CʻP=\Lambda .\supset . +&CʻP\cap \hat{x} \{pʻ\overrightarrow{P}ʻʻ(\alpha \cap CʻP)=\overrightarrow{P}ʻx\}=CʻP\cap \hat{x} (\text{V}=\overrightarrow{P}ʻx) &\qquad \text{(4)}\\ +\vdash .*50·24. \supset \vdash :\text{Hp}.&\supset .(x).x{\sim}\in \overrightarrow{P}ʻx.\\ +[*24·104] &\supset .(x).\overrightarrow{P}ʻx\neq \text{V} &\qquad \text{(5)}\\ +\vdash .(4).(5). \supset \vdash :\text{Hp}.\alpha \cap CʻP=\Lambda .\supset .CʻP\cap \hat{x} &\{pʻ\overrightarrow{P}ʻʻ(\alpha \cap CʻP)=\overrightarrow{P}ʻx\}=\Lambda \\ +[(3)] &=\overrightarrow{\text{seq}} _{P}ʻpʻ\overrightarrow{P}ʻʻ(\alpha \cap CʻP) &\qquad \text{(6)}\\ +\vdash .(2).(6).*206·53.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*206·54">*206·54</a>.</b> \[\begin{align}\vdash \colon\ldotp P\in \text{Ser}.\supset :&\text{E}!\text{seq}_{P}ʻpʻ\overrightarrow{P}ʻʻ(\alpha \cap CʻP).\equiv .\text{E}!\text{min}_{P}ʻ\alpha + :\\ +&\text{E}!\text{max}_{P}ʻpʻ\overrightarrow{P}ʻʻ(\alpha \cap CʻP).\equiv .\text{E}!\text{prec}_{P}ʻ\alpha\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*206·53. &\supset \vdash \colon\ldotp \text{Hp}.\supset :\text{E}!\text{seq}_{P}ʻpʻ\overrightarrow{P}ʻʻ(\alpha \cap CʻP).\equiv .\text{E}!\text{min}_{P}ʻ\alpha + &\qquad \text{(1)}\\ +\vdash .*206·13·101. &\supset \vdash :\text{E}!\text{max}_{P}ʻpʻ\overrightarrow{P}ʻʻ(\alpha \cap CʻP).\equiv .\text{E}!\text{prec}_{P}ʻ\alpha &\qquad \text{(2)}\\ +\vdash .(1).(2).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*206·55">*206·55</a>.</b> \[\begin{align}\vdash \colon\ldotp &P\in \text{Ser}.\supset :(\alpha ).\alpha \in \text{ᗡ}ʻ\text{min}_{P}\cup \text{ᗡ}ʻ\text{prec}_{P}.\equiv .\\ +&(\alpha ).pʻ\overrightarrow{P}ʻʻ(\alpha \cap CʻP)\in \text{ᗡ}ʻ\text{max}_{P}\cup \text{ᗡ}ʻ\text{seq}_{P} + \quad[*206·54·161.*205·32]\end{align}\]</p> + +<p><span class="pagenum" id="Page_591">[Pg 591]</span></p> + +<p class="nind"><b><a id="*206·551">*206·551</a>.</b> \[\begin{align}\vdash :P\in \text{Ser}.\supset .&\overrightarrow{\text{seq}} _{P}ʻ\alpha = \overrightarrow{\text{seq}} _{P}ʻpʻ\overrightarrow{P}ʻʻpʻ\overleftarrow{P}ʻʻ(\alpha + \cap CʻP).\\ +&\overrightarrow{\text{max}}_{P}ʻ\alpha = \overrightarrow{\text{max}}_{P}ʻpʻ\overrightarrow{P}ʻʻpʻ\overleftarrow{P}ʻʻ(\alpha \cap CʻP)\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*206·13. &\supset \vdash .\overrightarrow{\text{seq}} _{P}ʻ\alpha = \overrightarrow{\text{min}}_{P}ʻpʻ\overleftarrow{P}ʻʻ(\alpha \cap CʻP) &\qquad \text{(1)}\\ +\vdash .(1).*206·53.&\supset \vdash :\text{Hp}.\supset .\overrightarrow{\text{seq}} _{P}ʻ\alpha = \overrightarrow{\text{seq}} _{P}ʻpʻ\overrightarrow{P}ʻʻ{pʻ\overleftarrow{P}ʻʻ(\alpha + \cap CʻP)\cap CʻP} &\qquad \text{(2)}\\ +\vdash .(2).*200·54.&\supset \vdash :\text{Hp}.\dot{\exists} !P.\supset .\overrightarrow{\text{seq}} _{P}ʻ\alpha = \overrightarrow{\text{seq}} _{P}ʻpʻ\overrightarrow{P}ʻʻpʻ\overleftarrow{P}ʻʻ(\alpha + \cap CʻP) &\qquad \text{(3)}\\ +\vdash .*206·18. &\supset \vdash :P = \dot{\Lambda} .\supset .\overrightarrow{\text{seq}} _{P}ʻ\alpha = \Lambda .\overrightarrow{\text{seq}} _{P}ʻpʻ\overrightarrow{P}ʻʻpʻ\overleftarrow{P}ʻʻ(\alpha + \cap CʻP) = \Lambda &\qquad \text{(4)}\\ +\vdash .(3).(4). & \supset \vdash :\text{Hp}.\supset .\overrightarrow{\text{seq}} _{P}ʻ\alpha = \overrightarrow{\text{seq}} _{P}ʻpʻ\overrightarrow{P}ʻʻpʻ\overleftarrow{P}ʻʻ(\alpha + \cap CʻP) &\qquad \text{(5)}\\ +\vdash .*206·53. \supset \vdash :\text{Hp}.\supset .\overrightarrow{\text{max}}_{P}ʻ\alpha &= \overrightarrow{\text{prec}} _{P}ʻpʻ\overleftarrow{P}ʻʻ(\alpha \cap CʻP)\\ +[*206·13·101.*200·54] & = \overrightarrow{\text{max}}_{P}ʻpʻ\overrightarrow{P}ʻʻpʻ\overleftarrow{P}ʻʻ(\alpha \cap CʻP) &\qquad \text{(6)}\\ +\vdash .(5).(6).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*206·56">*206·56</a>.</b> \[\begin{align}\vdash \colon\ldotp P\in \text{Ser}.\supset :(\alpha ).pʻ\overrightarrow{P}ʻʻ(\alpha \cap CʻP)\in &\text{ᗡ}ʻ\text{max}_{P}\cup \text{ᗡ}ʻ\text{seq}_{P}. + \equiv .\\ +&(\alpha ).\alpha \in \text{ᗡ}ʻ\text{max}_{P}\cup \text{ᗡ}ʻ\text{seq}_{P}\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*10·1·11.\supset \vdash :(\alpha ).\alpha \in \text{ᗡ}ʻ\text{max}_{P}&\cup \text{ᗡ}ʻ\text{seq}_{P}.\supset .\\ +&(\alpha ).pʻ\overrightarrow{P}ʻʻ(\alpha \cap CʻP)\in \text{ᗡ}ʻ\text{max}_{P}\cup \text{ᗡ}ʻ\text{seq}_{P} &\qquad \text{(1)}\\ +\vdash .*10·1. \supset \vdash :(\alpha ).&pʻ\overrightarrow{P}ʻʻ(\alpha \cap CʻP)\in \text{ᗡ}ʻ\text{max}_{P}\cup \text{ᗡ}ʻ\text{seq}_{P}.\supset .\\ +&pʻ\overrightarrow{P}ʻʻpʻ\overleftarrow{P}ʻʻ(\beta \cap CʻP)\in \text{ᗡ}ʻ\text{max}_{P}\cup \text{ᗡ}ʻ\text{seq}_{P}.\\ +[*206·551] &\supset .\beta \in \text{ᗡ}ʻ\text{max}_{P}\cup \text{ᗡ}ʻ\text{seq}_{P} &\qquad \text{(2)}\\ +\vdash .(1).(2).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*206·57">*206·57</a>.</b> \[\begin{align}\vdash \colon\ldotp P\in \text{Ser}.\supset :&(\alpha ).\alpha \in \text{ᗡ}ʻ\text{min}_{P}\cup \text{ᗡ}ʻ\text{prec}_{P}. \equiv .\\ +&(\alpha ).\alpha \in \text{ᗡ}ʻ\text{max}_{P}\cup \text{ᗡ}ʻ\text{seq}_{P} \quad[*206·55·56]\end{align}\]</p> + +<p>This proposition is important, since it shows that when a serial +relation satisfies Dedekind's axiom, so does its converse. Thus if all +classes which have no maximum have an upper limit, then all classes +which have no minimum have a lower limit, and vice versa.</p> + +<p class="nind"><b><a id="*206·6">*206·6</a>.</b> \(\vdash :S\in P \overline{\,\text{smor}\,} Q.\supset .pʻ\overleftarrow{P}ʻʻ(\alpha \cap CʻP) = Sʻʻpʻ\overleftarrow{Q}ʻʻ\breve{S} ʻʻ\alpha\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*151·11.\supset \vdash :\text{Hp}.\supset .pʻ\overleftarrow{P}ʻʻ(\alpha \cap CʻP) &= pʻSʻʻʻ\breve{Q} ʻʻʻ\overleftarrow{S}ʻʻ(\alpha \cap \text{D}ʻS)\\ +[*72·341] &= Sʻʻpʻ\breve{Q} ʻʻʻ\overleftarrow{S}ʻʻ(\alpha \cap \text{D}ʻS)\\ +[*71·613] & = Sʻʻpʻ\overleftarrow{Q}ʻʻ\breve{S} ʻʻ\alpha :\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_592">[Pg 592]</span></p> + +<p class="nind"><b>*206·61.</b> \(\vdash :S\in P\,\overline{\,\text{smor}\,}\,Q.\supset .\overrightarrow{\text{seq}} _{P}ʻ\alpha =Sʻʻ\overrightarrow{\text{seq}} _{Q}ʻ\breve{S} ʻʻ\alpha\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*205·8.*206·6·13.\supset \vdash :\text{Hp}.\supset .\overrightarrow{\text{seq}} _{P}ʻ\alpha &=Sʻʻ\overrightarrow{\text{min}}_{Q}ʻ\breve{S} + ʻʻSʻʻpʻ\overleftarrow{Q}ʻʻ\breve{S} ʻʻ\alpha\\ +[*72·501.*151·11] &=Sʻʻ\overrightarrow{\text{min}}_{Q}ʻ(pʻ\overleftarrow{Q}ʻʻ\breve{S} ʻʻ\alpha \cap CʻQ)\\ +[*206·13.*205·15] &=Sʻʻ\overrightarrow{\text{seq}} _{Q}ʻ\breve{S} ʻʻ\alpha :\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*206·62.</b> \[\begin{align}&\vdash \colon\ldotp S\in P\,\overline{\,\text{smor}\,}\,Q.\supset :\text{E}!\text{seq}_{P}ʻ\alpha .\equiv .\text{E}!\text{seq}_{Q}ʻ\breve{S} + ʻʻ\alpha \\ +&[*206·61.*73·22·44.*53·3]\end{align}\]</p> + +<p class="nind"><b><a id="*206·63">*206·63</a>.</b> \[\begin{align}&\vdash :S\in P\,\overline{\,\text{smor}\,}\,Q.\text{E}!\text{seq}_{P}ʻ\alpha .\supset .\text{seq}_{P}ʻ\alpha =Sʻ\text{seq}_{Q}ʻ\breve{S} + ʻʻ\alpha \\ +&[*206·61·62.*53·31]\end{align}\]</p> + +<p class="nind"><b><a id="*206·7">*206·7</a>.</b> \[\begin{align}\vdash :P\in \text{trans}.\beta \subset CʻP.{\sim}(yPy).y{\sim}\in &\overrightarrow{\text{max}}_{P}ʻ\beta .\supset .\\ +&pʻ\overleftarrow{P}ʻʻ\beta =pʻ\overleftarrow{P}ʻʻ(\beta -{℩}ʻy)\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*51·222. \supset \vdash :y{\sim}\in \beta .&\supset .pʻ\overleftarrow{P}ʻʻ\beta =pʻ\overleftarrow{P}ʻʻ(\beta -{℩}ʻy) &\qquad \text{(1)}\\ +\vdash .*205·111.\supset \vdash \colon\ldotp \text{Hp}.y\in \beta .&\supset :y\in Pʻʻ\beta .{\sim}(yPy):\\ +[*37·1] &\supset :(\exists x).x\in \beta -{℩}ʻy.yPx:\\ +[*10·56.\text{Hp}] &\supset :z\in pʻ\overleftarrow{P}ʻʻ(\beta -{℩}ʻy).\supset .yPz:\\ +[*53·14.*51·221] &\supset :pʻ\overleftarrow{P}ʻʻ(\beta -{℩}ʻy)\subset pʻ\overleftarrow{P}ʻʻ\beta :\\ +[*40·16] &\supset :pʻ\overleftarrow{P}ʻʻ(\beta -{℩}ʻy)=pʻ\overleftarrow{P}ʻʻ\beta &\qquad \text{(2)}\\ +\vdash .(1).(2).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*206·71.</b> \(\vdash :P\in \text{trans}.\beta \subset CʻP.{\sim}(yPy).y{\sim}\in \overrightarrow{\text{max}}_{P}ʻ\beta .\supset .\overrightarrow{\text{seq}} _{P}ʻ\beta + =\overrightarrow{\text{seq}} _{P}ʻ(\beta -{℩}ʻy)\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*51·222. \supset \vdash :y{\sim}\in \beta .&\supset .\overrightarrow{\text{seq}} _{P}ʻ\beta =\overrightarrow{\text{seq}} _{P}ʻ(\beta -{℩}ʻy) &\qquad \text{(1)}\\ +\vdash .*205·111.\supset \vdash :\text{Hp}.y\in \beta .&\supset .y\in Pʻʻ\beta .{\sim}(yPy).\\ +[*37·1] & \supset .(\exists z).z\in \beta -{℩}ʻy.yPz &\qquad \text{(2)}\\ +\vdash .(2).*10·56.*201·1.&\supset \vdash :\text{Hp}.y\in \beta .\beta -{℩}ʻy\subset \overrightarrow{P}ʻx.\supset .yPx.\\ +[*32·18] &\supset .\beta \subset \overrightarrow{P}ʻx &\qquad \text{(3)}\\ +\vdash .(3).*206·7.&\supset \vdash \colon\ldotp \text{Hp}(2).\supset :\\ +&\beta \subset \overrightarrow{P}ʻx.\overrightarrow{P}ʻx\subset -pʻ\overrightarrow{P}ʻʻ\beta .\equiv .\beta -{℩}ʻy\subset \overrightarrow{P}ʻx.\overrightarrow{P}ʻx\subset -pʻ\overrightarrow{P}ʻʻ(\beta -{℩}ʻy):\\ +[*206·134]&\supset :x\text{seq}_{P}\beta .\equiv .x\text{seq}_{P}(\beta -{℩}ʻy) &\qquad \text{(4)}\\ +\vdash .(1).(4).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*206·72">*206·72</a>.</b> \(\vdash :P\in \text{trans}.{\sim}(yPy).y{\sim}\in \overrightarrow{\text{max}}_{P}ʻ\beta .\supset .\overrightarrow{\text{seq}} _{P}ʻ\beta + =\overrightarrow{\text{seq}} _{P}ʻ(\beta -{℩}ʻy)\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*206·71·131.*205·151.\supset \vdash :\text{Hp}.\supset .\overrightarrow{\text{seq}} _{P}ʻ\beta &=\overrightarrow{\text{seq}} _{P}ʻ(\beta \cap CʻP-{℩}ʻy)\\ +[*206·131] & =\overrightarrow{\text{seq}} _{P}ʻ(\beta -{℩}ʻy):\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_593">[Pg 593]</span></p> + +<p class="nind"><b><a id="*206·73">*206·73</a>.</b> \(\vdash :\exists !\gamma \cap CʻP.\text{E}!\text{prec}_{P}ʻ\gamma .\text{E}!\text{seq}_{P}ʻ\gamma .\supset .\text{prec}_{P}ʻ\gamma + P^{2}\text{seq}_{P}ʻ\gamma\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*206·211.\supset \vdash :\text{Hp}.&\supset .\gamma \cap CʻP\subset \overrightarrow{P}ʻ\text{seq}_{P}ʻ\gamma \cap \overleftarrow{P}ʻ\text{prec}_{P}ʻ\gamma + .\exists !\gamma \cap CʻP.\\ +[*34·11] &\supset .\text{prec}_{P}ʻ\gamma P^{2}\text{seq}_{P}ʻ\gamma :\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*206·731.</b> \(\vdash \colon\ldotp \exists !\gamma \cap CʻP:P\in \text{trans}.\lor.P^{2}\,\unicode{x2abd}\, J:\supset .{\sim}(\text{prec}_{P}ʻ\gamma =\text{seq}_{P}ʻ\gamma + )\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*206·73.\supset \\ +\vdash :\exists !\gamma \cap CʻP.\text{E}!\text{prec}_{P}ʻ\gamma .\text{E}!\text{seq}_{P}ʻ\gamma .P\in \text{trans}.&\supset .\text{prec}_{P}ʻ\gamma P\text{seq}_{P}ʻ\gamma + .\\ +[*206·133] &\supset .\text{prec}_{P}ʻ\gamma \neq \text{seq}_{P}ʻ\gamma &\qquad \text{(1)}\\ +\vdash .*206·73.\supset \\ +\vdash :\exists !\gamma \cap CʻP.\text{E}!\text{prec}_{P}ʻ\gamma .\text{E}!\text{seq}_{P}ʻ\gamma .P^{2}\,\unicode{x2abd}\, J.&\supset .\text{prec}_{P}ʻ\gamma + \neq \text{seq}_{P}ʻ\gamma &\qquad \text{(2)}\\ +\vdash .*14·21.\supset \vdash :{\sim}(\text{E}!\text{prec}_{P}ʻ\gamma .\text{E}!\text{seq}_{P}ʻ\gamma ).\supset .&{\sim}(\text{prec}_{P}ʻ\gamma =\text{seq}_{P}ʻ\gamma + ) &\qquad \text{(3)}\\ +\vdash .(1).(2).(3).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p>Note that "\(\text{prec}_{P}ʻ\gamma \neq \text{seq}_{P}ʻ\gamma\)" +is not the same proposition as \({\sim}(\text{prec}_{P}ʻ\gamma=\text{seq}_{P}ʻ\gamma)\). +The former involves \(\text{E}!\text{prec}_{P}ʻ\gamma.\text{E}!\text{seq}_{P}ʻ\gamma\), +while the latter does not, in virtue of the conventions as to +descriptive symbols explained in *14.</p> + +<p class="nind"><b><a id="*206·732">*206·732</a>.</b> \(\vdash \colon\ldotp P\in \text{trans}.\lor.P^{2}\,\unicode{x2abd}\, J:\supset .{\sim}(\text{prec}_{P}ʻ\gamma =\text{seq}_{P}ʻ\gamma )\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*206·14.\supset \vdash :\gamma \cap CʻP=\Lambda .&\supset .\overrightarrow{\text{prec}} _{P}ʻ\gamma =\overrightarrow{B}ʻ\breve{P} .\text{seq}_{P}ʻ\gamma + =\overrightarrow{B}ʻP.\\ +[*93·101] &\supset .\overrightarrow{\text{prec}} _{P}ʻ\gamma \cap \overrightarrow{\text{seq}} _{P}ʻ\gamma =\Lambda .\\ +[*53·4] &\supset .{\sim}(\text{prec}_{P}ʻ\gamma =\text{seq}_{P}ʻ\gamma ) &\qquad \text{(1)}\\ +\vdash .(1).*206·731.\supset \vdash .\text{Prop} +\end{array} +\]</p> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_594">[Pg 594]</span></p> +<h2 class="nobreak" id="*207">*207. LIMITS.</h2> +</div> + + +<p class="nind"><i>Summary of</i> *207.</p> + +<p>A term \(x\) is said to be the "upper limit" of \(\alpha\) in \(P\) +if \(\alpha\) has no maximum and \(x\) is the sequent of \(\alpha\). +In this case, \(x\) immediately follows the class \(\alpha\), though +there is no one member of \(\alpha\) which \(x\) immediately follows. +Sequents which are limits have special importance, and it is convenient +to have a special notation for them. We write "\(\text{lt}_Pʻ\alpha\)" +for the upper limit of \(\alpha\); or, if it is more convenient, +"\(\text{lt}(P)ʻ\alpha\)". (This is more convenient when \(P\) is +replaced by an expression consisting of several letters, or by a letter +with a suffix.) The <i>lower</i> limit of \(\alpha\) will be the +immediate predecessor of \(\alpha\) when \(\alpha\) has no minimum; +this we denote by \(\text{tl}_Pʻ\alpha\).</p> + +<p>The following propositions on limits for the most part follow +immediately from the propositions of <a href="#*206">*206</a> on sequents.</p> + +<p>Our definition is so framed that the limit of the null-class is +the first member of our series (if any). This departure from +usage is convenient in order that, whenever our series contains +any limiting point in the ordinary sense, the <i>series</i> +of limiting points may exist, <i>i.e.</i> in order that +\(P\unicode{x0294f}\text{D}ʻ\text{lt}_P\) may exist whenever there +are existent parts of CʻP which have upper limits. The series +\(P\unicode{x0294f}\text{D}ʻ\text{lt}_P\) is the "first derivative" of +\(P\). The definition of a limit is +\[ +\text{lt}_P=\text{seq}_{P}\upharpoonright (-\text{ᗡ}ʻ\text{max}_{P}) \quad\text{Df}. +\]</p> + +<p>Besides the limit, we require, for many purposes, a single notation for +the "limit or maximum." This we denote by "\(\text{limax}_P\)," putting +\[ +\text{limax}_P=\text{max}_{P}\unicode{x228d} \text{lt}_P \quad\text{Df}. +\] +Similarly for the lower limit or minimum we use "\(\text{limin}_{P}\)," +putting +\[ +\text{limin}_{P}=\text{min}_{P}\unicode{x228d} \text{tl}_P \qquad\text{Df}. +\] +We have \(\text{tl}_P=\text{lt}(\breve{P})\) (<a href="#*207·101">*207·101</a>) and +\(\text{limin}_{P}=\text{limax}(\breve{P})\) (<a href="#*207·401">*207·401</a>). Hence it is +unnecessary to prove propositions concerning lower limits, since they +result immediately from propositions concerning upper limits.</p> + +<p>In virtue of our definition of a limit, \(x\) limits \(\alpha\) +if \(x\) is a sequent of \(\alpha\) and \(\alpha\) has no maximum +(<a href="#*207·1">*207·1</a>). Thus if \(\alpha\) has a maximum, it has no limit (<a href="#*207·11">*207·11</a>), +but if it has no maximum, the class of its limits is the class of +its sequents (<a href="#*207·12">*207·12</a>). Thus the existence of the class of limits is +equivalent<span class="pagenum" id="Page_595">[Pg 595]</span> to the existence of the class of sequents combined with the +non-existence of the class of maxima, <i>i.e.</i></p> + +<p class="nind"><b>*207·13.</b> \(\vdash :\exists !\overrightarrow{\text{lt}} _{P}ʻ\alpha .\equiv .{\sim}\exists !\overrightarrow{\text{max}}_{P}ʻ\alpha + .\exists !\overrightarrow{\text{seq}} _{P}ʻ\alpha\)</p> + +<p><a href="#*207·2">*207·2</a>—<a href="#*207·232">·232</a> consist of various formulae for \(\overrightarrow{\text{lt}}_Pʻ\alpha\). We have</p> + +<p class="nind"><b>*207·2.</b> \(\vdash :P\in \text{connex} .x\,\text{lt}_P\,\alpha .\supset .\alpha \cap CʻP\subset \overrightarrow{P}ʻx.\overrightarrow{P}ʻx\subset Pʻʻ\alpha\)</p> + +<p><i>I.e.</i> the whole of \(\alpha \cap CʻP\) precedes \(x\), but any +predecessor of \(x\) precedes some member of \(\alpha\).</p> + +<p class="nind"><b>*207·231.</b> \(\vdash :P\in \text{Ser}.\exists !\overrightarrow{\text{lt}} _{P}ʻ\alpha .\supset .\overrightarrow{\text{lt}} _{P}ʻ\alpha + =CʻP\cap \hat{x} (\overrightarrow{P}ʻx=Pʻʻ\alpha )\)</p> + +<p><i>I.e.</i> the limit of \(\alpha\), if it exists, is the term whose +predecessors are identical with the predecessors of some part of +\(\alpha\).</p> + +<p>We have also</p> + +<p class="nind"><b>*207·232.</b> \(\vdash \colon\ldotp P\in \text{Ser}.\supset :x=\text{lt}_Pʻ\alpha .\equiv .x\in CʻP-\alpha .\overrightarrow{P}ʻx=Pʻʻ\alpha\)</p> + +<p>This proposition should be compared with <a href="#*205·54">*205·54</a>, which (slightly +re-written) is +\[ +\vdash \colon\ldotp P\in \text{Ser}.\supset :x=\text{max}_{P}ʻ\alpha .\equiv .x\in CʻP\cap \alpha .\overrightarrow{P}ʻx=Pʻʻ\alpha +\]</p> + +<p>From the two together we arrive at</p> + +<p class="nind"><b>*207·51.</b> \(\vdash \colon\ldotp P\in \text{Ser}.\supset :x=\text{limax}_Pʻ\alpha .\equiv .x\in CʻP.\overrightarrow{P}ʻx=Pʻʻ\alpha\)</p> + +<p class="nind">which serves to illustrate the utility of "\(\text{limax}_P\)."</p> + +<p>We have</p> + +<p class="nind"><b>*207·24.</b> \(\vdash :P\in \text{connex} .\supset .\overrightarrow{\text{lt}} _{P}ʻ\alpha \in 0\cup 1.\text{lt}_P\in 1\rightarrow \text{Cls}\)</p> + +<p><i>I.e.</i> if \(P\) is connected, a class cannot have more than one +limit; also</p> + +<p class="nind"><b>*207·25.</b> \(\vdash :P\in \text{trans}.\beta \subset Pʻʻ\alpha .\supset .\overrightarrow{\text{lt}} _{P}ʻ(\alpha \cup \beta )=\overrightarrow{\text{lt}} _{P}ʻ\alpha\)</p> + +<p><i>I.e.</i> any terms which have some \(\alpha\)'s beyond them may be +added to \(\alpha\) without altering the limit.</p> + +<p>We next have a set of propositions (<a href="#*207·251">*207·251</a>—<a href="#*207·27">·27</a>) proving that if a +class has a limit, any single term of the class may be removed without +altering the limit (<a href="#*207·261">*207·261</a>), and that in any case, provided the +class is not a unit class, its minimum (if any) may be removed without +altering the limit (<a href="#*207·27">*207·27</a>). We then prove (<a href="#*207·291">*207·291</a>) that if \(P\) is +a series, and \(\alpha\) is a class which has a limit, the predecessors of +the limit are the class \(P_{*}ʻʻ\alpha\).</p> + +<p>We then have a set of propositions (<a href="#*207·3">*207·3</a>—<a href="#*207·36">·36</a>) on the limit of +\(\overrightarrow{P}ʻx\) and kindred matters. If \(x\) has no immediate +predecessor, the limit of \(\overrightarrow{P}ʻx\) is \(x\), and vice +versa (<a href="#*207·32">*207·32</a><a href="#*207·33">·33</a>). Hence</p> + +<p class="nind"><b>*207·35.</b> \(\vdash :P\in \text{Rl}ʻJ\cap \text{connex} .\supset .\text{D}ʻ\text{lt}_P=CʻP-\text{ᗡ}ʻ(P\dot{-} P^{2})\)</p> + +<p><i>I.e.</i> the limit-points of \(P\) are those which have no immediate +predecessors.</p> + +<p><span class="pagenum" id="Page_596">[Pg 596]</span></p> + +<p>We next turn our attention to "\(\text{limax}_P\)." This again is +one-many, provided \(P\) is connected (<a href="#*207·41">*207·41</a>). We have by the +definition</p> + +<p class="nind"><b>*207·42.</b> \(\vdash :\exists !\overrightarrow{\text{max}}_{P}ʻ\alpha .\supset .\overrightarrow{\text{limax}} _{P}ʻ\alpha =\overrightarrow{\text{max}}_{P}ʻ\alpha\)</p> + +<p class="nind"><b>*207·43.</b> \(\vdash :\overrightarrow{\text{max}}_{P}ʻ\alpha =\Lambda .\supset .\overrightarrow{\text{limax}} _{P}ʻ\alpha =\overrightarrow{\text{seq}} _{P}ʻ\alpha + =\overrightarrow{\text{lt}} _{P}ʻ\alpha\)</p> + +<p class="nind"><b>*207·44.</b> \(\vdash .\text{ᗡ}ʻ\text{limax}_P=\text{ᗡ}ʻ\text{max}_{P}\cup \text{ᗡ}ʻ\text{lt}_P=\text{ᗡ}ʻ\text{max}_{P}\cup \text{ᗡ}ʻ\text{seq}_{P}\)</p> + +<p class="nind"><b>*207·45.</b> \(\vdash .\overrightarrow{\text{limax}} _{P}ʻ\alpha =\overrightarrow{\text{max}}_{P}ʻ\alpha \cup \overrightarrow{\text{lt}} _{P}ʻ\alpha\)</p> + +<p>Also we have</p> + +<p class="nind"><b>*207·46.</b> \(\vdash \colon\ldotp x=\text{limax}_Pʻ\alpha .\equiv :x=\text{max}_{P}ʻ\alpha .\lor.x=\text{lt}_Pʻ\alpha\)</p> + +<p class="nind">which is a very useful proposition, as is also <a href="#*207·51">*207·51</a> (given above).</p> + +<p>A useful proposition in dealing with classes of classes contained in a +series is</p> + +<p class="nind"><b>*207·54.</b> \(\vdash :P\in \text{Ser}.\kappa \subset \text{ᗡ}ʻ\text{lt}_P.\supset .\overrightarrow{\text{limax}} _{P}ʻ\text{lt}_Pʻʻ\kappa =\overrightarrow{\text{limax}} _{P}ʻsʻ\kappa + =\overrightarrow{\text{lt}} _{P}ʻsʻ\kappa\)</p> + +<p><i>I.e.</i> if every member of \(\kappa\) has a limit, the limit or +maximum (if any) of the limits is the limit or maximum, and in fact the +limit, of \(sʻ\kappa\).</p> + +<p>We have next a set of propositions (<a href="#*207·6">*207·6</a>—<a href="#*207·66">·66</a>) on correlations, +proving that the limit, or the \(\text{limax}\), of the correlates is +the correlate of the limit or \(\text{limax}\), <i>i.e.</i></p> + +<p class="nind"><b>*207·6.</b> \(\vdash :S\in P\,\overline{\,\text{smor}\,}\,Q.\supset .\overrightarrow{\text{lt}} _{P}ʻ\alpha =Sʻʻ\overrightarrow{\text{lt}} _{Q}ʻ\breve{S} ʻʻ\alpha\)</p> + +<p class="nind"><b>*207·64.</b> \(\vdash :S\in P\,\overline{\,\text{smor}\,}\,Q.\supset .\overrightarrow{\text{limax}} _{P}ʻ\alpha =Sʻʻ\overrightarrow{\text{limax}} _{Q}ʻ\breve{S} ʻʻ\alpha\)</p> + +<p>The last three propositions (<a href="#*207·7">*207·7</a>—<a href="#*207·72">·72</a>) are lemmas for use in the +theory of stretches (<a href="#*215·5">*215·5</a><a href="#*215·51">·51</a>).</p> + +<hr class="tb"> + +<p class="nind"><b>*207·01.</b> \(\text{lt}_P=\text{lt}(P)=\text{seq}_{P}\upharpoonright (-\text{ᗡ}ʻ\text{max}_{P}) \quad\text{Df}\)</p> + +<p class="nind"><b>*207·02.</b> \(\text{tl}_P=tl(P)=\text{prec}_{P}\upharpoonright (-\text{ᗡ}ʻ\text{min}_{P}) \quad\text{Df}\)</p> + +<p class="nind"><b>*207·03.</b> \(\text{limax}_P=\text{max}_{P}\unicode{x228d} \text{lt}_P \quad\text{Df}\)</p> + +<p class="nind"><b>*207·04.</b> \(\text{limin}_P=\text{min}_{P}\unicode{x228d} \text{tl}_P \quad\text{Df}\)</p> + +<p class="nind"><b><a id="*207·1">*207·1</a>.</b> \(\vdash :x\,\text{lt}_P\,\alpha .\equiv .x\text{seq}_{P}\alpha .{\sim}\exists !\overrightarrow{\text{max}}_{P}ʻ\alpha \quad[(*207·01)]\)</p> + +<p class="nind"><b><a id="*207·101">*207·101</a>.</b> \(\vdash .\text{tl}_P=\text{lt}(\breve{P} ) \quad[*205·102.*206·101.(*207·02)]\)</p> + +<p>We shall not give further propositions on lower limits, unless for some +special reason, since all of them result from propositions on upper +limits by means of <a href="#*207·101">*207·101</a>.</p> + +<p class="nind"><b><a id="*207·11">*207·11</a>.</b> \(\vdash :\exists !\overrightarrow{\text{max}}_{P}ʻ\alpha .\supset .\overrightarrow{\text{lt}} _{P}ʻ\alpha =\Lambda \quad[*207·1]\)</p> + +<p><span class="pagenum" id="Page_597">[Pg 597]</span></p> + +<p class="nind"><b><a id="*207·12">*207·12</a>.</b> \(\vdash :\overrightarrow{\text{max}}_{P}ʻ\alpha =\Lambda .\supset .\overrightarrow{\text{lt}} _{P}ʻ\alpha =\overrightarrow{\text{seq}} _{P}ʻ\alpha + \quad[*207·1]\)</p> + +<p class="nind"><b>*207·121.</b> \(\vdash :\alpha \cap CʻP\subset Pʻʻ\alpha .\supset .\overrightarrow{\text{lt}} _{P}ʻ\alpha =\overrightarrow{\text{seq}} _{P}ʻ\alpha \quad[*207·12.*205·123]\)</p> + +<p class="nind"><b>*207·13.</b> \(\vdash :\exists !\overrightarrow{\text{lt}} _{P}ʻ\alpha .\equiv .{\sim}\exists !\overrightarrow{\text{max}}_{P}ʻ\alpha + .\exists !\overrightarrow{\text{seq}} _{P}ʻ\alpha \quad[*207·1]\)</p> + +<p class="nind"><b>*207·14.</b> \[\begin{align}&\vdash \colon\ldotp \exists !\overrightarrow{\text{max}}_{P}ʻ\alpha .\lor.\exists !\overrightarrow{\text{seq}} _{P}ʻ\alpha + :\equiv :\exists !\overrightarrow{\text{max}}_{P}ʻ\alpha .\lor.\exists !\overrightarrow{\text{lt}} _{P}ʻ\alpha \\ +&[*207·13.*5·63]\end{align}\]</p> + +<p>The above proposition is important because +\[ +(\alpha ):\exists !\overrightarrow{\text{max}}_{P}ʻ\alpha .\lor.\exists !\overrightarrow{\text{lt}} _{P}ʻ\alpha +\] +is the characteristic of "Dedekindian" series, <i>i.e.</i> of such as +fulfil Dedekind's axiom.</p> + +<p class="nind"><b>*207·15.</b> + \[\begin{align}&\vdash :x\,\text{lt}_P\,\alpha .\equiv .x\in CʻP.\alpha \cap CʻP\subset Pʻʻ\alpha \cap \overrightarrow{P}ʻx.\overrightarrow{P}ʻx\subset -pʻ\overleftarrow{P}ʻʻ(\alpha \cap CʻP)\\ +&[*207·1.*205·123.*206·134]\end{align}\]</p> + +<p class="nind"><b>*207·16.</b> \(\vdash .\overrightarrow{\text{lt}} _{P}ʻ\alpha =\overrightarrow{\text{lt}} _{P}ʻ(\alpha \cap CʻP) \quad[*207·15.*37·265]\)</p> + +<p class="nind"><b>*207·17.</b> \(\vdash .\overrightarrow{\text{lt}} _{P}ʻ\Lambda =\overrightarrow{B}ʻP \quad[*207·12.*205·161.*206·14]\)</p> + +<p class="nind"><b>*207·18.</b> \(\vdash :\text{ᗡ}ʻP\subset \text{D}ʻ\text{lt}_P.\equiv .CʻP=\text{D}ʻ\text{lt}_P\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*207·17.\supset \vdash :\text{ᗡ}ʻP\subset \text{D}ʻ\text{lt}_P.&\equiv .\text{ᗡ}ʻP\cup \overrightarrow{B}ʻP\subset \text{D}ʻ\text{lt}_P.\\ +[*93·103] &\equiv .CʻP\subset \text{D}ʻ\text{lt}_P.\\ +[*207·15] &\equiv .CʻP=\text{D}ʻ\text{lt}_P:\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*207·2">*207·2</a>.</b> \[\begin{align}&\vdash :P\in \text{connex} .x\,\text{lt}_P\,\alpha .\supset .\alpha \cap CʻP\subset \overrightarrow{P}ʻx.\overrightarrow{P}ʻx\subset Pʻʻ\alpha \\ +&[*207·15.*202·503]\end{align}\]</p> + +<p class="nind"><b>*207·21.</b> \(\vdash :P^{2}\,\unicode{x2abd}\, J.x\in CʻP.\alpha \cap CʻP\subset \overrightarrow{P}ʻx.\overrightarrow{P}ʻx\subset Pʻʻ\alpha .\supset .x\,\text{lt}_P\,\alpha \)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*200·53.&\supset \vdash :P^{2}\,\unicode{x2abd}\, J.\supset .Pʻʻ\alpha \subset -pʻ\overleftarrow{P}ʻʻ(\alpha \cap CʻP) &\qquad \text{(1)}\\ +\vdash .(1). \supset \vdash :\text{Hp}.&\supset .x\in CʻP.\alpha \cap CʻP\subset \overrightarrow{P}ʻx.\overrightarrow{P}ʻx\subset -pʻ\overrightarrow{P}ʻʻ(\alpha \cap CʻP).\\ +[*206·134] & \supset .x\text{seq}_{P}\alpha &\qquad \text{(2)}\\ +\vdash .*22·44. \supset \vdash :\text{Hp}.&\supset .\alpha \cap CʻP\subset Pʻʻ\alpha .\\ +[*205·123] &\supset .\overrightarrow{\text{max}}_{P}ʻ\alpha =\Lambda &\qquad \text{(3)}\\ +\vdash .(2).(3).*207·1.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*207·22.</b> \[\begin{align}&\vdash :P\in \text{connex} .P^{2}\,\unicode{x2abd}\, J.\supset .\overrightarrow{\text{lt}} _{P}ʻ\alpha + =CʻP\cap \hat{x} (\alpha \cap CʻP\subset \overrightarrow{P}ʻx.\overrightarrow{P}ʻx\subset Pʻʻ\alpha )\\ +&[*207·2·21]\end{align}\]</p> + +<p><span class="pagenum" id="Page_598">[Pg 598]</span></p> + +<p>This is very often the most convenient form for +\(\overrightarrow{\text{lt}} _{P}\alpha\). It states that a limit of +\(\alpha\) is a member \(x\) of \(CʻP\) such that \(\alpha \cap CʻP\) +wholly precedes \(x\), but every predecessor of \(x\) precedes some +member of \(\alpha\).</p> + +<p class="nind"><b>*207·23.</b> \(\vdash :P\in \text{Ser}.\supset .\overrightarrow{\text{lt}} _{P}ʻ\alpha =CʻP\cap \hat{x} (\overrightarrow{P}ʻx=Pʻʻ\alpha .\alpha \cap CʻP\subset Pʻʻ\alpha )\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*13·12.*22·42.\supset \\ +\vdash :\overrightarrow{P}ʻx=Pʻʻ\alpha .\alpha \cap CʻP\subset Pʻʻ\alpha .&\supset .\alpha \cap CʻP\subset \overrightarrow{P}ʻx.\overrightarrow{P}ʻx\subset Pʻʻ\alpha &\qquad \text{(1)}\\ +\vdash .*201·501.*37·265.&\supset \vdash \colon\ldotp P\in \text{trans}.\supset :\alpha \cap CʻP\subset \overrightarrow{P}ʻx.\supset .Pʻʻ\alpha \subset \overrightarrow{P}ʻx:\\ +[\text{Fact}] &\supset :\alpha \cap CʻP\subset \overrightarrow{P}ʻx.\overrightarrow{P}ʻx\subset Pʻʻ\alpha .\supset .\overrightarrow{P}ʻx=Pʻʻ\alpha &\qquad \text{(2)}\\ +\vdash .*22·44.&\supset \vdash :\alpha \cap CʻP\subset \overrightarrow{P}ʻx.\overrightarrow{P}ʻx\subset Pʻʻ\alpha .\supset .\alpha \cap CʻP\subset Pʻʻ\alpha &\qquad \text{(3)}\\ +\vdash .(1).(2).(3).\supset \\ +\vdash \colon\ldotp P\in \text{trans}.&\supset :\alpha \cap CʻP\subset \overrightarrow{P}ʻx.\overrightarrow{P}ʻx\subset Pʻʻ\alpha .\equiv .\overrightarrow{P}ʻx=Pʻʻ\alpha .\alpha \cap CʻP\subset Pʻʻ\alpha &\qquad \text{(4)}\\ +\vdash .(4).*207·22.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*207·231.</b> \(\vdash :P\in \text{Ser}.\exists !\overrightarrow{\text{lt}} _{P}ʻ\alpha .\supset .\overrightarrow{\text{lt}} _{P}ʻ\alpha + =CʻP\cap \hat{x} (\overrightarrow{P}ʻx=Pʻʻ\alpha ) \quad[*207·23]\)</p> + +<p class="nind"><b><a id="*207·232">*207·232</a>.</b> \[\begin{align}&\vdash \colon\ldotp P\in \text{Ser}.\supset :x=\text{lt}_Pʻ\alpha .\equiv .x\in CʻP-\alpha .\overrightarrow{P}ʻx=Pʻʻ\alpha \\ +&[*206·28.*207·1]\end{align}\]</p> + +<p class="nind"><b>*207·24.</b> \(\vdash :P\in \text{connex} .\supset .\overrightarrow{\text{lt}} _{P}ʻ\alpha \in 0\cup 1.\text{lt}_P\in 1\rightarrow \text{Cls}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*206·161.*71·26.(*207·01).\supset \vdash :\text{Hp}.&\supset .\text{lt}_P\in 1\rightarrow \text{Cls}. &\qquad \text{(1)}\\ +[*71·12] &\supset .\text{lt}_Pʻ\alpha \in 0\cup 1 &\qquad \text{(2)}\\ +\vdash .(1).(2).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*207·25.</b> \(\vdash :P\in \text{trans}.\beta \subset Pʻʻ\alpha .\supset .\overrightarrow{\text{lt}} _{P}ʻ(\alpha \cup \beta )=\overrightarrow{\text{lt}} _{P}ʻ\alpha\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*205·193. &\supset \vdash :\text{Hp}.\exists !\overrightarrow{\text{max}}_{P}ʻ\alpha .\supset .\exists !\overrightarrow{\text{max}}_{P}ʻ(\alpha + \cup \beta ) &\qquad \text{(1)}\\ +\vdash .(1).*207·11.&\supset \vdash :\text{Hp}.\exists !\overrightarrow{\text{max}}_{P}ʻ\alpha .\supset .\overrightarrow{\text{lt}} _{P}ʻ\alpha + =\Lambda .\overrightarrow{\text{lt}} _{P}ʻ(\alpha \cup \beta )=\Lambda &\qquad \text{(2)}\\ +\vdash .*205·193.*207*12.\supset \\ +\vdash :\text{Hp}.\overrightarrow{\text{max}}_{P}ʻ\alpha =\Lambda .&\supset .\overrightarrow{\text{lt}} _{P}ʻ\alpha =\overrightarrow{\text{seq}} _{P}ʻ\alpha + .\overrightarrow{\text{lt}} _{P}ʻ(\alpha \cup \beta )=\overrightarrow{\text{seq}} _{P}ʻ(\alpha \cup \beta ).\\ +[*206·24] &\supset .\overrightarrow{\text{lt}} _{P}ʻ\alpha =\overrightarrow{\text{lt}} _{P}ʻ(\alpha \cup \beta ) &\qquad \text{(3)}\\ +\vdash .(2).(3).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*207·251">*207·251</a>.</b> \(\vdash :P\in \text{trans}.y\in Pʻʻ(\beta -{℩}ʻy).\supset .\overrightarrow{\text{lt}} _{P}ʻ\beta =\overrightarrow{\text{lt}} _{P}ʻ(\beta -{℩}ʻy)\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*51·222. &\supset \vdash :y{\sim}\in \beta .\supset .\overrightarrow{\text{lt}} _{P}ʻ\beta =\overrightarrow{\text{lt}} _{P}ʻ(\beta -{℩}ʻy) &\qquad \text{(1)}\\ +\vdash .*207·25. & \supset \vdash :\text{Hp}.\supset .\overrightarrow{\text{lt}} _{P}ʻ\{(\beta -{℩}ʻy)\cup {℩}ʻy\}=\overrightarrow{\text{lt}} _{P}ʻ(\beta + -{℩}ʻy) &\qquad \text{(2)}\\ +\vdash .(2).*51·21.&\supset \vdash :\text{Hp}.y\in \beta .\supset .\overrightarrow{\text{lt}} _{P}ʻ\beta =\overrightarrow{\text{lt}} _{P}ʻ(\beta -{℩}ʻy) (3) +\vdash .(1).(3).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_599">[Pg 599]</span></p> + +<p class="nind"><b>*207·26.</b> \[\begin{align}&\vdash :P\in \text{trans}.{\sim}(yPy).\exists !\overrightarrow{\text{lt}} _{P}ʻ\beta .\supset .\overrightarrow{\text{lt}} _{P}ʻ\beta + =\overrightarrow{\text{lt}} _{P}ʻ(\beta -\iota ʻy)\\ +&[*207·13·12.*206·72]\end{align}\]</p> + +<p class="nind"><b><a id="*207·261">*207·261</a>.</b> \[\begin{align}&\vdash :P\in \text{trans}.y\in \overrightarrow{\text{min}}_{P}ʻ\beta .\exists !\overrightarrow{\text{lt}} _{P}ʻ\beta + .\supset .\overrightarrow{\text{lt}} _{P}ʻ\beta =\overrightarrow{\text{lt}} _{P}ʻ(\beta -\iota ʻy)\\ +&[*207·26.*205·194]\end{align}\]</p> + +<p class="nind"><b>*207·262.</b> \[\begin{align}&\vdash :P\in \text{trans}\cap \text{connex} .\exists !\overrightarrow{\text{lt}} _{P}ʻ\beta .\supset .\overrightarrow{\text{lt}} _{P}ʻ\beta + =\overrightarrow{\text{lt}} _{P}ʻ(\beta -\overrightarrow{\text{min}}_{P}ʻ\beta )\\ +&[*207·261.*205·3]\end{align}\]</p> + +<p class="nind"><b>*207·263.</b> \[\begin{align}&\vdash :P\in \text{trans}\cap \text{connex} .\supset .\overrightarrow{\text{lt}} _{P}ʻ\beta \subset \overrightarrow{\text{lt}} _{P}ʻ(\beta + -\overrightarrow{\text{min}}_{P}ʻ\beta )\\ +&[*207·262.*24·12]\end{align}\]</p> + +<p class="nind"><b><a id="*207·27">*207·27</a>.</b> \(\vdash :P\in \text{trans}\cap \text{connex} .\beta \cap CʻP{\sim}\in 1.\supset .\overrightarrow{\text{lt}} _{P}ʻ\beta =\overrightarrow{\text{lt}} _{P}ʻ(\beta + -\overrightarrow{\text{min}}_{P}ʻ\beta )\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*24·26·101.&\supset \vdash :\overrightarrow{\text{min}}_{P}ʻ\beta =\Lambda .\supset .\overrightarrow{\text{lt}} _{P}ʻ\beta =\overrightarrow{\text{lt}} _{P}ʻ(\beta + -\overrightarrow{\text{min}}_{P}ʻ\beta ) &\qquad \text{(1)}\\ +\vdash .*52·181.\supset \\ +\vdash :\text{Hp}.\exists !\overrightarrow{\text{min}}_{P}ʻ\beta .&\supset .(\exists y).y\in \beta \cap CʻP.y\neq \text{min}_{P}ʻ\beta .\\ +[*205·2] &\supset .(\exists y).y\in (\beta \cap CʻP)-\iota ʻ\text{min}_{P}ʻ\beta .\text{min}_{P}ʻ\beta Py.\\ +[*37·1] &\supset .\text{min}_{P}ʻ\beta \in Pʻʻ(\beta -\iota ʻ\text{min}_{P}ʻ\beta ).\\ +[*207·251] &\supset .\overrightarrow{\text{lt}} _{P}ʻ\beta =\overrightarrow{\text{lt}} _{P}ʻ(\beta -\iota ʻ\text{min}_{P}ʻ\beta ) &\qquad \text{(2)}\\ +\vdash .(1).(2).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*207·28.</b> \(\vdash :P\in \text{trans}.\supset .\overrightarrow{\text{lt}} _{P}ʻ(\alpha \cup Pʻʻ\alpha )=\overrightarrow{\text{lt}} _{P}ʻ\alpha \quad[*207·25]\)</p> + +<p class="nind"><b>*207·281.</b> \[\begin{align}&\vdash :P\in \text{trans}.{\sim}\exists !\overrightarrow{\text{max}}_{P}ʻ\alpha .\supset .\overrightarrow{\text{lt}} _{P}ʻPʻʻ\alpha + =\overrightarrow{\text{lt}} _{P}ʻ\alpha \\ +&[*207·28·16.*205·123]\end{align}\]</p> + +<p class="nind"><b>*207·282.</b> \[\begin{align}&\vdash :P\in \text{trans}.{\sim}\exists !\overrightarrow{\text{max}}_{P}ʻ\alpha .{\sim}\exists !\overrightarrow{\text{max}}_{P}ʻ\beta + .Pʻʻ\alpha =Pʻʻ\beta .\supset .\overrightarrow{\text{lt}} _{P}ʻ\alpha =\overrightarrow{\text{lt}} _{P}ʻ\beta \\ +&[*207·281]\end{align}\]</p> + +<p class="nind"><b>*207·29.</b> \(\vdash :P\in \text{trans}.\supset .\overrightarrow{\text{lt}} _{P}ʻ\alpha =\overrightarrow{\text{lt}} _{P}ʻP_{*}ʻʻ\alpha\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*207·16·28.\supset \vdash :\text{Hp}.\supset .\overrightarrow{\text{lt}} _{P}ʻ\alpha &=\overrightarrow{\text{lt}} _{P}ʻ\{(\alpha \cup Pʻʻ\alpha )\cap CʻP\}\\ +[*201·52] &=\text{lt}_PʻP_{*}ʻʻ\alpha :\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*207·291">*207·291</a>.</b> \(\vdash :P\in \text{trans}\cap \text{connex} .\text{E}!\text{lt}_Pʻ\alpha .\supset .\overrightarrow{P}ʻ\text{lt}_Pʻ\alpha =P_{*}ʻʻ\alpha\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*207·29. &\supset \vdash :\text{Hp}.\supset .\overrightarrow{P}ʻ\text{lt}_Pʻ\alpha =\overrightarrow{P}ʻ\text{lt}_PʻP_{*}ʻʻ\alpha &\qquad \text{(1)}\\ +\vdash .*90·14·172. &\supset \vdash .P_{*}ʻʻ\alpha \subset CʻP.PʻʻP_{*}ʻʻ\alpha \subset P_{*}ʻʻ\alpha &\qquad \text{(2)}\\ +\vdash .*207·11·12. &\supset \vdash :\text{Hp}.\supset .\text{seq}_{P}ʻP_{*}ʻʻ\alpha =\text{lt}_PʻP_{*}ʻʻ\alpha &\qquad \text{(3)}\\ +\vdash .(2).(3).*206·3. &\supset \vdash :\text{Hp}.\supset .\overrightarrow{P}ʻ\text{seq}_{P}ʻP_{*}ʻʻ\alpha =P_{*}ʻʻ\alpha &\qquad \text{(4)}\\ +\vdash .(1).(3).(4).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_600">[Pg 600]</span></p> + +<p class="nind"><b><a id="*207·3">*207·3</a>.</b> \(\vdash :\alpha \cap CʻP=\Lambda .\supset .\overrightarrow{\text{lt}} _{P}ʻ\alpha =\overrightarrow{B}ʻP\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*205·151·161.&\supset \vdash :\text{Hp}.\supset .\overrightarrow{\text{max}}_{P}ʻ\alpha =\Lambda &\qquad \text{(1)}\\ +\vdash .*206·14.&\supset \vdash :\text{Hp}.\supset .\overrightarrow{\text{seq}} _{P}ʻ\alpha =\overrightarrow{B}ʻP &\qquad \text{(2)}\\ +\vdash .(1).(2).*207·12.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*207·31.</b> \(\vdash :P\,\unicode{x2abd}\, J.x\in CʻP-\text{ᗡ}ʻ(P\dot{-} P^{2}).\supset .x\text{lt}_P\overrightarrow{P}ʻx\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*206·41.&\supset \vdash :\text{Hp}.\supset .\overrightarrow{\text{max}}_{P}ʻ\overrightarrow{P}ʻx=\Lambda &\qquad \text{(1)}\\ +\vdash .*206·4. &\supset \vdash :\text{Hp}.\supset .x\text{seq}_{P}\overrightarrow{P}ʻx &\qquad \text{(2)}\\ +\vdash .(1).(2).*207·1.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*207·32">*207·32</a>.</b> \[\begin{align}&\vdash :P\in \text{Rl}ʻJ\cap \text{connex} .x\in CʻP-\text{ᗡ}ʻ(P\dot{-} P^{2}).\supset .x=\text{lt}_Pʻ\overrightarrow{P}ʻx\\ +&[*207·31·24]\end{align}\]</p> + +<p class="nind"><b><a id="*207·33">*207·33</a>.</b> \(\vdash :x\in \text{ᗡ}ʻ(P\dot{-} P^{2}).\supset .\overrightarrow{\text{lt}} _{P}ʻ\overrightarrow{P}ʻx=\Lambda \quad[*205·252.*207·11]\)</p> + +<p class="nind"><b>*207·34.</b> \(\vdash :P\in \text{connex} .x\,\text{lt}_P\,\alpha .\supset .x\text{lt}_P\overrightarrow{P}ʻx.x{\sim}\in \text{ᗡ}ʻ(P\dot{-} P^{2})\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*207·15. \supset \vdash :\text{Hp}.\supset .x\in CʻP.&\alpha \cap CʻP\subset Pʻʻ\alpha .\alpha \cap CʻP\subset \overrightarrow{P}ʻx.\\ +&\overrightarrow{P}ʻx\subset -pʻ\overleftarrow{P}ʻʻ(\alpha \cap CʻP) &\qquad \text{(1)}\\ +\vdash .*40·16. \supset \vdash :\alpha \cap CʻP\subset \overrightarrow{P}ʻx.&\supset .pʻ\overleftarrow{P}ʻʻ\overrightarrow{P}ʻx\subset pʻ\overleftarrow{P}ʻʻ(\alpha \cap CʻP).\\ +[*22·81] &\supset .-pʻ\overleftarrow{P}ʻʻ(\alpha \cap CʻP)\subset -pʻ\overleftarrow{P}ʻʻ\overrightarrow{P}ʻx &\qquad \text{(2)}\\ +\vdash .(1).(2). \supset \vdash :\text{Hp}.&\supset .x\in CʻP.\overrightarrow{P}ʻx\subset -pʻ\overleftarrow{P}ʻʻ\overrightarrow{P}ʻx.\\ +[*22·42] &\supset .x\in CʻP.\overrightarrow{P}ʻx\subset \overrightarrow{P}ʻx.\overrightarrow{P}ʻx\subset -pʻ\overleftarrow{P}ʻʻ\overrightarrow{P}ʻx &\qquad \text{(3)}\\ +\vdash .(1).*202·505. \supset \vdash :\text{Hp}.&\supset .\overrightarrow{P}ʻx\subset (\alpha \cap CʻP)\cup Pʻʻ\alpha .\alpha \cap CʻP\subset Pʻʻ\alpha .\\ +[*22·62] &\supset .\overrightarrow{P}ʻx\subset Pʻʻ\alpha &\qquad \text{(4)}\\ +\vdash .(1).*37·2·265. &\supset \vdash :\text{Hp}.\supset .Pʻʻ\alpha \subset Pʻʻ\overrightarrow{P}ʻx &\qquad \text{(5)}\\ +\vdash .(4).(5). &\supset \vdash :\text{Hp}.\supset .\overrightarrow{P}ʻx\subset Pʻʻ\overrightarrow{P}ʻx &\qquad \text{(6)}\\ +\vdash .(3).(6).*207·15.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*207·35.</b> \(\vdash :P\in R1ʻJ\cap \text{connex} .\supset .\text{D}ʻ\text{lt}_P=CʻP-\text{ᗡ}ʻ(P\dot{-} P^{2})\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*207·34.&\supset \vdash :\text{Hp}.\supset .\text{D}ʻ\text{lt}_P\subset -\text{ᗡ}ʻ(P\dot{-} P^{2}) &\qquad \text{(1)}\\ +\vdash .*207·15.&\supset \vdash .\text{D}ʻ\text{lt}_P\subset CʻP &\qquad \text{(2)}\\ +\vdash .*207·32.&\supset \vdash :\text{Hp}.\supset .CʻP-\text{ᗡ}ʻ(P\dot{-} P^{2})\subset \text{D}ʻ\text{lt}_P &\qquad \text{(3)}\\ +\vdash .(1).(2).(3).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_601">[Pg 601]</span></p> + +<p class="nind"><b><a id="*207·36">*207·36</a>.</b> \[\begin{align}\vdash :P\in \text{Rl}ʻJ\cap &\text{connex} .\supset .\\ +&\text{D}ʻ\text{lt}_P=\text{lt}_Pʻʻ\overrightarrow{P}ʻʻ{CʻP-\text{ᗡ}ʻ(P\dot{-} P^{2})}=\text{lt}_Pʻʻ\overrightarrow{P}ʻʻCʻP\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*207·32.&\supset \vdash :\text{Hp}.\supset .CʻP-\text{ᗡ}ʻ(P\dot{-} P^{2})=\text{lt}_Pʻʻ\overrightarrow{P}ʻʻ\{CʻP-\text{ᗡ}ʻ(P\dot{-} P^{2})\} &\qquad \text{(1)}\\ +\vdash .(1).*207·35.&\supset \vdash :\text{Hp}.\supset .\text{D}ʻ\text{lt}_P=\text{lt}_Pʻʻ\overrightarrow{P}ʻʻ\{CʻP-\text{ᗡ}ʻ(P\dot{-} P^{2})\} &\qquad \text{(2)}\\ +\vdash .*207·33. &\supset \vdash .\text{lt}_Pʻʻ\overrightarrow{P}ʻʻ\{CʻP\cap \text{ᗡ}ʻ(P\dot{-} P^{2})\}=\Lambda &\qquad \text{(3)}\\ +\vdash .(2).(3). &\supset \vdash :\text{Hp}.\supset .\text{D}ʻ\text{lt}_P=\text{lt}_Pʻʻ\overrightarrow{P}ʻʻCʻP &\qquad \text{(4)}\\ +\vdash .(2).(4).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p>In virtue of this proposition, all limits are limits of classes +of the form \(\overrightarrow{P}ʻx\). In this respect, limits (in +general) differ from segments. If we call \(Pʻʻ\alpha\) the segment +defined by \(\alpha\), there will in general be segments not of the +form \(\overrightarrow{P}ʻx\). These, however, will be the segments +which have no sequents, and therefore no limits; thus their existence +does not introduce limits not derivable from classes of the form +\(\overrightarrow{P}ʻx\).</p> + +<p class="nind"><b>*207·4.</b> \[\begin{align}\vdash \colon\ldotp x \text{limax}_P \alpha .&\equiv :x \text{max}_{P} \alpha .\lor.x \text{lt}_P \alpha :\\ +&\equiv :x \text{max}_{P} \alpha .\lor.{\sim}\exists !\overrightarrow{\text{max}}_{P}ʻ\alpha .x \text{seq}_{P} \alpha \quad[(*207·03)]\end{align}\]</p> + +<p class="nind"><b><a id="*207·401">*207·401</a>.</b> \(\vdash .\text{limin}_P=\text{limax}(\breve{P} ) \quad[(*207·04)]\)</p> + +<p class="nind"><b><a id="*207·41">*207·41</a>.</b> \[\begin{align}&\vdash :P\in \text{connex} .\supset .\text{limax}_P,\text{limin}_P\in 1\rightarrow \text{Cls}\\ +&[*71·24.*205·31.*207·24.(*207·03·04)]\end{align}\]</p> + +<p class="nind"><b>*207·42.</b> \(\vdash :\exists !\overrightarrow{\text{max}}_{P}ʻ\alpha .\supset .\overrightarrow{\text{limax}} _{P}ʻ\alpha =\overrightarrow{\text{max}}_{P}ʻ\alpha + \quad[*207·4]\)</p> + +<p class="nind"><b>*207·43.</b> \(\vdash :\overrightarrow{\text{max}}_{P}ʻ\alpha =\Lambda .\supset .\overrightarrow{\text{limax}} _{P}ʻ\alpha =\overrightarrow{\text{seq}} _{P}ʻ\alpha + =\overrightarrow{\text{lt}} _{P}ʻ\alpha \quad[*207·4]\)</p> + +<p class="nind"><b>*207·44.</b> \[\begin{align}&\vdash .\text{ᗡ}ʻ\text{limax}_P=\text{ᗡ}ʻ\text{max}_{P}\cup \text{ᗡ}ʻ\text{lt}_P=\text{ᗡ}ʻ\text{max}_{P}\cup \text{ᗡ}ʻ\text{seq}_{P}\\ +&[*207·14.(*207·03)]\end{align}\]</p> + +<p class="nind"><b>*207·45.</b> \(\vdash .\overrightarrow{\text{limax}} _{P}ʻ\alpha =\overrightarrow{\text{max}}_{P}ʻ\alpha \cup \overrightarrow{\text{lt}} _{P}ʻ\alpha + \quad[(*207·03)]\)</p> + +<p class="nind"><b>*207·46.</b> \(\vdash \colon\ldotp x=\text{limax}_Pʻ\alpha .\equiv :x=\text{max}_{P}ʻ\alpha .\lor.x=\text{lt}_Pʻ\alpha\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*207·45·11.&\supset \vdash \colon\ldotp \exists !\overrightarrow{\text{max}}_{P}ʻ\alpha .\supset :x=\text{limax}_Pʻ\alpha .\equiv .x=\text{max}_{P}ʻ\alpha + &\qquad \text{(1)}\\ +\vdash .*207·45·12.&\supset \vdash \colon\ldotp \overrightarrow{\text{max}}_{P}ʻ\alpha =\Lambda .\supset :x=\text{limax}_Pʻ\alpha .\equiv .x=\text{lt}_Pʻ\alpha &\qquad \text{(2)}\\ +\vdash .(1).(2).*5·32.\supset \\ +\vdash \colon\ldotp &\exists !\overrightarrow{\text{max}}_{P}ʻ\alpha .x=\text{limax}_Pʻ\alpha .\lor.\overrightarrow{\text{max}}_{P}ʻ\alpha + =\Lambda .x=\text{limax}_Pʻ\alpha :\equiv :\\ +&\exists !\overrightarrow{\text{max}}_{P}ʻ\alpha .x=\text{max}_{P}ʻ\alpha .\lor.\overrightarrow{\text{max}}_{P}ʻ\alpha + =\Lambda .x=\text{lt}_Pʻ\alpha &\qquad \text{(3)}\\ +\vdash .(3).*4·42.\supset \\ +\vdash \colon\ldotp x=\text{limax}_Pʻ\alpha .&\equiv :\exists !\overrightarrow{\text{max}}_{P}ʻ\alpha .x=\text{max}_{P}ʻ\alpha .\lor.\overrightarrow{\text{max}}_{P}ʻ\alpha + =\Lambda .x=\text{lt}_Pʻ\alpha :\\ +[*30·32] &\equiv :x=\text{max}_{P}ʻ\alpha .\lor.\overrightarrow{\text{max}}_{P}ʻ\alpha =\Lambda .x=\text{lt}_Pʻ\alpha :\\ +[*207·13] &\equiv :x=\text{max}_{P}ʻ\alpha .\lor.x=\text{lt}_Pʻ\alpha \colon\ldotp \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_602">[Pg 602]</span></p> + +<p class="nind"><b>*207·47.</b> \(\vdash : \exists ! \overrightarrow{\text{lt}} _{P}ʻ\alpha . \equiv . \exists ! \overrightarrow{\text{limax}} _{P}ʻ\alpha + . {\sim} \exists ! \overrightarrow{\text{max}}_{P}ʻ\alpha\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *207·45·11 . &\supset \vdash : \exists ! \overrightarrow{\text{lt}} _{P}ʻ\alpha . \supset . \exists ! \overrightarrow{\text{limax}} _{P}ʻ\alpha + . {\sim} \exists ! \overrightarrow{\text{max}}_{P}ʻ\alpha &\qquad \text{(1)}\\ +\vdash . *207·45 . &\supset \vdash : \exists ! \overrightarrow{\text{limax}} _{P}ʻ\alpha . {\sim} \exists ! \overrightarrow{\text{max}}_{P}ʻ\alpha + . \supset . \exists ! \overrightarrow{\text{lt}} _{P}ʻ\alpha &\qquad \text{(2)}\\ +\vdash . (1) . (2) . \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*207·48.</b> \(\vdash . \overrightarrow{\text{limax}} _{P}ʻ\alpha = \overrightarrow{\text{limax}} _{P}ʻ(\alpha \cap CʻP) \quad[*207·45 . *205·151 . *207·16]\)</p> + +<p class="nind"><b>*207·481.</b> \[\begin{align}&\vdash : P \in \text{trans} . \supset . \overrightarrow{\text{limax}} _{P}ʻ\alpha + = \overrightarrow{\text{limax}} _{P}ʻP_{*}ʻʻ\alpha\\ +&[*207·45 . *205·191 . *207·29]\end{align}\]</p> + +<p class="nind"><b>*207·482.</b> \(\vdash : P \in \text{Ser} . \alpha \subset CʻP . \alpha = \text{limax}_Pʻ\alpha . \supset . \alpha \subset \overrightarrow{P}_{*}ʻ\alpha\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *205·22 . *90·151 . &\supset \vdash : \text{Hp} . a = \text{max}_{P}ʻ\alpha . \supset . \alpha \subset \overrightarrow{P}_{*}ʻ\alpha &\qquad \text{(1)}\\ +\vdash . *207·291 . *90·151 . \supset \vdash : \text{Hp} . a = \text{lt}_Pʻ\alpha . &\supset . P_{*}ʻʻ\alpha \subset \overrightarrow{P}_{*}ʻa .\\ +[*90·21] &\supset . \alpha \subset \overrightarrow{P}_{*}ʻa &\qquad \text{(2)}\\ +\vdash . (1) . (2) . *207·46 . \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*207·5.</b> \[\begin{align}&\vdash : P \in \text{Ser} . \supset . \overrightarrow{\text{limax}} _{P}ʻ\alpha = \overrightarrow{\text{seq}} _{P}ʻPʻʻ\alpha + = \overrightarrow{\text{min}}_{P}ʻ(\overrightarrow{\text{max}}_{P}ʻ\alpha \cup \overrightarrow{\text{seq}}_{P}ʻ\alpha )\\ +&[*206·33·35·37]\end{align}\]</p> + +<p class="nind"><b><a id="*207·51">*207·51</a>.</b> \[\begin{align}&\vdash \colon\ldotp P \in \text{Ser} . \supset : x = \text{limax}_Pʻ\alpha . \equiv . x \in CʻP . \overrightarrow{P}ʻx = Pʻʻ\alpha \\ +&[*205·54 . *207·232·46]\end{align}\]</p> + +<p class="nind"><b>*207·52.</b> \(\vdash \colon\ldotp P \in \text{Ser} . \exists ! Pʻʻ\alpha . \supset : x = \text{limax}_Pʻ\alpha . \equiv . \overrightarrow{P}ʻx = Pʻʻ\alpha \quad[*207·51]\)</p> + +<p class="nind"><b>*207·521.</b> + \(\vdash \colon\ldotp P \in \text{Ser} . \supset : x = \text{lt}_Pʻ\alpha . \equiv . x \in CʻP . \overrightarrow{P}ʻx = Pʻʻ\alpha . {\sim} \text{E}! \text{max}_{P}ʻ\alpha\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *207·51 . \supset \vdash \colon\ldotp \text{Hp} . \supset :\\ +x \in CʻP . \overrightarrow{P}ʻx = Pʻʻ\alpha . {\sim} \text{E}! \text{max}_{P}ʻ\alpha . &\equiv . x = \text{limax}_Pʻ\alpha . {\sim} \text{E}! \text{max}_{P}ʻ\alpha .\\ +[*207·46] &\equiv . x = \text{lt}_Pʻ\alpha \colon\ldotp \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*207·53.</b> \(\vdash : P \in \text{Ser} . \kappa \subset \text{ᗡ}ʻ\text{limax}_P . \supset . \overrightarrow{\text{limax}} _{P}ʻ\text{limax}_Pʻʻ\kappa + = \overrightarrow{\text{limax}} _{P}ʻsʻ\kappa\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *207·51 . \supset \vdash \colon\ldotp \text{Hp} . &\supset : \alpha \in \kappa . \supset _\alpha . \overrightarrow{P}ʻ\text{limax}_Pʻ\alpha = Pʻʻ\alpha :\\ +[*37·68] &\supset : \overrightarrow{P}ʻʻ\text{limax}_Pʻʻ\kappa = Pʻʻʻ\kappa :\\ +[*40·5·38] &\supset : Pʻʻ\text{limax}_Pʻʻ\kappa = Pʻʻsʻ\kappa :\\ +[*207·51] &\supset : x = \text{limax}_Pʻ\text{limax}_Pʻʻ\kappa . \equiv . x = \text{limax}_Pʻsʻ\kappa \colon\ldotp \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_603">[Pg 603]</span></p> + +<p class="nind"><b>*207·54.</b> \(\vdash :P\in \text{Ser}.\kappa \subset \text{ᗡ}ʻ\text{lt}_P.\supset .\overrightarrow{\text{limax}} _{P}ʻ\text{lt}_Pʻʻ\kappa =\overrightarrow{\text{limax}} _{P}ʻsʻ\kappa + =\overrightarrow{\text{lt}} _{P}ʻsʻ\kappa\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*205·561.*207·13.\supset \vdash :\text{Hp}.&\supset .sʻ\kappa {\sim}\in \text{ᗡ}ʻ\text{max}_{P}.\\ +[*207·43] & \supset .\overrightarrow{\text{limax}} _{P}ʻsʻ\kappa =\overrightarrow{\text{lt}} _{P}ʻsʻ\kappa &\qquad \text{(1)}\\ +\vdash .*207·13·43.\supset \vdash :\text{Hp}.&\supset .\text{lt}_Pʻʻ\kappa =\text{limax}_Pʻʻ\kappa .\\ +[*207·53] &\supset .\overrightarrow{\text{limax}} _{P}ʻ\overrightarrow{\text{lt}} _{P}ʻʻ\kappa =\overrightarrow{\text{limax}} _{P}ʻsʻ\kappa &\qquad \text{(2)}\\ +\vdash .(1).(2).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*207·55.</b> + \[\begin{align}&\vdash :P\in \text{Ser}.\kappa \subset \text{ᗡ}ʻ\text{lt}_P.sʻ\kappa \in \text{ᗡ}ʻ\text{lt}_P.\supset .\text{limax}_Pʻ\text{lt}_Pʻʻ\kappa =\text{lt}_Pʻsʻ\kappa\\ +&[*207·54]\end{align}\]</p> + +<p class="nind"><b><a id="*207·6">*207·6</a>.</b> \(\vdash :S\in P\,\overline{\,\text{smor}\,}\,Q.\supset .\overrightarrow{\text{lt}} _{P}ʻ\alpha =Sʻʻ\overrightarrow{\text{lt}} _{Q}ʻ\breve{S} ʻʻ\alpha\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*205·8.*37·43.\supset \vdash \colon\ldotp \text{Hp}.&\supset :\exists !\overrightarrow{\text{max}}_{P}ʻ\alpha .\equiv .\exists !\overrightarrow{\text{max}}_{Q}ʻ\breve{S} + ʻʻ\alpha : &\qquad \text{(1)}\\ +[*207·11] &\supset :\exists !\overrightarrow{\text{max}}_{P}ʻ\alpha .\supset .\overrightarrow{\text{lt}} _{P}ʻ\alpha =\Lambda .\overrightarrow{\text{lt}} _{Q}ʻ\breve{S} + ʻʻ\alpha =\Lambda &\qquad \text{(2)}\\ +\vdash .(1).\text{Transp}.*207·12.\supset \\ +\vdash \colon\ldotp \text{Hp}.\overrightarrow{\text{max}}_{P}ʻ\alpha =\Lambda .&\supset :\overrightarrow{\text{lt}} _{P}ʻ\alpha =\overrightarrow{\text{seq}} _{P}ʻ\alpha + .\overrightarrow{\text{lt}} _{Q}ʻ\breve{S} ʻʻ\alpha =\overrightarrow{\text{seq}} _{Q}ʻ\breve{S} ʻʻ\alpha :\\ +[*206·61] &\supset :\overrightarrow{\text{lt}} _{P}ʻ\alpha =Sʻʻ\overrightarrow{\text{lt}} _{Q}ʻ\breve{S} ʻʻ\alpha &\qquad \text{(3)}\\ +\vdash .(2).(3).*37·29.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*207·61.</b> \(\vdash \colon\ldotp S\in P\,\overline{\,\text{smor}\,}\,Q.\supset :\text{E}!\text{lt}_Pʻ\alpha .\equiv .\text{E}!\text{lt}_{Q}ʻ\breve{S} ʻʻ\alpha \quad[*207·6.*53·3]\)</p> + +<p class="nind"><b>*207·62.</b> \(\vdash :S\in P\,\overline{\,\text{smor}\,}\,Q.\text{E}!\text{lt}_Pʻ\alpha .\supset .\text{lt}_Pʻ\alpha =Sʻ\text{lt}_{Q}ʻ\breve{S} ʻʻ\alpha \quad[*207·6.*53·31]\)</p> + +<p class="nind"><b>*207·63.</b> \(\vdash :S\in P\,\overline{\,\text{smor}\,}\,Q.\supset .\text{lt}_Pʻʻ\kappa =Sʻʻ\text{lt}_{Q}ʻʻ\breve{S} ʻʻʻ\kappa\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*207·6.*40·5.\supset \vdash :\text{Hp}.\supset .\text{lt}_Pʻʻ\kappa &=sʻSʻʻʻ\overrightarrow{\text{lt}} _{Q}ʻʻ\breve{S} ʻʻʻ\kappa \\ +[*40·38·5] &=Sʻʻ\text{lt}_{Q}ʻʻ\breve{S} ʻʻʻ\kappa :\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*207·64.</b> \[\begin{align}&\vdash :S\in P\,\overline{\,\text{smor}\,}\,Q.\supset .\overrightarrow{\text{limax}} _{P}ʻ\alpha =Sʻʻ\overrightarrow{\text{limax}} _{Q}ʻ\breve{S} + ʻʻ\alpha\\ +&[*205·8.*207·6·45]\end{align}\]</p> + +<p class="nind"><b>*207·65.</b> \[\begin{align}&\vdash \colon\ldotp S\in P\,\overline{\,\text{smor}\,}\,Q.\supset :\text{E}!\text{limax}_Pʻ\alpha .\equiv .\text{E}!\text{limax}_{Q}ʻ\breve{S} ʻʻ\alpha\\ +&[*207·64]\end{align}\]</p> + +<p class="nind"><b><a id="*207·66">*207·66</a>.</b> \[\begin{align}&\vdash :S\in P\,\overline{\,\text{smor}\,}\,Q.\text{E}!\text{limax}_Pʻ\alpha .\supset .\text{limax}_Pʻ\alpha =Sʻ\text{limax}_{Q}ʻ\breve{S} ʻʻ\alpha\\ +&[*207·64]\end{align}\]</p> + +<p><span class="pagenum" id="Page_604">[Pg 604]</span></p> + +<p class="nind"><b><a id="*207·7">*207·7</a>.</b> \[\begin{align}\vdash \colon\ldotp P\in \text{trans}.\lor.&P^{2}\,\unicode{x2abd}\, J:\supset :\\ +&\text{limin}_Pʻ\gamma =\text{limax}_Pʻ\gamma .\supset .\text{limin}_Pʻ\gamma =\text{min}_{P}ʻ\gamma =\text{max}_{P}ʻ\gamma\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*207·42·43.\supset \vdash :&\text{E}!\text{min}_{P}ʻ\gamma .\text{E}!\text{limax}_Pʻ\gamma .{\sim}\text{E}!\text{max}_{P}ʻ\gamma .\supset .\\ +&\text{limin}_Pʻ\gamma =\text{min}_{P}ʻ\gamma .\text{limax}_Pʻ\gamma =\text{seq}_{P}ʻ\gamma .\\ +[*205·11.*206·2] &\supset .\text{limin}_Pʻ\gamma \in \gamma .\text{limax}_Pʻ\gamma {\sim}\in \gamma .\\ +[*13·14] &\supset .\text{limin}_Pʻ\gamma \neq \text{limax}_Pʻ\gamma &\qquad \text{(1)}\\ +\text{Similarly}\\ +\vdash :\text{E}!\text{max}_{P}ʻ\gamma .\text{E}!&\text{limin}_Pʻ\gamma .{\sim}\text{E}!\text{min}_{P}ʻ\gamma + .\supset .\text{limin}_Pʻ\gamma \neq \text{limax}_Pʻ\gamma &\qquad \text{(2)}\\ +\vdash .*206·732.*207·43·12.\supset \\ +\vdash :\text{Hp}.{\sim}\text{E}&!\text{min}_{P}ʻ\gamma .{\sim}\text{E}!\text{max}_{P}ʻ\gamma .\supset .{\sim}{\text{limin}_Pʻ\gamma =\text{limax}_Pʻ\gamma } &\qquad \text{(3)}\\ +\vdash .(1).(2).(3).&\supset \vdash :\text{Hp}.\text{limin}_Pʻ\gamma =\text{limax}_Pʻ\gamma .\supset .\text{E}!\text{min}_{P}ʻ\gamma .\text{E}!\text{max}_{P}ʻ\gamma .\\ +[*207·42] & \supset .\text{limin}_Pʻ\gamma =\text{min}_{P}ʻ\gamma =\text{max}_{P}ʻ\gamma :\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*207·71.</b> \[\begin{align}\vdash \colon\ldotp P\in \text{connex} :P\in \text{trans}.\lor.P^{2}\,\unicode{x2abd}\, &J:\text{limin}_Pʻ\gamma =\text{limax}_Pʻ\gamma :\supset .\\ +&\gamma \cap CʻP\in 1.\gamma \cap CʻP={℩}ʻ\text{limax}_Pʻ\gamma \\ +&[*207·7.*205·73]\end{align}\]</p> + +<p class="nind"><b><a id="*207·72">*207·72</a>.</b> \[\begin{align}&\vdash \colon\ldotp P\in \text{connex} .P^{2}\,\unicode{x2abd}\, J.\supset :\text{limin}_Pʻ\gamma =\text{limax}_Pʻ\gamma .\equiv .\gamma \cap CʻP\in 1\\ +&[*207·71.*205·731·17.*207·42]\end{align}\]</p> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_605">[Pg 605]</span></p> +<h2 class="nobreak" id="208_THE_CORRELATION_OF_SERIES"> +*208. THE CORRELATION OF SERIES.</h2> +</div> + + +<p class="nind"><i>Summary of</i> *208.</p> + +<p>The propositions of this number are chiefly important on account of +their consequences in the theory of well-ordered series (*250 ff.) and +in the theory of vector-families (*330 ff.). When two well-ordered +series are ordinally similar, they have only one correlator; and a +well-ordered series is not ordinally similar to any of its segments. +Of these two propositions, the first is an immediate consequence of +<a href="#*208·41">*208·41</a>, and the second is an immediate consequence of <a href="#*208·47">*208·47</a>.</p> + +<p>Propositions concerning correlators of two relations \(P\) and \(Q\) +are obtained from propositions concerning correlators of \(P\) with +itself, by means of the fact that, if \(S\), \(T\) are two correlators +of \(P\) and \(Q\), \(S\mid \breve{T}\) is a correlator of \(P\) with +itself. Again, correlators of \(P\) with itself are considered, in +this number, as a special case of correlators of \(P\) with parts of +itself. This latter is a notion which will prove important for other +reasons than those for which it is used in our present context. If +\(P\) is connected, and \(S\) correlates \(P\) with part of itself (so +that \(S^{;}P\,\unicode{x2abd}\, P\)), \(CʻP\) will contain terms of three +kinds, (1) those for which \(Sʻx=x\), (2) those for which \((Sʻx)Px\), +(3) those for which \(xP(Sʻx)\). Our propositions result from the +non-existence (under certain circumstances) of maxima or minima of +classes (2) and (3).</p> + +<p>The following definition defines "correlations of P with parts (or the +whole) of itself." The letters "\(\text{cror}\)" stand for "ordinal +correlation." For a cardinal correlation, should occasion arise, we +should use "\(\text{cr}\)," <i>i.e.</i> we should put +\[ +\text{cr}ʻ\alpha =sʻ\,\overline{\text{ sm }}\, \alpha ʻʻ\text{Cl}ʻ\alpha \quad\text{Df}, +\] +so that \(S\in \text{cr}ʻ\alpha .\equiv .S\in 1\rightarrow 1.\text{ᗡ}ʻS=\alpha .\text{D}ʻS\subset \alpha\). +For the present, we are concerned with the corresponding ordinal +notion; thus we require +\[ +S\in \text{cror}ʻP.\equiv .S\in 1\rightarrow 1.\text{ᗡ}ʻS=CʻP.S^{;}P\,\unicode{x2abd}\, P. +\] +This is secured by putting +\[ +\text{cror}ʻP=sʻ\,\overline{\,\text{smor}\,}\,Pʻʻ\text{Rl}ʻP \quad\text{Df}. +\]</p> + +<p><span class="pagenum" id="Page_606">[Pg 606]</span></p> + +<p>It will be observed that if \(\alpha\) is what we called +a "non-reflexive" class (cf. <a href="#*124">*124</a>), \(\text{cr}ʻ\alpha={℩}ʻI\upharpoonright \alpha\), +and \(S\in \text{cr}ʻ\alpha .\supset.\text{D}ʻS=\alpha\). When \(CʻP\) +is non-reflexive, the same is true of \(P\); and when \(CʻP\) is +reflexive, \(P\) is also reflexive, in the sense that it contains +proper parts similar to itself, though if \(P\) is well-ordered, such +proper parts cannot be segments of \(P\), but must extend to the end of +\(CʻP\).</p> + +<p>The class of correlators of \(P\) with the whole of itself, <i>i.e.</i> +\(P\,\overline{\,\text{smor}\,}\,P\), is a sub-class of \(\text{cror}ʻP\), and +is specially important. This class differs widely in its properties +from the corresponding cardinal class. If \(\alpha\) has more than one +member, the class \(\alpha \,\overline{\text{ sm }}\, \alpha\) (which is the +"permutations" of \(\alpha\) in the usual elementary sense) always +has more than one member. But the class \(P\,\overline{\,\text{smor}\,}\,P\) +(which consists of such permutations of \(CʻP\) as keep the order +unchanged) will consist of the single term \(I\upharpoonright CʻP\), +unless \(CʻP\) contains classes which have neither a minimum nor a +maximum, in which case there will be many correlators of \(P\) with +itself. As a simple illustration, take the series of negative and +positive integers in their natural order. Then if \(\nu\) is any one +of these integers, \(+\nu\) is a correlator of the whole series with +itself. If we take only the positive integers, \(+\nu\) is no longer +a correlator of the whole series with itself, since all integers less +than \(\nu\) are omitted from the correlate.</p> + +<p>The first important use of the propositions of this number is in the +beginning of the theory of well-ordered series (*250). The propositions +there used are</p> + +<p class="nind"><b>*208·41.</b> \[\begin{align}\vdash :P\in \text{connex} .P^{2}\,\unicode{x2abd}\, J.\text{Cl ex}ʻCʻP\subset \text{ᗡ}ʻ&\text{min}_{P}\cup \text{ᗡ}ʻ\text{max}_{P}.\\ +&S,T\in P \,\,\text{smor}\, \,Q.\supset .(P \overline{\,\text{smor}\,}Q) \in 1\end{align}\]</p> + +<p><i>I.e.</i> if \(P\) is connected and asymmetrical, and every existent +sub-class of \(CʻP\) has either a minimum or a maximum, \(P\) and \(Q\) +cannot have more than one correlator.</p> + +<p class="nind"><b>*208·42.</b> \(\text{In the same circumstances},\, P\,\overline{\,\text{smor}\,}\,P={℩}ʻ(I\upharpoonright CʻP)\)</p> + +<p class="nind"><b>*208·43.</b> \(\vdash :\text{Cl ex}ʻCʻP\subset \text{ᗡ}ʻ\text{min}_{P}.S\in \text{cror}ʻP.\supset .{\sim}(\exists x).(Sʻx)Px\)</p> + +<p><i>I.e.</i> if every existent sub-class of \(CʻP\) has a minimum, a +correlator of \(P\) with part of itself can never move terms backwards. +Thus for example, to take a simple instance, an infinite series +consisting of some of the natural numbers in order of magnitude cannot +have its \(\mu\)th term less than \(\mu\).</p> + +<p class="nind"><b>*208·45.</b> \(\vdash :P\in \text{connex} .\text{Cl ex}ʻCʻP\subset \text{ᗡ}ʻ\text{min}_{P}\cap \text{ᗡ}ʻ\text{max}_{P}.\supset .\text{Rl}ʻP\cap \text{Nr}ʻP={℩}ʻP\)</p> + +<p><i>I.e.</i> if \(P\) is connected and every existent sub-class of +\(CʻP\) has both a maximum and a minimum, no proper part of \(P\) is +similar to \(P\). This proposition is important in the theory of finite +series and finite ordinals.</p> + +<p class="nind"><b>*208·46.</b> \(\vdash :\text{Cl ex}ʻCʻP\subset \text{ᗡ}ʻ\text{min}_{P}.S\in \text{cror}ʻP.\supset .CʻP\cap pʻ\overleftarrow{P}ʻʻ\text{D}ʻS=\Lambda\)</p> + +<p><span class="pagenum" id="Page_607">[Pg 607]</span></p> + +<p><i>I.e.</i> if every existent sub-class of \(CʻP\) has a minimum, a +part of \(P\) which is similar to \(P\) must go up to the end of \(P\), +<i>i.e.</i> must not wholly precede any member of \(CʻP\).</p> + +<p class="nind"><b>*208·47.</b> \(\vdash :\text{Cl ex}ʻCʻP\subset \text{ᗡ}ʻ\text{min}_{P}.Q\,\unicode{x2abd}\, P.\exists !CʻP\cap pʻ\overleftarrow{P}ʻʻCʻQ.\supset .{\sim}(Q\,\,\text{smor}\,\,P)\)</p> + +<p>This is an immediate consequence of <a href="#*208·46">*208·46</a>.</p> + +<p>The proof of the above propositions proceeds simply by showing that +if \(S\in \text{cror}ʻP\) and \((Sʻx)Px\), then \((SʻSʻx)P(Sʻx)\), +so that \(x\) is not the earliest term for which \((Sʻx)Px\), +since \(Sʻx\) is an earlier term for which the same thing holds. +Hence \(\hat{x}\{(Sʻx)Px\}\) can have no minimum; and similarly +\(\hat{x}\{xP(Sʻx)\}\) can have no maximum (<a href="#*208·14">*208·14</a>). So far we +require no hypothesis as to \(P\). Assuming now \(P\in \text{connex}.P^{2}\,\unicode{x2abd}\, J\), +we show similarly that if \(S\) correlates the whole of \(P\) with itself, +\(\hat{x}\{(Sʻx)Px\}\) can have no maximum and \(\hat{x}\{xP(Sʻx)\}\) +can have no minimum.</p> + +<p>Propositions about correlators of \(P\) with \(Q\) follow from the +above by taking two correlators \(S\) and \(T\), and applying the above +propositions to \(S\mid \breve{T}\), which is a correlator of \(P\) +with the whole of itself.</p> + +<hr class="tb"> + +<p class="nind"><b>*208·01.</b> \(\text{cror}ʻP=sʻ\,\overline{\,\text{smor}\,}\,Pʻʻ\text{Rl}ʻP \quad\text{Df}\)</p> + +<p class="nind"><b>*208·1.</b> \(\vdash :S\in \text{cror}ʻP.\equiv .S\in 1\rightarrow 1.\text{ᗡ}ʻS=CʻP.S^{;}P\,\unicode{x2abd}\, P\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*40·4.(*208·01).*151·11.\supset \\ +\vdash :S\in \text{cror}ʻP.&\equiv .(\exists Q).Q\,\unicode{x2abd}\, P.S\in 1\rightarrow 1.\text{ᗡ}ʻS=CʻP.Q=S^{;}P.\\ +[*13·195] &\equiv .S\in 1\rightarrow 1.\text{ᗡ}ʻS=CʻP.S^{;}P\,\unicode{x2abd}\, P:\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*208·11.</b> \(\vdash :S\in \text{cror}ʻP.\supset .S^{;}P\,\unicode{x2abd}\, P\unicode{x0294f}\text{D}ʻS\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*150·203.\supset \vdash \colon\ldotp \text{Hp}.&\supset :x(S^{;}P)y.\supset .x,y\in \text{D}ʻS &\qquad \text{(1)}\\ +\vdash .*208·1. &\supset \vdash \colon\ldotp \text{Hp}.\supset :x(S^{;}P)y.\supset .xPy &\qquad \text{(2)}\\ +\vdash .(1).(2). &\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*208·111">*208·111</a>.</b> \[\begin{align}&\vdash :S\in \text{cror}ʻP.\supset .\text{D}ʻS=CʻS^{;}P=SʻʻCʻP.\text{D}ʻS\subset \text{ᗡ}ʻS\\ +&[*150·22·23.*208·1.*33·265]\end{align}\]</p> + +<p class="nind"><b>*208·12.</b> \(\vdash :S\in \text{cror}ʻP.\supset .\breve{S} ^{;}S^{;}P=P.P\,\unicode{x2abd}\, \breve{S} ^{;}P \quad[*151·252·26.*208·1]\)</p> + +<p class="nind"><b>*208·13.</b> \(\vdash :S\in \text{cror}ʻP.(Sʻx)Px.\supset .(SʻSʻx)P(Sʻx)\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*208·12.\supset \vdash :\text{Hp}.&\supset .(Sʻx)(\breve{S} ^{;}P)x.\\ +[*150·41] & \supset .(SʻSʻx)P(Sʻx):\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*208·131.</b> \(\vdash :S\in \text{cror}ʻP.xP(Sʻx).\supset .(Sʻx)P(SʻSʻx) \quad[\text{Proof as in *208·13}]\)</p> + +<p><span class="pagenum" id="Page_608">[Pg 608]</span></p> + +<p class="nind"><b><a id="*208·14">*208·14</a>.</b> \(\vdash :S\in \text{cror}ʻP.\supset .\overrightarrow{\text{min}}_{P}ʻ\hat{x}\{(Sʻx)Px\}=\Lambda .\overrightarrow{\text{max}}_{P}ʻ\hat{x}\{xP(Sʻx)\}=\Lambda\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*208·13.*20·3.\supset \vdash \colon\ldotp \text{Hp}.\supset :x\in \hat{x}\{(Sʻx)Px\}.&\supset .Sʻx\in \hat{x}\{(Sʻx)Px\}.(Sʻx)Px.\\ +[*37·105] &\supset .x\in \breve{P} ʻʻ\hat{x} \{(Sʻx)Px\} &\qquad \text{(1)}\\ +\vdash .(1).*24·3. \supset \vdash :\text{Hp}.&\supset .\hat{x}\{(Sʻx)Px\}-\breve{P} ʻʻ\hat{x}\{(Sʻx)Px\}=\Lambda .\\ +[*205·11] &\supset .\overrightarrow{\text{min}}_{P}ʻ\hat{x}\{(Sʻx)Px\}=\Lambda&\qquad \text{(2)}\\ +\text{Similarly}\quad \vdash :\text{Hp}.&\supset .\overrightarrow{\text{max}}_{P}ʻ\hat{x}\{xP(Sʻx)\}=\Lambda &\qquad \text{(3)}\\ +\vdash .(2).(3).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p>Thus the proof that \(\hat{x}\{(Sʻx)Px\}\) has no minimum, and +\(\hat{x}\{xP(Sʻx)\}\) no maximum, requires no hypothesis as to +\(P\). The proof that \(\hat{x}\{(Sʻx)Px\}\) has no maximum, and +\(\hat{x}\{xP(Sʻx)\}\) no minimum, requires the hypothesis +\(P\in\text{connex} .P^{2}\,\unicode{x2abd}\, J\). This proof results from +the following propositions.</p> + +<p class="nind"><b><a id="*208·2">*208·2</a>.</b> \(\vdash :P\in \text{connex} .P^{2}\,\unicode{x2abd}\, J.S\in \text{cror}ʻP.\supset .P=\breve{S} ^{;}P.S^{;}P=P\unicode{x0294f}\text{D}ʻS\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*150·41.\supset \vdash \colon\ldotp \text{Hp}.\supset :x(\breve{S} ^{;}P)y.&\equiv .(Sʻx)P(Sʻy).\\ +[*50·43·45] &\supset .Sʻx\neq Sʻy.{\sim}\{(Sʻy)P(Sʻx)\}.\\ +[*30·37.*150·41] &\supset .x\neq y.{\sim}\{y(S^{;}P)x\}.\\ +[*208·12.\text{Transp}] &\supset .x\neq y.{\sim}(yPx) &\qquad \text{(1)}\\ +\vdash .*150·203.&\supset \vdash :x(\breve{S} ^{;}P)y.\supset .x,y\in \text{ᗡ}ʻS &\qquad \text{(2)}\\ +\vdash .(2).*208·1.&\supset \vdash \colon\ldotp \text{Hp}.\supset :x(\breve{S} ^{;}P)y.\supset .x,y\in CʻP &\qquad \text{(3)}\\ +\vdash .(1).(3).*202·103.&\supset \vdash \colon\ldotp \text{Hp}.\supset :x(\breve{S} ^{;}P)y.\supset .xPy &\qquad \text{(4)}\\ +\vdash .(4).*208·12. &\supset \vdash :\text{Hp}.\supset .P=\breve{S} ^{;}P &\qquad \text{(5)}\\ +\vdash .(5). \supset \vdash :\text{Hp}.\supset .S^{;}P&=S^{;}\breve{S} ^{;}P\\ +[*150·38] &=P\unicode{x0294f}\text{D}ʻS &\qquad \text{(6)}\\ +\vdash .(5).(6).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*208·21.</b> \(\vdash :P\in \text{connex} .P^{2}\,\unicode{x2abd}\, J.S\in \text{cror}ʻP.(Sʻx)Px.x\in \text{D}ʻS.\supset .xP(\breve{S} ʻx)\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*33·43.\supset \vdash :\text{Hp}.&\supset .(Sʻx)(P\unicode{x0294f}\text{D}ʻS)x.\\ +[*208·2] &\supset .(Sʻx)(S^{;}P)x.\\ +[*150·41] &\supset .(\breve{S} ʻSʻx)P(\breve{S} ʻx).\\ +[*72·241.*33·43] &\supset .xP(\breve{S} ʻx):\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*208·211.</b> \[\begin{align}&\vdash :P\in \text{connex} .P^{2}\,\unicode{x2abd}\, J.S\in \text{cror}ʻP.xP(Sʻx).x\in \text{D}ʻS.\supset .(\breve{S} ʻx)Px\\ +&[\text{Proof as in *208·21}]\end{align}\]</p> + +<p><span class="pagenum" id="Page_609">[Pg 609]</span></p> + +<p class="nind"><b>*208·22.</b> \[\begin{align}\vdash :P\in \text{connex} .P^{2}\,\unicode{x2abd}\, J.&S\in \text{cror}ʻP.\text{ᗡ}ʻS\subset \text{D}ʻS.\supset .\\ +&\overrightarrow{\text{max}}_{P}ʻ\hat{x}\{(Sʻx)Px\}=\Lambda .\overrightarrow{\text{min}}_{P}ʻ\hat{x}\{xP(Sʻx)\}=\Lambda\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*33·43. \supset \vdash \colon\ldotp \text{Hp}.&\supset :(Sʻx)Px.\supset .x\in \text{D}ʻS.x\in \text{ᗡ}ʻS.\\ +[*208·21] &\supset .xP(\breve{S} ʻx).x\in \text{ᗡ}ʻS.\\ +[*72·241] & \supset .xP(\breve{S} ʻx).\breve{S} ʻx\in \hat{x}\{(Sʻx)Px\}.\\ +[*37·1] &\supset .x\in Pʻʻ\hat{x}\{(Sʻx)Px\} &\qquad \text{(1)}\\ +\vdash .(1).*205·123.\supset &\vdash :\text{Hp}.\supset .\overrightarrow{\text{max}}_{P}ʻ\hat{x}\{(Sʻx)Px\}=\Lambda &\qquad \text{(2)}\\ +\text{Similarly}\quad &\vdash :\text{Hp}.\supset .\overrightarrow{\text{min}}_{P}ʻ\hat{x}\{xP(Sʻx)\}=\Lambda &\qquad \text{(3)}\\ +\vdash .(2).(3).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p>Observe that, in virtue of <a href="#*208·111">*208·111</a>, the above hypothesis gives +\(\text{D}ʻS=\text{ᗡ}ʻS=CʻP\), so that \(S\in P\overline{\,\text{smor}\,}P\). Hence we are led to <a href="#*208·3">*208·3</a>.</p> + +<p class="nind"><b><a id="*208·3">*208·3</a>.</b> \[\begin{align}\vdash :P&\in \text{connex} .P^{2}\,\unicode{x2abd}\, J.S\in P\,\overline{\,\text{smor}\,}\,P.\supset .\\ +&{\sim}\exists !\text{min}_{P}ʻ\hat{x}\{(Sʻx)Px\}.{\sim}\exists !\overrightarrow{\text{max}}_{P}ʻ\hat{x}\{(Sʻx)Px\}.\\ +&{\sim}\exists !\text{min}_{P}ʻ\hat{x}\{xP(Sʻx)\}.{\sim}\exists !\overrightarrow{\text{max}}_{P}ʻ\hat{x}\{xP(Sʻx)\}\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*151·11.*150·23.\supset \vdash :\text{Hp}.&\supset .S\in 1\rightarrow 1.\text{ᗡ}ʻS=CʻP.S^{;}P=P.\text{D}ʻS=CʻP.\\ +[*208·1] &\supset .S\in \text{cror}ʻP.\text{ᗡ}ʻS=\text{D}ʻS &\qquad \text{(1)}\\ +\vdash .(1).*208·14·22.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*208·31.</b> \(\vdash :S,T\in P\,\overline{\,\text{smor}\,}\,Q.\supset .S\mid \breve{T} \in P\,\overline{\,\text{smor}\,}\,P \quad[*151·131·141]\)</p> + +<p class="nind"><b>*208·32.</b> \[\begin{align}\vdash :P&\in \text{connex} .P^{2}\,\unicode{x2abd}\, J.S,T\in P\,\overline{\,\text{smor}\,}\,Q.\supset .\\ +&{\sim}\exists !\overrightarrow{\text{min}}_{P}ʻ\hat{x} \{(Sʻ\breve{T} ʻx)Px\}.{\sim}\exists !\overrightarrow{\text{max}}_{P}ʻ\hat{x}\{(Sʻ\breve{T} ʻx)Px\}.\\ +&{\sim}\exists !\overrightarrow{\text{min}}_{P}ʻ\hat{x}\{xP(Sʻ\breve{T} ʻx)\}.{\sim}\exists !\overrightarrow{\text{max}}_{P}ʻ\hat{x}\{xP(Sʻ\breve{T} ʻx)\}\\\ +&[*208·3·31.*34·41]\end{align}\]</p> + +<p class="nind"><b>*208·4.</b> \[\begin{align}\vdash :P\in \text{connex} .P^{2}\,\unicode{x2abd}\, J.\text{Cl ex}ʻCʻP\subset \text{ᗡ}ʻ&\text{min}_{P}\cup \text{ᗡ}ʻ\text{max}_{P}.\\ +&S,T\in P\,\overline{\,\text{smor}\,}\,Q.\supset .S=T\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*208·32. &\supset \vdash :\text{Hp}.\supset .\hat{x}\{(Sʻ\breve{T} ʻx)Px\}=\Lambda .\hat{x}\{xP(Sʻ\breve{T} ʻx)\}=\Lambda &\qquad \text{(1)}\\ +\vdash .*208·31.*34·41. &\supset \vdash \colon\ldotp \text{Hp}.\supset :x\in CʻP.\supset .Sʻ\breve{T} ʻx\in CʻP &\qquad \text{(2)}\\ +\vdash .(1).(2).*202·103.\supset \vdash \colon\ldotp \text{Hp}.&\supset :x\in CʻP.\supset .Sʻ\breve{T} ʻx=x.\\ +[*72·241] &\supset .\breve{T} ʻx=\breve{S} ʻx:\\ +[*150·23] &\supset :x\in \text{D}ʻS\cup \text{D}ʻT.\supset .\breve{T} ʻx=\breve{S} ʻx:\\ +[*33·46] &\supset :S=T\colon\ldotp \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*208·41">*208·41</a>.</b> \[\begin{align}\vdash :P\in \text{connex} .P^{2}\,\unicode{x2abd}\, J.\text{Cl ex}ʻCʻP\subset \text{ᗡ}ʻ&\text{min}_{P}\cup \text{ᗡ}ʻ\text{max}_{P}.\\ +&P\,\,\text{smor}\,\,Q.\supset .(P\,\overline{\,\text{smor}\,}\,Q)\in 1\\ +[*208·4.*151·12.*52·16]\end{align}\]</p> + +<p><span class="pagenum" id="Page_610">[Pg 610]</span></p> + +<p>The above proposition is of great importance in the theory of +well-ordered series.</p> + +<p class="nind"><b>*208·42.</b> \[\begin{align}\vdash :P\in \text{connex} .P^{2}\,\unicode{x2abd}\, J.\text{Cl ex}ʻCʻP\subset \text{ᗡ}ʻ\text{min}_{P}\cup &\text{ᗡ}ʻ\text{max}_{P}.\supset + .\\ +&P\,\overline{\,\text{smor}\,}\,P={℩}ʻ(I\upharpoonright CʻP)\\ +[*208·4.*51·141.*151·121]\end{align}\]</p> + +<p class="nind"><b>*208·43.</b> \(\vdash :\text{Cl ex}ʻCʻP\subset \text{ᗡ}ʻ\text{min}_{P}.S\in \text{cror}ʻP.\supset .{\sim}(\exists x).(Sʻx)Px \quad[*208·14]\)</p> + +<p class="nind"><b>*208·431.</b> \(\vdash :\text{Cl ex}ʻCʻP\subset \text{ᗡ}ʻ\text{max}_{P}.S\in \text{cror}ʻP.\supset .{\sim}(\exists x).xP(Sʻx) \quad[*208·14]\)</p> + +<p class="nind"><b>*208·44.</b> \[\begin{align}\vdash :P\in \text{connex} .\text{Cl ex}ʻCʻP\subset \text{ᗡ}ʻ\text{min}_{P}\cap &\text{ᗡ}ʻ\text{max}_{P}.S\in \text{cror}ʻP.\supset .\\ +&S=I\upharpoonright CʻP\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*208·43·431.*202·103.\supset \vdash \colon\ldotp \text{Hp}.&\supset :x\in CʻP.\supset .Sʻx=x.\\ +[*50·14.*35·7] &\supset .Sʻx=(I\upharpoonright CʻP)ʻx:\\ +[*208·1.*50·5·52] &\supset :x\in \text{ᗡ}ʻS\cup \text{ᗡ}ʻ(I\upharpoonright CʻP).\supset .Sʻx=(I\upharpoonright CʻP)ʻx:\\ +[*33·45] &\supset :S=I\upharpoonright CʻP\colon\ldotp \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p>In virtue of this proposition, if \(P\) is a finite series, no proper +part of \(P\) is ordinally similar to \(P\). (It will be shown later +that a finite series is one in which every existent contained class has +both a maximum and a minimum.) The following proposition gives a more +explicit form of the above result.</p> + +<p class="nind"><b>*208·45.</b> \(\vdash :P\in \text{connex} .\text{Cl ex}ʻCʻP\subset \text{ᗡ}ʻ\text{min}_{P}\cap \text{ᗡ}ʻ\text{max}_{P}.\supset .\text{Rl}ʻP\cap \text{Nr}ʻP={℩}ʻP\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*208·441. \supset \vdash \colon\ldotp \text{Hp}.&\supset :S\in 1\rightarrow 1.\text{ᗡ}ʻS=CʻP.S^{;}P\,\unicode{x2abd}\, P.\supset .S=I\upharpoonright CʻP.\\ +[*150·534] &\supset .S^{;}P=P &\qquad \text{(1)}\\ +\vdash .(1).*13·12.\supset \vdash \colon\ldotp \text{Hp}.&\supset :Q\,\unicode{x2abd}\, P.S\in 1\rightarrow 1.\text{ᗡ}ʻS=CʻP.Q=S^{;}P.\supset .Q=P:\\ +[*151·1] &\supset :Q\,\unicode{x2abd}\, P.Q\,\,\text{smor}\,\,P.\supset .Q=P:\\ +[*152·1] &\supset :\text{Rl}ʻP\cap \text{Nr}ʻP\subset {℩}ʻP &\qquad \text{(2)}\\ +\vdash .*61·34.*152·3.&\supset \vdash .{℩}ʻP\subset \text{Rl}ʻP\cap \text{Nr}ʻP &\qquad \text{(3)}\\ +\vdash .(2).(3).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p>The following propositions are useful in the theory of segments of +well-ordered series, since they show that a well-ordered series is +never ordinally similar to any of its segments.</p> + +<p class="nind"><b><a id="*208·46">*208·46</a>.</b> \(\vdash :\text{Cl ex}ʻCʻP\subset \text{ᗡ}ʻ\text{min}_{P}.S\in \text{cror}ʻP.\supset .CʻP\cap pʻ\overleftarrow{P}ʻʻ\text{D}ʻS=\Lambda\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*208·1. \supset \vdash \colon\ldotp S\in \text{cror}ʻP.&\supset :x\in CʻP\cap pʻ\overleftarrow{P}ʻʻ\text{D}ʻS.\supset .(Sʻx)Px:\\ +[\text{Transp}] &\supset :{\sim}\{(Sʻx)Px\}.\supset .x{\sim}\in CʻP\cap pʻ\overleftarrow{P}ʻʻ\text{D}ʻS &\qquad \text{(1)}\\ +\vdash .(1).*208·43.&\supset \vdash :\text{Hp}.\supset .(x).x{\sim}\in CʻP\cap pʻ\overleftarrow{P}ʻʻ\text{D}ʻS:\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_611">[Pg 611]</span></p> + +<p class="nind"><b>*208·461.</b> \[\begin{align}&\vdash :\text{Cl ex}ʻCʻP\subset \text{ᗡ}ʻ\text{min}_{P}.S\in \text{cror}ʻP.\dot{\exists} !P.\supset .pʻ\overleftarrow{P}ʻʻ\text{D}ʻS=\Lambda\\ +&[*208·46·1.*40·62]\end{align}\]</p> + +<p class="nind"><b><a id="*208·47">*208·47</a>.</b> \(\vdash :\text{Cl ex}ʻCʻP\subset \text{ᗡ}ʻ\text{min}_{P}.Q\,\unicode{x2abd}\, P.\exists !CʻP\cap pʻ\overleftarrow{P}ʻʻCʻQ.\supset .{\sim}(Q\,\,\text{smor}\,\,P)\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*208·46.(*208·01).\supset \\ +\vdash \colon\ldotp \text{Hp}.&\supset :Q\,\unicode{x2abd}\, P.S\in Q\,\overline{\,\text{smor}\,}\,P.\supset .CʻP\cap pʻ\overleftarrow{P}ʻʻ\text{D}ʻS=\Lambda &\qquad \text{(1)}\\ +\vdash .(1).\text{Transp}.*151·11.*150·23.\supset \\ +\vdash \colon\ldotp \text{Hp}.&\supset :Q\,\unicode{x2abd}\, P.\exists !CʻP\cap pʻ\overleftarrow{P}ʻʻCʻQ.\supset .(S).S{\sim}\in Q\,\overline{\,\text{smor}\,}\,P.\\ +[*151·12] &\supset .{\sim}(Q\,\,\text{smor}\,\,P)\colon\ldotp \supset \vdash .\text{Prop} +\end{array} +\]</p> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_612">[Pg 612]</span></p> +<h2 class="nobreak" id="SECTION_B_III">SECTION B.<br> +<span class="allsmcap">ON SECTIONS, SEGMENTS, STRETCHES, AND DERIVATIVES.</span></h2> +</div> + + +<p>In this section, our chief topic will be <i>sections</i> and +<i>segments</i>. This topic will occupy <a href="#*211">*211</a>, <a href="#*212">*212</a> and <a href="#*213">*213</a>, and +<a href="#*210">*210</a> will consist of propositions whose chief utility lies in their +application to segments. In <a href="#*214">*214</a>, we shall consider Dedekindian +series, which are intimately connected with segments, owing to the +fact that one of the chief propositions in the subject is that +the series of segments of a series is Dedekindian. In <a href="#*215">*215</a>, we +shall consider "stretches," which consist of any consecutive piece +of a series, and are constituted by the product of an upper and +lower section. Finally, in <a href="#*216">*216</a>, we shall consider the derivative +of a series, or of a class \(\alpha\) contained in a series: the +former is the series of limit-points of the series, <i>i.e.</i> +\(P\unicode{x0294f}\text{D}ʻ\text{lt}_P\), the latter is the class of +limits of existent sub-classes of \(\alpha \cap CʻP\), <i>i.e.</i> +\(\text{lt}_Pʻʻ\text{Cl ex}ʻ(\alpha \cap CʻP)\).</p> + +<p>A class is called a <i>section</i> of \(P\) when it is contained in +\(CʻP\), and contains all the predecessors of its members, <i>i.e.</i> +\(\alpha\) is a section of \(P\) if \(\alpha \subset CʻP.Pʻʻ\alpha\subset \alpha\). +Thus a section consists of all the field up to a certain point. +It may consist of all the predecessors of \(x\), <i>i.e.</i> it +may be of the form \(\overrightarrow{P}ʻx\); or again, it may +consist of these together with \(x\), in which case it is of the +form \(\overrightarrow{P}ʻx\cup {℩}ʻx\); or again, it may be not +definable by means of a single sequent or maximum, but be of the form +\(Pʻʻ\alpha\), where \(\alpha\) is a class without a limit or maximum. +The class of sections of \(P\) is denoted by \(\text{sect}ʻP\). A +section of \(\breve{P}\) will be called an "upper section" of \(P\).</p> + +<p>The idea of a <i>segment</i> is slightly less general than that of a +<i>section</i>. We define a segment of \(P\) as any class of the form +\(Pʻʻ\alpha\), <i>i.e.</i> as any member of \(\text{D}ʻP_{\in }\). +Provided \(P\) is transitive, segments are contained among sections. +But even in a series sections are not, in general, contained among +segments: if \(P\) is a series, and if \(x\) is a member of \(CʻP\) +which has no immediate successor, \(\overrightarrow{P}ʻx\cup {℩}ʻx\) +will be a section but not a segment.</p> + +<p>If a segment has a maximum, it must also have a sequent. Segments which +have no maximum form a specially important class of segments: these are +classes \(\alpha\) such that \(\alpha =Pʻʻ\alpha\); they form the class +<span class="pagenum" id="Page_613">[Pg 613]</span>\(\text{D}ʻ(P_{\in }\dot{\cap} I)\).</p> + +<p>The properties of sections and segments considered as classes of +classes are many and various: they are considered in <a href="#*211">*211</a>. In <a href="#*212">*212</a>, +we pass to the consideration of the series of sections and segments. +These series are \(P_{\text{lc}}\unicode{x0294f}\text{sect}ʻP\) +and \(P_{\text{lc}}\unicode{x0294f}\text{D}ʻP_{\in}\) (cf. +<a href="#*170">*170</a>). The series of such segments as have no maximum is +\(P_{\text{lc}}\unicode{x0294f}\text{D}ʻ(P_{\in }\dot{\cap} I)\). We put +\[ +\begin{aligned} +\varsigma ʻP&=P_{\text{lc}}\unicode{x0294f}\text{D}ʻP_{\in } \quad\text{Df},\\ +\text{sgm}ʻP&=P_{\text{lc}}\unicode{x0294f}\text{D}ʻ(P_{\in }\dot{\cap} I) \quad\text{Df}. +\end{aligned} +\] +It then appears that +\[ +\varsigma ʻP_{*}=\text{sgm}ʻP_{*}=P_{\text{lc}}\unicode{x0294f}\text{sect}ʻP, +\] +so that it is unnecessary to introduce a special notation for the +series of sections.</p> + +<p>Whenever \(P\) is connected and transitive, +\(P_{\text{lc}}\unicode{x0294f}\text{D}ʻP_{\in}\) turns out to be +equivalent to logical inclusion combined with diversity (with the field +limited to \(\text{D}ʻP_{\in }\)). That is to say (<a href="#*212·23">*212·23</a>), +\[ +\vdash :P\in \text{trans}\cap \text{connex} .\supset .\varsigma ʻP=\hat{\alpha} \hat{\beta}\{\alpha ,\beta \in \text{D}ʻP_{\in }.\alpha \subset \beta .\alpha \neq \beta\}. +\] +Hence it follows (<a href="#*212·24">*212·24</a>) that +\[ +\vdash :P_{*}\in \text{connex} .\supset .\varsigma ʻP_{*}=\hat{\alpha} \hat{\beta} \{\alpha ,\beta \in \text{sect}ʻP.\alpha \subset \beta .\alpha \neq \beta\}· +\] +We have also (<a href="#*211·6">*211·6</a><a href="#*211·17">·17</a>) +\[ +\vdash \colon\ldotp P_{*}\in \text{connex} .\alpha ,\beta \text{sect}ʻP.\supset :\alpha \subset \beta .\lor.\beta \subset \alpha +\] +Hence it easily follows that whenever \(P_{*}\) is connected, +\(\varsigma ʻP_{*}\) is a series. Similarly \(\varsigma ʻP\) will be a +series if \(P\) is transitive and connected.</p> + +<p>The fact of connection, which is required in order that \(\varsigmaʻP\) +or \(\varsigma ʻP_{*}\) may be a series, results from +\[ +\begin{aligned} +\alpha ,\beta \in \text{sect}ʻP.&\supset :\alpha \subset \beta .\lor.\beta \subset \alpha \\ +\text{or}\quad \alpha ,\beta \in \text{D}ʻP_{\in }.&\supset :\alpha \subset \beta .\lor.\beta \subset \alpha . +\end{aligned} +\] +In order to deal with such cases generally, we study, in a preliminary +number (<a href="#*210">*210</a>), the consequences to be deduced from the hypothesis +\[ +\alpha ,\beta \in \kappa .\supset _{\alpha ,\beta }:\alpha \subset \beta .\lor.\beta \subset \alpha . +\] +We find that, with this hypothesis, putting +\[ +Q=\hat{\alpha} \hat{\beta} (\alpha ,\beta \in \kappa .\alpha \subset \beta .\alpha \neq \beta ), +\] +\(Q=P_{\text{lc}}\unicode{x0294f}\kappa\) if \(\kappa \subset\text{Cl}ʻCʻP\) +(<a href="#*210·13">*210·13</a>), and thus in the same circumstances +\(P_{\text{lc}}\unicode{x0294f}\kappa\) is a series (<a href="#*210·14">*210·14</a>).</p> + +<p>The interesting point about such series is their behaviour with +regard to limits. Assuming that \(\kappa\) is not a unit class (so +as to insure \(\dot{\exists} !Q)\), if \(\lambda\) is any sub-class +of \(\kappa\), the logical product \(pʻ\lambda\) is the minimum of +\(\lambda\) if it is a member of \(\lambda\) (<a href="#*210·21">*210·21</a>), and the +lower limit of \(\lambda\) if it is a member of \(\kappa\) but not +of \(\lambda\) (<a href="#*210·23">*210·23</a>). Similarly \(sʻ\lambda\) is the maximum of +\(\lambda\) if it is a member of \(\lambda\) (<a href="#*210·211">*210·211</a>), and the upper +limit of \(\lambda\) if it is not a member of \(\lambda\) but is a +member of \(\kappa\) (<a href="#*210·231">*210·231</a>). Thus if \(\kappa\) is such that, +whenever \(\lambda \subset \kappa\), we have<span class="pagenum" id="Page_614">[Pg 614]</span> \(sʻ\lambda \in \kappa\), +it follows that every sub-class of \(\kappa\) has either a maximum or a +limit, <i>i.e.</i> the series \(P_{\text{lc}}\unicode{x0294f}\kappa\) +is Dedekindian. Now each of the three classes \(\text{sect}ʻP\), +\(\text{D}ʻP_{\in }\), \(\text{D}ʻ(P_{\in }\dot{\cap} I)\) verifies +this condition, <i>i.e.</i> the sum of any sub-class of any one of +these classes belongs to the class in question (<a href="#*211·63">*211·63</a><a href="#*211·64">·64</a><a href="#*211·65">·65</a>). (This +holds without any hypothesis as to \(P\).) Hence we arrive at the +result that \(\varsigma ʻP_{*}\) (<i>i.e.</i> the series of sections) +is a Dedekindian series whenever \(P_{*}\) is connected and \(P\) is +not null (<a href="#*214·32">*214·32</a>), while \(\varsigma ʻP\) (<i>i.e.</i> the series of +segments) is a Dedekindian series whenever \(P\) is transitive and +connected and not null (<a href="#*214·33">*214·33</a>), and \(\text{sgm}ʻP\) (the series of +segments having no maximum) is a Dedekindian series whenever it exists +and \(P\) is connected (<a href="#*214·34">*214·34</a>). These propositions are important, and +are the source of much of the utility of sections and segments.</p> + +<p>For many purposes, especially in ordinal arithmetic, it is necessary +to consider sections not as classes, but as series. That is to say, if +\(\alpha\) is a member of \(\text{sect}ʻP\), we want to deal with +\(P\unicode{x0294f} \alpha\) rather than with \(\alpha\). The series of +all such terms as \(P \unicode{x0294f} \alpha\) might be supposed to +be \(P \unicode{x0294f} ^{;}\varsigma ʻP_{*}\). But here a limitation is +necessary owing to the fact that, if \(BʻP\) exists, \(\Lambda\) and +\(\iota ʻBʻP\) are both sections, and \(P \unicode{x0294f} \Lambda\) +and \(P \unicode{x0294f} \iota ʻBʻP\) are both \(\dot{\Lambda}\), so +that \(P \unicode{x0294f} ^{;}\varsigma ʻP_{*}\) will be a relation which +\(\dot{\Lambda}\) will have to itself. In order to avoid this, we first +exclude \(\Lambda\) from the sections to be considered, and thus put +\[ +P_{\varsigma }=P\unicode{x0294f}^{;}(\varsigma ʻP_{*})\unicode{x0294f}(-\iota ʻ\Lambda ) \quad\text{Df}. +\] +Then \(P_{\varsigma }\) is the series of segments considered as series. +Provided \(P_{\text{po}}\) is a series, the relation \(P_{\varsigma}\) +holds between any two members \(Q\) and \(R\) of its field +when, and only when, \(Q\,\unicode{x2abd}\, R.Q\neq R\). The subject +of \(P_{\varsigma }\) is considered in <a href="#*213">*213</a>; the utility of the +propositions of this number will not appear until we come to ordinal +arithmetic.</p> + +<p>The subject of Dedekindian relations is next considered (<a href="#*214">*214</a>). We +define a Dedekindian relation as one such that every class has either +a maximum or a sequent. A Dedekindian series must have a first and a +last term, since the first term must be the sequent of \(\Lambda\), and +the last must be the maximum of the field. A Dedekindian series may be +discrete, or compact (<i>i.e.</i> such that there is a term between +any two, <i>i.e.</i> such that \(P^{2}=P\)), or partly one and partly +the other. A finite series must be Dedekindian: a well-ordered series +is Dedekindian if it has a last term. But the chief importance of the +Dedekindian property is in connection with compact series. A compact +Dedekindian series is said to possess "Dedekindian continuity"; such +series have many important properties. They are a wider class than +series possessing Cantorian continuity; these latter will be considered +in Section F of this Part.</p> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_615">[Pg 615]</span></p> +<h2 class="nobreak" id="*210"> +*210. ON SERIES OF CLASSES GENERATED BY THE RELATION +OF INCLUSION.</h2> +</div> + + +<p class="nind"><i>Summary of</i> *210.</p> + +<p>In the theory of series it frequently happens that we have to deal +with a class of classes such that, of any two, one is contained in the +other. <i>I.e.</i> if \(\kappa\) is the class of classes, we have +\[ +\alpha ,\beta \in \kappa .\supset _{\alpha ,\beta }:\alpha \subset \beta .\lor.\beta \subset \alpha . +\]</p> + +<p>Instances of this are afforded by the various classes of sections, to +be considered in <a href="#*211">*211</a>. When \(\kappa\) fulfils the above condition, +the classes composing \(\kappa\) can be arranged in a series by the +relation of inclusion (combined with inequality), <i>i.e.</i> by the +relation +\[ +\hat{\alpha} \hat{\beta} (\alpha ,\beta \in \kappa .\alpha \subset \beta .\alpha \neq \beta ), +\] +or, what comes to the same, +\[ +\hat{\alpha} \hat{\beta} (\alpha ,\beta \in \kappa .\exists !\beta -\alpha ). +\] +If \(P\) is any relation such that \(\kappa \subset \text{Cl}ʻCʻP\), +the above relation of inclusion is equal to +\[ +P_{\text{lc}}\unicode{x0294f}\kappa . +\] +(For the definition of \(P_{\text{lc}}\), see <a href="#*170">*170</a>.) Thus under the +above circumstances, \(P_{\text{lc}}\unicode{x0294f}\kappa\) is a +series, whatever \(P\) may be.</p> + +<p>The importance of such relations of inclusion, as generators of series, +is in connection with the existence of maxima and minima or limits. If +we put +\[ +Q=\hat{\alpha} \hat{\beta} (\alpha ,\beta \in \kappa .\exists !\beta -\alpha ). +\] +where \(\kappa\) satisfies the above condition, then if +\(\lambda\subset \kappa\) and if \(sʻ\lambda \in \kappa\), \(sʻ\lambda\) is +the maximum or the upper limit of \(\lambda\) with respect to \(Q\), +according as \(sʻ\lambda\) is a member of \(\lambda\) or not. Similarly +if \(pʻ\lambda \in \kappa\), p\(ʻ\lambda\) is the minimum or lower +limit of \(\lambda\), according as \(pʻ\lambda\) is a member of +\(\lambda\) or not. Hence if \(\kappa\) is such that the sum of any +sub-class of \(\kappa\) is a member of \(\kappa\), every sub-class of +\(\kappa\) has either a maximum or an upper limit; and if the product +of every sub-class of \(\kappa\) is a member of \(\kappa\), every +sub-class of \(\kappa\) has either a minimum or a lower limit.</p> + +<p>In order that every sub-class of \(\kappa\) should have a minimum or +a lower limit, it is sufficient that the sum of every sub-class of +\(\kappa\) should be a member<span class="pagenum" id="Page_616">[Pg 616]</span> of \(\kappa\). For, if \(\lambda\) is +any sub-class of \(\kappa\), consider those members of \(\kappa\) which +are contained in \(pʻ\lambda\), <i>i.e.</i> +\[ +\kappa \cap \text{Cl}ʻpʻ\lambda +\] +If \(pʻ\lambda \in \kappa\), the sum of these classes = \(pʻ\lambda\), +and is the lower limit or minimum of \(\kappa\). But if +\(pʻ\lambda{\sim}\in \kappa\), then every member of \(\kappa\) which is not +contained in \(sʻ(\kappa \cap \text{Cl}ʻpʻ\lambda )\) is also not +contained in \(pʻ\lambda\), and is therefore not contained in some +member of \(\lambda\). Hence \(sʻ(\kappa \cap \text{Cl}ʻpʻ\lambda )\) +is the lower limit of \(\lambda\).</p> + +<p>It is owing to these propositions that segments of series are of such +great importance in connection with limits.</p> + +<p>The hypothesis that if \(\lambda \subset \kappa\), \(pʻ\lambda\) is a +member of \(\kappa\), will usually fail to be verified in the case when +\(\lambda =\Lambda\), since in this case \(pʻ\lambda =\text{V}\). But +all the results desired can be obtained from the hypothesis that, if +\(\lambda \subset \kappa\), \((pʻ\lambda \cap sʻ\kappa )\in \kappa\). +This hypothesis is equivalent to the other except in the case of +\(\Lambda\), in which case it requires \(sʻ\kappa \in \kappa\), which +is much more often verified than \(\text{V}\in \kappa\), which was +required by the other hypothesis.</p> + +<p>The principal propositions of this number are the following:</p> + +<p class="nind"><b>*210·1.</b> \(\vdash \colon\colon\alpha ,\beta \in \kappa .\supset _{\alpha ,\beta }:\alpha + \subset \beta .\lor.\beta \subset \alpha \colon\ldotp \supset \colon\ldotp \alpha ,\beta \in \kappa .\supset :\alpha \subset \beta .\alpha \neq \beta .\equiv .\exists !\beta -\alpha\)</p> + +<p class="nind"><b>*210·11.</b> \(\vdash :Q=\hat{\alpha} \hat{\beta} (\alpha ,\beta \in \kappa .\alpha \subset \beta .\alpha \neq \beta ).\supset .Q\in \text{trans}\cap \text{Rl}ʻJ\)</p> + +<p class="nind"><b>*210·12.</b> \(\vdash :\text{Hp} *210·1·11.\supset .Q\in \text{Ser}\)</p> + +<p class="nind"><b>*210·13.</b> \(\vdash :\text{Hp} *210·12.\kappa \subset \text{Cl}ʻCʻP.\supset .Q=P_{\text{lc}}\unicode{x0294f}\kappa\)</p> + +<p class="nind"><b>*210·2.</b> \(\vdash :\text{Hp} *210·12.\kappa {\sim}\in 1.\supset .\overrightarrow{\text{min}}_{Q}ʻ\lambda =\lambda \cap \kappa \cap \iota ʻpʻ(\lambda \cap \kappa )\)</p> + +<p class="nind"><b>*210·21.</b> \(\vdash :\text{Hp} *210·2.\lambda \subset \kappa .pʻ\lambda \in \lambda .\supset .\text{min}_{Q}ʻ\lambda =pʻ\lambda\)</p> + +<p><a href="#*210·211">*210·211</a> gives an analogous proposition for \(sʻ\lambda\) and +\(\text{max}_{Q}\). We shall not here mention such analogues, unless +for some special reason.</p> + +<p class="nind"><b>*210·23.</b> \(\vdash :\text{Hp} *210·2.\lambda \subset \kappa .pʻ\lambda -\lambda .\supset .pʻ\lambda =\text{prec}_{Q}ʻ\lambda =\text{tl}_{Q}ʻ\lambda\)</p> + +<p class="nind"><b>*210·232.</b> \(\vdash :\text{Hp} *210·2.\lambda \subset \kappa .pʻ\lambda \in \kappa .\supset .pʻ\lambda =\text{limin}_Pʻ\lambda\)</p> + +<p class="nind"><b>*210·251.</b> \(\vdash :\text{Hp} *210·2:\lambda \subset \kappa .\supset _{\lambda }.sʻ\lambda + \in \kappa :\supset :\lambda \in \kappa .\supset .sʻ\lambda (\overrightarrow{\text{max}}_{Q}ʻ\lambda \cup \overrightarrow{\text{seq}} _{Q}ʻ\lambda )\)</p> + +<p class="nind"><b>*210·252.</b> \[\begin{align}\vdash :&\text{Hp} *210·2:\lambda \subset \kappa .\supset _{\lambda }.pʻ\lambda \cap sʻ\kappa :\supset :\\ +&\lambda \subset \kappa .\supset .pʻ\lambda \cap sʻ\lambda \in (\overrightarrow{\text{min}}_{Q}ʻ\lambda \cup \overrightarrow{\text{prec}} _{Q}ʻ\lambda + ).pʻ\lambda \cap sʻ\kappa =\text{limin}_{Q}ʻ\lambda\end{align}\]</p> + +<p class="nind"><b>*210·254.</b> \(\vdash :\text{Hp} *210·251.\supset .(\lambda ).\lambda \in \text{ᗡ}ʻ\text{max}_{Q}\cup \text{ᗡ}ʻ\text{seq}_{Q}\)</p> + +<p class="nind"><b>*210·26.</b> \[\begin{align}\vdash :\text{Hp} *210·2.\lambda \subset \kappa .pʻ\lambda {\sim}\in \lambda .&sʻ(\kappa \cap \text{Cl}ʻpʻ\lambda )\in \kappa .\supset .\\ +&sʻ(\kappa \cap \text{Cl}ʻpʻ\lambda )=\text{prec}_{Q}ʻ\lambda\end{align}\]</p> + +<p class="nind"><b>*210·28.</b> \[\begin{align}\vdash :\text{Hp} *210·2.sʻʻ&\text{Cl}ʻ\kappa \subset \kappa .\supset .\\ +&(\lambda ).\lambda \in (\text{ᗡ}ʻ\text{max}_{Q}\cup \text{ᗡ}ʻ\text{seq}_{Q})(\text{ᗡ}ʻ\text{min}_{Q}\cup \text{ᗡ}ʻ\text{prec}_{Q})\end{align}\]</p> + +<p>Thus if \(\kappa\) is a class of not less than two classes such that, +of any two of its members, one must be contained in the other, and if +\(Q\) is the relation<span class="pagenum" id="Page_617">[Pg 617]</span> \(\alpha \subset \beta .\alpha \neq \beta\) +confined to members of \(\kappa\), then \(Q\) is a series (<a href="#*210·12">*210·12</a>) +in which, provided the sums of sub-classes of \(\kappa\) are always +members of \(\kappa\), every class has either a maximum or an upper +limit, and every class has either a minimum or a lower limit (<a href="#*210·28">*210·28</a>).</p> + +<p>The reader will observe that, if \(\alpha\), +\(\beta \in \kappa.\supset _{\alpha ,\beta }:\alpha \subset \beta .\lor.\beta \subset\alpha\), +any <i>finite</i> sub-class of \(\kappa\) must contain its own sum and +product as members. For example, if we have two classes \(\alpha\) and +\(\beta\), if \(\alpha \subset \beta\), then \(\alpha =pʻ(\iota ʻ\alpha\cup \iota ʻ\beta )\) +and \(\beta =sʻ(\iota ʻ\alpha \cup \iota ʻ\beta)\); if we have three +classes \(\alpha\), \(\beta\), \(\gamma\), and \(\alpha \subset \beta.\beta \subset \gamma\), +then \(\alpha =pʻ(\iota ʻ\alpha \cup \iotaʻ\beta \cup \iota ʻ\gamma )\) +and \(\gamma =sʻ(\iota ʻ\alpha \cup\iota ʻ\beta \cup \iota ʻ\gamma )\); +and so on. Thus the hypothesis \(sʻʻ\text{Cl}ʻ\kappa \subset \kappa\) +is only required in order to enable us to deal with infinite +sub-classes of \(\kappa\).</p> + +<hr class="tb"> + +<p class="nind"><b><a id="*210·1">*210·1</a>.</b> \(\vdash \colon\colon\alpha ,\beta \in \kappa .\supset _{\alpha ,\beta }:\alpha + \subset \beta .\lor.\beta \subset \alpha \colon\ldotp \supset \colon\ldotp \alpha ,\beta \in \kappa .\supset :\alpha \subset \beta .\alpha \neq \beta .\equiv .\exists !\beta -\alpha\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*24·6. &\supset \vdash :\alpha \subset \beta .\alpha \neq \beta .\supset .\exists !\beta -\alpha &\qquad \text{(1)}\\ +\vdash .*24·55. &\supset \vdash :\exists !\beta -\alpha .\supset .{\sim}(\beta \subset \alpha ). &\qquad \text{(2)}\\ +[*22·42] &\supset .\alpha \neq \beta &\qquad \text{(3)}\\ +\vdash .*2·53. &\supset \vdash \colon\ldotp \text{Hp}.\alpha ,\beta \in \kappa .{\sim}(\beta \subset \alpha ).\supset .\alpha \subset \beta &\qquad \text{(4)}\\ +\vdash .(2).(3).(4). &\supset \vdash \colon\ldotp \text{Hp}.\alpha ,\beta \in \kappa .\supset :\exists !\beta -\alpha .\supset .\alpha \subset \beta .\alpha \neq \beta &\qquad \text{(5)}\\ +\vdash .(1).(5).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*210·11.</b> \(\vdash :Q=\hat{\alpha} \hat{\beta} (\alpha ,\beta \in \kappa .\alpha \subset \beta .\alpha \neq \beta ).\supset .Q\in \text{trans}\cap \text{Rl}ʻJ\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*50·11. &\supset \vdash :\text{Hp}.\supset .Q\in \text{Rl}ʻJ &\qquad \text{(1)}\\ +\vdash .*22·44. &\supset \vdash \colon\ldotp \text{Hp}.\supset :\alpha Q\beta .\beta Q\gamma .\supset .\alpha \subset \gamma &\qquad \text{(2)}\\ +\vdash .*24·6.*21·33. \supset \vdash \colon\ldotp \text{Hp}.&\supset :\alpha Q\beta .\beta Q\gamma .\supset .\exists !\beta -\alpha .\beta \subset \gamma .\\ +[*24·58] &\supset .\exists !\gamma -\alpha .\\ +[*24·21] &\supset .\alpha \neq \gamma &\qquad \text{(3)}\\ +\vdash .(2).(3).&\supset \vdash \colon\ldotp \text{Hp}.\supset :\alpha Q\beta .\beta Q\gamma .\supset .\alpha Q\gamma &\qquad \text{(4)}\\ +\vdash .(1).(4).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*210·12">*210·12</a>.</b> \(\vdash :\text{Hp}*210·1·11.\supset .Q\in \text{Ser}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*10·1. \supset \vdash \colon\ldotp \text{Hp}.\alpha ,\beta \in \kappa .&\supset :\alpha \subset \beta .\lor.\beta \subset \alpha :\\ +[*5·62] &\supset :\alpha \subset \beta .\alpha \neq \beta .\lor.\beta \subset \alpha .\beta \neq \alpha .\lor.\alpha =\beta &\qquad \text{(1)}\\ +\vdash .*21·33. \supset \vdash \colon\ldotp \text{Hp}.&\supset :\alpha Q\beta .\supset _{\alpha ,\beta }.\alpha ,\beta \in \kappa :\\ +[*33·352] &\supset :CʻQ\subset \kappa &\qquad \text{(2)}\\ +\vdash .(1).(2). &\supset \vdash \colon\colon\text{Hp}.\supset \colon\ldotp \alpha ,\beta \in CʻQ.\supset :\alpha Q\beta .\lor.\beta Q\alpha .\lor.\alpha =\beta &\qquad \text{(3)}\\ +\vdash .*210·11.(3).*204·12.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_618">[Pg 618]</span></p> + +<p class="nind"><b>*210·121.</b> \(\vdash : \text{Hp} *210·12 . \supset . \text{D}ʻQ = \kappa - \iota ʻsʻ\kappa . \text{ᗡ}ʻQ = \kappa - \iota ʻpʻ\kappa\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *21·33 . \supset \vdash \colon\colon \text{Hp} . \supset \colon\ldotp \alpha \in \text{D}ʻQ . &\equiv : \alpha \in \kappa : (\exists \beta ) . \beta \in \kappa . \alpha \subset \beta . \alpha \neq \beta :\\ +[*210·1] &\equiv : \alpha \in \kappa : (\exists \beta ) . \beta \in \kappa . \exists ! \beta - \alpha :\\ +[*40·151.\text{Transp}] &\equiv : \alpha \in \kappa . \exists ! sʻ\kappa - \alpha :\\ +[*24·55] &\equiv : \alpha \in \kappa . {\sim} (sʻ\kappa \subset \alpha ) :\\ +[*22·41 . *40·13] &\equiv : \alpha \in \kappa . \alpha \neq sʻ\kappa &\qquad \text{(1)}\\ +\vdash . *21·33 . \supset \vdash \colon\colon \text{Hp} . \supset \colon\ldotp \alpha \in \text{ᗡ}ʻQ . &\equiv : \alpha \in \kappa : (\exists \beta ) . \beta \in \kappa . \beta \subset \alpha . \beta \neq \alpha :\\ +[*210·1] &\equiv : \alpha \in \kappa : (\exists \beta ) . \beta \in \kappa . \exists ! \alpha - \beta :\\ +[*40·15 . \text{Transp}] &\equiv : \alpha \in \kappa . \exists ! \alpha - pʻ\kappa :\\ +[*24·55] &\equiv : \alpha \in \kappa . {\sim} (\alpha \subset pʻ\kappa ) :\\ +[*22·41 . *40·12] &\equiv : \alpha \in \kappa . \alpha \neq pʻ\kappa &\qquad \text{(2)}\\ +\vdash . (1) . (2) . \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*210·122.</b> \(\vdash : \text{Hp} *210·12 . \kappa {\sim} \in 1 . \supset . CʻQ = \kappa\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *52·181 . \supset \vdash \colon\colon \text{Hp} . &\supset \colon\ldotp \alpha \in \kappa . \supset : (\exists \beta ) . \beta \in \kappa . \beta \neq \alpha :\\ +[\text{Hp} . *10·1] &\supset : (\exists \beta ) : \beta \in \kappa . \beta \neq \alpha : \alpha \subset \beta . \lor . \beta \subset \alpha :\\ +[*21·33] &\supset : (\exists \beta ) : \beta \in \kappa : \alpha Q\beta . \lor . \beta Q\alpha :\\ +[*33·132] &\supset : \alpha \in CʻQ &\qquad \text{(1)}\\ +\vdash . *21·33 . \supset \vdash \colon\ldotp \text{Hp} . &\supset : \alpha Q\beta . \supset _{\alpha , \beta } . \alpha , \beta \in \kappa :\\ +[*33·352] &\supset : CʻQ \subset \kappa &\qquad \text{(2)}\\ +\vdash . (1) . (2) . \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*210·123.</b> \(\vdash : \text{Hp} *210·12 . \kappa \in 0 \cup 1 . \supset . Q = \dot{\Lambda}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *52·41 . \text{Transp} . \supset \vdash : \text{Hp} . &\supset . {\sim} (\exists \alpha , \beta ) . \alpha , \beta \in \kappa . \alpha \neq \beta .\\ +[*21·33] &\supset . {\sim} (\exists \alpha , \beta ) . \alpha Q\beta : \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*210·124.</b> \(\vdash \colon\ldotp \text{Hp} *210·12 . \supset : \alpha Q\beta . \equiv . \alpha , \beta \in \kappa . \exists ! \beta - \alpha \quad[*210·1]\)</p> + +<p class="nind"><b><a id="*210·13">*210·13</a>.</b> \(\vdash : \text{Hp} *210·12 . \kappa \subset \text{Cl}ʻCʻP . \supset . Q = P_{\text{lc}} \unicode{x0294f} \kappa\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *170·102 . \supset \vdash \colon\ldotp \text{Hp} . \supset : \alpha (P_{\text{lc}} + \unicode{x0294f} \kappa )\beta . &\equiv . \alpha , \beta \in \kappa . \exists ! \beta - \alpha - Pʻʻ(\alpha - \beta ) . &\qquad \text{(1)}\\ +[*210·124] &\supset . \alpha Q\beta &\qquad \text{(2)}\\ +\vdash . *210·1·124 . \supset \vdash \colon\ldotp \text{Hp} . \supset : \alpha Q\beta . &\supset . \alpha , \beta \in \kappa . \exists ! \beta - \alpha . \alpha \subset \beta .\\ +[*37·29] &\supset . \alpha , \beta \in \kappa . \exists ! \beta - \alpha . Pʻʻ(\alpha - \beta ) = \Lambda .\\ +[*24·23·313] &\supset . \alpha , \beta \in \kappa . \exists ! \beta - \alpha - Pʻʻ(\alpha - \beta ) .\\ +[(1)] &\supset . \alpha (P_{\text{lc}} \unicode{x0294f} \kappa ) \beta &\qquad \text{(3)}\\ +\vdash . (2) . (3) . \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_619">[Pg 619]</span></p> + +<p>Thus under the hypothesis of <a href="#*210·1">*210·1</a>, +\(P_{\text{lc}}\unicode{x0294f}\kappa\) does not depend upon \(P\), so +long as \(\kappa \subset \text{Cl}ʻCʻP\). Also we have</p> + +<p class="nind"><b><a id="*210·14">*210·14</a>.</b> \[\begin{align}&\vdash :\text{Hp}*210·1.\kappa \subset \text{Cl}ʻCʻP.\supset .P_{\text{lc}}\unicode{x0294f}\kappa \in \text{Ser}\\ +&[*210·12·13]\end{align}\]</p> + +<p class="nind"><b>*210·15.</b> \[\begin{align}&\vdash \colon\ldotp \text{Hp}*210·12.\alpha ,\beta \in \kappa .\supset :{\sim}(\alpha Q\beta ).\equiv .\beta \subset \alpha \\ +&[*210·124.*24·55]\end{align}\]</p> + +<p class="nind"><b>*210·16.</b> \[\begin{align}&\vdash \colon\colon\text{Hp}*210·1.\supset \colon\ldotp \\ +&\alpha \in \kappa .\lambda \subset \kappa .\supset :\alpha \subset pʻ\lambda .\lor.pʻ\lambda \subset \alpha :\alpha \subset sʻ\lambda .\lor.sʻ\lambda \subset \alpha\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*10·1.\supset \vdash \colon\colon\text{Hp}.\alpha \in \kappa .\lambda \subset \kappa .&\supset \colon\ldotp \beta \in \lambda .\supset _{\beta }:\alpha + \subset \beta .\lor.\beta \subset \alpha \colon\ldotp &\qquad \text{(1)}\\ +[*10·57] &\supset \colon\ldotp \beta \in \lambda .\supset _{\beta }.\alpha \subset \beta :\lor:(\exists \beta ).\beta \in \lambda .\beta \subset \alpha \colon\ldotp \\ +[*40·15·12]&\supset \colon\ldotp \alpha \subset pʻ\lambda .\lor.pʻ\lambda \subset \alpha &\qquad \text{(2)}\\ +\vdash .(1).*10·57.\supset \\ +\vdash \colon\colon\text{Hp}.\alpha \in \kappa .\lambda \subset \kappa .&\supset \colon\ldotp \beta \in \lambda .\supset _{\beta }.\beta + \subset \alpha :\lor:(\exists \beta ).\beta \in \kappa .\alpha \subset \beta \colon\ldotp \\ +[*40·151·13] &\supset \colon\ldotp sʻ\lambda \subset \alpha .\lor.\alpha \subset sʻ\lambda &\qquad \text{(3)}\\ +\vdash .(2).(3).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*210·17.</b> \[\begin{align}\vdash :\text{Hp}*210·12.&\lambda \subset \kappa .\supset .\\ +&\kappa -\breve{Q} ʻʻ\lambda =\kappa \cap \text{Cl}ʻpʻ\lambda .\kappa -Qʻʻ\lambda =\kappa \cap \hat{\gamma} (sʻ\lambda \subset y)\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*37·105.\text{Transp}.&\supset \vdash \colon\ldotp \alpha \in \kappa -\breve{Q} ʻʻ\lambda .\equiv :\alpha \in \kappa :\beta \in \lambda .\supset _{\beta }.{\sim}(\beta + Q\alpha ) &\qquad \text{(1)}\\ +\vdash .(1).*210·15.\supset \\ +\vdash \colon\colon\text{Hp}.\supset \colon\ldotp \alpha \in \kappa -\breve{Q} ʻʻ\lambda .&\equiv :\alpha \in \kappa :\beta \in \lambda .\supset _{\beta }.\alpha \subset \beta :\\ +[*40·15] &\equiv :\alpha \in \kappa \cap \text{Cl}ʻpʻ\lambda &\qquad \text{(2)}\\ +\text{Similarly}\quad \vdash \colon\ldotp \text{Hp}.&\supset :\alpha \in \kappa -Qʻʻ\lambda .\equiv .\alpha \in \kappa \cap \hat{\gamma} (sʻ\lambda \subset \gamma ) &\qquad \text{(3)}\\ +\vdash .(2).(3).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*210·2.</b> \(\vdash :\text{Hp}*210·12.\kappa {\sim}\in 1.\supset .\overrightarrow{\text{min}}_{Q}ʻ\lambda =\lambda \cap \kappa \cap \iota ʻpʻ(\lambda \cap \kappa )\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*205·15.*210·122.\supset \vdash :\text{Hp}.\supset .\overrightarrow{\text{min}}_{Q}ʻ\lambda &=\overrightarrow{\text{min}}_{Q}ʻ(\lambda \cap \kappa )\\ +[*205*11] & =\lambda \cap \kappa -\breve{Q} ʻʻ(\lambda \cap \kappa )\\ +[*210·17] & =\lambda \cap \kappa \cap \text{Cl}ʻpʻ(\lambda \cap \kappa ) &\qquad \text{(1)}\\ +\vdash .*40·12.\supset \vdash \colon\ldotp \alpha \in \lambda \cap \kappa .&\supset :pʻ(\lambda \cap \kappa )\subset \alpha :\\ +[*22·41] &\supset :\alpha \subset pʻ(\lambda \cap \kappa ).\equiv .\alpha =pʻ(\lambda \cap \kappa ) &\qquad \text{(2)}\\ +\vdash .(2).*5·32.&\supset \vdash .\lambda \cap \kappa \cap \text{Cl}ʻpʻ(\lambda \cap \kappa )=\lambda \cap \kappa \cap \iota ʻpʻ(\lambda \cap \kappa ) &\qquad \text{(3)}\\ +\vdash .(1).(3).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_620">[Pg 620]</span></p> + +<p>Observe that \(\lambda \cap \kappa \cap \iota ʻpʻ(\lambda \cap \kappa)\) +is either \(\iota ʻpʻ(\lambda \cap \kappa )\) or \(\Lambda\), +according as \(pʻ(\lambda \cap \kappa )\) is or is not a member of +\(\lambda \cap \kappa\).</p> + +<p class="nind"><b>*210·201.</b> \[\begin{align}&\vdash : \text{Hp} *210·2 . \lambda \subset \kappa . \supset . \overrightarrow{\text{min}}_{Q}ʻ\lambda = \lambda \cap \iota ʻp ʻ\lambda \\ +&[*210·2 . *22·621]\end{align}\]</p> + +<p class="nind"><b>*210·202.</b> \[\begin{align}&\vdash : \text{Hp} *210·2 . \supset . \overrightarrow{\text{max}}_{Q}ʻ\lambda = \lambda \cap \kappa \cap {℩}ʻsʻ(\lambda \cap \kappa )\\ +&[\text{Proof as in *210·2}]\end{align}\]</p> + +<p class="nind"><b>*210·203.</b> \[\begin{align}&\vdash : \text{Hp} *210·2 . \lambda \subset \kappa . \supset . \overrightarrow{\text{max}}_{Q}ʻ\lambda = \lambda \cap {℩}ʻsʻ\lambda \\ +&[*210·202 . *22·621]\end{align}\]</p> + +<p class="nind"><b><a id="*210·21">*210·21</a>.</b> \[\begin{align}&\vdash : \text{Hp} *210·2 . \lambda \subset \kappa . pʻ\lambda \in \lambda . \supset . \text{min}_{Q}ʻ\lambda = pʻ\lambda\\ +&[*210·201 . *51·31]\end{align}\]</p> + +<p class="nind"><b><a id="*210·211">*210·211</a>.</b> \[\begin{align}&\vdash : \text{Hp} *210·2 . \lambda \subset \kappa . sʻ\lambda \in \lambda . \supset . \text{max}_{Q}ʻ\lambda = sʻ\lambda\\ +&[*210·203 . *51·31]\end{align}\]</p> + +<p class="nind"><b>*210·22.</b> \[\begin{align}&\vdash : \text{Hp} *210·12 . \lambda \subset \kappa . pʻ\lambda {\sim} \in \lambda . \supset . {\sim} \exists ! \overrightarrow{\text{min}}_{Q}ʻ\lambda + \\ +&[*210·201·123 . *51·211]\end{align}\]</p> + +<p class="nind"><b>*210·221.</b> \[\begin{align}&\vdash : \text{Hp} *210·12 . \lambda \subset \kappa . sʻ\lambda {\sim} \in \lambda . \supset . {\sim} \exists ! \overrightarrow{\text{max}}_{Q}ʻ\lambda + \\ +&[*210·203·123 . *51·211]\end{align}\]</p> + +<p class="nind"><b>*210·222.</b> \[\begin{align}&\vdash \colon\ldotp \text{Hp} *210·2 . \lambda \subset \kappa . \supset : pʻ\lambda \in \lambda . \equiv . \text{E}! \text{min}_{Q}ʻ\lambda + \\ +&[*210·21·22]\end{align}\]</p> + +<p class="nind"><b>*210·223.</b> \[\begin{align}&\vdash \colon\ldotp \text{Hp} *210·2 . \lambda \subset \kappa . \supset : sʻ\lambda \in \lambda . \equiv . \text{E}! \text{max}_{Q}ʻ\lambda\\ +&[*210·211·221]\end{align}\]</p> + +<p class="nind"><b><a id="*210·23">*210·23</a>.</b> \(\vdash : \text{Hp} *210·2 . \lambda \subset \kappa . pʻ\lambda \in \kappa - \lambda . \supset . pʻ\lambda = \text{prec}_{Q}ʻ\lambda + = \text{tl}_{Q}ʻ\lambda\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *210·22 . &\supset \vdash : \text{Hp} . \supset . \overrightarrow{\text{min}}_{Q}ʻ\lambda = \Lambda . &\qquad \text{(1)}\\ +[*205·122 . *210·122] &\supset . \lambda \subset \breve{Q} ʻʻ\lambda &\qquad \text{(2)}\\ +\vdash . (2) . *210·12 . *206·174 . \supset \\ +\vdash : \text{Hp} . \supset . \overrightarrow{\text{prec}} _{Q}ʻ\lambda &= CʻQ \cap \hat{\alpha} (\overleftarrow{Q}ʻ\alpha = \breve{Q} ʻʻ\lambda )\\ +[*210·122] &= \kappa \cap \hat{\alpha} (\overleftarrow{Q}ʻ\alpha = \breve{Q} ʻʻ\lambda ) &\qquad \text{(3)}\\ +\vdash . *37·105 . *210·124 . \supset \\ +\vdash \colon\ldotp \text{Hp} . \supset : \beta \in \breve{Q} ʻʻ\lambda . &\equiv . (\exists \gamma ) . \gamma \in \lambda . \exists ! \beta - \gamma . \beta \in \kappa .\\ +[*40·15 . \text{Transp}] &\equiv . \exists ! \beta - pʻ\lambda . \beta \in \kappa .\\ +[*210·124] &\equiv . (pʻ\lambda ) Q\beta &\qquad \text{(4)}\\ +\vdash . (4) . \supset \vdash : \text{Hp} . &\supset . pʻ\lambda \in \kappa . \overleftarrow{Q}ʻpʻ\lambda = \breve{Q} ʻʻ\lambda .\\ +[(3)] &\supset . pʻ\lambda \in \overrightarrow{\text{prec}} _{Q}ʻ\lambda &\qquad \text{(5)}\\ +\vdash . (5) . *210·12 . *206·16 . &\supset \vdash : \text{Hp} . \supset . pʻ\lambda = \text{prec}_{Q}ʻ\lambda &\qquad \text{(6)}\\ +\vdash . (1) . (6) . *207·12 . &\supset \vdash : \text{Hp} . \supset . pʻ\lambda = \text{tl}_{Q}ʻ\lambda &\qquad \text{(7)}\\ +\vdash . (6) . (7) . \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_621">[Pg 621]</span></p> + +<p class="nind"><b><a id="*210·231">*210·231</a>.</b> \[\begin{align}&\vdash :\text{Hp}*210·2.\lambda \subset \kappa .sʻ\lambda \in \kappa -\lambda .\supset .sʻ\lambda =\text{seq}_{Q}ʻ\lambda =\text{lt}_{Q}ʻ\lambda + \\ +&[\text{Proof as in *210·23}]\end{align}\]</p> + +<p>In virtue of <a href="#*210·21">*210·21</a><a href="#*210·23">·23</a>, every class which is contained in \(\kappa\), +and whose product is a member of \(\kappa\), has either a minimum or +a lower limit; and in virtue of <a href="#*210·211">*210·211</a><a href="#*210·231">·231</a>, every class which is +contained in \(\kappa\), and whose sum is a member of \(\kappa\), has +either a maximum or an upper limit.</p> + +<p class="nind"><b>*210·232.</b> \(\vdash :\text{Hp}*210·2.\lambda \subset \kappa .pʻ\lambda \in \kappa .\supset .pʻ\lambda =\text{limin}_Pʻ\lambda \quad[*210·21·23]\)</p> + +<p class="nind"><b>*210·233.</b> \(\vdash :\text{Hp}*210·2.\lambda \subset \kappa .sʻ\lambda \in \kappa .\supset .sʻ\lambda =\text{limax}_Pʻ\lambda \quad[*210·211·231]\)</p> + +<p class="nind"><b>*210·24.</b> \[\begin{align}&\vdash :\text{Hp}*210·2.\supset .\kappa \cap {℩}ʻpʻ\kappa =\overrightarrow{B}ʻQ.\kappa \cap {℩}ʻsʻ\kappa =\overrightarrow{B}ʻ\breve{Q} \\ +&[*205·12·121.*210·201·203·122]\end{align}\]</p> + +<p class="nind"><b>*210·241.</b> \(\vdash :\text{Hp}*210·2.pʻ\kappa \in \kappa .\supset .pʻ\kappa =BʻQ \quad[*210·24]\)</p> + +<p class="nind"><b>*210·242.</b> \(\vdash :\text{Hp}*210·2.sʻ\kappa \in \kappa .\supset .sʻ\kappa =Bʻ\breve{Q} \quad [*210·24]\)</p> + +<p class="nind"><b><a id="*210·25">*210·25</a>.</b> \[\begin{align}\vdash \colon\ldotp \text{Hp}*210·2:\lambda \subset \kappa .\supset _{\lambda }.pʻ\lambda \in &\kappa :\supset :\\ +&\lambda \subset \kappa .\supset .pʻ\lambda \in (\overrightarrow{\text{min}}_{Q}ʻ\lambda \cup \overrightarrow{\text{prec}} _{Q}ʻ\lambda )\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*210·21.&\supset \vdash \colon\ldotp \text{Hp}.\supset :\lambda \subset \kappa .pʻ\lambda \in \lambda .\supset .pʻ\lambda \in \overrightarrow{\text{min}}_{Q}ʻ\lambda + &\qquad \text{(1)}\\ +\vdash .*210·23.&\supset \vdash \colon\ldotp \text{Hp}.\supset :\lambda \subset \kappa .pʻ\lambda {\sim}\in \lambda .\supset .pʻ\lambda \in \overrightarrow{\text{prec}} _{Q}ʻ\lambda + &\qquad \text{(2)}\\ +\vdash .(1).(2).&\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*210·251">*210·251</a>.</b> \[\begin{align}\vdash \colon\ldotp \text{Hp}*210·2:\lambda \subset \kappa .\supset _{\lambda }.sʻ\lambda \in &\kappa :\supset :\\ +&\lambda \supset \kappa .\supset .sʻ\lambda \in (\overrightarrow{\text{max}}_{Q}ʻ\lambda \cup \overrightarrow{\text{seq}} _{Q}ʻ\lambda )\\ +[\text{Proof as in *210·25}]\end{align}\]</p> + +<p class="nind"><b><a id="*210·252">*210·252</a>.</b> \[\begin{align}\vdash \colon\ldotp &\text{Hp}*210·2:\lambda \subset \kappa .\supset _{\lambda }.pʻ\lambda \cap sʻ\kappa \in \kappa :\supset :\\ +&\lambda \subset \kappa .\supset .pʻ\lambda \cap sʻ\kappa \in (\overrightarrow{\text{min}}_{Q}ʻ\lambda \cup \overrightarrow{\text{prec}} _{Q}ʻ\lambda + ).pʻ\lambda \cap sʻ\kappa =\text{limin}_Pʻ\lambda\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*40·23·161.\supset \vdash :\lambda \subset \kappa .\exists !\lambda .&\supset .pʻ\lambda \subset sʻ\kappa .\\ +[*22·621] &\supset .pʻ\lambda \cap sʻ\kappa =pʻ\lambda &\qquad \text{(1)}\\ +\vdash .(1).*210·21·23.&\supset \vdash :\text{Hp}.\lambda \subset \kappa .\exists !\lambda .\supset .pʻ\lambda \cap sʻ\kappa \in (\overrightarrow{\text{min}}_{Q}ʻ\lambda + \cup \overrightarrow{\text{prec}} _{Q}ʻ\lambda ) &\qquad \text{(2)}\\ +\vdash .*40·2. &\supset \vdash :{\sim}\exists !\lambda .\supset .pʻ\lambda \cap sʻ\kappa =sʻ\kappa &\qquad \text{(3)}\\ +\vdash .(3).*24·12.\supset \vdash :\text{Hp}.&\supset .sʻ\kappa \in \kappa .\\ +[*210·242] &\supset .sʻ\kappa =Bʻ\breve{Q} .\\ +[*206·14] &\supset .sʻ\kappa =\text{prec}_{Q}ʻ\Lambda &\qquad \text{(4)}\\ +\vdash .(3).(4).&\supset \vdash :\text{Hp}.{\sim}\exists !\lambda .\supset .pʻ\lambda \cap sʻ\kappa \in (\overrightarrow{\text{min}}_{Q}ʻ\lambda \cup \overrightarrow{\text{prec}} _{Q}ʻ\lambda + ) &\qquad \text{(5)}\\ +\vdash .(2).(5).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p>This proposition is more useful than <a href="#*210·25">*210·25</a>, because its hypothesis is +much oftener verified. In order that the hypothesis of *210·25 may be<span class="pagenum" id="Page_622">[Pg 622]</span> +verified, we must have \(\text{V}\in \kappa\), since +\(\Lambda \subset\kappa .pʻ\Lambda =\text{V}\); hence we must also +have \(sʻ\kappa =\text{V}\). But the hypothesis of <a href="#*210·252">*210·252</a> only +requires, as far as \(\Lambda\) is concerned, that we should have +\(sʻ\kappa \in \kappa\).</p> + +<p class="nind"><b>*210·253.</b> \[\begin{align}&\vdash :\text{Hp}*210·252.\supset .(\lambda ).\lambda \in \text{ᗡ}ʻ\text{min}_{Q}\cup \text{ᗡ}ʻ\text{prec}_{Q}\\ +&[*210·252.*205·15.*206·131]\end{align}\]</p> + +<p class="nind"><b>*210·254.</b> \[\begin{align}&\vdash :\text{Hp}*210·251.\supset .(\lambda ).\lambda \in \text{ᗡ}ʻ\text{max}_{Q}\cup \text{ᗡ}ʻ\text{seq}_{Q}\\ +&[\text{Proof as in *210·253}]\end{align}\]</p> + +<p class="nind"><b>*210·26.</b> \[\begin{align}\vdash :\text{Hp}*210·2.\lambda \subset \kappa .pʻ\lambda {\sim}\in \lambda .&sʻ(\kappa \cap \text{Cl}ʻpʻ\lambda )\in \kappa .\supset .\\ +&sʻ(\kappa \cap \text{Cl}ʻpʻ\lambda )=\text{prec}_{Q}ʻ\lambda \end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*210·22.\supset \vdash :\text{Hp}.&\supset .{\sim}\exists !\overrightarrow{\text{min}}_Qʻ\lambda .\\ +[*205·122] &\supset .\lambda \subset \breve{Q} ʻʻ\lambda &\qquad \text{(1)}\\ +\vdash .*60·2. & \supset \vdash :\beta \in \kappa \cap \text{Cl}ʻpʻ\lambda .\supset .\beta \subset pʻ\lambda :\\ +[*40·151] &\supset \vdash :sʻ(\kappa \cap \text{Cl}ʻpʻ\lambda )\subset pʻ\lambda &\qquad \text{(2)}\\ +\vdash .(2). \supset \vdash :\text{Hp}.&\supset .sʻ(\kappa \cap \text{Cl}ʻpʻ\lambda )\in \kappa \cap \text{Cl}ʻpʻ\lambda . &\qquad \text{(3)}\\ +[*210·211] \supset .sʻ(\kappa \cap \text{Cl}ʻpʻ\lambda )&=\text{max}_{Q}ʻ(\kappa \cap \text{Cl}ʻpʻ\lambda )\\ +[*210·17] &=\text{max}_{Q}ʻ(\kappa -\breve{Q} ʻʻ\lambda )\\ +[(1)]&=\text{max}_{Q}ʻ(\kappa -\lambda -\breve{Q} ʻʻ\lambda )\\ +[*210·122.*202·502.(3)]&=\text{max}_{Q}ʻpʻ\overrightarrow{Q}ʻʻ\lambda \\ +[*206·1·101] & =\text{prec}_{Q}ʻ\lambda :\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*210·261.</b> \[\begin{align}\vdash :\text{Hp}*210·2.&\lambda \subset k.sʻ\lambda {\sim}\in \lambda .pʻ\hat{\alpha} (\alpha \in \kappa .sʻ\lambda \subset \alpha )\in \kappa .\supset .\\ +&pʻ\hat{\alpha} (\alpha \in \kappa .sʻ\lambda \subset \alpha )=\text{seq}_{Q}ʻ\lambda \quad[\text{Proof as in *210·26}]\end{align}\]</p> + +<p class="nind"><b>*210·262.</b> + \[\begin{align}\vdash :\text{Hp}*210·2.\lambda \subset \kappa .sʻ\lambda {\sim}\in \lambda .&sʻ\kappa \cap pʻ\hat{\alpha} (\alpha \in \kappa .sʻ\lambda \subset \alpha )\in \kappa .\supset .\\ +&sʻ\kappa \cap pʻ\hat{\alpha} (\alpha \in \kappa .sʻ\lambda \subset \alpha )=\text{seq}_{Q}ʻ\lambda \end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*40·23·161.\supset \\ +\vdash :\text{Hp}.\exists !\hat{\alpha} (\alpha \in \kappa .sʻ\lambda \subset \alpha ).&\supset .pʻ\hat{\alpha} (\alpha \in \kappa .sʻ\lambda \subset \alpha )\subset sʻ\kappa .\\ +[*22·621] &\supset .sʻ\kappa \cap pʻ\hat{\alpha} (\alpha \in \kappa .sʻ\lambda \subset \alpha )=pʻ\hat{\alpha} (\alpha \in \kappa .sʻ\lambda \subset \alpha ).\\ +[*210·261] &\supset .sʻ\kappa \cap pʻ\hat{\alpha} (\alpha \in \kappa .sʻ\lambda \subset \alpha )=\text{seq}_{Q}ʻ\lambda &\qquad \text{(1)}\\ +\vdash .*10·51. &\supset \vdash \colon\ldotp \hat{\alpha} (\alpha \in \kappa .sʻ\lambda \subset \alpha )=\Lambda .\supset :\alpha \in \kappa .\supset _{\alpha }.{\sim}(sʻ\lambda + \subset \alpha ) &\qquad \text{(2)}\\ +\vdash .(2).*210·16.&\supset \vdash \colon\ldotp \text{Hp}.\hat{\alpha} (\alpha \in \kappa .sʻ\lambda \subset \alpha )=\Lambda .\supset :\alpha \in \kappa .\supset _{\alpha }.\alpha + \subset sʻ\lambda :\\ +[*40·151] & \supset :sʻ\kappa \subset sʻ\lambda :\\ +[*40·161] &\supset :sʻ\kappa =sʻ\lambda &\qquad \text{(3)}\\ +\vdash .*40·2.\supset \vdash :\text{Hp}(3).&\supset .sʻ\kappa \cap pʻ\hat{\alpha} (\alpha \in \kappa .sʻ\lambda \subset \alpha )=sʻ\kappa . &\qquad \text{(4)}\\ +[\text{Hp}.(3)] & \supset .sʻ\lambda \in \kappa .\\ +[*210·231] &\supset .sʻ\lambda =\text{seq}_{Q}ʻ\lambda .\\ +[(3).(4)] &\supset .sʻ\kappa \cap pʻ\hat{\alpha} (\alpha \in \kappa .sʻ\lambda \subset \alpha )=\text{seq}_{Q}ʻ\lambda &\qquad \text{(5)}\\ +\vdash .(1).(5).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p>The same remark applies to this proposition as to <a href="#*210·252">*210·252</a>.</p> + +<p><span class="pagenum" id="Page_623">[Pg 623]</span></p> + +<p class="nind"><b>*210·27.</b> \[\begin{align}\vdash \colon\ldotp \text{Hp}*210·2:&\lambda \subset \kappa .\supset _{\lambda }.sʻ\lambda \in \kappa :\supset :\\ +&\lambda \subset \kappa .\supset _{\lambda }.\exists !(\overrightarrow{\text{max}} _{Q}ʻ\lambda \cup \overrightarrow{\text{seq}} _{Q}ʻ\lambda + ).\exists !(\overrightarrow{\text{min}} _{Q}ʻ\lambda \cup \overrightarrow{\text{prec}} _{Q}ʻ\lambda )\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*210·251.&\supset \vdash \colon\ldotp \text{Hp}.\supset :\lambda \subset \kappa .\supset _{\lambda }.\exists !(\overrightarrow{\text{max}} _{Q}ʻ\lambda + \cup \overrightarrow{\text{seq}} _{Q}ʻ\lambda ) &\qquad \text{(1)}\\ +\vdash .*210·222.&\supset \vdash :\text{Hp}.\lambda \subset \kappa .pʻ\lambda \in \lambda .\supset .\exists !\overrightarrow{\text{min}} _{Q}ʻ\lambda &\qquad \text{(2)}\\ +\vdash .*10·1. \supset \vdash \colon\ldotp \text{Hp}.&\supset :sʻ(\kappa \cap \text{Cl}ʻpʻ\lambda )\in \kappa :\\ +[*210·26] &\supset :\lambda \subset \kappa .pʻ\lambda {\sim}\in \lambda .\supset .\exists !\overrightarrow{\text{prec}} _{P}ʻ\lambda &\qquad \text{(3)}\\ +\vdash .(2).(3). \supset \vdash \colon\ldotp \text{Hp}.&\supset :\lambda \subset \kappa .\supset .\exists !(\overrightarrow{\text{min}} _{Q}ʻ\lambda + \cup \overrightarrow{\text{prec}} _{Q}ʻ\lambda ) &\qquad \text{(4)}\\ +\vdash .(1).(4).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*210·271.</b> \[\begin{align}\vdash \colon\ldotp \text{Hp}*210·2:&\lambda \subset \kappa .\supset _{\lambda }.pʻ\lambda \in \kappa :\supset :\\ +&\lambda \subset \kappa .\supset _{\lambda }.\exists !(\overrightarrow{\text{max}} _{Q}ʻ\lambda \cup \overrightarrow{\text{seq}} _{Q}ʻ\lambda + ).\exists !(\overrightarrow{\text{min}} _{Q}ʻ\lambda \cup \overrightarrow{\text{prec}} _{Q}ʻ\lambda )\\ +&[\text{Proof as in *210·27}]\end{align}\]</p> + +<p class="nind"><b>*210·272.</b> \[\begin{align}\vdash \colon\ldotp \text{Hp}*210·2:&\lambda \subset \kappa .\supset _{\lambda }.pʻ\lambda \cap sʻ\kappa \in \kappa :\supset :\\ +&\lambda \subset \kappa .\supset _{\lambda }.\exists !(\overrightarrow{\text{max}} _{Q}ʻ\lambda \cup \overrightarrow{\text{seq}} _{Q}ʻ\lambda + ).\exists !(\overrightarrow{\text{min}} _{Q}ʻ\lambda \cup \overrightarrow{\text{prec}} _{Q}ʻ\lambda )\\ +&[\text{Proof as in *210·27, using *210·262}]\end{align}\]</p> + +<p class="nind"><b><a id="*210·28">*210·28</a>.</b> \[\begin{align}\vdash :\text{Hp}*210·2.sʻʻ&\text{Cl}ʻ\kappa \subset \kappa .\supset .\\ +&(\lambda ).\lambda \in (\text{ᗡ}ʻ\text{max}_{Q}\cup \text{ᗡ}ʻ\text{seq}_{Q})\cap (\text{ᗡ}ʻ\text{min}_{Q}\cup \text{ᗡ}ʻ\text{prec}_{Q})\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*37·61.&\supset \vdash \colon\ldotp \text{Hp}.\supset :\lambda \subset \kappa .\supset _{\lambda }.sʻ\lambda \in \kappa :\\ +[*210·27] \supset :\lambda \subset \kappa .\supset _{\lambda }.&\exists !(\overrightarrow{\text{max}} _{Q}ʻ\lambda \cup \overrightarrow{\text{seq}} _{Q}ʻ\lambda + ).\exists !(\overrightarrow{\text{min}} _{Q}ʻ\lambda \cup \overrightarrow{\text{prec}} _{Q}ʻ\lambda ) &\qquad \text{(1)}\\ +\vdash .(1).*22·43.\supset \\ +\vdash :\text{Hp}.\supset .(\lambda ).&\exists !\{\overrightarrow{\text{max}} _{Q}ʻ(\lambda \cap \kappa )\cup \overrightarrow{\text{seq}} _{Q}ʻ\lambda \cap \kappa )\}.\\ +&\exists !.\{\overrightarrow{\text{min}} _{Q}ʻ(\lambda \cap \kappa )\cup \overrightarrow{\text{prec}} _{Q}ʻ(\lambda \cap \kappa )\}.\\ +[*210·122]\supset .(\lambda ).&\exists !\{\overrightarrow{\text{max}} _{Q}ʻ(\lambda \cap CʻQ)\cup \overrightarrow{\text{seq}} _{Q}ʻ(\lambda \cap CʻQ)\}.\\ +&\exists !\{\overrightarrow{\text{min}} _{Q}ʻ(\lambda \cap CʻQ)\cup \overrightarrow{\text{prec}} _{Q}ʻ(\lambda \cap CʻQ)\}.\\ +[*205·15·151.*206·131]\supset .(\lambda ).&\exists !\{\overrightarrow{\text{max}} _{Q}ʻ\lambda \cup \overrightarrow{\text{seq}} _{Q}ʻ\lambda\}.\exists + !\{\overrightarrow{\text{min}} _{Q}ʻ\lambda \cup \overrightarrow{\text{prec}} _{Q}ʻ\lambda\}.\\ +[*33·41] &\supset .(\lambda ).\lambda \in \text{ᗡ}ʻ\text{max}_{Q}\cup \text{ᗡ}ʻ\text{seq}_{Q}.\lambda \in \text{ᗡ}ʻ\text{min}_{Q}\cup \text{ᗡ}ʻ\text{prec}_{Q}:\supset + \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*210·281">*210·281</a>.</b> \[\begin{align}\vdash :\text{Hp}*210·2.pʻʻ&\text{Cl}ʻ\kappa \subset \kappa .\supset .\\ +&(\lambda ).\lambda \in (\text{ᗡ}ʻ\text{max}_{Q}\cup \text{ᗡ}ʻ\text{seq}_{Q})\cap (\text{ᗡ}ʻ\text{min}_{Q}\cup \text{ᗡ}ʻ\text{prec}_{Q})\end{align}\]</p> + +<p class="nind"><b><a id="*210·282">*210·282</a>.</b> \[\begin{align}\vdash \colon\ldotp \text{Hp}*210·2:\lambda &\subset \kappa .\supset _{\lambda }.pʻ\lambda \cap sʻ\kappa \in \kappa :\supset .\\ +&(\lambda ).\lambda \in (\text{ᗡ}ʻ\text{max}_{Q}\cup \text{ᗡ}ʻ\text{seq}_{Q})\cap (\text{ᗡ}ʻ\text{min}_{Q}\cup \text{ᗡ}ʻ\text{prec}_{Q})\end{align}\]</p> + +<p>Thus when either of the hypotheses of <a href="#*210·281">*210·281</a><a href="#*210·282">·282</a> is fulfilled, the +series \(Q\) is Dedekindian both upwards and downwards.</p> + +<p class="nind"><b>*210·29.</b> \(\vdash :\text{Hp}*210·251.\supset .(\lambda ).\lambda \in \text{ᗡ}ʻ\text{limax}_P\cap \text{ᗡ}ʻ\text{limin}_P \quad[*210·28.*207·44]\)</p> + +<p class="nind"><b>*210·291.</b> \[\begin{align}\vdash :\text{Hp}*210·252.\supset .(\lambda ).\lambda \in \text{ᗡ}ʻ\text{limax}_P\cap \text{ᗡ}ʻ\text{limin}_P\\ +&[*210·282.*207·44]\end{align}\]</p> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_624">[Pg 624]</span></p> +<h2 class="nobreak" id="*211"> +*211. ON SECTIONS AND SEGMENTS.</h2> +</div> + + +<p class="nind"><i>Summary of</i> *211.</p> + +<p>The theory of the modes of separation of a series into two classes, +one of which wholly precedes the other, and which together make up the +whole series, is of fundamental importance. When one out of a pair +of such classes is given, the other is the rest of the series; we +may therefore, for most purposes, confine our attention to that one +of the two classes which comes first in the serial order. Any class +which can be the first of such a pair we shall call a <i>section</i> +of our series. If \(P\) is the series, we shall denote the class of +its sections by "\(\text{sect}ʻP\)." If \(\alpha\) is a section of +\(P\), we shall call \(CʻP - \alpha\) (which is the second class of our +pair) the <i>complement</i> of \(\alpha\). The class of complements of +sections is +\[ +(CʻP -)ʻʻ\text{sect}ʻP, +\] +which is identical with \(\text{sect}ʻ\breve{P}\) (<a href="#*211·75">*211·75</a>).</p> + +<p>In order that a class may be a section of \(P\), it is necessary and +sufficient that it should be contained in \(CʻP\) and should contain +all its own predecessors; thus we put +\[ +\text{sect}ʻP = \hat{\alpha} (\alpha \subset CʻP . Pʻʻ\alpha \subset \alpha ) \quad\text{Df}. +\] +We have also, by *90·23, +\[ +\text{sect}ʻP = \hat{\alpha} (\alpha = P_{*}ʻʻ\alpha ) \quad(*211·13). +\]</p> + +<p>Among sections, a specially important class consists of classes which +are composed of all the predecessors of some class, <i>i.e.</i> classes +of the form \(Pʻʻ\beta\), <i>i.e.</i> classes which are members of +\(\text{D}ʻP_{\in}\). Whenever \(P\) is transitive, \(PʻʻPʻʻ\beta\subset Pʻʻ\beta\); +hence \(Pʻʻ\beta\) is a section according to +the above definition. When \(P\) is a series, the complement of +\(Pʻʻ\beta\) (when \(\beta\) exists and is contained in \(CʻP\)) is +\[ +\overrightarrow{\text{max}} _Pʻ\beta \cup pʻ\overleftarrow{P}ʻʻ\beta . +\]</p> + +<p>The members of \(\text{D}ʻP_{\in}\) are called <i>segments</i> of the +series generated by P. In a series in which every sub-class has a +maximum or a sequent, \(\text{D}ʻP_{\in} = \overrightarrow{P}ʻʻCʻP\) +(<a href="#*211·38">*211·38</a>), <i>i.e.</i> the predecessors of a class are always the +predecessors of a single term, namely the maximum of the class if it +exists,<span class="pagenum" id="Page_625">[Pg 625]</span> or the sequent if no maximum exists. But if there are classes +which have neither a maximum nor a sequent, the predecessors of such +classes are not coextensive with the predecessors of any single term. +Thus in general the series of segments will be larger than the original +series. For example, if our original series is of the type of the +series of rationals in order of magnitude, the series of segments is +of the type of the series of real numbers, <i>i.e.</i> the type of the +continuum.</p> + +<p>Among segments, a specially important class consists of those which +have no maximum. In this case, if \(\alpha\) is such a segment, we have +\(\alpha \subset Pʻʻ\alpha\); and since (provided \(P\) is transitive) +we also have, for all segments, \(Pʻʻ\alpha \subset \alpha\), the +segments having no maximum are those for which \(\alpha = Pʻʻ\alpha\), +<i>i.e.</i> they are the class \(\text{D}ʻ(P_{\in} \dot{\cap} I)\). +In compact series, all segments belong to this latter class, but +in general only those segments belong to it which correspond to a +"Häufungsstelle." In all cases in which the existence of a limit is not +known, the segment fulfils the functions of a limit; that is to say, +in those places in the series where a limit might be expected, we have +a segment having no limit or maximum, which takes the same place in +the series of segments as would be taken by the limit in the original +series if the limit existed. Segments having no limit or maximum are +limiting points in the series of segments, and every class of segments +which has no maximum in the series of segments has a limit in that +series.</p> + +<p>We have thus three classes to deal with, namely +\[ +\begin{aligned} +&(1)\quad \text{sect}ʻP,\\ +&(2) \quad\text{D}ʻP_{\in },\\ +&(3) \quad\text{D}ʻ(P_{\in } \dot{\cap} I). +\end{aligned} +\] +Of these the second is contained in the first when \(P\) is transitive +(<a href="#*211·15">*211·15</a>), and the third is contained in the first and second +(<a href="#*211·14">*211·14</a>). The second consists of those members of the first which have +either a sequent or no maximum (<a href="#*211·32">*211·32</a>); the third consists of those +members of the first which have no maximum (<a href="#*211·41">*211·41</a>). If every member +of the third class has a limit, <i>i.e.</i> if +\[ +\text{D}ʻ(P_{\in } \dot{\cap} I) \subset \text{ᗡ}ʻ\text{seq}_{P}, +\] +then every class has either a sequent or a maximum, <i>i.e.</i> the +series is Dedekindian; and the converse also holds (<a href="#*211·47">*211·47</a>).</p> + +<p>When \(P\) is connected, of any two sections one must be contained +in the other (<a href="#*211·6">*211·6</a>). Moreover, if \(\lambda\) is contained in any +one of the three classes \(\text{sect}ʻP\), \(\text{D}ʻP_{\in }\), +\(\text{D}ʻ(P_{\in } \dot{\cap} I)\), then \(sʻ\lambda\) is a member +of that class (<a href="#*211·63">*211·63</a><a href="#*211·64">·64</a><a href="#*211·65">·65</a>). Hence the propositions of <a href="#*210">*210</a> become +available. It is thus that the existence of limits in series of +segments or sections is proved: the maximum or upper limit of any +class \(\lambda\) consisting of segments or sections is \(sʻ\lambda\), +and the minimum or lower limit is the sum of the segments that are +contained in every \(\lambda\).</p> + +<p><span class="pagenum" id="Page_626">[Pg 626]</span></p> + +<p>We begin, in this number, with elementary properties of +\(\text{sect}ʻP\). The sections of \(P\) are the segments of \(P_{*}\) +(<a href="#*211·13">*211·13</a>) and the sections of \(P_{\text{po}}\) (<a href="#*211·17">*211·17</a>). We have</p> + +<p class="nind"><b>*211·26.</b> \(\vdash .CʻP \in \text{sect}ʻP.sʻ\text{sect}ʻP=CʻP\)</p> + +<p>We then proceed to the elementary properties of segments, <i>i.e.</i> +of \(\text{D}ʻP_{\in }\) (<a href="#*211·3">*211·3</a>—<a href="#*211·38">·38</a>). We have</p> + +<p class="nind"><b>*211·3.</b> \(\vdash .\overrightarrow{P}ʻʻCʻP\subset \text{D}ʻP_{\in }\)</p> + +<p class="nind"><b>*211·301.</b> \(\vdash .\text{D}ʻP \in \text{D}ʻP_{\in }\)</p> + +<p class="nind"><b>*211·302.</b> \(\vdash :P \in \text{Ser}.\supset .\overrightarrow{P}ʻʻCʻP=\text{sect}ʻP\cap \text{ᗡ}ʻ\text{seq}_{P}\)</p> + +<p class="nind"><b>*211·351.</b> \(\vdash :P \in \text{Ser}.\supset .\text{sect}ʻP - \text{D}ʻP_{\in }=\overrightarrow{P}_{*}ʻʻ(CʻP - \text{D}ʻP_{1})\)</p> + +<p>We then proceed to elementary properties of segments having no maximum, +<i>i.e.</i> of \(\text{D}ʻ(P_{\in }\dot{\cap} I)\) (<a href="#*211·4">*211·4</a>—<a href="#*211·47">·47</a>). We +have</p> + +<p class="nind"><b>*211·42.</b> \(\vdash :P \in \text{trans}.\supset .\text{D}ʻ(P_{\in }\dot{\cap} I)=\text{D}ʻP_{\in } - \text{ᗡ}ʻ\text{max}_{P}\)</p> + +<p class="nind"><b>*211·44.</b> \(\vdash .\Lambda \in \text{D}ʻ(P_{\in }\dot{\cap} I).\Lambda \in \text{D}ʻP_{\in }.\Lambda \in \text{sect}ʻP\)</p> + +<p class="nind"><b>*211·451.</b> \(\vdash :\overrightarrow{P}ʻx\in \text{D}ʻ(P_{\in }\dot{\cap} I).\supset .x{\sim}\in \text{ᗡ}ʻ(P\dot{-} P^{2})\)</p> + +<p>Our next set of propositions (<a href="#*211·5">*211·5</a>—<a href="#*211·553">·553</a>) is concerned with compact +series, <i>i.e.</i> with the hypothesis \(P^{2}=P\). We have</p> + +<p class="nind"><b>*211·51.</b> \(\vdash :P^{2}=P.\supset .\text{D}ʻP_{\in }=\text{D}ʻ(P_{\in }\dot{\cap} I)\)</p> + +<p class="nind"><b>*211·551.</b> \(\vdash \colon\ldotp P \in \text{Ser}.\supset :\text{ᗡ}ʻ\text{max}_{P}\cap \text{ᗡ}ʻ\text{seq}_{P}=\Lambda .\equiv .P=P^{2}\)</p> + +<p><i>I.e.</i> a series is compact when, and only when, no class has both +a maximum and a sequent.</p> + +<p>We come next to the application of the propositions of <a href="#*210">*210</a> +(<a href="#*211·56">*211·56</a>—<a href="#*211·692">·692</a>). These propositions proceed from</p> + +<p class="nind"><b>*211·56.</b> \(\vdash \colon\ldotp P \in \text{connex} .\alpha ,\beta \in \text{sect}ʻP.\supset :\alpha \subset \beta .\lor.\beta \subset Pʻʻ\alpha\)</p> + +<p>(Here "\(P_{\text{po}} \in \text{connex}\)" may be substituted in the +hypothesis: cf. <a href="#*211·561">*211·561</a>.) The propositions of this set, which are very +important, have been already mentioned.</p> + +<p>Our next set of propositions (<a href="#*211·7">*211·7</a>—<a href="#*211·762">·762</a>) are concerned with the +complements of sections and segments. Some of these propositions have +been already mentioned; others of importance are:</p> + +<p class="nind"><b>*211·7.</b> \(\vdash :\alpha \in \text{sect}ʻP.\supset .CʻP-\alpha \in \text{sect}ʻ\breve{P}\)</p> + +<p class="nind"><b>*211·703.</b> \(\vdash :P \in \text{connex} .\alpha \in \text{sect}ʻP-\iota ʻCʻP.\supset .\exists !pʻ\overleftarrow{P}ʻʻ\alpha\)</p> + +<p class="nind"><b>*211·726.</b> \[\begin{align}\vdash :P \in \text{connex} \cap &\text{Rl}ʻJ.\alpha \in \text{sect}ʻP.\supset .\\ +&\overrightarrow{\text{max}}_{P}ʻ\alpha =\overrightarrow{\text{prec}} _{P}ʻ(CʻP-\alpha ).\overrightarrow{\text{seq}} _{P}ʻ\alpha + =\overrightarrow{\text{min}}_{P}ʻ(CʻP-\alpha )\end{align}\]</p> + +<p><span class="pagenum" id="Page_627">[Pg 627]</span></p> + +<p class="nind"><b>*211·727.</b> \[\begin{align}\vdash \colon\ldotp P\in \text{connex} \cap \text{Rl}ʻJ.\alpha \in &\text{sect}ʻP.\supset :\\ +&\text{E}!\text{limax}_Pʻ\alpha .\equiv .\text{E}!\text{limin}_Pʻ(CʻP-\alpha )\end{align}\]</p> + +<p class="nind"><b>*211·728.</b> \[\begin{align}\vdash \colon\ldotp P\in \text{connex} \cap &\text{Rl}ʻJ.\alpha \in \text{sect}ʻP:{\sim}\text{E}!\text{max}_{P}ʻ\alpha .\lor.\\ +&{\sim}\text{E}!\text{min}_{P}ʻ(CʻP-\alpha ):\supset .\overrightarrow{\text{limax}} _{P}ʻ\alpha =\overrightarrow{\text{limin}} _{P}ʻ(CʻP-\alpha )\end{align}\]</p> + +<p>The remaining propositions are mainly occupied with +relation-arithmetic. The most important of them is</p> + +<p class="nind"><b>*211·82.</b> \[\begin{align}\vdash \colon\colon P\in \text{Ser}.&Q\in \text{D}ʻP\unicode{x0294f}.\supset \colon\ldotp \\ +&CʻQ\in \text{sect}ʻP.\equiv :(\exists R).P=Q\unicode{x2909}R.\lor.(\exists x).P=Q\unicode{x21f8} x:\\ +&\equiv :(\exists R).P=Q\unicode{x2909}R.\lor.P=Q\unicode{x21f8} Bʻ\breve{P}\end{align}\]</p> + +<p>That is, given any series contained in \(P\), if something can be added +to make it into \(P\), its field is a section of \(P\), and vice versa.</p> + +<hr class="tb"> + +<p class="nind"><b>*211·01.</b> \(\text{sect}ʻP=\hat{\alpha} (\alpha \subset CʻP.Pʻʻ\alpha \subset \alpha ) \quad\text{Df}\)</p> + +<p class="nind"><b>*211·1.</b> \(\vdash :\alpha \in \text{sect}ʻP.\equiv .\alpha \subset CʻP.Pʻʻ\alpha \subset \alpha \quad[(*211·01)]\)</p> + +<p class="nind"><b>*211·11.</b> \(\vdash :\alpha \in \text{D}ʻP_{\in }.\equiv .(\exists \beta ).\alpha =Pʻʻ\beta \quad[*37·101]\)</p> + +<p class="nind"><b>*211·12.</b> \(\vdash :\alpha \in \text{D}ʻ(P_{\in }\dot{\cap} I).\equiv .\alpha =Pʻʻ\alpha\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*37·101.*50·1. \supset \vdash :\alpha \in \text{D}ʻ(P_{\in }\dot{\cap} I).&\equiv .(\exists \beta ).\alpha =Pʻʻ\beta .\alpha =\beta .\\ +[*13·195] &\equiv .\alpha =Pʻʻ\alpha :\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*211·13">*211·13</a>.</b> \(\vdash :\alpha \in \text{sect}ʻP.\equiv .\alpha =P_{*}ʻʻ\alpha .\equiv .\alpha \in \text{D}ʻ{(P_{*})_{\in }\dot{\cap} + I}.\equiv .\alpha \in \text{D}ʻ(P_{*})_{\in }\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*211·1.*90·23. &\supset \vdash :\alpha \in \text{sect}ʻP.\equiv .\alpha =P_{*}ʻʻ\alpha &\qquad \text{(1)}\\ +\vdash .*90·17. &\supset \vdash .P_{*}ʻʻP_{*}ʻʻ\beta =P_{*}ʻʻ\beta \\ +[*13·12] &\supset \vdash :\alpha =P_{*}ʻʻ\beta .\supset .P_{*}ʻʻ\alpha =\alpha :\\ +[*211·11] &\supset \vdash :\alpha \in \text{D}ʻ(P_{*})_{\in }.\supset .\alpha =P_{*}ʻʻ\alpha &\qquad \text{(2)}\\ +\vdash .*10·24.*211·11. &\supset \vdash :\alpha =P_{*}ʻʻ\alpha .\supset .\alpha \in \text{D}ʻ(P_{*})_{\in } &\qquad \text{(3)}\\ +\vdash .(1).(2).(3).*211·12.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p>In virtue of the above proposition, the properties of \(\text{sect}ʻP\) +can be deduced from those of \(\text{D}ʻP_{\in }\) or +\(\text{D}ʻ(P_{\in }\dot{\cap} I)\) by substituting \(P_{*}\) for \(P\).</p> + +<p class="nind"><b>*211·131.</b> \(\vdash :\alpha \in \text{sect}ʻP.\supset .Pʻʻ\alpha =P_{\text{po}}ʻʻ\alpha\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*211·13. \supset \vdash :\text{Hp}.\supset .Pʻʻ\alpha &=PʻʻP_{*}ʻʻ\alpha\\ +[*91·52] &=P_{\text{po}}ʻʻ\alpha :\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_628">[Pg 628]</span></p> + +<p class="nind"><b>*211·132.</b> \[\begin{align}\vdash :\alpha \in \text{sect}ʻP.\supset .&\text{D}ʻ(P\unicode{x0294f}\alpha )=\text{D}ʻ(P_{\text{po}}\unicode{x0294f}\alpha + ).\text{ᗡ}ʻ(P\unicode{x0294f}\alpha )=\text{ᗡ}ʻ(P_{\text{po}}\unicode{x0294f}\alpha ).\\ +&Cʻ(P\unicode{x0294f}\alpha )=Cʻ(P_{\text{po}}\unicode{x0294f}\alpha )\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*37·41.*211·131.\supset \vdash :\text{Hp}.\supset .\text{D}ʻ(P_{\text{po}}\unicode{x0294f}\alpha )&=\alpha \cap Pʻʻ\alpha \\ +[*37·41] &=\text{D}ʻ(P\unicode{x0294f}\alpha ) &\qquad \text{(1)}\\ +\vdash .*91·502. & \supset \vdash .\text{ᗡ}ʻ(P\unicode{x0294f}\alpha )\subset \text{ᗡ}ʻ(P_{\text{po}}\unicode{x0294f}\alpha ) &\qquad \text{(2)}\\ +\vdash .*37·41. \supset \vdash \colon\ldotp y\in \text{ᗡ}ʻ(P_{\text{po}}\unicode{x0294f}\alpha ).&\equiv :y\in \alpha \cap \breve{P} _{\text{po}}ʻʻ\alpha :\\ +[*91·57] &\equiv :y\in (\alpha \cap \breve{P} ʻʻ\alpha )\cup (\alpha \cap \breve{P} ʻʻ\breve{P} _{\text{po}}ʻʻ\alpha ) &\qquad \text{(3)}\\ +\vdash .*211·1. \supset \vdash :\text{Hp}.y\in \alpha \cap \breve{P} ʻʻ\breve{P} _{\text{po}}ʻʻ\alpha .&\supset .(\exists z).zPy.z\in \alpha .\\ +[*37·105] &\supset .y\in \breve{P} ʻʻ\alpha &\qquad \text{(4)}\\ +\vdash .(3).(4).\supset \vdash :\text{Hp}.&\supset .\text{ᗡ}ʻ(P_{\text{po}}\unicode{x0294f}\alpha )\subset \alpha \cap \breve{P} ʻʻ\alpha .\\ +[*37·41] &\supset .\text{ᗡ}ʻ(P_{\text{po}}\unicode{x0294f}\alpha )\subset \text{ᗡ}ʻ(P\unicode{x0294f}\alpha ) &\qquad \text{(5)}\\ +\vdash .(2).(5). \supset \vdash :\text{Hp}.&\supset .\text{ᗡ}ʻ(P_{\text{po}}\unicode{x0294f}\alpha )=\text{ᗡ}ʻ(P\unicode{x0294f}\alpha ) &\qquad \text{(6)}\\ +\vdash .(1).(6).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*211·133.</b> \(\vdash :P_{\text{po}}\in \text{connex} .\alpha \in \text{sect}ʻP-1.\supset .Cʻ(P\unicode{x0294f}\alpha )=\alpha \)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*202·55.\supset \vdash :\text{Hp}.&\supset .Cʻ(P_{\text{po}}\unicode{x0294f}\alpha )=\alpha .\\ +[*211·132] &\supset .Cʻ(P\unicode{x0294f}\alpha )=\alpha :\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*211·14">*211·14</a>.</b> \(\vdash .\text{D}ʻ(P_{\in }\dot{\cap} I)\subset \text{D}ʻP_{\in }.\text{D}ʻ(P_{\in }\dot{\cap} I)\subset \text{sect}ʻP\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*33·263. &\supset \vdash .\text{D}ʻ(P_{\in }\dot{\cap} I)\subset \text{D}ʻP_{\in } &\qquad \text{(1)}\\ +\vdash .*211·12.*22·42.&\supset \vdash :\alpha \in \text{D}ʻ(P_{\in }\dot{\cap} I).\supset .Pʻʻ\alpha \subset \alpha &\qquad \text{(2)}\\ +\vdash .*211·12.*37·15.&\supset \vdash :\alpha \in \text{D}ʻ(P_{\in }\dot{\cap} I).\supset .\alpha \subset CʻP &\qquad \text{(3)}\\ +\vdash .(2).(3).*211·1.&\supset \vdash :\alpha \in \text{D}ʻ(P_{\in }\dot{\cap} I).\supset .\alpha \in \text{sect}ʻP &\qquad \text{(4)}\\ +\vdash .(1).(4).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*211·15">*211·15</a>.</b> \(\vdash :P\in \text{trans}.\supset .\text{D}ʻP_{\in }\subset \text{sect}ʻP\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*211·11.*37·15.&\supset \vdash :\alpha \in \text{D}ʻP_{\in }.\supset .\alpha \subset CʻP &\qquad \text{(1)}\\ +\vdash .*211·11.*201·5.&\supset \vdash :P\in \text{trans}.\alpha \in \text{D}ʻP_{\in }.\supset .Pʻʻ\alpha \subset \alpha &\qquad \text{(2)}\\ +\vdash .(1).(2).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*211·16.</b> \(\vdash .P_{\text{po}}ʻʻ\alpha \in \text{sect}ʻP\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*91·504.*37·15.&\supset \vdash .P_{\text{po}}ʻʻ\alpha \subset CʻP &\qquad \text{(1)}\\ +\vdash .*91·51·511. &\supset \vdash .PʻʻP_{\text{po}}ʻʻ\alpha \subset P_{\text{po}}ʻʻ\alpha &\qquad \text{(2)}\\ +\vdash .(1).(2).*211·1.&\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_629">[Pg 629]</span></p> + +<p class="nind"><b><a id="*211·17">*211·17</a>.</b> \[\begin{align}&\vdash .\text{sect}ʻP=\text{sect}ʻP_{\text{po}}=\text{sect}ʻP_{*} +&[*211·13.*90·4.*91·602]\end{align}\]</p> + +<p>The following propositions are useful in dealing with +sectional relations, <i>i.e.</i> relations of the form +\(P\unicode{x0294f}\alpha\), where \(\alpha \in \text{sect}ʻP\). Unit +sections often need special treatment, owing to the fact that for them +we do not have \(CʻP\unicode{x0294f}\alpha =\alpha\).</p> + +<p class="nind"><b>*211·18.</b> \(\vdash :P_{\text{po}}\,\unicode{x2abd}\, J.\supset .\text{sect}ʻP\cap 1=\iotaʻʻ\overrightarrow{B}ʻP\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*211·13.\supset \vdash :\alpha \in \text{sect}ʻP\cap 1.&\equiv .\alpha =P_{*}ʻʻ\alpha .\alpha \in 1.\\ +[*52·1.*53·301] &\equiv .(\exists x).\alpha =\iota ʻx.\overrightarrow{P}_{*}ʻx=\iota ʻx.\\ +[*91·54.*90·12] &\equiv .(\exists x).\alpha =\iota ʻx.\overrightarrow{P}_{\text{po}}ʻx\subset \iota ʻx.x\in CʻP &\qquad \text{(1)}\\ +\vdash .(1).\supset \vdash \colon\ldotp \text{Hp}.\supset :\alpha \in \text{sect}ʻP\cap 1.&\equiv .(\exists x).\alpha =\iota ʻx.\overrightarrow{P}_{\text{po}}ʻx=\Lambda .x\in CʻP.\\ +[*91·504] &\equiv .(\exists x).\alpha =\iota ʻx.x{\sim}\in \text{ᗡ}ʻP.x\in CʻP.\\ +[*93·103] &\equiv .\alpha \in \iotaʻʻ\overrightarrow{B}ʻP\colon\ldotp \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*211·181.</b> \(\vdash :P_{\text{po}}\in \text{Ser}.\exists !\overrightarrow{B}ʻP.\supset .\text{sect}ʻP\cap 1=\iota ʻ\iota ʻBʻP\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*202·13·523.\supset \vdash :\text{Hp}.\supset .\overrightarrow{B}ʻP\in 1 &\qquad \text{(1)}\\ +\vdash .(1).*211·18.*53·3.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*211·182.</b> \(\vdash :P_{\text{po}}\in \text{Ser}.\overrightarrow{B}ʻP=\Lambda .\supset .\text{sect}ʻP\cap 1=\Lambda \quad[*211·18]\)</p> + +<p class="nind"><b>*211·2.</b> + \(\vdash :\alpha \in \text{sect}ʻP.\supset .\alpha =\alpha \cap CʻP=a\cup Pʻʻ\alpha =(\alpha \cap CʻP)\cup Pʻʻ\alpha =Pʻʻ\alpha \cup \overrightarrow{\text{max}}_{P}ʻ\alpha\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*211·1.*22·621·62.\supset \vdash :\text{Hp}.&\supset .\alpha =a\cap CʻP.\alpha =\alpha \cup Pʻʻ\alpha . &\qquad \text{(1)}\\ +[*13·12] \supset .\alpha &=(\alpha \cap CʻP)\cup Pʻʻ\alpha &\qquad \text{(2)}\\ +[*205·131] &=Pʻʻ\alpha \cup \overrightarrow{\text{max}}_{P}ʻ\alpha &\qquad \text{(3)}\\ +\vdash .(1).(2).(3).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*211·21.</b> \(\vdash \colon\ldotp \alpha \in \text{sect}ʻP.\supset :{\sim}\exists !\overrightarrow{\text{max}}_{P}ʻ\alpha .\equiv .\alpha \in \text{D}ʻ(P_{\in }\dot{\cap} + I)\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*211·2·12. &\supset \vdash :\alpha \in \text{sect}ʻP.{\sim}\exists !\overrightarrow{\text{max}}_{P}ʻ\alpha .\supset .\alpha \in \text{D}ʻ(P_{\in }\dot{\cap} + I) &\qquad \text{(1)}\\ +\vdash .*211·12.*205·111.&\supset \vdash :\alpha \in \text{D}ʻ(P_{\in }\dot{\cap} I).\supset .{\sim}\exists !\overrightarrow{\text{max}}_{P}ʻ\alpha &\qquad \text{(2)}\\ +\vdash .(1).(2).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_630">[Pg 630]</span></p> + +<p class="nind"><b>*211·22.</b> \(\vdash :P\in \text{connex} .\alpha \in \text{sect}ʻP.\supset .\alpha \cup \overrightarrow{\text{seq}} _{P}ʻ\alpha \in \text{sect}ʻP\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*24·24.*13·12. &\supset \vdash :\text{Hp}.\overrightarrow{\text{seq}} _{P}ʻ\alpha =\Lambda .\supset .\alpha \cup \overrightarrow{\text{seq}} _{P}ʻ\alpha + \in \text{sect}ʻP &\qquad \text{(1)}\\ +\vdash .*206·16.*53·3·31.\supset \vdash :\text{Hp}.\exists !\overrightarrow{\text{seq}} _{P}ʻ\alpha .\supset .&Pʻʻ(\alpha \cup \overrightarrow{\text{seq}} _{P}ʻ\alpha + )=Pʻʻ\alpha \cup \overrightarrow{P}ʻ\text{seq}_{P}ʻ\alpha \\ +[*206·213] &\subset Pʻʻ\alpha \cup (\alpha \cap CʻP)\cup Pʻʻ\alpha \\ +[*211·2] &\subset \alpha &\qquad \text{(2)}\\ +\vdash .*211·1.*206·18. &\supset \vdash :\text{Hp}.\supset .\alpha \cup \overrightarrow{\text{seq}} _{P}ʻ\alpha \subset CʻP &\qquad \text{(3)}\\ +\vdash .(1).(2).(3).*211·1.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*211·23.</b> \(\vdash :P\in \text{connex} .\alpha \in \text{sect}ʻP.\text{E}!\text{seq}_{P}ʻ\alpha .\supset .\alpha =Pʻʻ(\alpha \cup \overrightarrow{\text{seq}} _{P}ʻ\alpha + )=\overrightarrow{P}ʻ\text{seq}_{P}ʻ\alpha\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*206·211.*211·2.&\supset \vdash :\text{Hp}.\supset .\alpha \subset \overrightarrow{P}ʻ\text{seq}_{P}ʻ\alpha &\qquad \text{(1)}\\ +\vdash .*206·213.*211·2.&\supset \vdash :\text{Hp}.\supset .\overrightarrow{P}ʻ\text{seq}_{P}ʻ\alpha \subset \alpha &\qquad \text{(2)}\\ +\vdash .(1).(2). &\supset \vdash :\text{Hp}.\supset .\alpha =\overrightarrow{P}ʻ\text{seq}_{P}ʻ\alpha &\qquad \text{(3)}\\ +\vdash .*53·3·31. \supset \vdash :\text{Hp}.\supset .Pʻʻ(\alpha \cup \overrightarrow{\text{seq}} _{P}ʻ\alpha )&=Pʻʻ\alpha \cup \overrightarrow{P}ʻ\text{seq}_{P}ʻ\alpha \\ +[(3)] &=Pʻʻ\alpha \cup \alpha \\ +[*211·2] & =\alpha &\qquad \text{(4)}\\ +\vdash .(3).(4).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*211·24.</b> \(\vdash :P\in \text{connex} .\alpha \in \text{sect}ʻP\cap (\text{ᗡ}ʻ\text{seq}_{P}\cup -\text{ᗡ}ʻ\text{max}_{P}).\supset + .\alpha \in \text{D}ʻP_{\in }\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*211·23·11.&\supset \vdash :P\in \text{connex} .\alpha \in \text{sect}ʻP\cap \text{ᗡ}ʻ\text{seq}_{P}.\supset .\alpha \in \text{D}ʻP_{\in } &\qquad \text{(1)}\\ +\vdash .*211·21·14.&\supset \vdash :\alpha \in \text{sect}ʻP-\text{ᗡ}ʻ\text{max}_{P}.\supset .\alpha \in \text{D}ʻP_{\in } &\qquad \text{(2)}\\ +\vdash .(1).(2).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*211·26.</b> \(\vdash .CʻP\in \text{sect}ʻP.sʻ\text{sect}ʻP=CʻP\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*22·42.*37·15. &\supset \vdash .CʻP\subset CʻP.PʻʻCʻP\subset CʻP.\\ +[*211·1] &\supset \vdash .CʻP\in \text{sect}ʻP &\qquad \text{(1)}\\ +\vdash .(1).*40·13.&\supset \vdash .CʻP\subset sʻ\text{sect}ʻP &\qquad \text{(2)}\\ +\vdash .*40·151.*211·1. &\supset \vdash .sʻ\text{sect}ʻP\subset CʻP &\qquad \text{(3)}\\ +\vdash .(2).(3).&\supset \vdash .sʻ\text{sect}ʻP=CʻP &\qquad \text{(4)}\\ +\vdash .(1).(4).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*211·27.</b> \(\vdash :P\in \text{trans}.\supset .(\alpha \cap CʻP)\cup Pʻʻ\alpha \in \text{sect}ʻP\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*22·43.*37·15.&\supset \vdash .(\alpha \cap CʻP)\cup Pʻʻ\alpha \subset CʻP &\qquad \text{(1)}\\ +\vdash .*37·22·265. &\supset \vdash .Pʻʻ\{(\alpha \cap CʻP)\cup Pʻʻ\alpha\}=Pʻʻ\alpha \cup PʻʻPʻʻ\alpha &\qquad \text{(2)}\\ +\vdash .(2).*201·5. &\supset \vdash :\text{Hp}.\supset .Pʻʻ\{(\alpha \cap CʻP)\cup Pʻʻ\alpha\}=Pʻʻ\alpha &\qquad \text{(3)}\\ +\vdash .(1).(3).*211·1.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_631">[Pg 631]</span></p> + +<p class="nind"><b>*211·271.</b> \(\vdash :P\in \text{trans}.\supset .(\exists \beta ).\beta \in \text{sect}ʻP.\overrightarrow{\text{max}}_{P}ʻ\alpha =\overrightarrow{\text{max}}_{P}ʻ\beta + .\overrightarrow{\text{seq}} _{P}ʻ\alpha =\overrightarrow{\text{seq}} _{P}ʻ\beta\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*205·15·19.\supset \\ +\vdash :\text{Hp}.&\supset .\overrightarrow{\text{max}}_{P}ʻ\alpha =\overrightarrow{\text{max}}_{P}ʻ\{(\alpha \cap CʻP)\cup Pʻʻ\alpha\} &\qquad \text{(1)}\\ +\vdash .*206·131·25.\supset \\ +\vdash :\text{Hp}.&\supset .\overrightarrow{\text{seq}} _{P}ʻ\alpha =\overrightarrow{\text{seq}} _{P}ʻ\{(\alpha \cap CʻP)\cup Pʻʻ\alpha\} &\qquad \text{(2)}\\ +\vdash .(1).(2).*211·27.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*211·272.</b> \[\begin{align}\vdash \colon\ldotp P\in &\text{trans}.\supset :\\ +&(\alpha ).\alpha \in \text{ᗡ}ʻ\text{max}_{P}\cup \text{ᗡ}ʻ\text{seq}_{P}.\equiv .\text{sect}ʻP\subset \text{ᗡ}ʻ\text{max}_{P}\cup + \text{ᗡ}ʻ\text{seq}_{P}\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*24·11·14.&\supset \vdash :(\alpha ).\alpha \in \text{ᗡ}ʻ\text{max}_{P}\cup \text{ᗡ}ʻ\text{seq}_{P}.\supset .\text{sect}ʻP\subset \text{ᗡ}ʻ\text{max}_{P}\cup + \text{ᗡ}ʻ\text{seq}_{P} &\qquad \text{(1)}\\ +\vdash .*33·41.&\supset \vdash \colon\ldotp \text{sect}ʻP\subset \text{ᗡ}ʻ\text{max}_{P}\cup \text{ᗡ}ʻ\text{seq}_{P}.\supset :\\ +&\beta \in \text{sect}ʻP.\supset _{\beta }.\exists !(\overrightarrow{\text{max}}_{P}ʻ\beta \cup \overrightarrow{\text{seq}} _{P}ʻ\beta ):\\ +[*13·12]\supset :\beta \in \text{sect}ʻP.\overrightarrow{\text{max}}_{P}ʻ\alpha &=\overrightarrow{\text{max}}_{P}ʻ\beta .\overrightarrow{\text{seq}} _{P}ʻ\alpha + =\overrightarrow{\text{seq}} _{P}ʻ\beta .\supset _{\alpha ,\beta }.\\ +&\exists !(\overrightarrow{\text{max}}_{P}ʻ\alpha \cup \overrightarrow{\text{seq}} _{P}ʻ\alpha ):\\ +[*10·23]\supset :(\exists \beta ).\beta \in \text{sect}ʻP.\overrightarrow{\text{max}}_{P}ʻ\alpha &=\overrightarrow{\text{max}}_{P}ʻ\beta .\overrightarrow{\text{seq}} _{P}ʻ\alpha + =\overrightarrow{\text{seq}} _{P}ʻ\beta .\supset _{\alpha }.\\ +&\exists !(\overrightarrow{\text{max}}_{P}ʻ\alpha \cup \overrightarrow{\text{seq}} _{P}ʻ\alpha ) &\qquad \text{(2)}\\ +\vdash .(2).*211·271.\supset \\ +\vdash \colon\ldotp \text{Hp}.\supset :\text{sect}ʻP\subset \text{ᗡ}ʻ\text{max}_{P}\cup \text{ᗡ}ʻ\text{seq}_{P}.&\supset .(\alpha ).\exists !(\overrightarrow{\text{max}}_{P}ʻ\alpha + \cup \overrightarrow{\text{seq}} _{P}ʻ\alpha ).\\ +[*33·41] &\supset .(\alpha ).\alpha \in \text{ᗡ}ʻ\text{max}_{P}\cup \text{ᗡ}ʻ\text{seq}_{P} &\qquad \text{(3)}\\ +\vdash .(1).(3).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*211·28.</b> \[\begin{align}\vdash \colon\ldotp P\in \text{Ser}.\alpha \subset CʻP.\alpha {\sim}\in 1.&(CʻP-\alpha ){\sim}\in 1.\supset :\\ +&\alpha \in \text{sect}ʻP.\equiv .P =P \unicode{x0294f}\alpha \unicode{x2909}P \unicode{x0294f}(CʻP-\alpha )\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*204·45.&\supset \vdash :\text{Hp}.\alpha \in \text{sect}ʻP.\supset .P=P\unicode{x0294f}\alpha \unicode{x2909}P \unicode{x0294f}(CʻP -\alpha ) &\qquad \text{(1)}\\ +\vdash .*160·1.*202·55.&\supset \vdash \colon\ldotp \text{Hp}.P=P\unicode{x0294f}\alpha \unicode{x2909}P\unicode{x0294f}(CʻP-\alpha ).\supset :\\ +&x\in \alpha .y\in CʻP-\alpha .\supset .xPy:\\ +[\text{Transp}.*204·3] &\supset :x\in \alpha .yPx.\supset .y\in \alpha :\\ +[*211·1] &\supset :\alpha \in \text{sect}ʻP &\qquad \text{(2)}\\ +\vdash .(1).(2).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*211·281.</b> \(\vdash :P\in \text{Ser}.CʻQ\cap CʻR=\Lambda .P=Q\unicode{x2909}R.\supset .CʻQ\in \text{sect}ʻP\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*160·1.\supset \vdash \colon\ldotp \text{Hp}.&\supset :x\in CʻQ.y\in CʻR.\supset .xPy:\\ +[\text{Transp}.*204·3] &\supset :x\in CʻQ.yPx.\supset .y\in CʻQ:\\ +[*211·1] &\supset :CʻQ\in \text{sect}ʻP\colon\ldotp \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_632">[Pg 632]</span></p> + +<p class="nind"><b>*211·282.</b> \[\begin{align}\vdash \colon\ldotp P\in \text{Ser}.Q\in &\text{D}ʻP\unicode{x0294f}.CʻP-CʻQ{\sim}\in I.\supset :\\ +&CʻQ\in \text{sect}ʻP.\equiv .(\exists R).CʻQ\cap CʻR=\Lambda .P=Q\unicode{x2909}R\\ +&[*211·28·281.*200·12]\end{align}\]</p> + +<p class="nind"><b>*211·283.</b> \(\vdash :P\,\unicode{x2abd}\, J.P=Q\unicode{x2909}R.\supset .CʻQ\cap CʻR=\Lambda\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*160·1.\supset \vdash :\text{Hp}.&\supset .CʻQ\uparrow CʻR\,\unicode{x2abd}\, J.\\ +[*200·32] & \supset .CʻQ\cap CʻR=\Lambda :\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p>The following propositions are concerned with \(\text{D}ʻP_{\in }\). +This is to be compared with two other classes, namely \(\text{sect}ʻP\) +and \(\overrightarrow{P}ʻʻCʻP\). The members of \(\text{sect}ʻP\) which +do not belong to \(\text{D}ʻP_{\in }\) are those which have a maximum +but no sequent, <i>i.e.</i> (if \(P\) is a series), those classes which +consist of a term \(x\) together with all its predecessors, where x +has no immediate successor. In series in which every term except the +last has an immediate successor, \(CʻP\) will be the only member of +\(\text{sect}ʻP-\text{D}ʻP_{\in }\), if the series has a last term; if +the series has no last term, \(\text{sect}ʻP=\text{D}ʻP_{\in }\).</p> + +<p>The members of \(\text{D}ʻP_{\in }\) which are not members of +\(\overrightarrow{P}ʻʻCʻP\) are those that have no sequent, <i>i.e.</i> +those that have no upper limit (for a member of \(\text{D}ʻP_{\in}\) +which has no sequent has also no maximum). These are the members +of \(\text{D}ʻP_{\in }\) corresponding to a "gap," <i>i.e.</i> to a +Dedekind section in which neither the earlier terms have a maximum +nor the later terms a minimum. Hence in a Dedekindian series, +\(\text{D}ʻP_{\in }=\overrightarrow{P}ʻʻCʻP\); and conversely, +if \(\text{D}ʻP_{\in }=\overrightarrow{P}ʻʻCʻP\), the series is +Dedekindian. These properties of \(\text{D}ʻP_{\in }\) are proved in +the following propositions.</p> + +<p class="nind"><b><a id="*211·3">*211·3</a>.</b> \(\vdash .\overrightarrow{P}ʻʻCʻP\subset \text{D}ʻP_{\in } \quad[*53·301.*211·11]\)</p> + +<p class="nind"><b>*211·301.</b> \(\vdash .\text{D}ʻP\in \text{D}ʻP_{\in } \quad[*37·25.*211·11]\)</p> + +<p class="nind"><b>*211·302.</b> \(\vdash :P\in \text{Ser}.\supset .\overrightarrow{P}ʻʻCʻP=\text{sect}ʻP\cap \text{ᗡ}ʻ\text{seq}_{P}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*206·4. &\supset \vdash :\text{Hp}.\supset .\overrightarrow{P}ʻʻCʻP\subset \text{ᗡ}ʻ\text{seq}_{P} &\qquad \text{(1)}\\ +\vdash .*211·3·15.&\supset \vdash :\text{Hp}.\supset .\overrightarrow{P}ʻʻCʻP\subset \text{sect}ʻP &\qquad \text{(2)}\\ +\vdash .*211·23. &\supset \vdash :\text{Hp}.\supset .\text{sect}ʻP\cap \text{ᗡ}ʻ\text{seq}_{P}\subset \overrightarrow{P}ʻʻCʻP &\qquad \text{(3)}\\ +\vdash .(1).(2).(3).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*211·31">*211·31</a>.</b> \[\begin{align}&\vdash \colon\ldotp P\in \text{trans}\cap \text{connex} .\alpha \in \text{D}ʻP_{\in }.\supset :\text{E}!\text{seq}_{P}ʻ\alpha + .\lor.{\sim}\text{E}!\text{max}_{P}ʻ\alpha \\ +&[*206·52.*211·11]\end{align}\]</p> + +<p class="nind"><b>*211·311.</b> \[\begin{align}&\vdash :P\in \text{trans}\cap \text{connex} .\alpha \in \text{D}ʻP_{\in }.\text{E}!\text{seq}_{P}ʻ\alpha .\supset .\alpha =\overrightarrow{P}ʻ\text{seq}_{P}ʻ\alpha + \\ +&[*206·31.*211·11]\end{align}\]</p> + +<p><span class="pagenum" id="Page_633">[Pg 633]</span></p> + +<p class="nind"><b>*211·312.</b> \(\vdash :P\in \text{trans}\cap \text{connex} .\alpha \in \text{D}ʻP_{\in }.\supset .\alpha =Pʻʻ(\alpha \cup \overrightarrow{\text{seq}} _{P}ʻ\alpha )\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*211·15·23.\supset \vdash :\text{Hp}.\text{E}!\text{seq}_{P}ʻ\alpha .&\supset .\alpha =Pʻʻ(\alpha \cup \overrightarrow{\text{seq}} _{P}ʻ\alpha ) &\qquad \text{(1)}\\ +\vdash .*211·31. \supset \vdash :\text{Hp}.{\sim}\text{E}!\text{seq}_{P}ʻ\alpha .&\supset .{\sim}\text{E}!\text{max}_{P}ʻ\alpha .\\ +[*211·21·15·12]&\supset .\alpha =Pʻʻ\alpha .\\ +[*24·24.\text{Hp}] &\supset .\alpha =Pʻʻ(\alpha \cup \overrightarrow{\text{seq}} _{P}ʻ\alpha ) &\qquad \text{(2)}\\ +\vdash .(1).(2).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*211·313.</b> \(\vdash :\alpha \in \text{sect}ʻP\cap \text{D}ʻP_{\in }.\supset .(\exists \beta ).\beta \in \text{sect}ʻP.\alpha =Pʻʻ\beta\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*211·1·11.\supset \vdash \colon\ldotp \text{Hp}.&\supset :Pʻʻ\alpha \subset \alpha :(\exists \beta ).\alpha =Pʻʻ\beta :\\ +[*37·265] &\supset :Pʻʻ\alpha \subset \alpha :(\exists \beta ).\beta \subset CʻP.\alpha =Pʻʻ\beta :\\ +[*22·62] &\supset :(\exists \beta ).\beta \subset CʻP.\alpha =Pʻʻ(\alpha \cup \beta ):\\ +[*22·58] &\supset :(\exists \beta ).\beta \subset CʻP.Pʻʻ(\alpha \cup \beta )\subset \alpha \cup \beta .\alpha =Pʻʻ(\alpha \cup \beta ):\\ +[*37·15] &\supset :(\exists \beta ).\alpha \cup \beta \subset CʻP.Pʻʻ(\alpha \cup \beta )\subset \alpha \cup \beta .\alpha =Pʻʻ(\alpha \cup \beta ):\\ +[*211·1] & \supset :(\exists \beta ).\alpha \cup \beta \in \text{sect}ʻP.\alpha =Pʻʻ(\alpha \cup \beta )\colon\ldotp \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*211·314">*211·314</a>.</b> \(\vdash :P\in \text{Rl}ʻJ\cap \text{connex} .\alpha \in \text{sect}ʻP\cap \text{D}ʻP_{\in }.\text{E}!\text{max}_{P}ʻ\alpha + .\supset .\text{E}!\text{seq}_{P}ʻ\alpha\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*211·313.*205·7.\supset \\ +\vdash :\text{Hp}.&\supset .(\exists \beta ).\beta \in \text{sect}ʻP.\alpha =Pʻʻ\beta .\text{E}!\text{max}_{P}ʻ\beta &\qquad \text{(1)}\\ +\vdash .*37·18.\supset \\ +\vdash :\beta \in \text{sect}ʻP.\alpha =Pʻʻ\beta .\text{E}!\text{max}_{P}ʻ\beta .&\supset .\overrightarrow{P}ʻ\text{max}_{P}ʻ\beta \subset \alpha &\qquad \text{(2)}\\ +\vdash .*211·1.*205·111.\supset \\ +\vdash :\text{Hp}(2).P\in \text{connex} .y\in Pʻʻ\beta .&\supset .y\in \beta -\iota ʻ\text{max}_{P}ʻ\beta .\\ +[*205·21] &\supset .yP\text{max}_{P}ʻ\beta &\qquad \text{(3)}\\ +\vdash .(2).(3).\supset \vdash :\text{Hp}(2).P\in \text{connex} .&\supset .\alpha =\overrightarrow{P}ʻ\text{max}_{P}ʻ\beta .\\ +[*206·4] &\supset .\text{max}_{P}ʻ\beta \text{seq}_{P}\alpha &\qquad \text{(4)}\\ +\vdash .(1).(4).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_634">[Pg 634]</span></p> + +<p>The above proposition and the two following propositions enable us +in certain cases to prove propositions concerning the relations of +\(\text{sect}ʻP\) and \(\text{D}ʻP_{\in }\) without assuming that \(P\) +is transitive. An example of the use of these propositions occurs +in <a href="#*211·754">*211·754</a>, where the hypothesis assumes \(P_{\in }\text{Rl}ʻJ\cap\text{connex}\). +If we used <a href="#*211·31">*211·31</a> and its consequences instead of +<a href="#*211·314">*211·314</a> and its consequences, the hypothesis of *211·754 would have to +assume \(P\in \text{Ser}\).</p> + +<p class="nind"><b>*211·315.</b> \[\begin{align}\vdash \colon\ldotp P\in \text{Rl}ʻJ\cap \text{connex} .\alpha \in &\text{sect}ʻP.\supset :\\ +&\alpha \in \text{D}ʻP_{\in }.\equiv .\alpha \in \text{ᗡ}ʻ\text{seq}_{P}\cup -\text{ᗡ}ʻ\text{max}_{P}\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*211·314. \supset \vdash \colon\ldotp \text{Hp}.\supset :\alpha \in \text{D}ʻP_{\in }.\supset .\alpha \in \text{ᗡ}ʻ\text{seq}_{P}\cup -\text{ᗡ}ʻ\text{max}_{P} + &\qquad \text{(1)}\\ +\vdash .(1).*211·24.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*211·316.</b> \[\begin{align}&\vdash :P\in \text{Rl}ʻJ\cap \text{connex} .\supset .\text{sect}ʻP-\text{D}ʻP_{\in }=\text{sect}ʻP\cap + \text{ᗡ}ʻ\text{max}_{P}-\text{ᗡ}ʻ\text{seq}_{P}\\ +&[*211·315.\text{Transp}]\end{align}\]</p> + +<p class="nind"><b>*211·317.</b> \(\vdash :P\in \text{trans}.\supset .\text{D}ʻP_{\in }=P_{\in }ʻʻ\text{sect}ʻP\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*211·15·313. \supset \vdash :\text{Hp}.\supset .\text{D}ʻP_{\in }\subset P_{\in }ʻʻ\text{sect}ʻP &\qquad \text{(1)}\\ +\vdash .(1).*37·15.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*211·32">*211·32</a>.</b> \[\begin{align}&\vdash :P\in \text{trans}\cap \text{connex} .\supset .\text{D}ʻP_{\in }=\text{sect}ʻP\cap (\text{ᗡ}ʻ\text{seq}_{P}\cup + -\text{ᗡ}ʻ\text{max}_{P})\\ +&[*211·24·15·31]\end{align}\]</p> + +<p class="nind"><b>*211·321.</b> \[\begin{align}&\vdash :P\in \text{trans}\cap \text{connex} .\supset .\text{sect}ʻP-\text{D}ʻP_{\in }=\text{sect}ʻP\cap + \text{ᗡ}ʻ\text{max}_{P}-\text{ᗡ}ʻ\text{seq}_{P}\\ +&[*211·32]\end{align}\]</p> + +<p class="nind"><b>*211·33.</b> \[\begin{align}\vdash \colon\ldotp P\in &\text{Ser}.\alpha \in \text{sect}ʻP.\supset :\\ +&\alpha {\sim}\in \text{D}ʻP_{\in }.\supset .\text{E}!\text{seq}_{P}ʻPʻʻ\alpha .{\sim}\text{E}!\text{seq}_{P}ʻ\overrightarrow{\text{seq}} _{P}ʻPʻʻ\alpha\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*211·321. \supset \vdash :\text{Hp}.\alpha {\sim}\in \text{D}ʻP_{\in }.&\supset .\text{E}!\text{max}_{P}ʻ\alpha . &\qquad \text{(1)}\\ +[*206·35] & \supset .\text{E}!\text{seq}_{P}ʻPʻʻ\alpha .\text{seq}_{P}ʻPʻʻ\alpha =\text{max}_{P}ʻ\alpha &\qquad \text{(2)}\\ +\vdash .(1).*206·46. \supset \vdash :\text{Hp}.\alpha {\sim}\in \text{D}ʻP_{\in }.\supset .\overrightarrow{\text{seq}} _{P}ʻ\alpha + &=\overrightarrow{\text{seq}} _{P}ʻ\overrightarrow{\text{max}}_{P}ʻ\alpha \\ +[(2)] & =\overrightarrow{\text{seq}} _{P}ʻ\overrightarrow{\text{seq}} _{P}ʻPʻʻ\alpha &\qquad \text{(3)}\\ +\vdash .*211·321. \supset \vdash :\text{Hp}.\alpha {\sim}\in \text{D}ʻP_{\in }.&\supset .{\sim}\text{E}!\text{seq}_{P}ʻ\alpha .\\ +[(3)] &\supset .{\sim}\text{E}!\text{seq}_{P}ʻ\overrightarrow{\text{seq}} _{P}ʻPʻʻ\alpha &\qquad \text{(4)}\\ +\vdash .(2).(4). &\supset \vdash :\text{Hp}.\alpha {\sim}\in \text{D}ʻP_{\in }.\supset .\\ +&\text{E}!\text{seq}_{P}ʻPʻʻ\alpha .{\sim}\text{E}!\text{seq}_{P}ʻ\overrightarrow{\text{seq}} _{P}ʻPʻʻ\alpha :\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*211·34.</b> \[\begin{align}\vdash \colon\ldotp P\in \text{Ser}.\supset :\alpha \in &\text{sect}ʻP-\text{D}ʻP_{\in }.\equiv .\\ +&\alpha =Pʻʻ\alpha \cup \iota ʻ\text{seq}_{P}ʻPʻʻ\alpha .{\sim}\text{E}!\text{seq}_{P}ʻ\overrightarrow{\text{seq}} _{P}ʻPʻʻ\alpha\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*211·321. \supset \vdash :\text{Hp}.\alpha \in \text{sect}ʻP-\text{D}ʻP_{\in }.&\supset .\text{E}!\text{max}_{P}ʻ\alpha .\\ +[*206·35] &\supset .\text{max}_{P}ʻ\alpha =\text{seq}_{P}ʻPʻʻ\alpha . &\qquad \text{(1)}\\ +[*211·2] & \supset .\alpha =Pʻʻ\alpha \cup \iota ʻ\text{seq}_{P}ʻPʻʻ\alpha &\qquad \text{(2)}\\ +\vdash .*211·321. \supset \vdash :\text{Hp}.\alpha \in \text{sect}ʻP-\text{D}ʻP_{\in }.&\supset .{\sim}\text{E}!\text{seq}_{P}ʻ\alpha .\\ +[*206·46.(1)] &\supset .{\sim}\text{E}!\text{seq}_{P}ʻ\overrightarrow{\text{seq}} _{P}ʻPʻʻ\alpha &\qquad \text{(3)}\\ +\vdash .*206·21.*205·111.\supset \vdash :\text{Hp}.\alpha =Pʻʻ\alpha \cup \iota ʻ&\text{seq}_{P}ʻPʻʻ\alpha .\supset .\\ +&\text{seq}_{P}ʻPʻʻ\alpha =\text{max}_{P}ʻ\alpha &\qquad \text{(4)}\\ +\vdash . (4) . *206·46 . \supset \vdash : \text{Hp} . \alpha = Pʻʻ\alpha \cup {℩}ʻ\text{seq}_{P}ʻPʻʻ\alpha . {\sim} \text{E}! &\text{seq}_{P}ʻ\overrightarrow{\text{seq}} + _{P}ʻPʻʻ\alpha . \supset .\\ +&{\sim} \text{E}! \text{seq}_{P}ʻ\alpha &\qquad \text{(5)}\\ +\vdash . *206·18 . *22·58 . \supset \vdash : &\alpha = Pʻʻ\alpha \cup {℩}ʻ\text{seq}_{P}ʻPʻʻ\alpha + . \supset . \alpha \subset CʻP . Pʻʻ\alpha \subset \alpha &\qquad \text{(6)}\\ +\vdash . (4) . (5) . (6) . *211·321 . \supset \\ +\vdash : \text{Hp} . \alpha = Pʻʻ\alpha \cup {℩}ʻ\text{seq}_{P}ʻPʻʻ\alpha . {\sim} &\text{E}! \text{seq}_{P}ʻ\overrightarrow{\text{seq}} _{P}ʻPʻʻ\alpha + . \supset . \alpha \in \text{sect}ʻP - \text{D}ʻP_{\in } &\qquad \text{(7)}\\ +\vdash . (2) . (3) . (7) . \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_635">[Pg 635]</span></p> + +<p class="nind"><b>*211·35.</b> \[\begin{align}&\vdash \colon\ldotp P \in \text{Ser} . \supset : \alpha \in \text{sect}ʻP - \text{D}ʻP_{\in } . \equiv .\\ +&(\exists x) . x \in CʻP . \alpha = \overrightarrow{P}ʻx \cup {℩}ʻx . {\sim} \text{E}! \breve{P} _{1}ʻx\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *211·34 . \supset \vdash \colon\ldotp \text{Hp} . \supset :\\ +\alpha \in \text{sect}ʻP - \text{D}ʻP_{\in } . &\equiv . (\exists x) . x = \text{seq}_{P}ʻPʻʻ\alpha + . \alpha = Pʻʻ\alpha \cup {℩}ʻx . {\sim} \text{E}! \text{seq}_{P}ʻ{℩}ʻx\\ +[*206·21 . *205·111] \equiv . (\exists x) . x = \text{seq}_{P}ʻPʻʻ\alpha . x &= \text{max}_{P}ʻ\alpha .\\ +\alpha &= Pʻʻ\alpha \cup {℩}ʻx . {\sim} \text{E}! \text{seq}_{P}ʻ{℩}ʻx .\\ +[*206·35] &\equiv . (\exists x) . x = \text{max}_{P}ʻ\alpha . \alpha = Pʻʻ\alpha \cup {℩}ʻx . {\sim} \text{E}! \text{seq}_{P}ʻ{℩}ʻx .\\ +[*205·22] &\equiv . (\exists x) . x = \text{max}_{P}ʻ\alpha . \alpha = \overrightarrow{P}ʻx \cup {℩}ʻx . {\sim} \text{E}! \text{seq}_{P}ʻ{℩}ʻx .\\ +[*205·197] &\equiv . (\exists x) . x \in CʻP . \alpha = \overrightarrow{P}ʻx \cup {℩}ʻx . {\sim} \text{E}! \text{seq}_{P}ʻ{℩}ʻx .\\ +[*206·44] &\equiv . (\exists x) . x \in CʻP . \alpha = \overrightarrow{P}ʻx \cup {℩}ʻx . {\sim} \in ! \breve{P} _{1}ʻx \colon\ldotp \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*211·351.</b> \(\vdash : P \in \text{Ser} . \supset . \text{sect}ʻP - \text{D}ʻP_{\in } = \overrightarrow{P}_{*}ʻʻ(CʻP - \text{D}ʻP_{1})\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *204·7 . *211·35 . \supset \\ +\vdash \colon\ldotp \text{Hp} . \supset : \alpha \in \text{sect}ʻP - \text{D}ʻP_{\in } . &\equiv . (\exists x) . x \in CʻP - \text{D}ʻP_{1} + . \alpha = \overrightarrow{P}ʻx \cup {℩}ʻx .\\ +[*201·521] &\equiv . (\exists x) . x \in CʻP - \text{D}ʻP_{1} . \alpha = \overrightarrow{P}_{*}ʻx .\\ +[*37·7] &\equiv . \alpha \in \overrightarrow{P}_{*}ʻʻ(CʻP - \text{D}ʻP_{1}) \colon\ldotp \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*211·36.</b> \(\vdash \colon\ldotp P \in \text{Ser} . \text{D}ʻP_{1} = \text{D}ʻP . \supset : \alpha \in \text{sect}ʻP - \text{D}ʻP_{\in } + . \equiv . \alpha = CʻP . \text{E}! Bʻ\breve{P}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *211·351 . \supset \vdash : \text{Hp} . &\supset . \text{sect}ʻP - \text{D}ʻP_{\in } = \overrightarrow{P}_{*}ʻʻ\overrightarrow{B}ʻ\breve{P} &\qquad \text{(1)}\\ +\vdash . (1) . *202·52 . &\supset \vdash \colon\ldotp \text{Hp} . \supset :\\ +&\alpha \in \text{sect}ʻP - \text{D}ʻP_{\in } . \equiv . (\exists x) . x = Bʻ\breve{P} . \alpha = \overrightarrow{P}_{*}ʻx .\\ +[*204·11 . *201·521] &\equiv . (\exists x) . x = \beta ʻ\breve{P} . \alpha = CʻP .\\ +[*14·204] &\equiv . \alpha = CʻP . \text{E}! Bʻ\breve{P} \colon\ldotp \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_636">[Pg 636]</span></p> + +<p class="nind"><b>*211·361.</b> \(\vdash :P\in \text{Ser}.\text{D}ʻP_{1}=CʻP.\supset .\text{sect}ʻP=\text{D}ʻP_{\in }\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*201·63. \supset \vdash :\text{Hp}.&\supset .\text{D}ʻP_{1}\subset \text{D}ʻP.\\ +[*93·103] &\supset .\overrightarrow{B}ʻ\breve{P} =\Lambda &\qquad \text{(1)}\\ +\vdash .(1).*211·36.&\supset \vdash :\text{Hp}.\supset .\text{sect}ʻP-\text{D}ʻP_{\in }=\Lambda &\qquad \text{(2)}\\ +\vdash .(2).*211·15.&\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*211·371.</b> \[\begin{align}&\vdash \colon\ldotp P\in \text{trans}\cap \text{connex} :(\alpha ).\alpha \in \text{ᗡ}ʻ\text{max}_{P}\cup \text{ᗡ}ʻ\text{seq}_{P}:\supset + .\text{D}ʻP_{\in }\subset \text{ᗡ}ʻ\text{seq}_{P}\\ +&[*211·32]\end{align}\]</p> + +<p class="nind"><b>*211·372.</b> \(\vdash \colon\ldotp P\in \text{trans}\cap \text{connex} :(\alpha ).\alpha \in \text{ᗡ}ʻ\text{max}_{P}\cup \text{ᗡ}ʻ\text{seq}_{P}:\supset + .\text{D}ʻP_{\in }=\overrightarrow{P}ʻʻCʻP\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*211·371.\supset \vdash \colon\ldotp \text{Hp}.&\supset :\alpha \in \text{D}ʻP_{\in }.\supset .\text{E}!\text{seq}_{P}ʻ\alpha .\\ +[*206·3.*211·1·15] &\supset .\alpha =\overrightarrow{P}ʻ\text{seq}_{P}ʻ\alpha .\\ +[*206·18] &\supset .\alpha \in \overrightarrow{P}ʻʻCʻP &\qquad \text{(1)}\\ +\vdash .(1).*211·3.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*211·38">*211·38</a>.</b> \(\vdash \colon\ldotp P\in \text{Ser}.\supset :(\alpha ).\alpha \in \text{ᗡ}ʻ\text{max}_{P}\cup \text{ᗡ}ʻ\text{seq}_{P}.\equiv + .\text{D}ʻP_{\in }=\overrightarrow{P}ʻʻCʻP\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*211·11.&\supset \vdash \colon\ldotp \text{D}ʻP_{\in }=\overrightarrow{P}ʻʻCʻP.\equiv :(\beta ):(\exists x).Pʻʻ\beta =\overrightarrow{P}ʻx.x\in CʻP &\qquad \text{(1)}\\ +\vdash .*206·174.*205·111.\supset \\ +\vdash :P\in \text{Ser}.{\sim}&\exists !\overrightarrow{\text{max}}_{P}ʻ\beta .\supset .\overrightarrow{\text{seq}} _{P}ʻ\beta + =CʻP\cap \hat{x} (Pʻʻ\beta =\overrightarrow{P}ʻx) &\qquad \text{(2)}\\ +\vdash .(1).(2).&\supset \vdash \colon\ldotp P\in \text{Ser}.\text{D}ʻP_{\in }=\overrightarrow{P}ʻʻCʻP.\supset :{\sim}\exists !\overrightarrow{\text{max}}_{P}ʻ\beta + .\supset .\exists !\overrightarrow{\text{seq}} _{P}ʻ\beta :\\ +[*33·41] &\supset :\beta \in \text{ᗡ}ʻ\text{max}_{P}\cup \text{ᗡ}ʻ\text{seq}_{P} &\qquad \text{(3)}\\ +\vdash .(3).*211·372.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p>The following propositions are concerned with \(\text{D}ʻ(P_{\in}\dot{\cap} I)\), +<i>i.e.</i> with those sections of \(P\) which +have no maximum. If \(P\) is compact (<i>i.e.</i> if \(P^{2}=P\)), +\(\text{D}ʻ(P_{\in }\dot{\cap} I)=\text{D}ʻP_{\in }\). If \(P\) +is also a Dedekindian series, \(\text{D}ʻ(P_{\in }\dot{\cap}I)=\overrightarrow{P}ʻʻCʻP\). +This is the mark of Dedekindian continuity, since it states that, if +\(Pʻʻ\alpha\) has no maximum, there is an \(x\) for which +\(Pʻʻ\alpha=\overrightarrow{P}ʻx\), and this \(x\) is the upper limit of +\(\overrightarrow{P}ʻx\) has no maximum, so that the series is compact.</p> + +<p class="nind"><b><a id="*211·4">*211·4</a>.</b> \(\vdash .\text{D}ʻ(P_{\in }\dot{\cap} I)\subset -\text{ᗡ}ʻ\text{max}_{P}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*211·12.\supset \vdash :\alpha \in \text{D}ʻ(P_{\in }\dot{\cap} I).&\supset .\alpha -Pʻʻ\alpha =\Lambda .\\ +[*205·111] &\supset .\overrightarrow{\text{max}}_{P}ʻ\alpha =\Lambda :\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_637">[Pg 637]</span></p> + +<p class="nind"><b><a id="*211·41">*211·41</a>.</b> \(\vdash .\text{D}ʻ(P_{\in }\dot{\cap} I)=\text{sect}ʻP-\text{ᗡ}ʻ\text{max}_{P}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*211·1.*205·111.\supset \\ +\vdash :\alpha \in \text{sect}ʻP-\text{ᗡ}ʻ\text{max}_{P}.&\equiv .\alpha \subset CʻP.Pʻʻ\alpha \subset a.\alpha \subset Pʻʻ\alpha .\\ +[*22·41] &\equiv .\alpha \subset CʻP.\alpha =Pʻʻ\alpha .\\ +[*37·15.*211·12] &\equiv .\alpha \in \text{D}ʻ(P_{\in }\dot{\cap} I):\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*211·411.</b> \(\vdash :P\in \text{trans}.\alpha =Pʻʻ\beta .\alpha \subset Pʻʻ\alpha .\supset .\alpha =Pʻʻ\alpha\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*30·37. &\supset \vdash :\text{Hp}.\supset .Pʻʻ\alpha =PʻʻPʻʻ\beta \\ +[*201·5] &\subset Pʻʻ\beta \\ +[\text{Hp}] &\subset \alpha &\qquad \text{(1)}\\ +\vdash .(1).*22·41.&\supset \vdash :\text{Hp}.\supset .\alpha =Pʻʻ\alpha :\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*211·42.</b> \(\vdash :P\in \text{trans}.\supset .\text{D}ʻ(P_{\in }\dot{\cap} I)=\text{D}ʻP_{\in }-\text{ᗡ}ʻ\text{max}_{P}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*211·14·4. & \supset \vdash .\text{D}ʻ(P_{\in }\dot{\cap} I)\subset \text{D}ʻP_{\in }-\text{ᗡ}ʻ\text{max}_{P} &\qquad \text{(1)}\\ +\vdash .*211·411·11.*205·111.&\supset \vdash :\text{Hp}.\alpha \in \text{D}ʻP_{\in }-\text{ᗡ}ʻ\text{max}_{P}.\supset .\alpha \in \text{D}ʻ(P_{\in }\dot{\cap} + I) &\qquad \text{(2)}\\ +\vdash .(1).(2).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*211·43.</b> \(\vdash :P\in \text{trans}\cap \text{connex} .\supset .\text{D}ʻP_{\in }-\text{ᗡ}ʻ\text{seq}_{P}\subset \text{D}ʻ(P_{\in }\dot{\cap} I)\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*211·312.\supset \vdash \colon\ldotp \text{Hp}.\supset :\alpha \in \text{D}ʻP_{\in }.\overrightarrow{\text{seq}} _{P}ʻ\alpha =\Lambda .&\supset .\alpha =Pʻʻ\alpha .\\ +[*211·12] &\supset .\alpha \in \text{D}ʻ(P_{\in }\dot{\cap} I)\colon\ldotp \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*211·431.</b> \[\begin{align}\vdash :P\in \text{trans}\cap &\text{connex} .\supset .\\ +&\text{D}ʻP_{\in }-\text{D}ʻ(P_{\in }\dot{\cap} I)=\text{sect}ʻP\cap \text{ᗡ}ʻ\text{max}_{P}\cap \text{ᗡ}ʻ\text{seq}_{P}\\ +[*211·32·41]\end{align}\]</p> + +<p class="nind"><b>*211·44.</b> \[\begin{align}&\vdash .\Lambda \in \text{D}ʻ(P_{\in }\dot{\cap} I).\Lambda \in \text{D}ʻP_{\in }.\Lambda \in \text{sect}ʻP\\ +&[*37·29.*211·12·14]\end{align}\]</p> + +<p class="nind"><b>*211·45.</b> \(\vdash :P\in \text{trans}.x{\sim}\in \text{ᗡ}ʻ(P\dot{-} P^{2}).\supset .\overrightarrow{P}ʻx\in \text{D}ʻ(P_{\in}\dot{\cap} I)\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*201·501. &\supset \vdash :\text{Hp}.\supset .Pʻʻ\overrightarrow{P}ʻx\subset \overrightarrow{P}ʻx &\qquad \text{(1)}\\ +\vdash .*33·41.*32·3·34.&\supset \vdash :\text{Hp}.\supset .\overrightarrow{P}ʻx-Pʻʻ\overrightarrow{P}ʻx=\Lambda &\qquad \text{(2)}\\ +\vdash .(1).(2). &\supset \vdash :\text{Hp}.\supset .\overrightarrow{P}ʻx=Pʻʻ\overrightarrow{P}ʻx &\qquad \text{(3)}\\ +\vdash .(3).*211·12.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*211·451.</b> \(\vdash :\overrightarrow{P}ʻx\in \text{D}ʻ(P_{\in }\dot{\cap} I).\supset .x{\sim}\in \text{ᗡ}ʻ(P\dot{-} P^{2})\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*211·12.\supset \vdash \colon\ldotp \text{Hp}.&\supset :\overrightarrow{P}ʻx=Pʻʻ\overrightarrow{P}ʻx:\\ +[*37·3] &\supset :yPx.\equiv _{y}.yP^{2}x:\\ +[*10·51]&\supset :{\sim}(\exists y).yPx.{\sim}(yP^{2}x)\colon\ldotp \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_638">[Pg 638]</span></p> + +<p class="nind"><b>*211·452.</b> \[\begin{align}&\vdash \colon\ldotp P\in \text{trans}.\supset :\overrightarrow{P}ʻx\in \text{D}ʻ(P_{\in }\dot{\cap} + I).\equiv .x{\sim}\in \text{ᗡ}ʻ(P\dot{-} P^{2})\\ +&[*211*·45·451]\end{align}\]</p> + +<p class="nind"><b>*211·46.</b> \[\begin{align}\vdash \colon\ldotp P\in \text{trans}\cap \text{connex} :(\alpha ).\alpha \in &\text{ᗡ}ʻ\text{max}_{P}\cup \text{ᗡ}ʻ\text{seq}_{P}:\supset .\\ +&\text{D}ʻ(P_{\in }\dot{\cap} I)=\overrightarrow{P}ʻʻ\{CʻP-\text{ᗡ}ʻ(P\dot{-} P^{2})\}\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*211·452.&\supset \vdash :\text{Hp}.\supset .\overrightarrow{P}ʻʻ{CʻP-\text{ᗡ}ʻ(P\dot{-} P^{2})}\subset \text{D}ʻ(P_{\in }\dot{\cap} I) &\qquad \text{(1)}\\ +\vdash .*211·372·14.\supset \\ +\vdash \colon\ldotp \text{Hp}.\supset :\alpha \in \text{D}ʻ(P_{\in }\dot{\cap} I).&\supset .(\exists x).x\in CʻP.\alpha =\overrightarrow{P}ʻx.\overrightarrow{P}ʻx\in \text{D}ʻ(P_{\in }\dot{\cap} + I).\\ +[*211·452] &\supset .(\exists x).x\in .CʻP.\alpha =.\overrightarrow{P}ʻx.x{\sim}\in \text{ᗡ}ʻ(P\dot{-} P^{2}).\\ +[*37·7] &\supset .\alpha \in \overrightarrow{P}ʻʻ\{CʻP-\text{ᗡ}ʻ(P\dot{-} P^{2})\} &\qquad \text{(2)}\\ +\vdash .(1).(2).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*211·47">*211·47</a>.</b> \(\vdash \colon\ldotp P\in \text{trans}.\supset :(\alpha ).\alpha \in \text{ᗡ}ʻ\text{max}_{P}\cup \text{ᗡ}ʻ\text{seq}_{P}.\equiv .\text{D}ʻ(P_{\in }\dot{\cap} + I)\subset \text{ᗡ}ʻ\text{seq}_{P}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*211·272.*24·43.\supset \\ +\vdash \colon\ldotp \text{Hp}.\supset :(\alpha ).\alpha \in \text{ᗡ}ʻ\text{max}_{P}\cup \text{ᗡ}ʻ\text{seq}_{P}.&\equiv .\text{sect}ʻP-\text{ᗡ}ʻ\text{max}_{P}\subset + \text{ᗡ}ʻ\text{seq}_{P}.\\ +[*211·41] &\equiv .\text{D}ʻ(P_{\in }\dot{\cap} I)\subset \text{ᗡ}ʻ\text{seq}_{P}\colon\ldotp \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p>The following propositions are concerned with certain consequences of +the hypothesis \(P^{2}=P\). This hypothesis is important because it is +the defining characteristic of compact series.</p> + +<p class="nind"><b><a id="*211·5">*211·5</a>.</b> \(\vdash :P^{2}=P.\alpha =Pʻʻ\beta .\supset .\alpha =Pʻʻ\alpha\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*37·33.\supset \vdash :\text{Hp}.&\supset .Pʻʻ\beta =PʻʻPʻʻ\beta .\\ +[\text{Hp}.*13·12] &\supset .\alpha =Pʻʻ\alpha :\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*211·51.</b> \(\vdash :P^{2}=P.\supset .\text{D}ʻP_{\in }=\text{D}ʻ(P_{\in }\dot{\cap} I) \quad[*211·5·11·12]\)</p> + +<p>Thus in compact series there is no distinction between the two sorts of +segments.</p> + +<p class="nind"><b>*211·52.</b> \(\vdash \colon\ldotp P^{2}=P.P\in \text{connex} .\supset :\text{E}!\text{max}_{P}ʻ\alpha .\supset .{\sim}\text{E}!\text{seq}_{P}ʻ\alpha\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*206·5.\supset \vdash :P^{2}\,\unicode{x2abd}\, P.P\in \text{connex} .&\text{E}!\text{max}_{P}ʻ\alpha .\text{E}!\text{seq}_{P}ʻ\alpha + .\supset .\dot{\exists} !(P\dot{-} P^{2}) &\qquad \text{(1)}\\ +\vdash .(1).\text{Transp}.\supset \vdash :P^{2}=P.P\in \text{connex} .&\text{E}!\text{max}_{P}ʻ\alpha .\supset .{\sim}\text{E}!\text{seq}_{P}ʻ\alpha :\\ + &\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*211·53.</b> \[\begin{align}\vdash \colon\colon P^{2}=P.P\in \text{connex} .\supset \colon\ldotp \text{E}!\text{max}_{P}ʻ\alpha .&\lor.\text{E}!\text{seq}_{P}ʻ\alpha + :\equiv :\\ +&\text{E}!\text{max}_{P}ʻ\alpha .\equiv .{\sim}\text{E}!\text{seq}_{P}ʻ\alpha\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*4·64.\supset \vdash \colon\ldotp \text{E}!\text{max}_{P}ʻ\alpha .\lor.\text{E}!\text{seq}_{P}ʻ\alpha :&\equiv :{\sim}\text{E}!\text{seq}_{P}ʻ\alpha + .\supset .\text{E}!\text{max}_{P}ʻ\alpha &\qquad \text{(1)}\\ +\vdash .*4·73.*211·52.\supset \\ +\vdash \colon\colon\text{Hp}.\supset \colon\ldotp {\sim}\text{E}!\text{seq}_{P}ʻ\alpha .\supset .\text{E}!\text{max}_{P}ʻ\alpha :&\equiv :\text{E}!\text{max}_{P}ʻ\alpha + .\equiv .{\sim}\text{E}!\text{seq}_{P}ʻ\alpha &\qquad \text{(2)}\\ +\vdash .(1).(2).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_639">[Pg 639]</span></p> + +<p>The condition \((\alpha ):\text{E}!\text{max}_{P}ʻ\alpha .\equiv.{\sim}\text{E}!\text{seq}_{P}ʻ\alpha\) +is the Dedekindian definition of continuity. In virtue of the above +proposition, this is equivalent, in a series, to compactness combined +with Dedekind's axiom, namely +\[ +(\alpha ):\text{E}!\text{max}_{P}ʻ\alpha .\lor.\text{E}!\text{seq}_{P}ʻ\alpha . +\]</p> + +<p class="nind"><b>*211*54.</b> \(\vdash \colon\ldotp P\,\unicode{x2abd}\, J:\exists !\overrightarrow{\text{max}}_{P}ʻ\alpha .\supset _{\alpha }.{\sim}\exists !\overrightarrow{\text{seq}} _{P}ʻ\alpha + :\supset .P\,\unicode{x2abd}\, P^{2}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*10·1.\supset \vdash \colon\ldotp \text{Hp}.&\supset :\exists !\overrightarrow{\text{max}}_{P}ʻ\iota ʻx.\supset .{\sim}\exists !\text{seq}_{P}ʻ\iota ʻx:\\ +[*205·18] &\supset :x\in CʻP.\supset .{\sim}\exists !\text{seq}_{P}ʻ\iota ʻx.\\ +[*206·42] &\supset .{\sim}\exists !\overleftarrow{P\dot{-} P^{2}}ʻx.\\ +[*33·4] &\supset .x{\sim}\in \text{D}ʻ(P\dot{-} P^{2}) &\qquad \text{(1)}\\ +\vdash .*33·263.&\supset \vdash :x{\sim}\in CʻP.\supset .x{\sim}\in \text{D}ʻ(P\dot{-} P^{2}) &\qquad \text{(2)}\\ +\vdash .(1).(2).\supset \vdash :\text{Hp}.&\supset .\text{D}ʻ(P\dot{-} P^{2})=\Lambda .\\ +[*33·241.*25·3] &\supset .P\,\unicode{x2abd}\, P^{2}:\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*211·541.</b> \(\vdash \colon\ldotp P\in \text{Rl}ʻJ\cap \text{trans}:\exists !\overrightarrow{\text{max}}_{P}ʻ\alpha .\supset _{\alpha }.{\sim}\exists + !\overrightarrow{\text{seq}} _{P}ʻ\alpha :\supset .P=P^{2}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*201·1.&\supset \vdash :\text{Hp}.\supset .P^{2}\,\unicode{x2abd}\, P &\qquad \text{(1)}\\ +\vdash .*211·54.&\supset \vdash :\text{Hp}.\supset .P\,\unicode{x2abd}\, P^{2} &\qquad \text{(2)}\\ +\vdash .(1).(2).&\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*211·55.</b> \[\begin{align}&\vdash \colon\colon P\in \text{Ser}.\supset \colon\ldotp \exists !\overrightarrow{\text{max}}_{P}ʻ\alpha .\supset _{\alpha }.{\sim}\exists + !\overrightarrow{\text{seq}} _{P}ʻ\alpha :\equiv .P=P^{2}\\ +&[*211·52·541]\end{align}\]</p> + +<p class="nind"><b>*211·551.</b> \[\begin{align}&\vdash \colon\ldotp P\in \text{Ser}.\supset :\text{ᗡ}ʻ\text{max}_{P}\cap \text{ᗡ}ʻ\text{seq}_{P}=\Lambda .\equiv .P=P^{2}\\ +&[*211·55.*33·41]\end{align}\]</p> + +<p class="nind"><b>*211·552.</b> \[\begin{align}\vdash \colon\colon P\in \text{Ser}.\supset \colon\ldotp \text{E}!\text{max}_{P}ʻ\alpha .&\equiv _{\alpha }.{\sim}\text{E}!\text{seq}_{P}ʻ\alpha + :\equiv :\\ +&P=P^{2}:(\alpha ):\text{E}!\text{max}_{P}ʻ\alpha .\lor.\text{E}!\text{seq}_{P}ʻ\alpha \\ +&[*211·55]\end{align}\]</p> + +<p class="nind"><b><a id="*211·553">*211·553</a>.</b> \[\begin{align}\vdash \colon\colon P\in \text{Ser}.\supset \colon\ldotp \text{ᗡ}ʻ\text{max}_{P}=-&\text{ᗡ}ʻ\text{seq}_{P}.\equiv :\\ +&P=P^{2}:(\alpha ).\alpha \in \text{ᗡ}ʻ\text{max}_{P}\cup \text{ᗡ}ʻ\text{seq}_{P}\\ +[*211·552.*71·163]\end{align}\]</p> + +<p>The following propositions are concerned in showing that +\(\text{sect}ʻP\), \(\text{D}ʻP_{\in }\), and \(\text{D}ʻ(P_{\in}\dot{\cap} I)\) +all verify the hypotheses of <a href="#*210">*210</a>, if taken as the +\(\kappa\) of that number.</p> + +<p class="nind"><b><a id="*211·56">*211·56</a>.</b> \(\vdash \colon\ldotp P\in \text{connex} .\alpha ,\beta \in \text{sect}ʻP.\supset :\alpha \subset \beta .\lor.\beta \subset Pʻʻ\alpha\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*211·2.\supset \vdash \colon\ldotp \text{Hp}.\exists !\alpha -\beta .&\supset .\exists !\alpha \cap CʻP-\beta -Pʻʻ\beta \\. +[*202·501] &\supset .\exists !\alpha \cap pʻ\overleftarrow{P}ʻʻ\beta .\\ +[*40·682] &\supset .\beta \subset Pʻʻ\supset \alpha &\qquad \text{(1)}\\ +\vdash .(1).*24·55.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_640">[Pg 640]</span></p> + +<p class="nind"><b><a id="*211·561">*211·561</a>.</b> \[\begin{align}&\vdash \colon\ldotp P_{\text{po}}\in \text{connex} .\alpha ,\beta \in \text{sect}ʻP.\supset :\alpha \subset \beta .\lor.\beta \subset Pʻʻ\alpha \\ +&[*211·56·17·131]\end{align}\]</p> + +<p class="nind"><b>*211·562.</b> \(\vdash \colon\ldotp P_{\text{po}}\in \text{connex} .\alpha ,\beta \in \text{sect}ʻP.\supset :\alpha \subset \beta .\lor.\beta \subset \alpha \quad[*211·561·1]\)</p> + +<p class="nind"><b><a id="*211·6">*211·6</a>.</b> \(\vdash \colon\ldotp P\in \text{connex} .\alpha ,\beta \in \text{sect}ʻP.\supset :\alpha \subset \beta .\lor.\beta \subset \alpha \quad[*211·56·1]\)</p> + +<p class="nind"><b><a id="*211·61">*211·61</a>.</b> \[\begin{align}&\vdash \colon\ldotp P\in \text{trans}\cap \text{connex} .\alpha ,\beta \in \text{D}ʻP_{\in }.\supset :\alpha \subset \beta .\lor.\beta \subset \alpha \\ +&[*211·15·6]\end{align}\]</p> + +<p class="nind"><b>*211·62.</b> \(\vdash \colon\ldotp P\in \text{connex} .\alpha ,\beta \in \text{D}ʻ(P_{\in }\dot{\cap} I).\supset :\alpha \subset \beta .\lor.\beta \subset \alpha \quad[*211·14·6]\)</p> + +<p>In the hypothesis of <a href="#*211·61">*211·61</a>, it is necessary that \(P\) should be +transitive as well as connected. Take, for example, +\[ +P=x\downarrow y\unicode{x228d} y\downarrow z\unicode{x228d} z\downarrow x (x\neq y.x\neq z.y\neq z). +\] +Then \(P\) is connected, but not transitive; also we have +\[ +\overrightarrow{P}ʻy=\iota ʻx.\overrightarrow{P}ʻz=\iota ʻy. +\] +Hence \(\iota ʻx,\iota ʻy\in \text{D}ʻP_{\in }.{\sim}(\iota ʻx\subset\iota ʻy).{\sim}(\iota ʻy\subset \iota ʻx)\). +Thus connection is not sufficient in the hypothesis of <a href="#*211·61">*211·61</a>.</p> + +<p class="nind"><b><a id="*211·63">*211·63</a>.</b> \(\vdash :\lambda \subset \text{sect}ʻP.\supset .sʻ\lambda \in \text{sect}ʻP\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*211·1.\supset \vdash \colon\ldotp \text{Hp}.&\supset :\alpha \in \lambda .\supset _{\alpha }.\alpha \subset CʻP:\\ +[*40·151] &\supset :sʻ\lambda \subset CʻP &\qquad \text{(1)}\\ +\vdash .*211·1.\supset \vdash \colon\ldotp \text{Hp}.&\supset :\alpha \in \lambda .\supset _{\alpha }.Pʻʻ\alpha \subset \alpha :\\ +[*40·8] &\supset :Pʻʻsʻ\lambda \subset sʻ\lambda &\qquad \text{(2)}\\ +\vdash .(1).(2).*211·1.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p>This proposition shows that \(\text{sect}ʻP\) verifies the hypothesis +of <a href="#*210·251">*210·251</a>, with the exception of \(\text{sect}ʻP{\sim}\in 1\), which +requires \(\dot{\exists} !P\).</p> + +<p class="nind"><b>*211·631.</b> \(\vdash :\lambda \subset \text{sect}ʻP.\supset .pʻ\lambda \cap CʻP\in \text{sect}ʻP\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*22·43.&\supset \vdash .pʻ\lambda \cap CʻP\subset CʻP &\qquad \text{(1)}\\ +\vdash .*211·1.\supset \vdash \colon\ldotp \text{Hp}.&\supset :\alpha \in \lambda .\supset_\alpha.Pʻʻ\alpha \subset \alpha :\\ +[*40·81] &\supset :Pʻʻpʻ\lambda \subset pʻ\lambda :\\ +[*37·265·15] &\supset :Pʻʻ(pʻ\lambda \cap CʻP)\subset pʻ\lambda \cap CʻP &\qquad \text{(2)}\\ +\vdash .(1).(2).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*211·632.</b> \(\vdash :\lambda \subset \text{sect}ʻP.\exists !\lambda .\supset .pʻ\lambda \in \text{sect}ʻP\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*40·23.\supset \vdash :\text{Hp}.&\supset .pʻ\lambda \subset sʻ\lambda .\\ +[*211·63·1] &\supset .pʻ\lambda \subset CʻP &\qquad \text{(1)}\\ +\vdash .(1).*211·631.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*211·633.</b> \(\vdash :\lambda \subset \text{sect}ʻP.\supset .pʻ\lambda \cap sʻ\text{sect}ʻP\in \text{sect}ʻP \quad[*211·631·26]\)</p> + +<p><span class="pagenum" id="Page_641">[Pg 641]</span></p> + +<p>This proposition shows that \(\text{sect}ʻP\) verifies the hypothesis of +<a href="#*210·252">*210·252</a>, with the exception of \(\text{sect}ʻP{\sim}\in 1\), which +requires \(\dot{\exists} !P\).</p> + +<p class="nind"><b><a id="*211·64">*211·64</a>.</b> \(\vdash :\lambda \subset \text{D}ʻP_{\in }.\supset .sʻ\lambda \in \text{D}ʻP_{\in }\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*72·504.\supset \vdash :\text{Hp}.\supset .sʻ\lambda &=sʻP_{\in }ʻʻ\breve{P} _{\in }ʻʻ\lambda \\ +[*40·38] &=Pʻʻsʻ\breve{P} _{\in }ʻʻ\lambda &\qquad \text{(1)}\\ +\vdash .(1).*211·11.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*211·65">*211·65</a>.</b> \(\vdash :\lambda \subset \text{D}ʻ(P_{\in }\dot{\cap} I).\supset .sʻ\lambda \in \text{D}ʻ(P_{\in }\dot{\cap} I)\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*211·12.\supset \vdash \colon\ldotp \text{Hp}.&\supset :\alpha \in \lambda .\supset _{\alpha }.\alpha =P_{\in }ʻ\alpha :\\ +[*50·17] &\supset :\lambda =P_{\in }ʻʻ\lambda :\\ +[*40·38] &\supset :sʻ\lambda =Pʻʻsʻ\lambda :\\ +[*211·12] &\supset :sʻ\lambda \in \text{D}ʻ(P_{\in }\dot{\cap} I)\colon\ldotp \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*211·66.</b> \(\vdash :\dot{\exists} !P.\supset .\text{sect}ʻP,\text{D}ʻP_{\in }{\sim}\in 1\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*211·44·26. &\supset \vdash .\Lambda ,CʻP\in \text{sect}ʻP &\qquad \text{(1)}\\ +\vdash .*33·24.&\supset \vdash :\text{Hp}.\supset .\Lambda \neq CʻP &\qquad \text{(2)}\\ +\vdash .(1).(2).*52·41. &\supset \vdash :\text{Hp}.\supset .\text{sect}ʻP{\sim}\in 1 &\qquad \text{(3)}\\ +\vdash .*211·44·301. &\supset \vdash :\text{Hp}.\supset .\Lambda ,\text{D}ʻP\in \text{D}ʻP_{\in } &\qquad \text{(4)}\\ +\vdash .*33·24. &\supset \vdash :\text{Hp}.\supset .\Lambda \neq \text{D}ʻP &\qquad \text{(5)}\\ +\vdash .(4).(5).*52·41. &\supset \vdash :\text{Hp}.\supset .\text{D}ʻP_{\in }{\sim}\in 1 &\qquad \text{(6)}\\ +\vdash .(3).(6).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*211·661.</b> \(\vdash :P\in \text{trans}.\exists !\text{Cl ex}ʻCʻP-\text{ᗡ}ʻ\text{max}_{P}.\supset .\text{D}ʻ(P_{\in }\dot{\cap} I){\sim}\in 1\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*205·111.\supset \vdash :\alpha \in \text{Cl ex}ʻCʻP-\text{ᗡ}ʻ\text{max}_{P}.&\supset .\exists !\alpha .\alpha \subset CʻP.\alpha \subset Pʻʻ\alpha .\\ +[*24·58.*37·2]&\supset .\exists !Pʻʻ\alpha .Pʻʻ\alpha \subset PʻʻPʻʻ\alpha &\qquad \text{(1)}\\ +\vdash .(1).*201·5.\supset \\ +\vdash \colon\ldotp P\in \text{trans}.\supset :\alpha \in \text{Cl ex}ʻCʻP-\text{ᗡ}ʻ\text{max}_{P}.&\supset .Pʻʻ\alpha =PʻʻPʻʻ\alpha .\exists !Pʻʻ\alpha .\\ +[*211·12] & \supset .Pʻʻ\alpha \in \text{D}ʻ(P_{\in }\dot{\cap} I).\exists !Pʻʻ\alpha .\\ +[*10·24] &\supset .\exists !\text{D}ʻ(P_{\in }\dot{\cap} I)-\iota ʻ\Lambda &\qquad \text{(2)}\\ +\vdash .(2).*211·44.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p>The following propositions sum up the above results in relation to +the hypotheses of <a href="#*210">*210</a>. The relation \(P_{\text{lc}}\) with its +field limited to sections or segments, which occurs in the following +propositions, is important, and will be considered at length in the +following number.</p> + +<p class="nind"><b>*211·67.</b> \[\begin{align}&\vdash :P\in \text{connex} .\kappa =\text{sect}ʻP.Q=P_{\text{lc}}\unicode{x0294f}\kappa .\supset .\text{Hp}*210·12\\ +&[*211·6.*210·13]\end{align}\]</p> + +<p><span class="pagenum" id="Page_642">[Pg 642]</span></p> + +<p class="nind"><b>*211·671.</b> \[\begin{align}\vdash : P \in \text{connex} . \kappa = &\text{sect}ʻP . Q = P_{\text{lc}} \unicode{x0294f} \kappa . \dot{\exists} ! P . \supset .\\ +&\text{Hp} *210·251 . \text{Hp} *210·252 \quad[*211·67·66·63·633]\end{align}\]</p> + +<p class="nind"><b>*211·68.</b> \[\begin{align}&\vdash : P \in \text{trans} \cap \text{connex} . \kappa = \text{D}ʻP_{\in } . Q = P_{\text{lc}} + \unicode{x0294f} \kappa . \supset . \text{Hp} *210·12\\ +&[*211·61 . *210·13]\end{align}\]</p> + +<p class="nind"><b>*211·681.</b> \[\begin{align}&\vdash : P \in \text{trans} \cap \text{connex} . \kappa = \text{D}ʻP_{\in } . Q = P_{\text{lc}} + \unicode{x0294f} \kappa . \dot{\exists} ! P . \supset . \text{Hp} *210·251\\ +&[*211·68·66·64]\end{align}\]</p> + +<p class="nind"><b>*211·69.</b> \[\begin{align}&\vdash : P \in \text{connex} . \kappa = \text{D}ʻ(P_{\in } \dot{\cap} I) . Q = P_{\text{lc}} \unicode{x0294f} \kappa . \supset . \text{Hp} *210·12\\ +&[*211·62 . *210·13]\end{align}\]</p> + +<p class="nind"><b>*211·691.</b> \[\begin{align}\vdash : P \in \text{connex} . \kappa = \text{D}ʻ(P_{\in } \dot{\cap} I) . Q = P_{\text{lc}} + &\unicode{x0294f} \kappa . \text{D}ʻ(P_{\in } \dot{\cap} I) {\sim} \in 1 . \supset .\\ +&\text{Hp} *210·251 \quad[*211·69·65]\end{align}\]</p> + +<p class="nind"><b><a id="*211·692">*211·692</a>.</b> \[\begin{align}\vdash : P \in &\text{trans} \cap \text{connex} . \kappa = \text{D}ʻ(P_{\in } \dot{\cap} I) . Q = P_{\text{lc}} + \unicode{x0294f} \kappa .\\ +&\exists ! \text{Cl ex}ʻCʻP - \text{ᗡ}ʻ\text{max}_{P} . \supset . \text{Hp} *210·251 \quad[*211·691·661]\end{align}\]</p> + +<p>The following propositions are concerned with the relations +of sections and segments of \(P\) to sections and segments of +\(\breve{P}\). When \(\alpha \in \text{sect}ʻP\), \(CʻP - \alpha \in\text{sect}ʻ\breve{P}\), +and vice versa. Also, if \(P\) is connected, the maximum of \(\alpha\) +(if any) is the precedent with respect to \(P\) (<i>i.e.</i> the +sequent with respect to \(\breve{P}\) of \(CʻP - \alpha\), and the +sequent of \(\alpha\) (if any) is the minimum with respect to \(P\) +(<i>i.e.</i> the maximum with respect to \(\breve{P}\)) of \(CʻP -\alpha\). +Hence the relations to be proved follow easily.</p> + +<p class="nind"><b><a id="*211·7">*211·7</a>.</b> \(\vdash : \alpha \in \text{sect}ʻP . \supset . CʻP - \alpha \in \text{sect}ʻ\breve{P}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *22·43 . &\supset \vdash . CʻP - \alpha \subset CʻP &\qquad \text{(1)}\\ +\vdash . *211·1 . *37·1 . \supset \vdash \colon\ldotp \text{Hp} . &\supset : x \in \alpha . yPx . \supset . y \in \alpha :\\ +[\text{Transp}] &\supset : x \in \alpha . y {\sim} \in \alpha . \supset . {\sim} (yPx) :\\ +[*37·1 . \text{Transp}] &\supset : x \in \alpha . \supset . x {\sim} \in \breve{P} ʻʻ(- \alpha ) :\\ +[*37·265] &\supset : \alpha \subset - \breve{P} ʻʻ(CʻP - \alpha ) :\\ +[\text{Transp}] &\supset : \breve{P} ʻʻ(CʻP - \alpha ) \subset - \alpha :\\ +[*37·15] &\supset : \breve{P} ʻʻ(CʻP - \alpha ) \subset CʻP - \alpha &\qquad \text{(2)}\\ +\vdash . (1) . (2) . *211·1 . \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*211·701.</b> \(\vdash : \alpha \in \text{sect}ʻP . \text{E}! \text{max}_{P}ʻ\alpha . \supset . pʻ\overleftarrow{P}ʻʻ\alpha \subset \overleftarrow{P}ʻ\text{max}_{P}ʻ\alpha + \subset CʻP - \alpha\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *40·12 . \supset \vdash : \text{Hp} . &\supset . pʻ\overleftarrow{P}ʻʻ\alpha \subset \overleftarrow{P}ʻ\text{max}_{P}ʻ\alpha &\qquad \text{(1)}\\ +\vdash . *205·101 . \supset \vdash : \text{Hp} . &\supset . \text{max}_{P}ʻ\alpha {\sim} \in Pʻʻ\alpha .\\ +[*37·1 . \text{Transp} . *32·181] &\supset . \overleftarrow{P}ʻ\text{max}_{P}ʻ\alpha \subset - \alpha .\\ +[*33·152] &\supset . \overleftarrow{P}ʻ\text{max}_{P}ʻ\alpha \subset CʻP - \alpha &\qquad \text{(2)}\\ +\vdash . (1) . (2) . \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_643">[Pg 643]</span></p> + +<p class="nind"><b>*211·702.</b> \(\vdash : P \in \text{connex} . \alpha \in \text{sect}ʻP . \supset . CʻP - \alpha \subset pʻ\overleftarrow{P}ʻʻ\alpha \quad[*202·501 . *211·1]\)</p> + +<p class="nind"><b>*211·703.</b> \[\begin{align}&\vdash : P \in \text{connex} . \alpha \in \text{sect}ʻP - {℩}ʻCʻP . \supset . \exists ! pʻ\overleftarrow{P}ʻʻ\alpha \\ +&[*211·702·1 . *24·58]\end{align}\]</p> + +<p class="nind"><b>*211·71.</b> \[\begin{align}\vdash : P \in \text{connex} . \alpha \in \text{sect}ʻP . \text{E}! &\text{max}_{P}ʻ\alpha . \supset .\\ +&pʻ\overleftarrow{P}ʻʻ\alpha = \overleftarrow{P}ʻ\text{max}_{P}ʻ\alpha = CʻP - \alpha\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *202·501 . *211·2 . \supset \vdash : \text{Hp} . &\supset . CʻP - \alpha \subset pʻ\overleftarrow{P}ʻʻ\alpha &\qquad \text{(1)}\\ +\vdash . (1) . *211·701 . \supset \vdash : \text{Hp} . &\supset . CʻP - \alpha = pʻ\overleftarrow{P}ʻʻ\alpha &\qquad \text{(2)}\\ +\vdash . (2) . *211·701 . \supset \vdash : \text{Hp} . &\supset . \overleftarrow{P}ʻ\text{max}_{P}ʻ\alpha \subset pʻ\overleftarrow{P}ʻʻ\alpha .\\ +[*211·701] &\supset . \overleftarrow{P}ʻ\text{max}_{P}ʻ\alpha = pʻ\overleftarrow{P}ʻʻ\alpha &\qquad \text{(3)}\\ +\vdash . (2) . (3) . \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p>If \(\alpha\) is a section of \(P\), we shall call \(CʻP - \alpha\) +the complement of \(\alpha\). By the above proposition, if \(\alpha\) +is a section of \(P\) having a maximum, its complement is a section of +\(\breve{P}\) which is a member of \(\overleftarrow{P}ʻʻCʻP\).</p> + +<p class="nind"><b>*211·711.</b> \[\begin{align}\vdash : P &\in \text{connex} . P^{2} \,\unicode{x2abd}\, J . \alpha \in \text{sect}ʻP . \supset .\\ +&\alpha = CʻP - pʻ\overleftarrow{P}ʻʻ\alpha . CʻP \cap pʻ\overleftarrow{P}ʻʻ\alpha = CʻP - \alpha \quad[*202·503 . *211·2]\end{align}\]</p> + +<p class="nind"><b>*211·712.</b> \(\vdash : P \in \text{connex} . \alpha \in \text{sect}ʻP . \text{E}! \text{min}_{P}ʻ(CʻP - \alpha ) . \supset . \alpha = \overrightarrow{P}ʻ\text{min}_{P}ʻ(CʻP + - \alpha )\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *211·71 \frac{\breve{P}}{P}. \supset \\ +\vdash : P \in \text{connex} . \beta \in \text{sect}ʻ\breve{P} . \text{E}! \text{min}_{P}ʻ\beta . \supset . \overrightarrow{P}ʻ\text{min}_{P}ʻ\beta + = CʻP - \beta &\qquad \text{(1)}\\ +\vdash . *211·7 . *24·492 . \supset \vdash : \alpha \in \text{sect}ʻP . \beta = CʻP - \alpha . \supset . \beta \in \text{sect}ʻ\breve{P} . \alpha = CʻP - \beta &\qquad \text{(2)}\\ +\vdash . (1) . (2) . \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*211·713.</b> \(\vdash : P \in \text{connex} . \alpha \in \text{sect}ʻP - \text{D}ʻP_{\in } . \supset . \text{E}! \text{max}_{P}ʻ\alpha + . {\sim} \text{E}! \text{min}_{P}ʻ(CʻP - \alpha )\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *211·24 . \text{Transp} . &\supset \vdash : \text{Hp} . \supset . \text{E}! \text{max}_{P}ʻ\alpha &\qquad \text{(1)}\\ +\vdash . *211·712·3 . &\supset \vdash : P \in \text{connex} . \alpha \in \text{sect}ʻP . \text{E}! \text{min}_{P}ʻ(CʻP - \alpha ) . \supset . \alpha \in \text{D}ʻP_{\in } + &\qquad \text{(2)}\\ +\vdash . (2) . \text{Transp} . &\supset \vdash : \text{Hp} . \supset . {\sim} \text{E}! \text{min}_{P}ʻ(CʻP - \alpha ) &\qquad \text{(3)}\\ +\vdash . (1) . (3) . \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_644">[Pg 644]</span></p> + +<p class="nind"><b><a id="*211·714">*211·714</a>.</b> \(\vdash :P\in \text{connex} .\alpha \in \text{sect}ʻP.\supset .\overrightarrow{\text{seq}} _{P}ʻ\alpha \subset \overrightarrow{\text{min}}_{P}ʻ(CʻP-\alpha )\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*206·18·2.&\supset \vdash :x\in \overrightarrow{\text{seq}} _{P}ʻ\alpha .\supset .x\in CʻP-\alpha &\qquad \text{(1)}\\ +\vdash .*206·134. &\supset \vdash :x\in \overrightarrow{\text{seq}} _{P}ʻ\alpha .\supset .\overrightarrow{P}ʻx\subset CʻP-pʻ\overleftarrow{P}ʻʻ\alpha &\qquad \text{(2)}\\ +\vdash .(2).*202·501.*211·2.\supset \\ +\vdash \colon\ldotp \text{Hp}.\supset :x\in \overrightarrow{\text{seq}} _{P}ʻ\alpha .&\supset .\overrightarrow{P}ʻx\subset \alpha .\\ +[*37·462] &\supset .x{\sim}\in \breve{P} ʻʻ(CʻP-\alpha ) &\qquad \text{(3)}\\ +\vdash .(1).(3).*205·11.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p>The above hypothesis is not sufficient to secure +\(\overrightarrow{\text{seq}}_{P}ʻ\alpha =\text{min}_{P}ʻ(CʻP-\alpha)\), +as may be seen by putting +\[ +P=\alpha \uparrow (\alpha \cup \iota ʻx),\,\text{where}\,\exists !\alpha .x{\sim}\in \alpha . +\] +We then have \(P\in \text{connex} .Pʻʻ\alpha =\alpha .CʻP-\alpha=\iota ʻx.pʻ\overleftarrow{P}ʻʻ\alpha =\alpha \cup \iota ʻx\). +Thus \(\overrightarrow{\text{min}}_{P}ʻ(CʻP-\alpha )=\iotaʻx.\overrightarrow{\text{seq}}_{P}ʻ\alpha =\Lambda\). +It will be seen that \(\alpha \uparrow (\alpha \cup \iota ʻx)\in\text{trans}\), +so that it is useless to add \(P\in \text{trans}\) to +the hypothesis of <a href="#*211·714">*211·714</a>. A sufficient addition is +\(P\,\unicode{x2abd}\,J\), as is proved in the following proposition.</p> + +<p class="nind"><b>*211·715.</b> \(\vdash :P\in \text{connex} \cap \text{Rl}ʻJ.\alpha \in \text{sect}ʻP.\supset .\overrightarrow{\text{seq}} _{P}ʻ\alpha =\overrightarrow{\text{min}}_{P}ʻ(CʻP-\alpha + )\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*205·14.&\supset \vdash :x\text{min}_{P}(CʻP-\alpha ).\supset .x\in CʻP-\alpha .\overrightarrow{P}ʻx\cap (CʻP-\alpha )=\Lambda &\qquad \text{(1)}\\ +\vdash .(1).*33·152.*211·2.\supset \\ +\vdash \colon\ldotp \text{Hp}.\supset :x\text{min}_{P}(CʻP-\alpha ).&\supset .x\in CʻP-\alpha -Pʻʻ\alpha .\overrightarrow{P}ʻx\subset \alpha .\\ +[*202·501] &\supset .x\in CʻP\cap pʻ\overleftarrow{P}ʻʻ\alpha.\overrightarrow{P}ʻx\subset \alpha .\\ +[*200·5] &\supset .x\in CʻP\cap pʻ\overleftarrow{P}ʻʻ\alpha .\overrightarrow{P}ʻx\subset -pʻ\overleftarrow{P}ʻʻ\alpha .\\ +[*37·1.\text{Transp}] &\supset .x\in CʻP\cap pʻ\overleftarrow{P}ʻʻ\alpha -\breve{P} ʻʻpʻ\overleftarrow{P}ʻʻ\alpha .\\ +[*206·11] &\supset .x\text{seq}_{P}\alpha &\qquad \text{(2)}\\ +\vdash .(2).*211·714.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*211·72.</b> \[\begin{align}\vdash :&P\in \text{connex} .\alpha \in \text{sect}ʻP-\text{D}ʻP_{\in }.\supset .\\ +&CʻP-\alpha =\breve{P} ʻʻ(CʻP-\alpha ).CʻP-\alpha \in \text{D}ʻ\{(\breve{P} )_{\in }\dot{\cap} I\} \quad[*211·21·7·713]\end{align}\]</p> + +<p class="nind"><b>*211·721.</b> \[\begin{align}\vdash :P\in \text{connex} .\alpha \in \text{sect}ʻP\cap (\text{ᗡ}ʻ\text{max}_{P}&\cup \text{ᗡ}ʻ\text{seq}_{P}).\supset .\\ +&\overrightarrow{\text{seq}} _{P}ʻ\alpha =\overrightarrow{\text{min}}_{P}ʻ(CʻP-\alpha )\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*211·71.\supset \vdash :P\in \text{connex} .\alpha \in \text{sect}ʻP\cap \text{ᗡ}ʻ\text{max}_{P}.&\supset .pʻ\overleftarrow{P}ʻʻ\alpha =CʻP-\alpha .\\ +[*206·13] &\supset .\overrightarrow{\text{seq}} _{P}ʻ\alpha =\overrightarrow{\text{min}}_{P}ʻ(CʻP-\alpha + ) &\qquad \text{(1)}\\ +\vdash .*211·714.\supset +\vdash \colon\ldotp P\in \text{connex} .\alpha \in \text{sect}ʻP.&\supset :\overrightarrow{\text{seq}} _{P}ʻ\alpha \subset \overrightarrow{\text{min}}_{P}ʻ(CʻP-\alpha ):\\ +[*205·3.*206·16] &\supset :\exists !\overrightarrow{\text{seq}} _{P}ʻ\alpha .\supset .\overrightarrow{\text{seq}} _{P}ʻ\alpha =\overrightarrow{\text{min}}_{P}ʻ(CʻP-\alpha + ) &\qquad \text{(2)}\\ +\vdash .(2).&\supset \vdash :P\in \text{connex} .\alpha \in \text{sect}ʻP\cap \text{ᗡ}ʻ\text{seq}_{P}.\supset .\overrightarrow{\text{seq}} _{P}ʻ\alpha + =\overrightarrow{\text{min}}_{P}ʻ(CʻP-\alpha ) &\qquad \text{(3)}\\ +\vdash .(1).(3).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_645">[Pg 645]</span></p> + +<p class="nind"><b>*211·722.</b> \[\begin{align}\vdash :P\in \text{connex} .\alpha \in \text{sect}ʻP.\text{E}!\text{max}_{P}ʻ\alpha .\text{E}!&\text{seq}_{P}ʻ\alpha .\supset .\\ +&\text{max}_{P}ʻ\alpha =\text{prec}_{P}ʻ(CʻP-\alpha )\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*211·721·7.\supset \vdash :\text{Hp}.&\supset .CʻP-\alpha \in \text{sect}ʻ\breve{P} .\text{E}!\text{min}_{P}ʻ(CʻP-\alpha ).\\ +\left[*211·721 \frac{\breve{P},\,CʻP-\alpha}{P,\,\,\alpha}\right] \supset .\overrightarrow{\text{prec}} _{P}ʻ(CʻP-\alpha )&=\overrightarrow{\text{max}}_{P}ʻ\{CʻP-(CʻP-\alpha )\}\\ +[*24·492] &=\overrightarrow{\text{max}}_{P}ʻ\alpha \\ +[\text{Hp}] & =\iota ʻ\text{max}_{P}ʻ\alpha :\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p>We have always, if \(P\in \text{connex} .\alpha \in \text{sect}ʻP\), +\[ +\overrightarrow{\text{prec}} _{P}ʻ(CʻP-\alpha )\subset \overrightarrow{\text{max}}_{P}ʻ\alpha . +\] +The converse inclusion does not always hold, as appears (on writing +\(\breve{P}\) in place of \(P\)) from the note to <a href="#*211·714">*211·714</a>. To +secure the converse implication, it is sufficient to assume +\(P\,\unicode{x2abd}\, J\) or \(\text{E}!\text{seq}_{P}ʻ\alpha\) or +\({\sim}\text{E}!\text{max}_{P}ʻ\alpha\).</p> + +<p class="nind"><b>*211·723.</b> \(\vdash :P\in \text{connex} .\alpha \in \text{sect}ʻP.\supset .\overrightarrow{\text{prec}} _{P}ʻ(CʻP-\alpha )\subset \overrightarrow{\text{max}}_{P}ʻ\alpha\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*202·11.*211·7.\supset \vdash :\text{Hp}.\supset .\breve{P} \in \text{connex} .CʻP-\alpha \in \text{sect}ʻ\breve{P} .\\ +\left[*211·714 \frac{\breve{P}}{P}.*205·102.*206·101\right] \supset .\overrightarrow{\text{prec}} _{P}ʻ(CʻP-\alpha )\subset \overrightarrow{\text{max}}_{P}ʻ\alpha + :\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*211·724.</b> \[\begin{align}\vdash :P\in \text{connex} .\alpha \in \text{sect}ʻP\cap (\text{ᗡ}ʻ\text{seq}_{P}\cup -\text{ᗡ}ʻ&\text{max}_{P}).\supset .\\ +&\overrightarrow{\text{max}}_{P}ʻ\alpha =\overrightarrow{\text{prec}} _{P}ʻ(CʻP-\alpha )\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*211·722.\supset \vdash :P\in \text{connex} .\alpha \in \text{sect}ʻP\cap \text{ᗡ}ʻ\text{seq}_{P}&\cap \text{ᗡ}ʻ\text{max}_{P}.\supset .\\ +&\overrightarrow{\text{max}}_{P}ʻ\alpha =\overrightarrow{\text{prec}} _{P}ʻ(CʻP-\alpha ) &\qquad \text{(1)}\\ +\vdash .*211·723.*24·13.\supset \vdash :P\in \text{connex} .\alpha \in \text{sect}ʻP&-\text{ᗡ}ʻ\text{max}_{P}.\supset .\\ +&\overrightarrow{\text{max}}_{P}ʻ\alpha =\overrightarrow{\text{prec}} _{P}ʻ(CʻP-\alpha ) &\qquad \text{(2)}\\ +\vdash .(1).(2).*22·91.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*211·725.</b> \[\begin{align}&\vdash :P\in \text{connex} .\alpha \in \text{sect}ʻP\cap \text{ᗡ}ʻ\text{seq}_{P}.\supset .\\ +&\overrightarrow{\text{max}}_{P}ʻ\alpha =\overrightarrow{\text{prec}} _{P}ʻ(CʻP-\alpha ).\overrightarrow{\text{seq}} _{P}ʻ\alpha + =\overrightarrow{\text{min}}_{P}ʻ(CʻP-\alpha ) \quad[*211·721·724]\end{align}\]</p> + +<p><span class="pagenum" id="Page_646">[Pg 646]</span></p> + +<p class="nind"><b>*211·726.</b> \[\begin{align}\vdash :P\in \text{connex} \cap \text{Rl}ʻ&J.\alpha \in \text{sect}ʻP.\supset .\\ +&\overrightarrow{\text{max}}_{P}ʻ\alpha = \overrightarrow{\text{prec}} _{P}ʻ(CʻP-\alpha ).\overrightarrow{\text{seq}} _{P}ʻ\alpha = \overrightarrow{\text{min}}_{P}ʻ(CʻP-\alpha + )\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*200·11.*202·11.*211·7.\supset \vdash :\text{Hp}.&\supset .\breve{P} \in \text{connex} \cap \text{Rl}ʻJ.CʻP-\alpha \in \text{sect}ʻ\breve{P}\\ . +[*211·715.*205·102.*206·101] &\supset .\overrightarrow{\text{prec}} _{P}ʻ(CʻP-\alpha ) = \overrightarrow{\text{max}}_{P}ʻ\alpha &\qquad \text{(1)}\\ +\vdash .(1).*211·715.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*211·727.</b> \[\begin{align}\vdash \colon\ldotp &P\in \text{connex} \cap \text{Rl}ʻJ.\alpha \in \text{sect}ʻP.\supset :\\ +&\text{E}!\text{limax}_Pʻ\alpha . \equiv .\text{E}!\text{limin}_Pʻ(CʻP-\alpha ) \quad[*211·726.*207·44]\end{align}\]</p> + +<p class="nind"><b>*211·728.</b> \[\begin{align}\vdash \colon\ldotp P\in \text{connex} &\cap \text{Rl}ʻJ.\alpha \in \text{sect}ʻP:{\sim}\text{E}!\text{max}_{P}ʻ\alpha .\lor.\\ +&{\sim}\text{E}!\text{min}_{P}ʻ(CʻP-\alpha ):\supset .\overrightarrow{\text{limax}} _{P}ʻ\alpha = \overrightarrow{\text{limin}} _{P}ʻ(CʻP-\alpha )\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*211·726.*207·43·12.\supset \vdash :\text{Hp}.{\sim}\text{E}!&\text{max}_{P}ʻ\alpha .\supset .\\ +\overrightarrow{\text{limax}} _{P}ʻ\alpha &= \overrightarrow{\text{min}}_{P}ʻ(CʻP-\alpha )\\ +[*207·46.*211·726] &= \overrightarrow{\text{limin}} _{P}ʻ(CʻP-\alpha ) &\qquad \text{(1)}\\ +\text{Similarly}\quad \vdash :\text{Hp}.{\sim}\text{E}!\text{min}_{P}ʻ(CʻP-\alpha ).&\supset .\overrightarrow{\text{limax}} _{P}ʻ\alpha = \overrightarrow{\text{limin}} _{P}ʻ(CʻP-\alpha + ) &\qquad \text{(2)}\\ +\vdash .(1).(2).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*211·729.</b> \[\begin{align}\vdash :P\in \text{connex} \cap \text{Rl}ʻJ.&\alpha \in \text{sect}ʻP-(\text{ᗡ}ʻ\text{max}_{P}\cap \text{ᗡ}ʻ\text{seq}_{P}).\supset .\\ +&\overrightarrow{\text{limax}} _{P}ʻ\alpha = \overrightarrow{\text{limin}} _{P}ʻ(CʻP-\alpha ) \quad[*211·728·726]\end{align}\]</p> + +<p class="nind"><b>*211·73.</b> \[\begin{align}\vdash :P\in \text{connex} .&\alpha \in \text{sect}ʻP-\text{D}ʻ(P_{\in }\dot{\cap} I).\supset .\\ +&CʻP-\alpha \in \text{D}ʻ\{(\breve{P} )_{\in }\dot{\cap} I\}-\text{ᗡ}ʻ\text{prec}_{P}\cup \{\text{sect}ʻP-\text{D}ʻ(\breve{P} )_{\in }\}\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*211·21.\supset \vdash :\text{Hp}.&\supset .\alpha \in \text{sect}ʻP-\text{ᗡ}ʻ\text{max}_{P}.\\ +[*211·7·723] &\supset .CʻP-\alpha \in \text{sect}ʻ\breve{P} -\text{ᗡ}ʻ\text{prec}_{P}.\\ +[*24·41] \supset .CʻP-\alpha \in &(\text{sect}ʻ\breve{P} -\text{ᗡ}ʻ\text{max}_{P}-\text{ᗡ}ʻ\text{prec}_{P})\cup \\ +&(\text{sect}ʻ\breve{P} \cap \text{ᗡ}ʻ\text{max}_{P}-\text{ᗡ}ʻ\text{prec}_{P}).\\ +[*211·31·21] &\supset .CʻP-\alpha \in {\text{D}ʻ(\breve{P} )_{\in }\dot{\cap} I}-\text{ᗡ}ʻ\text{prec}_{P}\cup \{\text{sect}ʻP-\text{D}ʻ(\breve{P} )_{\in }\}:\\ +\supset \vdash .\text{Prop} +\end{array}\]</p> + +<p class="nind"><b>*211·74.</b> \[\begin{align}\vdash :P\in \text{trans} \cap \text{connex} .\alpha \in \text{D}ʻP_{\in }-&\text{D}ʻ(P_{\in }\dot{\cap} I).\supset .\\ +&CʻP-\alpha \in \text{D}ʻ(\breve{P} )_{\in }-\text{D}ʻ\{(\breve{P} )_{\in }\dot{\cap} I\}\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*211·431.&\supset \vdash :\text{Hp}.\supset .\alpha \in \text{sect}ʻP\cap \text{ᗡ}ʻ\text{max}_{P}\cap \text{ᗡ}ʻ\text{seq}_{P}.\\ +[*211·7·725] &\supset .CʻP-\alpha \in \text{sect}ʻ\breve{P} \cap \text{ᗡ}ʻ\text{prec}_{P}\cap \text{ᗡ}ʻ\text{min}_{P}.\\ +\left[*211·431 \frac{\breve{P}}{P}\right] &\supset .CʻP-\alpha \in \text{D}ʻ(\breve{P} )_{\in }-\text{D}ʻ\{(\breve{P} )_{\in }\dot{\cap} I\}:\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_647">[Pg 647]</span></p> + +<p>The following propositions sum up our previous results.</p> + +<p class="nind"><b><a id="*211·75">*211·75</a>.</b> \(\vdash \colon\ldotp \alpha \subset CʻP.Q=\breve{P} .\supset :\alpha \in \text{sect}ʻP.\equiv .CʻP-\alpha \in \text{sect}ʻQ \quad[*211·7]\)</p> + +<p class="nind"><b>*211·751.</b> \[\begin{align}\vdash \colon\ldotp P\in \text{Ser}.\alpha &\subset CʻP.Q=\breve{P} .\supset :\\ +&\alpha \in \text{D}ʻP_{\in }.\equiv .CʻP-\alpha \in \text{sect}ʻQ\cap (\text{ᗡ}ʻ\text{max}_{Q}\cup -\text{ᗡ}ʻ\text{seq}_{Q})\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*211·32.\supset \vdash \colon\ldotp \text{Hp}.&\supset :\alpha \in \text{D}ʻP_{\in }.\equiv .\alpha \in \text{sect}ʻP\cap (\text{ᗡ}ʻ\text{seq}_{P}\cup + -\text{ᗡ}ʻ\text{max}_{P}).\\ +[*211·75·726] &\equiv .CʻP-\alpha \in \text{sect}ʻQ\cap (\text{ᗡ}ʻ\text{max}_{Q}\cup -\text{ᗡ}ʻ\text{seq}_{Q})\colon\ldotp \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p>In the above proposition, "\(P \in \text{trans}\)" is necessary in +order that \(\text{D}ʻP_{\in }\) may be contained in \(\text{sect}ʻP\), +and "\(P \in \text{Rl}ʻJ\)" is necessary in order that "\((CʻP-\alpha){\sim}\in \text{ᗡ}ʻ\text{seq}_{Q}\)" +may imply "\(\alpha {\sim}\in\text{ᗡ}ʻ\text{max}_{P}\)." Hence the +full hypothesis "\(P\in\text{Ser}\)" becomes necessary.</p> + +<p class="nind"><b>*211·752.</b> \[\begin{align}\vdash \colon\ldotp P\in \text{connex} .\alpha \subset &CʻP.Q=\breve{P} .\supset :\\ +&\alpha \in \text{D}ʻ(P_{\in }\dot{\cap} I).\supset .CʻP-\alpha \in \text{sect}ʻQ-\text{ᗡ}ʻ\text{seq}_{Q}\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*211·41. &\supset \vdash :\alpha \in \text{D}ʻ(P_{\in }\dot{\cap} I).\equiv .\alpha \in \text{sect}ʻP-\text{ᗡ}ʻ\text{max}_{P} &\qquad \text{(1)}\\ +\vdash .(1).*211·7·723.&\supset \vdash :\text{Hp}.\alpha \in \text{D}ʻ(P_{\in }\dot{\cap} I).\supset .\\ +&CʻP-\alpha \in \text{sect}ʻQ-\text{ᗡ}ʻ\text{seq}_{Q}:\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*211·753.</b> \[\begin{align}\vdash \colon\ldotp P&\in \text{Rl}ʻJ\cap \text{connex} .\alpha \subset CʻP.Q=\breve{P} .\supset :\\ +&\alpha \in \text{D}ʻ(P_{\in }\dot{\cap} I).\equiv .CʻP-\alpha \in \text{sect}ʻQ- \text{ᗡ}ʻ\text{seq}_{Q} \quad[*211·41·7·726]\end{align}\]</p> + +<p class="nind"><b><a id="*211·754">*211·754</a>.</b> \[\begin{align}\vdash \colon\ldotp P \in\text{Rl}ʻJ\cap &\text{connex} .\alpha \subset CʻP.Q=\breve{P} .\supset :\\ +&\alpha \in \text{sect}ʻP-\text{D}ʻP_{\in }.\equiv .CʻP-\alpha \in \text{D}ʻ(Q_{\in }\dot{\cap} I)\cap \text{ᗡ}ʻ\text{seq}_{Q}\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*211·316.\supset \\ +\vdash \colon\ldotp \text{Hp}.\supset :\alpha \in \text{sect}ʻP-\text{D}ʻP_{\in }.&\equiv .\alpha \in \text{sect}ʻP\cap (\text{ᗡ}ʻ\text{max}_{P}\cap + -\text{ᗡ}ʻ\text{seq}_{P}).\\ +[*211·7·726] &\equiv .CʻP-\alpha \in \text{sect}ʻQ\cap (\text{ᗡ}ʻ\text{seq}_{Q}\cap -\text{ᗡ}ʻ\text{max}_{Q}).\\ +[*211·41] & \equiv .CʻP-\alpha \in \text{D}ʻ(Q_{\in }\dot{\cap} I)\cap \text{ᗡ}ʻ\text{seq}_{Q}\colon\ldotp \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*211·755.</b> \[\begin{align}&\vdash \colon\ldotp P\in \text{trans}\cap \text{connex} .\alpha \subset CʻP.Q=\breve{P} .\supset :\\ +&\alpha \in \text{D}ʻP_{\in }-\text{D}ʻ(P_{\in }\dot{\cap} I).\equiv .CʻP-\alpha \in \text{D}ʻQ_{\in }-\text{D}ʻ(Q_{\in }\dot{\cap} + I) \quad[*211·74]\end{align}\]</p> + +<p class="nind"><b>*211·756.</b> \[\begin{align}&\vdash \colon\ldotp P\in \text{ᗡ}ʻʻʻJ\cap \text{connex} .\alpha \subset CʻP.Q=\breve{P} .\supset :\\ +&\alpha \in \text{sect}ʻP-\text{D}ʻ(P_{\in }\dot{\cap} I).\equiv .CʻP-\alpha \in \text{sect}ʻQ\cap \text{ᗡ}ʻ\text{seq}_{Q} \quad[*211·41·7·726]\end{align}\]</p> + +<p class="nind"><b>*211·757.</b> \[\begin{align}\vdash \colon\ldotp P&\in \text{Ser}.\alpha \subset CʻP.\supset :\\ +&\alpha \in \text{sect}ʻP- \text{D}ʻ(P_{\in }\dot{\cap} + I).\equiv .CʻP-\alpha \in \overleftarrow{P}ʻʻCʻP \quad[*211·756·302]\end{align}\]</p> + +<p><span class="pagenum" id="Page_648">[Pg 648]</span></p> + +<p class="nind"><b>*211·76.</b> \(\vdash :P\in \text{Ser}.\supset .\text{D}ʻP_{\in }=(CʻP-)ʻʻ(\text{sect}ʻ\breve{P} -\text{ᗡ}ʻ\text{tl}_P)\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*207·13.\text{Transp}.&\supset \vdash .-\text{ᗡ}ʻ\text{tl}_P=\text{ᗡ}ʻ\text{min}_{P}\cup -\text{ᗡ}ʻ\text{seq}_{P} &\qquad \text{(1)}\\ +\vdash .(1).*211·751.\supset \vdash \colon\ldotp \text{Hp}.\supset :\alpha \in \text{D}ʻP_{\in }.&\equiv .\alpha \subset CʻP.CʻP-\alpha \in \text{sect}ʻ\breve{P} -\text{ᗡ}ʻ\text{tl}_P.\\ +[*24·492] &\equiv .(\exists \beta ).\beta \in \text{sect}ʻ\breve{P} -\text{ᗡ}ʻ\text{tl}_P.\alpha =CʻP-\beta .\\ +[*38·13] &\equiv .\alpha \in (CʻP-)ʻʻ(\text{sect}ʻ\breve{P} -\text{ᗡ}ʻ\text{tl}_P)\colon\ldotp \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*211·761.</b> \[\begin{align}&\vdash :P\in \text{Ser}.\supset .\text{sect}ʻP\cap \text{ᗡ}ʻ\text{lt}_P=(CʻP-)ʻʻ\{\text{sect}ʻ\breve{P} -\text{D}ʻ(\breve{P} )_{\in }\}\\ +&[\text{Proof as in *211·76}]\end{align}\]</p> + +<p class="nind"><b><a id="*211·762">*211·762</a>.</b> \(\vdash :P\in \text{Ser}.\supset .\text{D}ʻ(P_{\in }\dot{\cap} I)=(CʻP-)ʻʻ(\text{sect}ʻ\breve{P} -\overleftarrow{P}ʻʻCʻP)\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*211·757.\text{Transp}.\supset \\ +\vdash \colon\ldotp \text{Hp}.\supset :\alpha \in \text{sect}ʻP.CʻP-\alpha {\sim}\in \overleftarrow{P}ʻʻCʻP.\equiv .\alpha \in \text{D}ʻ(P_{\in }\dot{\cap} I) &\qquad \text{(1)}\\ +\vdash .(1).*24·492.*38·13.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*211·8.</b> \[\begin{align}&\vdash :P_{\text{po}}\in \text{Ser}.\alpha \in \text{sect}ʻP.\supset .\\ +&\overrightarrow{\text{max}}_{P}ʻ\alpha =\overrightarrow{\text{max}}(P_{\text{po}})ʻ\alpha .\overrightarrow{\text{min}}_{P}ʻ(CʻP-\alpha + )=\overrightarrow{\text{min}}(P_{\text{po}})ʻ(CʻP-\alpha )=\overrightarrow{\text{seq}} (P_{\text{po}})ʻ\alpha\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*211·13.*91·602. &\supset \vdash :\text{Hp}.\supset .\alpha \in \text{sect}ʻP_{\text{po}} &\qquad \text{(1)}\\ +\vdash .*211·131.*205·111. &\supset \vdash :\text{Hp}.\supset .\overrightarrow{\text{max}}_{P}ʻ\alpha =\overrightarrow{\text{max}}(P_{\text{po}})ʻ\alpha &\qquad \text{(2)}\\ +\vdash .(2) \frac{\breve{P}}{P}.*211·7.(1). \supset \vdash :\text{Hp}.\supset .\overrightarrow{\text{min}}_{P}ʻ(CʻP-\alpha )&=\overrightarrow{\text{min}}(P_{\text{po}})ʻ(CʻP-\alpha + ) &\qquad \text{(3)}\\ +[*211·726]&=\overrightarrow{\text{seq}} (P_{\text{po}})ʻ\alpha &\qquad \text{(4)}\\ +\vdash .(2).(3).(4).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p>The above proposition is used in <a href="#*232·352">*232·352</a> and <a href="#*234·242">*234·242</a>.</p> + +<p>The following propositions lead up to <a href="#*211·82">*211·82</a>, which is used in <a href="#*213·4">*213·4</a>. +<a href="#*211·83">*211·83</a><a href="#*211·841">·841</a><a href="#*211·9">·9</a> are also used in <a href="#*213">*213</a>.</p> + +<p class="nind"><b>*211·81.</b> \[\begin{align}\vdash :P\in \text{Ser}.\alpha \in \text{sect}ʻP.\alpha &{\sim}\in 1.CʻP-\alpha \in 1.\supset .\\ +&CʻP-\alpha =\iota ʻBʻ\breve{P} .P=P\unicode{x0294f}\alpha \unicode{x21f8} Bʻ\breve{P} .\alpha =\text{D}ʻP\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*211·7·181·182 \frac{\breve{P}}{P}.&\supset \vdash :\text{Hp}.\supset .CʻP-\alpha =\iota ʻBʻ\breve{P} &\qquad \text{(1)}\\ +\vdash .*204·461. &\supset \vdash :\text{Hp}.\supset .P= P\unicode{x0294f}\text{D}ʻP\unicode{x21f8} Bʻ\breve{P} &\qquad \text{(2)}\\ +\vdash .(1).*211·1. &\supset \vdash :\text{Hp}.\supset .\alpha =\text{D}ʻP &\qquad \text{(3)}\\ +\vdash .(1).(2).(3).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_649">[Pg 649]</span></p> + +<p class="nind"><b>*211·811.</b> \(\vdash :P\in \text{Ser}-\iota ʻ\dot{\Lambda} .P=Q\unicode{x21f8} x.\supset .CʻQ\in \text{sect}ʻP.x=Bʻ\breve{P} .CʻQ=\text{D}ʻP\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*161·11.\supset \vdash \colon\ldotp \text{Hp}.&\supset :y\in CʻQ.\supset _{y}.yPx:\\ +[*204·1] &\supset :x{\sim}\in CʻQ:\\ +[*161·15] &\supset :x=Bʻ\breve{P} &\qquad \text{(1)}\\ +\vdash .*161·13.\supset \vdash :\text{Hp}.&\supset .CʻQ=\text{D}ʻP &\qquad \text{(2)}\\ +\vdash .(1).(2).*211·1.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*211·812.</b> \[\begin{align}&\vdash \colon\ldotp P\in \text{Ser}-\iota ʻ\dot{\Lambda} .Q\in \text{D}ʻP\unicode{x0294f}.\supset :\\ +&CʻQ\in \text{sect}ʻP.CʻP-CʻQ\in 1.\equiv .(\exists x).P=Q\unicode{x21f8} x.\equiv .P=Q\unicode{x21f8} Bʻ\breve{P}\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*204·4.*201·12. &\supset \vdash :\text{Hp}.\supset .CʻQ{\sim}\in 1 &\qquad \text{(1)}\\ +\vdash .*204·41.&\supset \vdash :\text{Hp}.\supset .Q=P\unicode{x0294f}CʻQ &\qquad \text{(2)}\\ +\vdash .(1).(2).*211·81.\supset \vdash :\text{Hp}.CʻQ\in &\text{sect}ʻP.CʻP-CʻQ\in 1.\supset .\\ +&CʻP-CʻQ=\iota ʻBʻ\breve{P} .P=Q\unicode{x21f8} Bʻ\breve{P} &\qquad \text{(3)}\\ +\vdash .*211·811.&\supset \vdash \colon\ldotp \text{Hp}.\supset :(\exists x).P=Q\unicode{x21f8} x.\equiv .P=Q\unicode{x21f8} Bʻ\breve{P} &\qquad \text{(4)}\\ +\vdash .*211·811.&\supset \vdash :\text{Hp}.P=Q\unicode{x21f8} x.\supset .CʻQ\in \text{sect}ʻP.CʻP-CʻQ\in 1 &\qquad \text{(5)}\\ +\vdash .(3).(4).(5).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*211·82">*211·82</a>.</b> \[\begin{align}\vdash \colon\colon P\in \text{Ser}.&Q\in \text{D}ʻP\unicode{x0294f}.\supset \colon\ldotp \\ +CʻQ\in \text{sect}ʻP.&\equiv :(\exists R).P=Q\unicode{x2909}R.\lor.(\exists x).P=Q\unicode{x21f8} x:\\ +&\equiv :(\exists R).P=Q\unicode{x2909}R.\lor.P=Q\unicode{x21f8} Bʻ\breve{P} \\ +&[*211·282·283·812.*160·22.*161·2]\end{align}\]</p> + +<p class="nind"><b><a id="*211·83">*211·83</a>.</b> \(\vdash :\dot{\exists} !P.x{\sim}\in CʻP.\supset .\text{sect}ʻ(P\unicode{x21f8} x)=\text{sect}ʻP\cup \iota ʻ(CʻP\cup \iota ʻx)\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*211·1.\supset \\ +\vdash \colon\ldotp \text{Hp}.\supset :\alpha \in \text{sect}ʻ(P\unicode{x21f8} x).&\equiv .\alpha \subset CʻP\cup \iota ʻx.(P\unicode{x21f8} x)ʻʻ\alpha \subset \alpha &\qquad \text{(1)}\\ +\vdash .(1).*161·11.\supset \\ +\vdash \colon\ldotp \text{Hp}.\supset :\alpha \in \text{sect}ʻ(P\unicode{x21f8} x).x\in \alpha .&\equiv .\alpha \subset CʻP\cup \iota ʻx.Pʻʻ\alpha \cup CʻP\subset \alpha .x\in \alpha .\\ +[*22·41] &\equiv .\alpha =CʻP\cup \iota ʻx &\qquad \text{(2)}\\ +\vdash .(1).*161·11.\supset \\ +\vdash \colon\ldotp \text{Hp}.\supset :\alpha \in \text{sect}ʻ(P\unicode{x21f8} x).x{\sim}\in \alpha .&\equiv .\alpha \subset CʻP\cup \iota ʻx.Pʻʻ\alpha \subset \alpha .x{\sim}\in \alpha .\\ +[*51·25] &\equiv .\alpha \subset CʻP.Pʻʻ\alpha \subset \alpha .\\ +[*211·1] &\equiv .\alpha \in \text{sect}ʻP &\qquad \text{(3)}\\ +\vdash .(2).(3).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_650">[Pg 650]</span></p> + +<p class="nind"><b>*211·84.</b> \[\begin{align}\vdash :CʻP\cap CʻQ=\Lambda .\supset .\text{sect}ʻ(P\unicode{x2909}Q)&=\text{sect}ʻP\cup (CʻP\cup )ʻʻ\text{sect}ʻQ\\ +&=\text{sect}ʻP\cup (CʻP\cup )ʻʻ(\text{sect}ʻQ-\iota ʻ\Lambda )\end{align} +\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*211·1. &\supset \vdash :\alpha \in \text{sect}ʻ(P\unicode{x2909}Q).\equiv .\alpha \subset CʻP\cup CʻQ.(P\unicode{x2909}Q)ʻʻ\alpha \subset \alpha &\qquad \text{(1)}\\ +\vdash .(1).*160·11.\supset \\ +\vdash \colon\ldotp \text{Hp}.\supset :\alpha \in \text{sect}ʻ(P\unicode{x2909}Q).\alpha \subset CʻP.&\equiv .\alpha CʻP.Pʻʻ\alpha \subset \alpha .\\ +[*211·1] &\equiv .\alpha \in \text{sect}ʻP &\qquad \text{(2)}\\ +\vdash .(1).*160·11.\supset \\ +\vdash \colon\ldotp \text{Hp}.\supset :\alpha \in &\text{sect}ʻ(P\unicode{x2909}Q).\exists !\alpha \cap CʻQ.\equiv .\\ +&\alpha \subset CʻP\cup CʻQ.CʻP\cup Qʻʻ\alpha \subset \alpha .\exists !\alpha \cap CʻQ.\\ +[*24·43·491] &\equiv .\alpha -CʻP\subset CʻQ.CʻP\subset \alpha .Qʻʻ\alpha \subset \alpha -CʻP.\exists !\alpha -CʻP.\\ +[*24·491.*37·265]&\equiv .\alpha -CʻP\subset CʻQ.CʻP\subset \alpha .Qʻʻ(\alpha -CʻP)\subset \alpha -CʻP.\exists !\alpha -CʻP.\\ +[*211·1]&\equiv .\alpha -CʻP\in \text{sect}ʻQ-\iota ʻ\Lambda .CʻP\subset \alpha .\\ +[*22*92]&\equiv .\alpha \in (CʻP\cup )ʻʻ(\text{sect}ʻQ-\iota ʻ\Lambda ) &\qquad \text{(3)}\\ +\vdash .*211·26·44.&\supset \vdash .CʻP\in \text{sect}ʻP.CʻP\in (CʻP\cup )ʻʻ(\text{sect}ʻQ\cap \iota ʻ\Lambda ) &\qquad \text{(4)}\\ +\vdash .(2).(3).\supset \vdash :\text{Hp}.\supset .\text{sect}ʻ(P\unicode{x2909}Q)&=\text{sect}ʻP\cup (CʻP\cup )ʻʻ(\text{sect}ʻQ-\iota ʻ\Lambda )\\ +[(4)] & =\text{sect}ʻP\cup (CʻP\cup )ʻʻ\text{sect}ʻQ:\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*211·841">*211·841</a>.</b> \[\begin{align}&\vdash :CʻP\cap CʻQ=\Lambda .\supset .\\ +&\text{sect}ʻ(P\unicode{x2909}Q)-\iota ʻ\Lambda =(\text{sect}ʻP-\iota ʻ\Lambda )\cup (CʻP\cup )ʻʻ(\text{sect}ʻQ-\iota ʻ\Lambda ) \quad[*211·84]\end{align}\]</p> + +<p class="nind"><b><a id="*211·9">*211·9</a>.</b> \(\vdash .\text{sect}ʻ(x\downarrow y)=\iota ʻ\Lambda \cup \iota ʻ\iota ʻx\cup \iota ʻ(\iota ʻx\cup \iota ʻy)\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*211·1·26.&\supset \vdash .\Lambda \in \text{sect}ʻ(x\downarrow y).\iota ʻx\cup \iota ʻy\in \text{sect}ʻ(x\downarrow y) &\qquad \text{(1)}\\ +\vdash .*55·13. &\supset \vdash :x\neq y.\supset .(x\downarrow y)ʻʻ\iota ʻx=\Lambda &\qquad \text{(2)}\\ +\vdash .*55·13. &\supset \vdash :x=y.\supset .(x\downarrow y)ʻʻ\iota ʻx=\iota ʻx &\qquad \text{(3)}\\ +\vdash .(2).(3). &\supset \vdash .(x\downarrow y)ʻʻ\iota ʻx\subset \iota ʻx.\\ +[*211·1] &\supset \vdash .\iota ʻx\in \text{sect}ʻ(x\downarrow y) &\qquad \text{(4)}\\ +\vdash .*211·1.*54·4.\supset \\ +\vdash \colon\ldotp \beta \in \text{sect}ʻ(x\downarrow y).&\supset :\beta =\Lambda .\lor.\beta =\iota ʻx.\lor.\beta =\iota ʻy.\lor.\beta =\iota ʻx\cup \iota ʻy &\qquad \text{(5)}\\ +\vdash .*55·13. \supset \vdash :x\neq y.&\supset .x\in (x\downarrow y)ʻʻ\iota ʻy-\iota ʻy\\ +[*211·1] &\supset .\iota ʻy{\sim}\in \text{sect}ʻP &\qquad \text{(6)}\\ +\vdash .*51·23. & \supset \vdash :x=y.\supset .\iota ʻy=\iota ʻx &\qquad \text{(7)}\\ +\vdash .(5).(6).(7).&\supset \vdash \colon\ldotp \beta \in \text{sect}ʻ(x\downarrow y).\supset :\beta =\Lambda .\lor.\beta =\iota ʻx.\lor.\beta =\iota ʻx\cup \iota ʻy &\qquad \text{(8)}\\ +\vdash .(1).(4).(8).\supset \vdash .\text{Prop} +\end{array} +\]</p> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_651">[Pg 651]</span></p> +<h2 class="nobreak" id="*212"> +*212. THE SERIES OF SEGMENTS.</h2> +</div> + + +<p class="nind"><i>Summary of</i> *212.</p> + +<p>The series of segments or sections of a series may be ordered by the +relation of inclusion, after the manner considered in <a href="#*210">*210</a>. Since, as +was shown in <a href="#*211">*211</a>, sections and segments have the properties assigned +to \(\kappa\) in the hypothesis of *210, the resulting series are +such that every class has either a maximum or a sequent, and either a +minimum or a precedent; <i>i.e.</i> the series of segments or sections +are Dedekindian. Most of the properties of the series of sections and +of the series of segments which have no maximum, only require that the +original relation should be connected. The properties of the series +of segments in general \((\text{D}ʻP_{\in })\) require also that the +original relation should be transitive.</p> + +<p>We denote the series of segments by \(\varsigma ʻP\), putting +\[ +\varsigma ʻP=P_{\text{lc}}\unicode{x0294f}\text{D}ʻP_{\in } \quad\text{Df}. +\]</p> + +<p>We then have, in virtue of <a href="#*210·13">*210·13</a> and <a href="#*211·61">*211·61</a>,</p> + +<p class="nind"><b>*212·23.</b> \(\vdash :P\in \text{trans}\cap \text{connex} .\supset .\varsigma ʻP=\hat{\alpha} \hat{\beta} \{\alpha ,\beta \in \text{D}ʻP_{\in }.\alpha + \subset \beta .\alpha \neq \beta\}\)</p> + +<p>In like manner, for the series of segments which have no maximum, we put +\[ +\text{sgm}ʻP=P_{\text{lc}}\unicode{x0294f}\text{D}ʻ(P_{\in }\dot{\cap} I) \quad\text{Df}, +\] +and we have</p> + +<p class="nind"><b>*212·22.</b> \(\vdash :P\in \text{connex} .\supset .\text{sgm}ʻP=\hat{\alpha} \hat{\beta} \{\alpha ,\beta \in \text{D}ʻ(P_{\in }\dot{\cap} + I).\alpha \subset \beta .\alpha \neq \beta\}\)</p> + +<p>We do not need a special notation for the series of sections, since, in +virtue of <a href="#*211·13">*211·13</a>, it is \(\varsigmaʻP_{*}\) or \(\text{sgm}ʻP_{*}\). +Thus, by <a href="#*212·23">*212·23</a>,</p> + +<p class="nind"><b>*212·24.</b> \(\vdash :P_{*}\in \text{connex} .\supset .\varsigma ʻP_{*}=\hat{\alpha} \hat{\beta} \{\alpha ,\beta \in \text{sect}ʻP.\alpha \subset \beta .a\neq \beta\}\)</p> + +<p>We begin the number with various propositions on the fields, etc. of +these relations, and on the conditions for their existence. We have</p> + +<p class="nind"><b>*212·132.</b> \(\vdash .\text{D}ʻ\varsigma ʻP=\text{D}ʻP_{\in }-\iota ʻ\text{D}ʻP.\text{ᗡ}ʻ\varsigma ʻP=\text{D}ʻP_{\in }-\iota ʻ\Lambda\)</p> + +<p class="nind"><b>*212·133.</b> \(\vdash :\dot{\exists} !P.\supset .Cʻ\varsigma ʻP=\text{D}ʻP_{\in }.Bʻ\varsigma ʻP=\Lambda .Bʻ\text{Cnv}ʻ\varsigma ʻP=\text{D}ʻP\)</p> + +<p class="nind"><b>*212·14.</b> \(\vdash :\dot{\exists} !P.\equiv .\dot{\exists} !\varsigma ʻP\)</p> + +<p><span class="pagenum" id="Page_652">[Pg 652]</span></p> + +<p class="nind"><b>*212·152.</b> \(\vdash .\text{ᗡ}ʻ\text{sgm}ʻP=\text{D}ʻ(P_{\in }\dot{\cap} I)-\iota ʻ\Lambda\)</p> + +<p class="nind"><b>*212·17.</b> \(\vdash :\dot{\exists} !\varsigma ʻP_{*}.\equiv .\exists !\text{sect}ʻP-\iota ʻ\Lambda .\equiv .\text{sect}ʻP{\sim}\in 1.\equiv .\dot{\exists} !P\)</p> + +<p class="nind"><b>*212·172.</b> \(\vdash :\dot{\exists} !P.\supset .Cʻ\varsigma ʻP_{*}=\text{sect}ʻP.Bʻ\varsigma ʻP_{*}=\Lambda .Bʻ\text{Cnv}ʻ\varsigma ʻP_{*}=CʻP\)</p> + +<p>Of the next set of propositions (<a href="#*212·2">*212·2</a>—<a href="#*212·25">·25</a>), several have already +been mentioned. An important proposition is</p> + +<p class="nind"><b>*212·25.</b> \(\vdash :P\in \text{Ser}.\supset .\overrightarrow{P}^{;}P=(\varsigma ʻP)\unicode{x0294f}\overrightarrow{P}ʻʻCʻP\)</p> + +<p class="nind">for this shows that the series of segments contains a series similar to +\(P\).</p> + +<p>We take up next the application of the propositions of *210 to +the series of sections and segments. We show that if \(P\in\text{connex}\), +\(\text{sgm}ʻP\) and \(\varsigma ʻP_{*}\) are series +(<a href="#*212·3">*212·3</a>), and that if \(P\) is also transitive, \(\varsigma ʻP\) is a +series (<a href="#*212·31">*212·31</a>). We have</p> + +<p class="nind"><b>*212·322.</b> \(\vdash :P\in \text{connex} .\dot{\exists} !P.\lambda \subset \text{sect}ʻP.\supset .sʻ\lambda =\text{limax}(\varsigma ʻP_{*})ʻ\lambda \)</p> + +<p class="nind"><b>*212·34.</b> \(\vdash :P\in \text{connex} .\dot{\exists} !P.\lambda \subset \text{sect}ʻP.\supset .pʻ\lambda \cap CʻP=\text{limin}(\varsigma ʻP_{*})ʻ\lambda\)</p> + +<p class="nind">so that every class of sections has both an upper limit or maximum and +a lower limit or minimum (<a href="#*212·35">*212·35</a>).</p> + +<p>We then prove similar propositions for \(\varsigma ʻP\) and +\(\text{sgm}ʻP\), except that in place of <a href="#*212·34">*212·34</a> we have</p> + +<p class="nind"><b>*212·431.</b> \[\begin{align}\vdash :P\in \text{trans}\cap \text{connex} .\dot{\exists} !&P.\lambda \subset \text{D}ʻP_{\in }.\supset .\\ +&sʻ(\text{D}ʻP_{\in }\cap \text{Cl}ʻpʻ\lambda )=\text{limin}(\varsigma ʻP)ʻ\lambda\end{align}\]</p> + +<p class="nind"><b>*212·53.</b> \[\begin{align}\vdash :P\in \text{connex} .\dot{\exists} !\text{sgm}ʻP.\lambda \subset \text{D}ʻ&(P_{\in }\dot{\cap} I).\supset .\\ +&sʻ{\text{D}ʻ(P_{\in }\dot{\cap} I)\cap \text{Cl}ʻpʻ\lambda }=\text{limin}(\text{sgm}ʻP)ʻ\lambda \end{align}\]</p> + +<p>The reason of the difference from *212·34 is that the product of +an existent class of segments may not be a segment. Suppose, for +example, the segments are all those that contain a given term \(x\), +where \(x\) has no immediate successor; then their logical product is +\(\overrightarrow{P}ʻx\cup \iota ʻx\), which is a section but not a +segment.</p> + +<p>We have next (<a href="#*212·6">*212·6</a>—<a href="#*212·667">·667</a>) a number of propositions on the limits and +maxima of sub-classes of \(\overrightarrow{P}ʻʻCʻP\) in the series +\(\varsigma ʻP\). The interest of this subject lies in its relation to +irrationals. If \(\alpha\) is a class contained in \(CʻP\) and having +no limit or maximum, \(\overrightarrow{P}ʻʻ\alpha\) is contained in +\(Cʻ\varsigma ʻP\), and has a limit in \(\varsigma ʻP\). We may call +this limit an irrational segment. There is no irrational term in +\(CʻP\), because in \(P\) there is no limit to \(\alpha\); but the +limit, in \(\varsigma ʻP\), of \(\overrightarrow{P}ʻʻ\alpha\) may be +called irrational, because it corresponds to no term in \(CʻP\). It +should be observed that (as will be proved in Section F) if \(P\) is +similar to the series of rationals, \(\varsigma ʻP\) is similar to the +series of real numbers.</p> + +<p>The most useful propositions in this subject are:</p> + +<p><b>*212·6.</b> \[\begin{align}\vdash :P\in \text{Ser}.\alpha \subset CʻP.&\supset .\\ +&\overrightarrow{\text{max}}(\varsigma ʻP)ʻ\overrightarrow{P}ʻʻ\alpha =\overrightarrow{\text{max}}(\overrightarrow{P}^{;}P)ʻ\overrightarrow{P}ʻʻ\alpha + =\overrightarrow{P}ʻʻ\overrightarrow{\text{max}}_{P}ʻ\alpha \end{align}\]</p> + +<p><span class="pagenum" id="Page_653">[Pg 653]</span></p> + +<p class="nind"><b>*212·601.</b> \[\begin{align}\vdash \colon\ldotp P\in &\text{Ser}.\alpha \subset CʻP.\supset :\\ +&\text{E}!\text{max}_{P}ʻ\alpha .\equiv .\text{E}!\text{max}(\overrightarrow{P}^{;}P)ʻ\overrightarrow{P}ʻʻ\alpha + .\equiv .\text{E}!\text{max}(\varsigma ʻP)ʻ\overrightarrow{P}ʻʻ\alpha \end{align}\]</p> + +<p class="nind"><b>*212·602.</b> \(\vdash \colon\ldotp P\in \text{Ser}.\dot{\exists} !P.\alpha \subset CʻP.\supset :\text{E}!\text{max}_{P}ʻ\alpha .\equiv .Pʻʻ\alpha \in \overrightarrow{P}ʻʻ\alpha\)</p> + +<p class="nind"><b>*212·61.</b> \(\vdash :P\in \text{trans}\cap \text{connex} .\dot{\exists} !P.\supset .\text{limax}(\varsigma ʻP)ʻ\overrightarrow{P}ʻʻ\alpha =Pʻʻ\alpha\)</p> + +<p class="nind"><b>*212·632.</b> + \(\vdash :P\in \text{Ser}.\dot{\exists} !P.\alpha \subset CʻP.Pʻʻ\alpha {\sim}\in \overrightarrow{P}ʻʻCʻP.\supset .Pʻʻ\alpha =\text{lt}(\varsigma ʻP)ʻ\overrightarrow{P}ʻʻ\alpha\)</p> + +<p class="nind"><b>*212·661.</b> \[\begin{align}\vdash :P\in \text{Ser}.\kappa \subset \text{D}ʻP_{\in }.\text{E}!\text{lt}(\varsigma ʻP)ʻ&\kappa .\supset .\\ +&\text{lt}(\varsigma ʻP)ʻ\kappa =\text{lt}(\varsigma ʻP)ʻ\overrightarrow{P}ʻʻsʻ\kappa =sʻ\kappa\end{align}\]</p> + +<p>This shows that every limit in the series of segments is a limit of +a class of what we may call <i>rational</i> segments (<i>i.e.</i> +segments of the form \(\overrightarrow{P}ʻx\)), namely it is the limit +of \(\overrightarrow{P}ʻʻsʻ\kappa\).</p> + +<p class="nind"><b>*212·667.</b> \(\vdash :P\in \text{Ser}.\supset .\text{D}ʻ\text{lt}(\varsigma ʻP)-\iota ʻ\Lambda =\text{ᗡ}ʻ\text{sgm}ʻP\)</p> + +<p>This shows that the segments (other than \(\Lambda\)) which are limits +of classes of segments are the segments (other than \(\Lambda\)) which +have no maximum in \(P\).</p> + +<p>The number ends with a set of propositions (<a href="#*212·7">*212·7</a>—<a href="#*212·72">·72</a>) on the +relations of the sections and segments of two correlated series. If +\(S\) is a correlator of \(P\) with \(Q\), then \(S_{\in }\) (with +its converse domain limited) is a correlator of \(\varsigma ʻP_{*}\) +with \(\varsigma ʻQ_{*}\), \(\varsigma ʻP\) with \(\varsigma ʻQ\) and +\(\text{sgm}ʻP\) with \(\text{sgm}ʻQ\) (<a href="#*212·71">*212·71</a><a href="#*212·711">·711</a><a href="#*212·712">·712</a>). Hence</p> + +<p class="nind"><b>*212·72.</b> \(\vdash :P\,\,\text{smor}\,\,Q.\supset .\varsigma ʻP_{*}\,\text{smor}\,\varsigma ʻQ_{*}.\varsigma ʻP\,\text{smor}\,\varsigma ʻQ.\text{sgm}ʻP\,\text{smor}\,\text{sgm}ʻQ\)</p> + +<p>This proposition is used in the next number, and also in *271.</p> + +<hr class="tb"> + +<p class="nind"><b>*212·01.</b> \(\varsigma ʻP=P_{\text{lc}}\unicode{x0294f}\text{D}ʻP_{\in } \quad\text{Df}\)</p> + +<p class="nind"><b>*212·02.</b> \(\text{sgm}ʻP=P_{\text{lc}}\unicode{x0294f}\text{D}ʻ(P_{\in }\dot{\cap} I) \quad\text{Df}\)</p> + +<p class="nind"><b>*212·1.</b> \[\begin{align}&\vdash :\alpha (\varsigma ʻP)\beta .\equiv .\alpha ,\beta \in \text{D}ʻP_{\in }.\exists !\beta -\alpha -Pʻʻ(\alpha -\beta )\\ +&[*170·102.*37·15]\end{align}\]</p> + +<p class="nind"><b>*212·11.</b> \(\vdash :\alpha (\text{sgm}ʻP)\beta .\equiv .\alpha ,\beta \in \text{D}ʻ(P_{\in }\dot{\cap} I).\exists !\beta -\alpha\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*170·102.*37·15.\supset\\ +\vdash :\alpha (\text{sgm}ʻP)\beta .&\equiv .\alpha ,\beta \in \text{D}ʻ(P_{\in }\dot{\cap} I).\exists !\beta -\alpha -Pʻʻ(\alpha -\beta ) &\qquad \text{(1)}\\ +\vdash .*211·12.\supset \vdash :\alpha \in \text{D}ʻ(P_{\in }\dot{\cap} I).&\supset .-\alpha =-Pʻʻ\alpha .\\ +[*37·2.\text{Transp}] &\supset .-\alpha \subset -Pʻʻ(\alpha -\beta ).\\ +[*22·621] &\supset .-\alpha -Pʻʻ(\alpha -\beta )=-\alpha &\qquad \text{(2)}\\ +\vdash .(1).(2).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*212·12.</b> \(\vdash :\alpha (\text{sgm}ʻP_{*})\beta .\equiv .\alpha ,\beta \in \text{sect}ʻP.\exists !\beta -\alpha \quad[*211·13.*212·11]\)</p> + +<p>Thus \(\text{sgm}ʻP_{*}\) has the same connection with +\(\text{sect}ʻP\) as \(\text{sgm}ʻP\) has with \(\text{D}ʻ(P_{\in}\dot{\cap} I)\). +When \(P\) is transitive, \(\text{sgm}ʻP_{*}\) also +has the same connection<span class="pagenum" id="Page_654">[Pg 654]</span> with \(\text{sect}ʻP\) as \(\varsigma ʻP\) has +with \(\text{D}ʻP_{\in }\). The following proposition makes these facts +more explicit.</p> + +<p class="nind"><b>*212·121.</b> \(\vdash .\text{sgm}ʻP_{*}=\varsigma ʻP_{*}=P_{\text{lc}}\unicode{x0294f}\text{sect}ʻP\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*211·13. &\supset \vdash .\text{sgm}ʻP_{*}=P_{\text{lc}}\unicode{x0294f}\text{sect}ʻP &\qquad \text{(1)}\\ +\vdash .*212·1.*211·13. &\supset \vdash :\alpha (\varsigma ʻP_{*})\beta .\equiv .\alpha ,\beta \in \text{sect}ʻP.\exists !\beta -\alpha -P_{*}ʻʻ(\alpha + -\beta ) &\qquad \text{(2)}\\ +\vdash .*211·13. \supset \vdash :\alpha \in \text{sect}ʻP.&\supset .\alpha =P_{*}ʻʻ\alpha .\\ +[*37·2] &\supset .P_{*}ʻʻ(\alpha -\beta )\subset \alpha .\\ +[\text{Transp}.*22·621] &\supset .-\alpha -P_{*}ʻʻ(\alpha -\beta )=-\alpha &\qquad \text{(3)}\\ +\vdash .(2).(3).\supset \vdash :\alpha (\varsigma ʻP_{*})\beta .&\equiv .\alpha ,\beta \in \text{sect}ʻP.\exists !\beta -\alpha .\\ +[*211·12] &\equiv.\alpha (\text{sgm}ʻP_{*})\beta &\qquad \text{(4)}\\ +\vdash .(1).(4).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*212·122.</b> \(\vdash .\varsigma ʻP,\text{sgm}ʻP\in \text{ᗡ}ʻʻʻJ \quad[*170·17]\)</p> + +<p class="nind"><b>*212·123.</b> \(\vdash .Cʻ\varsigma ʻP,Cʻ\text{sgm}ʻP{\sim}\in 1 \quad[*200·12.*212·122]\)</p> + +<p class="nind"><b>*212·13.</b> \(\vdash :\Lambda (\varsigma ʻP)\beta .\equiv .\beta \in \text{D}ʻP_{\in }-\iota ʻ\Lambda \quad[*170·6]\)</p> + +<p class="nind"><b>*212·131.</b> \(\vdash :\alpha (\varsigma ʻP)(\text{D}ʻP).\equiv .\alpha \in \text{D}ʻP_{\in }-\iota ʻ\text{D}ʻP\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*212·1. \supset \vdash :\alpha (\varsigma ʻP)(\text{D}ʻP).&\equiv .\alpha ,\text{D}ʻP\in \text{D}ʻP_{\in }.\exists !\text{D}ʻP-\alpha -Pʻʻ(\alpha -\text{D}ʻP).\\ +[*211·301.*37·15] &\equiv .\alpha \in \text{D}ʻP_{\in }.\alpha \subset \text{D}ʻP.\exists !\text{D}ʻP-\alpha .\\ +[*24·55.*22·41] &\equiv .\alpha \in \text{D}ʻP_{\in }.\alpha \subset \text{D}ʻP.\alpha \neq \text{D}ʻP.\\ +[*37·15] &\equiv .\alpha \in \text{D}ʻP_{\in }.\alpha \neq \text{D}ʻP:\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*212·132.</b> \(\vdash .\text{D}ʻ\varsigma ʻP=\text{D}ʻP_{\in }-\iota ʻ\text{D}ʻP.\text{ᗡ}ʻ\varsigma ʻP= \text{D}ʻP_{\in }-\iota ʻ\Lambda\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*212·13·131.&\supset \vdash .\text{D}ʻP_{\in }-\iota ʻ\text{D}ʻP\subset \text{D}ʻ\varsigma ʻP.\text{D}ʻP_{\in }-\iota + ʻ\Lambda \subset \text{ᗡ}ʻ\varsigma ʻP &\qquad \text{(1)}\\ +\vdash .*212·1.&\supset \vdash .\text{D}ʻ\varsigma ʻP\subset \text{D}ʻP_{\in }.\text{ᗡ}ʻ\varsigma ʻP\subset \text{D}ʻP_{\in } &\qquad \text{(2)}\\ +\vdash .*212·1.\supset \vdash :\alpha \in \text{D}ʻ\varsigma ʻP.&\supset .(\exists \beta ).\beta \in \text{D}ʻP_{\in }.\exists !\beta -\alpha .\\ +[*37·15] & \supset .\exists !\text{D}ʻP-\alpha &\qquad \text{(3)}\\ +\vdash .*212·1.\supset \vdash :\beta \in \text{ᗡ}ʻ\varsigma ʻP.&\supset .(\exists \alpha ).\exists !\beta -\alpha .\\ +[*24·561] &\supset .\exists !\beta &\qquad \text{(4)}\\ +\vdash .(3).(4).&\supset \vdash .\text{D}ʻ\varsigma ʻP\subset -\iota ʻ\text{D}ʻP.\text{ᗡ}ʻ\varsigma ʻP\subset -\iota ʻ\Lambda &\qquad \text{(5)}\\ +\vdash .(1).(2).(5).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*212·133.</b> \(\vdash :\dot{\exists} !P.\supset .Cʻ\varsigma ʻP=\text{D}ʻP_{\in }.Bʻ\varsigma ʻP=\Lambda .Bʻ\text{Cnv}ʻ\varsigma ʻP= \text{D}ʻP\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*33·24.\supset \vdash :\text{Hp}.&\supset .\Lambda \neq \text{D}ʻP.\\ +[*212·132] &\supset .\Lambda \in \text{D}ʻ\varsigma ʻP.\text{D}ʻP\in \text{ᗡ}ʻ\varsigma ʻP.\\ +[*51·221] \supset .\text{D}ʻ\varsigma ʻP&=\{(\text{D}ʻP_{\in }-\iota ʻ\text{D}ʻP)-\iota ʻ\Lambda\}\cup \iota ʻ\Lambda .\\ +[*212·132] \supset .Cʻ\varsigma ʻP&=\{(\text{D}ʻP_{\in }-\iota ʻ\Lambda )-\iota ʻ\text{D}ʻP\}\cup \iota ʻ\Lambda \cup (\text{D}ʻP_{\in }-\iota ʻ\Lambda )\\ +[*22·63] & =(\text{D}ʻP_{\in }-\iota ʻ\Lambda )\cup \iota ʻ\Lambda \\ +[*51·221] & =\text{D}ʻP_{\in } &\qquad \text{(1)}\\ +\vdash .(1).*93·103.*212·132.\supset \vdash :\text{Hp}.\supset .\overrightarrow{B}ʻ\varsigma ʻP&=\text{D}ʻP_{\in }-(\text{D}ʻP_{\in }-\iota ʻ\Lambda )\\ +[*211·44] &=\iota ʻ\Lambda &\qquad \text{(2)}\\ +\vdash .(1).*93·103.*212·132.\supset \vdash :\text{Hp}.\supset .\overrightarrow{B}ʻ\text{Cnv}ʻ\varsigma ʻP&=\text{D}ʻP_{\in }-(\text{D}ʻP_{\in }-\iota ʻ\text{D}ʻP)\\ +[*211·301] & =\iota ʻ\text{D}ʻP &\qquad \text{(3)}\\ +\vdash .(1).(2).(3).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_655">[Pg 655]</span></p> + +<p class="nind"><b>*212·134.</b> \(\vdash :P=\dot{\Lambda} .\supset .\varsigma ʻP=\dot{\Lambda} \quad[*170·35]\)</p> + +<p class="nind"><b>*212·14.</b> \(\vdash :\dot{\exists} !P.\equiv .\dot{\exists} !\varsigma ʻP\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*212·133.*211·301.\supset \vdash :\dot{\exists} !P.&\supset .\exists !Cʻ\varsigma ʻP.\\ +[*33·24] &\supset .\dot{\exists} !\varsigma ʻP &\qquad \text{(1)}\\ +\vdash .(1).*212·134.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*212·141.</b> \(\vdash :\alpha \in Cʻ\varsigma ʻP.\equiv .\alpha \in \text{D}ʻP_{\in }.\dot{\exists} !P\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*10·24. \supset \vdash :\alpha \in Cʻ\varsigma ʻP.&\supset .\exists !Cʻ\varsigma ʻP.\\ +[*33·24.*212·14] &\supset .\dot{\exists} !P. &\qquad \text{(1)}\\ +[*212·133.\text{Hp}] &\supset .\alpha \in \text{D}ʻP_{\in } &\qquad \text{(2)}\\ +\vdash .*212·133.\supset \vdash :\alpha \in \text{D}ʻP_{\in }.\dot{\exists} !P.&\supset .\alpha \in Cʻ\varsigma ʻP &\qquad \text{(3)}\\ +\vdash .(1).(2).(3).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*212·142.</b> \(\vdash :\dot{\exists} !\varsigma ʻP.\equiv .\text{D}ʻP_{\in }{\sim}\in 1\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*211·66.*212·14. &\supset \vdash :\dot{\exists} !\varsigma ʻP.\supset .\text{D}ʻP_{\in }{\sim}\in 1 &\qquad \text{(1)}\\ +\vdash .*212·132.*211·44.\supset \vdash :\text{D}ʻP_{\in }{\sim}\in 1.&\supset .\exists !\text{ᗡ}ʻ\varsigma ʻP.\\ +[*33·24] &\supset .\dot{\exists} !\varsigma ʻP &\qquad \text{(2)}\\ +\vdash .(1).(2).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*212·15.</b> \(\vdash :\Lambda (\text{sgm}ʻP)\beta .\equiv .\beta \in \text{D}ʻ(P_{\in }\dot{\cap} I)-\iota ʻ\Lambda \quad[\text{Proof as in *212·13}]\)</p> + +<p class="nind"><b>*212·151.</b> \(\vdash :P=\dot{\Lambda} .\supset .\text{sgm}ʻP=\dot{\Lambda} \quad[*170·35]\)</p> + +<p>The converse implication does not hold in this case. For the existence +of \(\text{sgm}ʻP\), it is necessary that \(CʻP\) should contain +classes having no maximum.</p> + +<p><span class="pagenum" id="Page_656">[Pg 656]</span></p> + +<p class="nind"><b>*212·152.</b> \(\vdash .\text{ᗡ}ʻ\text{sgm}ʻP=\text{D}ʻ(P_{\in }\dot{\cap} I)-\iota ʻ\Lambda \quad[\text{Proof as in *212·132}]\)</p> + +<p class="nind"><b>*212·153.</b> \(\vdash :\dot{\exists} !\text{sgm}ʻP.\equiv .\exists !\text{D}ʻ(P_{\in }\dot{\cap} I)-\iota ʻ\Lambda .\equiv .\text{D}ʻ(P_{\in }\dot{\cap} I){\sim}\in 1\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*212·15. &\supset \vdash :\exists !\text{D}ʻ(P_{\in }\dot{\cap} I)-\iota ʻ\Lambda .\supset .\dot{\exists} !\text{sgm}ʻP &\qquad \text{(1)}\\ +\vdash .*212·152.&\supset \vdash :\dot{\exists} !\text{sgm}ʻP.\supset .\exists !\text{D}ʻ(P_{\in }\dot{\cap} I)-\iota ʻ\Lambda &\qquad \text{(2)}\\ +\vdash .*212·11. \supset \vdash :\dot{\exists} !\text{sgm}ʻP.&\supset .(\exists \alpha ,\beta ).\alpha ,\beta \in \text{D}ʻ(P_{\in }\dot{\cap} I).\alpha \neq \beta .\\ +[*52·16.\text{Transp}] &\supset .\text{D}ʻ(P_{\in }\dot{\cap} I){\sim}\in 1 &\qquad \text{(3)}\\ +\vdash .*211·44.*52·181.\supset \\ +\vdash :\text{D}ʻ(P_{\in }\dot{\cap} I){\sim}\in 1.&\supset .(\exists \beta ).\beta \in \text{D}ʻ(P_{\in }\dot{\cap} I).\beta \neq \Lambda .\\ +[*212·15] & \supset .\dot{\exists} !\text{sgm}ʻP &\qquad \text{(4)}\\ +\vdash .(1).(2).(3).(4).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*212·154.</b> \(\vdash :\dot{\exists} !\text{sgm}ʻP.\supset .Cʻ\text{sgm}ʻP=\text{D}ʻ(P_{\in }\dot{\cap} I)\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*212·153·15.\supset \vdash :\text{Hp}.&\supset .\Lambda \in \text{D}ʻ\text{sgm}ʻP.\\ +[*212·152] &\supset .\text{D}ʻ(P_{\in }\dot{\cap} I)\subset Cʻ\text{sgm}ʻP &\qquad \text{(1)}\\ +\vdash .*212·11.&\supset \vdash .Cʻ\text{sgm}ʻP\subset \text{D}ʻ(P_{\in }\dot{\cap} I) &\qquad \text{(2)}\\ +\vdash .(1).(2).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*212·155.</b> \(\vdash :\dot{\exists} !\text{sgm}ʻP.\supset .\Lambda =Bʻ\text{sgm}ʻP \quad[*212·152·154.*93·103]\)</p> + +<p class="nind"><b>*212·156.</b> \[\begin{align}\vdash :\alpha \in Cʻ\text{sgm}ʻP.&\equiv .\alpha \in \text{D}ʻ(P_{\in }\dot{\cap} I).\dot{\exists} !\text{sgm}ʻP.\\ +&\equiv .\alpha \in \text{D}ʻ(P_{\in }\dot{\cap} I).\text{D}ʻ(P_{\in }\dot{\cap} I){\sim}\in 1\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*212·154.&\supset \vdash :\alpha \in \text{D}ʻ(P_{\in }\dot{\cap} I).\dot{\exists} !\text{sgm}ʻP.\supset .\alpha \in Cʻ\text{sgm}ʻP &\qquad \text{(1)}\\ +\vdash .*10·24.*33·24.\supset \vdash :\alpha \in Cʻ\text{sgm}ʻP.&\supset .\dot{\exists} !\text{sgm}ʻP. &\qquad \text{(2)}\\ +[*212·154] &\supset .\alpha \in \text{D}ʻ(P_{\in }\dot{\cap} I) &\qquad \text{(3)}\\ +\vdash .(1).(2).(3).*212·153.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*212·16.</b> \(\vdash :\text{ᗡ}ʻP\subset \text{D}ʻP.\supset .\text{D}ʻP\in \text{D}ʻ(P_{\in }\dot{\cap} I)\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*37·27.\supset \vdash :\text{Hp}.\supset .Pʻʻ\text{D}ʻP=\text{D}ʻP &\qquad \text{(1)}\\ +\vdash .(1).*211·12.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*212·161.</b> \(\vdash :\text{ᗡ}ʻP\subset \text{D}ʻP.\dot{\exists} !P.\supset .\dot{\exists} !\text{sgm}ʻP\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*33·24.*212·16.\supset \vdash :\text{Hp}.&\supset .\text{D}ʻP\in \text{D}ʻ(P_{\in }\dot{\cap} I)-\iota ʻ\Lambda .\\ +[*212·15] &\supset .\Lambda (\text{sgm}ʻP)(\text{D}ʻP).\\ +[*11·36] & \supset .\dot{\exists} !\text{sgm}ʻP:\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*212·162.</b> \[\begin{align}\vdash :\text{ᗡ}ʻP\subset \text{D}ʻ&P.\dot{\exists} !P.\supset .\\ +&\text{D}ʻP=Bʻ\text{Cnv}ʻ\text{sgm}ʻP.\text{D}ʻ\text{sgm}ʻP=\text{D}ʻ(P_{\in }\dot{\cap} I)-\iota ʻ\text{D}ʻP\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*212·16·152.*33·24.&\supset \vdash :\text{Hp}.\supset .\text{D}ʻP\in \text{ᗡ}ʻ\text{sgm}ʻP &\qquad \text{(1)}\\ +\vdash .*212·11.*37·24.&\supset \vdash :\text{Hp}.\alpha \in \text{D}ʻ(P_{\in }\dot{\cap} I)-\iota ʻ\text{D}ʻP.\supset .\alpha (\text{sgm}ʻP)(\text{D}ʻP) &\qquad \text{(2)}\\ +\vdash .*37·24. \supset \vdash :\alpha \in \text{D}ʻ(P_{\in }\dot{\cap} I).&\supset .{\sim}\exists !(\alpha -\text{D}ʻP).\\ +[*212·11] & \supset .{\sim}\{(\text{D}ʻP)(\text{sgm}ʻP)\alpha\} &\qquad \text{(3)}\\ +\vdash .(2).(3).\supset \vdash :\text{Hp}.&\supset .\text{D}ʻ(P_{\in }\dot{\cap} + I)-\iota ʻ\text{D}ʻP\subset \text{D}ʻ\text{sgm}ʻP.\text{D}ʻP{\sim}\in \text{D}ʻ\text{sgm}ʻP &\qquad \text{(4)}\\ +\vdash .(1).(4).*212·154.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_657">[Pg 657]</span></p> + +<p class="nind"><b>*212·17.</b> \(\vdash :\dot{\exists} !\varsigma ʻP_{*}.\equiv .\exists !\text{sect}ʻP-\iota ʻ\Lambda .\equiv .\text{sect}ʻP{\sim}\in 1.\equiv .\dot{\exists} !P\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*212·132.*211·13.&\supset \vdash :\dot{\exists} !\varsigma ʻP_{*}.\equiv .\exists !\text{sect}ʻP-\iota ʻ\Lambda &\qquad \text{(1)}\\ +\vdash .*212·142.*211·13.&\supset \vdash :\dot{\exists} !\varsigma ʻP_{*}.\equiv .\text{sect}ʻP{\sim}\in 1 &\qquad \text{(2)}\\ +\vdash .*212·14. \supset \vdash :\dot{\exists} !\varsigma ʻP_{*}.&\equiv .\dot{\exists} !P_{*}.\\ +[*90·141] &\equiv .\dot{\exists} !P &\qquad \text{(3)}\\ +\vdash .(1).(2).(3).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*212·171.</b> \[\begin{align}&\vdash .\text{D}ʻ\varsigma ʻP_{*}=\text{sect}ʻP-\iota ʻCʻP.\text{ᗡ}ʻ\varsigma ʻP_{*}=\text{sect}ʻP-\iota ʻ\Lambda \\ +&[*212·132.*211·13.*90·14]\end{align}\]</p> + +<p class="nind"><b>*212·172.</b> \[\begin{align}&\vdash :\dot{\exists} !P.\supset .Cʻ\varsigma ʻP_{*}=\text{sect}ʻP.Bʻ\varsigma ʻP_{*}=\Lambda .Bʻ\text{Cnv}ʻ\varsigma ʻP_{*}=CʻP\\ +&[*212·133.*211·13.*90·141]\end{align}\]</p> + +<p class="nind"><b>*212·173.</b> \[\begin{align}&\vdash :\alpha \in Cʻ\varsigma ʻP_{*}.\equiv .\alpha \in \text{sect}ʻP.\dot{\exists} !P.\equiv .\alpha \in \text{sect}ʻP.\text{sect}ʻP{\sim}\in 1\\ +&[*212·141·142·14.*211·13]\end{align}\]</p> + +<p class="nind"><b>*212·18.</b> \(\vdash.\varsigmaʻ\breve{P}_{*}=(CʻP-)^{;}\text{Cnv}ʻ\varsigma ʻP_{*}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*212·12·121.&\supset \vdash :\alpha (\varsigma ʻ\breve{P} _{*})\beta .\equiv .\alpha ,\beta \in \text{sect}ʻ\breve{P} .\exists !\beta -\alpha .\\ +[*211·7] &\equiv .(\exists \gamma ,\delta ).\gamma ,\delta \in \text{sect}ʻP.\alpha =CʻP-\gamma .\beta =CʻP-\delta .\exists !\beta -\alpha .\\ +[*24·55] & \equiv .(\exists \gamma ,\delta ).\gamma ,\delta \in \text{sect}ʻP.\alpha =CʻP-\gamma .\beta =CʻP-\delta .{\sim}(\beta \subset \alpha ).\\ +[*211·1.*24·492]&\equiv .(\exists \gamma ,\delta ).\gamma ,\delta \in \text{sect}ʻP.\alpha =CʻP-\gamma .\beta =CʻP-\delta .{\sim}(\gamma \subset \delta ).\\ +[*212·12.*24·55]&\equiv .\alpha {(CʻP-)^{;}\text{Cnv}ʻ\varsigma ʻP_{*}}\beta :\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*212·181.</b> \(\vdash .(\varsigma ʻ\breve{P} _{*})\,\text{smor}\,(\text{Cnv}ʻ\varsigma ʻP_{*}) \quad[*212·18]\)</p> + +<p>The above proposition is used in *252·43.</p> + +<p class="nind"><b><a id="*212·2">*212·2</a>.</b> \(\vdash .\text{sgm}ʻP\,\unicode{x2abd}\, \varsigma ʻP.\text{sgm}ʻP\,\unicode{x2abd}\, \varsigma ʻP_{*} \quad[*211·14.*212·1·11·12]\)</p> + +<p class="nind"><b>*212·21.</b> \(\vdash :P\in \text{trans}.\supset .\varsigma ʻP\,\unicode{x2abd}\, \varsigma ʻP_{*} \quad[*211·15.*212·12]\)</p> + +<p class="nind"><b>*212·22.</b> \[\begin{align}&\vdash :P\in \text{connex} .\supset .\text{sgm}ʻP=\hat{\alpha} \hat{\beta} \{\alpha ,\beta \in \text{D}ʻ(P_{\in }\dot{\cap} + I).\alpha \subset \beta .\alpha \neq \beta\}\\ +&[*211·62.*210·1.*212·11]\end{align}\]</p> + +<p class="nind"><b><a id="*212·23">*212·23</a>.</b> \[\begin{align}&\vdash :P\in \text{trans}\cap \text{connex} .\supset .\varsigma ʻP=\hat{\alpha} \hat{\beta} \{\alpha ,\beta \in \text{D}ʻP_{\in }.\alpha + \subset \beta .\alpha \neq \beta\}\\ +&[*210·13.*211·61.(*212·01)]\end{align}\]</p> + +<p><span class="pagenum" id="Page_658">[Pg 658]</span></p> + +<p class="nind"><b><a id="*212·24">*212·24</a>.</b> \[\begin{align}&\vdash : P_{*} \in \text{connex} . \supset . \varsigma ʻP_{*} + = \hat{\alpha}\hat{\beta} \{\alpha , \beta \in \text{sect}ʻP . \alpha \subset \beta . \alpha \neq \beta\}\\ +&[*212·121·22 . *211·13]\end{align}\]</p> + +<p class="nind"><b><a id="*212·25">*212·25</a>.</b> \(\vdash : P \in \text{Ser} . \supset . \overrightarrow{P}^{;}P = (\varsigma ʻP) \unicode{x0294f} \overrightarrow{P}ʻʻCʻP\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *204·33·331 . \supset \\ +\vdash \colon\ldotp \text{Hp} . \supset : \alpha (\overrightarrow{P}^{;}P)\beta + . &\equiv . \alpha , \beta \in \overrightarrow{P}ʻʻCʻP . \alpha \subset \beta . \alpha \neq \beta .\\ +[*212·23 . *211·3] &\equiv . \alpha , \beta \in \overrightarrow{P}ʻʻCʻP . \alpha (\varsigma ʻP)\beta \colon\ldotp \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p>The following propositions, down to <a href="#*212·55">*212·55</a>, consist of applications +of the propositions of <a href="#*210">*210</a>, where the \(\kappa\) of that number +is replaced by \(\text{sect}ʻP\), \(\text{D}ʻP_{\in }\), or +\(\text{D}ʻ(P_{\in } \dot{\cap} 1)\), and the \(Q\) is replaced by +\(P_{\text{lc}} \unicode{x0294f} \kappa\), <i>i.e.</i> by \(\varsigmaʻP_{*}\), +\(\varsigma ʻP\), or \(\text{sgm}ʻP\). The propositions +which follow are important, since the use of segments, especially in +connection with continuity, depends largely upon them.</p> + +<p class="nind"><b><a id="*212·3">*212·3</a>.</b> \[\begin{align}&\vdash : P \in \text{connex} . \supset . \text{sgm}ʻP, \varsigma ʻP_{*} \in \text{Ser}\\ +&[*211·67 . *210·14 . *212·121]\end{align}\]</p> + +<p class="nind"><b><a id="*212·31">*212·31</a>.</b> \[\begin{align}&\vdash : P \in \text{trans} \cap \text{connex} . \supset . \varsigma ʻP \in \text{Ser}\\ +&[*211·68 . *210·14 . (*212·01)]\end{align}\]</p> + +<p class="nind"><b>*212·32.</b> + \[\begin{align}&\vdash : P \in \text{connex} . \dot{\exists} ! P . \lambda \in \text{sect}ʻP . sʻ\lambda \in \lambda . \supset . sʻ\lambda = \text{max}(\varsigma ʻP_{*})ʻ\lambda + \\ +&[*210·211.*211·67.*212·17]\end{align}\]</p> + +<p>We write \(\text{max}(\varsigma ʻP_{*})ʻ\lambda\), instead of putting +\(\varsigma ʻP_{*}\) below the line, because, when we have to deal with +an expression not consisting of a single letter, it is inconvenient to +write it as a suffix, especially when it contains a suffix itself, as +in this case.</p> + +<p class="nind"><b>*212·321.</b> + \[\begin{align}\vdash : P \in \text{connex} . \dot{\exists} ! P . \lambda \subset \text{sect}ʻP . sʻ\lambda {\sim} \in \lambda . \supset . sʻ\lambda &= \text{seq}(\varsigma ʻP_{*})ʻ\lambda \\ +&= \text{lt}(\varsigma ʻP_{*})ʻ\lambda \\ +[*210·231 . *211·67 . *212·17 . *211·63]\end{align}\]</p> + +<p class="nind"><b>*212·322.</b> \[\begin{align}&\vdash : P \in \text{connex} . \dot{\exists} ! P . \lambda \subset \text{sect}ʻP . \supset . sʻ\lambda = \text{limax}(\varsigma ʻP_{*})ʻ\lambda + \\ +&[*212·32·321.*207·46]\end{align}\]</p> + +<p class="nind"><b>*212·33.</b> \[\begin{align}\vdash : P \in \text{connex} . \dot{\exists} ! P . \lambda \subset \text{sect}ʻP . &pʻ\lambda \cap CʻP \in \lambda . \supset .\\ +pʻ\lambda \cap CʻP = \text{min}(\varsigma ʻP_{*})ʻ\lambda\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *211·671 . *210·252 . *211·26 . \supset \\ +\vdash : \text{Hp} . &\supset . pʻ\lambda \cap CʻP \in \overrightarrow{\text{min}}(\varsigma ʻP_{*})ʻ\lambda \cup \overrightarrow{\text{prec}} (\varsigma ʻP_{*})ʻ\lambda + &\qquad \text{(1)}\\ +\vdash . *206·2 . &\supset \vdash : pʻ\lambda \cap CʻP \in \lambda . \supset . pʻ\lambda \cap CʻP {\sim} e \overrightarrow{\text{prec}} (\varsigma ʻP_{*})ʻ\lambda + &\qquad \text{(2)}\\ +\vdash . (1) . (2) . &\supset \vdash : \text{Hp} . \supset . pʻ\lambda \cap CʻP \in \overrightarrow{\text{min}}(\varsigma ʻP_{*})ʻ\lambda &\qquad \text{(3)}\\ +\vdash . (3) . *205·31 . \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_659">[Pg 659]</span></p> + +<p class="nind"><b>*212·331.</b> \[\begin{align}\vdash :P\in \text{connex} .\dot{\exists} !P.\lambda \subset \text{sect}ʻP.&pʻ\lambda \cap CʻP{\sim}\in \lambda .\supset .\\ +&pʻ\lambda \cap CʻP = \text{prec} (\varsigma ʻP_{*})ʻ\lambda = \text{tl}(\varsigma ʻP_{*})ʻ\lambda\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*211·671.*210·252.*211·26.\supset \\ +\vdash :\text{Hp}.\supset .pʻ\lambda \cap CʻP = \text{limin}(\varsigma ʻP_{*})ʻ\lambda &\qquad \text{(1)}\\ +\vdash .*205·1.\text{Transp}.\supset \vdash :pʻ\lambda \cap CʻP{\sim}\in \lambda .\supset .pʻ\lambda \cap CʻP{\sim}\in \overrightarrow{\text{min}}(\varsigma ʻP_{*})ʻ\lambda + &\qquad \text{(2)}\\ +\vdash .(1).(2).*206·161.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*212·34">*212·34</a>.</b> \[\begin{align}&\vdash :P\in \text{connex} .\dot{\exists} !P.\lambda \subset \text{sect}ʻP.\supset .pʻ\lambda \cap CʻP = \text{limin}(\varsigma ʻP_{*})ʻ\lambda \\ +&[*212·33·331.*207·46]\end{align}\]</p> + +<p class="nind"><b><a id="*212·35">*212·35</a>.</b> \[\begin{align}&\vdash :P\in \text{connex} .\dot{\exists} !P.\supset .\\ +&(\lambda ).\lambda \in \{\text{ᗡ}ʻ\text{max}(\varsigma ʻP_{*})\cup \text{ᗡ}ʻ\text{seq}(\varsigma ʻP_{*})\}\cap \{\text{ᗡ}ʻ\text{min}(\varsigma ʻP_{*})\cup + \text{ᗡ}ʻ\text{prec}(\varsigma ʻP_{*})\}\\ +&[*210·28.*211·671.*212·121]\end{align}\]</p> + +<p class="nind"><b>*212·36.</b> \(\vdash \colon\ldotp P\in \text{connex} .\supset :\lambda \in Cʻ\text{sgm}ʻ\varsigma ʻP_{*}.\supset .\text{E}!\text{seq}(\varsigma ʻP_{*})ʻ\lambda\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*211·47.*212·35·3.\supset \\ +\vdash \colon\ldotp \text{Hp}.\dot{\exists} !P.\supset :\lambda \in Cʻ\text{sgm}ʻ\varsigma ʻP_{*}.&\supset .\text{E}!\text{seq}(\varsigma ʻP_{*})ʻ\lambda &\qquad \text{(1)}\\ +\vdash .*33·24.\supset \vdash :\lambda \in Cʻ\text{sgm}ʻ\varsigma ʻP_{*}.&\supset .\dot{\exists} !\text{sgm}ʻ\varsigma ʻP_{*}.\\ +[*212·151.\text{Transp}] &\supset .\dot{\exists} !\varsigma ʻP_{*}.\\ +[*212·17] &\supset .\dot{\exists} !P &\qquad \text{(2)}\\ +\vdash .(1).(2).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*212·4.</b> \[\begin{align}&\vdash :P\in \text{trans} \cap \text{connex} .\dot{\exists} !P.\lambda \subset \text{D}ʻP_{\in }.sʻ\lambda + \in \lambda .\supset .sʻ\lambda = \text{max}(\varsigma ʻP)ʻ\lambda\\ +&[*211·68·66.*210·211]\end{align}\]</p> + +<p class="nind"><b>*212·401.</b> \[\begin{align}\vdash :P\in \text{trans} \cap \text{connex} .\dot{\exists} !P.\lambda \subset \text{D}ʻ&P_{\in }.sʻ\lambda {\sim}\in \lambda .\supset .\\ +&sʻ\lambda = \text{seq}(\varsigma ʻP)ʻ\lambda = \text{lt}(\varsigma ʻP)ʻ\lambda \\ +[*211·68·66·64.*210·231]\end{align}\]</p> + +<p class="nind"><b>*212·402.</b> \[\begin{align}&\vdash :P\in \text{trans} \cap \text{connex} .\dot{\exists} !P.\lambda \subset \text{D}ʻP_{\in }.\supset + .sʻ\lambda = \text{limax}(\varsigma ʻP)ʻ\lambda \\ +&[*212·4·401.*207·46]\end{align}\]</p> + +<p class="nind"><b>*212·41.</b> \[\begin{align}&\vdash :P\in \text{trans} \cap \text{connex} .\dot{\exists} !P.\lambda \subset \text{D}ʻP_{\in }.pʻ\lambda + \in \lambda .\supset .pʻ\lambda = \text{min}(\varsigma ʻP)ʻ\lambda \\ +&[*211·68·66.*210·21]\end{align}\]</p> + +<p class="nind"><b><a id="*212·411">*212·411</a>.</b> \[\begin{align}\vdash :P\in \text{trans} \cap \text{connex} .\dot{\exists} !P.\lambda \subset \text{D}ʻ&P_{\in }.pʻ\lambda \in \text{D}ʻP_{\in }-\lambda + .\supset .\\ +&pʻ\lambda = \text{prec}(\varsigma ʻP)ʻ\lambda = \text{tl}(\varsigma ʻP)ʻ\lambda \\ +[*211·68·66.*210·23]\end{align}\]</p> + +<p class="nind"><b><a id="*212·42">*212·42</a>.</b> \[\begin{align}&\vdash :P\in \text{trans} \cap \text{connex} .\dot{\exists} !P.\lambda \subset \text{D}ʻP_{\in }.pʻ\lambda {\sim}\in \lambda .\supset .\\ +&sʻ(\text{D}ʻP_{\in }\cap \text{Cl}ʻpʻ\lambda ) = \text{prec}(\varsigma ʻP)ʻ\lambda = \text{tl}(\varsigma ʻP)ʻ\lambda \\ +&[*210·26·22.*211·68·66·64]\end{align}\]</p> + +<p><span class="pagenum" id="Page_660">[Pg 660]</span></p> + +<p>The cases considered in <a href="#*212·411">*212·411</a> and <a href="#*212·42">*212·42</a> are not mutually +exclusive, since if \(pʻ\lambda \in \text{D}ʻP_{\in }\), we have +\(sʻ(\text{D}ʻP_{\in } \cap \text{Cl}ʻpʻ\lambda ) = pʻ\lambda\).</p> + +<p class="nind"><b>*212·421.</b> \[\begin{align}\vdash : P \in \text{trans} \cap \text{connex} . \dot{\exists} ! P . \lambda \subset \text{D}ʻ&P_{\in } + . pʻ\lambda {\sim} \in \text{D}ʻP_{\in } . \supset .\\ +&sʻ(\text{D}ʻP_{\in } \cap \text{Cl}ʻpʻ\lambda ) = Pʻʻpʻ\lambda\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*211·15·1. \supset \vdash \colon\ldotp \text{Hp} . &\supset : \alpha \in \lambda . \supset _{\alpha } . Pʻʻ\alpha \subset \alpha :\\ +[*40·81] &\supset : Pʻʻpʻ\lambda \subset pʻ\lambda &\qquad \text{(1)}\\ +\vdash .(1).*211·11. \supset \vdash : \text{Hp} . &\supset . Pʻʻpʻ\lambda \in \text{D}ʻP_{\in } \cap \text{Cl}ʻpʻ\lambda .\\ +[*40·13] &\supset . Pʻʻpʻ\lambda \subset sʻ(\text{D}ʻP_{\in } \cap \text{Cl}ʻpʻ\lambda ) &\qquad \text{(2)}\\ +\vdash .*13·196.*60·2. \supset \vdash \colon\ldotp \text{Hp} . &\supset : \alpha \in \text{D}ʻP_{\in } \cap \text{Cl}ʻpʻ\lambda . \supset _{\alpha } + . \alpha \subset pʻ\lambda . \alpha \neq pʻ\lambda :\\ +[*211·56·15·632] &\supset : \exists ! \lambda . \alpha \in \text{D}ʻP_{\in } \cap \text{Cl}ʻpʻ\lambda . \supset _{\alpha } . \alpha \subset Pʻʻpʻ\lambda :\\ +[*40·151] &\supset : \exists ! \lambda . \supset . sʻ(\text{D}ʻP_{\in } \cap \text{Cl}ʻpʻ\lambda ) \subset Pʻʻpʻ\lambda &\qquad \text{(3)}\\ +\vdash .*40·2.*37·24. \supset \vdash : \lambda = \Lambda .&\supset . sʻ(\text{D}ʻP_{\in } + \cap \text{Cl}ʻpʻ\lambda ) \subset \text{D}ʻP . Pʻʻpʻ\lambda = \text{D}ʻP &\qquad \text{(4)}\\ +\vdash .(3).(4). \supset \vdash : \text{Hp} . &\supset . sʻ(\text{D}ʻP_{\in } \cap \text{Cl}ʻpʻ\lambda ) \subset Pʻʻpʻ\lambda &\qquad \text{(5)}\\ +\vdash .(2).(5).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*212·43.</b> \[\begin{align}\vdash : P \in \text{trans} \cap &\text{connex} . \dot{\exists} ! P . \lambda \subset \text{D}ʻP_{\in } + . pʻ\lambda {\sim} \in \text{D}ʻP_{\in } . \supset .\\ +&Pʻʻpʻ\lambda = \text{prec}(\varsigma ʻP)ʻ\lambda = \text{tl}(\varsigma ʻP)ʻ\lambda \quad[*212·42·421]\end{align}\]</p> + +<p>Thus with regard to the lower end of a class chosen out of +\(Cʻ\varsigma ʻP\), we have three cases to distinguish: (1) if +\(pʻ\lambda \in \lambda\), \(pʻ\lambda\) is the minimum; (2) if +\(pʻ\lambda \in \text{D}ʻP_{\in } - \lambda\), \(pʻ\lambda\) is the +lower limit; (3) if \(pʻ\lambda {\sim} \in \text{D}ʻP_{\in }\), +\(Pʻʻpʻ\lambda\) is the lower limit.</p> + +<p class="nind"><b><a id="*212·431">*212·431</a>.</b> \[\begin{align}\vdash : P \in \text{trans} \cap \text{connex} . \dot{\exists} ! P . \lambda &\subset \text{D}ʻP_{\in } . \supset .\\ +&sʻ(\text{D}ʻP_{\in } \cap \text{Cl}ʻpʻ\lambda ) = \text{limin}(\varsigma ʻP)ʻ\lambda\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*212·42. &\supset \vdash : \text{Hp} . pʻ\lambda {\sim} \in \lambda . \supset . sʻ(\text{D}ʻP_{\in } + \cap \text{Cl}ʻpʻ\lambda ) = tl(\varsigma ʻP)ʻ\lambda &\qquad \text{(1)}\\ +\vdash .*22·441. \supset \vdash : \text{Hp} . pʻ\lambda \in \lambda . &\supset . pʻ\lambda \in (\text{D}ʻP_{\in } \cap \text{Cl}ʻpʻ\lambda ) .\\ +[*40·13] &\supset . pʻ\lambda \subset sʻ(\text{D}ʻP_{\in } \cap \text{Cl}ʻpʻ\lambda ) &\qquad \text{(2)}\\ +\vdash .*60·2. &\supset \vdash : \alpha \in \text{D}ʻP_{\in } \cap \text{Cl}ʻpʻ\lambda . \supset . \alpha \subset pʻ\lambda :\\ +[*40·151] &\supset \vdash . sʻ(\text{D}ʻP_{\in } \cap \text{Cl}ʻpʻ\lambda ) \subset pʻ\lambda &\qquad \text{(3)}\\ +\vdash .(2).(3). \supset \vdash : \text{Hp} . pʻ\lambda \in \lambda . &\supset . sʻ(\text{D}ʻP_{\in } \cap \text{Cl}ʻpʻ\lambda ) = pʻ\lambda .\\ +[*212·41] &\supset . sʻ(\text{D}ʻP_{\in } \cap \text{Cl}ʻpʻ\lambda ) = min(\varsigma ʻP)ʻ\lambda &\qquad \text{(4)}\\ +\vdash .(1).(4).*207·46. \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*212·44.</b> \[\begin{align}\vdash : &P \in \text{trans} \cap \text{connex} . \dot{\exists} ! P . \supset .\\ +&(\lambda ) . \lambda \in \{\text{ᗡ}ʻ\text{max}(\varsigma ʻP) \cup \text{ᗡ}ʻ\text{seq}(\varsigma ʻP)\} \cap \{\text{ᗡ}ʻ\text{min}(\varsigma ʻP) \cup \text{ᗡ}ʻ\text{prec}(\varsigma ʻP)\}\\ +&[*211·681.*210·28]\end{align}\]</p> + +<p><span class="pagenum" id="Page_661">[Pg 661]</span></p> + +<p class="nind"><b>*212·45.</b> \(\vdash \colon\ldotp P\in \text{trans}\cap \text{connex} .\supset :\lambda \in Cʻ\text{sgm}ʻ\varsigma ʻP.\supset .\text{E}!\text{seq}(\varsigma ʻP)ʻ\lambda \)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*211·47.*212·44·31.\supset \\ +\vdash \colon\ldotp \text{Hp}.\dot{\exists} !P.\supset :\lambda \in Cʻ\text{sgm}ʻ\varsigma ʻP.&\supset .\text{E}!\text{seq}(\varsigma ʻP)ʻ\lambda &\qquad \text{(1)}\\ +\vdash .*33·24.\supset \vdash :\lambda \in Cʻ\text{sgm}ʻ\varsigma ʻP.&\supset .\dot{\exists} !\text{sgm}ʻ\varsigma ʻP.\\ +[*212·151.\text{Transp}] &\supset .\dot{\exists} !\varsigma ʻP.\\ +[*212·14] &\supset .\dot{\exists} !P &\qquad \text{(2)}\\ +\vdash .(1).(2).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p>The proofs of the following propositions are exactly analogous to those +of the corresponding propositions on \(\varsigma ʻP\).</p> + +<p class="nind"><b>*212·5.</b> \[\begin{align}\vdash :P\in \text{connex} .\dot{\exists} !\text{sgm}ʻP.\lambda \subset \text{D}ʻ(P_{\in }\dot{\cap} I).&sʻ\lambda \in \lambda .\supset .\\ +&sʻ\lambda =\text{max}(\text{sgm}ʻP)ʻ\lambda\end{align}\]</p> + +<p class="nind"><b>*212·501.</b> \[\begin{align}\vdash :P\in \text{connex} .\dot{\exists} !\text{sgm}ʻP.\lambda &\subset \text{D}ʻ(P_{\in }\dot{\cap} I).sʻ\lambda {\sim}\in \lambda .\supset .\\ +&sʻ\lambda =\text{seq}(\text{sgm}ʻP)ʻ\lambda =\text{lt}(\text{sgm}ʻP)ʻ\lambda\end{align}\]</p> + +<p class="nind"><b>*212·502.</b> \[\begin{align}&\vdash :P\in \text{connex} .\dot{\exists} !\text{sgm}ʻP.\lambda \subset \text{D}ʻ(P_{\in }\dot{\cap} + I).\supset .sʻ\lambda =\text{limax}(\text{sgm}ʻP)ʻ\lambda\\ +&[*212·5·501]\end{align}\]</p> + +<p class="nind"><b>*212·51.</b> \[\begin{align}\vdash :P\in \text{connex} .\dot{\exists} !\text{sgm}ʻP.\lambda \subset \text{D}ʻ(P_{\in }\dot{\cap} I).&pʻ\lambda \in \lambda .\supset .\\ +&pʻ\lambda =\text{min}(\text{sgm}ʻP)ʻ\lambda \end{align}\]</p> + +<p class="nind"><b><a id="*212·511">*212·511</a>.</b> \[\begin{align}\vdash :P\in \text{connex} .\dot{\exists} !\text{sgm}ʻP.\lambda &\subset \text{D}ʻ(P_{\in }\dot{\cap} I).pʻ\lambda \in \text{D}ʻ(P_{\in }\dot{\cap} + I)-\lambda .\supset .\\ +&pʻ\lambda =\text{prec}(\text{sgm}ʻP)ʻ\lambda =\text{tl}(\text{sgm}ʻP)ʻ\lambda\end{align}\]</p> + +<p class="nind"><b>*212·52.</b> \[\begin{align}\vdash :P\in &\text{connex} .\dot{\exists} !\text{sgm}ʻP.\lambda \subset \text{D}ʻ(P_{\in }\dot{\cap} I).pʻ\lambda {\sim}\in \lambda .\supset .\\ +&sʻ\{\text{D}ʻ(P_{\in }\dot{\cap} I)\cap \text{Cl}ʻpʻ\lambda \}=\text{prec}(\text{sgm}ʻP)ʻ\lambda =\text{tl}(\text{sgm}ʻP)ʻ\lambda\end{align}\]</p> + +<p>This proposition includes <a href="#*212·511">*212·511</a>, since, if +\(pʻ\lambda \in\text{D}ʻ(P_{\in }\dot{\cap} I)\), we have +\[ +sʻ\{\text{D}ʻ(P_{\in }\dot{\cap} I)\cap \text{Cl}ʻpʻ\lambda\}=pʻ\lambda . +\]</p> + +<p class="nind"><b>*212·53.</b> \[\begin{align}\vdash :P\in &\text{connex} .\dot{\exists} !\text{sgm}ʻP.\lambda \subset \text{D}ʻ(P_{\in }\dot{\cap} I).\supset .\\ +&sʻ\{\text{D}ʻ(P_{\in }\dot{\cap} I)\cap \text{Cl}ʻpʻ\lambda\}=\text{limin}(\text{sgm}ʻP)ʻ\lambda \quad[*212·51·52]\end{align}\]</p> + +<p>The proof proceeds as in <a href="#*212·431">*212·431</a>.</p> + +<p class="nind"><b>*212·54.</b> \[\begin{align}&\vdash :P\in \text{connex} .\dot{\exists} !\text{sgm}ʻP.\supset .\\ +&(\lambda ).\lambda \in \{\text{ᗡ}ʻ\text{max}(\text{sgm}ʻP)\cup \text{ᗡ}ʻ\text{seq}(\text{sgm}ʻP)\}\cap \{\text{ᗡ}ʻ\text{min}(\text{sgm}ʻP)\cap\text{ᗡ}ʻ\text{prec}(\text{sgm}ʻP)\}\end{align}\]</p> + +<p class="nind"><b><a id="*212·55">*212·55</a>.</b> \(\vdash \colon\ldotp P\in \text{connex} .\supset :\lambda \in Cʻ\text{sgm}ʻ\text{sgm}ʻP.\supset .\text{E}!\text{seq}(\text{sgm}ʻP)ʻ\lambda\)</p> + +<p>The following propositions are concerned with the relations of maxima, +limits and sequents in \(P\) and \(\varsigma ʻP\) respectively. The +series \(\overrightarrow{P}^{;}P\), which is ordinally similar to \(P\), +is contained in \(\varsigma ʻP\); and if \(\alpha\) has a maximum or +limit in \(P\), the maximum or limit of \(\overrightarrow{P}ʻʻ\alpha\) +in \(\varsigma ʻP\) is \(\overrightarrow{P}ʻ\text{max}_{P}ʻ\alpha\) or +\(\overrightarrow{P}ʻ\text{lt}_Pʻ\alpha\). In this way, a series +(namely \(\overrightarrow{P}^{;}P\)) which has the same ordinal +properties as \(P\) can be placed in a certain Dedekindian series +(namely \(\varsigma ʻP\)) in such a way that the classes which have +limits in \(P\) are those whose correlates have<span class="pagenum" id="Page_662">[Pg 662]</span> limits which are +members of \(\overrightarrow{P}ʻʻCʻP\), while those whose correlates +have limits which are not members of \(\overrightarrow{P}ʻʻCʻP\) +are those which have neither a maximum nor a limit in \(P\). These +relations are important in many connections. For example, if \(P\) is +of the type of the rationals, \(\varsigma ʻP\) is of the type of the +real numbers: \(Cʻ\varsigma ʻP-\overrightarrow{P}ʻʻCʻP\) corresponds to +the irrationals, and classes contained in \(\overrightarrow{P}ʻʻCʻP\) +but having a limit not belonging to \(\overrightarrow{P}ʻʻCʻP\) +correspond to series of rationals having an irrational limit. +In the original series \(P\), there are no irrational limits; +but if \(\alpha\) is a class in \(CʻP\) and having no limit, +\(\overrightarrow{P}ʻʻ\alpha\) has an irrational limit in \(\varsigmaʻP\).</p> + +<p class="nind"><b><a id="*212·6">*212·6</a>.</b> \[\begin{align}\vdash :P\in \text{Ser}.\alpha \subset Cʻ&P.\supset .\\ +&\overrightarrow{\text{max}}(\varsigma ʻP)ʻ\overrightarrow{P}ʻʻ\alpha =\overrightarrow{\text{max}}(\overrightarrow{P}^{;}P)ʻ\overrightarrow{P}ʻʻ\alpha + =\overrightarrow{P}ʻʻ\overrightarrow{\text{max}}_{P}ʻ\alpha\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*205·9.*200·12.\supset \\ +\vdash :\text{Hp}.&\supset .\overrightarrow{\text{max}}(\varsigma ʻP)ʻ\overrightarrow{P}ʻʻ\alpha =\overrightarrow{\text{max}}(\overrightarrow{P}^{;}P)ʻ\overrightarrow{P}ʻʻ\alpha + &\qquad \text{(1)}\\ +\vdash .*204·35.*205·8.\supset \\ +\vdash :\text{Hp}.&\supset .\overrightarrow{\text{max}}(\overrightarrow{P}^{;}P)ʻ\overrightarrow{P}ʻʻ\alpha =\overrightarrow{P}ʻʻ\overrightarrow{\text{max}}_{P}ʻ\alpha + &\qquad \text{(2)}\\ +\vdash .(1).(2).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*212·601.</b> \[\begin{align}\vdash \colon\ldotp P\in &\text{Ser}.\alpha \subset CʻP.\supset :\\ +&\text{E}!\text{max}_{P}ʻ\alpha .\equiv .\text{E}!\text{max}(\overrightarrow{P}^{;}P)ʻ\overrightarrow{P}ʻʻ\alpha + .\equiv .\text{E}!\text{max}(\varsigma ʻP)ʻ\overrightarrow{P}ʻʻ\alpha \\ +[*212·6]\end{align}\]</p> + +<p class="nind"><b>*212·602.</b> \(\vdash \colon\ldotp P\in \text{Ser}.\dot{\exists} !P.\alpha \subset CʻP.\supset :\text{E}!\text{max}_{P}ʻ\alpha .\equiv .Pʻʻ\alpha \in \overrightarrow{P}ʻʻ\alpha\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*212·601.*210·223.\supset \\ +\vdash \colon\ldotp \text{Hp}.\supset :\text{E}!\text{max}_{P}ʻ\alpha .&\equiv .sʻ\overrightarrow{P}ʻʻ\alpha \in \overrightarrow{P}ʻʻ\alpha .\\ +[*40·5] & \equiv .Pʻʻ\alpha \in \overrightarrow{P}ʻʻ\alpha \colon\ldotp \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*212·61.</b> \[\begin{align}&\vdash :P\in \text{trans}\cap \text{connex} .\dot{\exists} !P.\supset .\text{limax}(\varsigma ʻP)ʻ\overrightarrow{P}ʻʻ\alpha =Pʻʻ\alpha\\ +&[*212·402.*40·5]\end{align}\]</p> + +<p class="nind"><b>*212·62.</b> \[\begin{align}\vdash \colon\ldotp P\in \text{Ser}.\dot{\exists} !&P.\supset :\\ +&\text{E}!\text{limax}_Pʻ\alpha .\equiv .\text{E}!\text{limax}(\overrightarrow{P}^{;}P)ʻ\overrightarrow{P}ʻʻ\alpha .\\ +&\equiv .\text{limax}(\varsigma ʻP)ʻ\overrightarrow{P}ʻʻ\alpha =\overrightarrow{P}ʻ\text{limax}_Pʻ\alpha .\\ +&\equiv .\text{limax}(\varsigma ʻP)ʻ\overrightarrow{P}ʻʻ\alpha \in \overrightarrow{P}ʻʻCʻP\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*204·35.*207·65.\supset \vdash \colon\ldotp \text{Hp}.\supset :\text{E}!\text{limax}_Pʻ\alpha .&\equiv .\text{E}!\text{limax}(\overrightarrow{P}^{;}P)ʻ\overrightarrow{P}ʻʻ\alpha + &\qquad \text{(1)}\\ +\vdash .*207·51.\supset \vdash \colon\ldotp \text{Hp}.\supset :\overrightarrow{P}ʻ\text{limax}_Pʻ\alpha =Pʻʻ\alpha .&\equiv .\text{limax}_Pʻ\alpha =\text{limax}_Pʻ\alpha .\\ +[*14·28] &\equiv .\text{E}!\text{limax}_Pʻ\alpha &\qquad \text{(2)}\\ +\vdash .(2).*212·61.\supset \\ +\vdash \colon\ldotp \text{Hp}.\supset :\text{E}!\text{limax}_Pʻ\alpha .&\equiv .\text{limax}(\varsigma ʻP)ʻ\overrightarrow{P}ʻʻ\alpha =\overrightarrow{P}ʻ\text{limax}_{ +P}ʻ\alpha &\qquad \text{(3)}\\ +\vdash .*207·51.*14·204.\supset \\ +\vdash \colon\ldotp \text{Hp}.\supset :\text{E}!\,\text{limax}_{P}ʻ\alpha .&\equiv .(\exists x).x\in CʻP.\overrightarrow{P}ʻx=Pʻʻ\alpha .\\ +[*37·7] &\equiv .Pʻʻ\alpha \in \overrightarrow{P}ʻʻCʻP &\qquad \text{(4)}\\ +\vdash .(1).(3).(4).\supset \vdash .\text{Prop} +\end{array} +\]</p> +<p><span class="pagenum" id="Page_663">[Pg 663]</span></p> + +<p class="nind"><b>*212·621.</b> + \(\vdash \colon\ldotp P\in \text{Ser}.\beta \subset CʻP.\supset :\text{limax}_Pʻ\alpha \in \beta .\equiv .\text{limax}(\varsigma ʻP)ʻ\overrightarrow{P}ʻʻ\alpha \in \overrightarrow{P}ʻʻ\beta\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*33·24. \supset \vdash :\text{Hp}.\text{limax}_Pʻ\alpha \in \beta .&\supset .\dot{\exists} !P.\text{limax}_Pʻ\alpha \in \beta .\\ +[*14·21] &\supset .\dot{\exists} !P.\text{E}!\text{limax}_Pʻ\alpha .\text{limax}_Pʻ\alpha \in \beta .\\ +[*212·62] &\supset .\text{limax}(\varsigma ʻP)ʻ\overrightarrow{P}ʻʻ\alpha \in \overrightarrow{P}ʻʻ\beta &\qquad \text{(1)}\\ +\vdash .*33·24.*22·621.&\supset \vdash :\text{Hp}.\text{limax}(\varsigma ʻP)ʻ\overrightarrow{P}ʻʻ\alpha \in \overrightarrow{P}ʻʻ\beta .\supset .\\ +&\dot{\exists} !P.\text{limax}(\varsigma ʻP)ʻ\overrightarrow{P}ʻʻ\alpha \in \overrightarrow{P}ʻʻ\beta \cap \overrightarrow{P}ʻʻCʻP.\\ +[*212·62] &\supset .\text{limax}(\varsigma ʻP)ʻ\overrightarrow{P}ʻʻ\alpha \in \overrightarrow{P}ʻʻ\beta .\text{limax}(\varsigma ʻP)ʻ\overrightarrow{P}ʻʻ\beta =\overrightarrow{P}ʻ\text{limax}_Pʻ\alpha .\\ +[*72·512.*204·34] &\supset .\text{limax}_Pʻ\alpha \in \beta &\qquad \text{(2)}\\ +\vdash .(1).(2).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*212·63.</b> \[\begin{align}&\vdash :P\in \text{Ser}.\dot{\exists} !P.\alpha \subset CʻP.{\sim}\text{E}!\text{max}_{P}ʻ\alpha + .\supset .\text{lt}(\varsigma ʻP)ʻ\overrightarrow{P}ʻʻ\alpha =Pʻʻ\alpha \\ +&[*212·61·601.*207·43]\end{align}\]</p> + +<p class="nind"><b>*212·631.</b> + \[\begin{align}\vdash \colon\ldotp P\in \text{Ser}.\dot{\exists} !P.\alpha \subset CʻP.\supset :\text{E}!\text{lt}_Pʻ\alpha .&\equiv .\text{lt}(\varsigma ʻP)ʻ\overrightarrow{P}ʻʻ\alpha =\overrightarrow{P}ʻ\text{lt}_Pʻ\alpha .\\ +&\equiv .\text{lt}(\varsigma ʻP)ʻ\overrightarrow{P}ʻʻ\alpha \in \overrightarrow{P}ʻʻCʻP\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*207·47. \supset \vdash :\text{E}!\text{lt}_Pʻ\alpha .&\equiv .\text{E}!\text{limax}_Pʻ\alpha .{\sim}\text{E}!\text{max}_{P}ʻ\alpha &\qquad \text{(1)}\\ +\vdash .(1).*212·62·601.\supset \\ +\vdash \colon\ldotp \text{Hp}.\supset :\text{E}!\text{lt}_Pʻ\alpha .&\equiv .\text{limax}(\varsigma ʻP)ʻ\overrightarrow{P}ʻʻ\alpha =\overrightarrow{P}ʻ\text{limax}_Pʻ\alpha .{\sim}\text{E}!\text{max}(\varsigma ʻP)ʻ\overrightarrow{P}ʻʻ\alpha .\\ +[*207·43·11] &\equiv .\text{lt}(\varsigma ʻP)ʻ\overrightarrow{P}ʻʻ\alpha =\overrightarrow{P}ʻ\text{lt}_Pʻ\alpha &\qquad \text{(2)}\\ +\vdash .(1).*212·62·601.\supset \\ +\vdash \colon\ldotp \text{Hp}.\supset :\text{E}!\text{lt}_Pʻ\alpha .&\equiv .\text{limax}(\varsigma ʻP)ʻ\overrightarrow{P}ʻʻ\alpha \in \overrightarrow{P}ʻʻCʻP.{\sim}\text{E}!\text{max}(\varsigma ʻP)ʻ\overrightarrow{P}ʻʻ\alpha .\\ +[*207·43·11] &\equiv .\text{lt}(\varsigma ʻP)ʻ\overrightarrow{P}ʻʻ\alpha \in \overrightarrow{P}ʻʻCʻP &\qquad \text{(3)}\\ +\vdash .(2).(3).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*212·632.</b> + \(\vdash :P\in \text{Ser}.\dot{\exists} !P.\alpha \subset CʻP.Pʻʻ\alpha {\sim}\in \overrightarrow{P}ʻʻCʻP.\supset .Pʻʻ\alpha =\text{lt}(\varsigma ʻP)ʻ\overrightarrow{P}ʻʻ\alpha\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*212·602.\supset \vdash :\text{Hp}.&\supset .{\sim}\text{E}!\text{max}_{P}ʻ\alpha .\\ +[*212·601] &\supset .{\sim}\text{E}!\text{max}(\varsigma ʻP)ʻ\overrightarrow{P}ʻʻ\alpha .\\ +[*212·61] &\supset .\text{lt}(\varsigma ʻP)ʻ\overrightarrow{P}ʻʻ\alpha =Pʻʻ\alpha :\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_664">[Pg 664]</span></p> + +<p class="nind"><b>*212·633.</b> \[\begin{align}\vdash \colon\ldotp P\in \text{Ser}.\dot{\exists} !P.x\in CʻP.\beta \subset &CʻP.\supset :\\ +&x=\text{lt}_Pʻ\beta .\equiv .\overrightarrow{P}ʻx=\text{lt}(\varsigma ʻP)ʻ\overrightarrow{P}ʻʻ\beta\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*212·631.*14·21.\supset \\ +\vdash \colon\ldotp \text{Hp}.\supset :x&=\text{lt}_Pʻ\beta .\supset .\overrightarrow{P}ʻx=\text{lt}(\varsigma P)ʻ\overrightarrow{P}ʻʻ\beta &\qquad \text{(1)}\\ +\vdash .*212·402.\supset \vdash :\text{Hp}.\overrightarrow{P}ʻx&=\text{lt}(\varsigma ʻP)ʻ\overrightarrow{P}ʻʻ\beta .\supset .\overrightarrow{P}ʻx=Pʻʻ\beta &\qquad \text{(2)}\\ +\vdash .*206·2. \supset \vdash :\text{Hp}.\overrightarrow{P}ʻx&=\text{lt}(\varsigma ʻP )ʻ\overrightarrow{P} ʻʻ\beta .\supset .\overrightarrow{P}ʻx{\sim}\in \overrightarrow{P}ʻʻ\beta .\\ +[*72·512.*204·34] &\supset .x{\sim}\in \beta &\qquad \text{(3)}\\ +\vdash .(2).(3).*207·232.\supset \vdash :\text{Hp}.\overrightarrow{P}ʻx=\text{lt}(\varsigma ʻP)ʻ\overrightarrow{P}ʻʻ\beta .\supset .x&=\text{lt}_Pʻ\beta &\qquad \text{(4)}\\ +\vdash .(1).(4).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*212·65.</b> \(\vdash \colon\ldotp P\in \text{Ser}.\alpha \subset CʻP.\supset :\text{E}!\text{seq}_{P}ʻ\alpha .\equiv .\overrightarrow{P}ʻ\text{seq}_{P}ʻ\alpha + =\text{seq}(\varsigma ʻP)ʻ\overrightarrow{P}ʻʻ\alpha\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*206·17.*210·15.*211·3.\supset \\ +\vdash \colon\colon\text{Hp}.\supset \colon\ldotp &\overrightarrow{P}ʻ\text{seq}_{P}ʻ\alpha =\text{seq}(\varsigma ʻP )ʻ\overrightarrow{P} ʻʻ\alpha .\equiv :\\ +&y\in \alpha \cap CʻP,\supset _{y}.\overrightarrow{P}ʻy\subset \overrightarrow{P}ʻ\text{seq}_{P}ʻ\alpha .\overrightarrow{P}ʻy\neq \overrightarrow{P}ʻ\text{seq}_{P}ʻ\alpha + :\\ +&\gamma \in \text{D}ʻP_{\in }.\gamma \subset \overrightarrow{P}ʻ\text{seq}_{P}ʻ\alpha .\gamma \neq \overrightarrow{P}ʻ\text{seq}_{P}ʻ\alpha .\supset _{\gamma }.(\exists + z).z\in \alpha .\gamma \subset \overrightarrow{P}ʻz :\\ +[*204·33.*206·22]&\equiv :y\in \alpha \cap CʻP.\supset _{y}.yP\text{seq}_{P}ʻ\alpha :\\ +&\gamma \in \text{D}ʻP_{\in }.\gamma \subset (\alpha \cap CʻP)\cup Pʻʻ\alpha .\gamma \neq (\alpha \cap CʻP)\cup Pʻʻ\alpha .\supset _{\gamma }.(\exists + z).z\in \alpha .\gamma \subset \overrightarrow{P}ʻz &\qquad \text{(1)}\\ +\vdash .*211·56.&\supset \vdash :\text{Hp}.\gamma \in \text{D}ʻP_{\in }.z\in CʻP-\gamma .\supset .\gamma \subset \overrightarrow{P}ʻz &\qquad \text{(2)}\\ +\vdash .(2).&\supset \vdash :\text{Hp}.\gamma \in \text{D}ʻP_{\in }.\gamma \subset (\alpha \cap CʻP)\cup Pʻʻ\alpha .\gamma \neq (\alpha \cap CʻP)\cup Pʻʻ\alpha .\supset .\\ +&(\exists z).z\in \alpha .\gamma \subset \overrightarrow{P}ʻz &\qquad \text{(3)}\\ +\vdash .(1).(3).\supset \\ +\vdash \colon\colon\text{Hp}.\supset \colon\ldotp \overrightarrow{P}ʻ\text{seq}_{P}ʻ\alpha + =\text{seq}(\varsigma ʻP )ʻ\overrightarrow{P} ʻʻ\alpha .&\equiv :y\in \alpha \cap CʻP.\supset _{y}.yP\text{seq}_{P}ʻ\alpha :\\ +[*206·211.*14·21] &\equiv :\text{E}!\text{seq}_{P}ʻ\alpha \colon\colon\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*212·651.</b> \[\begin{align}\vdash \colon\ldotp &P\in \text{Ser}.\alpha \subset CʻP.\supset :\\ +&\text{E}!\text{seq}_{P}ʻ\alpha + .\equiv .\text{seq}(\varsigma ʻP)ʻ\overrightarrow{P}ʻʻ\alpha \in \overrightarrow{P} ʻʻCʻP .\equiv .\text{E}!\text{seq}(\overrightarrow{P}^{;}P)ʻ\overrightarrow{P}ʻʻ\alpha\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*212·65.&\supset \vdash \colon\ldotp \text{Hp}.\supset :\text{E}!\text{seq}_{P}ʻ\alpha + .\supset .\text{seq}(\varsigma ʻP)ʻ\overrightarrow{P}ʻʻ\alpha \in \overrightarrow{P}ʻʻCʻP &\qquad \text{(1)}\\ +\vdash .*206·17.*210·15.*211·3.\supset \\ +\vdash \colon\colon\text{Hp}.\supset \colon\ldotp &\text{seq}(\varsigma ʻP)ʻ\overrightarrow{P}ʻʻ\alpha =\overrightarrow{P}ʻw.w\in CʻP.\equiv :\\ +&y\in \alpha \cap CʻP.\supset _{y}.\overrightarrow{P}ʻy\subset \overrightarrow{P}ʻw.\overrightarrow{P}ʻy\neq \overrightarrow{P}ʻw:\\ +&\gamma \in \text{D}ʻP_{\in }.\gamma \subset \overrightarrow{P}ʻw.\gamma \neq \overrightarrow{P}ʻw.\supset _{\gamma }.(\exists + z).z\in \alpha .\gamma \subset \overrightarrow{P}ʻz:w\in CʻP:\\ +[*204·33.*211·3]&\supset :\alpha \cap CʻP\subset \overrightarrow{P}ʻw:yPw.\supset _{y}.(\exists z).z\in \alpha .\overrightarrow{P}ʻy\subset \overrightarrow{P}ʻz:w\in CʻP :\\ +[*204·32] &\supset :\alpha \cap CʻP\subset \overrightarrow{P}ʻw.\overrightarrow{P}ʻw\subset \alpha \cup Pʻʻ\alpha .w\in CʻP:\\ +[*206·171.*33·15] &\supset :w=\text{seq}_{P}ʻ\alpha &\qquad \text{(2)}\\ +\vdash . (2) . *37·7 . *14·204 . \supset \\ +\vdash \colon\ldotp \text{Hp} . &\supset : \text{seq}(\varsigma ʻP)ʻ\overrightarrow{P}ʻʻ\alpha \in \overrightarrow{P} ʻʻCʻP . \supset . \text{E}! \text{seq}_{P}ʻ\alpha + &\qquad \text{(3)}\\ +\vdash . (1) . (3) . *206·62 . \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_665">[Pg 665]</span></p> + +<p class="nind"><b><a id="*212·652">*212·652</a>.</b> \[\begin{align}\vdash : P \in \text{Ser} . \alpha \subset CʻP . \text{E}! \text{max}_{P}ʻ\alpha + . \text{E}! &\text{seq}(\varsigma ʻP )ʻ\overrightarrow{P} ʻʻ\alpha . \supset .\\ +&\text{seq}(\varsigma ʻP)ʻ\overrightarrow{P}ʻʻ\alpha = \alpha \cup Pʻʻ\alpha\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *212·6·601 . *206·46 . &\supset \vdash : \text{Hp} . \supset . \text{seq}(\varsigma ʻP)ʻ\overrightarrow{P}ʻʻ\alpha = \text{seq}(\varsigma ʻP)ʻ\iota ʻ\overrightarrow{P}ʻ\text{max}_{P}ʻ\alpha + &\qquad \text{(1)}\\ +\vdash . *206·17 . *210·15 . *211·3 . \supset \\ +\vdash \colon\colon \text{Hp} . &\supset \colon\ldotp \beta = \text{seq}(\varsigma ʻP)ʻ\iota ʻ\overrightarrow{P}ʻ\text{max}_{P}ʻ\alpha . \equiv :\\ +&\beta \in \text{D}ʻP_{\in } . \overrightarrow{P}ʻ\text{max}_{P}ʻ\alpha \subset \beta . \overrightarrow{P}ʻ\text{max}_{P}ʻ\alpha \neq \beta :\\ +&\gamma \in \text{D}ʻP_{\in } . \gamma \subset \beta . \gamma \neq \beta . \supset _{\gamma } . \gamma \subset \overrightarrow{P}ʻ\text{max}_{P}ʻ\alpha :\\ +[*201·55 . *210·1] \supset : \beta \in \text{D}ʻP_{\in } . \exists ! &\beta - Pʻʻ(\overrightarrow{P}ʻ\text{max}_{P}ʻ\alpha \cup \iota ʻ\text{max}_{P}ʻ\alpha ) :\\ +&\gamma \in \text{D}ʻP_{\in } . \gamma \subset \beta . \gamma \neq \beta . \supset _{\gamma } . \gamma \subset \overrightarrow{P}ʻ\text{max}_{P}ʻ\alpha :\\ +[*211·56] &\supset : \beta \in \text{D}ʻP_{\in } . \overrightarrow{P}ʻ\text{max}_{P}ʻ\alpha \cup \iota ʻ\text{max}_{P}ʻ\alpha \subset \beta :\\ +&\gamma \in \text{D}ʻP_{\in } . \gamma \subset \beta . \gamma \neq \beta . \supset _{\gamma } . \gamma \subset \overrightarrow{P}ʻ\text{max}_{P}ʻ\alpha :\\ +[*211·3] &\supset : \beta \in \text{D}ʻP_{\in } . \overrightarrow{P}ʻ\text{max}_{P}ʻ\alpha \cup \iota ʻ\text{max}_{P}ʻ\alpha + \subset \beta : x \in \beta . \supset _{x} . \overrightarrow{P}ʻx \subset \overrightarrow{P}ʻ\text{max}_{P}ʻ\alpha :\\ +[*40·5] &\supset : \beta \in \text{D}ʻP_{\in } . \overrightarrow{P}ʻ\text{max}_{P}ʻ\alpha \cup \iota ʻ\text{max}_{P}ʻ\alpha + \subset \beta . P ʻʻ\beta \subset \overrightarrow{P}ʻ\text{max}_{P}ʻ\alpha :\\ +[*202·56] &\supset : \beta \in \text{D}ʻP_{\in } . \overrightarrow{P}ʻ\text{max}_{P}ʻ\alpha \cup \iota ʻ\text{max}_{P}ʻ\alpha = \beta :\\ +[*205·131·22] &\supset : \beta = \alpha \cup Pʻʻ\alpha &\qquad \text{(2)}\\ +\vdash . (1) . (2) . \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*212·653.</b> \(\vdash \colon\ldotp P \in \text{Ser} . \text{E}! \text{max}_{P}ʻ\alpha . \alpha \subset CʻP . \supset : \text{E}! \text{seq}_{P}ʻ\alpha + . \equiv . \text{E}! \text{seq}(\varsigma ʻP) +ʻ\overrightarrow{P}ʻʻ\alpha\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *212·652 . &\supset \vdash \colon\ldotp \text{Hp} . \text{E}! \text{seq}(\varsigma ʻP)ʻ\overrightarrow{P}ʻʻ\alpha . \supset . \alpha \cup Pʻʻ\alpha \in \text{D}ʻP_{\in } + &\qquad \text{(1)}\\ +\vdash . *205·191 . &\supset \vdash : \text{Hp} . \supset . \text{E}! \text{max}_{P}ʻ(\alpha \cup Pʻʻ\alpha ) &\qquad \text{(2)}\\ +\vdash . (1) . (2) . *211·31 . \supset \vdash : \text{Hp} (1) . &\supset . \text{E}! \text{seq}_{P}ʻ(\alpha \cup Pʻʻ\alpha ) .\\ +[*206·25] &\supset . \text{E}! \text{seq}_{P}ʻ\alpha &\qquad \text{(3)}\\ +\vdash . *212·65 . &\supset \vdash : \text{Hp} . \text{E}! \text{seq}_{P}ʻ\alpha . \supset . \text{E}! \text{seq}(\varsigma ʻP)ʻ\overrightarrow{P}ʻʻ\alpha &\qquad \text{(4)}\\ +\vdash . (3) . (4) . \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*212·66.</b> \(\vdash : P \in \text{trans} \cap \text{connex} . \kappa \subset \text{D}ʻP_{\in } + . {\sim} \text{E}! \text{max}(\varsigma ʻP )ʻ\kappa . \supset . {\sim} \text{E}! \text{max}_{P}ʻsʻ\kappa\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *210·1 . *212·23 . \supset \\ +\vdash \colon\ldotp \text{Hp} . &\supset : \beta \in \kappa . \supset _{\beta } + . (\exists \gamma ) . \gamma \in \kappa . \beta \subset \gamma . \exists ! \gamma - \beta :\\ +[*201·5] &\supset : \beta \in \kappa . x \in \beta . \supset _{\beta ,x} . (\exists \gamma ) . \gamma \in \kappa . \exists ! \gamma - \overrightarrow{P} ʻx - \iota ʻx :\\ +[*202·101] \supset : x \in sʻ\kappa . &\supset _{x} . (\exists \gamma ) . \gamma \in \kappa . \exists ! \gamma \cap \overleftarrow{P}ʻx .\\ +[*37·46] &\supset _{x} . x \in Pʻʻsʻ\kappa \colon\ldotp \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_666">[Pg 666]</span></p> + +<p class="nind"><b>*212·661.</b> \(\vdash : P \in \text{Ser}.\kappa \subset \text{D}ʻP_{\in } + . \text{E}! \text{lt}(\varsigma ʻP)ʻ\kappa .\supset .\text{lt}(\varsigma ʻP)ʻ\kappa = \text{lt}(\varsigma ʻP)ʻ\overrightarrow{P}ʻʻsʻ\kappa = sʻ\kappa\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *212·402. \supset \vdash : \text{Hp}.\supset .\text{lt}(\varsigma ʻP)ʻ\kappa &= sʻ\kappa &\qquad \text{(1)}\\ +\vdash . *212·402. \supset \vdash : \text{Hp}.\supset .\text{limax}(\varsigma ʻP)ʻ\overrightarrow{P}ʻʻsʻ\kappa &= Pʻʻsʻ\kappa \\ +[*212·66] & = sʻ\kappa &\qquad \text{(2)}\\ +\vdash . *212·601·66 .\supset \vdash :\text{Hp}.&\supset .{\sim}\text{E}!\text{max}(\varsigma ʻP)ʻ\overrightarrow{P}ʻʻsʻ\kappa &\qquad \text{(3)}\\ +\vdash .(2).(3). \supset \vdash :\text{Hp}.\supset .\text{lt}(\varsigma ʻP)ʻ\overrightarrow{P}ʻʻsʻ\kappa &= sʻ\kappa &\qquad \text{(4)}\\ +\vdash .(1).(4).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*212·662.</b> \[\begin{align}\vdash : P \in \text{Ser}. \kappa \subset \text{D}ʻP_{\in } . \text{E}! &\text{lt}(\varsigma ʻP)ʻ\kappa .\supset .\\ +&(\exists \lambda ).\lambda \subset \overrightarrow{P}ʻʻCʻP.\text{lt}(\varsigma ʻP)ʻ\kappa = \text{lt}(\varsigma ʻP)ʻ\lambda \\ +[*212·661]\end{align}\]</p> + +<p class="nind"><b>*212·663.</b> \[\begin{align}\vdash : P \in \text{Ser}. x \in CʻP.\overrightarrow{P}ʻx \in \text{D}ʻ&\text{lt}(\varsigma ʻP).\supset .\\ +&\overrightarrow{P}ʻx = \text{lt}(\varsigma ʻP)ʻ\overrightarrow{P}ʻʻ\overrightarrow{P}ʻx.x = \text{lt}_Pʻ\overrightarrow{P}ʻx\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *212·661 . \supset \vdash : P \in \text{Ser}. x \in CʻP.\overrightarrow{P}ʻx &= lt (\varsigma ʻP)ʻ\kappa .\supset .\\ +\overrightarrow{P}ʻx = sʻ\kappa .\overrightarrow{P}ʻx &= lt (\varsigma ʻP)ʻ\overrightarrow{P}ʻʻsʻ\kappa .\\ +[*13·12.*212·66] \supset . \overrightarrow{P}ʻx &= \text{lt}(\varsigma ʻP)ʻ\overrightarrow{P}ʻʻ\overrightarrow{P}ʻx.{\sim}\text{E}! \text{max}_{P}ʻ\overrightarrow{P}ʻx.\\ +[*206·4] \supset . \overrightarrow{P}ʻx = \text{lt}(\varsigma ʻP)ʻ\overrightarrow{P}ʻʻ\overrightarrow{P}ʻx.x &= \text{lt}_Pʻ\overrightarrow{P}ʻx : \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*212·664.</b> \(\vdash \colon\ldotp P \in \text{Ser}.x \in CʻP.\supset :x \in \text{D}ʻ\text{lt}_P . \equiv .\overrightarrow{P}ʻx \in \text{D}ʻ\text{lt}(\varsigma ʻP)\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *212.631 .\supset \vdash : \text{Hp} . x = \text{lt}_Pʻ\alpha .\supset .\overrightarrow{P}ʻx = \text{lt}_Pʻ\overrightarrow{P}ʻʻ\alpha &\qquad \text{(1)}\\ +\vdash . (1) . *212.663 .\supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*212·665.</b> \(\vdash : P \in \text{Ser}. \dot{\exists} ! P . \alpha \in \text{D}ʻ(P_{\in } + \dot{\cap} I) .\supset . \text{lt}(\varsigma ʻP)ʻ\overrightarrow{P}ʻʻ\alpha = \alpha\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*211·4 .\supset \vdash : \text{Hp} .&\supset . {\sim} \text{E}! \text{max}_{P}ʻ\alpha .\\ +[*212·601·44] \supset .\text{lt}(\varsigma ʻP)ʻ\overrightarrow{P}ʻʻ\alpha &= \text{limax}(\varsigma ʻP)ʻ\overrightarrow{P}ʻʻ\alpha \\ +[*212·402·*40·5] &= Pʻʻ\alpha \\ +[*211·12] & = \alpha :\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*212·666.</b> \(\vdash : P \in \text{Ser}. \dot{\exists} ! P .\supset . \text{D}ʻ\text{lt}(\varsigma ʻP) = \text{D}ʻ(P_{\in } \dot{\cap} I)\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*212·66·661 .\supset \vdash : \text{Hp} . \kappa \subset \text{D}ʻP_{\in }.\gamma + = \text{lt}(\varsigma ʻP)ʻ\kappa .&\supset . \gamma = sʻ\kappa . {\sim} \text{E}!\text{max}_{P}ʻ\gamma .\\ +[*211·64·42] &\supset .\gamma \in \text{D}ʻ(P_{\in } \dot{\cap} I) &\qquad \text{(1)}\\ +\vdash .(1).*212·665 .\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*212·667">*212·667</a>.</b> \(\vdash : P \in \text{Ser}.\supset . \text{D}ʻ\text{lt}(\varsigma ʻP) - \iota ʻ\Lambda = \text{ᗡ}ʻ\text{sgm}ʻP \quad[*212·152·151·666]\)</p> + +<p><span class="pagenum" id="Page_667">[Pg 667]</span></p> + +<p class="nind"><b><a id="*212·7">*212·7</a>.</b> \(\vdash :S\in P\,\overline{\,\text{smor}\,}\,Q.\supset .\text{sect}ʻP = S_{\in }ʻʻ\text{sect}ʻQ.Cʻ\varsigma ʻP_{*} = S_{\in }ʻʻCʻ\varsigma ʻQ_{*}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*151·11·131. &\supset \vdash :\text{Hp}.\beta \subset CʻQ.\supset .Sʻʻ\beta \subset CʻP &\qquad \text{(1)}\\ +\vdash .*37·2.& \supset \vdash :Qʻʻ\beta \subset \beta .\supset .SʻʻQʻʻ\beta \subset Sʻʻ\beta &\qquad \text{(2)}\\ +\vdash .(2).*72·503. \supset \vdash :\text{Hp}.\beta \subset CʻQ.Qʻʻ\beta \subset \beta .&\supset .SʻʻQʻʻ\breve{S} ʻʻSʻʻ\beta \subset Sʻʻ\beta .\\ +[*151·11] &\supset .PʻʻSʻʻ\beta \subset Sʻʻ\beta &\qquad \text{(3)}\\ +\vdash .(1).(3).*211·1.&\supset \vdash :\text{Hp}.\beta \in \text{sect}ʻQ.\supset .Sʻʻ\beta \in \text{sect}ʻP &\qquad \text{(4)}\\ +\vdash .(4).*151·131. \supset \vdash :\text{Hp}.\alpha \in \text{sect}ʻP.&\supset .\breve{S} ʻʻ\alpha \in \text{sect}ʻQ.\\ +[*72·502] &\supset .\alpha \in S_{\in }ʻʻ\text{sect}ʻQ &\qquad \text{(5)}\\ +\vdash (4).(5). &\supset \vdash :\text{Hp}.\supset .\text{sect}ʻP = S_{\in }ʻʻ\text{sect}ʻQ &\qquad \text{(6)}\\ +\vdash .(6).*212·17·172.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*212·701.</b> \[\begin{align}&\vdash :S\in P\,\overline{\,\text{smor}\,}\,Q.\supset .\text{D}ʻP_{\in } = S_{\in }ʻʻ\text{D}ʻQ_{\in }.Cʻ\varsigma ʻP = S_{\in }ʻʻCʻ\varsigma + ʻQ\\ +&[\text{Proof as in *212·7}]\end{align}\]</p> + +<p class="nind"><b>*212·702.</b> \[\begin{align}\vdash :S\in P\overline{\,\text{smor}\,} &Q.\supset .\\ +&\text{D}ʻ(P_{\in }\dot{\cap} I) = S_{\in }ʻʻ\text{D}ʻ(Q_{\in }\dot{\cap} I).Cʻ\text{sgm}ʻP = S_{\in }ʻʻCʻ\text{sgm}ʻQ\\ +&[\text{Proof as in *212·7}]\end{align}\]</p> + +<p class="nind"><b><a id="*212·71">*212·71</a>.</b> \(\vdash :S\in P\,\overline{\,\text{smor}\,}\,Q.\supset .S_{\in }\upharpoonright Cʻ\varsigma ʻQ_{*}\in (\varsigma ʻP_{*})\overline{\,\text{smor}\,} + (\varsigma ʻQ_{*})\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*71·381.\supset \vdash \colon\colon\text{Hp}.\supset \colon\ldotp \alpha ,\beta \in \text{sect}ʻQ.\supset :\exists !\beta -\alpha . &\equiv .\exists !Sʻʻ\beta -Sʻʻ\alpha &\qquad \text{(1)}\\ +\vdash .(1).*212·7.\supset \vdash \colon\ldotp \text{Hp}.\alpha ,\beta \in \text{sect}ʻQ.\supset :\alpha (\varsigma ʻQ_{*})\beta . &\equiv .Sʻʻ\alpha (\varsigma ʻP_{*})Sʻʻ\beta + .\\ +[*150·41] & \equiv .\alpha \{\breve{S} _{\in }^{;}(\varsigma ʻP_{*})\}\beta &\qquad \text{(2)}\\ +\vdash .(2).*212·172.&\supset \vdash :\text{Hp}.\supset .\varsigma ʻQ_{*}\,\unicode{x2abd}\, \breve{S} _{\in }^{;}(\varsigma ʻP_{*}) &\qquad \text{(3)}\\ +\text{Similarly}\quad &\vdash :\text{Hp}.\supset .\varsigma ʻP_{*}\,\unicode{x2abd}\, S_{\in }^{;}(\varsigma ʻQ_{*}) &\qquad \text{(4)}\\ +\vdash .*72·451. &\supset \vdash :\text{Hp}.\supset .S_{\in }\upharpoonright Cʻ\varsigma ʻQ_{*}\in 1\rightarrow 1 &\qquad \text{(5)}\\ +\vdash .(3).(4).(5).*151·27.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*212·711">*212·711</a>.</b> \[\begin{align}&\vdash :S\in P\,\overline{\,\text{smor}\,}\,Q.\supset .S_{\in }\upharpoonright Cʻ\varsigma ʻQ\in (\varsigma ʻP)\overline{\,\text{smor}\,} (\varsigma ʻQ)\\ +&[\text{Proof as in *212·71}]\end{align}\]</p> + +<p class="nind"><b><a id="*212·712">*212·712</a>.</b> \[\begin{align}&\vdash :S\in P\,\overline{\,\text{smor}\,}\,Q.\supset .S_{\in }\upharpoonright Cʻ\text{sgm}ʻP\in (\text{sgm}ʻP)\overline{\,\text{smor}\,} (\text{sgm}ʻQ)\\ +&[\text{Proof as in *212·7}]\end{align}\]</p> + +<p class="nind"><b><a id="*212·72">*212·72</a>.</b> \[\begin{align}&\vdash :P \,\,\text{smor}\, \,Q.\supset .\varsigma ʻP_{*}\,\text{smor}\, \varsigma ʻQ_{*}.\varsigma + ʻP \,\,\text{smor}\, \varsigma ʻQ.\text{sgm}ʻP \,\,\text{smor}\, \text{sgm}ʻ Q\\ +&[*212·71·711·712]\end{align}\]</p> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_668">[Pg 668]</span></p> +<h2 class="nobreak" id="*213"> +*213. SECTIONAL RELATIONS.</h2> +</div> + + +<p class="nind"><i>Summary of</i> *213.</p> + +<p>If \(\alpha\) is a section of \(P\), \(P\unicode{x0294f}\alpha\) is +called a <i>sectional relation</i> of \(P\); and if \(\alpha\) is a +segment of \(P\), \(P\unicode{x0294f}\alpha\) is called a <i>segmental +relation</i> of \(P\). If \(P_{\text{po}}\) is serial, sectional +relations may be arranged in a series by the relation of inclusion +(<a href="#*213·153">*213·153</a>). That is, if we call the series of sectional relations +\(P_{\varsigma}\), we shall so define \(P_{\varsigma }\) as to secure +that if \(P_{\text{po}}\) is serial, +\[ +QP_{\varsigma }R.\equiv .Q,R\in P\unicode{x0294f}ʻʻ(\text{sect}ʻP-\iota ʻ\Lambda ).Q\,\unicode{x2abd}\, R.Q\neq R \quad(*213·21). +\] +The natural definition to take would be +\[ +P_{\varsigma }=P\unicode{x0294f}^{;}\varsigma ʻP_{*}. +\] +But this has the disadvantage that if \(xBP\), +\[ +P\unicode{x0294f}\iota ʻx=P\unicode{x0294f}\Lambda .\Lambda ,\iota ʻx\in \text{sect}ʻP. +\] +Thus \(P\unicode{x0294f}\alpha =P\unicode{x0294f}\beta\) +does not imply \(\alpha =\beta\); and when \(P\) is serial, +\(P\unicode{x0294f}^{;}\varsigma ʻP_{*}\) is not serial, because +\(\dot{\Lambda} (P\unicode{x0294f}^{;}\varsigma ʻP_{*})\dot{\Lambda}\). +In order to obviate this inconvenience, we confine ourselves to +sections which are not null, putting +\[ +P_{\varsigma }=P\unicode{x0294f}^{;}(\varsigma ʻP_{*})\unicode{x0294f}(-\iota ʻ\Lambda ) \quad\text{Df}. +\] +With the above definition, we have (<a href="#*213·151">*213·151</a><a href="#*213·152">·152</a>), if +\(P_{\text{po}}\in \text{Ser}\), +\[ +(P\unicode{x0294f})\upharpoonright Cʻ{(\varsigma ʻP_{*})\unicode{x0294f}-\iota ʻ\Lambda }\in 1\rightarrow 1 +\] +and \(P_{\varsigma }\,\text{smor}\,(\varsigmaʻP_{*})\unicode{x0294f}(-\iota ʻ\Lambda )\).</p> + +<p>The relation \(P_{\varsigma }\) is very useful in dealing with +well-ordered series; in this case, we have (as will be shown later) +\[ +P_{\varsigma }=P\unicode{x0294f}^{;}\overrightarrow{P}^{;}P\unicode{x0294f}\text{ᗡ}ʻP\unicode{x21f8}P. +\] +It will be seen that, if \(P_{\text{po}}\in \text{Ser}\), whenever +\(P\) exists, \(P=Bʻ\breve{P} _{\varsigma }\) (<a href="#*213·158">*213·158</a>); and whenever +\(\overrightarrow{B}ʻP\) exists, \(\dot{\Lambda} =BʻP_{\varsigma }\) +(<a href="#*213·155">*213·155</a>).</p> + +<p>We have, if \(P_{\text{po}}\in \text{Ser}\), +\[ +QP_{\varsigma }R.\equiv .R\in CʻP_{\varsigma }.Q\in \text{D}ʻR_{\varsigma } \quad(*213·245). +\] +Hence \(R\in CʻP_{\varsigma }.\supset .\overrightarrow{P}_{\varsigma}ʻR=\text{D}ʻR_{\varsigma }.R_{\varsigma }=P_{\varsigma}\unicode{x0294f}CʻR_{\varsigma }\) + (<a href="#*213·246">*213·246</a><a href="#*213·242">·242</a>).</p> + +<p>If \(P\) is serial, the sectional relations of \(P\) are all relations +such that by adding something to them they become \(P\), <i>i.e.</i> +they are +\[ +\hat{Q} \{(\exists R).P=Q\unicode{x2909} R.\lor.(\exists x).P=Q\unicode{x21f8}x\} \quad(*213·4). +\]<span class="pagenum" id="Page_669">[Pg 669]</span> +Hence their relation-numbers are those that can be made equal to that +of \(P\) by being added to. This fact is important in connection with +the theory of greater and less among relation-numbers.</p> + +<p>The propositions of this number are rendered complicated by the +necessity of taking account of the possibility of a section being a +unit class. This necessitates a good many propositions which are merely +lemmas; but in the end the complications mostly disappear.</p> + +<p>We begin with propositions on the field, etc., of \(P_{\varsigma }\). +We have</p> + +<p class="nind"><b>*213·141.</b> \(\vdash .\text{D}ʻP_{\varsigma }=P\unicode{x0294f}ʻʻ(\text{sect}ʻP-\iota ʻ\Lambda -\iota ʻCʻP)\)</p> + +<p class="nind"><b>*213·142.</b> \(\vdash :P_{\text{po}}\,\unicode{x2abd}\, J.\supset .CʻP_{\varsigma }=P\unicode{x0294f}ʻʻ(\text{sect}ʻP-\iota ʻ\Lambda )\)</p> + +<p class="nind"><b>*213·16.</b> \(\vdash .\text{D}ʻP_{\varsigma }=P\unicode{x0294f}ʻʻ(\text{sect}ʻP-\iota ʻ\Lambda )-\iota ʻP\)</p> + +<p class="nind"><b>*213·161.</b> \(\vdash :P_{\text{po}}\,\unicode{x2abd}\, + J.\exists !\overrightarrow{B}ʻP.\supset . P\unicode{x0294f}ʻʻ\text{sect}ʻP= P\unicode{x0294f}ʻʻ(\text{sect}ʻP-\iota ʻ\Lambda )=CʻP_{\varsigma }\)</p> + +<p class="nind"><b>*213·162.</b> \(\vdash :P_{\text{po}}\in \text{Ser}.\supset .\text{ᗡ}ʻP_{\varsigma }=P\unicode{x0294f}ʻʻ\text{sect}ʻP-\iota ʻ\dot{\Lambda}\)</p> + +<p>We then prove:</p> + +<p class="nind"><b>*213·17.</b> \[\begin{align}\vdash :P_{\text{po}}\in \text{Ser}.\supset .\text{Nr}ʻ\varsigma ʻP_{*}=\dot{1} \dot{+} &\text{Nr}ʻP_{\varsigma }.\\ +&\text{Nr}ʻ(\varsigma ʻP_{*})\unicode{x0294f}(\text{ᗡ}ʻ\varsigma ʻP_{*})=\text{Nr}ʻP_{\varsigma }\end{align}\]</p> + +<p>If \(P\) is finite, it follows from the above that \(\varsigma ʻP_{*}\) +is not similar to \(P_{\varsigma }\); but if \(P\) is infinite and has +a beginning and is well-ordered, we find +\[ +\text{Nr}ʻ\varsigma ʻP_{*}=\text{Nr}ʻP_{\varsigma }. +\]</p> + +<p class="nind"><b>*213·172.</b> \(\vdash :P_{\text{po}},Q_{\text{po}}\in \text{Ser}.P \,\,\text{smor}\, \,Q.\supset .P_{\varsigma } \,\text{smor}\, Q_{\varsigma }\)</p> + +<p>We then have a set of propositions (<a href="#*213·2">*213·2</a>—<a href="#*213·251">·251</a>) chiefly concerned +with the sections of \(R\), where \(R\in CʻP_{\varsigma }\). Besides +those already mentioned, the following are important:</p> + +<p class="nind"><b>*213·24.</b> \(\vdash :\beta \in \text{sect}ʻP.R=P\unicode{x0294f}\beta .\supset .\text{sect}ʻR=\text{sect}ʻP\subset \text{Cl}ʻCʻR\)</p> + +<p class="nind"><b>*213·243.</b> \(\vdash .\overrightarrow{P}_{\varsigma }ʻP=\text{D}ʻP_{\varsigma }\)</p> + +<p class="nind"><b>*213·25.</b> \(\vdash \colon\ldotp P_{\text{po}}\in \text{Ser}.Q,R\in CʻP_{\varsigma }.\supset :Q\in \text{D}ʻR_{\varsigma }.\lor.R\in + \text{D}Q_{\varsigma }.\lor.Q=R\)</p> + +<p>Our next set (<a href="#*213·3">*213·3</a>—<a href="#*213·32">·32</a>) is concerned with \(\dot{\Lambda}\) and \(x\downarrow y\). We have</p> + +<p class="nind"><b>*213·3.</b> \(\vdash :P=\dot{\Lambda} .\supset .P_{\varsigma }=\dot{\Lambda}\)</p> + +<p class="nind"><b>*213·32.</b> \(\vdash :P\in 2_{r}.\supset .P_{\varsigma }=\dot{\Lambda} \downarrow P.P_{\varsigma }\in 2_{r}\)</p> + +<p>We then have three propositions (<a href="#*213·4">*213·4</a><a href="#*213·41">·41</a><a href="#*213·42">·42</a>) showing that a sectional +relation of \(P\) is one which becomes \(P\) by being added to. We +proceed to a set of propositions (<a href="#*213·5">*213·5</a>—<a href="#*213·58">·58</a>) on \((P\unicode{x21f8}x)_{\varsigma }\), +and \((P\unicode{x2909}Q)_{\varsigma }\), leading to</p> + +<p class="nind"><b>*213·57.</b> \(\vdash :P_{\text{po}}\,\unicode{x2abd}\, J.\text{Nr}ʻQ=\text{Nr}ʻP\dot{+} \dot{1} .\supset .\text{Nr}ʻQ_{\varsigma }=\text{Nr}ʻP_{\varsigma }\dot{+} + \dot{1}\)</p> + +<p class="nind"><b>*213·58.</b> \[\begin{align}\vdash :P_{\text{po}}\,\unicode{x2abd}\, J.Q_{\text{po}}\in \text{Ser}.CʻP\cap &CʻQ=\Lambda .\supset .\\ +&\text{Nr}ʻ(P\unicode{x2909}Q)_{\varsigma }=\text{Nr}ʻP_{\varsigma }\dot{+} \text{Nr}ʻQ_{\varsigma }\end{align}\]</p> + +<p><span class="pagenum" id="Page_670">[Pg 670]</span></p> + +<hr class="tb"> + +<p class="nind"><b>*213·01.</b> \(P_{\varsigma }=P\unicode{x0294f}^{;}(\varsigma ʻP_{*})\unicode{x0294f}(-\iota ʻ\Lambda ) \quad\text{Df}\)</p> + +<p class="nind"><b>*213·1.</b> \[\begin{align}\vdash :QP_{\varsigma }&R.\equiv .\\ +&(\exists \alpha ,\beta ).\alpha ,\beta \in \text{sect}ʻP-\iota ʻ\Lambda .\exists !\beta -\alpha .Q=P\unicode{x0294f}\alpha .R=P\unicode{x0294f}\beta \\ +&[*212·12·121.(*213·01)]\end{align}\]</p> + +<p class="nind"><b>*213·11.</b> \[\begin{align}\vdash \colon\ldotp &P_{\text{po}}\in \text{connex} .\supset :QP_{\varsigma }R.\equiv .\\ +&(\exists \alpha ,\beta ).\alpha ,\beta \in \text{sect}ʻP-\iota ʻ\Lambda .\alpha \subset \beta .\alpha \neq \beta .Q=P\unicode{x0294f}\alpha .R=P\unicode{x0294f}\beta \\ +&[*213·1.*211·6·17.*210·1]\end{align}\]</p> + +<p class="nind"><b>*213·12.</b> \(\vdash .\text{D}ʻ(\varsigma ʻP_{*})\unicode{x0294f}(-\iota ʻ\Lambda )=\text{sect}ʻP-\iota ʻ\Lambda -\iota ʻCʻP\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*212·12. \supset \vdash :\alpha \in \text{D}ʻ(\varsigma ʻP_{*})\unicode{x0294f}(-\iota + ʻ\Lambda ).&\equiv .(\exists \beta ).\beta \in \text{sect}ʻP.\exists !\beta -\alpha .\alpha \neq \Lambda .\\ +[*212·12] &\equiv .\alpha \neq \Lambda .\in \text{D}ʻ\varsigma ʻP_{*}.\\ +[*212·171] &\equiv .\alpha \in \text{sect}ʻP-\iota ʻ\Lambda -\iota ʻCʻP:\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*213·121.</b> \(\vdash :P_{\text{po}}\in \text{Ser}.\supset .\overrightarrow{B}ʻ(\varsigma ʻP_{*})\unicode{x0294f}(-\iota + ʻ\Lambda )=\text{sect}ʻP\cap 1=\iota ʻʻ\overrightarrow{B}ʻP\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*212·12.*213·12.&\supset \vdash \colon\ldotp \beta \in \overrightarrow{B}ʻ(\varsigma ʻP_{*})\unicode{x0294f}(-\iota ʻ\Lambda ).\equiv :\\ +&\beta \in \text{sect}ʻP-\iota ʻ\Lambda -\iota ʻCʻP:\alpha \in \text{sect}ʻP.\exists !\beta -\alpha .\supset _{\alpha }.\alpha =\Lambda &\qquad \text{(1)}\\ +\vdash .*211·3·13·1.*37·18.\supset \\ +\vdash :\beta \in \text{sect}ʻP.x\in \beta .&\supset .\overrightarrow{P}_{*}ʻx\in \text{sect}ʻP.\overrightarrow{P}_{*}ʻx\subset \beta .\exists !\overrightarrow{P}_{*}ʻx + &\qquad \text{(2)}\\ +\vdash .(1).\text{Transp}.(2). &\supset \vdash \colon\ldotp \beta \in \overrightarrow{B}ʻ(\varsigma ʻP_{*})\unicode{x0294f}(-\iota ʻ\Lambda ).\supset :\\ +&\beta \in \text{sect}ʻP-\iota ʻ\Lambda -\iota ʻCʻP:x\in \beta .\supset _{x}.\overrightarrow{P}_{*}ʻx=\beta &\qquad \text{(3)}\\ +\vdash .*200·391.\supset \vdash \colon\ldotp \text{Hp}.&\beta \in \text{sect}ʻP-\iota ʻ\Lambda :x\in \beta .\supset _{x}.\overrightarrow{P}_{*}ʻx=\beta .\supset :\\ +&\beta \in \text{sect}ʻP-\iota ʻ\Lambda :x,y\in \beta .\supset _{x,y}.x=y:\\ +[*52·16] &\supset :\beta \in \text{sect}ʻP\cap 1 &\qquad \text{(4)}\\ +\vdash .(3).(4). &\supset \vdash :\text{Hp}.\supset .\overrightarrow{B} ʻ(\varsigma ʻP_{*})\unicode{x0294f}(-\iota ʻ\Lambda )\subset \text{sect}ʻP\cap 1 &\qquad \text{(5)}\\ +\vdash .*213·12.*200·12. &\supset \vdash :\text{Hp}.\supset .\text{sect}ʻP\cap 1\subset \text{D}ʻ(\varsigma ʻP_{*})\unicode{x0294f}(-\iota ʻ\Lambda ) &\qquad \text{(6)}\\ +\vdash .*51·401. &\supset \vdash \colon\ldotp \beta \in \text{sect}ʻP\cap 1.\supset :\alpha \subset \beta .\alpha \neq \beta .\supset .\alpha =\Lambda &\qquad \text{(7)}\\ +\vdash .(7).*212·22·121. &\supset \vdash :\text{Hp}.\supset .\text{sect}ʻP\cap 1\subset -\text{ᗡ}ʻ(\varsigma ʻP_{*})\unicode{x0294f}(-\iota ʻ\Lambda ) &\qquad \text{(8)}\\ +\vdash .(5).(6).(8).*211·18.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*213·122.</b> \[\begin{align}&\vdash :P_{\text{po}}\in \text{Ser}.\exists !\overrightarrow{B}ʻP.\text{D}.Bʻ(\varsigma ʻP)\unicode{x0294f}(-\iota ʻ\Lambda )=\iota ʻBʻP\\ +&[*213·121.*211·181]\end{align}\]</p> + +<p class="nind"><b>*213·123.</b> \[\begin{align}&\vdash :P_{\text{po}}\in + \text{Ser}.\overrightarrow{B}ʻP=\Lambda .\supset .\overrightarrow{B}ʻ(\varsigma ʻP)\unicode{x0294f}(-\iota ʻ\Lambda )=\Lambda \\ +&[*213·121]\end{align}\]</p> + +<p class="nind"><b>*213·124.</b> \[\begin{align}&\vdash \colon\ldotp P_{\text{po}}\in \text{Ser}.\supset :\text{E}!Bʻ(\varsigma ʻP)\unicode{x0294f}(-\iota ʻ\Lambda ).\equiv .\text{E}!BʻP\\ +&[*213·122·123]\end{align}\]</p> + +<p><span class="pagenum" id="Page_671">[Pg 671]</span></p> + +<p class="nind"><b>*213·125.</b> \(\vdash :P_{\text{po}}\,\unicode{x2abd}\, J.\supset .Cʻ\varsigma ʻP_{*}-\iota ʻ\Lambda {\sim}\in 1\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*212·17. \supset \vdash :P=\dot{\Lambda} .&\supset .Cʻ\varsigma ʻP_{*}=\Lambda .\\ +[*52·21] &\supset .Cʻ\varsigma ʻP_{*}-\iota ʻ\Lambda {\sim} \in 1 &\qquad \text{(1)}\\ +\vdash .*212·172.&\supset \vdash :\dot{\exists} !P.\supset .Cʻ\varsigma ʻP_{*}=\text{sect}ʻP.CʻP\in Cʻ\varsigma ʻP_{*}-\iota ʻ\Lambda &\qquad \text{(2)}\\ +\vdash .*211·13·3.*200·39.&\supset \vdash :\text{Hp}.x\in \text{D}ʻP.\supset .\overrightarrow{P}_{*}ʻx\in \text{sect}ʻP.\exists !CʻP-\overrightarrow{P}_{*}ʻx + &\qquad \text{(3)}\\ +\vdash .(2).(3). &\supset \vdash :\text{Hp}.\dot{\exists} !P.\supset .Cʻ\varsigma ʻP_{*}-\iota ʻ\Lambda {\sim}\in 1 &\qquad \text{(4)}\\ +\vdash .(1).(4).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p>The hypothesis \(P_{\text{po}}\,\unicode{x2abd}\, J\), in the above +proposition, restricts \(P\) more than is necessary for the truth +of the conclusion. What we really require is \(P = \dot{\Lambda}.\lor.(\exists x).x\in CʻP .\overrightarrow{P}_{*}ʻx\neq CʻP\), +<i>i.e.</i> \(\overrightarrow{P}_{*}ʻʻCʻP\neq \iota ʻCʻP\). This holds +if either (1) the field of \(P\) does not consist of a single family, +or (2) there is a member of \(CʻP\) which does not have the relation +\(P_{\text{po}}\) to itself. Thus the only case excluded is that of a +single cyclic family. The hypothesis \(\overrightarrow{P}_{*}ʻʻCʻP\neq\iota ʻCʻP\) +may be substituted for \(P_{\text{po}}\,\unicode{x2abd}\,J\) in most of the +subsequent propositions of this number in which +\(P_{\text{po}}\,\unicode{x2abd}\, J\) occurs in the hypothesis. We have, +however, preferred the hypothesis \(P_{\text{po}}\,\unicode{x2abd}\,J\), +as it gives a more immediate application to the case of \(P\in\text{Ser}\), +which is the case in which the propositions of the present number are +important.</p> + +<p class="nind"><b>*213·126.</b> \(\vdash :P_{\text{po}}\,\unicode{x2abd}\, J.\dot{\exists} !P.\supset .\exists !\text{sect}ʻP-\iota ʻ\Lambda -\iota ʻCʻP\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*213·125.*212·172.&\supset \vdash :\text{Hp}.\supset .\text{sect}ʻP-\iota ʻ\Lambda {\sim}\in 1 &\qquad \text{(1)}\\ +\vdash .*211·26.*33*·24. &\supset \vdash :\text{Hp}.\supset .CʻP\in \text{sect}ʻP-\iota ʻ\Lambda &\qquad \text{(2)}\\ +\vdash .(1).(2).*52·181.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*213·13.</b> \(\vdash :P_{\text{po}}\,\unicode{x2abd}\, J.\supset .Cʻ(\varsigma ʻP_{*})\unicode{x0294f}(-\iota ʻ\Lambda )=\text{sect}ʻP-\iota ʻ\Lambda\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash *213·125.\supset \\ +\vdash \colon\colon\text{Hp}.&\supset \colon\ldotp \alpha \in \text{sect}ʻP-\iota ʻ\Lambda .\supset :(\exists \beta ):\beta \in \text{sect}ʻP-\iota ʻ\Lambda :\exists !\alpha -\beta .\lor.\exists !\beta -\alpha :\\ +[*212·12] &\supset :(\exists \beta ):\alpha \{(\varsigma ʻP_{*})\unicode{x0294f}(-\iota ʻ\Lambda )\}\beta .\lor.\beta \{(\varsigma ʻP_{*})\unicode{x0294f}(-\iota + ʻ\Lambda )\}\alpha :\\ +[*33·132] &\supset :\alpha \in Cʻ(\varsigma ʻP_{*})\unicode{x0294f}(-\iota ʻ\Lambda ) &\qquad \text{(1)}\\ +\vdash .(1).*212·172. &\supset \vdash :\text{Hp}.\dot{\exists} !P.\supset .Cʻ(\varsigma ʻP_{*})\unicode{x0294f}(-\iota ʻ\Lambda )=\text{sect}ʻP-\iota ʻ\Lambda &\qquad \text{(2)}\\ +\vdash .*212·17.*211·1.&\supset \vdash :P=\dot{\Lambda} .\supset .Cʻ(\varsigma ʻP_{*})\unicode{x0294f}(-\iota ʻ\Lambda )=\text{sect}ʻP-\iota ʻ\Lambda &\qquad \text{(3)}\\ +\vdash .(2).(3).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*213·131.</b> \[\begin{align}&\vdash :P_{\text{po}}\in \text{Ser}.\supset .\text{ᗡ}ʻ(\varsigma ʻP_{*})\unicode{x0294f}(-\iota + ʻ\Lambda )=\text{sect}ʻP-\iota ʻ\Lambda -\iota ʻʻ\overrightarrow{B}ʻP\\ +&[*213·13·121]\end{align}\]</p> + +<p class="nind"><b>*213·132.</b> \[\begin{align}&\vdash :P_{\text{po}}\in \text{Ser}.\exists !\overrightarrow{B}ʻP.\supset .\text{ᗡ}ʻ(\varsigma ʻP_{*})\unicode{x0294f}(-\iota + ʻ\Lambda )=\text{sect}ʻP-\iota ʻ\Lambda -\iota ʻ\iota ʻBʻP\\ +&[*213·13·122]\end{align}\]</p> + +<p><span class="pagenum" id="Page_672">[Pg 672]</span></p> + +<p class="nind"><b>*213·133.</b> \[\begin{align}&\vdash :P_{\text{po}}\in \text{Ser}.\overrightarrow{B}ʻP=\Lambda .\supset .\text{ᗡ}ʻ(\varsigma ʻP_{*})\unicode{x0294f}(-\iota + ʻ\Lambda )=\text{sect}ʻP-\iota ʻ\Lambda\\ +&[*213·13·123]\end{align}\]</p> + +<p class="nind"><b>*213·134.</b> \(\vdash :P_{\text{po}}\,\unicode{x2abd}\, J.\dot{\exists} !P.\supset .Bʻ\text{Cnv}ʻ(\varsigma ʻP_{*})\unicode{x0294f}(-\iota ʻ\Lambda )=CʻP \quad[*213·12·13]\)</p> + +<p class="nind"><b>*213·14.</b> \[\begin{align}\vdash .&\text{D}ʻP_{\varsigma }=P\unicode{x0294f}ʻʻ\text{D}ʻ(\varsigma ʻP_{*})\unicode{x0294f}(-\iota ʻ\Lambda ).\text{ᗡ}ʻP_{\varsigma }=P\unicode{x0294f}ʻʻ\text{ᗡ}ʻ(\varsigma + ʻP_{*})\unicode{x0294f}(-\iota ʻ\Lambda ).\\ +&CʻP_{\varsigma }=P\unicode{x0294f}ʻʻCʻ(\varsigma ʻP_{*})\unicode{x0294f}(-\iota ʻ\Lambda )\\ +&[*150·21·211·22]\end{align}\]</p> + +<p class="nind"><b>*213·141.</b> \[\begin{align}&\vdash .\text{D}ʻP_{\varsigma }=P\unicode{x0294f}ʻʻ(\text{sect}ʻP-\iota ʻ\Lambda -\iota ʻCʻP)\\ +&[*213·12·14]\end{align}\]</p> + +<p class="nind"><b>*213·142.</b> \[\begin{align}&\vdash :P_{\text{po}}\,\unicode{x2abd}\, J.\supset .CʻP_{\varsigma }=P\unicode{x0294f}ʻʻ(\text{sect}ʻP-\iota ʻ\Lambda )\\ +&[*213·13·14]\end{align}\]</p> + +<p class="nind"><b>*213·143.</b> \[\begin{align}&\vdash :P_{\text{po}}\in \text{Ser}.\supset .\text{ᗡ}ʻP_{\varsigma }=P\unicode{x0294f}ʻʻ(\text{sect}ʻP-\iota + ʻ\Lambda -\iotaʻʻ\overrightarrow{B}ʻP)\\ +&[*213·131·14]\end{align}\]</p> + +<p class="nind"><b>*213·144.</b> \[\begin{align}&\vdash :P_{\text{po}}\in \text{Ser}.\exists !\overrightarrow{B}ʻP.\supset .\text{ᗡ}ʻP_{\varsigma }=P\unicode{x0294f}ʻʻ(\text{sect}ʻP-\iota + ʻ\Lambda -\iota ʻ\iota ʻBʻP)\\ +&[*213·132·14]\end{align}\]</p> + +<p class="nind"><b>*213·145.</b> \[\begin{align}&\vdash :P_{\text{po}}\in \text{Ser}.\overrightarrow{B}ʻP=\Lambda .\supset .\text{ᗡ}ʻP_{\varsigma }=P\unicode{x0294f}ʻʻ(\text{sect}ʻP-\iota + ʻ\Lambda )\\ +&[*213·143]\end{align}\]</p> + +<p class="nind"><b>*213·146.</b> \(\vdash :P\,\unicode{x2abd}\, J.\supset .P\unicode{x0294f}ʻʻ\text{sect}ʻP=P\unicode{x0294f}ʻʻ(\text{sect}ʻP-1)\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*37·22. &\supset \vdash .P\unicode{x0294f}ʻʻ\text{sect}ʻP=P\unicode{x0294f}ʻʻ(\text{sect}ʻP-1)\cup P\unicode{x0294f}ʻʻ(\text{sect}ʻP\cap 1) &\qquad \text{(1)}\\ +\vdash .*200·35.\supset \vdash :Q\in P\unicode{x0294f}ʻʻ(\text{sect}ʻP\cap 1).&\supset .Q=\dot{\Lambda} .\\ +[*36·27] &\supset .Q=P\unicode{x0294f}\Lambda .\\ +[*211·44] &\supset .Q\in P\unicode{x0294f}ʻʻ(\text{sect}ʻP-1) &\qquad \text{(2)}\\ +\vdash .(1).(2).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*213·15.</b> \(\vdash \colon\ldotp P_{\text{po}}\in \text{Ser}.\alpha \in \text{sect}ʻP-\iota ʻ\Lambda .\supset :P\unicode{x0294f}\alpha =\dot{\Lambda} .\equiv .\alpha \in 1\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*200·35.&\supset \vdash :\text{Hp}.\alpha \in 1.\supset .P\unicode{x0294f}\alpha =\dot{\Lambda} &\qquad \text{(1)}\\ +\vdash .*52·41. \supset \vdash \colon\ldotp \text{Hp}.\alpha {\sim}\in 1.&\supset :(\exists x,y).x,y\in \alpha .x\neq y:\\ +[*211·1.*202·103] &\supset :(\exists x,y):x(P_{\text{po}}\unicode{x0294f}\alpha )y.\lor.y(P_{\text{po}}\unicode{x0294f}\alpha )x:\\ +[*11·7] &\supset :\dot{\exists} !P_{\text{po}}\unicode{x0294f}\alpha :\\ +[*37·41] &\supset :\exists !\alpha \cap P_{\text{po}}ʻʻ\alpha :\\ +[*211·131] &\supset :\exists !\alpha \cap Pʻʻ\alpha :\\ +[*37·41] &\supset :\dot{\exists} !P\unicode{x0294f}\alpha &\qquad \text{(2)}\\ +\vdash .(1).(2).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_673">[Pg 673]</span></p> + +<p class="nind"><b><a id="*213·151">*213·151</a>.</b> \(\vdash :P_{\text{po}}\in \text{Ser}.\supset .(P\unicode{x0294f})\upharpoonright (\text{sect}ʻP-{℩}ʻ\Lambda )\in 1\rightarrow 1\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*213·15.\supset \\ +\vdash :\text{Hp}.\alpha \in \text{sect}ʻP-{℩}ʻ\Lambda -1.\beta \in \text{sect}ʻP-{℩}ʻ&\Lambda .P\unicode{x0294f}\alpha =P\unicode{x0294f}\beta .\supset .\beta {\sim}\in 1.\\ +[*211·133] & \supset .CʻP\unicode{x0294f}\beta =\beta .CʻP\unicode{x0294f}\alpha =\alpha .\\ +[\text{Hp}] &\supset .\alpha =\beta &\qquad \text{(1)}\\ +\vdash .*213·15.\supset \\ +\vdash :\text{Hp}.&\alpha \in \text{sect}ʻP\cap 1.\beta \in \text{sect}ʻP-{℩}ʻ\Lambda .P\unicode{x0294f}\alpha =P\unicode{x0294f}\beta .\supset .\beta \in 1 &\qquad \text{(2)}\\ +\vdash .(2).*211·18.\supset \vdash :\text{Hp}(2).&\supset .\alpha ,\beta \in {℩}ʻʻ\overrightarrow{B}ʻP.\\ +[*202·523·13] & \supset .\alpha =\beta &\qquad \text{(3)}\\ +\vdash .(1).(3).\supset \vdash \colon\ldotp \text{Hp}.&\supset :\alpha ,\beta \in \text{sect}ʻP-{℩}ʻ\Lambda .P\unicode{x0294f}\alpha =P\unicode{x0294f}\beta .\supset .\alpha =\beta \colon\ldotp \\ +&\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*213·152">*213·152</a>.</b> \(\vdash :P_{\text{po}}\in \text{Ser}.\supset .P_{\varsigma }\,\text{smor}\,(\varsigma ʻP_{*})\unicode{x0294f}(-{℩}ʻ\Lambda ) \quad[*213·151·13]\)</p> + +<p class="nind"><b><a id="*213·153">*213·153</a>.</b> \(\vdash :P_{\text{po}}\in \text{Ser}.\supset .P_{\varsigma }\in \text{Ser} \quad[*213·152.*212·3.*204·4·21]\)</p> + +<p class="nind"><b>*213·154.</b> \(\vdash :P_{\text{po}}\in \text{Ser}.\supset .\overrightarrow{B}ʻP_{\varsigma }=P\unicode{x0294f}ʻʻ{℩}ʻʻ\overrightarrow{B}ʻP \quad[*213·151·121.*151·5]\)</p> + +<p class="nind"><b><a id="*213·155">*213·155</a>.</b> \(\vdash :P_{\text{po}}\in \text{Ser}.\exists !\overrightarrow{B}ʻP.\supset .BʻP_{\varsigma }=\dot{\Lambda}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*213·151·122.*151·5.\supset \\ +\vdash :\text{Hp}.\supset .BʻP_{\varsigma }&=P\unicode{x0294f}({℩}ʻBʻP)\\ +[*200·35] &=\dot{\Lambda} :\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*213·156.</b> \(\vdash :P_{\text{po}}\in \text{Ser}.\overrightarrow{B}ʻP=\Lambda .\supset .\overrightarrow{B}ʻP_{\varsigma }=\Lambda \quad[*213·154]\)</p> + +<p class="nind"><b>*213·157.</b> \(\vdash \colon\ldotp P_{\text{po}}\in \text{Ser}.\supset :\text{E}!BʻP.\equiv .\text{E}!BʻP_{\varsigma } \quad[*213·155·156]\)</p> + +<p class="nind"><b><a id="*213·158">*213·158</a>.</b> \(\vdash :P_{\text{po}}\in \text{Ser}.\dot{\exists} !P.\supset .Bʻ\breve{P} _{\varsigma }=P\)</p> + +<p><i>Dem.</i> +\[ +\vdash .*213·151·134.*151·5.\supset \vdash :\text{Hp}.\supset .Bʻ\breve{P} _{\varsigma }=P\unicode{x0294f}CʻP:\supset \vdash .\text{Prop} +\]</p> + +<p class="nind"><b>*213·16.</b> \(\vdash .\text{D}ʻP_{\varsigma }=P\unicode{x0294f}ʻʻ(\text{sect}ʻP-{℩}ʻ\Lambda )-{℩}ʻP\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*213·141.\supset \\ +\vdash :Q\in \text{D}ʻP_{\varsigma }.&\equiv .(\exists \alpha ).\alpha \in \text{sect}ʻP-{℩}ʻ\Lambda .Q=P\unicode{x0294f}\alpha .\alpha \neq CʻP &\qquad \text{(1)}\\ +\vdash .*211·1.\supset \vdash \colon\ldotp \alpha \in \text{sect}ʻP.Q=P\unicode{x0294f}\alpha .\supset :\alpha \neq CʻP.&\equiv .\exists !CʻP-\alpha .\\ +[*36·25.\text{Transp}] &\equiv .Q\neq P &\qquad \text{(2)}\\ +\vdash .(1).(2).\supset \vdash :Q\in \text{D}ʻP_{\varsigma }.&\equiv .(\exists \alpha ).\alpha \in \text{sect}ʻP-{℩}ʻ\Lambda .Q=P\unicode{x0294f}\alpha .Q\neq P:\\ +&\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_674">[Pg 674]</span></p> + +<p class="nind"><b>*213·161.</b> \(\vdash :P_{\text{po}}\,\unicode{x2abd}\, + J.\exists !\overrightarrow{B}ʻP.\supset .P\unicode{x0294f}ʻʻ\text{sect}ʻP=P\unicode{x0294f}ʻʻ(\text{sect}ʻP-\iota ʻ\Lambda )=CʻP_{\varsigma }\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*211·18.\supset \vdash :\text{Hp}.&\supset .\iotaʻʻ\overrightarrow{B}ʻP\subset \text{sect}ʻP\cap 1.\exists !\iotaʻʻ\overrightarrow{B}ʻP.\\ +[*37·2·45] &\supset .P\unicode{x0294f}ʻʻ\iotaʻʻ\overrightarrow{B}ʻP\subset P\unicode{x0294f}ʻʻ(\text{sect}ʻP-\iota ʻ\Lambda ).\exists !P\unicode{x0294f}ʻʻ\iotaʻʻ\overrightarrow{B}ʻP.\\ +[*200·35] &\supset .\dot{\Lambda} \in P\unicode{x0294f}ʻʻ(\text{sect}ʻP-\iota ʻ\Lambda ).\\ +[*36·27] &\supset .P\unicode{x0294f}\Lambda \in P\unicode{x0294f}ʻʻ(\text{sect}ʻP-\iota ʻ\Lambda ).\\ +[*37·22] \supset .P\unicode{x0294f}ʻʻ\text{sect}ʻP&=P\unicode{x0294f}ʻʻ(\text{sect}ʻP-\iota ʻ\Lambda )\\ +[*213·142] &=CʻP_{\varsigma }:\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*213·162.</b> \(\vdash :P_{\text{po}}\in \text{Ser}.\supset .\text{ᗡ}ʻP_{\varsigma }=P\unicode{x0294f}ʻʻ\text{sect}ʻP-\iota ʻ\dot{\Lambda}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*213·143.\supset \vdash \colon\ldotp \text{Hp}.&\supset :Q\in \text{ᗡ}ʻP_{\varsigma }.\equiv .\\ +&(\exists \alpha ).\alpha \in \text{sect}ʻP-\iota ʻ\Lambda -\iotaʻʻ\overrightarrow{B}ʻP.Q=P\unicode{x0294f}\alpha . &\qquad \text{(1)}\\ +[*213·15.*211·18]&\supset .Q\in P\unicode{x0294f}ʻʻ\text{sect}ʻP-\iota ʻ\dot{\Lambda} &\qquad \text{(2)}\\ +\vdash .*213·15. \supset \vdash \colon\ldotp &\text{Hp}.\supset :Q\in P\unicode{x0294f}ʻʻ\text{sect}ʻP-\iota ʻ\dot{\Lambda} .\supset .\\ +&(\exists \alpha ).\alpha \in \text{sect}ʻP-\iota ʻ\Lambda -\iotaʻʻ\overrightarrow{B}ʻP.Q=P\unicode{x0294f}\alpha .\\ +[(1)] & \supset .Q\in \text{ᗡ}ʻP_{\varsigma } &\qquad \text{(3)}\\ +\vdash .(2).(3).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*213·163.</b> \(\vdash :P_{\text{po}}\in \text{Ser}.\overrightarrow{B}ʻP=\Lambda .\supset .CʻP_{\varsigma }=P\unicode{x0294f}ʻʻ\text{sect}ʻP_\iota ʻ\dot{\Lambda} \)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*213·156.\supset \vdash :\text{Hp}.\supset .CʻP_{\varsigma }=\text{ᗡ}ʻP_{\varsigma } &\qquad \text{(1)}\\ +\vdash .(1).*213·162.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*213·164.</b> \[\begin{align}&\vdash :P_{\text{po}}\in \text{Ser}.\overrightarrow{B}ʻP=\Lambda .\supset .\text{D}ʻP_{\varsigma }=P\unicode{x0294f}ʻʻ\text{sect}ʻP-\iota + ʻ\dot{\Lambda} -\iota ʻP\\ +&[*213·142·163·16]\end{align}\]</p> + +<p class="nind"><b>*213·17.</b> \[\begin{align}\vdash :P_{\text{po}}\in \text{Ser}.\supset .\text{Nr}ʻ\varsigma ʻP_{*}=\dot{1} \dot{+} &\text{Nr}ʻP_{\varsigma }.\\ +&\text{Nr}ʻ(\varsigma P_{*})\unicode{x0294f}(\text{ᗡ}ʻ\varsigma ʻP_{*})=\text{Nr}ʻP_{\varsigma }\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*212·171·172. \supset \vdash :\dot{\exists} !P.&\supset .Bʻ\varsigma ʻP_{*}=\Lambda .\text{ᗡ}ʻ\varsigma ʻP_{*}=\text{sect}ʻP-\iota ʻ\Lambda .\\ +&(\varsigma ʻP_{*})\unicode{x0294f}(-\iota ʻ\Lambda )=(\varsigma ʻP_{*})\unicode{x0294f}(\text{ᗡ}ʻ\varsigma ʻP_{*}) &\qquad \text{(1)}\\ +\vdash .(1).*213·125. &\supset \vdash :\text{Hp}.\dot{\exists} !P.\supset .\text{ᗡ}ʻ\varsigma ʻP_{*}{\sim}\in 1 &\qquad \text{(2)}\\ +\vdash .*212·3.*91·602.&\supset \vdash :\text{Hp}.\supset .\varsigma ʻP_{*}\in \text{connex} &\qquad \text{(3)}\\ +\vdash .(1).*213·152. &\supset \vdash :\text{Hp}.\dot{\exists} !P.\supset .\text{Nr}ʻ(\varsigma ʻP_{*})\unicode{x0294f}(\text{ᗡ}ʻ\varsigma ʻP_{*})=\text{Nr}ʻP_{\varsigma } + &\qquad \text{(4)}\\ +\vdash .(1).(2).(3).*204·46.\supset \\ +\vdash :\text{Hp}.\dot{\exists} !P.\supset .\text{Nr}ʻ\varsigma ʻP_{*}&=\dot{1} \dot{+} \text{Nr}ʻ(\varsigma ʻP_{*})\unicode{x0294f}(\text{ᗡ}ʻ\varsigma ʻP_{*})\\ +[(4)] &=\dot{1} \dot{+} \text{Nr}ʻP_{\varsigma } &\qquad \text{(5)}\\ +\vdash .*212·17.*150·42.&\supset \vdash :P=\dot{\Lambda} .\supset .\varsigma ʻP_{*}=\dot{\Lambda} .P_{\varsigma }=\dot{\Lambda} .\\ +[*161·201]&\supset .\text{Nr}ʻ\varsigma ʻP_{*}=\dot{1} \dot{+} \text{Nr}ʻP_{\varsigma }.\text{Nr}ʻ(\varsigma ʻP_{*})\unicode{x0294f}(\text{ᗡ}ʻ\varsigma + ʻP_{*})=\text{Nr}ʻP_{\varsigma } &\qquad \text{(6)}\\ +\vdash .(4).(5).(6).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_675">[Pg 675]</span></p> + +<p class="nind"><b>*213·171.</b> \(\vdash \colon\ldotp P_{\text{po}},Q_{\text{po}}\in \text{Ser}.\supset :P_{\varsigma }\,\text{smor}\, Q_{\varsigma }. \equiv .\varsigma ʻP_{*} + \,\text{smor}\, \varsigma ʻQ_{*}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*212·172.&\supset \vdash \colon\ldotp \text{Hp}.\dot{\exists} !P.\dot{\exists} !Q.\supset :\text{E}!Bʻ\varsigma ʻP_{*}.\text{E}!Bʻ\varsigma ʻQ_{*}:\\ +[*204·47.*91·602.*212·3] +\supset :\varsigma ʻP_{*} \,\text{smor}\, \varsigma ʻQ_{*}. &\equiv .(\varsigma ʻP_{*})\unicode{x0294f}(\text{ᗡ}ʻ\varsigma ʻP_{*})\,\text{smor}\,(\varsigma ʻQ_{*})\unicode{x0294f}(\text{ᗡ}ʻ\varsigma + ʻQ_{*}).\\ +[*213·17] &\equiv .P_{\varsigma } \,\text{smor}\, Q_{\varsigma } &\qquad \text{(1)}\\ +\vdash .*213·158. \supset \vdash :\text{Hp}.\dot{\exists} !P.P_{\varsigma } \,\text{smor}\, Q_{\varsigma }.&\supset .\dot{\exists} !Q_{\varsigma }.\\ +[*212·17.*150·42] &\supset .\dot{\exists} !Q &\qquad \text{(2)}\\ +\vdash .*212·17. &\supset \vdash :\text{Hp}.\dot{\exists} !P.\varsigma ʻP_{*} \,\text{smor}\, \varsigma ʻQ_{*}.\supset .\dot{\exists} !Q &\qquad \text{(3)}\\ +\vdash .(1).(2).(3).&\supset \vdash \colon\ldotp \text{Hp}.\dot{\exists} !P.\supset :\varsigma ʻP_{*} \,\text{smor}\, \varsigma ʻQ_{*}. \equiv .P_{\varsigma} \,\text{smor}\, + Q_{\varsigma } &\qquad \text{(4)}\\ +\vdash .*212·17. \supset \vdash :\text{Hp}.P = \dot{\Lambda} .\varsigma ʻP_{*} \,\text{smor}\, \varsigma ʻQ_{*}.&\supset .\varsigma ʻP_{*} + = \dot{\Lambda} .\varsigma ʻQ_{*} = \dot{\Lambda} .\\ +[*150·42] &\supset .P_{\varsigma } = \dot{\Lambda} .Q_{\varsigma } = \dot{\Lambda} &\qquad \text{(5)}\\ +\vdash .*213·17. &\supset \vdash :\text{Hp}.P_{\varsigma } \,\text{smor}\, Q_{\varsigma }.\supset .\varsigma ʻP_{*} \,\text{smor}\, \varsigma ʻQ_{*} + &\qquad \text{(6)}\\ +\vdash .(5).(6). &\supset \vdash \colon\ldotp \text{Hp}.P = \dot{\Lambda} .\supset :\varsigma ʻP_{*} \,\text{smor}\, \varsigma ʻQ_{*}. \equiv .P_{\varsigma } + \,\text{smor}\, Q_{\varsigma } &\qquad \text{(7)}\\ +\vdash .(4).(7). &\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*213·172.</b> \(\vdash :P_{\text{po}},Q_{\text{po}}\in \text{Ser}.P \,\,\text{smor}\, \,Q.\supset .P_{\varsigma } \,\text{smor}\, Q_{\varsigma } + \quad[*212·72.*213·171]\)</p> + +<p class="nind"><b>*213·18.</b> \(\vdash :P\in \text{connex} .R\in \text{D}ʻP_{\varsigma }.\supset .\exists !CʻP\cap pʻ\overleftarrow{P}ʻʻCʻR\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*213·1.\supset \vdash :R\in \text{D}ʻP_{\varsigma }.&\supset .(\exists \alpha ).\alpha \in \text{sect}ʻP-\iota ʻCʻP.R = P\unicode{x0294f}\alpha .\\ +[*37·41] &\supset .(\exists \alpha ).\alpha \in \text{sect}ʻP-\iota ʻCʻP.CʻR\subset \alpha .\\ +[*40·16] &\supset .(\exists \alpha ).\alpha \in \text{sect}ʻP-\iota ʻCʻP.pʻ\overleftarrow{P}ʻʻ\alpha \subset pʻ\overleftarrow{P}ʻʻCʻR &\qquad \text{(1)}\\ +\vdash .*211·703.&\supset \vdash :\text{Hp}.\alpha \in \text{sect}ʻP-\iota ʻCʻP.\supset .\exists !pʻ\overleftarrow{P}ʻʻ\alpha &\qquad \text{(2)}\\ +\vdash .*211·1. &\supset \vdash :\alpha \in \text{sect}ʻP-\iota ʻCʻP.\supset .\exists !CʻP-\alpha .\\ +[*33·24] &\supset .\dot{\exists} !P &\qquad \text{(3)}\\ +\vdash .(2).(3).*40·69.&\supset \vdash :\text{Hp}.\alpha \in \text{sect}ʻP-\iota ʻCʻP.\supset .\exists !CʻP\cap pʻ\overleftarrow{P}ʻʻ\alpha &\qquad \text{(4)}\\ +\vdash .(1).(4).\supset \\ +\vdash :\text{Hp}.R\in \text{D}ʻP_{\varsigma }.&\supset + .(\exists \alpha ).\exists !CʻP\cap pʻ\overleftarrow{P}ʻʻ\alpha .pʻ\overleftarrow{P}ʻʻ\alpha \subset pʻ\overleftarrow{P}ʻʻCʻR:\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*213·2">*213·2</a>.</b> \[\begin{align}\vdash \colon\ldotp &P_{\text{po}}\in + \text{Ser}.\alpha ,\beta \in \text{sect}ʻP-\iota ʻ\Lambda .Q = P\unicode{x0294f}\alpha .R = P\unicode{x0294f}\beta .\supset :\\ +&\exists !\beta -\alpha . \equiv .\dot{\exists} !R\dot{-} Q. \equiv .Q\,\unicode{x2abd}\, R.Q \neq R. \equiv .\alpha \subset \beta .\alpha \neq \beta\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*36·24. &\supset \vdash :\alpha \subset \beta .\supset .P\unicode{x0294f}\alpha \,\unicode{x2abd}\, P\unicode{x0294f}\beta &\qquad \text{(1)}\\ +\vdash .*211·133. &\supset \vdash :\text{Hp}.\alpha ,\beta {\sim}\in 1.P\unicode{x0294f}\alpha \,\unicode{x2abd}\, P\unicode{x0294f}\beta .\supset .\alpha \subset \beta&\qquad \text{(2)}\\ +\vdash .*211·181·182.\supset \vdash :\text{Hp}.\alpha \in 1.&\supset .\alpha = \iota ʻBʻP.\\ +[*202·521] &\supset .\alpha \subset \beta &\qquad \text{(3)}\\ +\vdash .*213·15. &\supset \vdash :\text{Hp}.\beta \in 1.\alpha {\sim}\in 1.\supset .{\sim}(P\unicode{x0294f}\alpha \,\unicode{x2abd}\, P\unicode{x0294f}\beta ) &\qquad \text{(4)}\\ +\vdash .(2).(3).(4). &\supset \vdash :\text{Hp}.P\unicode{x0294f}\alpha \,\unicode{x2abd}\, P\unicode{x0294f}\beta .\supset .\alpha \subset \beta &\qquad \text{(5)}\\) +\vdash .(1).(5).\supset \vdash \colon\ldotp \text{Hp}.&\supset :\alpha \subset \beta .\equiv .Q\,\unicode{x2abd}\, R: &\qquad \text{(6)}\\ +[\text{Transp}] &\supset :\exists !\alpha -\beta .\equiv .\dot{\exists} !Q\dot{-} R &\qquad \text{(7)}\\ +\vdash .(6).*213·151.\supset \\ +\vdash \colon\ldotp \text{Hp}.&\supset :\alpha \subset \beta .\alpha \neq \beta .\equiv .Q\,\unicode{x2abd}\, R.Q\neq R &\qquad \text{(8)}\\ +\vdash .(7).(8).*210·1.*211·562.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_676">[Pg 676]</span></p> + +<p class="nind"><b>*213·21.</b> \[\begin{align}\vdash \colon\ldotp P_{\text{po}}\in \text{Ser}.\supset :QP_{\varsigma }R.&\equiv + .Q,R\in P\unicode{x0294f}ʻʻ(\text{sect}ʻP-{℩}ʻ\Lambda ).\dot{\exists} !R\dot{-} Q.\\ +&\equiv .Q,R\in P\unicode{x0294f}ʻʻ(\text{sect}ʻP-{℩}ʻ\Lambda ).Q\,\unicode{x2abd}\, R.Q\neq R\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*213·1·2.\supset \vdash \colon\ldotp \text{Hp}.\supset :\\ +QP_{\varsigma }R.&\equiv .(\exists \alpha ,\beta ).\alpha ,\beta \in \text{sect}ʻP-{℩}ʻ\Lambda .Q=P\unicode{x0294f}\alpha .R=P\unicode{x0294f}\beta .\dot{\exists} !R\dot{-} Q.\\ +&\equiv .(\exists \alpha ,\beta ).\alpha ,\beta \in \text{sect}ʻP-{℩}ʻ\Lambda .Q=P\unicode{x0294f}\alpha .R=P\unicode{x0294f}\beta .Q\,\unicode{x2abd}\, R.Q\neq R &\qquad \text{(1)}\\ +\vdash .(1).*37·6.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*213·22.</b> \[\begin{align}&\vdash \colon\ldotp P_{\text{po}}\in \text{Ser}.\exists !\overrightarrow{B}ʻP.\supset :\\ +&QP_{\varsigma }R.\equiv .Q,R\in P\unicode{x0294f}ʻʻ\text{sect}ʻP.\dot{\exists} !R\dot{-} Q.\equiv .Q,R\in P\unicode{x0294f}ʻʻ\text{sect}ʻP.Q\,\unicode{x2abd}\, R.Q\neq R\\ +&[*213·21·161]\end{align}\]</p> + +<p class="nind"><b>*213·23.</b> \[\begin{align}&\vdash \colon\ldotp P_{\text{po}}\in \text{connex} .Q,R\in CʻP_{\varsigma }.\supset :Q\,\unicode{x2abd}\, R.\lor.R\,\unicode{x2abd}\, Q\\ +&[*213·1.*211·6·17.*36·24]\end{align}\]</p> + +<p class="nind"><b>*213·24.</b> \(\vdash :\beta \in \text{sect}ʻP.R=P\unicode{x0294f}\beta .\supset .\text{sect}ʻR=\text{sect}ʻP\cap \text{Cl}ʻCʻR\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*36·29.\supset \vdash \colon\ldotp \text{Hp}.&\supset :R\,\unicode{x2abd}\, P:&\qquad \text{(1)}\\ +[*211·1] &\supset :\alpha \in \text{sect}ʻP\cap \text{Cl}ʻCʻR.\supset .\alpha \subset CʻR.Rʻʻ\alpha \subset \alpha .\\ +&\supset .\alpha \in \text{sect}ʻR &\qquad \text{(2)}\\ +\vdash .(1).*211·1.\supset \\ +\vdash \colon\ldotp \text{Hp}.&\supset :\alpha \in \text{sect}ʻR.\supset .\alpha \subset CʻP.\alpha \subset CʻP.(P\unicode{x0294f}\beta )ʻʻ\alpha \subset \alpha &\qquad \text{(3)}\\ +\vdash .(3).*37·41·413.\supset \\ +\vdash :\text{Hp}.\alpha \in \text{sect}ʻR.&\supset .\alpha \subset \beta .\beta \cap Pʻʻ(\alpha \cap \beta )\subset \alpha .\\ +[*22·621.*37·2] &\supset .\beta \cap Pʻʻ\alpha \subset \alpha .Pʻʻ\alpha \subset Pʻʻ\beta .\\ +[*211·1] &\supset .\beta \cap Pʻʻ\alpha \subset \alpha .Pʻʻ\alpha \subset \beta .\\ +[*22·621] &\supset .Pʻʻ\alpha \subset \alpha &\qquad \text{(4)}\\ +\vdash .(3).(4).\supset \vdash \colon\ldotp \text{Hp}.&\supset :\alpha \in \text{sect}ʻR.\supset .\alpha \subset CʻR.\alpha \in \text{sect}ʻP &\qquad \text{(5)}\\ +\vdash .(2).(5).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_677">[Pg 677]</span></p> + +<p class="nind"><b>*213·241.</b> \(\vdash :R\in P\unicode{x0294f}ʻʻ\text{sect}ʻP.\supset .R_{\varsigma }\,\unicode{x2abd}\, P_{\varsigma }\unicode{x0294f}CʻR_{\varsigma }\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*213·1.\supset \\ +\vdash \colon\ldotp \text{Hp}.\supset :QR_{\varsigma }Qʻ.&\equiv .(\exists \alpha ,\alpha').\alpha ,\alpha'\in \text{sect}ʻR-\iota ʻ\Lambda .\\ +&Q=R\unicode{x0294f}\alpha .Qʻ=R\unicode{x0294f}\alpha'.\exists !\alpha '-\alpha .\\ +[*213·24] &\equiv .(\exists \alpha ,\alpha').\alpha ,\alpha'\in \text{sect}ʻP\cap \text{Cl}ʻCʻR-\iota ʻ\Lambda .\\ +&Q=R\unicode{x0294f}\alpha .Qʻ=R\unicode{x0294f}\alpha'.\exists !\alpha '-\alpha .\\ +[*213·1] &\supset .QP_{\varsigma }Qʻ &\qquad \text{(1)}\\ +\vdash .(1).*33·17.\supset \vdash :\text{Hp}.\supset .R_{\varsigma }\,\unicode{x2abd}\, P_{\varsigma }\unicode{x0294f}CʻR_{\varsigma }:\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*213·242">*213·242</a>.</b> \(\vdash :P_{\text{po}}\in + \text{Ser}.R\in P\unicode{x0294f}ʻʻ\text{sect}ʻP.\supset .R_{\varsigma }=P_{\varsigma }\unicode{x0294f}CʻR_{\varsigma }\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*213·1.*211·1.&\supset \vdash \colon\ldotp Q(P_{\varsigma }\unicode{x0294f}CʻR_{\varsigma })Qʻ.\supset :\\ +&(\exists \alpha , \alpha').\alpha ,\alpha'\in \text{sect}ʻP-\iota ʻ\Lambda .Q=P\unicode{x0294f}\alpha .Qʻ=P\unicode{x0294f}\alpha'.\exists !\alpha '-\alpha :\\ +&(\exists \gamma ,\gamma').\gamma ,\gamma'\in \text{sect}ʻR-\iota ʻ\Lambda .Q=R\unicode{x0294f}\gamma .Qʻ=R\unicode{x0294f}\gamma' &\qquad \text{(1)}\\ +\vdash .*213·24·151.\supset \\ +\vdash \colon\ldotp \text{Hp}.&\supset :\alpha \in \text{sect}ʻP.\gamma \in \text{sect}ʻR.Q=P\unicode{x0294f}\alpha =P\unicode{x0294f}\gamma .\supset .\alpha =\gamma &\qquad \text{(2)}\\ +\vdash .(1).(2).&\supset \vdash \colon\ldotp \text{Hp}.\supset :Q(P_{\varsigma }\unicode{x0294f}CʻR_{\varsigma })Qʻ.\supset .\\ +&(\exists \gamma ,\gamma').\gamma ,\gamma'\in \text{sect}ʻR-\iota ʻ\Lambda .Q=R\unicode{x0294f}\gamma .Qʻ=R\unicode{x0294f}\gamma'.\exists !\gamma '-\gamma . +[*213·1] \supset . QR_{\varsigma }Qʻ &\qquad \text{(3)}\\ +\vdash . (3) . *213·241 . \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*213·243.</b> \(\vdash . \overrightarrow{P}_{\varsigma }ʻP = \text{D}ʻP_{\varsigma }\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*213·1.\supset \vdash :R\in \overrightarrow{P}_{\varsigma }ʻP.&\equiv .(\exists \alpha ,\beta ).\alpha ,\beta \in \text{sect}ʻP-\iota ʻ\Lambda .\\ +&R=P\unicode{x0294f}\alpha .P=P\unicode{x0294f}\beta .\exists !\beta -\alpha &\qquad \text{(1)}\\ +\vdash .*37·41.&\supset \vdash .Cʻ(P\unicode{x0294f}\beta )\subset \beta &\qquad \text{(2)}\\ +\vdash .(2). \supset \vdash :\exists !CʻP-\beta .&\supset .\exists !CʻP-Cʻ(P\unicode{x0294f}\beta ).\\ +[*13·14] &\supset .P\neq P\unicode{x0294f}\beta &\qquad \text{(3)}\\ +\vdash .(3).\text{Transp}.&\supset \vdash :\beta \in \text{sect}ʻP.P=P\unicode{x0294f}\beta .\supset .CʻP=\beta &\qquad \text{(4)}\\ +\vdash .(1).(4).\supset \vdash :R\in \overrightarrow{P}_{\varsigma }ʻP.&\equiv + .(\exists \alpha ).\alpha \in \text{sect}ʻP-\iota ʻ\Lambda .R=P\unicode{x0294f}\alpha .\exists !CʻP-\alpha .\\ +[*211·1] &\equiv .(\exists \alpha ).\alpha \in \text{sect}ʻP-\iota ʻ\Lambda -\iota ʻCʻP.R=P\unicode{x0294f}\alpha .\\ +[*213·141] &\equiv .R\in \text{D}ʻP_{\varsigma }:\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*213·244.</b> \(\vdash :R\in CʻP_{\varsigma }.Q\in \text{D}ʻR_{\varsigma }.\supset .QP_{\varsigma }R\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*213·243.\supset \vdash \colon\ldotp R\in CʻP_{\varsigma }.\supset :Q\in \text{D}ʻR_{\varsigma }.&\supset .QR_{\varsigma }R.\\ +[*213·241] &\supset .QP_{\varsigma }R:\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_678">[Pg 678]</span></p> + +<p class="nind"><b>*213·245.</b> \(\vdash \colon\ldotp P_{\text{po}}\in \text{Ser}.\supset :QP_{\varsigma }R.\equiv .R\in CʻP_{\varsigma }.Q\in \text{D}ʻR_{\varsigma }\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*213·11.\supset \vdash \colon\ldotp \text{Hp}.\supset :\\ +QP_{\varsigma }R.&\equiv + .(\exists \alpha ,\beta ).\alpha ,\beta \in \text{sect}ʻP-\iota ʻ\Lambda .Q=P\unicode{x0294f}\alpha .R=P\unicode{x0294f}\beta .\alpha \subset \beta .\alpha \neq \beta .\\ +[*213·24] &\equiv .(\exists \alpha ,\beta ).\beta \in \text{sect}ʻP-\iota ʻ\Lambda .R=P\unicode{x0294f}\beta .\alpha \in \text{sect}ʻR-\iota ʻ\Lambda .\\ +&\alpha \subset \beta .\alpha \neq \beta .Q=P\unicode{x0294f}\alpha .\\ +[*213·142.*211·133.*36·21] +&\equiv .(\exists \alpha ).R\in CʻP_{\varsigma }.\alpha \in \text{sect}ʻR-\iota ʻ\Lambda .\alpha \subset CʻR.\alpha \neq CʻR.Q=R\unicode{x0294f}\alpha .\\ +[*213·141]&\equiv .R\in CʻP_{\varsigma }.Q\in \text{D}ʻR_{\varsigma }\colon\ldotp \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*213·246">*213·246</a>.</b> \(\vdash :P_{\text{po}}\in \text{Ser}.R\in CʻP_{\varsigma }.\supset .\overrightarrow{P}_{\varsigma }ʻR=\text{D}ʻR_{\varsigma } + \quad[*213·245]\)</p> + +<p class="nind"><b>*213·247.</b> \[\begin{align}&\vdash \colon\ldotp P_{\text{po}}\in \text{Ser}.\supset :Q(P_{\varsigma }\unicode{x0294f}\text{D}ʻP_{\varsigma })R.\equiv + .R\in \text{D}ʻP_{\varsigma }.Q\in \text{D}ʻR_{\varsigma }\\ +&[*213·245]\end{align}\]</p> + +<p class="nind"><b>*213·25.</b> \(\vdash \colon\ldotp P_{\text{po}}\in \text{Ser}.Q,R\in CʻP_{\varsigma }.\supset :Q\in \text{D}ʻR_{\varsigma }.\lor.R\in + \text{D}ʻQ_{\varsigma }.\lor.Q=R\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*213·153.\supset \vdash \colon\ldotp \text{Hp}.&\supset :QP_{\varsigma }R.\lor.RP_{\varsigma }Q.\lor.Q=R:\\ +[*213·245] &\supset :Q\in \text{D}ʻR_{\varsigma }.\lor.R\in \text{D}ʻQ_{\varsigma }.\lor.Q=R\colon\ldotp \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*213·251">*213·251</a>.</b> \[\begin{align}\vdash \colon\ldotp P_{\text{po}}\in \text{Ser}.Q,R\in CʻP_{\varsigma }.{\sim}&(Q=\dot{\Lambda} .R=\dot{\Lambda} ).\supset :\\ +&Q\in CʻR_{\varsigma }.\lor.R\in \text{D}ʻQ_{\varsigma }\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*213·158. &\supset \vdash :\text{Hp}.\dot{\exists} !R.Q=R.\supset .Q\in CʻR_{\varsigma } &\qquad \text{(1)}\\ +\vdash .(1).*13·12. &\supset \vdash :\text{Hp}.\dot{\exists} !Q.Q=R.\supset .Q\in CʻR_{\varsigma } &\qquad \text{(2)}\\ +\vdash .(1).(2). &\supset \vdash :\text{Hp}.Q=R.\supset .Q\in CʻR_{\varsigma } &\qquad \text{(3)}\\ +\vdash .(3).*213·25.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*213·3">*213·3</a>.</b> \(\vdash :P=\dot{\Lambda} .\supset .P_{\varsigma }=\dot{\Lambda}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*212·17.\supset \vdash :\text{Hp}.&\supset .\varsigma ʻP_{*}=\dot{\Lambda} .\\ +[*150·42] & \supset .P_{\varsigma }=\dot{\Lambda} :\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*213·301.</b> \(\vdash :\exists !\text{sect}ʻP-\iota ʻ\Lambda -\iota ʻCʻP.\supset .\dot{\exists} !P_{\varsigma } \quad[*213·141]\)</p> + +<p class="nind"><b>*213·302.</b> \(\vdash \colon\ldotp P_{\text{po}}\,\unicode{x2abd}\, J.\supset :\dot{\exists} !P.\equiv .\dot{\exists} !P_{\varsigma }\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*213·126·301.\supset \vdash :\text{Hp}.\dot{\exists} !P.\supset .\dot{\exists} !P_{\varsigma } &\qquad \text{(1)}\\ +\vdash .(1).*213·3.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*213·31.</b> \(\vdash :x\neq y.\supset .(x\downarrow y)_{\varsigma }=\dot{\Lambda} \downarrow (x\downarrow y)\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*211·9.\supset \\ +\vdash :\text{Hp}.&\supset .\text{sect}ʻ(x\downarrow y)-\iota ʻ\Lambda =\iota ʻ\iota ʻx\cup \iota ʻ(\iota ʻx\cup \iota ʻy).\exists !(\iota ʻx\cup \iota ʻy)-\iota ʻx.\\ +[*213·1·141] &\supset .\{(x\downarrow y)\unicode{x0294f}{℩}ʻx\}(x\downarrow y)_{\varsigma }\{(x\downarrow y)\unicode{x0294f}({℩}ʻx\cup {℩}ʻy)\}.\\ +&\text{D}ʻ(x\downarrow y)_{\varsigma }={℩}ʻ(x\downarrow y)\unicode{x0294f}{℩}ʻx.\\ +[*200·35.*55·15]&\supset .\dot{\Lambda} (x\downarrow y)_{\varsigma }(x\downarrow y).\text{D}ʻ(x\downarrow y)_{\varsigma }=\iota ʻ\dot{\Lambda} &\qquad \text{(1)}\\ +\vdash .*213·153.*204·25.&\supset \vdash :\text{Hp}.\supset .(x\downarrow y)_{\varsigma }\in \text{Ser}&\qquad \text{(2)}\\ +\vdash .(1).(2).*204·27.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_679">[Pg 679]</span></p> + +<p class="nind"><b><a id="*213·32">*213·32</a>.</b> \(\vdash :P\in 2_{r}.\supset .P_{\varsigma }=\dot{\Lambda} \downarrow P.P_{\varsigma }\in 2_{r} \quad[*213·31]\)</p> + +<p class="nind"><b><a id="*213·4">*213·4</a>.</b> \[\begin{align}\vdash :P\in &\text{Ser}.\supset .\\ +&P\unicode{x0294f}ʻʻ\text{sect}ʻP=\hat{Q} \{(\exists R).P=Q\unicode{x2909}R.\lor.(\exists x).P=Q\unicode{x21f8} x\}\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*211·82.*5·32.\supset \\ +\vdash \colon\colon\text{Hp}.&\supset \colon\ldotp Q\in P\unicode{x0294f}ʻʻ\text{sect}ʻP.\equiv :\\ +&Q\in \text{D}ʻP\unicode{x0294f}:(\exists R).P=Q\unicode{x2909}R.\lor.(\exists x).P=Q\unicode{x21f8} x &\qquad \text{(1)}\\ +\vdash .*211·283.*160·5.&\supset \vdash :\text{Hp}.P=Q\unicode{x2909}R.\supset .Q\in \text{D}ʻP\unicode{x0294f} &\qquad \text{(2)}\\ +\vdash .*161·11.&\supset \vdash :\text{Hp}.P=Q\unicode{x21f8} x.\supset .Q=P\unicode{x0294f}CʻP &\qquad \text{(3)}\\ +\vdash .(2).(3).&\supset \vdash \colon\ldotp \text{Hp}:(\exists R).P=Q\unicode{x2909}R.\lor.(\exists x).P=Q\unicode{x21f8} x:\supset .\\ +&Q\in \text{D}ʻP\unicode{x0294f} &\qquad \text{(4)}\\ +\vdash .(1).(4).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*213·41">*213·41</a>.</b> \[\begin{align}\vdash :&P\in \text{Ser}.\exists !\overrightarrow{B}ʻP.\supset .\\ +&CʻP_{\varsigma }=\hat{Q} \{(\exists R).P=Q\unicode{x2909}R.\lor.(\exists x).P=Q\unicode{x21f8} x\} \quad[*213·4·161]\end{align}\]</p> + +<p class="nind"><b><a id="*213·42">*213·42</a>.</b> \[\begin{align}&\vdash :P\in \text{Ser}.\overrightarrow{B}ʻP=\Lambda .\supset .\\ +&CʻP_{\varsigma }=\hat{Q} \{(\exists R).P=Q\unicode{x2909}R.\lor.(\exists x).P=Q\unicode{x21f8} x\}-\iota ʻ\dot{\Lambda} \quad[*213·4·163]\end{align}\]</p> + +<p class="nind"><b><a id="*213·5">*213·5</a>.</b> \(\vdash :P_{\text{po}}\,\unicode{x2abd}\, J.x{\sim}\in CʻP.\supset .\text{D}ʻ(P\unicode{x21f8} x)_{\varsigma }=CʻP_{\varsigma }\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*213·141.*211·83.\supset \\ +\vdash :\text{Hp}.\dot{\exists} !P.\supset .\text{D}ʻ(P\unicode{x21f8} x)_{\varsigma }&=(P\unicode{x21f8} x)\unicode{x0294f}ʻʻ(\text{sect}ʻP-\iota ʻ\Lambda )\\ +[*36·4.*161·1] &=P\unicode{x0294f}ʻʻ(\text{sect}ʻP-\iota ʻ\Lambda )\\ +[*213·142] &= CʻP_{\varsigma } &\qquad \text{(1)}\\ +\vdash .*213·3.*161·2.\supset \vdash :P&=\dot{\Lambda} .\supset .\text{D}ʻ(P\unicode{x21f8} x)_{\varsigma }=\Lambda .CʻP_{\varsigma }=\Lambda &\qquad \text{(2)}\\ +\vdash .(1).(2).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*213·51.</b> \(\vdash :P_{\text{po}}\,\unicode{x2abd}\, J.x{\sim}\in CʻP.\supset .(P\unicode{x21f8} x)_{\varsigma }=P_{\varsigma }\unicode{x21f8} (P\unicode{x21f8} x)\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*213·1.*211·83.\supset \vdash \colon\colon\text{Hp}.\dot{\exists} !&P.\supset \colon\ldotp Q(P\unicode{x21f8} x)_{\varsigma }R.\equiv :\\ +&(\exists \alpha , \beta ).\alpha ,\beta \in \text{sect}ʻP-\iota ʻ\Lambda \cup \iota ʻ(CʻP\cup \iota ʻx).\\ +&\exists !\beta -\alpha .Q=(P\unicode{x21f8} x)\unicode{x0294f}\alpha .R=(P\unicode{x21f8} x)\unicode{x0294f}\beta .\\ +[*211·1.*36·4] &\equiv :(\exists \alpha ,\beta ).\alpha ,\beta \in \text{sect}ʻP-\iota ʻ\Lambda .\exists !\beta -\alpha .Q=P\unicode{x0294f}\alpha .R=P\unicode{x0294f}\beta .\lor.\\ +&(\exists \alpha ).\alpha \in \text{sect}ʻP-\iota ʻ\Lambda .Q=P\unicode{x0294f}\alpha .R=P\unicode{x21f8} x:\\ +[*213·1·142] &\equiv :QP_{\varsigma }R.\lor.Q\in CʻP_{\varsigma }.R=P\unicode{x21f8} x:\\ +[*161·11] &\equiv :Q{P_{\varsigma }\unicode{x21f8} (P\unicode{x21f8} x)}R &\qquad \text{(1)}\\ +\vdash .*213·3.*161·2.& \supset \vdash :P=\dot{\Lambda} .\supset .(P\unicode{x21f8} x)_{\varsigma }=\dot{\Lambda} .P_{\varsigma }\unicode{x21f8} + (P\unicode{x21f8} x)=\dot{\Lambda} &\qquad \text{(2)}\\ +\vdash .(1).(2).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_680">[Pg 680]</span></p> + +<p class="nind"><b>*213·52.</b> \[\begin{align}&\vdash \colon\ldotp Q_{\text{po}}\in \text{connex} .CʻP\cap CʻQ=\Lambda .\supset :\\ +&(\exists \beta ).\beta \cap CʻQ{\sim}\in 1.\beta \in (CʻP\cup )ʻʻ(\text{sect}ʻQ-\iota ʻ\Lambda ).S=(P\unicode{x2909}Q)\unicode{x0294f}\beta .\equiv .\\ +&(\exists \gamma ).\gamma \in \text{sect}ʻQ-\iota ʻ\Lambda -1.S=P\unicode{x2909}Q\unicode{x0294f}\gamma\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*37·6. \supset \vdash :&\beta \in (CʻP\cup )ʻʻ(\text{sect}ʻQ-\iota ʻ\Lambda ).S=(P\unicode{x2909}Q)\unicode{x0294f}\beta .\equiv .\\ +&(\exists \gamma ).\gamma \in \text{sect}ʻQ-\iota ʻ\Lambda .\beta =CʻP\cup \gamma .S=(P\unicode{x2909}Q)\unicode{x0294f}(CʻP\cup \gamma ) &\qquad \text{(1)}\\ +\vdash .*160·11.\supset \vdash \colon\colon\text{Hp}.\gamma \in \text{sect}ʻQ.\supset \colon\ldotp &x\{(P\unicode{x2909}Q)\unicode{x0294f}(CʻP\cup \gamma )\}y.\equiv :\\ +&xPy.\lor.x\in CʻP.y\in \gamma .\lor.x(Q\unicode{x0294f}\gamma )y\colon\ldotp \\ +[*211·133.*160·11]& \supset \colon\ldotp \gamma {\sim}\in 1.\supset .(P\unicode{x2909}Q)\unicode{x0294f}(CʻP\cup \gamma )=P\unicode{x2909}Q\unicode{x0294f}\gamma &\qquad \text{(2)}\\ +\vdash .*24·24. &\supset \vdash :\text{Hp}.\beta =CʻP\cup \gamma .\supset .\beta \cap CʻQ=\gamma \cap CʻQ &\qquad \text{(3)}\\ +\vdash .(1).(2).(3).\supset \\ +\vdash \colon\ldotp \text{Hp}.\supset :\beta \cap &CʻQ{\sim}\in 1.\beta \in (CʻP\cup )ʻʻ(\text{sect}ʻQ-\iota ʻ\Lambda ).S=(P\unicode{x2909}Q)\unicode{x0294f}\beta .\equiv .\\ +&(\exists \gamma ).\gamma \in \text{sect}ʻQ-\iota ʻ\Lambda -1.S=P\unicode{x2909}Q\unicode{x0294f}\gamma .\beta =CʻP\cup \gamma &\qquad \text{(4)}\\ +\vdash .(4).*10·281.*13·19.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*213·53.</b> \[\begin{align}\vdash :P_{\text{po}}\,\unicode{x2abd}\, J.Q_{\text{po}}\in \text{Ser}.\overrightarrow{B}ʻQ=\Lambda .CʻP\cap &CʻQ=\Lambda .\supset .\\ +&(P\unicode{x2909}Q)_{\varsigma }=P_{\varsigma }\unicode{x2909}(P\unicode{x2909}^{;}Q_{\varsigma })\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*213·1.*211·841.&\supset \vdash \colon\colon\text{Hp}.\supset \colon\ldotp R(P\unicode{x2909}Q)_{\varsigma }S.\equiv :\\ +&(\exists \alpha ,\beta ).\alpha ,\beta \in \text{sect}ʻP-\iota ʻ\Lambda \cup (CʻP\cup )ʻʻ(\text{sect}ʻQ-\iota ʻ\Lambda ).\\ +&\exists !\beta -\alpha .R=(P\unicode{x2909}Q)\unicode{x0294f}\alpha .S=(P\unicode{x2909}Q)\unicode{x0294f}\beta :\\ +[*211·182]\equiv :(\exists \alpha ,\beta ).\alpha ,\beta \in \text{sect}ʻ&P-\iota ʻ\Lambda \cup (CʻP\cup )ʻʻ(\text{sect}ʻQ-1-\iota ʻ\Lambda ).\\ +&\exists !\beta -\alpha .R=(P\unicode{x2909}Q)\unicode{x0294f}\alpha .S=(P\unicode{x2909}Q)\unicode{x0294f}\beta :\\ +[*160·1.*213·52] +&\equiv :(\exists \alpha ,\beta ).\alpha ,\beta \in \text{sect}ʻP-\iota ʻ\Lambda .\exists !\beta -\alpha .R=P\unicode{x0294f}\alpha .S=P\unicode{x0294f}\beta .\lor.\\ +&(\exists \alpha ,\gamma ).\alpha \in \text{sect}ʻP-\iota ʻ\Lambda .\gamma \in \text{sect}ʻQ-\iota ʻ\Lambda .R=P\unicode{x0294f}\alpha .S=P\unicode{x2909}Q\unicode{x0294f}\gamma .\lor.\\ +&(\exists \gamma ,\delta ).\gamma ,\delta \in \text{sect}ʻQ-\iota ʻ\Lambda .\exists !\delta -\gamma .R=P\unicode{x2909}Q\unicode{x0294f}\gamma .S=P\unicode{x2909}Q\unicode{x0294f}\delta :\\ +[*213·1·142]&\equiv :RP_{\varsigma }S.\lor.R\in CʻP_{\varsigma }.S\in + CʻP\unicode{x2909}^{;}Q_{\varsigma }.\lor.R(P\unicode{x2909}^{;}Q_{\varsigma })S:\\ +[*160·11]. &\equiv :R\{P_{\varsigma }\unicode{x2909}(P\unicode{x2909}^{;}Q_{\varsigma })\}S\colon\colon\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_681">[Pg 681]</span></p> + +<p class="nind"><b>*213·531.</b> \[\begin{align}&\vdash \colon\colon Q_{\text{po}}\in \text{Ser}.\exists !\overrightarrow{B}ʻQ.CʻP\cap CʻQ=\Lambda .\supset \colon\ldotp \\ +&(\exists \beta ).\beta \in (CʻP\cup )ʻʻ(\text{sect}ʻQ-\iota ʻ\Lambda ).S=(P\unicode{x2909}Q)\unicode{x0294f}\beta .\equiv :\\ +&S=P\unicode{x21f8} BʻQ.\lor.(\exists \gamma ).\gamma \in \text{sect}ʻQ-\iota ʻ\Lambda -\iota ʻ\iota ʻBʻQ.S=P\unicode{x2909}Q\unicode{x0294f}\gamma\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*213·52.\supset \\ +\vdash \colon\colon\text{Hp}.\supset \colon\ldotp &(\exists \beta ).\beta \in (CʻP\cup )ʻʻ(\text{sect}ʻQ-\iota ʻ\Lambda ).S=(P\unicode{x2909}Q)\unicode{x0294f}\beta .\equiv :\\ +R(\exists \beta ).\beta \in (CʻP\cup )ʻʻ(\text{sect}ʻQ\cap 1).S=(P\unicode{x2909}Q)\unicode{x0294f}\beta .\lor. +&(\exists \gamma ).\gamma \in \text{sect}ʻQ-\iota ʻ\Lambda -1.S=P\unicode{x2909}Q\unicode{x0294f}\gamma :\\ +[*211·181]&\equiv :(\exists \beta ).\beta =CʻP\cup \iota ʻBʻQ.S=(P\unicode{x2909}Q)\unicode{x0294f}\beta .\lor.\\ +&(\exists \gamma ).\gamma \in \text{sect}ʻQ-\iota ʻ\Lambda -\iota ʻ\iota ʻBʻQ.S=P\unicode{x2909}Q\unicode{x0294f}\gamma &\qquad \text{(1)}\\ +\vdash .*160·11.\supset \vdash \colon\colon\text{Hp}.\supset \colon\ldotp &x\{(P\unicode{x2909}Q)\unicode{x0294f}(CʻP\cup \iota ʻBʻQ)\}y.\equiv :\\ +&xPy.\lor.x\in CʻP.y=BʻQ:\\ +[*161·11] &\equiv :x(P\unicode{x21f8} BʻQ)y &\qquad \text{(2)}\\ +\vdash .(1).(2).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*213·54.</b> \[\begin{align}\vdash :\dot{\exists} !P.P_{\text{po}}\,\unicode{x2abd}\, J.&Q_{\text{po}}\in + \text{Ser}.\exists !\overrightarrow{B}ʻQ.CʻP\cap CʻQ=\Lambda .\text{ᗡ}ʻQ_{\varsigma }{\sim}\in 1.\supset .\\ +&(P\unicode{x2909}Q)_{\varsigma }=P_{\varsigma }\unicode{x21f8} + (P\unicode{x21f8} BʻQ)\unicode{x2909}\{P\unicode{x2909}^{;}(Q_{\varsigma }\unicode{x0294f}\text{ᗡ}ʻQ_{\varsigma })\}\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*213·1.*211·841.&\supset \vdash \colon\colon\text{Hp}.\supset \colon\ldotp R(P\unicode{x2909}Q)_{\varsigma }S.\equiv :\\ +&(\exists \alpha ,\beta ).\alpha ,\beta \in \text{sect}ʻP-\iota ʻ\Lambda \cup (CʻP\cup )ʻʻ(\text{sect}ʻQ-\iota ʻ\Lambda ).\exists !\beta -\alpha .\\ +&R=(P\unicode{x2909}Q)\unicode{x0294f}\alpha .S=(P\unicode{x2909}Q)\unicode{x0294f}\beta :\\ +[*213·531] +&\equiv :(\exists \alpha ,\beta ).\alpha ,\beta \in \text{sect}ʻP-\iota ʻ\Lambda .\exists !\beta -\alpha .R=P\unicode{x0294f}\alpha .Q=P\unicode{x0294f}\beta .\lor.\\ +&(\exists \alpha ).\alpha \in \text{sect}ʻP-\iota ʻ\Lambda .R=P\unicode{x0294f}\alpha .S=P\unicode{x21f8} BʻQ.\lor.\\ +&(\exists \alpha ,\gamma ).\alpha \in \text{sect}ʻP-\iota ʻ\Lambda .R=P\unicode{x0294f}\alpha .\beta \in \text{sect}ʻQ-\iota ʻ\Lambda -\iota ʻ\iota ʻBʻQ.\\ +&S=P\unicode{x2909}Q\unicode{x0294f}\gamma .\lor.\\ +&(\exists \gamma ).R=P\unicode{x21f8} BʻQ.\beta \in \text{sect}ʻQ-\iota ʻ\Lambda -\iota ʻ\iota ʻBʻQ.S=P\unicode{x2909}Q\unicode{x0294f}\gamma .\lor.\\ +&(\exists \gamma ,\delta ).\gamma ,\delta \in \text{sect}ʻQ-\iota ʻ\Lambda -\iota ʻ\iota ʻBʻQ.\exists !\delta -\gamma .R=P\unicode{x2909}Q\unicode{x0294f}\gamma .\\ +&S=P\unicode{x2909}Q\unicode{x0294f}\delta :\\ +[*213·1·142·132] +\equiv :RP_{\varsigma }S.&\lor.R\in CʻP_{\varsigma }.S=P\unicode{x21f8} BʻQ.\lor.R=P\unicode{x21f8} BʻQ.S\in P\unicode{x2909}ʻʻ\text{ᗡ}ʻQ_{\varsigma }.\\ +&\lor.R,S\in (P\unicode{x2909}ʻʻ)\text{ᗡ}ʻQ_{\varsigma }.R(P\unicode{x2909}^{;}Q)S.\lor.R\in CʻP_{\varsigma }.S\in P\unicode{x2909}ʻʻ\text{ᗡ}ʻQ_{\varsigma }:\\ +[*161·11.*211·133.*160·11] +&\equiv :R{P_{\varsigma }\unicode{x21f8} (P\unicode{x21f8} BʻQ)\unicode{x2909}(P\unicode{x2909}^{;}Q_{\varsigma }\unicode{x0294f}\text{ᗡ}ʻQ_{\varsigma })}S\colon\colon\supset + \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_682">[Pg 682]</span></p> + +<p class="nind"><b>*213·541.</b> \(\vdash :P_{\text{po}}\in \text{Ser}.\exists !\overrightarrow{B}ʻP.\text{ᗡ}ʻP_{\varsigma }\in 1.\supset .P\in 2_{r}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*213·144.*211·26.\supset \vdash \colon\ldotp \text{Hp}.&\supset :P\unicode{x0294f}ʻʻ(\text{sect}ʻP-\iota ʻ\Lambda -\iota ʻ\iota ʻBʻP)=\iota ʻP:\\ +[*211·3·13] \supset :x\in \text{ᗡ}ʻP.&\supset .P\unicode{x0294f}\overrightarrow{P}_{*}ʻx=P.\\ +[*202·55] &\supset .\overrightarrow{P}_{*}ʻx=CʻP.\\ +[*200·39] & \supset .\overleftarrow{P}_{\text{po}}ʻx=\Lambda .\\ +[*202·522·523] & \supset .x=Bʻ\breve{P} :\\ +[*204·271] &\supset :P_{\text{po}}\in 2_{r}:\\ +[*56·111.*91·504] &\supset :P\in 2_{r}\colon\ldotp \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*213·55.</b> \[\begin{align}\vdash :\dot{\exists} !P.P_{\text{po}}\,\unicode{x2abd}\, J.Q\in 2_{r}.&CʻP\cap CʻQ=\Lambda .\supset .\\ +&(P\unicode{x2909}Q)_{\varsigma }=P_{\varsigma }\unicode{x21f8} (P\unicode{x21f8} BʻQ)\unicode{x21f8} (P\unicode{x2909}Q)\end{align}\]</p> + +<p><i>Dem.</i></p> + +<p>As in *213·54,</p> + +<p>\[ +\begin{array}{l} +\vdash \colon\colon\text{Hp}.&\supset \colon\ldotp R(P\unicode{x2909}Q)_{\varsigma }S.\\ +\equiv :RP_{\varsigma }S.&\lor.R\in CʻP_{\varsigma }.S=P\unicode{x21f8} BʻQ.\lor.R\in CʻP_{\varsigma }.S\in P\unicode{x2909}ʻʻ\text{ᗡ}ʻQ_{\varsigma }.\\ +&\lor.R=P\unicode{x21f8} BʻQ.S\in P\unicode{x2909}ʻʻ\text{ᗡ}ʻQ_{\varsigma }.\lor.R,S\in P\unicode{x2909}ʻʻ\text{ᗡ}ʻQ_{\varsigma }.\\ +&R(P\unicode{x2909}^{;}Q_{\varsigma })S:\\ +[*213·32]&\equiv :RP_{\varsigma }S.\lor.R\in CʻP_{\varsigma }.S=P\unicode{x21f8} BʻQ.\lor.R\in CʻP_{\varsigma }.S=P\unicode{x2909}Q.\\ +&\lor.R=P\unicode{x21f8} BʻQ.S=P\unicode{x2909}Q.\lor.R=P\unicode{x2909}Q.S=P\unicode{x2909}Q.\\ +&R(P\unicode{x2909}^{;}Q_{\varsigma })S:\\ +[*213·32]\equiv :RP_{\varsigma }S.&\lor.R\in CʻP_{\varsigma }.S=P\unicode{x21f8} BʻQ.\lor.R\in CʻP_{\varsigma }.S=P\unicode{x2909}Q.\\ +&\lor.R=P\unicode{x21f8} BʻQ.S=P\unicode{x2909}Q:\\ +[*161·11]&\equiv :R\{P_{\varsigma }\unicode{x21f8} (P\unicode{x21f8} BʻQ)\unicode{x21f8} (P\unicode{x2909}Q)\}S\colon\colon\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*213·56.</b> \[\begin{align}\vdash \colon\ldotp &P_{\text{po}}\,\unicode{x2abd}\, J.Q_{\text{po}}\in \text{Ser}.CʻP\cap CʻQ=\Lambda .\supset :\\ +&\overrightarrow{B}ʻQ=\Lambda .\supset .(P\unicode{x2909}Q)_{\varsigma }=P_{\varsigma }\unicode{x2909}(P\unicode{x2909}^{;}Q_{\varsigma }):\\ +&\dot{\exists} !P.\exists !\overrightarrow{B}ʻQ.Q{\sim}\in 2_{r}.\supset .\\ +&(P\unicode{x2909}Q)_{\varsigma }=P_{\varsigma }\unicode{x21f8} + (P\unicode{x21f8} BʻQ)\unicode{x2909}\{P\unicode{x2909}^{;}(Q_{\varsigma }\unicode{x0294f}\text{ᗡ}ʻQ_{\varsigma })\}:\\ +&\dot{\exists} !P.Q\in 2_{r}.\supset .(P\unicode{x2909}Q)_{\varsigma }=P_{\varsigma }\unicode{x21f8} (P\unicode{x21f8} BʻQ)\unicode{x21f8} (P\unicode{x2909}Q):\\ +&P=\dot{\Lambda} .\supset .(P\unicode{x2909}Q)_{\varsigma }=Q_{\varsigma } \quad[*213·53·54·541·55.*160·22]\end{align}\]</p> + +<p class="nind"><b><a id="*213·561">*213·561</a>.</b> \(\vdash :CʻP\cap CʻQ=\Lambda .\supset .(P\unicode{x2909})\unicode{x0294f}CʻQ_{\varsigma }\in 1\rightarrow 1\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*213·1.\supset \vdash :R\in CʻQ_{\varsigma }.&\supset .CʻR\subset CʻQ &\qquad \text{(1)}\\ +\vdash .(1).\supset \vdash \colon\ldotp \text{Hp}.R,S\in CʻQ_{\varsigma }.&\supset :CʻP\cap CʻR=\Lambda .CʻP\cap CʻS=\Lambda :\\ +[*160·52] & \supset :P\unicode{x2909}R=P\unicode{x2909}S.\supset .R=S\colon\ldotp \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_683">[Pg 683]</span></p> + +<p class="nind"><b>*213·57.</b> \(\vdash :P_{\text{po}}\,\unicode{x2abd}\, J.\text{Nr}ʻQ=\text{Nr}ʻP\dot{+} \dot{1} .\supset .\text{Nr}ʻQ_{\varsigma }=\text{Nr}ʻP_{\varsigma }\dot{+} + \dot{1}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*181·2·12.(*181·01).\supset \\ +\vdash :\text{Hp}.&\supset .(\exists R,x).R\,\,\text{smor}\,\,P.x{\sim}\in CʻR.Q=R\unicode{x21f8} x.\\ +[*213·51] &\supset .(\exists R,x).R\,\,\text{smor}\,\,P.x{\sim}\in CʻR.Q_{\varsigma }=R_{\varsigma }\unicode{x21f8} (R\unicode{x21f8} x).\\ +[*181·32] &\supset .(\exists R).R\,\,\text{smor}\,\,P.\text{Nr}ʻQ_{\varsigma }=\text{Nr}ʻR_{\varsigma }\dot{+} \dot{1} .\\ +[*213·172] &\supset .\text{Nr}ʻQ_{\varsigma }=\text{Nr}ʻP_{\varsigma }\dot{+} \dot{1} :\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*213·58">*213·58</a>.</b> \(\vdash :P_{\text{po}}\,\unicode{x2abd}\, J.Q_{\text{po}}\in + \text{Ser}.CʻP\cap CʻQ=\Lambda .\supset .\text{Nr}ʻ(P\unicode{x2909}Q)_{\varsigma }=\text{Nr}ʻP_{\varsigma }\dot{+} \text{Nr}ʻQ_{\varsigma }\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *213·53·561 .\supset \\ +\vdash :\text{Hp}.\overrightarrow{B}ʻQ=\Lambda .&\supset .(P\unicode{x2909}Q)_{\varsigma }=P_{\varsigma }\unicode{x2909}(P\unicode{x2909}^{;}Q_{\varsigma }).\text{Nr}ʻP\unicode{x2909}^{;}Q_{\varsigma }=\text{Nr}ʻQ_{\varsigma }.\\ +[*180·32] &\supset .\text{Nr}ʻ(P\unicode{x2909}Q)_{\varsigma }=\text{Nr}ʻP_{\varsigma }\dot{+} \text{Nr}ʻQ_{\varsigma } &\qquad \text{(1)}\\ +\vdash .*213·54·561.*181·32.\supset\\ +\vdash :\text{Hp}.\dot{\exists} !P.\exists !\overrightarrow{B}ʻQ.\text{ᗡ}ʻQ_{\varsigma }{\sim}\in 1.\supset .\text{Nr}ʻ(P\unicode{x2909}Q)_{\varsigma }&=\text{Nr}ʻP_{\varsigma }\dot{+} + \dot{1} \dot{+} \text{Nr}ʻQ_{\varsigma }\unicode{x0294f}\text{ᗡ}ʻQ_{\varsigma }\\ +[*204·46.*213·157] &=\text{Nr}ʻP_{\varsigma }\dot{+} \text{Nr}ʻQ_{\varsigma } &\qquad \text{(2)}\\ +\vdash .*213·541·55.*181·32.\supset \\ +\vdash :\text{Hp}.\dot{\exists} !P.\text{ᗡ}ʻQ_{\varsigma }\in 1.&\supset .Q\in 2_{r}.\text{Nr}ʻ(P\unicode{x2909}Q)_{\varsigma }=\text{Nr}ʻP_{\varsigma }\dot{+} + \dot{1} \dot{+} \dot{1} .\\ +[*181·56] &\supset .Q\in 2_{r}.\text{Nr}ʻ(P\unicode{x2909}Q)_{\varsigma }=\text{Nr}ʻP_{\varsigma }\dot{+} 2_{r}.\\ +[*213·32] &\supset .\text{Nr}ʻ(P\unicode{x2909}Q)_{\varsigma }=\text{Nr}ʻP_{\varsigma }\dot{+} \text{Nr}ʻQ_{\varsigma } &\qquad \text{(3)}\\ +\vdash .*160·22.*213·3.&\supset \vdash :P=\dot{\Lambda} .\supset .\text{Nr}ʻ(P\unicode{x2909}Q)_{\varsigma }=\text{Nr}ʻP_{\varsigma }\dot{+} \text{Nr}ʻQ_{\varsigma } + &\qquad \text{(4)}\\ +\vdash .(1).(2).(3).(4).\supset \vdash .\text{Prop} +\end{array} +\]</p> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_684">[Pg 684]</span></p> +<h2 class="nobreak" id="*214"> +*214. DEDEKINDIAN RELATIONS.</h2> +</div> + + +<p class="nind"><i>Summary of</i> *214.</p> + +<p>We call a relation "Dedekindian" when it is such that every class +has either a maximum or a sequent with respect to it. As a rule, the +hypothesis that a relation is Dedekindian is only important in the case +of serial relations. Dedekindian series have considerable importance, +especially in connection with limits.</p> + +<p>When \(P\) is transitive, the hypothesis that \(P\) is Dedekindian +is equivalent to the hypothesis that every section of \(P\) has a +maximum or a sequent (<a href="#*214·13">*214·13</a>); it is also equivalent to the assumption +that every segment of \(P\) has a maximum or a sequent (<a href="#*214·131">*214·131</a>), +<i>i.e.</i> to the assumption that every segment of \(P\) which has no +maximum has a limit, <i>i.e.</i> to +\[ +\text{D}ʻ(P_{\in }\dot{\cap} I)\subset \text{ᗡ}ʻ\text{lt}_P. +\] +When \(P\) is a series, the hypothesis that it is Dedekindian is +equivalent to the hypothesis that every segment has a sequent +(<a href="#*214·15">*214·15</a>), <i>i.e.</i> to the hypothesis that the class of segments +is the class \(\overrightarrow{P}ʻʻCʻP\) (<a href="#*214·151">*214·151</a>). If \(P\) is a +Dedekindian series, so is \(\breve{P}\), and vice versa (<a href="#*214·14">*214·14</a>). +Whenever \(P\) is connected and not null, \(\varsigma ʻP_{*}\) is +a Dedekindian series (<a href="#*214·32">*214·32</a>), and so is \(\text{sgm}ʻP\) if it +exists (<a href="#*214·34">*214·34</a>); whenever \(P\) is transitive and connected and not +null, \(\varsigma ʻP\) is a Dedekindian series (<a href="#*214·33">*214·33</a>). All these +propositions have been virtually proved already: almost the only thing +new in the present number is the definition, which is +\[ +\text{Ded}=\hat{P}\{(\alpha ).\alpha \in \text{ᗡ}ʻ\text{max}_{P}\cup \text{ᗡ}ʻ\text{seq}_{P}\} \quad\text{Df}. +\]</p> + +<p><a href="#*214·4">*214·4</a>—<a href="#*214·43">·43</a> give properties of series which have Dedekindian +continuity. We have</p> + +<p class="nind"><b>*214·4.</b> \(\vdash \colon\ldotp P^{2}=P.P\in \text{connex} .\supset :P\in \text{Ded}.\equiv .\text{ᗡ}ʻ\text{max}_{P}=-\text{ᗡ}ʻ\text{seq}_{P}\)</p> + +<p class="nind"><b>*214·41.</b> \(\vdash \colon\ldotp P\in \text{Ser}.\supset :P^{2}=P.P\in \text{Ded}.\equiv .\text{ᗡ}ʻ\text{max}_{P}=-\text{ᗡ}ʻ\text{seq}_{P}\)</p> + +<p><span class="pagenum" id="Page_685">[Pg 685]</span></p> + +<p><i>I.e.</i> in a series, Dedekindian continuity is equivalent to the +assumption that the classes which have a maximum are the same as the +classes which have no sequent.</p> + +<p class="nind"><b>*214·42.</b> \(\vdash :P\in \text{Ser} \cap \text{Ded}.P^{2} = P.\alpha \in \text{sect}ʻP.\supset .\text{limax}_Pʻ\alpha = \text{limin}_Pʻ(CʻP-\alpha )\)</p> + +<p>This proposition is important in dealing with Dedekind "cuts."</p> + +<p class="nind"><b>*214·43.</b> \[\begin{align}\vdash \colon\ldotp P\in &\text{Ser} \cap \text{Ded}.\alpha \in \text{sect}ʻP.\supset :\\ +&\text{limax}_Pʻ\alpha = \text{limin}_Pʻ(CʻP-\alpha ).\lor.\text{max}_{P}ʻ\alpha P_{1} \text{min}_{P}ʻ(CʻP-\alpha )\end{align}\]</p> + +<p><a href="#*214·5">*214·5</a> shows that a Dedekindian relation has a beginning and an end; +the following propositions deal with \(P\dot{\cap} J\) when \(P\) is +Dedekindian.</p> + +<p><a href="#*214·6">*214·6</a> shows that a relation which is similar to a Dedekindian relation +is Dedekindian.</p> + +<p>We call a relation "semi-Dedekindian" if it becomes Dedekindian by the +addition of one term at the end; the definition is</p> + +<p class="nind"><b>*214·02.</b> \(\text{semi Ded} = \hat{P} (\text{sect}ʻP-\iota ʻCʻP\subset \text{ᗡ}ʻ\text{max}_{P}\cup \text{ᗡ}ʻ\text{seq}_{P}) \quad\text{Df}\)</p> + +<hr class="tb"> + +<p class="nind"><b>*214·01.</b> \(\text{Ded} = \hat{P}\{(\alpha ).\alpha \in \text{ᗡ}ʻ\text{max}_{P}\cup \text{ᗡ}ʻ\text{seq}_{P}\} \quad\text{Df}\)</p> + +<p class="nind"><b>*214·02.</b> \(\text{semi Ded} = \hat{P} (\text{sect}ʻP-\iota ʻCʻP\subset \text{ᗡ}ʻ\text{max}_{P}\cup \text{ᗡ}ʻ\text{seq}_{P}) \quad\text{Df}\)</p> + +<p class="nind"><b>*214·1.</b> \(\vdash :P\in \text{Ded}. \equiv .(\alpha ).\alpha \in \text{ᗡ}ʻ\text{max}_{P}\cup \text{ᗡ}ʻ\text{seq}_{P} \quad[(*214·01)]\)</p> + +<p class="nind"><b>*214·101.</b> \[\begin{align}& \vdash :P\in \text{Ded}. \equiv .-\text{ᗡ}ʻ\text{max}_{P}\subset \text{ᗡ}ʻ\text{seq}_{P}. \equiv .-\text{ᗡ}ʻ\text{max}_{P}\subset + \text{ᗡ}ʻ\text{lt}_P\\ +&[*214·1.*24·312.*207·12]\end{align}\]</p> + +<p class="nind"><b>*21·411.</b> \[\begin{align}& \vdash :P\in \text{Ded}. \equiv .(\alpha ).\alpha \in \text{ᗡ}ʻ\text{max}_{P}\cup + \text{ᗡ}ʻ\text{lt}_P. \equiv .(\alpha ).\alpha \in \text{ᗡ}ʻ\text{limax}_P\\ +&[*214·1.*207·14·44]\end{align}\]</p> + +<p class="nind"><b>*214·12.</b> \[\begin{align}& \vdash \colon\ldotp P\in \text{Ded}. \equiv :\alpha \subset CʻP.\supset _{\alpha }.\alpha \in \text{ᗡ}ʻ\text{max}_{P}\cup + \text{ᗡ}ʻ\text{seq}_{P}\\ +&[*214·1.*205·151.*206·131]\end{align}\]</p> + +<p class="nind"><b><a id="*214·13">*214·13</a>.</b> \[\begin{align}& \vdash \colon\ldotp P\in \text{trans}.\supset :P\in \text{Ded}. \equiv .\text{sect}ʻP\subset \text{ᗡ}ʻ\text{max}_{P}\cup + \text{ᗡ}ʻ\text{seq}_{P}\\ +&[*211·272.*214·1]\end{align}\]</p> + +<p class="nind"><b><a id="*214·131">*214·131</a>.</b> \(\vdash \colon\ldotp P\in \text{trans}.\supset :P\in \text{Ded}. \equiv .\text{D}(ʻP_{\in }\dot{\cap} I)\subset \text{ᗡ}ʻ\text{seq}_{P} + \quad[*211·47.*214·1]\)</p> + +<p class="nind"><b>*214·132.</b> \[\begin{align}& \vdash \colon\ldotp P\in \text{trans}.\supset :P\in \text{Ded}. \equiv .\text{D}ʻP_{\in }\subset \text{ᗡ}ʻ\text{max}_{P}\cup + \text{ᗡ}ʻ\text{seq}_{P}\\ +&[*214·131.*211·42]\end{align}\]</p> + +<p class="nind"><b><a id="*214·14">*214·14</a>.</b> \(\vdash \colon\ldotp P\in \text{Ser}.\supset :P\in \text{Ded}. \equiv .\breve{P} \in \text{Ded} \quad[*206·57.*214·1]\)</p> + +<p class="nind"><b>*214·141.</b> + \[\begin{align}& \vdash \colon\ldotp P\in \text{Ser}.\supset :P\in \text{Ded}. \equiv .(\alpha ).pʻ\overrightarrow{P}ʻʻ(\alpha \cap CʻP)\in \text{ᗡ}ʻ\text{max}_{P}\text{ᗡ}ʻ\text{seq}_{P}\\ +&[*206·56.*214·1]\end{align}\]</p> + +<p class="nind"><b><a id="*214·15">*214·15</a>.</b> \[\begin{align}& \vdash \colon\ldotp P\in \text{Ser}.\supset :P\in \text{Ded}. \equiv .\text{D}ʻP_{\in }\subset \text{ᗡ}ʻ\text{seq}_{P}\\ +&[*206·36.*214·1.*211·11]\end{align}\]</p> + +<p class="nind"><b><a id="*214·151">*214·151</a>.</b> \(\vdash P\in \text{Ser}.\supset :P\in \text{Ded}. \equiv .\text{D}ʻP\in = \overrightarrow{P}ʻʻCʻP \quad[*211·38.*214·1]\)</p> + +<p><span class="pagenum" id="Page_686">[Pg 686]</span></p> + +<p class="nind"><b>*214·2.</b> \(\vdash :P\in \text{trans}\cap \text{connex} \cap \text{Ded}.\supset .\text{D}ʻP_{\in }\subset \text{ᗡ}ʻ\text{seq}_{P} \quad[*211·371]\)</p> + +<p class="nind"><b>*214·21.</b> \(\vdash :P\in \text{trans}\cap \text{connex} \cap \text{Ded}.\supset .\text{D}ʻP_{\in }=\overrightarrow{P}ʻʻCʻP \quad [*211·372]\)</p> + +<p class="nind"><b>*214·22.</b> \[\begin{align}& \vdash :P\in \text{trans}\cap \text{connex} \cap \text{Ded}.\supset .\text{D}ʻ(P_{\in }\dot{\cap} + I)=\overrightarrow{P}ʻʻ\{CʻP-\text{ᗡ}ʻ(P\dot{-} P^{2})\}\\ +&[*211·46]\end{align}\]</p> + +<p class="nind"><b>*214·23.</b> \[\begin{align}&\vdash :P\in \text{trans}\cap \text{connex} \cap \text{Ded}.{\sim}\text{E}!\text{max}_{P}ʻ\alpha .\supset .\\ +&\text{seq}_{P}ʻ\alpha =\text{max}_{P}ʻ(\alpha \cup \iota ʻ\text{seq}_{P}ʻ\alpha ).\text{E}!\text{max}_{P}ʻ(\alpha \cup \iota ʻ\text{seq}_{P}ʻ\alpha + )\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*214·101.\supset \vdash :\text{Hp}.&\supset .\text{E}!\text{seq}_{P}ʻ\alpha .\\ +[*206·47] &\supset .\text{seq}_{P}ʻ\alpha =\text{max}_{P}ʻ(\alpha \cup \iota ʻ\text{seq}_{P}ʻ\alpha ). &\qquad \text{(1)}\\ +[*14·21] &\supset .\text{E}!\text{max}_{P}ʻ(\alpha \cup \iota ʻ\text{seq}_{P}ʻ\alpha ) &\qquad \text{(2)}\\ +\vdash .(1).(2).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*214·24.</b> \[\begin{align}& \vdash :P\in \text{connex} \cap \text{Ded}.\alpha \in \text{sect}ʻP.\supset .\overrightarrow{\text{seq}} _{P}ʻ\alpha + =\overrightarrow{\text{min}}_{P}ʻ(CʻP-\alpha )\\ +&[*211·721]\end{align}\]</p> + +<p class="nind"><b>*214·241.</b> \[\begin{align}& \vdash :P\in \text{connex} .\breve{P} \in \text{Ded}.\alpha \in \text{sect}ʻP.\supset .\overrightarrow{\text{max}}_{P}ʻ\alpha + =\overrightarrow{\text{prec}} _{P}ʻ(CʻP-\alpha )\\ +&\left[*214·24 \frac{\breve{P}}{P}. *211·7\right]\end{align}\]</p> + +<p class="nind"><b>*214·3.</b> \[\begin{align}\vdash \colon\colon\alpha ,&\beta \in \kappa .\supset _{\alpha ,\beta }:\alpha \subset \beta .\lor.\beta \subset \alpha \colon\ldotp\\ +&\kappa {\sim}\in 1.Q=\hat{\alpha} \hat{\beta} (\alpha ,\beta \in \kappa .\alpha \subset \beta .\alpha \neq \beta )\colon\ldotp \supset \colon\ldotp\\ +&\lambda \subset \kappa .\supset _{\lambda }.sʻ\lambda \in \kappa :\supset .Q\in \text{Ser}\cap \text{Ded}\\ +&[*210·12·253]\end{align}\]</p> + +<p class="nind"><b>*214·31.</b> \[\begin{align}& \vdash \colon\ldotp \text{Hp}*214·3:\lambda \subset \kappa .\supset _{\lambda }.pʻ\lambda + \cap sʻ\kappa \in \kappa :\supset .Q\in \text{Ser}\cap \text{Ded}\\ +&[*210·12·254]\end{align}\]</p> + +<p class="nind"><b><a id="*214·32">*214·32</a>.</b> \(\vdash :P\in \text{connex} .\dot{\exists} !P.\supset .\varsigma ʻP_{*}\in \text{Ser}\cap \text{Ded} \quad[*212·3·35]\)</p> + +<p class="nind"><b><a id="*214·33">*214·33</a>.</b> \(\vdash :P\in \text{trans}\cap \text{connex} .\dot{\exists} !P.\supset .\varsigma ʻP\in \text{Ser}\cap \text{Ded} \quad[*212·31·44]\)</p> + +<p class="nind"><b><a id="*214·34">*214·34</a>.</b> \(\vdash :P\in \text{connex} .\dot{\exists} !\text{sgm}ʻP.\supset .\text{sgm}ʻP\in \text{Ser}\cap \text{Ded} \quad[*212·3·54]\)</p> + +<p class="nind"><b><a id="*214·4">*214·4</a>.</b> \[\begin{align}& \vdash \colon\ldotp P^{2}=P.P\in \text{connex} .\supset :P\in \text{Ded}.\equiv .\text{ᗡ}ʻ\text{max}_{P}=-\text{ᗡ}ʻ\text{seq}_{P}\\ +&[*211·53]\end{align}\]</p> + +<p class="nind"><b>*214·41.</b> \[\begin{align}& \vdash \colon\ldotp P\in \text{Ser}.\supset :P^{2}=P.P\in \text{Ded}.\equiv .\text{ᗡ}ʻ\text{max}_{P}=-\text{ᗡ}ʻ\text{seq}_{P}\\ +&[*211·552]\end{align}\]</p> + +<p class="nind"><b>*214·42.</b> \(\vdash :P\in \text{Ser}\cap \text{Ded}.P^{2}=P.\alpha \in \text{sect}ʻP.\supset .\text{limax}_Pʻ\alpha =\text{limin}_Pʻ(CʻP-\alpha )\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*211·721.\supset \vdash \colon\ldotp \text{Hp}.&\supset :\overrightarrow{\text{seq}} _{P}ʻ\alpha =\overrightarrow{\text{min}}_{P}ʻ(CʻP-\alpha ):\\ +[*214·101] &\supset :{\sim}\text{E}!\text{max}_{P}ʻ\alpha .\supset .\text{lt}_Pʻ\alpha =\text{min}_{P}ʻ(CʻP-\alpha ) &\qquad \text{(1)}\\ +\vdash .*211·726.&\supset \vdash :\text{Hp}.\text{E}!\text{max}_{P}ʻ\alpha .\supset .\text{max}_{P}ʻ\alpha =\text{prec}_{P}ʻ(CʻP-\alpha + ) &\qquad \text{(2)}\\ +\vdash .*214·14·41.&\supset \vdash :\text{Hp}.\text{E}!\text{prec}_{P}ʻ(CʻP-\alpha ).\supset .{\sim}\text{E}!\text{max}_{P}ʻ(CʻP-\alpha ).\\ +[*207·12] &\supset .\text{prec}_{P}ʻ(CʻP-\alpha )=\text{tl}_Pʻ(CʻP-\alpha ) &\qquad \text{(3)}\\ +\vdash .(2).(3). &\supset \vdash :\text{Hp}.\text{E}!\text{max}_{P}ʻ\alpha .\supset .\text{max}_{P}ʻ\alpha =\text{tl}_Pʻ(CʻP-\alpha ) &\qquad \text{(4)}\\ +\vdash .(1).(4).*207·46.\supset \\ +\vdash \colon\ldotp \text{Hp}.&\supset :\text{limax}_Pʻ\alpha =\text{min}_{P}ʻ(CʻP-\alpha ).\lor.\text{limax}_Pʻ\alpha =\text{tl}_Pʻ(CʻP-\alpha ):\\ +[*207·46] &\supset :\text{limax}_Pʻ\alpha =\text{limin}_Pʻ(CʻP-\alpha )\colon\ldotp \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_687">[Pg 687]</span></p> + +<p class="nind"><b><a id="*214·43">*214·43</a>.</b> \[\begin{align}\vdash \colon\ldotp P\in &\text{Ser}\cap \text{Ded}.\alpha \in \text{sect}ʻP.\supset :\\ +&\text{limax}_Pʻ\alpha =\text{limin}_Pʻ(CʻP-\alpha ).\lor.\text{max}_{P}ʻ\alpha P_{1}\text{min}_{P}ʻ(CʻP-\alpha )\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*214·11. \supset \vdash \colon\ldotp \text{Hp}.\supset :{\sim}\text{E}!\text{max}_{P}ʻ\alpha .\supset .\text{limax}_Pʻ\alpha &=\text{seq}_{P}ʻ\alpha \\ +[*211·715] &=\text{min}_{P}ʻ(CʻP-\alpha ) &\qquad \text{(1)}\\ +\vdash .*211·726.\supset \vdash :\text{E}!\text{max}_{P}ʻ\alpha .{\sim}\text{E}!\text{min}_{P}ʻ(CʻP-\alpha ).\supset .\\ +&\text{limax}_Pʻ\alpha =\text{tl}_Pʻ(CʻP-\alpha ) &\qquad \text{(2)}\\ +\vdash .(1).(2).*207·46.&\supset \vdash \colon\ldotp \text{Hp}:{\sim}\text{E}!\text{max}_{P}ʻ\alpha .\lor.{\sim}\text{E}!\text{min}_{P}ʻ(CʻP-\alpha ):\supset .\\ +&\text{limax}_Pʻ\alpha =\text{limin}_Pʻ(CʻP-\alpha ) &\qquad \text{(3)}\\ +\vdash .*211·726.\supset \vdash :\text{Hp}.\text{E}!&\text{max}_{P}ʻ\alpha .\text{E}!\text{min}_{P}ʻ(CʻP-\alpha ).\supset .\\ +&\text{E}!\text{max}_{P}ʻ\alpha .\text{E}!\text{seq}_{P}ʻ\alpha .\text{seq}_{P}ʻ\alpha =\text{min}_{P}ʻ(CʻP-\alpha ).\\ +[*206·5] &\supset .\text{max}_{P}ʻ\alpha P_{1}\text{min}_{P}ʻ(CʻP-\alpha ) &\qquad \text{(4)}\\ +\vdash .(3).(4).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p>The following propositions are no longer mere restatements of previous +results.</p> + +<p class="nind"><b><a id="*214·5">*214·5</a>.</b> \(\vdash :P\in \text{Ded}.\supset .\exists !\overrightarrow{B}ʻP.\exists !\overrightarrow{B}ʻ\breve{P} .\overrightarrow{B}ʻP=\overrightarrow{\text{seq}} _{P}ʻ\Lambda + .\overrightarrow{B}ʻ\breve{P} =\overrightarrow{\text{max}}_{P}ʻCʻP\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*205·161.*214·101.\supset \vdash :\text{Hp}.&\supset .\exists !\overrightarrow{\text{seq}} _{P}ʻ\Lambda .\\ +[*206·14] &\supset .\exists !\overrightarrow{B}ʻP.\overrightarrow{B}ʻP=\overrightarrow{\text{seq}} _{P}ʻ\Lambda &\qquad \text{(1)}\\ +\vdash .*206·18·2. &\supset \vdash .\overrightarrow{\text{seq}} _{P}ʻCʻP=\Lambda &\qquad \text{(2)}\\ +\vdash .(2).*214·1.\supset \vdash :\text{Hp}.&\supset .\exists !\overrightarrow{\text{max}}_{P}ʻCʻP.\\ +[*93·117] &\supset .\exists !\overrightarrow{B}ʻ\breve{P} .\overrightarrow{B}ʻ\breve{P} =\overrightarrow{\text{max}}_{P}ʻCʻP &\qquad \text{(3)}\\ +\vdash .(1).(3).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*214·51.</b> \(\vdash \colon\ldotp P\in \text{Ded}.\supset :{\sim}(xPx).\lor.x\in \text{D}ʻ(P\dot{-} P^{2})\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*214·1.\supset \vdash \colon\ldotp \text{Hp}.&\supset :\exists !\overrightarrow{\text{max}}_{P}ʻ\iota ʻx.\lor.\exists !\overrightarrow{\text{seq}} _{P}ʻ\iota ʻx:\\ +[*53·301.*206·42] &\supset :\exists !\iota ʻx-\overrightarrow{P}ʻx.\lor.\exists ! \overleftarrow{P\dot{-} P^{2}}ʻx:\\ +[*51·31.*33·4] &\supset :{\sim}(xPx).\lor.x\in \text{D}ʻ(P\dot{-} P^{2})\colon\ldotp \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_688">[Pg 688]</span></p> + +<p class="nind"><b>*214·52.</b> \(\vdash :P\in \text{Ded}.P\,\unicode{x2abd}\, P^{2}.\supset .P\,\unicode{x2abd}\, J \quad[*214·51]\)</p> + +<p class="nind"><b>*214·53.</b> \(\vdash :P\in \text{Ded}.\supset .\text{D}ʻP=\text{D}ʻ(P\dot{\cap} J)\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*214·51.\supset \vdash :\text{Hp}.xPx.&\supset .x\in \text{D}ʻ(P\dot{-} P^{2}).\\ +[*33·13] &\supset .(\exists y).xPy.x\dot{-} P^{2}y.\\ +[*34·54.\text{Transp}] &\supset .(\exists y).xPy.x\neq y &\qquad \text{(1)}\\ +\vdash .(1).*13·195.\supset \vdash \colon\ldotp \text{Hp}.&\supset :(\exists y).xPy.\supset .(\exists y).xPy.x\neq y:\\ +[*33·13] &\supset :\text{D}ʻP\subset \text{D}ʻ(P\dot{\cap} J):\\ +[*33·25] &\supset :\text{D}ʻP=\text{D}ʻ(P\dot{\cap} J)\colon\ldotp \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*214·531.</b> \(\vdash :P\in \text{Ded}.\supset .CʻP=Cʻ(P\dot{\cap} J)\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*93·12.\supset \vdash \colon\ldotp x\in \overrightarrow{B}ʻ\breve{P} .&\supset :x{\sim}\in \text{D}ʻP:(\exists y).yPx:\\ +[*13·14] &\supset :(\exists y).yPx.x\neq y:\\ +[*33·13] &\supset :x\in \text{ᗡ}ʻ(P\dot{\cap} J) &\qquad \text{(1)}\\ +\vdash .(1).*214·53. \supset \vdash :\text{Hp}.&\supset .\text{D}ʻP\cup \overrightarrow{B}ʻ\breve{P} \subset Cʻ(P\dot{\cap} J).\\ +[*93·12] &\supset .CʻP\subset Cʻ(P\dot{\cap} J).\\ +[*33·252] &\supset .CʻP=Cʻ(P\dot{\cap} J):\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*214·532.</b> \(\vdash :P\in \text{Ded}.\supset .\text{ᗡ}ʻP=\text{ᗡ}ʻ(P\dot{\cap} J)\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*34·54. \supset \vdash :\overrightarrow{P}ʻx=\iota ʻx.&\supset .\overrightarrow{P}ʻx=Pʻʻ\overrightarrow{P}ʻx.\\ +[*205·123] &\supset .\overrightarrow{\text{max}}_{P}ʻ\iota ʻx=\Lambda &\qquad \text{(1)}\\ +\vdash .*206·134.&\supset \vdash :\overrightarrow{P}ʻx=\iota ʻx.\supset .\\ +&\overrightarrow{\text{seq}} _{P}ʻ\overrightarrow{P}ʻx=CʻP\cap \hat{y} \{\iota ʻx\subset \overrightarrow{P}ʻy.\overrightarrow{P}ʻy\subset -pʻ\overleftarrow{P}ʻʻ\iota ʻx\}\\ +[*53·301·01] &=CʻP\cap \hat{y} \{\iota ʻx\subset \overrightarrow{P}ʻy.\overrightarrow{P}ʻy\subset -\overleftarrow{P}ʻx\}\\ +[\text{Hp}] &=CʻP \cap \hat{y} \{\iota ʻx\subset \overrightarrow{P}ʻy.\overrightarrow{P}ʻy\subset -\iota ʻx\}\\ +[*51·161] & =\Lambda &\qquad \text{(2)}\\ +\vdash .(1).(2). &\supset \vdash :\overrightarrow{P}ʻx=\iota ʻx.\supset .\overrightarrow{\text{max}}_{P}ʻ\overrightarrow{P}ʻx=\Lambda .\overrightarrow{\text{seq}} _{P}ʻ\overrightarrow{P}ʻx=\Lambda + &\qquad \text{(3)}\\ +\vdash .(3).\text{Transp}.\supset \vdash \colon\ldotp \text{Hp}.&\supset :(x).\overrightarrow{P}ʻx\neq \iota ʻx:\\ +[*51·401.\text{Transp}] &\supset :(x):\exists !\overrightarrow{P} ʻx.\supset .\exists !\overrightarrow{P}ʻx-\iota ʻx:\\ +[*33·41] &\supset :x\in \text{ᗡ}ʻP.\supset .x\in \text{ᗡ}ʻ(P\dot{\cap} J) &\qquad \text{(4)}\\ +\vdash .(4).*33·251.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_689">[Pg 689]</span></p> + +<p class="nind"><b>*214·54.</b> \(\vdash :P\in \text{Ded}.\supset .P\dot{\cap} J\in \text{Ded}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*205·111·195.\supset \vdash .\overrightarrow{\text{max}}_{P}ʻ\alpha &\subset \alpha \cap Cʻ(P\dot{\cap} J)-Pʻʻ\alpha \\ +[*37·201] &\subset \alpha \cap Cʻ(P\dot{\cap} J)-(P\dot{\cap} J)ʻʻ\alpha \\ +[*205·111] &\subset \overrightarrow{\text{max}}(P\dot{\cap} J)ʻ\alpha .\\ +[*24·59] &\supset \vdash :{\sim}\exists !\overrightarrow{\text{max}}(P\dot{\cap} J)ʻ\alpha .\supset .{\sim}\exists !\overrightarrow{\text{max}}_{P}ʻ\alpha &\qquad \text{(1)}\\ +\vdash .(1).*214·1. &\supset \vdash :\text{Hp}.{\sim}\exists !\overrightarrow{\text{max}}(P\dot{\cap} J)ʻ\alpha .\supset .\exists !\overrightarrow{\text{seq}} _{P}ʻ\alpha + &\qquad \text{(2)}\\ +\vdash .*206·2·17.\supset \\ +\vdash \colon\ldotp x\text{seq}_{P}\alpha .&\equiv :y\in \alpha \cap CʻP.\supset _{y}.yPx.y\neq x:x\in CʻP:\\ +&yPx.\supset _{y}.(\exists z).z\in \alpha .{\sim}(zPy):\\ +[*214·531] &\equiv :y\in \alpha \cap Cʻ(P\dot{\cap} J).\supset _{y}.y(P\dot{\cap} J)x:x\in Cʻ(P\dot{\cap} J):\\ +&yPx.\supset _{y}.(\exists z).z\in \alpha .{\sim}(zPy):\\ +[*23·43.*3·14]\supset :y\in \alpha \cap Cʻ&(P\dot{\cap} J).\supset _{y}.y(P\dot{\cap} J)x:x\in Cʻ(P\dot{\cap} J):\\ +&y(P\dot{\cap} J)x.\supset _{y}.(\exists z).z\in \alpha .{\sim}\{z(P\dot{\cap} J)y\}:\\ +[*206·17] &\supset :x\text{seq}(P\dot{\cap} J)\alpha &\qquad \text{(3)}\\ +\vdash .(2).(3).&\supset \vdash :\text{Hp}.{\sim}\exists !\overrightarrow{\text{max}}(P\dot{\cap} J)ʻ\alpha .\supset .\exists !\text{seq}(P\dot{\cap} J)ʻ\alpha &\qquad \text{(4)}\\ +\vdash .(4).*214·1.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*214·6">*214·6</a>.</b> \(\vdash :P\in \text{Ded}.P\,\,\text{smor}\,\,Q.\supset .Q\in \text{Ded}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*207·65.*214·11.\supset \\ +\vdash :P\in \text{Ded}.S\in P\,\overline{\,\text{smor}\,}\,Q.&\supset .(\alpha ).\breve{S} ʻʻ\alpha \in \text{ᗡ}ʻ\text{limax}_{Q}.\\ +[*71·481] &\supset .\text{Cl}ʻ\text{ᗡ}ʻS\subset \text{ᗡ}ʻ\text{limax}_{Q} .\\ +[*151·11.*214·12]&\supset .Q\in \text{Ded}:\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*214·7.</b> \[\begin{align}& \vdash \colon\ldotp P \in \text{semi Ded}.\equiv :\alpha \in \text{sect}ʻP.\alpha \neq CʻP.\supset _{\alpha }.\exists + !(\overrightarrow{\text{max}}_{P}ʻ\alpha \cup \overrightarrow{\text{seq}} _{P}ʻ\alpha )\\ +&[(*214·02)]\end{align}\]</p> + +<p class="nind"><b>*214·71.</b> \(\vdash . \text{Ded} \subset \text{semi Ded} \quad[*214·1·7]\)</p> + +<p class="nind"><b>*214·72.</b> \[\begin{align}& \vdash \colon\ldotp P \in \text{trans}.\supset :P \in \text{Ded}.\equiv .P \in \text{semi Ded}.\exists !\overrightarrow{B}ʻ\breve{P}\\ +&[*214·7·13.*205·121]\end{align}\]</p> + +<p class="nind"><b>*214·73.</b> \(\vdash . \text{semi Ded}-\iota ʻ\dot{\Lambda} \subset \text{ᗡ}ʻB \quad[*206·14.*211·44.*214·7]\)</p> + +<p><span class="pagenum" id="Page_690">[Pg 690]</span></p> + +<p>The proof of the following proposition is given in a somewhat +compressed form, since, if given with the usual fullness, it would +require various lemmas not required elsewhere.</p> + +<p class="nind"><b>*214·74.</b> \(\vdash :P\in \text{Ser}\cap \text{semi Ded}.\supset .P\unicode{x0294f}\overleftarrow{P}_{*}ʻx\in \text{semi Ded}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*214·7.&\supset \vdash :\text{Hp}.\alpha \in \text{sect}ʻP.\alpha \neq CʻP.\supset .\exists !(\overrightarrow{\text{max}}_{P}ʻ\alpha \cup \overrightarrow{\text{seq}} _{P}ʻ\alpha + ) &\qquad \text{(1)}\\ +\vdash .*205·261.\supset \\ +\vdash :\text{Hp}(1).\overleftarrow{P}_{*}ʻx{\sim}\in 1.x\in \alpha .\supset .\overrightarrow{\text{max}}(P\unicode{x0294f}\overleftarrow{P}_{*}ʻx)ʻ\alpha + &=\overrightarrow{\text{max}}_{P}ʻ(\alpha \cap \overleftarrow{P}_{*}ʻx)\\ +[*205·262] &=\overrightarrow{\text{max}}_{P}ʻ\alpha &\qquad \text{(2)}\\ +\vdash .*211·75·56.&\supset \vdash :\text{Hp}(2).\supset .CʻP-\alpha \subset \overleftarrow{P}_{*}ʻx &\qquad \text{(3)}\\ +\vdash .(3).*211·715.\supset \vdash :\text{Hp}(2).Q=P\unicode{x0294f}\overleftarrow{P}_{*}ʻx.\supset .\overrightarrow{\text{seq}} _{P}ʻ\alpha + &=\overrightarrow{\text{min}}_{P}ʻ(\overleftarrow{P}_{*}ʻx-\alpha )\\ +[*205·261] &=\overrightarrow{\text{min}}_{Q}ʻ(-\alpha )\\ +[*211·715.*206·25] &=\overrightarrow{\text{seq}} _{Q}ʻ(\alpha \cap \overleftarrow{P}_{*}ʻx) &\qquad \text{(4)}\\ +\vdash .(2).(4).\supset \\ +\vdash :\text{Hp}(4).\supset .\overrightarrow{\text{max}}_{P}ʻ\alpha \cup \overrightarrow{\text{seq}} _{P}ʻ\alpha &=\overrightarrow{\text{max}}_{Q}ʻ(\alpha + \cap \overleftarrow{P}_{*}ʻx)\cup \overrightarrow{\text{seq}} _{Q}ʻ(\alpha \cap \overleftarrow{P}_{*}ʻx) &\qquad \text{(5)}\\ +\vdash .(1).(5).\supset \vdash :\text{Hp}.\alpha \in \text{sect}ʻP.\alpha &\neq CʻP.\overleftarrow{P}_{*}ʻx{\sim}\in 1.x\in \alpha .Q=P\unicode{x0294f}\overleftarrow{P}_{*}ʻx.\supset + .\\ +&\exists !\{\overrightarrow{\text{max}}_{Q}ʻ(\alpha \cap \overleftarrow{P}_{*}ʻx)\cup \overrightarrow{\text{seq}} _{Q}ʻ(\alpha \cap \overleftarrow{P}_{*}ʻx)\} + &\qquad \text{(6)}\\ +\vdash .*211·715.\supset \vdash :\text{Hp}.\overleftarrow{P}_{*}ʻx{\sim}\in 1.\alpha &=\overrightarrow{P}ʻx.\supset .\overrightarrow{\text{seq}} _{P}ʻ\alpha + =\overrightarrow{\text{min}}_{P}ʻ\overleftarrow{P}_{*}ʻx\\ +[*205·261]&=\overrightarrow{\text{min}}(P\unicode{x0294f}\overleftarrow{P}_{*}ʻx)ʻ\overleftarrow{P}_{*}ʻx\\ +[*206·14] &=\overrightarrow{\text{seq}} (P\unicode{x0294f}\overleftarrow{P}_{*}ʻx)ʻ\Lambda &\qquad \text{(7)}\\ +\vdash .(7).*206·401&.\supset \vdash :\text{Hp}.\overleftarrow{P}_{*}ʻx{\sim}\in 1.\supset .\exists !\overrightarrow{\text{seq}} (P\unicode{x0294f}\overleftarrow{P}_{*}ʻx)ʻ\Lambda + &\qquad \text{(8)}\\ +\vdash .(6).(8).\supset \vdash \colon\ldotp \text{Hp}.\overleftarrow{P}_{*}ʻx&{\sim}\in 1.Q=P\unicode{x0294f}\overleftarrow{P}_{*}ʻx.\supset :\\ +&\beta \in \text{sect}ʻQ-\iota ʻCʻQ.\supset _{\beta }.\exists !(\overrightarrow{\text{max}}_{Q}ʻ\beta \cup \overrightarrow{\text{seq}} _{Q}ʻ\beta ):\\ +[*214·7] &\supset :Q\in \text{semi Ded} &\qquad \text{(9)}\\ +\vdash .*214·7.*200·35.&\supset \vdash :\text{Hp}.\overleftarrow{P}_{*}ʻx\in 1.\supset .P\unicode{x0294f}\overleftarrow{P}_{*}ʻx\in \text{semi Ded} &\qquad \text{(10)}\\ +\vdash .(9).(10).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*214·75.</b> \[\begin{align}& \vdash :P\in \text{semi Ded}.P\,\,\text{smor}\,\,Q.\supset .Q\in \text{semi Ded}\\ +&[*205·8.*206·61.*212·7]\end{align}\]</p> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_691">[Pg 691]</span></p> +<h2 class="nobreak" id="*215">*215. STRETCHES.</h2> +</div> + + +<p class="nind"><i>Summary of</i> *215.</p> + +<p>A <i>stretch</i> of a series is any piece taken out of it, and not +having any gaps; that is, it is a class contained in the series, and +containing all terms which come between any two of its terms. Thus it +is defined as +\[ +\hat{\alpha} (\alpha \subset CʻP.Pʻʻ\alpha \cap \breve{P} ʻʻ\alpha \subset \alpha ). +\]</p> + +<p>We denote the class of stretches by "\(\text{str}ʻP\)," where +"\(\text{str}\)" stands for "stretch" or "Strecke." A stretch which +has no predecessors is a section of \(P\); one which has no successors +is a section of \(\breve{P}\). The properties of stretches are chiefly +important in connection with compact series. In discrete series, +stretches are the same as intervals.</p> + +<p>If \(P\) is transitive, stretches of \(P\) are the products of sections +of \(P\) and sections of \(\breve{P}\), <i>i.e.</i> of upper and lower +sections of \(P\) (<a href="#*215·16">*215·16</a>). If \(P\) is connected, and \(\alpha\) is +a lower section, \(\beta\) an upper section, then if the two have a +stretch \(\alpha \cap \beta\) in common, we have +\[ +\alpha =Pʻʻ(\alpha \cap \beta )\cup (\alpha \cap \beta ).\beta =\breve{P} ʻʻ(\alpha \cap \beta )\cup (\alpha \cap \beta ) \quad(*215·161). +\] +A slightly more general form of this proposition is</p> + +<p class="nind"><b>*215·165.</b> \[\begin{align}&\vdash :P_{\text{po}}\in \text{connex} .\alpha \in \text{sect}ʻP.\beta \in \text{sect}ʻ\breve{P} .\exists !\alpha \cap \beta .\supset .\\ +&\alpha =P_{*}ʻʻ(\alpha \cap \beta ).\beta =\breve{P} _{*}ʻʻ(\alpha \cap \beta ).Pʻʻ\alpha =P_{\text{po}}ʻʻ(\alpha \cap \beta ).\breve{P} ʻʻ\beta =\breve{P} _{\text{po}}ʻʻ(\alpha + \cap \beta )\end{align}\]</p> + +<p>A specially important case is when \(\alpha\) and \(\beta\) have just one term +in common. In this case we have</p> + +<p class="nind"><b>*215·166.</b> \[\begin{align}&\vdash :P_{\text{po}}\in \text{Ser}.\alpha \in \text{sect}ʻP.\beta \in \text{sect}ʻ\breve{P} .\alpha \cap \beta \in 1.\supset .\\ +&\alpha \cap \beta =\iota ʻ\text{max}_{P}ʻ\alpha =\iota ʻ\text{min}_{P}ʻ\beta\end{align}\]</p> + +<p><span class="pagenum" id="Page_692">[Pg 692]</span></p> + +<p>When \(\alpha \cap \beta\) has more than one term, if the upper limit +or maximum of \(\alpha\) and the lower limit or minimum of \(\beta\) +both exist, the latter precedes the former (<a href="#*215·52">*215·52</a>); if \(\alpha\) +and \(\beta\) have no common part, but together exhaust the field of +\(P\), we have either \(\text{limax}_Pʻ\alpha =\text{limin}_Pʻ\beta\) +or \(\text{limax}_Pʻ\alpha P_{1}\text{limin}_Pʻ\beta\), assuming +\(\text{E}!\text{limax}_Pʻ\alpha .\text{E}!\text{limin}_Pʻ\beta\) +(<a href="#*215·54">*215·54</a>). Hence if \(\text{limax}_Pʻ\alpha\) has no immediate +successor, it must be identical with \(\text{limin}_Pʻ\beta\). Thus we +have</p> + +<p class="nind"><b>*215·543.</b> \[\begin{align}\vdash :P\in \text{Ser}.&\alpha \in \text{sect}ʻP.\beta \in \text{sect}ʻ\breve{P} .\alpha \cup \beta =CʻP.\alpha \cap \beta \in 0\cup 1.\\ +&\text{E}!\text{limax}_Pʻ\alpha .\text{limax}_Pʻ\alpha {\sim}\in \text{D}ʻP_{1}.\supset .\text{limax}_Pʻ\alpha =\text{limin}_Pʻ\beta\end{align}\]</p> + +<p>The above propositions will be useful in Section C (<a href="#*231">*231</a> and <a href="#*233">*233</a>).</p> + +<hr class="tb"> + +<p class="nind"><b>*215·01.</b> \(\text{str}ʻP=\hat{\alpha} (\alpha \subset CʻP.Pʻʻ\alpha \cap \breve{P} ʻʻ\alpha \subset \alpha ) \quad\text{Df}\)</p> + +<p class="nind"><b>*215·1.</b> \(\vdash :\alpha \in \text{str}ʻP.\equiv .\alpha \subset CʻP.Pʻʻ\alpha \cap \breve{P} ʻʻ\alpha \subset \alpha \quad[(*215·01)]\)</p> + +<p class="nind"><b>*215·11.</b> \(\vdash .\text{str}ʻP=\text{str}ʻ\breve{P} \quad[(*215·01).*33·22]\)</p> + +<p class="nind"><b>*215·13.</b> \(\vdash .\text{sect}ʻP\subset \text{str}ʻP.\text{sect}ʻ\breve{P} \subset \text{str}ʻP \quad[*215·1.*211·1]\)</p> + +<p class="nind"><b>*215·14.</b> \(\vdash :\alpha \in \text{sect}ʻP.\beta \in \text{sect}ʻ\breve{P} .\supset .\alpha \cap \beta \in \text{str}ʻP\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*211·1.\supset \vdash :\text{Hp}.&\supset .\alpha \subset CʻP.Pʻʻ\alpha \subset \alpha .\breve{P} ʻʻ\beta \subset \beta .\\ +[*22·43.*37·21] &\supset .\alpha \cap \beta \subset CʻP.Pʻʻ(\alpha \cap \beta )\subset \alpha .\breve{P} ʻʻ(\alpha \cap \beta )\subset \beta .\\ +[*22·49] &\supset .\alpha \cap \beta \subset CʻP.Pʻʻ(\alpha \cap \beta )\cap \breve{P} ʻʻ(\alpha \cap \beta )\subset \alpha \cap \beta .\\ +[*215·1] &\supset .\alpha \cap \beta \in \text{str}ʻP:\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*215·15.</b> \[\begin{align}&\vdash :P\in \text{trans}.\alpha \in \text{str}ʻP.\supset .\alpha \cup Pʻʻ\alpha \in \text{sect}ʻP.\alpha \cup \breve{P} ʻʻ\alpha \in \text{sect}ʻ\breve{P} .\\ +&\alpha =(\alpha \cup Pʻʻ\alpha )\cap (\alpha \cup \breve{P} ʻʻ\alpha )\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*211·27.*215·1.\supset \vdash :\text{Hp}.&\supset .\alpha \cup Pʻʻ\alpha \in \text{sect}ʻP.\alpha \cup \breve{P} ʻʻ\alpha \in \text{sect}ʻ\breve{P} &\qquad \text{(1)}\\ +\vdash .*215·1.*22·62. \supset \vdash :\text{Hp}.\supset .\alpha &=\alpha \cup (Pʻʻ\alpha \cap \breve{P} ʻʻ\alpha )\\ +[*22·69] &=(\alpha \cup Pʻʻ\alpha )\cap (\alpha \cup \breve{P} ʻʻ\alpha ) &\qquad \text{(2)}\\ +\vdash .(1).(2).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*215·16">*215·16</a>.</b> + \[\begin{align}\vdash :P\in \text{trans}.\supset .\text{str}ʻP=\hat{\gamma} \{(\exists \alpha ,\beta ).\alpha \in \text{sect}ʻP.\beta \in \text{sect}ʻ\breve{P} .\gamma &=\alpha \cap \beta\}\\ +&=sʻ\{(\text{sect}ʻP)\cap_{,,} ʻʻ\text{sect}ʻ\breve{P}\}\\ +[*215·14·15.*40·7]\end{align}\]</p> + +<p class="nind"><b>*215·161.</b> \[\begin{align}\vdash :P\in \text{connex} .&\alpha \in \text{sect}ʻP.\beta \in \text{sect}ʻ\breve{P} .\exists !\alpha \cap \beta .\supset .\\ +&\alpha =Pʻʻ(\alpha \cap \beta )\cup (\alpha \cap \beta ).\beta =\breve{P} ʻʻ(\alpha \cap \beta )\cup (\alpha \cap \beta )\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*211·1.*37·2.&\supset \vdash :\text{Hp}.\supset .Pʻʻ(\alpha \cap \beta )\cup (\alpha \cap \beta )\subset \alpha &\qquad \text{(1)}\\ +\vdash .*211·702. \supset \vdash \colon\ldotp \text{Hp}.x\in \alpha -\beta .&\supset :y\in \beta .\supset .xPy:\\ +[*37·1] &\supset :\exists !(\alpha \cap \beta ).\supset .x\in Pʻʻ(\alpha \cap \beta ) &\qquad \text{(2)}\\ +\vdash .(2). & \supset \vdash :\text{Hp}.x\in \alpha .\supset .x\in Pʻʻ(\alpha \cap \beta )\cup (\alpha \cap \beta ) &\qquad \text{(3)}\\ +\vdash .(1).(3).&\supset \vdash :\text{Hp}.\supset .\alpha =Pʻʻ(\alpha \cap \beta )\cup (\alpha \cap \beta ) &\qquad \text{(4)}\\ +\vdash .(4)\frac{\breve{P}}{P}.&\supset \vdash :\text{Hp}.\supset \beta =\breve{P} ʻʻ(\alpha \cap \beta )\cup (\alpha \cap \beta ) &\qquad \text{(5)}\\ +\vdash .(4).(5).&\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_693">[Pg 693]</span></p> + +<p class="nind"><b>*215·162.</b> \[\begin{align}\vdash :P\in \text{trans}\cap \text{connex} .\alpha \in &\text{sect}ʻP.\beta \in \text{sect}ʻ\breve{P} .\exists !\alpha \cap \beta .\supset .\\ +&Pʻʻ\alpha =Pʻʻ(\alpha \cap \beta ).\breve{P} ʻʻ\beta =\breve{P} ʻʻ(\alpha \cap \beta )\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*215·161.\supset \vdash :\text{Hp}.\supset .Pʻʻ\alpha &=PʻʻPʻʻ(\alpha \cap \beta )\cup Pʻʻ(\alpha \cap \beta )\\ +[*201·5] & =Pʻʻ(\alpha \cap \beta ) &\qquad \text{(1)}\\ +\text{Similarly}\quad &\vdash :\text{Hp}.\supset .\breve{P} ʻʻ\beta =\breve{P} ʻʻ(\alpha \cap \beta ) &\qquad \text{(2)}\\ +\vdash .(1).(2).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*215·163.</b> \[\begin{align}\vdash :P\in \text{trans}\cap \text{connex} .\alpha \in \text{sect}ʻP.\beta \in &\text{sect}ʻ\breve{P} .\exists !\alpha \cap \beta .\supset .\\ +&pʻ\overleftarrow{P}ʻʻ\alpha =pʻ\overleftarrow{P}ʻʻ(\alpha \cap \beta )\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*40·16. &\supset \vdash .pʻ\overleftarrow{P}ʻʻ\alpha \subset pʻ\overleftarrow{P}ʻʻ(\alpha \cap \beta ) &\qquad \text{(1)}\\ +\vdash .*10·56.*37·1.&\supset \vdash \colon\ldotp \text{Hp}:y\in \alpha \cap \beta .\supset _{y}.yPx:z\in Pʻʻ(\alpha \cap \beta ):\supset .zPx &\qquad \text{(2)}\\ +\vdash .(2).*215·161.&\supset \vdash \colon\ldotp \text{Hp}:y\in \alpha \cap \beta .\supset _{y}.yPx:\supset :z\in \alpha .\supset _{z}.zPx &\qquad \text{(3)}\\ +\vdash .(1).(3).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*215·164.</b> \[\begin{align}\vdash :\text{Hp}*215·162.\supset .&\overrightarrow{\text{min}}_{P}ʻ\beta =\overrightarrow{\text{min}}_{P}ʻ(\alpha + \cap \beta ).\overrightarrow{\text{max}}_{P}ʻ\alpha =\overrightarrow{\text{max}}_{P}ʻ(\alpha \cap \beta ).\\ +&\overrightarrow{\text{seq}} _{P}ʻ\alpha =\overrightarrow{\text{seq}} _{P}ʻ(\alpha \cap \beta ).\overrightarrow{\text{prec}} _{P}ʻ\beta + =\overrightarrow{\text{prec}} _{P}ʻ(\alpha \cap \beta ).\\ +&\overrightarrow{\text{lt}} _{P}ʻ\alpha =\overrightarrow{\text{lt}} _{P}ʻ(\alpha \cap \beta ).\overrightarrow{\text{limax}} _{P}ʻ\alpha + =\overrightarrow{\text{limax}} _{P}ʻ(\alpha \cap \beta )\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*215·162. \supset \vdash :\text{Hp}.\supset .\overrightarrow{\text{max}} _{P}ʻ\alpha &=\alpha -Pʻʻ(\alpha \cap \beta )\\ +[*215·161] &=\alpha \cap \beta -Pʻʻ(\alpha \cap \beta )\\ +[*205·111] &=\overrightarrow{\text{max}}_{P}ʻ(\alpha \cup \beta ) &\qquad \text{(1)}\\ +\text{Similarly}\quad &\vdash :\text{Hp}.\supset .\overrightarrow{\text{min}}_{P}ʻ\beta =\overrightarrow{\text{min}}_{P}ʻ(\alpha \cap \beta ) &\qquad \text{(2)}\\ +\vdash .*215·163.*206·13. &\supset \vdash :\text{Hp}.\supset .\overrightarrow{\text{seq}} _{P}ʻ\alpha =\overrightarrow{\text{seq}} _{P}ʻ(\alpha + \cap \beta ) &\qquad \text{(3)}\\ +\text{Similarly}\quad & \vdash :\text{Hp}.\supset .\overrightarrow{\text{prec}} _{P}ʻ\beta =\overrightarrow{\text{prec}} _{P}ʻ(\alpha \cap \beta ) &\qquad \text{(4)}\\ +\vdash .(1).(3).*207·11·12.&\supset \vdash :\text{Hp}.\supset .\overrightarrow{\text{lt}} _{P}ʻ\alpha =\overrightarrow{\text{lt}} _{P}ʻ(\alpha + \cap \beta ) &\qquad \text{(5)}\\ +\vdash .(1).(5).*207·45. &\supset \vdash :\text{Hp}.\supset .\overrightarrow{\text{limax}} _{P}ʻ\alpha =\overrightarrow{\text{limax}} _{P}ʻ(\alpha + \cap \beta ) &\qquad \text{(6)}\\ +\vdash .(1).(2).(3).(4).(5).(6).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*215·165.</b> \[\begin{align}&\vdash :P_{\text{po}}\in \text{connex} .\alpha \in \text{sect}ʻP.\beta \in \text{sect}ʻ\breve{P} .\exists !\alpha \cap \beta .\supset .\\ +&\alpha =P_{*}ʻʻ(\alpha \cap \beta ).\beta =\breve{P} _{*}ʻʻ(\alpha \cap \beta ).Pʻʻ\alpha =P_{\text{po}}ʻʻ(\alpha \cap \beta ).\breve{P} ʻʻ\beta =\breve{P} _{\text{po}}ʻʻ(\alpha + \cap \beta )\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*211·17.\supset \vdash :\text{Hp}.&\supset .\alpha \in \text{sect}ʻP_{\text{po}}.\beta \in \text{sect}ʻ\breve{P} _{\text{po}}.\exists !\alpha \cap \beta .\\ +[*215·161] &\supset .\alpha =P_{*}ʻʻ(\alpha \cap \beta ).\beta =\breve{P} _{*}ʻʻ(\alpha \cap \beta ). &\qquad \text{(1)}\\ +[*91·52] & \supset .Pʻʻ\alpha =P_{\text{po}}ʻʻ(\alpha \cap \beta ).\breve{P} ʻʻ\beta =\breve{P} _{\text{po}}ʻʻ(\alpha \cap \beta )&\qquad \text{(2)}\\ +\vdash .(1).(2).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_694">[Pg 694]</span></p> + +<p class="nind"><b>*215·166.</b> \[\begin{align}\vdash :P_{\text{po}}\in \text{Ser}.\alpha \in \text{sect}ʻP.\beta \in &\text{sect}ʻ\breve{P} .\alpha \cap \beta \in 1.\supset .\\ +&\alpha \cap \beta = \iota ʻ\text{max}_{P}ʻ\alpha = \iota ʻ\text{min}_{P}ʻ\beta\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*215·161.*211·17.\supset \vdash :\text{Hp}.&\supset .\alpha = (\alpha \cap \beta )\cup P_{\text{po}}ʻʻ(\alpha \cap \beta ).\\ +[*215·165] &\supset .\alpha -Pʻʻ\alpha = (\alpha \cap \beta )-P_{\text{po}}ʻʻ(\alpha \cap \beta ).\\ +[*205·11] \supset .\overrightarrow{\text{max}}_{P}ʻ\alpha &= \overrightarrow{\text{max}}(P_{\text{po}})ʻ(\alpha \cap \beta )\\ +[*205·17] & = \alpha \cap \beta &\qquad \text{(1)}\\ +\text{Similarly}\quad &\vdash :\text{Hp}.\supset .\overrightarrow{\text{min}}_{P}ʻ\beta = \alpha \cap \beta &\qquad \text{(2)}\\ +\vdash .(1).(2).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*215·17.</b> \(\vdash :P\in \text{trans}.\supset .\breve{P} ʻʻ\alpha \cap Pʻʻ\beta \in \text{str}ʻP\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*211·15·11.\supset \vdash :\text{Hp}.\supset .Pʻʻ\beta \in \text{sect}ʻP.\breve{P} ʻʻ\alpha \in \text{sect}ʻ\breve{P} &\qquad \text{(1)}\\ +\vdash .(1).*215·14.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*215·18.</b> \(\vdash .P(x\vdash\dashv y),P(x\unicode{x27dd} y),P(x\dashv y),P(x-y)\in \text{str}ʻP\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*211·13·3.&\supset \vdash .\overrightarrow{P}_{*}ʻy\in \text{sect}ʻP.\overleftarrow{P}_{*}ʻx\in \text{sect}ʻ\breve{P} &\qquad \text{(1)}\\ +\vdash .*211·16. &\supset \vdash .\overrightarrow{P}_{\text{po}}ʻy\in \text{sect}ʻP.\overleftarrow{P}_{\text{po}}ʻx\in \text{sect}ʻ\breve{P} &\qquad \text{(2)}\\ +\vdash .(1).(2).*215·14.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*215·19.</b> \(\vdash :P^{2}\,\unicode{x2abd}\, J.x\in CʻP.\supset .\iota ʻx\in \text{str}ʻP\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*53·301. &\supset \vdash .Pʻʻ\iota ʻx\cap \breve{P} ʻʻ\iota ʻx = \overrightarrow{P}ʻx\cap \overleftarrow{P}ʻx &\qquad \text{(1)}\\ +\vdash .(1).*50·43.&\supset \vdash .:\text{Hp}.\supset .Pʻʻ\iota ʻx\cap \breve{P} ʻʻ\iota ʻx = \Lambda &\qquad \text{(2)}\\ +\vdash .(2).*215·1.&\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*215·2.</b> \[\begin{align}\vdash :P\in \text{connex} .\alpha \in \text{str}ʻP.x\in \alpha .\supset .&Pʻʻ\alpha = \alpha -\overrightarrow{\text{max}}_{P}ʻ\alpha + \cup \overrightarrow{P}ʻx.\\ +&\breve{P} ʻʻ\alpha = \alpha -\overrightarrow{\text{min}}_{P}ʻ\alpha \cup \overleftarrow{P}ʻx\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*205·111.&\supset \vdash .\alpha -\overrightarrow{\text{max}}_{P}ʻ\alpha \subset Pʻʻ\alpha &\qquad \text{(1)}\\ +\vdash .*37·18 . &\supset \vdash :\text{Hp}.\supset .\overrightarrow{P}ʻx\subset Pʻʻ\alpha &\qquad \text{(2)}\\ +\vdash .(1).(2). &\supset \vdash :\text{Hp}.\supset .\alpha -\overrightarrow{\text{max}}_{P}ʻ\alpha \cup \overrightarrow{P}ʻx\subset Pʻʻ\alpha &\qquad \text{(3)}\\ +\vdash .*202·103.&\supset \vdash \colon\ldotp \text{Hp}.y\in Pʻʻ\alpha .\supset :y\in \overrightarrow{P}ʻx\cup \iota ʻx\cup \overleftarrow{P}ʻx:\\ +[*37·181] &\supset :y\in \overrightarrow{P}ʻx\cup \iota ʻx.\lor.y\in \breve{P} ʻʻ\alpha :\\ +[*4·73] &\supset :y\in \overrightarrow{P}ʻx\cup \iota ʻx.\lor.y\in Pʻʻ\alpha \cap \breve{P} ʻʻ\alpha :\\ +[*215·1] &\supset :y\in \overrightarrow{P}ʻx\cup \iota ʻx\cup \alpha :\\ +[\text{Hp}] &\supset :y\in \overrightarrow{P}ʻx\cup \alpha &\qquad \text{(4)}\\ +\vdash .*205·111.&\supset \vdash .y\in Pʻʻ\alpha .\supset .y{\sim}\in \overrightarrow{\text{max}}_{P}ʻ\alpha &\qquad \text{(5)}\\ +\vdash .(4).(5). &\supset \vdash :\text{Hp}.y\in Pʻʻ\alpha .\supset .y\in \alpha -\overrightarrow{\text{max}}_{P}ʻ\alpha \cup \overrightarrow{P}ʻx &\qquad \text{(6)}\\ +\vdash .(3).(6). &\supset \vdash :\text{Hp}.\supset .Pʻʻ\alpha =\alpha -\overrightarrow{\text{max}}_{P}ʻ\alpha \cup \overrightarrow{P}ʻx &\qquad \text{(7)}\\ +\text{Similarly}\quad &\vdash :\text{Hp}.\supset .\breve{P} ʻʻ\alpha =\alpha -\overrightarrow{\text{min}}_{P}ʻ\alpha \cup \overleftarrow{P}ʻx &\qquad \text{(8)}\\ +\vdash .(7).(8).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_695">[Pg 695]</span></p> + +<p class="nind"><b>*215·21.</b> \(\vdash :P\in \text{connex} .\alpha ,\beta \in \text{str}ʻP.\exists !\alpha \cap \beta .\supset .\alpha \cap \beta \in \text{str}ʻP\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*215·2.&\supset \vdash :\text{Hp}.\supset .(\exists x).x\in \alpha \cap \beta .Pʻʻ\alpha \subset \alpha \cup \overrightarrow{P}ʻx.Pʻʻ\beta \subset \beta \cup \overrightarrow{P}ʻx.\\ +&\breve{P} ʻʻ\alpha \subset \alpha \cup \overleftarrow{P}ʻx.\breve{P} ʻʻ\beta \subset \beta \cup \overleftarrow{P}ʻx.\\ +[*22·68]&\supset .(\exists x).x\in \alpha \cap \beta .Pʻʻ\alpha \cap Pʻʻ\beta \subset (\alpha \cap \beta )\cup \overrightarrow{P}ʻx.\\ +&\breve{P} ʻʻ\alpha \cap \breve{P} ʻʻ\beta \subset (\alpha \cap \beta )\cup \overleftarrow{P}ʻx.\\ +[*37·21]&\supset .(\exists x).x\in \alpha \cap \beta .Pʻʻ(\alpha \cap \beta )\subset (\alpha \cap \beta )\cup \overrightarrow{P}ʻx.\breve{P} ʻʻ(\alpha \cap \beta )\subset (\alpha \cap \beta )\cup \overleftarrow{P}ʻx.\\ +[*22·69]&\supset .(\exists x).x\in \alpha \cap \beta .Pʻʻ(\alpha \cap \beta )\cap \breve{P} ʻʻ(\alpha \cap \beta )\subset (\alpha \cap \beta )\cup (\overrightarrow{P}ʻx\cap \overleftarrow{P}ʻx) &\qquad \text{(1)}\\ +\vdash .*37·18.\supset \vdash :x\in \alpha \cap \beta .&\supset .\overrightarrow{P}ʻx\subset Pʻʻ\alpha \cap Pʻʻ\beta .\overleftarrow{P}ʻx\subset \breve{P} ʻʻ\alpha \cap \breve{P} ʻʻ\beta .\\ +[*22·49] &\supset .\overrightarrow{P}ʻx\cap \overleftarrow{P}ʻx\subset Pʻʻ\alpha \cap \breve{P} ʻʻ\alpha \cap Pʻʻ\beta \cap \breve{P} ʻʻ\beta &\qquad \text{(2)}\\ +\vdash .(2).*215·1.&\supset \vdash \colon\ldotp \text{Hp}.\supset :x\in \alpha \cap \beta .\supset .\overrightarrow{P}ʻx\cap \overleftarrow{P}ʻx\subset \alpha \cap \beta &\qquad \text{(3)}\\ +\vdash .(1).(3). \supset \vdash :\text{Hp}.&\supset .(\exists x).x\in \alpha \cap \beta .Pʻʻ(\alpha \cap \beta )\cap \breve{P} ʻʻ(\alpha \cap \beta )\subset \alpha \cap \beta .\\ +[*215·1] &\supset .\alpha \cap \beta \in \text{str}ʻP:\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*215·22.</b> \(\vdash :\alpha ,\beta \in \text{str}ʻP.\supset .\alpha \cap \beta \in \text{str}ʻP\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*215·1.\supset \vdash :\text{Hp}.&\supset .\alpha \subset CʻP.\beta \subset CʻP.Pʻʻ\alpha \cap \breve{P} ʻʻ\alpha \subset \alpha .Pʻʻ\beta \cap \breve{P} ʻʻ\beta \subset \beta .\\ +[*22·47·49] &\supset .\alpha \cap \beta \subset CʻP.Pʻʻ\alpha \cap Pʻʻ\beta \cap \breve{P} ʻʻ\alpha \cap \breve{P} ʻʻ\beta \subset \alpha \cap \beta .\\ +[*37·21] & \supset .\alpha \cap \beta \subset CʻP.Pʻʻ(\alpha \cap \beta )\cap \breve{P} ʻʻ(\alpha \cap \beta )\subset \alpha \cap \beta .\\ +[*215·1] &\supset .\alpha \cap \beta \in \text{str}ʻP:\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*215·23.</b> \(\vdash :P\in \text{connex} .\mu \subset \text{str}ʻP.\exists !pʻ\mu .\supset .sʻ\mu \in \text{str}ʻP\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*215·2.\supset \vdash \colon\ldotp \text{Hp}.x\in pʻ\mu .&\supset :\alpha \in \mu .\supset _{\alpha }.Pʻʻ\alpha + \subset \alpha \cup \overrightarrow{P}ʻx.\breve{P} ʻʻ\alpha \subset \alpha \cup \overleftarrow{P}ʻx:\\ +[*40·13] &\supset :\alpha \in \mu .\supset _{\alpha }.Pʻʻ\alpha \subset sʻ\mu \cup \overrightarrow{P}ʻx.\breve{P} ʻʻ\alpha \subset sʻ\mu \cup \overleftarrow{P}ʻx:\\ +[*40·43·38] &\supset :Pʻʻsʻ\mu \subset sʻ\mu \cup \overrightarrow{P}ʻx.\breve{P} ʻʻsʻ\mu \subset sʻ\mu \cup \overleftarrow{P}ʻx:\\ +[*22·49·69] &\supset :Pʻʻsʻ\mu \cap \breve{P} ʻʻsʻ\mu \subset sʻ\mu \cup (\overrightarrow{P}ʻx\cap \overleftarrow{P}ʻx) &\qquad \text{(1)}\\ +\vdash .*40·14.\supset \vdash :\text{Hp}.x\in pʻ\mu .\alpha \in \mu .&\supset .x\in \alpha .\alpha \in \text{str}ʻP.\\ +[*37·18] &\supset .\overrightarrow{P}ʻx\cap \overleftarrow{P}ʻx\subset Pʻʻ\alpha \cap \breve{P} ʻʻ\alpha .\alpha \in \text{str}ʻP.\\ +[*215·1] &\supset .\overrightarrow{P}ʻx\cap \overleftarrow{P}ʻx\subset \alpha .\\ +[*40·13] &\supset .\overrightarrow{P}ʻx\cap \overleftarrow{P}ʻx\subset sʻ\mu &\qquad \text{(2)}\\ +\vdash .(1).(2).&\supset \vdash :\text{Hp}.\exists !\mu .\supset .P ʻʻsʻ\mu \cap \breve{P} ʻʻsʻ\mu \subset sʻ\mu &\qquad \text{(3)}\\ +\vdash .*37·29. &\supset \vdash :\mu =\Lambda .\supset .P ʻʻsʻ\mu \cap \breve{P} ʻʻsʻ\mu \subset sʻ\mu &\qquad \text{(4)}\\ +\vdash .(3).(4).&\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_696">[Pg 696]</span></p> + +<p class="nind"><b>*215·24.</b> \(\vdash :\mu \subset \text{str}ʻP.\supset .CʻP\cap pʻ\mu \in \text{str}ʻP\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*37·265.&\supset \vdash .P ʻʻ(pʻ\mu \cap CʻP)\cap \breve{P} ʻʻ(pʻ\mu \cap CʻP)=P ʻʻpʻ\mu \cap \breve{P} ʻʻpʻ\mu &\qquad \text{(1)}\\ +\vdash .*37·2. &\supset \vdash :\alpha \in \mu .\supset .Pʻʻpʻ\mu \cap \breve{P} ʻʻpʻ\mu \subset P ʻʻ\alpha \cap \breve{P} ʻʻ\alpha &\qquad \text{(2)}\\ +\vdash .(2).*215·1.\supset \vdash \colon\ldotp \text{Hp}.&\supset :\alpha \in \mu .\supset .Pʻʻpʻ\mu \cap \breve{P} ʻʻpʻ\mu \subset \alpha :\\ +[*40·15] &\supset :Pʻʻpʻ\mu \cap \breve{P} ʻʻpʻ\mu \subset pʻ\mu &\qquad \text{(3)}\\ +\vdash .(1).(3).*215·1.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*215·25.</b> \(\vdash :\mu \subset \text{str}ʻP.\exists !\mu .\supset .pʻ\mu \in \text{str}ʻP\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*40·24.*215·1.\supset \vdash :\text{Hp}.\supset .pʻ\mu \subset CʻP &\qquad \text{(1)}\\ +\vdash .(1).*215·24.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*215·3.</b> \[\begin{align}\vdash \colon\ldotp P\in &\text{connex} .\alpha ,\beta \in \text{str}ʻP-{℩}ʻ\Lambda .\alpha \cap \beta =\Lambda .\supset :\\ +&\alpha \subset Pʻʻ\beta .\equiv .\alpha \subset pʻ\overrightarrow{P}ʻʻ\beta .\equiv .\beta \subset pʻ\overleftarrow{P}ʻʻ\alpha .\equiv .\beta \subset \breve{P} ʻʻ\alpha\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*215·1.&\supset \vdash :\text{Hp}.\supset .\alpha \subset CʻP-\beta &\qquad \text{(1)}\\ +\vdash .*22·48.&\supset \vdash :\alpha \subset P ʻʻ\beta .\supset .\alpha \cap \breve{P} ʻʻ\beta \subset Pʻʻ\beta \cap \breve{P} ʻʻ\beta :\\ +[*215·1] \supset \vdash :\text{Hp}.\alpha \subset Pʻʻ\beta .&\supset .\alpha \cap \breve{P} ʻʻ\beta \subset \beta .\\ +[*22·621.\text{Hp}] &\supset .\alpha \cap \breve{P} ʻʻ\beta =\Lambda &\qquad \text{(2)}\\ +\vdash .(1).(2).\supset \vdash :\text{Hp}.\alpha \subset P ʻʻ\beta .&\supset .\alpha \subset CʻP-\beta -\breve{P} ʻʻ\beta .\\ +[*202·501] &\supset .\alpha \subset pʻ\overrightarrow{P}ʻʻ\beta &\qquad \text{(3)}\\ +\vdash .*40·61. &\supset \vdash \colon\ldotp \text{Hp}.\alpha \subset pʻ\overrightarrow{P} ʻʻ\beta .\supset .\alpha \subset Pʻʻ\beta &\qquad \text{(4)}\\ +\vdash .(3).(4).\supset \vdash \colon\ldotp \text{Hp}.\supset :\alpha \subset Pʻʻ\beta .&\equiv .\alpha \subset pʻ\overrightarrow{P}ʻʻ\beta . &\qquad \text{(5)}\\ +[*40·67] &\equiv .\beta \subset pʻ\overleftarrow{P}ʻʻ\alpha .&\qquad \text{(6)}\\ +\left[(5) \frac{\breve{P}}{P}\right] & \equiv .\beta \subset \breve{P} ʻʻ\alpha &\qquad \text{(7)}\\ +\vdash .(5).(6).(7).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_697">[Pg 697]</span></p> + +<p class="nind"><b>*215·31.</b> \[\begin{align}\vdash :P\in \text{trans} \cap \text{connex} .\alpha \in \text{str}ʻP.\text{E}!&\text{min}_{P}ʻ\alpha .\text{E}!\text{max}_{P}ʻ\alpha + .\supset .\\ +&\alpha = P(\text{min}_{P}ʻ\alpha \vdash\dashv \text{max}_{P}ʻ\alpha )\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*205·2.*90·15·151.&\supset \vdash :\text{Hp}.y\in \alpha .\supset .\text{min}_{P}ʻ\alpha P_{*}y &\qquad \text{(1)}\\ +\vdash .(1)\frac{\breve{P}}{P}.*205·102. &\supset \vdash :\text{Hp}.y\in \alpha .\supset .yP_{*}\text{max}_{P}ʻ\alpha &\qquad \text{(2)}\\ +\vdash .(1).(2).*121·103. &\supset \vdash :\text{Hp}.\supset .\alpha \subset P(\text{min}_{P}ʻ\alpha \vdash\dashv \text{max}_{P}ʻ\alpha ) &\qquad \text{(3)}\\ +\vdash .*121·242.*201·19.*205·2.\supset \vdash :\text{Hp}.\supset .\\ +P(\text{min}_{P}ʻ\alpha \vdash\dashv \text{max}_{P}ʻ\alpha )&=\iota ʻ\text{min}_{P}ʻ\alpha \cup (\overleftarrow{P}ʻ\text{min}_{P}ʻ\alpha + \cap \overrightarrow{P}ʻ\text{max}_{P}ʻ\alpha )\cup \iota ʻ\text{max}_{P}ʻ\alpha\\ +[*37·18] &\subset \iota ʻ\text{min}_{P}ʻ\alpha \cup (\breve{P} ʻʻ\alpha \cap Pʻʻ\alpha )\cup \iota ʻ\text{max}_{P}ʻ\alpha \\ +[*205·11·111.*215·1] &\subset \alpha &\qquad \text{(4)}\\ +\vdash .(3).(4).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*215·32.</b> \[\begin{align}\vdash :P\in \text{trans} \cap \text{connex} .\alpha \in \text{str}ʻP.\text{E}!&\text{min}_{P}ʻ\alpha .\text{E}!\text{seq}_{P}ʻ\alpha + .\supset .\\ +&\alpha=P(\text{min}_{P}ʻ\alpha \vdash \text{seq}_{P}ʻ\alpha )\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*206·211.*205·2.\supset \vdash :\text{Hp}.&\supset .\alpha \subset \overleftarrow{P}_{*}ʻ\text{min}_{P}ʻ\alpha \cap \overrightarrow{P}ʻ\text{seq}_{P}ʻ\alpha + &\qquad \text{(1)}\\ +\vdash .*206·22.*205·22.\supset \vdash :\text{Hp}.&\supset .\overleftarrow{P}ʻ\text{min}_{P}ʻ\alpha \cap \overrightarrow{P}ʻ\text{seq}_{P}ʻ\alpha + =\breve{P} ʻʻ\alpha \cap (\alpha \cup Pʻʻ\alpha )\\ +[*215·1] &\subset \alpha .\\ +[*201·19.*121·241] &\supset .P(\text{min}_{P}ʻ\alpha \vdash \text{seq}_{P}ʻ\alpha )\subset \alpha &\qquad \text{(2)}\\ +\vdash .(1).(2).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*215·33.</b> \[\begin{align}\vdash :P\in \text{trans} \cap \text{connex} .&\alpha \in \text{str}ʻP.\text{E}!\text{prec}_{P}ʻ\alpha .\text{E}!\text{seq}_{P}ʻ\alpha + .\supset .\\ +&\alpha =P(\text{prec}_{P}ʻ\alpha -\text{seq}_{P}ʻ\alpha ) \quad[*206·22.*215·1]\end{align}\]</p> + +<p class="nind"><b>*215·4.</b> \(\vdash :P\in \text{connex} .\mu \in \text{Cl excl}ʻ(\text{str}ʻP-\iota ʻ\Lambda ).\supset .P_{\text{cl}}\unicode{x0294f}\mu =P_{\text{lc}}\unicode{x0294f}\mu + \)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*84·12.\supset \vdash \colon\ldotp \text{Hp}.&\supset :\alpha ,\beta \in \mu .\alpha \neq \beta .\supset .\alpha \cap \beta =\Lambda : &\qquad \text{(1)}\\ +[*170·1] \supset :\alpha (P_{\text{cl}}\unicode{x0294f}\mu )\beta .&\equiv .\alpha ,\beta \in \mu .\exists !\alpha -\breve{P} ʻʻ\beta .\\ +[*215·3.\text{Transp}] &\equiv .\alpha ,\beta \in \mu .\exists !\beta -Pʻʻ\alpha .\\ +[(1).*170·102] &\equiv .\alpha ,\beta \in \mu .\alpha P_{\text{lc}}\mu \colon\ldotp \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*215·41.</b> \(\vdash :P\in \text{trans} \cap \text{connex} .\mu \in \text{Cl excl}ʻ(\text{str}ʻP-\iota ʻ\Lambda ).\supset .P_{\text{lc}}\unicode{x0294f}\mu \in \text{Ser}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*84·12.*170·102.&\supset \vdash \colon\ldotp \text{Hp}.\supset :\alpha (P_{\text{lc}}\unicode{x0294f}\mu )\beta .\equiv .\exists !\beta -Pʻʻ\alpha &\qquad \text{(1)}\\ +\vdash .*215·3.\supset \\ +\vdash :\text{Hp}.\alpha ,\beta \in \mu .\alpha \subset Pʻʻ\beta .\beta \subset Pʻʻ\alpha .&\supset .\alpha \subset Pʻʻ\beta .\alpha \subset \breve{P} ʻʻ\beta .\\ +[*215·1] &\supset .\alpha \subset \beta &\qquad \text{(2)}\\ +\text{Similarly}\quad &\vdash :\text{Hp}(2).\supset .\beta \subset \alpha &\qquad \text{(3)}\\ +\vdash .(2).(3).\supset \vdash \colon\colon\text{Hp}.\alpha ,\beta \in \mu .&\supset \colon\ldotp \alpha \subset Pʻʻ\beta .\beta \subset Pʻʻ\alpha .\supset .\alpha =\beta \colon\ldotp \\ +[\text{Transp}.(1)] & \supset \colon\ldotp \alpha \neq \beta .\supset :\alpha (P_{\text{lc}}\unicode{x0294f}\mu )\beta .\lor.\beta (P_{\text{lc}}\unicode{x0294f}\mu + )\alpha &\qquad \text{(4)}\\ +\vdash .*37·1.\supset \\ +\vdash \colon\colon \text{Hp}.\beta \cap \gamma =\Lambda .{\sim}(\gamma \subset Pʻʻ\beta ).&\supset \colon\ldotp (\exists z):z\in \gamma :y\in \beta .\supset _{y}.{\sim}(zPy).z\neq + y\colon\ldotp \\ +[*202·103] &\supset \colon\ldotp (\exists z):z\in \gamma :y\in \beta .\supset _{y}.yPz\colon\ldotp \\ +[*11·61] & \supset \colon\ldotp y\in \beta .\supset _{y}.(\exists z).z\in \gamma .yPz\colon\ldotp \\ +[*37·1] &\supset \colon\ldotp \beta \subset Pʻʻ\gamma \colon\ldotp \\ +[*201·5.*37·2] &\supset \colon\ldotp \gamma \subset Pʻʻ\alpha .\supset .\beta \subset Pʻʻ\alpha \colon\ldotp \\ +[\text{Transp}] &\supset \colon\ldotp \exists !\beta -Pʻʻ\alpha .\supset .\exists !\gamma -Pʻʻ\alpha &\qquad \text{(5)}\\ +\vdash .(5).(1).&\supset \vdash \colon\ldotp \text{Hp}.\supset :\alpha (P_{\text{lc}}\unicode{x0294f}\mu )\beta ·\beta (P_{\text{lc}}\unicode{x0294f}\mu + )\gamma .\supset .\alpha (P_{\text{lc}}\unicode{x0294f}\mu )\gamma &\qquad \text{(6)}\\ +\vdash .(4).(6).*170·17.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_698">[Pg 698]</span></p> + +<p class="nind"><b>*215·42.</b> \[\begin{align}&\vdash :P\in \text{trans}\cap \text{connex} .\mu \in \text{Cl excl}ʻ(\text{str}ʻP-{℩}ʻ\Lambda ).\mu {\sim}\in 1.\supset .CʻP_{\text{lc}}\unicode{x0294f}\mu + =\mu \\ +&[*202·55.*215·41]\end{align}\]</p> + +<p class="nind"><b><a id="*215·5">*215·5</a>.</b> \[\begin{align}&\vdash \colon\ldotp P\in \text{trans}\cap \text{connex} .\alpha \in \text{sect}ʻP.\beta \in \text{sect}ʻ\breve{P} .\supset :\\ +&\exists !\alpha \cap \beta .\text{limax}_Pʻ\alpha =\text{limin}_Pʻ\beta .\supset .\alpha \cap \beta \in 1 \quad[*207·71.*215·164]\end{align}\]</p> + +<p class="nind"><b><a id="*215·51">*215·51</a>.</b> \[\begin{align}\vdash :P\in \text{Ser}.\alpha \in &\text{sect}ʻP.\beta \in \text{sect}ʻ\breve{P} .\alpha \cap \beta \in 1.\supset .\\ +&\text{limax}_Pʻ\alpha =\text{limin}_Pʻ\beta =\breve{{℩}} ʻ(\alpha \cap \beta ) \quad[*207·72.*215·164]\end{align}\]</p> + +<p class="nind"><b><a id="*215·52">*215·52</a>.</b> \[\begin{align}\vdash :\text{Hp}*215·5.\alpha \cap \beta {\sim}\in 0\cup 1.\text{E}!\text{limax}_Pʻ\alpha .\text{E}!&\text{limin}_Pʻ\beta .\supset .\\ +&\text{limin}_Pʻ\beta P\text{limax}_Pʻ\alpha\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*215·164.\supset \vdash \colon\ldotp \text{Hp}.\supset :\text{limax}_Pʻ\alpha &=\text{max}_{P}ʻ(\alpha \cap \beta ).\lor.\text{limax}_Pʻ\alpha =\text{seq}_{P}ʻ(\alpha + \cap \beta ):\\ +&\text{limin}_Pʻ\beta =\text{min}_{P}ʻ(\alpha \cap \beta ).\lor.\text{limin}_Pʻ\beta =\text{prec}_{P}ʻ(\alpha \cap \beta ) &\qquad \text{(1)}\\ +\vdash .*205·732.\supset \vdash :\text{Hp}.\text{limax}_Pʻ\alpha =\text{max}_{P}ʻ(\alpha \cap \beta ).&\text{limin}_Pʻ\beta =\text{min}_{P}ʻ(\alpha \cap \beta ).\supset .\\ +&\text{limin}_Pʻ\beta P\text{limax}_Pʻ\alpha &\qquad \text{(2)}\\ +\vdash .*206·15. \supset \vdash :\text{Hp}.\text{limax}_Pʻ\alpha =\text{seq}_{P}ʻ(\alpha \cap \beta ).&\text{limin}_Pʻ\beta =\text{min}_{P}ʻ(\alpha \cap \beta ).\supset .\\ +&\text{limin}_Pʻ\beta P\text{limax}_Pʻ\alpha &\qquad \text{(3)}\\ +\vdash .(3)\frac{\breve{P},\,\beta,\,\alpha}{P,\,\alpha,\,\beta}.\supset \vdash :\text{Hp}.\text{limax}_Pʻ\alpha =\text{max}_{P}ʻ(\alpha + \cap \beta ).&\text{limin}_Pʻ\beta =\text{prec}_{P}ʻ(\alpha \cap \beta ).\supset .\\ +&\text{limin}_Pʻ\beta P\text{limax}_Pʻ\alpha &\qquad \text{(4)}\\ +\vdash .*206·73. \supset \vdash :\text{Hp}.\text{limax}_Pʻ\alpha =\text{seq}_{P}ʻ(\alpha \cap \beta ).&\text{limin}_Pʻ\beta =\text{prec}_{P}ʻ(\alpha \cap \beta ).\supset .\\ +&\text{limin}_Pʻ\beta P\text{limax}_Pʻ\alpha &\qquad \text{(5)}\\ +\vdash .(1).(2).(3).(4).(5).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_699">[Pg 699]</span></p> + +<p class="nind"><b>*215·53.</b> \[\begin{align}\vdash :\text{Hp}*215·5.\alpha \cap \beta =\Lambda .\text{E}!\text{limax}_Pʻ\alpha .\text{E}!&\text{limin}_Pʻ\beta .\supset .\\ +&\text{limax}_Pʻ\alpha P_{*}\text{limin}_Pʻ\beta\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*207·2.*205·22.\supset \vdash :\text{Hp}.&\supset .\overrightarrow{P}ʻ\text{limax}_Pʻ\alpha \subset Pʻʻ\alpha .\overleftarrow{P}ʻ\text{limin}_Pʻ\beta \subset \breve{P} ʻʻ\beta .\\ +[*211·1] &\supset .\overrightarrow{P}ʻ\text{limax}_Pʻ\alpha \subset Pʻʻ\alpha .\overleftarrow{P}ʻ\text{limin}_Pʻ\beta \subset \beta &\qquad \text{(1)}\\ +\vdash .(1).*37·1.\supset \vdash :\text{Hp}.\text{limin}_Pʻ\beta P\text{limax}_Pʻ\alpha .&\supset .(\exists x).x\in \alpha .\text{limin}_Pʻ\beta Px.\\ +[(1)] &\supset .\exists !\alpha \cap \beta &\qquad \text{(2)}\\ +\vdash .(2).\text{Transp}.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*215·54">*215·54</a>.</b> \[\begin{align}\vdash :P&\in \text{Ser}.\alpha \in \text{sect}ʻP.\beta \in \text{sect}ʻ\breve{P} .\alpha \cap \beta =\Lambda .\alpha \cup \beta =CʻP.\\ +&\text{E}!\text{limax}_Pʻ\alpha .\text{E}!\text{limin}_Pʻ\beta .\supset :\text{limax}_Pʻ\alpha =\text{limin}_Pʻ\beta .\lor.\\ +&\text{limax}_Pʻ\alpha P_{1}\text{limin}_Pʻ\beta\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*211·726.\supset \vdash :\text{Hp}.\text{E}!&\text{max}_{P}ʻ\alpha .\text{E}!\text{min}_{P}ʻ\beta .\supset .\\ +&\text{limax}_Pʻ\alpha =\text{max}_{P}ʻ\alpha .\text{limin}_Pʻ\beta =\text{seq}_{P}ʻ\alpha .\\ +[*206·5] &\supset .\text{limax}_Pʻ\alpha P_{1}\text{limin}_Pʻ\beta &\qquad \text{(1)}\\ +\vdash .*211·726.\supset \vdash :\text{Hp}.{\sim}\text{E}!&\text{max}_{P}ʻ\alpha .\supset .\\ +&\text{limax}_Pʻ\alpha =\text{min}_{P}ʻ\beta .\text{limin}_Pʻ\beta =\text{min}_{P}ʻ\beta &\qquad \text{(2)}\\ +\vdash .*211·726.\supset \vdash :\text{Hp}.{\sim}\text{E}!\text{min}_{P}ʻ&\beta .\supset .\\ +&\text{limax}_Pʻ\alpha =\text{max}_{P}ʻ\alpha .\text{limin}_Pʻ\beta =\text{max}_{P}ʻ\alpha &\qquad \text{(3)}\\ +\vdash .(1).(2).(3).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*215·541.</b> \[\begin{align}&\vdash \colon\colon P\in \text{Ser}.\alpha \in \text{sect}ʻP.\beta \in \text{sect}ʻ\breve{P} .\alpha \cup \beta =CʻP.\supset \colon\ldotp \\ +&\alpha \cap \beta \in 0\cup 1.\supset :\text{E}!\text{limax}_Pʻ\alpha .\equiv .\text{E}!\text{limin}_Pʻ\beta \quad[*211·727.*215·51]\end{align}\]</p> + +<p class="nind"><b>*215·542.</b> \[\begin{align}\vdash :\text{Hp}*215·541.\alpha \cap \beta =&\Lambda .\text{E}!\text{limax}_Pʻ\alpha .\text{limax}_Pʻ\alpha {\sim}\in \text{D}ʻP_{1}.\supset .\\ +&\text{limax}_Pʻ\alpha =\text{limin}_Pʻ\beta \quad [*215·54.*211·727]\end{align}\]</p> + +<p class="nind"><b>*215·543.</b> \[\begin{align}&\vdash :P\in \text{Ser}.\alpha \in \text{sect}ʻP.\beta \in \text{sect}ʻ\breve{P} .\alpha \cup \beta =CʻP.\alpha \cap \beta \in 0\cup 1.\\ +&\text{E}!\text{limax}_Pʻ\alpha .\text{limax}_Pʻ\alpha {\sim}\in \text{D}ʻP_{1}.\supset + .\text{limax}_Pʻ\alpha =\text{limin}_Pʻ\beta \quad[*215·542·51]\end{align}\]</p> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_700">[Pg 700]</span></p> +<h2 class="nobreak" id="*216">*216. DERIVATIVES.</h2> +</div> + + +<p class="nind"><i>Summary of</i> *216.</p> + +<p>If \(\alpha\) is any class, and \(P\) is any series, the +<i>derivative</i> (or <i>first derivative</i>) of \(\alpha\) with +respect to \(P\) is the class of limits of existent sub-classes of +\(\alpha \cap CʻP\), <i>i.e.</i> \(\text{lt}_{p}ʻʻ\text{Cl ex}ʻ(\alpha\cap CʻP)\). +That is, a term \(x\) belongs to the derivative of \(\alpha\) if a set +of terms exists which is contained both in \(\alpha\) and in \(CʻP\), +and has \(x\) for its limit. The derivative of \(\alpha\) with respect +to \(P\) will be denoted by \(\delta _{p}ʻ\alpha\).</p> + +<p>In general, there will be members of \(\alpha\) not contained in +\(\delta_{p}ʻ\alpha\), and members of \(\delta _{p}ʻ\alpha\) not +contained in \(\alpha\). \(\alpha\) is said to be <i>dense</i> in \(P\) +if all its terms except the first (if there is a first) belong to +\(\delta _{p}ʻ\alpha\), that is, if all its terms except the first are +limits of existent classes contained in \(\alpha\). \(\alpha\) is said +to be <i>closed</i> in \(P\) if every existent sub-class of \(\alpha\) +which has no maximum has a limit which belongs to \(\alpha\), +<i>i.e.</i> if every existent sub-class of \(\alpha\) has a limit or a +maximum, and the derivative of \(\alpha\) is contained in \(\alpha\). +If \(\alpha\) is both dense and closed, it is called <i>perfect</i>. +In this case, all its terms are limits of classes chosen out of +\(\alpha\), and every class chosen out of \(\alpha\) has a limit or +maximum in \(\alpha\).</p> + +<p>The second derivative of \(\alpha\) is \(\delta _{p}ʻ\delta_{p}ʻ\alpha\), +<i>i.e.</i> \(\delta _{p}^{2}ʻ\alpha\), and so on. +(Derivatives of infinite order cannot be dealt with till a later +stage.) If \(P\) is serial, the second derivative of \(\alpha\) is +always contained in the first (<a href="#*216·14">*216·14</a>).</p> + +<p>If \(P\) is a Dedekindian series, \(\alpha\) is closed whenever +\(\delta _{p}ʻ\alpha \subset \alpha\). In order to secure a Dedekindian +series, it is sometimes convenient to replace \(P\) by the ordinally +similar series \(\overrightarrow{P}^{;}P\), which is contained in +the Dedekindian series \(\varsigma ʻP\). Then \(\alpha\) is replaced +by \(\overrightarrow{P}ʻʻ\alpha\), and \(\alpha\) is closed if +the derivative of \(\overrightarrow{P}ʻʻ\alpha\) with respect to +\(\varsigma ʻP\) is contained in \(\overrightarrow{P}ʻʻ\alpha\). The +relation of the derivative of \(\alpha\) in \(P\) to the derivative of +\(\overrightarrow{P}ʻʻ\alpha\) in \(\varsigma ʻP\) has been treated +in <a href="#*212·6">*212·6</a> and following propositions. This subject is resumed below +(<a href="#*216·5">*216·5</a> ff.).</p> + +<p>The derivative of the series P will be defined as the series of its +limit-points, and denoted by \(\nabla ʻP\). Thus we put +\[ +\nabla ʻP=P\unicode{x0294f}\text{D}ʻ\text{lt}_{p}. +\]</p> + +<p><span class="pagenum" id="Page_701">[Pg 701]</span></p> + +<p>If \(P\) is a series, the derivative of a class \(\alpha\) consists of +those members \(x\) of \(\text{ᗡ}ʻP\) which are such that members of +\(\alpha\) exist in every interval which ends in \(x\), <i>i.e.</i></p> + +<p class="nind"><b>*216·13.</b> \(\vdash \colon\colon P\in \text{Ser}.\supset \colon\ldotp x\in \delta _{p}ʻ\alpha .\equiv :x\in \text{ᗡ}ʻP:yPx.\supset _{y}.\exists + !\alpha \cap \overleftarrow{P}ʻy\cap \overrightarrow{P}ʻx\)</p> + +<p>We have</p> + +<p class="nind"><b>*216·2.</b> \(\vdash .\delta _{p}ʻCʻP=\text{D}ʻ\text{lt}_{p}-\overrightarrow{B}ʻP\)</p> + +<p class="nind"><b>*216·3.</b> \(\vdash :\alpha \in \text{dense}ʻP.\equiv .\alpha -\overrightarrow{\text{min}}_{p}ʻ\alpha \subset \delta _{p}ʻ\alpha\)</p> + +<p class="nind"><b>*216·32.</b> \(\vdash :\alpha \in \text{closed}ʻP.\equiv .\text{Cl ex}ʻ(\alpha \cap CʻP)\subset \text{ᗡ}ʻ\text{limax}_{p}.\delta _{p}ʻ\alpha \subset \alpha\)</p> + +<p>We prove (<a href="#*216·4">*216·4</a>—<a href="#*216·412">·412</a>) that the properties of \(\alpha\) with respect +to \(P\), as regards being dense, closed, or perfect, belong to +\(\breve{S} ʻʻ\alpha\) with respect to \(Q\) if \(S\) is a correlator +of \(P\) with \(Q\).</p> + +<p>We next consider the relation of \(\alpha\) in \(P\) to +\(\overrightarrow{P}ʻʻ\alpha\) in \(\varsigma ʻP\) (<a href="#*216·5">*216·5</a>—<a href="#*216·56">·56</a>). +The point of these propositions is that \(\varsigma ʻP\) is +Dedekindian, so that a class is closed in \(\varsigma ʻP\) if +it contains its first derivative. (It is usual to <i>define</i> +a class as closed whenever it contains its first derivative; +but this involves the tacit assumption that the series \(P\) is +Dedekindian. If \(P\) is the series of real numbers, this assumption +is of course verified.) We prove (<a href="#*216·52">*216·52</a>) that the derivative of +\(\overrightarrow{P}ʻʻ\alpha\) in \(\varsigma ʻP\) is \(Pʻʻʻ(\text{Cl ex}ʻ\alpha -\text{ᗡ}ʻ\text{max}_{p})\), +<i>i.e.</i> is the class of segments defined by such existent +sub-classes of \(\alpha\) as have no maximum; we show that +\(\alpha\) is dense, closed, or perfect in \(P\) according +as \(\overrightarrow{P}ʻʻ\alpha\) is dense, closed, or +perfect in \(\varsigma ʻP\) (<a href="#*216·53">*216·53</a><a href="#*216·54">·54</a><a href="#*216·56">·56</a>), and that +\(\alpha\) and \(\overrightarrow{P}ʻʻ\alpha\) are closed if +\(\overrightarrow{P}ʻʻ\alpha\) contains its first derivative (<a href="#*216·54">*216·54</a>).</p> + +<p>We end with various propositions on \(\nabla ʻP\) (<a href="#*216·6">*216·6</a>—<a href="#*216·621">·621</a>), of which +the chief is</p> + +<p class="nind"><b>*216·611.</b> \(\vdash :P\in \text{Ser}.\dot{\exists} !\nabla ʻP.\supset .Cʻ\nabla ʻP=CʻP-\text{ᗡ}ʻP_{1}=\delta _{p}ʻCʻP\cup \overrightarrow{B}ʻP\)</p> + +<p>This subject will be resumed in connection with well-ordered series in +*264.</p> + +<hr class="tb"> + +<p class="nind"><b>*216·01.</b> \(\delta _{p}ʻ\alpha =\text{lt}_{p}ʻʻ\text{Cl ex}ʻ(\alpha \cap CʻP)\quad\text{Df}\)</p> + +<p class="nind"><b>*216·02.</b> \(\text{dense}ʻP=\hat{\alpha} (\alpha -\overrightarrow{\text{min}}_{p}ʻ\alpha \subset \delta _{p}ʻ\alpha ) \quad\text{Df}\)</p> + +<p class="nind"><b>*216·03.</b> \(\text{closed}ʻP=\hat{\alpha} \{\text{Cl ex}ʻ(\alpha \cap CʻP)\subset \text{ᗡ}ʻ\text{limax}_{p}.\delta _{p}ʻ\alpha \subset \alpha\} \quad\text{Df}\)</p> + +<p class="nind"><b>*216·04.</b> \(\text{perf}ʻP=\text{dense}ʻP\cap \text{closed}ʻP \quad\text{Df}\)</p> + +<p class="nind"><b>*216·05.</b> \(\nablaʻP=P\unicode{x0294f}\text{D}ʻ\text{lt}_{p} \quad\text{Df}\)</p> + +<p><span class="pagenum" id="Page_702">[Pg 702]</span></p> + +<p class="nind"><b>*216·1.</b> \(\vdash :x\in \delta _{p}ʻ\alpha .\equiv .(\exists \beta ).\beta \subset \alpha \cap CʻP.\exists !\beta .x\text{lt}_{p}\beta \quad[(*216·01)]\)</p> + +<p class="nind"><b>*216·101.</b> \(\vdash :x\in \delta _{P}ʻ\alpha .\equiv .(\exists \beta ).\beta \subset \alpha .\exists !\beta .\beta \subset Pʻʻ\beta .x\text{seq}_{P}\beta\)</p> + + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*216·1.*207·1.\supset \\ +\vdash :x\in \delta _{P}ʻ\alpha .&\equiv .(\exists \beta ).\beta \subset \alpha \cap CʻP.\exists !\beta .\beta \cap CʻP\subset Pʻʻ\beta .x\text{seq}_{P}\beta .\\ +[*37·15] &\equiv .(\exists \beta ).\beta \subset \alpha .\exists !\beta .\beta \subset Pʻʻ\beta .x\text{seq}_{P}\beta :\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*216·11.</b> \(\vdash .\delta _{P}ʻ\alpha \subset \breve{P} ʻʻ\alpha\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*216·101.*206·142.\supset \vdash :x\in \delta _{P}ʻ\alpha .&\supset .(\exists \beta ).\beta \subset \alpha .\exists !\beta .x\in \breve{P} ʻʻ\beta .\\ +[*37·2] &\supset .x\in \breve{P} ʻʻ\alpha :\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*216·111.</b> \(\vdash .\delta _{P}ʻ\alpha \subset \text{ᗡ}ʻP \quad[*216·11.*37·16]\)</p> + +<p class="nind"><b>*216·12.</b> \(\vdash .\delta _{P}ʻ\alpha =\delta _{P}ʻ(\alpha \cap CʻP) \quad[*22·5.(*216·01)]\)</p> + +<p class="nind"><b>*216·13.</b> \(\vdash \colon\colon P\in \text{Ser}.\supset \colon\ldotp x\in \delta _{P}ʻ\alpha .\equiv :x\in \text{ᗡ}ʻP:yPx.\supset _{y}.\exists + !\alpha \cap \overleftarrow{P}ʻy\cap \overrightarrow{P}ʻx\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*206·173.*216·101.\supset \vdash \colon\colon P\in \text{connex} .P^{2}\,\unicode{x2abd}\, J.\supset \colon\ldotp \\ +x\in \delta _{P}ʻ\alpha .&\equiv :(\exists \beta ).\beta \subset \alpha .\exists !\beta .\beta \subset \overrightarrow{P}ʻx.\overrightarrow{P}ʻx\subset P ʻʻ\beta :\\ +[*37·46]&\equiv :(\exists \beta ):\beta \subset \alpha .\exists !\beta .\beta \subset \overrightarrow{P}ʻx:yPx.\supset _{y}.\exists !\beta \cap \overleftarrow{P}ʻy:\\ +[*24·58]&\supset :yPx.\supset _{y}.\exists !\alpha \cap \overrightarrow{P}ʻx\cap \overleftarrow{P}ʻy &\qquad \text{(1)}\\ +\vdash .*33·41·152.\supset \\ +\vdash \colon\ldotp x\in \text{ᗡ}ʻP:yPx.\supset _{y}.\exists + !\alpha \cap \overrightarrow{P}ʻx\cap \overleftarrow{P}ʻy:&\supset :\exists !\alpha \cap \overrightarrow{P}ʻx.\alpha \cap \overrightarrow{P}ʻx\subset \alpha \cap CʻP:\\ +[*216·1] &\supset :x\text{lt}_P(\alpha \cap \overrightarrow{P}ʻx).\supset .x\in \delta _{P}ʻ\alpha &\qquad \text{(2)}\\ +\vdash .*37·2.*201·501.&\supset \vdash :\text{Hp}.\supset .Pʻʻ(\alpha \cap \overrightarrow{P}ʻx)\subset \overrightarrow{P}ʻx &\qquad \text{(3)}\\ +\vdash .*50·24. &\supset \vdash :\text{Hp}.\supset .x{\sim}\in (\alpha \cap \overrightarrow{P}ʻx) &\qquad \text{(4)}\\ +\vdash .(3).(4).*207·232.\supset\\ +\vdash \colon\colon\text{Hp}.x\in \text{ᗡ}ʻP.\supset \colon\ldotp x\text{lt}_P(\alpha \cap \overrightarrow{P}ʻx).&\equiv :\overrightarrow{P}ʻx\subset Pʻʻ(\alpha \cap \overrightarrow{P}ʻx):\\ +[*37·46] &\equiv :yPx.\supset _{y}.\exists !\alpha \cap \overrightarrow{P}ʻx\cap \overleftarrow{P}ʻy &\qquad \text{(5)}\\ +\vdash .(2).(5).&\supset \vdash \colon\ldotp \text{Hp}.x\in \text{ᗡ}ʻP:yPx.\supset _{y}.\exists !\alpha \cap \overrightarrow{P}ʻx\cap \overleftarrow{P}ʻy:\supset .x\in \delta _{P}ʻ\alpha + &\qquad \text{(6)}\\ +\vdash .(1).(6).*216·111.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*216·14">*216·14</a>.</b> \(\vdash :P\in \text{Ser}.\supset .\delta _{^{2}}ʻ\alpha \subset \delta _{P}ʻ\alpha\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*71·47.\supset \vdash \colon\ldotp \text{Hp}.\supset :\beta &\subset \text{lt}_Pʻʻ\text{Cl ex}ʻ\alpha .\exists !\beta .\supset .\\ +&(\exists \kappa ).\kappa \subset \text{Cl ex}ʻ\alpha .\beta =\text{lt}_Pʻʻ\kappa .\exists !\beta .\\ +[*37·26] &\supset .(\exists \lambda ).\lambda \subset \text{Cl ex}ʻ\alpha \cap \text{ᗡ}ʻ\text{lt}_P.\beta =\text{lt}_Pʻʻ\lambda .\exists !\beta .\\ +[*207·54]&\supset .(\exists \lambda ).\lambda \subset \text{Cl ex}ʻ\alpha \cap \text{ᗡ}ʻ\text{lt}_P.\beta =\text{lt}_Pʻʻ\lambda .\exists !\beta .\overrightarrow{\text{limax}} _{P}ʻ\beta + =\overrightarrow{\text{lt}} _{P}ʻsʻ\lambda .\\ +[*216·1.*37·29.*53·24.\text{Transp}]&\supset .\overrightarrow{\text{limax}} _{P}ʻ\beta \subset \delta _{P}ʻ\alpha .\\ +[*207·45] &\supset .\overrightarrow{\text{lt}} _{P}ʻ\beta \subset \delta _{P}ʻ\alpha &\qquad \text{(1)}\\ +\vdash .(1).(*216·01).\supset \vdash \colon\ldotp \text{Hp}.&\supset :\beta \in \text{Cl ex}ʻ\delta _{P}ʻ\alpha .\supset .\overrightarrow{\text{lt}} _{P}ʻ\beta + \subset \delta _{P}ʻ\alpha :\\ +[*40·43·5] &\supset :\text{lt}_Pʻʻ\text{Cl ex}ʻ\delta _{P}ʻ\alpha \subset \delta _{P}ʻ\alpha \colon\ldotp \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_703">[Pg 703]</span></p> + +<p class="nind"><b>*216·15.</b> \(\vdash :\alpha \subset \beta .\supset .\delta _{P}ʻ\alpha \subset \delta _{P}ʻ\beta \quad[*37·2.(*216·01)]\)</p> + +<p class="nind"><b>*216·16.</b> \(\vdash :P\in \text{trans}\cap \text{connex} .\supset .\delta _{P}ʻ\alpha =\delta _{P}ʻ(\alpha -\overrightarrow{\text{min}}_{P}ʻ\alpha )\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*24·26·101.&\supset \vdash :\overrightarrow{\text{min}}_{P}ʻ\alpha =\Lambda .\supset .\delta _{P}ʻ\alpha =\delta _{P}ʻ(\alpha -\overrightarrow{\text{min}}_{P}ʻ\alpha + ) &\qquad \text{(1)}\\ +\vdash .*51·36.\supset \vdash :\beta \subset \alpha .\exists !\beta .\text{E}!\text{min}_{P}ʻ\alpha .\text{min}_{P}ʻ&\alpha {\sim}\in \beta .\supset .\\ +&\beta \in \text{Cl ex}ʻ(\alpha -\iota ʻ\text{min}_{P}ʻ\alpha ).\\ +[*37·18] &\supset .\overrightarrow{\text{lt}} _{P}ʻ\beta \subset \delta _{P}ʻ(\alpha -\iota ʻ\text{min}_{P}ʻ\alpha ) &\qquad \text{(2)}\\ +\vdash .*205·5.\supset \vdash :\text{Hp}.\beta \subset \alpha .\exists !\beta .\text{min}_{P}ʻ\alpha \in \beta .&\supset .\text{min}_{P}ʻ\alpha =\text{min}_{P}ʻ\beta .\\ +[*207·262] &\supset .\overrightarrow{\text{lt}} _{P}ʻ\beta \subset \overrightarrow{\text{lt}} _{P}ʻ(\beta -\iota ʻ\text{min}_{P}ʻ\alpha ) &\qquad \text{(3)}\\ +\vdash .(3).*37·18.&\supset \vdash :\text{Hp}(3).\beta \neq \iota ʻ\text{min}_{P}ʻ\alpha .\supset .\overrightarrow{\text{lt}} _{P}ʻ\beta \subset \delta _{P}ʻ(\alpha + -\iota ʻ\text{min}_{P}ʻ\alpha ) &\qquad \text{(4)}\\ +\vdash .*205·194·8.\supset \vdash :\text{E}!\text{min}_{P}ʻ\alpha .&\supset .\text{min}_{P}ʻ\alpha =\text{max}_{P}ʻ\iota ʻ\text{min}_{P}ʻ\alpha .\\ +[*207·11] &\supset .\overrightarrow{\text{lt}} _{P}ʻ\iota ʻ\text{min}_{P}ʻ\alpha =\Lambda &\qquad \text{(5)}\\ +\vdash .(5).*24·12.&\supset \vdash :\text{E}!\text{min}_{P}ʻ\alpha .\beta =\iota ʻ\text{min}_{P}ʻ\alpha .\supset .\overrightarrow{\text{lt}} _{P}ʻ\beta \subset \delta _{P}ʻ(\alpha + -\iota ʻ\text{min}_{P}ʻ\alpha ) &\qquad \text{(6)}\\ +\vdash .(4).(6).&\supset \vdash :\text{Hp}(3).\supset .\overrightarrow{\text{lt}} _{P}ʻ\beta \subset \delta _{P}ʻ(\alpha -\iota ʻ\text{min}_{P}ʻ\alpha + ) &\qquad \text{(7)}\\ +\vdash .(2).(7).&\supset \vdash :\text{Hp}.\beta \subset \alpha .\exists !\beta .\text{E}!\text{min}_{P}ʻ\alpha .\supset .\overrightarrow{\text{lt}} _{P}ʻ\beta + \subset \delta _{P}ʻ(\alpha -\iota ʻ\text{min}_{P}ʻ\alpha ) &\qquad \text{(8)}\\ +\vdash .(8).*40·5·43.&\supset \vdash :\text{Hp}.\text{E}!\text{min}_{P}ʻ\alpha .\supset .\delta _{P}ʻ\alpha \subset \delta _{P}ʻ(\alpha -\iota ʻ\text{min}_{P}ʻ\alpha + ) &\qquad \text{(9)}\\ +\vdash .(9).*216·15.&\supset \vdash :\text{Hp}.\text{E}!\text{min}_{P}ʻ\alpha .\supset .\delta _{P}ʻ\alpha =\delta _{P}ʻ(\alpha -\iota ʻ\text{min}_{P}ʻ\alpha + ) &\qquad \text{(10)}\\ +\vdash .(1).(10).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*216·2.</b> \(\vdash .\delta_{P}ʻCʻP=\text{D}ʻ\text{lt}_P-\overrightarrow{B}ʻP\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*37·15.*216·111.&\supset \vdash .\delta _{P}ʻCʻP\subset \text{D}ʻ\text{lt}_P-\overrightarrow{B}ʻP &\qquad \text{(1)}\\ +\vdash .*216·1. \supset \vdash :x\in \text{D}ʻ\text{lt}_P-\delta _{P}ʻCʻP.&\supset .x\text{lt}_P\Lambda .\\ +[*207·3] &\supset .x\in \overrightarrow{B}ʻP &\qquad \text{(2)}\\ +\vdash .(2).\text{Transp}. &\supset \vdash .\text{D}ʻ\text{lt}_P-\overrightarrow{B}ʻP\subset \delta _{P}ʻCʻP &\qquad \text{(3)}\\ +\vdash .(1).(3).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*216·21.</b> \[\begin{align}&\vdash :P\in \text{ᗡ}ʻʻʻJ\cap \text{connex} .\supset .\delta _{P}ʻCʻP=\text{ᗡ}ʻP-\text{ᗡ}ʻ(P\dot{-} P^{2})\\ +&[*207·35.*216·2]\end{align}\]</p> + +<p><span class="pagenum" id="Page_704">[Pg 704]</span></p> + +<p class="nind"><b>*216·22.</b> \(\vdash :P\in \text{ᗡ}ʻʻʻJ\cap \text{connex} .P\,\unicode{x2abd}\, P^{2}.\supset .\delta _{P}ʻCʻP=\text{ᗡ}ʻP \quad[*216·21]\)</p> + +<p class="nind"><b>*216·23.</b> \(\vdash :P\in \text{trans}.\supset .\delta _{P}ʻCʻP=\text{seq}_{P}ʻʻ\text{ᗡ}ʻ\text{sgm}ʻP=\text{lt}_Pʻʻ\text{ᗡ}ʻ\text{sgm}ʻP\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*206·25.*216·101.\supset \\ +\vdash \colon\ldotp \text{Hp}.\supset :x\in \delta _{P}ʻCʻP .&\equiv .(\exists \beta ).\exists !\beta .\beta \subset Pʻʻ\beta .x\text{seq}_{P}(Pʻʻ\beta ).\\ +[*24·58.*37·29] &\equiv .(\exists \beta ).\beta \subset Pʻʻ\beta .\exists !Pʻʻ\beta .x\text{seq}_{P}(Pʻʻ\beta ).\\ +[*201·55] &\supset .(\exists \beta ).PʻʻPʻʻ\beta =Pʻʻ\beta .\exists !Pʻʻ\beta .x\text{seq}_{P}(Pʻʻ\beta ).\\ +[*212·152] &\supset .x\in \text{seq}_{P}ʻʻ\text{ᗡ}ʻ\text{sgm}ʻP &\qquad \text{(1)}\\ +\vdash .*211·4.&\supset \vdash .\text{seq}_{P}ʻʻ\text{ᗡ}ʻ\text{sgm}ʻP=\text{lt}_Pʻʻ\text{ᗡ}ʻ\text{sgm}ʻP &\qquad \text{(2)}\\ +\vdash .*212·152.(*216·01).&\supset \vdash .\text{lt}_Pʻʻ\text{ᗡ}ʻ\text{sgm}ʻP\subset \delta _{P}ʻCʻP &\qquad \text{(3)}\\ +\vdash .(1).(2).(3).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*216·3.</b> \(\vdash :\alpha \in \text{dense}ʻP.\equiv .\alpha -\overrightarrow{\text{min}}_{P}ʻ\alpha \subset \delta _{P}ʻ\alpha \quad[(*216·02)]\)</p> + +<p class="nind"><b>*216·31.</b> \(\vdash :\alpha \in \text{dense}ʻP.\equiv .\alpha \subset CʻP.\alpha \cap \breve{P} ʻʻ\alpha \subset \delta _{P}ʻ\alpha\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*216·3·111.\supset \vdash :\alpha \in \text{dense}ʻP.&\supset .\alpha -\overrightarrow{\text{min}}_{P}ʻ\alpha \subset \text{ᗡ}ʻP.\\ +[*205·11] &\supset .\alpha \subset CʻP. &\qquad \text{(1)}\\ +[*205·11] &\supset .\alpha -\overrightarrow{\text{min}}_{P}ʻ\alpha =\alpha \cap \breve{P} ʻʻ\alpha &\qquad \text{(2)}\\ +\vdash .(1).(2).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*216·32.</b> \[\begin{align}&\vdash :\alpha \in \text{closed}ʻP.\equiv .\text{Cl ex}ʻ(\alpha \cap CʻP)\subset \text{ᗡ}ʻ\text{limax}_P.\delta _{P}ʻ\alpha \subset \alpha \\ +&[(*216·03)]\end{align}\]</p> + +<p class="nind"><b>*216·33.</b> \(\vdash \colon\ldotp \alpha \in \text{closed}ʻP.\equiv :\beta \subset \alpha .\exists !\beta .\beta \subset Pʻʻ\beta .\supset _{\beta }.\exists + !\overrightarrow{\text{lt}} _{P}ʻ\beta .\overrightarrow{\text{lt}} _{P}ʻ\beta \subset \alpha\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*207·45.*205·123.&\supset \vdash \colon\ldotp \text{Cl ex}ʻ(\alpha \cap CʻP)\subset \text{ᗡ}ʻ\text{limax}_P.\equiv :\\ +&\beta \subset \alpha .\exists !\beta .\beta \subset CʻP.\beta \subset Pʻʻ\beta .\supset _{\beta }.\exists !\overrightarrow{\text{lt}} _{P}ʻ\beta :\\ +[*37·15] & \equiv :\beta \subset \alpha .\exists !\beta .\beta \subset Pʻʻ\beta .\supset _{\beta }.\exists !\overrightarrow{\text{lt}} _{P}ʻ\beta &\qquad \text{(1)}\\ +\vdash .*40·43·5.\supset \vdash \colon\ldotp \delta _{P}ʻ\alpha \subset \alpha .&\equiv :\beta \subset \alpha .\exists !\beta .\beta \subset CʻP.\supset _{\beta }.\overrightarrow{\text{lt}} + _{P}ʻ\beta \subset \alpha :\\ +[*207·11.*24·12] &\equiv :\beta \subset \alpha .\exists !\beta .\beta \subset CʻP.\overrightarrow{\text{max}}_{P}ʻ\beta =\Lambda .\supset _{\beta }.\overrightarrow{\text{lt}} + _{P}ʻ\beta \subset \alpha :\\ +[*205·123.*37·15]&\equiv :\beta \subset \alpha .\exists !\beta .\beta \subset Pʻʻ\beta .\supset _{\beta }.\overrightarrow{\text{lt}} _{P}ʻ\beta + \subset \alpha &\qquad \text{(2)}\\ +\vdash .(1).(2).*216·32.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*216·34.</b> \[\begin{align}&\vdash \colon\colon P\in \text{connex} .\supset \colon\ldotp \alpha \in \text{closed}ʻP.\equiv :\\ +&\beta \subset \alpha .\exists !\beta .\beta \subset Pʻʻ\beta .\supset _{\beta }.\text{lt}_Pʻ\beta + \in \alpha \quad[*216·33.*71·332.*207·24]\end{align}\]</p> + +<p><span class="pagenum" id="Page_705">[Pg 705]</span></p> + +<p class="nind"><b>*216·35.</b> \(\vdash :P\in \text{Ser}.\text{Cl ex}ʻ\alpha \subset \text{ᗡ}ʻ\text{limax}_P.\supset .\text{Cl ex}ʻ\delta _{P}ʻ\alpha \subset \text{ᗡ}ʻ\text{limax}_P\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*71·47.*37·26.\supset \\ +\vdash \colon\ldotp \text{Hp}.&\supset :\beta \in \text{Cl ex}ʻ\delta _{P}ʻ\alpha + .\supset .(\exists \lambda ).\lambda \subset \text{Cl ex}ʻ\alpha \cap \text{ᗡ}ʻ\text{lt}_P.\beta =\text{lt}_Pʻʻ\lambda .\exists !\beta .\\ +[*207·54]&\supset .(\exists \lambda ).\lambda \subset \text{Cl ex}ʻ\alpha \cap \text{ᗡ}ʻ\text{lt}_P.\beta =\text{lt}_Pʻʻ\lambda .\exists !\beta .\overrightarrow{\text{limax}} _{P}ʻ\beta + =\overrightarrow{\text{limax}} _{P}ʻsʻ\lambda .\\ +[*37·29.\text{Transp}]&\supset .(\exists \lambda ).\lambda \subset \text{Cl ex}ʻ\alpha \cap \text{ᗡ}ʻ\text{lt}_P.\beta =\text{lt}_Pʻʻ\lambda .\exists !\lambda .\overrightarrow{\text{limax}} _{P}ʻ\beta =\overrightarrow{\text{limax}} _{P}ʻsʻ\lambda .\\ +[*53·24.\text{Transp}] &\supset .(\exists \lambda ).sʻ\lambda \in \text{Cl ex}ʻ\alpha .\overrightarrow{\text{limax}} _{P}ʻ\beta =\overrightarrow{\text{limax}} _{P}ʻsʻ\lambda .\\ +[\text{Hp}] & \supset .\exists !\overrightarrow{\text{limax}} _{P}ʻ\beta \colon\ldotp \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*216·36.</b> \(\vdash :\alpha \in \text{perf}ʻP.\equiv .\alpha \in \text{dense}ʻP\cap \text{closed}ʻP \quad[(*216·04)]\)</p> + +<p class="nind"><b>*216·37.</b> \[\begin{align}&\vdash :\alpha \in \text{perf}ʻP.\equiv .\text{Cl ex}ʻ\alpha \subset \text{ᗡ}ʻ\text{limax}_P.\delta _{P}ʻ\alpha + =\alpha -\overrightarrow{\text{min}}_{P}ʻ\alpha \\ +&[*216·3·32·36]\end{align}\]</p> + +<p class="nind"><b>*216·371.</b> \[\begin{align}&\vdash :\alpha \in \text{perf}ʻP.\equiv .\text{Cl ex}ʻ\alpha \subset \text{ᗡ}ʻ\text{limax}_P.\alpha \subset CʻP.\delta _{P}ʻ\alpha + =\alpha \cap \breve{P} ʻʻ\alpha \\ +&[*216·31·32·11·36]\end{align}\]</p> + +<p class="nind"><b>*216·38.</b> \(\vdash :P\in \text{trans}\cap \text{connex} .\alpha \in \text{dense}ʻP.\supset .\delta _{P}ʻ\alpha \in \text{dense}ʻP.\delta _{P}ʻ\alpha \subset \delta _{P}ʻ\delta + _{P}ʻ\alpha\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*216·3·15.\supset \vdash :\text{Hp}.&\supset .\delta _{P}ʻ(\alpha -\overrightarrow{\text{min}}_{P}ʻ\alpha )\subset \delta _{P}ʻ\delta _{P}ʻ\alpha .\\ +[*216·16] &\supset .\delta _{P}ʻ\alpha \subset \delta _{P}ʻ\delta _{P}ʻ\alpha .\\ +[*216·3] &\supset .\delta _{P}ʻ\alpha \in \text{dense}ʻP:\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*216·381.</b> \[\begin{align}&\vdash :P\in \text{Ser}.\alpha \in \text{dense}ʻP.\supset .\delta _{P}ʻ\alpha =\delta _{P}ʻ\delta _{P}ʻ\alpha + .\overrightarrow{\text{min}}_{P}ʻ\delta _{P}ʻ\alpha =\Lambda\\ +&[*216·38·14·11]\end{align}\]</p> + +<p class="nind"><b>*216·382.</b> \[\begin{align}&\vdash :P\in \text{Ser}.\alpha \in \text{dense}ʻP.\text{Cl ex}ʻ\alpha \subset \text{ᗡ}ʻ\text{limax}_P.\supset .\delta _{P}ʻ\alpha \in \text{perf}ʻP\\ +&[*216·35·381·37]\end{align}\]</p> + +<p class="nind"><b><a id="*216·4">*216·4</a>.</b> \(\vdash :S\in P\,\overline{\,\text{smor}\,}\,Q.\supset .\delta _{P}ʻ\alpha =Sʻʻ\delta _{Q}ʻ\breve{S} ʻʻ\alpha .\breve{S} ʻʻ\delta _{P}ʻ\alpha =\delta _{Q}ʻ\breve{S} + ʻʻ\alpha\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*207·63.\supset \vdash :\text{Hp}.\supset .\delta _{P}ʻ\alpha &=Sʻʻ\text{lt}_{Q}ʻʻ\breve{S} ʻʻʻ\text{Cl ex}ʻ\alpha\\ +[*71·491] &=Sʻʻ\text{lt}_{Q}ʻʻ\text{Cl ex}ʻ\breve{S} ʻʻ\alpha \\ +[(*216·01)] &=Sʻʻ\delta _{Q}ʻ\breve{S} ʻʻ\alpha &\qquad \text{(1)}\\ +\vdash .(1).*72·52.*216·111.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*216·401.</b> \(\vdash :S\in P\,\overline{\,\text{smor}\,}\,Q.\supset .P\unicode{x0294f}\delta _{P}ʻ\alpha =S^{;}(Q\unicode{x0294f}\delta _{Q}ʻ\breve{S} ʻʻ\alpha )\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*150·37.\supset \vdash :\text{Hp}.\supset .S^{;}(Q\unicode{x0294f}\delta _{Q}ʻ\breve{S} ʻʻ\alpha )&=(S^{;}Q)\unicode{x0294f}Sʻʻ\delta _{Q}ʻ\breve{S} + ʻʻ\alpha \\ +[*216·4.*151·11] & =P\unicode{x0294f}\delta _{P}ʻ\alpha :\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_706">[Pg 706]</span></p> + +<p class="nind"><b>*216·41.</b> \(\vdash \colon\ldotp S\in P\,\overline{\,\text{smor}\,}\,Q.\alpha \subset CʻP.\supset :\alpha \in \text{dense}ʻP.\equiv .\breve{S} ʻʻ\alpha \in \text{dense}ʻQ\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*216·3.*37·2.\supset \\ +\vdash \colon\ldotp \text{Hp}.&\supset :\alpha \in \text{dense}ʻP.\supset .\breve{S} ʻʻ(\alpha -\overrightarrow{\text{min}}_{P}ʻ\alpha )\subset \breve{S} ʻʻ\delta _{P}ʻ\alpha .\\ +[*71·38.*205·8] & \supset .\breve{S} ʻʻ\alpha -\overrightarrow{\text{min}}_{Q}ʻ\breve{S} ʻʻ\alpha \subset \breve{S} ʻʻ\delta _{P}ʻ\alpha .\\ +[*216·4] &\supset .\breve{S} ʻʻ\alpha -\overrightarrow{\text{min}}_{Q}ʻ\breve{S} ʻʻ\alpha \subset \delta _{Q}ʻ\breve{S} ʻʻ\alpha .\\ +[*216·3] &\supset .\breve{S} ʻʻ\alpha \in \text{dense}ʻQ &\qquad \text{(1)}\\ +\vdash .(1)\frac{Q,\,P,\,\breve{S}ʻʻ\alpha}{P,\,Q,\,\alpha}.\supset \\ +\vdash \colon\ldotp \text{Hp}.&\supset :\breve{S} ʻʻ\alpha \in \text{dense}ʻQ.\supset .Sʻʻ\breve{S} ʻʻ\alpha \in \text{dense}ʻP.\\ +[*72·502] &\supset .\alpha \in \text{dense}ʻP &\qquad \text{(2)}\\ +\vdash .(1).(2).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*216·411.</b> \(\vdash \colon\ldotp S\in P\,\overline{\,\text{smor}\,}\,Q.\alpha \subset CʻP.\supset :\alpha \in \text{closed}ʻP.\equiv .\breve{S} ʻʻ\alpha \in \text{closed}ʻQ\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*207·64.*37·431.\supset \\ +\vdash \colon\ldotp \text{Hp}.\supset :\beta \subset CʻP.\exists !\beta .\beta \in \text{ᗡ}ʻ&\text{limax}_P.\supset .\\ +&\breve{S} ʻʻ\beta \subset CʻQ.\exists !\breve{S} ʻʻ\beta .\breve{S} ʻʻ\beta \in \text{ᗡ}ʻ\text{limax}_{Q}:\\ +[*71·49]\supset :\text{Cl ex}ʻ\alpha \subset \text{ᗡ}ʻ&\text{limax}_P.\supset .\text{Cl ex}ʻ\breve{S} ʻʻ\alpha \subset \text{ᗡ}ʻ\text{limax}_{Q} &\qquad \text{(1)}\\ +\vdash .*37·2.*216·4.&\supset \vdash \colon\ldotp \text{Hp}.\supset :\delta _{P}ʻ\alpha \subset \alpha .\supset .\delta _{Q}ʻ\breve{S} + ʻʻ\alpha \subset \breve{S} ʻʻ\alpha &\qquad \text{(2)}\\ +\vdash .(1).(2).*216·32. &\supset \vdash \colon\ldotp \text{Hp}.\supset :\alpha \in \text{closed}ʻP.\supset .\breve{S} ʻʻ\alpha \in \text{closed}ʻQ &\qquad \text{(3)}\\ +\vdash .(3) \frac{Q,\,P,\,\breve{S}ʻʻ\alpha}{P,\,Q,\,\alpha}. \supset \vdash \colon\ldotp \text{Hp}.&\supset :\breve{S} ʻʻ\alpha \in \text{closed}ʻQ.\supset .Sʻʻ\breve{S} ʻʻ\alpha \in \text{closed}ʻP.\\ +[*72·502] & \supset .\alpha \in \text{closed}ʻP &\qquad \text{(4)}\\ +\vdash .(3).(4).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*216·412">*216·412</a>.</b> \[\begin{align}&\vdash \colon\ldotp S\in P\,\overline{\,\text{smor}\,}\,Q.\alpha \subset CʻP.\supset :\alpha \in \text{perf}ʻP.\equiv .\breve{S} ʻʻ\alpha \in \text{perf}ʻQ\\ +&[*216·41·411·36]\end{align}\]</p> + +<p class="nind"><b><a id="*216·5">*216·5</a>.</b> \(\vdash :P\in \text{Ser}.\supset .\text{ᗡ}ʻ\varsigma ʻP-\overrightarrow{P}ʻʻCʻP\subset \delta (\varsigma ʻP)ʻ\overrightarrow{P}ʻʻCʻP\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*212·134.*216·111.\supset \\ +\vdash :\text{Hp}.P=\dot{\Lambda} .\supset .\text{ᗡ}ʻ\varsigma ʻP-\overrightarrow{P}ʻʻCʻP=\Lambda .\delta (\varsigma ʻP)ʻ\overrightarrow{P}ʻʻCʻP=\Lambda &\qquad \text{(1)}\\ +\vdash .*212·632.\supset \\ +\vdash \colon\ldotp \text{Hp}.\dot{\exists} !P.Pʻʻ\alpha {\sim}\in \overrightarrow{P}ʻʻCʻP.\supset :Pʻʻ\alpha =\text{lt}(\varsigma ʻP)ʻ\overrightarrow{P}ʻʻ\alpha :\\ +[*216·1]\supset :\exists !\alpha .\alpha \subset CʻP.\supset .Pʻʻ\alpha \in \delta (\varsigma ʻP)ʻ\overrightarrow{P}ʻʻCʻP &\qquad \text{(2)}\\ +\vdash .(2).*212·132.*37·265.\supset \\ +\vdash :\text{Hp}.\dot{\exists} !P.\beta \in \text{ᗡ}ʻ\varsigma ʻP-\overrightarrow{P}ʻʻCʻP.\supset .\beta \in \delta (\varsigma ʻP)ʻ\overrightarrow{P}ʻʻCʻP &\qquad \text{(3)}\\ +\vdash .(1).(3).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_707">[Pg 707]</span></p> + +<p class="nind"><b>*216·51.</b> \[\begin{align}&\vdash :P\in \text{Ser} .\supset .\\ +&\delta (\varsigma ʻP)ʻ\overrightarrow{P}ʻʻCʻP=\delta (\varsigma ʻP)ʻCʻ\varsigma ʻP=\text{D}ʻ\text{lt}(\varsigma ʻP)-\iota ʻ\Lambda =\text{ᗡ}ʻ\text{sgm}ʻP\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*212·661.&\supset \vdash :\text{Hp}.\kappa \subset \text{D}ʻP_{\in }.x + = \text{lt}(\varsigma ʻP)ʻ\kappa .\supset .x = \text{lt}(\varsigma ʻP)ʻ\overrightarrow{P}ʻʻsʻ\kappa &\qquad \text{(1)}\\ +\vdash .*207·13.*212·133.\supset \\ +\vdash :\text{Hp}.\kappa \subset \text{D}ʻP_{\in }.x = \text{lt}(\varsigma ʻP)ʻ\kappa .&\supset .\kappa \neq \iota ʻ\Lambda &\qquad \text{(2)}\\ +\vdash .(1).(2).*40·26.\supset \\ +\vdash :\text{Hp}.\kappa \subset \text{D}ʻP_{\in }.\exists + !\kappa .x = \text{lt}(\varsigma ʻP)ʻ\kappa .&\supset .\exists !sʻ\kappa .x = \text{lt}(\varsigma ʻP)ʻ\overrightarrow{P}ʻʻsʻ\kappa .\\ +[*216·1] &\supset .x\in \delta (\varsigma ʻP)ʻ\overrightarrow{P}ʻʻCʻP &\qquad \text{(3)}\\ +\vdash .(3).*216·1. &\supset \vdash :\text{Hp}.\supset .\delta (\varsigma ʻP)ʻCʻ\varsigma ʻP \subset \delta (\varsigma ʻP)ʻ\overrightarrow{P}ʻʻCʻP &\qquad \text{(4)}\\ +\vdash .*211·3.*216·15.&\supset \vdash .\delta (\varsigma ʻP)ʻ\overrightarrow{P}ʻʻCʻP\subset \delta (\varsigma ʻP)ʻCʻ\varsigma ʻP &\qquad \text{(5)}\\ +\vdash .(4).(5). \supset \vdash :\text{Hp}.\supset .\delta (\varsigma ʻP)ʻ\overrightarrow{P}ʻʻCʻP &= \delta (\varsigma ʻP)ʻCʻ\varsigma ʻP &\qquad \text{(6)}\\ +[*216·2.*212·133] &= \text{D}ʻ\text{lt}(\varsigma ʻP)-\iota ʻ\Lambda &\qquad \text{(7)}\\ +\vdash .(6).(7).*212·667.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*216·52">*216·52</a>.</b> + \(\vdash :P\in \text{Ser} .\dot{\exists} !P.\alpha \subset CʻP.\supset .\delta (\varsigma ʻP)ʻ\overrightarrow{P}ʻʻ\alpha = Pʻʻʻ(\text{Cl ex}ʻ\alpha -\text{ᗡ}ʻ\text{max}_{P})\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*216·1.\supset \vdash \colon\ldotp &\text{Hp}.\supset :\gamma \in \delta (\varsigma ʻP)ʻ\overrightarrow{P}ʻʻ\alpha .\equiv .(\exists \kappa ).\kappa \subset \overrightarrow{P}ʻʻ\alpha .\exists !\kappa .\gamma = \text{lt}(\varsigma ʻP)ʻ\kappa .\\ +[*212·402] &\equiv .(\exists \kappa ).\kappa \subset \overrightarrow{P}ʻʻ\alpha .\exists !\kappa .{\sim}\in !\text{max}(\varsigma ʻP)ʻ\kappa .\gamma = sʻ\kappa .\\ +[*71·47.*37·2] &\equiv .(\exists \beta ).\beta \subset \alpha .\exists !\beta .{\sim}\text{E}!\text{max}(\varsigma ʻP)ʻ\overrightarrow{P}ʻʻ\beta .\gamma = sʻ\overrightarrow{P}ʻʻ\beta .\\ +[*40·5.*212·601]&\equiv .(\exists \beta ).\beta \subset \alpha .\exists !\beta .{\sim}\text{E}!\text{max}_{P}ʻ\beta .\gamma = Pʻʻ\beta .\\ +[*37·6] &\equiv .\gamma \in Pʻʻʻ(\text{Cl ex}ʻ\alpha -\text{ᗡ}ʻ\text{max}_{P})\colon\ldotp \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*216·521.</b> \(\vdash :P\in \text{Ser} .\alpha \subset CʻP.\supset .\overrightarrow{P}ʻʻ(\alpha -\overrightarrow{\text{min}}_{P}ʻ\alpha + ) = \overrightarrow{P}ʻʻ\alpha -\overrightarrow{\text{min}}(\varsigma ʻP)ʻ\overrightarrow{P}ʻʻ\alpha\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*71·381.*204·34.\supset \vdash :\text{Hp}.\supset .\overrightarrow{P}ʻʻ(\alpha -\overrightarrow{\text{min}}_{P}ʻ\alpha + ) &= \overrightarrow{P}ʻʻ\alpha -\overrightarrow{P}ʻʻ\text{min}_{P}ʻ\alpha \\ +[*212·6] &= \overrightarrow{P}ʻʻ\alpha -\overrightarrow{\text{min}}(\varsigma ʻP)ʻ\overrightarrow{P}ʻʻ\alpha :\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_708">[Pg 708]</span></p> + +<p class="nind"><b><a id="*216·53">*216·53</a>.</b> + \(\vdash \colon\ldotp P\in \text{Ser} .\dot{\exists} !P.\alpha \subset CʻP.\supset :\alpha \in \text{dense}ʻP.\equiv .\overrightarrow{P}ʻʻ\alpha \in \text{dense}ʻ\varsigma ʻP\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *216·52·3.\supset \\ +\vdash \colon\colon\text{Hp}.\supset \colon\ldotp &\overrightarrow{P}ʻʻ\alpha \in \text{dense}ʻ\varsigma ʻP.\equiv :\\ +&\overrightarrow{P}ʻʻ\alpha -\overrightarrow{\text{min}}(\varsigma ʻP)ʻ\overrightarrow{P}ʻʻ\alpha \subset Pʻʻʻ(\text{Cl ex}ʻ\alpha -\text{ᗡ}ʻ\text{max}_{P}):\\ +[*216·521] &\equiv :\overrightarrow{P}ʻʻ(\alpha -\overrightarrow{\text{min}}_{P}ʻ\alpha )\subset Pʻʻʻ(\text{Cl ex}ʻ\alpha -\text{ᗡ}ʻ\text{max}_{P}):\\ +[*37·6] &\equiv :x\in \alpha -\overrightarrow{\text{min}}_{P}ʻ\alpha .\supset _{x}.(\exists \beta ).\beta \subset \alpha .\exists !\beta .{\sim}\text{E}!\text{max}_{P}ʻ\beta + .\overrightarrow{P}ʻx = Pʻʻ\beta :\\ +[*207·521] &\equiv :x\in \alpha -\overrightarrow{\text{min}}_{P}ʻ\alpha .\supset _{x}.(\exists \beta ).\beta \subset \alpha .\exists !\beta .x = \text{lt}_Pʻ\beta :\\ +[*216·1] &\equiv :x\in \alpha -\overrightarrow{\text{min}}_{P}ʻ\alpha .\supset _{x}.x\in \delta _{P}ʻ\alpha :\\ +[*216·3] &\equiv :\alpha \in \text{dense}ʻP\colon\colon\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*216·54">*216·54</a>.</b> + \(\vdash \colon\ldotp P\in \text{Ser} .\dot{\exists} !P.\alpha \subset CʻP.\supset :\alpha \in \text{closed}ʻP.\equiv .\delta (\varsigma ʻP)ʻ\overrightarrow{P}ʻʻ\alpha \subset \overrightarrow{P}ʻʻ\alpha\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*216·52.\supset \\ +\vdash \colon\colon\text{Hp}.&\supset \colon\ldotp \delta (\varsigma ʻP)ʻ\overrightarrow{P}ʻʻ\alpha \subset \overrightarrow{P}ʻʻ\alpha .\equiv :Pʻʻʻ(\text{Cl ex}ʻ\alpha -\text{ᗡ}ʻ\text{max}_{P})\subset + \overrightarrow{P}ʻʻ\alpha :\\ +[*37·6] &\equiv :\beta \subset \alpha .\exists !\beta .{\sim}\text{E}!\text{max}_{P}ʻ\beta .\supset _{\beta }.(\exists x).x\in \alpha .Pʻʻ\beta = \overrightarrow{P}ʻx:\\ +[*207·521] &\equiv :\beta \subset \alpha .\exists !\beta .{\sim}\text{E}!\text{max}_{P}ʻ\beta .\supset _{\beta }.\text{lt}_Pʻ\beta \in \alpha :\\ +[*216·34] &\equiv :\alpha \in \text{closed}ʻP\colon\colon\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*216·55.</b> + \(\vdash \colon\ldotp P\in \text{Ser} .\dot{\exists} !P.\alpha \subset CʻP.\supset :\alpha \in \text{closed}ʻP.\equiv .\overrightarrow{P}ʻʻ\alpha \in \text{closed}ʻ\varsigma ʻP\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*212·44.\supset \vdash :\text{Hp}.\supset .\overrightarrow{P}ʻʻ\alpha \subset \text{ᗡ}ʻ\text{limax}(\varsigma ʻP) &\qquad \text{(1)}\\ +\vdash .(1).*212·54·32.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*216·56">*216·56</a>.</b> + \[\begin{align}\vdash \colon\ldotp P\in \text{Ser} .\dot{\exists} !P.\alpha &\subset CʻP.\supset :\alpha \in \text{perf}ʻP.\equiv .\overrightarrow{P}ʻʻ\alpha \in \text{perf}ʻ\varsigma ʻP.\\ +&\equiv .\delta (\varsigma ʻP)ʻ\overrightarrow{P}ʻʻ\alpha = \overrightarrow{P} ʻʻ\alpha - \overrightarrow{\text{min}}(\varsigma ʻP)ʻ\overrightarrow{P}ʻʻ\alpha\\ +&[*216·53·54·55·36·37.*212·44]\end{align}\]</p> + +<p class="nind"><b><a id="*216·6">*216·6</a>.</b> \(\vdash :x(\nabla ʻP)y.\equiv .x,y\in \text{D}ʻ\text{lt}_P.xPy \quad[(*216·05)]\)</p> + +<p class="nind"><b>*216·601.</b> \(\vdash :x\in \text{D}ʻ\text{lt}_P\cap \text{ᗡ}ʻP.P\in \text{connex} .\text{E}!BʻP.\supset .(BʻP)(\nabla ʻP)x\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*206·14. &\supset \vdash :\text{Hp}.\supset .BʻP\in \text{D}ʻ\text{lt}_P &\qquad \text{(1)}\\ +\vdash .*202·524.&\supset \vdash :\text{Hp}.\supset .(BʻP)Px &\qquad \text{(2)}\\ +\vdash .(1).(2).*216·6.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_709">[Pg 709]</span></p> + +<p class="nind"><b>*216·602.</b> \(\vdash :P\in \text{connex} .\text{E}!BʻP.\supset .\text{ᗡ}ʻ\nabla ʻP=\text{D}ʻ\text{lt}_P-\overrightarrow{B}ʻP=\delta _{P}ʻCʻP\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*216·601.&\supset \vdash :\text{Hp}.\supset .\text{D}ʻ\text{lt}_P-\overrightarrow{B}ʻP\subset \text{ᗡ}ʻ\nabla ʻP &\qquad \text{(1)}\\ +\vdash .*216·6. &\supset \vdash .\text{ᗡ}ʻ\nabla ʻP\subset \text{D}ʻ\text{lt}_P-\overrightarrow{B}ʻP &\qquad \text{(2)}\\ +\vdash .(1).(2).*216·2.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*216·603.</b> \(\vdash :P\in \text{connex} .\dot{\exists} !\nabla ʻP.\supset .Cʻ\nabla ʻP=\text{D}ʻ\text{lt}_P\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*200·35.\supset \vdash :\text{Hp}.&\supset .\text{D}ʻ\text{lt}_P{\sim}\in 1.\\ +[*202·55] &\supset .Cʻ\nabla ʻP=\text{D}ʻ\text{lt}_P:\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*216·61.</b> \(\vdash :P\in \text{Ser}.\text{E}!BʻP.\supset .\text{ᗡ}ʻ\nabla ʻP=\text{ᗡ}ʻP-\text{ᗡ}ʻP_{1} \quad[*216·602·21]\)</p> + +<p class="nind"><b>*216·611.</b> \(\vdash :P\in \text{Ser}.\dot{\exists} !\nabla ʻP.\supset .Cʻ\nabla ʻP=CʻP-\text{ᗡ}ʻP_{1}=\delta _{P}ʻCʻP\cup \overrightarrow{B}ʻP\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*216·603.*206·14.\supset \vdash :\text{Hp}.\supset .Cʻ\nabla ʻP&=(\text{D}ʻ\text{lt}_P-\overrightarrow{B}ʻP)\cup \overrightarrow{B}ʻP\\ +[*216·2] & =\delta _{P}ʻCʻP\cup \overrightarrow{B}ʻP &\qquad \text{(1)}\\ +[*216·21] &=(\text{ᗡ}ʻP-\text{ᗡ}ʻP_{1})\cup \overrightarrow{B}ʻP\\ +[*93·103.*24·412] &=CʻP-\text{ᗡ}ʻP_{1} &\qquad \text{(2)}\\ +\vdash .(1).(2).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*216·612.</b> \(\vdash :P\in \text{Ser}.\supset .\text{ᗡ}ʻ\nabla ʻP\subset \text{ᗡ}ʻP-\text{ᗡ}ʻP_{1}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*216·6. &\supset \vdash .\text{ᗡ}ʻ\nabla ʻP\subset \text{D}ʻ\text{lt}_P-\overrightarrow{B}ʻP &\qquad \text{(1)}\\ +\vdash .*216·2·21. &\supset \vdash :\text{Hp}.\supset .\text{D}ʻ\text{lt}_P-\overrightarrow{B}ʻP=\text{ᗡ}ʻP-\text{ᗡ}ʻP_{1} &\qquad \text{(2)}\\ +\vdash .(1).(2).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*216·62.</b> \(\vdash :P\in \text{Ser}.\dot{\exists} !\nabla ʻP.\supset .Cʻ\nabla ʻP=\text{seq}_{P}ʻʻCʻ\text{sgm}ʻP=\text{lt}_PʻʻCʻ\text{sgm}ʻP\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*216·611.\supset \vdash :\text{Hp}.\supset .Cʻ\nabla ʻP&=\delta _{P}ʻCʻP\cup \overrightarrow{B}ʻP\\ +[*216·23] &=\text{seq}_{P}ʻʻ\text{ᗡ}ʻ\text{sgm}ʻP\cup \overrightarrow{B}ʻP &\qquad \text{(1)}\\ +[*206·14] &=\text{seq}_{P}ʻʻ(\text{ᗡ}ʻ\text{sgm}ʻP\cup \iota ʻ\Lambda ) &\qquad \text{(2)}\\ +\vdash .*211·45. \supset \vdash :\text{Hp}.\exists !\text{ᗡ}ʻP-\text{ᗡ}ʻP_{1}.&\supset .\exists !\text{D}ʻ(P_{\in }\dot{\cap} I)-\iota ʻ\Lambda .\\ +[*212·153] &\supset .\dot{\exists} !\text{sgm}ʻP.\\ +[*212·155] &\supset .\text{ᗡ}ʻ\text{sgm}ʻP\cup \iota ʻ\Lambda =Cʻ\text{sgm}ʻP &\qquad \text{(3)}\\ +\vdash .(1).*216·23.*207·17.&\supset \vdash :\text{Hp}.\supset .Cʻ\nabla ʻP=\text{lt}_Pʻʻ(\text{ᗡ}ʻ\text{sgm}ʻP\cup \iota ʻ\Lambda ) &\qquad \text{(4)}\\ +\vdash .(2).(3).(4).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*216·621">*216·621</a>.</b> \(\vdash :P\in \text{Ser}.\dot{\exists} !\nabla ʻP.\supset .\dot{\exists} !\text{sgm}ʻP.\exists !\text{ᗡ}ʻP-\text{ᗡ}ʻP_{1} \quad[*216·62·612]\)</p> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_710">[Pg 710]</span></p> +<h2 class="nobreak" id="217_ON_SEGMENTS_OF_SUMS_AND_CONVERSES"> +*217. ON SEGMENTS OF SUMS AND CONVERSES.</h2> +</div> + + +<p class="nind"><i>Summary of</i> *217.</p> + +<p>The purpose of the present number is to prove <a href="#*217·43">*217·43</a>, which is +required in the theory of real numbers (Part VI, Section A), where +\(Q\) will be the series of positive ratios including zero, \(P\) will +be the series of negative ratios in the order from zero to -\(\infty\) +(both excluded), \(\alpha\) the real number zero, and \(Z\) and \(W\) +two different series either of which may be taken as the series of +negative and positive real numbers. In virtue of *217·43, these two +series are ordinally similar.</p> + +<hr class="tb"> + +<p class="nind"><b>*217·1.</b> \(\vdash :\alpha \cap CʻQ=\Lambda .\supset .(P\unicode{x2909} Q)ʻʻ\alpha =Pʻʻ\alpha \quad[*160·1]\)</p> + +<p class="nind"><b>*217·11.</b> \(\vdash :\exists !\alpha \cap CʻQ.\supset .(P\unicode{x2909} Q)ʻʻ\alpha =CʻP\cup Qʻʻ\alpha \quad[*160·1]\)</p> + +<p class="nind"><b>*217·12.</b> \(\vdash .\text{D}ʻ(P\unicode{x2909} Q)_{\in }\subset \text{D}ʻP_{\in }\cup (CʻP\cup )ʻʻ\text{D}ʻQ_{\in } \quad[*217·1·11.*211·11]\)</p> + +<p class="nind"><b>*217·13.</b> \(\vdash :CʻP\cap CʻQ=\Lambda .\supset .Pʻʻ\alpha =(P\unicode{x2909} Q)ʻʻ(\alpha -CʻQ) \quad[*217·1]\)</p> + +<p class="nind"><b>*217·14.</b> \(\vdash :\exists !Qʻʻ\alpha .\supset .CʻP\cup Qʻʻ\alpha =(P\unicode{x2909} Q)ʻʻ\alpha \quad[*217·11]\)</p> + +<p class="nind"><b>*217·15.</b> \[\begin{align}&\vdash :CʻP\cap CʻQ=\Lambda .\supset .\text{D}ʻP_{\in }\cup (CʻP\cup )ʻʻ(\text{D}ʻQ_{\in }-{℩}ʻ\Lambda + )\subset \text{D}ʻ(P\unicode{x2909} Q)_{\in }\\ +&[*217·13·14]\end{align}\]</p> + +<p class="nind"><b>*217·16.</b> \(\vdash \colon\ldotp CʻP\cap CʻQ=\Lambda :{\sim}\exists !\overrightarrow{B}ʻ\breve{P} .\lor.\exists !\overrightarrow{B}ʻQ:\supset .CʻP_{\in }\text{D}ʻ(P\unicode{x2909} + Q)_{\in }\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*211·301. &\supset \vdash :{\sim}\exists !\overrightarrow{B}ʻ\breve{P} .\supset .CʻP\in \text{D}ʻP_{\in } &\qquad \text{(1)}\\ +\vdash .(1).*217·15.&\supset \vdash :\text{Hp}.{\sim}\exists !\overrightarrow{B}ʻ\breve{P} .\supset .CʻP\in \text{D}ʻ(P\unicode{x2909} Q)_{\in } &\qquad \text{(2)}\\ +\vdash .*217·11. &\supset \vdash :\exists !\overrightarrow{B}ʻQ.\supset .(P\unicode{x2909} Q)ʻʻ\overrightarrow{B}ʻQ=CʻP &\qquad \text{(3)}\\ +\vdash .(2).(3).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*217·17.</b> \[\begin{align}\vdash \colon\ldotp CʻP\cap &CʻQ=\Lambda :{\sim}\exists !\overrightarrow{B}ʻ\breve{P} .\lor.\exists !\overrightarrow{B}ʻQ:\supset .\\ +&\text{D}ʻ(P\unicode{x2909} Q)_{\in }=\text{D}ʻP_{\in }\cup (CʻP\cup )ʻʻ\text{D}ʻQ_{\in } \quad[*217·12·15·16]\end{align}\]</p> + +<p class="nind"><b>*217·18.</b> \[\begin{align}\vdash \colon\ldotp CʻP\cap CʻQ=\Lambda :{\sim}&\exists !\overrightarrow{B}ʻ\breve{P} .\lor.{\sim}\exists !\overrightarrow{B}ʻQ:\supset .\\ +&\text{D}ʻ(P\unicode{x2909} Q)_{\in }=\text{D}ʻP_{\in }\cup (CʻP\cup )ʻʻ(\text{D}ʻQ_{\in }-{℩}ʻ\Lambda )\end{align}\]</p> + +<p><span class="pagenum" id="Page_711">[Pg 711]</span></p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*211·301.&\supset \vdash :{\sim}\exists !\overrightarrow{B}ʻ\breve{P} .\supset .(CʻP\cup )ʻʻ{℩}ʻ\Lambda \subset \text{D}ʻP_{\in } + &\qquad \text{(1)}\\ +\vdash .(1).*217·17.\supset \vdash :\text{Hp}.{\sim}&\exists !\overrightarrow{B}ʻ\breve{P} .\supset .\\ +&\text{D}ʻ(P\unicode{x2909}Q)_{\in }=\text{D}ʻP_{\in }\cup (CʻP\cup )ʻʻ(\text{D}ʻQ_{\in }-{℩}ʻ\Lambda ) &\qquad \text{(2)}\\ +\vdash .*217·11. &\supset \vdash :{\sim}\exists !\overrightarrow{B}ʻQ.\exists !\alpha \cap CʻQ.\supset .\exists !(P\unicode{x2909}Q)ʻʻ\alpha \cap CʻQ &\qquad \text{(3)}\\ +\vdash .*217·1. &\supset \vdash :{\sim}\exists !\overrightarrow{B}ʻQ.\exists !\overrightarrow{B}ʻ\breve{P} .\alpha \cap CʻQ=\Lambda .\supset .(P\unicode{x2909}Q)ʻʻ\alpha \neq CʻP &\qquad \text{(4)}\\ +\vdash .(3).(4). &\supset \vdash :\text{Hp}.{\sim}\exists !\overrightarrow{B}ʻQ.\exists !\overrightarrow{B}ʻ\breve{P} .\supset .CʻP{\sim}\in \text{D}ʻ(P\unicode{x2909}Q)_{\in } + &\qquad \text{(5)}\\ +\vdash .(5).*217·12·15.\supset \\ +\vdash :\text{Hp}(5).&\supset .\text{D}ʻ(P\unicode{x2909}Q)_{\in }=\text{D}ʻP_{\in }\cup (CʻP\cup )ʻʻ(\text{D}ʻQ_{\in }-{℩}ʻ\Lambda )&\qquad \text{(6)}\\ +\vdash .(2).(6).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*217·2.</b> \(\vdash :CʻP\cap CʻQ=\Lambda .\supset .\text{D}ʻP_{\in }\cap (CʻP\cup )ʻʻ(\text{D}ʻQ_{\in }-{℩}ʻ\Lambda )=\Lambda\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*211·11.\supset \\ +\vdash :\text{D}ʻP_{\in }\subset \text{Cl}ʻCʻP:\alpha \in (CʻP\cup )ʻʻ(\text{D}ʻQ_{\in }-{℩}ʻ\Lambda ).\supset .\exists !\alpha \cap CʻQ:\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*217·21.</b> \(\vdash :\exists !\overrightarrow{B}ʻ\breve{P} .\supset .\text{D}ʻP_{\in }\cap (CʻP\cup )ʻʻ\text{D}ʻQ_{\in }=\Lambda\)</p> + +<p><i>Dem.</i> +\[ +\vdash .*211·11.\supset \vdash :\text{Hp}.\alpha \in \text{D}ʻP_{\in }.\supset .\exists !CʻP-\alpha :\supset \vdash .\text{Prop} +\]</p> + +<p class="nind"><b>*217·22.</b> \[\begin{align}\vdash :P,Q\in \text{trans}\cap \text{connex} .CʻP\cap CʻQ&=\Lambda .\exists !\overrightarrow{B}ʻ\breve{P} .\exists!\overrightarrow{B}ʻQ.\supset .\\ +&\varsigma ʻ(P\unicode{x2909}Q)=\varsigma ʻP\unicode{x2909}(CʻP\cup )^;\varsigma ʻQ\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*201·401.*202·401.&\supset \vdash :\text{Hp}.\supset .P\unicode{x2909}Q_{\in }\text{trans}\cap \text{connex} &\qquad \text{(1)}\\ +\vdash .(1).*212·23.\supset \\ +\vdash \colon\colon\text{Hp}.\supset \colon\ldotp \alpha \{\varsigma ʻ(P\unicode{x2909}Q)\}\beta .&\equiv :\alpha ,\beta \in \text{D}ʻ(P\unicode{x2909}Q)_{\in }.\alpha + \subset \beta .\alpha \neq \beta :\\ +[*217·17·21]&\equiv :\alpha ,\beta \in \text{D}ʻP_{\in }.\alpha \subset \beta .\alpha \neq \beta .\lor.\alpha \in \text{D}ʻP_{\in }.\beta + \in (CʻP\cup )ʻʻ\text{D}ʻQ_{\in }.\\ +&\lor.\alpha ,\beta \in (CʻP\cup )ʻʻ\text{D}ʻQ_{\in }.\alpha \subset \beta .\alpha \neq \beta :\\ +[*212·23] &\equiv :\alpha (\varsigma ʻP)\beta .\lor.\alpha \in Cʻ\varsigma ʻP.\beta \in Cʻ(CʻP\cup )^{;}\varsigma ʻQ.\\ +&\lor.\alpha \{(CʻP\cup )^{;}\varsigma ʻQ\}\beta :\\ +[*160·11] &\equiv :\alpha \{\varsigma ʻP\unicode{x2909}(CʻP\cup )^{;}\varsigma ʻQ\}\beta \colon\colon\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*217·23.</b> + \[\begin{align}\vdash \colon\ldotp P,Q\in \text{trans}\cap \text{connex} .&CʻP\cap CʻQ=\Lambda :{\sim}\exists !\overrightarrow{B}ʻ\breve{P} .\lor.{\sim}\exists !\overrightarrow{B}ʻQ:\supset .\\ +&\varsigma ʻ(P\unicode{x2909}Q)=\varsigma ʻP\unicode{x2909}(CʻP\cup )^{;}(\varsigma ʻQ)\unicode{x0294f}(-{℩}ʻ\Lambda )\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*201·401.*202·401.*212·23.\supset \\ +\vdash \colon\colon\text{Hp}.\supset \colon\ldotp \alpha\{\varsigma ʻ(P\unicode{x2909}Q)\}\beta .&\equiv :\alpha ,\beta \in \text{D}ʻ(P\unicode{x2909}Q)_{\in }.\alpha + \subset \beta .\alpha \neq \beta :\\ +[*217·18·2]&\equiv :\alpha ,\beta \in \text{D}ʻP_{\in }.\alpha \subset \beta .\alpha \neq \beta .\lor.\alpha \in \text{D}ʻP_{\in }.\beta \in (CʻP\cup )ʻʻ(\text{D}ʻQ_{\in }-{℩}ʻ\Lambda + ).\\ +&\lor.\alpha ,\beta \in (CʻP\cup )ʻʻ(\text{D}ʻQ_{\in }-{℩}ʻ\Lambda ).\alpha \subset \beta .\alpha \neq \beta :\\ +[*212·23.*160·11]&\equiv :\alpha \{\varsigma ʻP\unicode{x2909}(CʻP\cup )^{;}(\varsigma ʻQ)\unicode{x0294f}(-{℩}ʻ\Lambda )\}\beta \colon\colon\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*217·24.</b> \(\vdash :\alpha \cap \beta =\Lambda .\supset .(\alpha \cup )\upharpoonright ʻ\text{Cl}ʻ\beta \in 1\rightarrow 1 \quad[*24·481]\)</p> + +<p class="nind"><b>*217·25.</b> \[\begin{align}&\vdash :CʻP\cap CʻQ=\Lambda .\supset .(CʻP\cup )\upharpoonright Cʻ\varsigma ʻQ\in \{(CʻP\cup )^{;}\varsigma ʻQ\}\overline{\,\text{smor}\,} (\varsigma ʻQ)\\ +&[*217·24]\end{align}\]</p> + +<p><span class="pagenum" id="Page_712">[Pg 712]</span></p> + +<p class="nind"><b>*217·3.</b> \[\begin{align}&\vdash :P\in \text{Ser}.\supset .\text{ᗡ}ʻP_{\in }=\overrightarrow{P}ʻʻCʻP\cup \text{D}ʻ(P_{\in }\dot{\cap} I)-\text{ᗡ}ʻ\text{seq}_{P}\\ +&[*211·32·302·41]\end{align}\]</p> + +<p class="nind"><b>*217·301.</b> \(\vdash :P\in \text{Ser}.\gamma \in \text{D}ʻ(P_{\in }\dot{\cap} I)-\text{ᗡ}ʻ\text{seq}_{P}.\supset .\gamma =CʻP-\breve{P} ʻʻ(CʻP-\gamma )\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*211·727.\supset \vdash :\text{Hp}.&\supset .{\sim}\text{E}!\,\text{limin}_Pʻ(CʻP-\gamma ).\\ +[*207·44.*211·7] &\supset .CʻP-\gamma \in \text{sect}ʻ\breve{P} -\text{ᗡ}ʻ\text{min}_{P}.\\ +[*211·41·12] &\supset .CʻP-\gamma =\breve{P} ʻʻ(CʻP-\gamma ):\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*217·31.</b> \(\vdash :P\in \text{Ser}.\gamma \in \text{D}ʻP_{\in }.\supset .(\exists \beta ).\gamma =Pʻʻ(CʻP-\breve{P} ʻʻ\beta )\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*201·53.\supset \\ +\vdash :\text{Hp}.\gamma =\overrightarrow{P}ʻx.\beta =\overleftarrow{P}_{*}ʻx.&\supset .CʻP-\breve{P} ʻʻ\beta =\overrightarrow{P}_{*}ʻx.\\ +[*201·53] &\supset .Pʻʻ(CʻP-\breve{P} ʻʻ\beta )=\gamma &\qquad \text{(1)}\\ +\vdash .(1).*217·3·301.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*217·32.</b> \(\vdash :P\in \text{Ser}.\supset .\text{D}ʻ(\breve{P} )_{\in }=(\breve{P} )_{\in }ʻʻ(CʻP-)ʻʻ\text{D}ʻP_{\in }\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*217·31 \frac{\breve{P}}{P}.\supset \vdash :\text{Hp}.\supset .\text{D}ʻ(\breve{P} )_{\in }\subset (\breve{P} )_{\in }ʻʻ(CʻP-)ʻʻ\text{D}ʻP_{\in } + &\qquad \text{(1)}\\ +\vdash .(1).*37·16.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*217·33.</b> \(\vdash .(\alpha -)\upharpoonright \text{Cl}ʻ\alpha \in 1\rightarrow 1\)</p> + +<p><i>Dem.</i> +\[ +\vdash .*24·492.\supset \vdash :\beta \subset \alpha .\gamma \subset \alpha .\alpha -\beta =\alpha -\gamma .\supset .\beta =\gamma :\supset \vdash .\text{Prop} +\]</p> + +<p class="nind"><b>*217·34.</b> \(\vdash :P\in \text{Ser}.\supset .P_{\in }\upharpoonright (\text{sect}ʻP-\text{ᗡ}ʻ\text{lt}_P)\in 1\rightarrow 1\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*211·1.&\supset \vdash :\alpha ,\beta \in \text{sect}ʻP.Pʻʻ\alpha =Pʻʻ\beta .\exists !\beta -\alpha .\supset .\exists !\beta -Pʻʻ\beta &\qquad \text{(1)}\\ +\vdash .(1).*205·111. & \supset \vdash :\text{Hp}.\text{Hp}(1).\supset .\text{E}!\text{max}_{P}ʻ\beta &\qquad \text{(2)}\\ +\vdash .*211·56. \supset \vdash :\text{Hp}(2).&\supset .\alpha \subset Pʻʻ\beta . &\qquad \text{(3)}\\ +[*205·111.(2)] &\supset .\text{max}_{P}ʻ\beta {\sim}\in \alpha &\qquad \text{(4)}\\ +\vdash .(3). \supset \vdash :\text{Hp}(2).&\supset .\alpha =Pʻʻ\beta .\\ +[*205·22.(2).\text{Hp}] &\supset .\alpha =\overrightarrow{P}ʻ\text{max}_{P}ʻ\beta =Pʻʻ\alpha &\qquad \text{(5)}\\ +\vdash .(4).(5).*207·232.&\supset \vdash :\text{Hp}(2).\supset .\text{max}_{P}ʻ\beta =\text{lt}_Pʻ\alpha &\qquad \text{(6)}\\ +\vdash .(6).\text{Transp}.&\supset \vdash :\text{Hp}.\alpha ,\beta \in \text{sect}ʻP.Pʻʻ\alpha =Pʻʻ\beta .{\sim}\text{E}!\text{lt}_Pʻ\alpha .\supset .\beta \subset \alpha &\qquad \text{(7)}\\ +\text{Similarly}\quad &\vdash :\text{Hp}.\alpha ,\beta \in \text{sect}ʻP.Pʻʻ\alpha =Pʻʻ\beta .{\sim}\text{E}!\text{lt}_Pʻ\beta .\supset .\alpha \subset \beta &\qquad \text{(8)}\\ +\vdash .(7).(8).&\supset \vdash :\text{Hp}.\alpha ,\beta \in \text{sect}ʻP-\text{ᗡ}ʻ\text{lt}_P.Pʻʻ\alpha =Pʻʻ\beta .\supset .\alpha =\beta :\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_713">[Pg 713]</span></p> + +<p class="nind"><b>*217·35.</b> \(\vdash :P\in \text{Ser}.\supset .(\breve{P} )_{\in }\mid (CʻP-)\upharpoonright \text{D}ʻP_{\in }\in 1\rightarrow 1\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*217·33.&\supset \vdash .(CʻP-)\upharpoonright \text{D}ʻP_{\in }\in 1\rightarrow 1 &\qquad \text{(1)}\\ +\vdash .*211·76.\supset \vdash :\text{Hp}.&\supset .(CʻP-)ʻʻ\text{D}ʻP_{\in }=\text{sect}ʻ\breve{P} -\text{ᗡ}ʻ\text{tl}_P.\\ +[*217·34] &\supset .(\breve{P} )_{\in }\upharpoonright (CʻP-)ʻʻ\text{D}ʻP_{\in }\in 1\rightarrow 1 &\qquad \text{(2)}\\ +\vdash .(1).(2).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*217·36.</b> \(\vdash :P\in \text{Ser}.\supset .\varsigma ʻ\breve{P} =(\breve{P} )_{\in }^{;}(CʻP-)^{;}\text{Cnv}ʻ\varsigma ʻP\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*212·23.\supset \\ +\vdash \colon\ldotp \text{Hp}.&\supset :\beta (\varsigma ʻP)\alpha .\gamma =\breve{P} ʻʻ(CʻP-\alpha ).\delta =\breve{P} ʻʻ(CʻP-\beta ).\supset .\\ +&\beta \subset \alpha .\alpha \neq \beta .CʻP-\alpha \subset CʻP-\beta .\\ +[*37·2.*217·35]&\supset .\gamma \subset \delta .\gamma \neq \delta .\\ +[*212·23] &\supset .\gamma (\varsigma ʻ\breve{P} )\delta &\qquad \text{(1)}\\ +\vdash .(1).\supset \vdash :\text{Hp}.&\supset .(\breve{P} )_{\in }^{;}(CʻP-)^{;}\text{Cnv}ʻ\varsigma ʻP\,\unicode{x2abd}\, \varsigma ʻ\breve{P} &\qquad \text{(2)}\\ +\vdash .(1).\text{Transp}.\supset \\ +\vdash :\text{Hp}.&\delta (\varsigma ʻ\breve{P} )\gamma .\gamma =\breve{P} ʻʻ(CʻP-\alpha ).\delta =\breve{P} ʻʻ(CʻP-\beta ).\alpha ,\beta \in \text{D}ʻP_{\in }.\supset + .\alpha \subset \beta &\qquad \text{(3)}\\ +\vdash .*217·35. &\supset \vdash :\text{Hp}(3).\supset .\alpha \neq \beta &\qquad \text{(4)}\\ +\vdash .(3).(4).*212·23.&\supset \vdash :\text{Hp}(3).\supset .\delta {(\breve{P} )_{\in }^{;}(CʻP-)^{;}\text{Cnv}ʻ\varsigma ʻP}\gamma &\qquad \text{(5)}\\ +\vdash .*217·31.\supset \vdash :&\text{Hp}.\delta (\varsigma ʻ\breve{P} )\gamma .\supset .\\ +&(\exists \alpha ,\beta ).\gamma =\breve{P} ʻʻ(CʻP-\alpha ).\delta =\breve{P} ʻʻ(CʻP-\beta ).\alpha ,\beta \in \text{D}ʻP_{\in } &\qquad \text{(6)}\\ +\vdash .(5).(6).&\supset \vdash :\text{Hp}.\supset .\varsigma ʻ\breve{P} \,\unicode{x2abd}\, (\breve{P} )_{\in }^{;}(CʻP-)^{;}\text{Cnv}ʻ\varsigma ʻP &\qquad \text{(7)}\\ +\vdash .(2).(7).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*217·37.</b> \[\begin{align}&\vdash :P\in \text{Ser}.\supset .(\breve{P} )_{\in }\mid (CʻP-)\upharpoonright \text{D}ʻP_{\in }\in + (\varsigma ʻ\breve{P} )\overline{\,\text{smor}\,} (\text{Cnv}ʻ\varsigma ʻP)\\ +&[*217·35·36]\end{align}\]</p> + +<p class="nind"><b>*217·38.</b> \(\vdash :P\in \text{Ser}.\supset .(\varsigma ʻ\breve{P} )\,\text{smor}\,(\text{Cnv}ʻ\varsigma ʻP) \quad[*217·37]\)</p> + +<p class="nind"><b>*217·4.</b> \[\begin{align}\vdash :&P,Q\in \text{Ser}.CʻP\cap CʻQ=\Lambda .\text{E}!BʻP.\text{E}!BʻQ.\supset .\\ +&\varsigma ʻ(\breve{P} \unicode{x2909} Q)=(\breve{P} )_{\in }^{;}(CʻP-)^{;}\text{Cnv}ʻ\varsigma ʻP\unicode{x2909} (CʻP\cup )^{;}\varsigma + ʻQ \quad[*217·22·36]\end{align}\]</p> + +<p class="nind"><b>*217·41.</b> \[\begin{align}\vdash \colon\ldotp &P,Q\in \text{Ser}.CʻP\cap CʻQ=\Lambda :{\sim}\text{E}!BʻP.\lor.{\sim}\text{E}!BʻQ:\supset .\\ +&\varsigma ʻ(\breve{P} \unicode{x2909} Q)=(\breve{P} )_{\in }^{;}(CʻP-)^{;}\text{Cnv}ʻ\varsigma ʻP\unicode{x2909} (CʻP\cup )^{;}(\varsigma + ʻQ)\unicode{x0294f}(-{℩}ʻ\Lambda ) +&[*217·23·36]\end{align}\]</p> + +<p><span class="pagenum" id="Page_714">[Pg 714]</span></p> + +<p class="nind"><b>*217·411.</b> \[\begin{align}\vdash :\text{Hp}&*217·41.\supset .{\varsigma ʻ(\breve{P} \unicode{x2909} Q)}\unicode{x0294f}(-{℩}ʻ\Lambda )=\\ +&(\breve{P} )_{\in }^{;}(CʻP-)^{;}\text{Cnv}ʻ(\varsigma ʻP)\unicode{x0294f}(-{℩}ʻ\text{D}ʻP)\unicode{x2909} (CʻP\cup )^{;}(\varsigma + ʻQ)\unicode{x0294f}(-{℩}ʻ\Lambda )\\ +&[*217·41]\end{align}\]</p> + +<p class="nind"><b>*217·42.</b> \[\begin{align}\vdash :\text{Hp}*217·41.&\supset .\{\varsigma ʻ(\breve{P} \unicode{x2909} Q)\}\unicode{x0294f}\{-{℩}ʻ\Lambda -{℩}ʻ\text{D}ʻ(P\unicode{x2909} Q)\}=\\ +&(\breve{P} )_{\in }^{;}(CʻP-)^{;}\text{Cnv}ʻ(\varsigma ʻP)\unicode{x0294f}(-{℩}ʻ\Lambda -{℩}ʻ\text{D}ʻP)\unicode{x21f8}\text{ᗡ}ʻP\\ +&\unicode{x2909} (CʻP\cup )^{;}(\varsigma ʻQ)\unicode{x0294f}(-{℩}ʻ\Lambda -{℩}ʻ\text{D}ʻQ) \quad[*217·411]\end{align}\]</p> + +<p class="nind"><b><a id="*217·43">*217·43</a>.</b> \[\begin{align}&\vdash \colon\ldotp P,Q\in \text{Ser}.CʻP\cap CʻQ=\Lambda :{\sim}\text{E}!BʻP.\lor.{\sim}\text{E}!BʻQ:\\ +&X=(\varsigma ʻP)\unicode{x0294f}(-{℩}ʻ\Lambda -{℩}ʻ\text{D}ʻP).Y=(\varsigma ʻQ)\unicode{x0294f}(-{℩}ʻ\Lambda -{℩}ʻ\text{D}ʻQ).\\ +&Z=\{\varsigma ʻ(\breve{P} \unicode{x2909} Q)\}\unicode{x0294f}\{-{℩}ʻ\Lambda -{℩}ʻ\text{D}ʻ(\breve{P} \unicode{x2909} Q)\}.\\ +&W=\breve{X} \unicode{x21f8}\alpha \unicode{x2909} Y.\alpha {\sim}\in CʻX\cup CʻY.\supset .\\ +&(\breve{P} )_{\in }\mid (CʻP-)\upharpoonright (\text{D}ʻP_{\in }-{℩}ʻ\Lambda -{℩}ʻ\text{D}ʻP)\unicode{x228d} (\text{D}ʻP)\downarrow \alpha \\ +&\unicode{x228d} (CʻP\cup )\upharpoonright (\text{D}ʻQ_{\in }-{℩}ʻ\Lambda + -{℩}ʻ\text{D}ʻQ)\in Z\,\overline{\text{smor}}\,\,W \quad[*217·37·25·42]\end{align}\]</p> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_715">[Pg 715]</span></p> +<h2 class="nobreak" id="SECTION_C_III">SECTION C.<br> +<span class="allsmcap">ON CONVERGENCE, AND THE LIMITS OF FUNCTIONS.</span></h2> +</div> + + +<p>The purpose of this section is to express in a general form the +definitions of convergence, the limits of functions, the continuity +of functions, and kindred notions, and to give such elementary +consequences of these definitions as may seem illustrative.</p> + +<p>In the definitions usually given in treatises on analysis, it is +assumed that both the arguments and the values of the function are +numbers of some kind, generally real numbers, and limits are taken with +respect to the order of magnitude. There is, however, nothing essential +in the definitions to demand so narrow a hypothesis. What is essential +is that the arguments should be given as belonging to a series, and +that the values should also be given as belonging to a series, which +need not be the same series as that to which the arguments belong. In +what follows, therefore, we assume that all the possible arguments +to our function, or at any rate all the arguments which we consider, +belong to the field of a certain relation \(Q\), which, in cases where +our definitions are useful, will be a serial relation; we assume +similarly that the values of our function, at least for arguments +belonging to \(CʻQ\), belong to the field of a relation \(P\), which, +in all important cases, will be a serial relation. The function itself +we represent by the relation of the value to the argument; that is, the +relation of \(f(x)\) to \(x\) is to be \(R\), so that, if the function +is one-valued, \(f(x) = Rʻx\). (If the function is not one-valued, +\(f(x)\) is any member of \(\overrightarrow{R}ʻx\).) Thus we may speak +of \(R\) as the function, \(Q\) as the argument-series, and \(P\) as +the value-series.</p> + +<p>To take an illustration: Suppose we are given a set of real numbers +\(x_{1}\), \(x_{2}\), ... \(x_{\nu }\), ..., where \(\nu\) may be +any finite integer. Here \(x_{\nu }\) is a function of \(\nu\); the +argument-series is that of the finite integers in order of magnitude, +the value-series is that of the real numbers (or any part of this +series which contains all the values \(x_{1}\), \(x_{2}\), ... +\(x_{\nu }\), ...). The function \(R\) is the relation of \(x_{\nu}\) +to \(\nu\), so that \(x_{\nu } = Rʻ\nu\). In this case, calling +the argument-series \(Q\) and the value-series \(P\) (as will be done +throughout this section), we have \(\text{ᗡ}ʻR = CʻQ = \text{the finite integers}\), +\(RʻʻCʻQ = \text{D}ʻR = \text{the class}\,\, x_{1}\), +<span class="pagenum" id="Page_716">[Pg 716]</span>\(x_{2}\), ... \(x_{\nu }\), ..., and \(R^{;}Q\)= the <i>series</i> +\(x_{1}\), \(x_{2}\), ... \(x_{\nu }\), .... The series which arranges +\(x_{1}\), \(x_{2}\), ... \(x_{\nu }\), ... in the order of their +own magnitudes, instead of the order of magnitude of their suffixes, +is \(P\unicode{x0294f}\text{D}ʻR\) or \(P\unicode{x0294f}RʻʻCʻQ\). +This will not be equal to \(R^{;}Q\) unless the function is one which +continually increases, <i>i.e.</i> one for which \(\mu \lt \nu .\supset.x_{\mu } \lt x_{\nu }\).</p> + +<p>In general, the propositions of the present section are only +important when \(P\) and \(Q\) are series. If our assertions are +not to be trivial, we must have \(\exists !CʻQ\cap \text{D}ʻR\) and +\(\exists !CʻP\cap RʻʻCʻQ\), <i>i.e.</i> there must be arguments +in \(CʻQ\) which lead to values in \(CʻP\). It will also generally +happen that the function is one-valued, <i>i.e.</i> that \(R\in 1\rightarrow \text{Cls}\). +But the above conditions, though necessary to the <i>importance</i> of +our propositions, are in general much narrower than the hypotheses that +are necessary for the truth of our propositions.</p> + +<p>The present section is wholly self-contained, that is to say, its +propositions are not referred to in the sequel. We have, in this +section, carried the subject as far as seemed suitable for the present +work; its further development belongs to treatises on analysis.</p> + +<p>We begin (<a href="#*230">*230</a>) with a general conception which is involved in the +notion of convergency. We shall say that the values of a function +converge (or, simply, that the function itself converges) into the +class \(\alpha\), if for late enough arguments the values always belong +to the class \(\alpha\), <i>i.e.</i> if there is a term \(y\) such +that, if \(yQ_{*}z\), \(Rʻz\in \alpha\), or, to avoid assuming that +\(R\) is one-valued, \(\overrightarrow{R}ʻz\subset \alpha\). Thus the +values of the function converge into the class \(\alpha\) if +\[ +(\exists y).y\in CʻQ\cap \text{ᗡ}ʻR.Rʻʻ\overleftarrow{Q}_{*}ʻy\subset \alpha . +\] +If a term \(y\) is one such that, from \(y\) onward, all values belong +to \(\alpha\), we write \(y\in R\overline{Q}_{\text{cn}}\alpha\) (where +"\(\text{cn}\)" stands for "convergent"), <i>i.e.</i> we put +\[ +R\overline{Q}_{\text{cn}}\alpha =\hat{y} \{y\in CʻQ\cap \text{ᗡ}ʻR.Rʻʻ\overleftarrow{Q}_{*}ʻy\subset \alpha\} \quad\text{Df}. +\] +When there is such a \(y\), <i>i.e.</i> when the function converges +into the class \(\alpha\), we write "\(RQ_{\text{cn}}\alpha\)," +<i>i.e.</i> we put +\[ +Q_{\text{cn}} = \hat{R} \hat{\alpha} (\exists !R\overline{Q}_{\text{cn}}\alpha ) \quad\text{Df}. +\] +"\(RQ_{\text{cn}}\alpha\)" may be read "\(R\) is \(Q\)-convergent +into \(\alpha\)." This means that for arguments sufficiently late in +the \(Q\)-series, the value of the function is always a member of +\(\alpha\). Thus <i>e.g.</i> if \(Rʻx = \dfrac{1}{x}\), and +\(\alpha= \hat{y} (y \lt 1)\), \(RQ_{\text{cn}}\alpha\), and if +\(z \gt 1\), \(z\in R\overline{Q}_{\text{cn}}\alpha\).</p> + +<p>We next consider (<a href="#*231">*231</a>) <i>limiting sections</i> and <i>ultimate +oscillations</i> of functions. For this purpose, we proceed as +follows. If \(RQ_{\text{cn}}\alpha\), then \(P_{*}ʻʻ\alpha\) is +a section of the \(P\)-series such that, for sufficiently late +arguments, the values of the function must belong to \(P_{*}ʻʻ\alpha\). +Hence if we take all possible values of \(\alpha\) for which +\(RQ_{\text{cn}}\alpha\), and take the logical product of all the +resulting sections \(P_{*}ʻʻ\alpha\), we get a section containing all +the "ultimate" values of the<span class="pagenum" id="Page_717">[Pg 717]</span> function; moreover this is obviously +the smallest section which has this property, because, if we take any +section \(\beta\) which contains all the "ultimate" values, we have +\(RQ_{\text{cn}}\beta\), and \(P_{*}ʻʻ\beta = \beta\), and therefore +the logical product in question is contained in \(\beta\). The logical +product in question is +\[ +pʻP_{*}ʻʻʻ\overleftarrow{Q}_{\text{cn}}ʻR. +\] +In order to avoid trivial exceptions which arise when +\(CʻQ\cap\text{ᗡ}ʻR = \Lambda\), we define the "limiting section" as +\[ +pʻP_{*}ʻʻʻ\overleftarrow{Q}_{\text{cn}}ʻR\cap CʻP. +\] +This "limiting section" we denote by \(P\overline{R}_{\text{sc}}Q\), where +the letters "\(\text{sc}\)" stand for "section." Thus we put +\[ +P\overline{R}_{\text{sc}}Q = pʻP_{*}ʻʻʻ\overleftarrow{Q}_{\text{cn}}ʻR\cap CʻP \quad\text{Df}. +\]</p> + +<p>\(P\overline{R}_{\text{sc}}Q\) is the class of those members \(x\) +of the series \(P\) which are such that, given any argument however +late, there are still arguments as late or later for which the value +of the function is not less than \(x\). In like manner, \(\breve{P}\overline{R}_{\text{sc}}Q\), +which we will call the "limiting upper section," consists of those +members \(x\) of the series \(P\) which are such that, given any +argument however late, there are still arguments as late or later +for which the value of the function is not greater than \(x\). +Thus the product of \(P\overline{R}_{\text{sc}}Q\) and \(\breve{P}\overline{R}_{\text{sc}}Q\) +is the smallest stretch which contains all the "ultimate" values of +the function, <i>i.e.</i> it is the stretch consisting of those terms +\(x\) which are such that, however late an argument we take, there +are arguments as late or later for which the value of the function is +not greater than \(x\), and also arguments for which it is not less +than \(x\). Thus the product of \(P\overline{R}_{\text{sc}}Q\) and +\(\breve{P} \overline{R}_{\text{sc}}Q\) represents what we may call +the "ultimate oscillation" of the function. We shall denote it by +\(P\overline{R}_{\text{os}}Q\), putting +\[ +P\overline{R}_{\text{os}}Q = P\overline{R}_{\text{sc}}Q\cap \breve{P} \overline{R}_{\text{sc}}Q \quad\text{Df}. +\] +We may express \(P\overline{R}_{\text{sc}}Q\) in a form not involving +\(Q_{\text{cn}}\), namely (<a href="#*231·12">*231·12</a>) +\[ +P\overline{R}_{\text{sc}}Q = pʻP_{*}ʻʻʻRʻʻʻ\overleftarrow{Q}_{*}ʻʻ(CʻQ\cap \text{ᗡ}ʻR)\cap CʻP. +\]</p> + +<p>This formula for \(P\overline{R}_{\text{sc}}Q\) may be elucidated +by the following considerations. If \(y\) is any member of +\(CʻQ\), then \(\text{ᗡ}ʻR\cap \overleftarrow{Q}_{*}ʻy\) consists +of all arguments from \(y\) onwards. Hence \(Rʻʻ(\text{ᗡ}ʻR\cap\overleftarrow{Q}_{*}ʻy)\). +<i>i.e.</i> \(Rʻʻ\overleftarrow{Q}_{*}ʻy\), +consists of all values of the function for arguments from \(y\) +onwards. Hence \(P_{*}ʻʻRʻʻ\overleftarrow{Q}_{*}ʻy\) consists of all +members of the \(P\)-series which are equalled or surpassed by values +of the function for arguments equal to or later than \(y\). Now if a +term \(x\) belongs to the class \(P_{*}ʻʻRʻʻ\overleftarrow{Q}_{*}ʻy\) +for every argument \(y\), it is a term such that, however far up the +argument-series \(Q\) we go, we shall still find values as great as +or greater than \(x\). When this is the case, we may say that \(x\) +is<span class="pagenum" id="Page_718">[Pg 718]</span> \(P\)-persistent. In this case, \(x\) may be regarded as not +greater than the "ultimate" values of the function. Now the class of +arguments concerned is \(CʻQ\cap \text{ᗡ}ʻR\). Hence the class of +\(P\)-persistent terms is +\[ +pʻP_{*}ʻʻʻRʻʻʻ\overleftarrow{Q}_{*}ʻʻ(CʻQ\cap \text{ᗡ}ʻR), +\] +where the factor \(CʻP\) may be added in order to accommodate the +formula to the trivial case where \(CʻQ\cap \text{ᗡ}ʻR = \Lambda\) (the +only case in which the factor \(CʻP\) makes any difference). Thus the +class of \(P\)-persistent terms is the limiting section. Similarly the +\(\breve{P}\)-persistent terms are the limiting upper section. These +are the terms which are not less than the "ultimate" values of the +function. Thus the product \(P\overline{R}_{\text{os}}Q\) is the terms +which are neither greater than all ultimate values, nor less; hence it +is the class of ultimate values, which may be appropriately called the +"ultimate oscillation."</p> + +<p>It will be seen that \(P\overline{R}_{\text{os}}Q\), being the product +of an upper and lower section, is itself a stretch: we may call it +(alternatively) the "limiting stretch." It consists of all members +\(x\) of the \(P\)-series such that the function does not, however +great we make the argument, become and remain less than \(x\), nor yet +become and remain greater than \(x\). If \(P\overline{R}_{\text{os}}Q\) +consists of a single term, that term is the limit of the function +as the argument travels up the series \(Q\). (This is, of course, +in general different from the limit of the values of the function +considered simply as a class of members of \(CʻP\), <i>i.e.</i> it is +different from \(\text{lt}_PʻRʻʻCʻQ\).) If \(P\overline{R}_{\text{os}}Q\) +does not consist of a single term or none, we shall have two limits +to consider, namely \(\text{limax}_PʻP\overline{R}_{\text{os}}Q\) and +\(\text{limin}_PʻP\overline{R}_{\text{os}}Q\), which give the two +boundaries of the ultimate values of the function. When the class +\(P\overline{R}_{\text{os}}Q\) is null, the function may be regarded as +having a definite limit: in this case, \(P\overline{R}_{\text{sc}}Q\) +and \(\breve{P} \overline{R}_{\text{sc}}Q\) are the two parts +of an "irrational" Dedekind cut, <i>i.e.</i> a cut in which the +first portion has no maximum and the second no minimum. Thus +\(P\overline{R}_{\text{os}}Q\in 0\cup 1\) is the condition for a +definite limit of the function as the argument grows indefinitely.</p> + +<p>The above gives the generalization of the limit of a function when +the argument may be any member of \(CʻQ\cap \text{ᗡ}ʻR\). In order to +obtain limits for other classes of arguments, it is only necessary, +as a rule, to limit the field of \(Q\) to the class of arguments in +question, <i>i.e.</i> to replace \(Q\) by \(Q\unicode{x0294f}\alpha\) +(cf.<a href="#*232">*232</a>). In order, however, to avoid vexatious and trivial +exceptions arising when \(\alpha \in 1\), it is more convenient +to replace \(Q\) by \(Q_{*}\unicode{x0294f}\alpha\). Thus the +section of \(P\) defined by the class of arguments \(\alpha\) is +\(P\overline{R}_{\text{sc}}(Q_{*}\unicode{x0294f}\alpha\)). We put +\[ +(P\overline{R}Q)_{\text{sc}}ʻ\alpha = P\overline{R}_{\text{sc}} (Q_{*}\unicode{x0294f}\alpha ) \quad\text{Df}. +\] +This definition is useful because we very often wish to be able to +exhibit the limiting section defined by \(\alpha\) as a function of +\(\alpha\). The section \((P\overline{R}Q)_{\text{sc}}ʻ\alpha\) is such +that, if \(x\) is any member of it, and \(y\) is any argument belonging +to \(\alpha\), there is in \(\alpha\) an argument equal to or later +than \(y\), for which the function<span class="pagenum" id="Page_719">[Pg 719]</span> has a value equal to or later than +\(x\). Thus \(x\) is such that the function does not ultimately become +less than \(x\) as the argument increases in the class \(\alpha\). The +limit or maximum of such terms as \(x\) is the limit or maximum of the +ultimate values of the function as the argument approaches the top of +\(\alpha\). The class of ultimate values is +\[ +(P\overline{R}Q)_{\text{sc}}ʻ\alpha \cap (\breve{P} \overline{R}Q)_{\text{sc}}ʻ\alpha ,\, \text{which we call}\, (P\overline{R}Q)_{\text{os}}ʻ\alpha . +\] +If the function has a definite limit as the argument increases in +\(\alpha\), the class of ultimate values must not contain more than one +term.</p> + +<p>Our next number (<a href="#*233">*233</a>) deals with the limit of a function for a +given argument. The limit or maximum of the class of ultimate values +is not necessarily the value for the limit of \(\alpha\). It will +be found, however, that, with a suitable hypothesis, the limiting +section \((P\overline{R}Q)_{\text{sc}}ʻ\alpha\) depends only upon +\(Q_{*}ʻʻ(\alpha \cap \text{ᗡ}ʻR)\), and if \(\alpha \cap \text{ᗡ}ʻR\) +has no maximum, it depends only upon \(Qʻʻ(\alpha \cap \text{ᗡ}ʻR)\). +Thus if \(\alpha \cap \text{ᗡ}ʻR\) and \(\beta \cap \text{ᗡ}ʻR\) both +have the same limit, they define the same limiting section. Hence if +\(a\) is the limit of \(\alpha\), the limiting section of \(\alpha\) is +\((P\overline{R}Q)_{\text{sc}}ʻ\overrightarrow{Q}ʻ\alpha\). The upper +limit of this is the upper limit of the ultimate values as the argument +approaches \(a\) from below. We put +\[ +R (PQ)ʻa = \text{limax}_Pʻ(P\overline{R}Q)_{\text{sc}}ʻ\overrightarrow{Q}ʻa \quad\text{Df}. +\] +We have thus four limits of the function as the argument approaches +\(a\), namely +\[ +R(PQ)ʻa,\quad R(\breve{P} Q)ʻa,\quad R(P\breve{Q} )ʻa,\quad R(\breve{P} \breve{Q} )ʻa . +\] +If \(R\) is a continuous function, these four are all equal to \(Rʻa\); +but in general they are different from each other and from \(Rʻa\). +The subject of the continuity of functions is dealt with in <a href="#*234">*234</a>. When +\(R(PQ)ʻa= R(\breve{P} Q)ʻa\), each is the limit of the function for +the argument \(a\) for approaches from below. It should be observed +that if \(R\) is defined for a set of arguments which are dense in +\(Q\), <i>i.e.</i> if \(\delta_{Q}ʻ\text{ᗡ}ʻR = CʻQ\), then \(R(PQ)ʻa\) +and \(R(\breve{P} Q)ʻa\) are defined for all arguments in \(CʻQ\).</p> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_720">[Pg 720]</span></p> +<h2 class="nobreak" id="*230"> +*230. ON CONVERGENTS.</h2> +</div> + + +<p class="nind"><i>Summary of</i> *230.</p> + +<p>In the present number, we have to consider the notion of a function +converging into a given class, or, as we may express it, the notion +that the value of the function "ultimately" belongs to the given class. +If \(R\) is the function in question, \(\alpha\) the given class, +and \(Q\) a series to which the arguments belong, we say that "\(R\) +is \(Q\)-convergent into \(\alpha\)" if there is an argument \(y\) +such that, for all arguments from \(y\) onward (in the \(Q\)-order), +the value of the function is an \(\alpha\). That is, \(R\) is +\(Q\)-convergent into \(\alpha\) if +\[ +(\exists y) . y\in CʻQ \cap \text{ᗡ}ʻR.Rʻʻ\overleftarrow{Q}_{*}ʻy \subset \alpha . +\] +A term \(y\) which is of this nature is said to belong to the class +\(R\overline{Q} _{\text{cn}}\alpha\). Thus \(R\) is \(Q\)-convergent +into \(\alpha\) if the class \(R\overline{Q} _{\text{cn}}\alpha\) is +not null. Hence we have the following pair of definitions: +\[ +\begin{aligned} +R\overline{Q} _{\text{cn}}\alpha &= CʻQ \cap \text{ᗡ}ʻR \cap \hat{y} (Rʻʻ\overleftarrow{Q}_{*}ʻy \subset \alpha ) \quad\text{Df},\\ +Q_{\text{cn}} &= \hat{R} \hat{\alpha} (\exists ! R\overline{Q} _{\text{cn}}\alpha ) \quad\text{Df}. +\end{aligned} +\]</p> + +<p>In all the cases that have any importance, \(R\) will be a one-valued +function (<i>i.e.</i> a one-many relation), \(Q\) will be a series, +and \(CʻQ \cap \text{ᗡ}ʻR\) will be a class having no maximum in +\(Q\). For, if \(CʻQ \cap \text{ᗡ}ʻR\) has a maximum in \(Q\), then +the classes into which \(R\) converges are simply those to which the +value for this maximum belongs. The following propositions, though +only <i>important</i> under the above circumstances, are in general +<i>true</i> under much wider hypotheses.</p> + +<p>It is possible to generalize still further the notion of convergence, +so as to apply to any property which belongs to \(R\) when confined to +sufficiently late arguments. For this purpose, we have to consider +\(R\unicode{x0294f}\overleftarrow{Q}_{*}ʻz\) where \(z\) is to be confined +to terms later than or equal to some term \(y\). If, under these +circumstances, \(R \unicode{x0294f} \overleftarrow{Q}_{*}ʻz\) always +belongs to the class \(\lambda\), we may say that \(R\) ultimately +becomes a \(\lambda\). We may put +\[ +\begin{aligned} +R\overline{Q} _{\text{cng}} \lambda &= \hat{y}\{y \in CʻQ \cap \text{ᗡ}ʻR : yQ_{*}z .\supset _{z} . R \unicode{x0294f} \overleftarrow{Q}_{*}ʻz + \in \lambda\} \quad\text{Df},\\ +Q_{\text{cng}} &= \hat{R} \hat{\lambda} (\exists ! R\overline{Q} _{\text{cng}}\lambda ) \quad\text{Df}. +\end{aligned} +\]</p> + +<p><span class="pagenum" id="Page_721">[Pg 721]</span></p> + +<p>This is the general conception of which \(Q_{\text{cn}}\) is a +particular case; in fact, +\[ +\vdash :RQ_{\text{cn}}\alpha .\equiv .RQ_{\text{cng}}(\breve{\text{D}} ʻʻ\text{Cl}ʻ\alpha ). +\] +\(Q_{\text{cng}}\) will have to be used when the ultimate properties +of the function with which we are concerned are not properties of its +values; but when they are properties of its values, \(Q_{\text{cn}}\) +enables us to deal with them more easily than \(Q_{\text{cng}}\).</p> + +<p>In this number, we prove the following propositions among others:</p> + +<p class="nind"><b>*230·171.</b> \(\vdash :y\in R\overline{Q} _{\text{cn}}(\overleftarrow{P}_{*}ʻx).\supset .x\in P_{*}ʻʻRʻʻ\overleftarrow{Q}_{*}ʻy\)</p> + +<p class="nind"><b>*230·211.</b> \(\vdash \colon\ldotp \alpha \subset \beta .\supset :RQ_{\text{cn}}\alpha .\supset .RQ_{\text{cn}}\beta\)</p> + +<p class="nind"><b>*230·253.</b> \[\begin{align}\vdash \colon\ldotp RʻʻCʻQ\subset \alpha .\supset :RQ_{\text{cn}}\alpha .&\equiv .\exists !CʻQ\cap \text{ᗡ}ʻR.\\ +&\equiv .\exists !RʻʻCʻQ.\equiv .\dot{\exists} !(R\upharpoonright CʻQ)\end{align}\]</p> + +<p class="nind"><b>*230·4.</b> \(\vdash .R\overline{Q} _{\text{cn}}\alpha =\text{ᗡ}ʻR\cap \breve{Q} _{*}ʻʻ(R\overline{Q} _{\text{cn}}\alpha )\)</p> + +<p class="nind"><b>*230·42.</b> \(\vdash \colon\ldotp Q_{*}\in \text{connex} .\supset :RQ_{\text{cn}}\alpha .RQ_{\text{cn}}\beta .\equiv .RQ_{\text{cn}}(\alpha \cap \beta )\)</p> + +<p class="nind"><b>*230·53.</b> \(\vdash \colon\ldotp Q\in \text{trans}\cap \text{connex} .\text{E}!\text{max}_{Q}ʻ\text{ᗡ}ʻR.\supset :RQ_{\text{cn}}\alpha + .\equiv .\overrightarrow{R}ʻ\text{max}_{Q}ʻ\text{ᗡ}ʻR\subset \alpha\)</p> + +<p>In virtue of this proposition, the case when +\(\text{E}!\text{max}_{Q}ʻ\text{ᗡ}ʻR\) is uninteresting, and in +order to obtain interesting interpretations of our propositions, +it is necessary to suppose that \(\text{ᗡ}ʻR\) has no maximum. +Similarly when, in later numbers, we consider \(\text{ᗡ}ʻR\cap\overrightarrow{Q}ʻx\), +we shall only obtain interesting results when this has no maximum, +which requires that \(Q\) should be a compact series \((Q^{2}=Q)\) +and \(\text{ᗡ}ʻR\) should be dense in \(Q\). These assumptions are, +however, not usually required for the <i>truth</i> of our propositions.</p> + +<hr class="tb"> + +<p class="nind"><b>*230·01.</b> \(R\overline{Q} _{\text{cn}}\alpha =CʻQ\cap \text{ᗡ}ʻR\cap \hat{y} (Rʻʻ\overleftarrow{Q}_{*}ʻy\subset \alpha ) \quad\text{Df}\)</p> + +<p class="nind"><b>*230·02.</b> \(Q_{\text{cn}}=\hat{R} \hat{\alpha} (\exists !R\overline{Q} _{\text{cn}}\alpha ) \quad\text{Df}\)</p> + +<p class="nind"><b>*230·1.</b> \(\vdash :y\in R\overline{Q} _{\text{cn}}\alpha .\equiv .y\in CʻQ\cap \text{ᗡ}ʻR.Rʻʻ\overleftarrow{Q}_{*}ʻy\subset \alpha \quad[(*230·01)]\)</p> + +<p class="nind"><b>*230·11.</b> \[\begin{align}&\vdash :RQ_{\text{cn}}\alpha .\equiv .\exists !R\overline{Q} _{\text{cn}}\alpha + .\equiv .(\exists y).y\in CʻQ\cap \text{ᗡ}ʻR.Rʻʻ\overleftarrow{Q}_{*}ʻy\subset \alpha \\ +&[(*230·02)]\end{align}\]</p> + +<p class="nind"><b>*230·12.</b> \(\vdash :y\in R\overline{Q} _{\text{cn}}\alpha .\supset .\overleftarrow{Q}_{*}ʻy\cap \text{ᗡ}ʻR\subset R\overline{Q} _{\text{cn}}\alpha\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*230·1.*201·14·15.\supset \\ +\vdash :y\in R\overline{Q} _{\text{cn}}\alpha .yQ_{*}z.z\in \text{ᗡ}ʻR.&\supset .Rʻʻ\overleftarrow{Q}_{*}ʻy\subset \alpha .z\in CʻQ\cap \text{ᗡ}ʻR.\overleftarrow{Q}_{*}ʻz\subset + \overleftarrow{Q}_{*}ʻy.\\ +[*37·2] &\supset .z\in CʻQ\cap \text{ᗡ}ʻR.Rʻʻ\overleftarrow{Q}_{*}ʻz\subset \alpha .\\ +[*230·1]&\supset .z\in R\overline{Q} _{\text{cn}}\alpha :\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_722">[Pg 722]</span></p> + +<p class="nind"><b>*230·13.</b> \(\vdash .R\overline{Q} _{\text{cn}}\alpha =(R\upharpoonright CʻQ)\overline{Q} _{\text{cn}}\alpha\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*35·64. &\supset \vdash .CʻQ\cap \text{ᗡ}ʻR=CʻQ\cap \text{ᗡ}ʻ(R\upharpoonright CʻQ) &\qquad \text{(1)}\\ +\vdash .*37·421.&\supset \vdash .Rʻʻ\overleftarrow{Q}_{*}ʻy=(R\upharpoonright CʻQ)ʻʻ\overleftarrow{Q}_{*}ʻy &\qquad \text{(2)}\\ +\vdash .(1).(2).*230·1.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*230·131.</b> \(\vdash :R\upharpoonright CʻQ=T\upharpoonright CʻQ.\supset .R\overline{Q} _{\text{cn}}\alpha =T\overline{Q} _{\text{cn}}\alpha \quad[*230·13]\)</p> + +<p class="nind"><b>*230·14.</b> \(\vdash :y\in R\overline{Q} _{\text{cn}}\alpha .\supset .\exists !CʻQ\cap \text{ᗡ}ʻR.\exists !\alpha \cap \text{D}ʻR\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*230·1.&\supset \vdash :\text{Hp}.\supset .y\in CʻQ\cap \text{ᗡ}ʻR.\overrightarrow{R}ʻy\subset \alpha.\\ +[*33·41] &\supset .y\in CʻQ\cap \text{ᗡ}ʻR.\exists !\overrightarrow{R}ʻy.\overrightarrow{R}ʻy\subset \alpha .\\ +[*22·621.*33·15] &\supset .\exists !CʻQ\cap \text{ᗡ}ʻR.\exists !\alpha \cap \text{D}ʻR:\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*230·141.</b> \(\vdash .R\overline{Q} _{\text{cn}}\lambda =\Lambda \quad[*230·14.\text{Transp}]\)</p> + +<p class="nind"><b>*230·142.</b> \(\vdash \colon\ldotp R=\dot{\Lambda} .\lor.Q=\dot{\Lambda} :\supset .R\overline{Q} _{\text{cn}}\alpha =\Lambda \quad[*230·14.\text{Transp}.*33·24]\)</p> + +<p class="nind"><b>*230·15.</b> \(\vdash :RQ_{\text{cn}}\alpha .\supset .\exists !CʻQ\cap \text{ᗡ}ʻR.\exists !\alpha \cap \text{D}ʻR \quad[*230·14·11]\)</p> + +<p class="nind"><b>*230·151.</b> \(\vdash :RQ_{\text{cn}}\alpha .\supset .\dot{\exists} !R.\dot{\exists} !Q.\exists !\alpha \quad[*230·15]\)</p> + +<p class="nind"><b>*230·152.</b> \(\vdash \colon\ldotp R=\dot{\Lambda} .\lor.Q=\dot{\Lambda} .\lor.\alpha =\Lambda :\supset .{\sim}(RQ_{\text{cn}}\alpha ) \quad[*230·151.\text{Transp}]\)</p> + +<p class="nind"><b>*230·16.</b> \(\vdash .R\overline{Q}_{\text{cn}}\alpha =R(\overline{Q_{*}\unicode{x0294f}\text{ᗡ}ʻR})_{\text{cn}}\alpha\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*230·14. &\supset \vdash :CʻQ\cap \text{ᗡ}ʻR=\Lambda .\supset .R\overline{Q} _{\text{cn}}\alpha =\Lambda .R(\overline{Q_{*}\unicode{x0294f}\text{ᗡ}ʻR} + )_[cn]\alpha =\Lambda &\qquad \text{(1)}\\ +\vdash .*90·41. &\supset \vdash :\exists !CʻQ\cap \text{ᗡ}ʻR.\supset .Cʻ(Q_{*}\unicode{x0294f}\text{ᗡ}ʻR)=CʻQ\cap \text{ᗡ}ʻR &\qquad \text{(2)}\\ +\vdash .*37·26. &\supset \vdash .Rʻʻ\overleftarrow{Q}_{*}ʻy=Rʻʻ(\overleftarrow{Q}_{*}ʻy\cap \text{ᗡ}ʻR) &\qquad \text{(3)}\\ +\vdash .(3).*35·102.&\supset \vdash :y\in \text{ᗡ}ʻR.\supset .Rʻʻ\overleftarrow{Q}_{*}ʻy=Rʻʻ\overleftarrow{Q_{*}\unicode{x0294f}\text{ᗡ}ʻR}ʻy &\qquad \text{(4)}\\ +\vdash .(2).(4).*230·1.\supset \\ +\vdash \colon\ldotp \exists !CʻQ\cap \text{ᗡ}ʻR.\supset :y\in R\overline{Q} _{\text{cn}}\alpha .&\equiv .y\in Cʻ(Q_{*}\unicode{x0294f}\text{ᗡ}ʻR)\cap + \text{ᗡ}ʻR.Rʻʻ\overleftarrow{Q_{*}\unicode{x0294f}\text{ᗡ}ʻR}ʻy\subset \alpha .\\ +[*230·1] &\equiv .y\in R(\overline{Q_{*}\unicode{x0294f}\text{ᗡ}ʻR} )_{\text{cn}}\alpha &\qquad \text{(5)}\\ +\vdash .(1).(5).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*230·161.</b> \(\vdash :Q_{*}\unicode{x0294f}\text{ᗡ}ʻR=S_{*}\unicode{x0294f}\text{ᗡ}ʻR.\supset .R\overline{Q} _{\text{cn}}\alpha =R\overline{S} _{\text{cn}}\alpha + \quad[*230·16]\)</p> + +<p class="nind"><b>*230·17.</b> \(\vdash :y\in CʻQ\cap \text{ᗡ}ʻR.\supset .\exists !Rʻʻ\overleftarrow{Q}_{*}ʻy\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*90·12.*33·41.&\supset \vdash :\text{Hp}.\supset .y\in \overleftarrow{Q}_{*}.\exists !\overrightarrow{R}ʻy.\\ +[*37·18]&\supset .\exists !Rʻʻ\overleftarrow{Q}_{*}ʻy:\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_723">[Pg 723]</span></p> + +<p class="nind"><b>*230·171.</b> \(\vdash :y\in R\overline{Q}_{\text{cn}}(\overleftarrow{P}_{*}ʻx).\supset .x\in P_{*}ʻʻRʻʻ\overleftarrow{Q}_{*}ʻy\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*230·1·17.\supset \vdash :\text{Hp}.&\supset .\exists !Rʻʻ\overleftarrow{Q}_{*}ʻy.Rʻʻ\overleftarrow{Q}_{*}ʻy\subset \overleftarrow{P}_{*}ʻx.\\ +[*22·621] &\supset .\exists !Rʻʻ\overleftarrow{Q}_{*}ʻy\cap \overleftarrow{P}_{*}ʻx.\\ +[*37·46] &\supset .x\in P_{*}ʻʻRʻʻ\overleftarrow{Q}_{*}ʻy:\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*230·21.</b> \(\vdash :\alpha \subset \beta .\supset .R\overline{Q}_{\text{cn}}\alpha \subset R\overline{Q}_{\text{cn}}\beta \quad[*230·1.*22·44]\)</p> + +<p class="nind"><b>*230·211.</b> \(\vdash \colon\ldotp \alpha \subset \beta .\supset :RQ_{\text{cn}}\alpha .\supset .RQ_{\text{cn}}\beta \quad[*230·21·11]\)</p> + +<p class="nind"><b>*230·22.</b> \(\vdash .R\overline{Q}_{\text{cn}}\alpha \cup R\overline{Q}_{\text{cn}}\beta \subset R\overline{Q}_{\text{cn}}(\alpha \cup \beta ) \quad[*230·21]\)</p> + +<p class="nind"><b>*230·221.</b> \(\vdash \colon\ldotp RQ_{\text{cn}}\alpha .\lor.RQ_{\text{cn}}\beta :\supset .RQ_{\text{cn}}(\alpha \cup \beta ) \quad[*230·211]\)</p> + +<p class="nind"><b>*230·23.</b> \(\vdash .R\overline{Q}_{\text{cn}}\alpha \cap R\overline{Q}_{\text{cn}}\beta =R\overline{Q}_{\text{cn}}(\alpha \cap \beta )\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*230·1.\supset \vdash :y\in R\overline{Q}_{\text{cn}}\alpha \cap R\overline{Q}_{\text{cn}}\beta .&\equiv .y\in CʻQ\cap \text{ᗡ}ʻR.Rʻʻ\overleftarrow{Q}_{*}ʻy\subset + \alpha .Rʻʻ\overleftarrow{Q}_{*}ʻy\subset \beta .\\ +[Comp.*230·1] &\equiv .y\in R\overline{Q}_{\text{cn}}(\alpha \cap \beta ):\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*230·231.</b> \(\vdash :RQ_{\text{cn}}(\alpha \cap \beta ).\supset .RQ_{\text{cn}}\alpha .RQ_{\text{cn}}\beta \quad[*230·211]\)</p> + +<p class="nind"><b>*230·24.</b> \(\vdash .R\overline{Q}_{\text{cn}}\alpha \cap R\overline{Q}_{\text{cn}}(\beta -\alpha )=\Lambda \quad[*230·23·141]\)</p> + +<p class="nind"><b>*230·25.</b> \(\vdash .R\overline{Q}_{\text{cn}}\alpha =R\overline{Q}_{\text{cn}}(\alpha \cap \text{D}ʻR)=R\overline{Q}_{\text{cn}}(\alpha + \cap Rʻʻ\breve{Q} _{*}ʻʻ\text{ᗡ}ʻR)=R\overline{Q}_{\text{cn}}(\alpha \cap RʻʻCʻQ)\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*37·15.&\supset \vdash :Rʻʻ\overleftarrow{Q}_{*}ʻy\subset \alpha .\equiv .Rʻʻ\overleftarrow{Q}_{*}ʻy\subset \alpha \cap \text{D}ʻR &\qquad \text{(1)}\\ +\vdash .*37·18.\supset \vdash \colon\ldotp y\in \text{ᗡ}ʻR.&\supset :Rʻʻ\overleftarrow{Q}_{*}ʻy\subset Rʻʻ\breve{Q} _{*}ʻʻ\text{ᗡ}ʻR:\\ +[\text{Comp}] &\supset :Rʻʻ\overleftarrow{Q}_{*}ʻy\subset \alpha .\equiv .Rʻʻ\overleftarrow{Q}_{*}ʻy\subset \alpha \cap Rʻʻ\breve{Q} _{*}ʻʻ\text{ᗡ}ʻR + &\qquad \text{(2)}\\ +\vdash .*37·2·18.&\supset \vdash .Rʻʻ\overleftarrow{Q}_{*}ʻy\subset RʻʻCʻQ.\\ +[\text{Comp}] &\supset \vdash :Rʻʻ\overleftarrow{Q}_{*}ʻy\subset \alpha .\equiv .Rʻʻ\overleftarrow{Q}_{*}ʻy\subset \alpha \cap RʻʻCʻQ &\qquad \text{(3)}\\ +\vdash .(1).(2).(3).*230·1.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*230·251.</b> \(\vdash .R\overline{Q}_{\text{cn}}(RʻʻCʻQ)=CʻQ\cap \text{ᗡ}ʻR\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*33·15.*37·2.\supset \vdash .(y).Rʻʻ\overleftarrow{Q}_{*}ʻy\subset RʻʻCʻQ &\qquad \text{(1)}\\ +\vdash .(1).*230·1.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*230·252.</b> \(\vdash :RʻʻCʻQ\subset \alpha .\supset .R\overline{Q}_{\text{cn}}\alpha =CʻQ\cap \text{ᗡ}ʻR \quad[*230·25·251]\)</p> + +<p class="nind"><b>*230·253.</b> \[\begin{align}\vdash \colon\ldotp &RʻʻCʻQ\subset \alpha .\supset:RQ_{\text{cn}}\alpha.\equiv.\exists !CʻQ\cap \text{ᗡ}ʻR.\equiv .\\ +&\exists !RʻʻCʻQ.\equiv .\dot{\exists} !(R\upharpoonright CʻQ) \quad[*230·11·252.*37·401.*35·64]\end{align}\]</p> + +<p><span class="pagenum" id="Page_724">[Pg 724]</span></p> + +<p class="nind"><b>*230·31.</b> \(\vdash .sʻR\overline{Q} _{\text{cn}}ʻʻ\kappa \subset R\overline{Q} _{\text{cn}}(sʻ\kappa )\)</p> + +<p><i>Dem.</i> +\[ +\vdash .*230·21.\supset \vdash :\alpha \in \kappa .\supset .R\overline{Q} _{\text{cn}}\alpha \subset R\overline{Q} _{\text{cn}}(sʻ\kappa ):\supset \vdash .\text{Prop} +\]</p> + +<p class="nind"><b>*230·311.</b> \(\vdash .Q_{\text{cn}}ʻʻ\kappa \subset \overrightarrow{Q}_{\text{cn}}ʻsʻ\kappa\)</p> + +<p><i>Dem.</i> +\[ +\vdash .*230·211.\supset \vdash :\alpha \in \kappa .RQ_{\text{cn}}\alpha .\supset .RQ_{\text{cn}}(sʻ\kappa ):\supset \vdash .\text{Prop} +\]</p> + +<p class="nind"><b>*230·32.</b> \(\vdash .R\overline{Q} _{\text{cn}}(pʻ\kappa )=CʻQ\cap \text{ᗡ}ʻR\cap pʻR\overline{Q} _{\text{cn}}ʻʻ\kappa\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*230·1.\supset \\ +\vdash \colon\ldotp y\in R\overline{Q} _{\text{cn}}(pʻ\kappa ).&\equiv :y\in CʻQ\cap \text{ᗡ}ʻR.Rʻʻ\overleftarrow{Q}_{*}ʻy\subset pʻ\kappa :\\ +[*40·15] &\equiv :y\in CʻQ\cap \text{ᗡ}ʻR:\alpha \in \kappa .\supset _{\alpha }.Rʻʻ\overleftarrow{Q}_{*}ʻy\subset \alpha :\\ +[*4·73]&\equiv :y\in CʻQ\cap \text{ᗡ}ʻR:\alpha \in \kappa .\supset _{\alpha }.y\in CʻQ\cap \text{ᗡ}ʻR.Rʻʻ\overleftarrow{Q}_{*}ʻy\subset \alpha :\\ +[*230·1] &\equiv :y\in CʻQ\cap \text{ᗡ}ʻR\cap pʻR\overline{Q} _{\text{cn}}ʻʻ\kappa \colon\ldotp \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*230·321.</b> \(\vdash :\exists !k.\supset .R\overline{Q} _{\text{cn}}(pʻ\kappa )=pʻR\overline{Q} _{\text{cn}}ʻʻ\kappa .pʻR\overline{Q} _{\text{cn}}ʻʻ\kappa + \subset CʻQ\cap \text{ᗡ}ʻR\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*230·1. &\supset \vdash :\alpha \in \kappa .\supset _{\alpha }.R\overline{Q} _{\text{cn}}\alpha \subset CʻQ\cap \text{ᗡ}ʻR &\qquad \text{(1)}\\ +\vdash .(1).*40·23·151.&\supset \vdash :\text{Hp}.\supset .pʻR\overline{Q} _{\text{cn}}ʻʻ\kappa \subset CʻQ\cap \text{ᗡ}ʻR &\qquad \text{(2)}\\ +\vdash .(2).*230·32.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*230·4.</b> \(\vdash .R\overline{Q} _{\text{cn}}\alpha =\text{ᗡ}ʻR\cap \breve{Q} _{*}ʻʻ(R\overline{Q} _{\text{cn}}\alpha )\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*230·11.*90·21.&\supset \vdash .R\overline{Q} _{\text{cn}}\alpha \subset \text{ᗡ}ʻR.R\overline{Q} _{\text{cn}}\alpha \subset \breve{Q} _{*}ʻʻ(R\overline{Q} + _{\text{cn}}\alpha ) &\qquad \text{(1)}\\ +\vdash .*201·14·15. &\supset \vdash :Rʻʻ\overleftarrow{Q}_{*}ʻy\subset \alpha .yQ_{*}z.\supset .Rʻʻ\overleftarrow{Q}_{*}ʻz\subset \alpha &\qquad \text{(2)}\\ +\vdash .(2).*230·1. &\supset \vdash :y\in R\overline{Q} _{\text{cn}}\alpha .yQ_{*}z,z\in \text{ᗡ}ʻR.\supset .z\in R\overline{Q} _{\text{cn}}\alpha :\\ +[*37·105] &\supset \vdash .\text{ᗡ}ʻR\cap \breve{Q} _{*}ʻʻ(R\overline{Q} _{\text{cn}}\alpha )\subset R\overline{Q} _{\text{cn}}\alpha &\qquad \text{(3)}\\ +\vdash .(1).(3).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*230·41.</b> \(\vdash \colon\ldotp Q_{*}\in \text{connex} .\supset :R\overline{Q} _{\text{cn}}\alpha \subset R\overline{Q} _{\text{cn}}\beta + .\lor.R\overline{Q} _{\text{cn}}\beta \subset R\overline{Q} _{\text{cn}}\alpha\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*211·61.*201·15.\supset \\ +\vdash \colon\ldotp \text{Hp}.\supset :\breve{Q} _{*}ʻʻ(R\overline{Q} _{\text{cn}}\alpha )&\subset \breve{Q} _{*}ʻʻ(R\overline{Q} _{\text{cn}}\beta + ).\lor.\breve{Q} _{*}ʻʻ(R\overline{Q} _{\text{cn}}\beta )\subset \breve{Q} _{*}ʻʻ(R\overline{Q} _{\text{cn}}\alpha ) :\\ +[\text{Fact}.*230·4]\supset :R\overline{Q} _{\text{cn}}\alpha &\subset R\overline{Q} _{\text{cn}}\beta .\lor.R\overline{Q} _{\text{cn}}\beta + \subset R\overline{Q} _{\text{cn}}\alpha \colon\ldotp \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*230·42.</b> \(\vdash \colon\ldotp Q_{*}\in \text{connex} .\supset :RQ_{\text{cn}}\alpha .RQ_{\text{cn}}\beta .\equiv .RQ_{\text{cn}}(\alpha + \cap \beta )\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*230·41.\supset \vdash \colon\ldotp \text{Hp}.&\supset :R\overline{Q} _{\text{cn}}\alpha \cap R\overline{Q} _{\text{cn}}\beta =R\overline{Q} _{\text{cn}}\alpha + .\lor.R\overline{Q} _{\text{cn}}\alpha \cap R\overline{Q} _{\text{cn}}\beta =R\overline{Q} _{\text{cn}}\beta :\\ +[*230·23] &\supset :R\overline{Q} _{\text{cn}}(\alpha \cap \beta )=R\overline{Q} _{\text{cn}}\alpha .\lor.R\overline{Q} _{\text{cn}}(\alpha + \cap \beta )=R\overline{Q} _{\text{cn}}\beta :\\ +[*230·11] &\supset :RQ_{\text{cn}}\alpha .RQ_{\text{cn}}\beta .\supset .RQ_{\text{cn}}(\alpha \cap \beta ) &\qquad \text{(1)}\\ +\vdash .(1).*230·231.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_725">[Pg 725]</span></p> + +<p class="nind"><b>*230·421.</b> \(\vdash :Q_{*}\in \text{connex} .\alpha \cap \beta =\Lambda .\supset .{\sim}\{RQ_{\text{cn}}\alpha .RQ_{\text{cn}}\beta\} \quad[*230·42·141]\)</p> + +<p class="nind"><b>*230·51.</b> \(\vdash :RQ_{\text{cn}}\alpha .\supset .pʻ\overleftarrow{Q}_{*}ʻʻCʻQ\cap \text{ᗡ}ʻR\subset R\overline{Q} _{\text{cn}}\alpha\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*201·14.&\supset \vdash :y\in CʻQ\cap \text{ᗡ}ʻR.Rʻʻ\overleftarrow{Q}_{*}ʻy\subset \alpha .z\in pʻ\overleftarrow{Q}_{*}ʻʻCʻQ.\supset .Rʻʻ\overleftarrow{Q}_{*}ʻz\subset + \alpha &\qquad \text{(1)}\\ +\vdash .*230·151.*40·62. &\supset \vdash :\text{Hp}.\supset .pʻ\overleftarrow{Q}_{*}ʻʻCʻQ\subset CʻQ &\qquad \text{(2)}\\ +\vdash .(1).(2).*230·1. &\supset \vdash :\text{Hp}.y\in R\overline{Q} _{\text{cn}}\alpha .z\in &pʻ\overleftarrow{Q}_{*}ʻʻCʻQ\cap \text{ᗡ}ʻR.\supset .\\ +&z\in CʻQ\cap \text{ᗡ}ʻR.Rʻʻ\overleftarrow{Q}_{*}ʻz\subset \alpha &\qquad \text{(3)}\\ +\vdash .(3).*230·1·11.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*230·511.</b> \(\vdash :y\in pʻ\overleftarrow{Q}_{*}ʻʻCʻQ.\supset .\overleftarrow{Q}_{*}ʻy=pʻ\overleftarrow{Q}_{*}ʻʻCʻQ\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*40·12.&\supset \vdash :\text{Hp}.\supset .pʻ\overleftarrow{Q}_{*}ʻʻCʻQ\subset \overleftarrow{Q}_{*}ʻy &\qquad \text{(1)}\\ +\vdash .*40·53.\supset \vdash \colon\ldotp \text{Hp}.z\in \overleftarrow{Q}_{*}ʻy.&\supset :x\in CʻQ.\supset _{x}.xQ_{*}y:yQ_{*}z:\\ +[*201·15] &\supset :x\in CʻQ.\supset _{x}.xQ_{*}z:\\ +[*40·53] &\supset :z\in pʻ\overleftarrow{Q}_{*}ʻʻCʻQ &\qquad \text{(2)}\\ +\vdash .(1).(2).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*230·512.</b> \(\vdash :\text{ᗡ}ʻR\cap pʻ\overleftarrow{Q}_{*}ʻʻCʻQ\subset R\overline{Q} _{\text{cn}}\alpha .\supset .Rʻʻpʻ\overleftarrow{Q}_{*}ʻʻCʻQ\subset \alpha\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*230·1.\supset \vdash \colon\ldotp \text{Hp}.\supset :y\in \text{ᗡ}ʻR\cap pʻ\overleftarrow{Q}_{*}ʻʻCʻQ.&\supset .Rʻʻ\overleftarrow{Q}_{*}ʻy\subset \alpha .\\ +[*230·511]&\supset .Rʻʻpʻ\overleftarrow{Q}_{*}ʻʻCʻQ\subset \alpha &\qquad \text{(1)}\\ +\vdash .(1).*10·23.&\supset \vdash :\text{Hp}.\exists !\text{ᗡ}ʻR\cap pʻ\overleftarrow{Q}_{*}ʻʻCʻQ.\supset .Rʻʻpʻ\overleftarrow{Q}_{*}ʻʻCʻQ\subset \alpha &\qquad \text{(2)}\\ +\vdash .*37·26·29. \supset \vdash :\text{ᗡ}ʻR\cap pʻ\overleftarrow{Q}_{*}ʻʻCʻQ=\Lambda .&\supset .Rʻʻpʻ\overleftarrow{Q}_{*}ʻʻCʻQ=\Lambda .\\ +[*24·12] &\supset .Rʻʻpʻ\overleftarrow{Q}_{*}ʻʻCʻQ\subset \alpha &\qquad \text{(3)}\\ +\vdash .(2).(3).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*230·513.</b> \(\vdash \colon\ldotp \dot{\exists} !Q.\supset :Rʻʻpʻ\overleftarrow{Q}_{*}ʻʻCʻQ\subset \alpha .\equiv .\text{ᗡ}ʻR\cap pʻ\overleftarrow{Q}_{*}ʻʻCʻQ\subset + R\overline{Q} _{\text{cn}}\alpha\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*230·511.\supset \vdash \colon\ldotp y\in \text{ᗡ}ʻR\cap &pʻ\overleftarrow{Q}_{*}ʻʻCʻQ.\supset :\\ +&Rʻʻpʻ\overleftarrow{Q}_{*}ʻʻCʻQ\subset \alpha .\supset .y\in \text{ᗡ}ʻR.Rʻʻ\overleftarrow{Q}_{*}ʻy\subset \alpha &\qquad \text{(1)}\\ +\vdash .(1).*40·62.*230·1.\supset \\ +&\vdash :\dot{\exists} !Q.y\in \text{ᗡ}ʻR\cap pʻ\overleftarrow{Q}_{*}ʻʻCʻQ.Rʻʻpʻ\overleftarrow{Q}_{*}ʻʻCʻQ.\subset \alpha .\supset .y\in R\overline{Q} _{\text{cn}}\alpha + &\qquad \text{(2)}\\ +\vdash .(2).\text{Comm}.&\supset \vdash \colon\ldotp \dot{\exists} !Q.\supset :\\ +&Rʻʻpʻ\overleftarrow{Q}_{*}ʻʻCʻQ\subset \alpha .\supset .\text{ᗡ}ʻR\cap pʻ\overleftarrow{Q}_{*}ʻʻCʻQ\subset R\overline{Q} _{\text{cn}}\alpha &\qquad \text{(3)}\\ +\vdash .(3).*230·512.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_726">[Pg 726]</span></p> + +<p class="nind"><b>*230·514.</b> \(\vdash : \dot{\exists} ! Q. \exists !pʻ\overleftarrow{Q}_{*}ʻʻCʻQ \cap \text{ᗡ}ʻR . Rʻʻpʻ\overleftarrow{Q}_{*}ʻʻCʻQ\subset + \alpha . \supset . RQ_{\text{cn}}\alpha\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *230·513. \supset \vdash : \text{Hp}. &\supset . \exists ! pʻ\overleftarrow{Q}_{*}ʻʻCʻQ \cap \text{ᗡ}ʻR . pʻ\overleftarrow{{Q}_{*}} + ʻʻ CʻQ \cap \text{ᗡ}ʻR \subset R\overline{Q} _{\text{cn}}\alpha .\\ +[*24·58.*230·11] &\supset . RQ_{\text{cn}} \alpha : \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*230·52.</b> \(\vdash : \exists ! CʻQ \cap \text{ᗡ}ʻR . \exists ! \text{ᗡ}ʻR \cap pʻ\overleftarrow{Q}_{*} ʻʻsʻR\overline{Q} _{\text{cn}}ʻʻ\kappa + .\kappa \subset \overleftarrow{Q}_{\text{cn}}ʻR .\supset . pʻ\kappa \in \overleftarrow{Q}_{\text{cn}}ʻR\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *40·16 . &\supset \vdash : \alpha \in \kappa . \supset . pʻ\overleftarrow{Q}_{*}ʻʻsʻR\overline{Q} _{\text{cn}}ʻʻ\kappa \subset pʻ\overleftarrow{Q}_{*}ʻʻR\overline{Q} + _{\text{cn}}\alpha &\qquad \text{(1)}\\ +\vdash . (1). *40·61 . \supset \\ +\vdash \colon\ldotp \text{Hp} . \supset : \alpha \in \kappa . &\supset . pʻ\overleftarrow{Q}_{*}ʻʻsʻR\overline{Q} _{\text{cn}}ʻʻ\kappa \subset \breve{Q} _{*}ʻʻR\overline{Q} + _{\text{cn}}\alpha .\\ +[\text{Fact}.*230·4] &\supset . \text{ᗡ}ʻR \cap pʻ\overleftarrow{Q}_{*}ʻʻsʻR\overline{Q} _{\text{cn}}ʻʻ\kappa \subset R\overline{Q} _{\text{cn}}\alpha :\\ +[*40·44] &\supset : \text{ᗡ}ʻR \cap pʻ\overleftarrow{Q}_{*}ʻʻsʻR\overline{Q} _{\text{cn}}ʻʻ\kappa \subset pʻR\overline{Q} _{\text{cn}}ʻʻ\kappa :\\ +[*230·321] \supset :\exists !\kappa .&\supset. \text{ᗡ}ʻR \cap pʻ\overleftarrow{Q}_{*}ʻʻsʻR\overline{Q} _{\text{cn}}ʻʻ\kappa \subset R\overline{Q} _{\text{cn}}(pʻ\kappa + ).\\ +[\text{Hp}.*24·58] &\supset . \exists ! R\overline{Q} _{\text{cn}}(pʻ\kappa ).\\ +[*230·11] &\supset . pʻ\kappa \in \overleftarrow{Q}_{\text{cn}}ʻR &\qquad \text{(2)}\\ +\vdash . *230·253.*40·2 . &\supset \vdash : \exists ! CʻQ \cap \text{ᗡ}ʻR . \kappa = \Lambda . \supset . P ʻ\kappa \in \overleftarrow{Q}_{\text{cn}}ʻR &\qquad \text{(3)}\\ +\vdash . (2) . (3). \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*230·53">*230·53</a>.</b> \(\vdash \colon\ldotp Q \in \text{trans} \cap \text{connex} . \text{E}! \text{max}_Qʻ\text{ᗡ}ʻR . \supset : RQ_{\text{cn}}\alpha + . \equiv . \overrightarrow{R}ʻ\text{max}_Qʻ\text{ᗡ}ʻR \subset \alpha\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *205·111. &\supset \vdash : \text{Hp}. \supset . \text{max}_Q ʻ\text{ᗡ}ʻR \in CʻQ \cap \text{ᗡ}ʻR &\qquad \text{(1)}\\ +\vdash . *205·141.*201·18. \supset \vdash : \text{Hp}. &\supset . \overleftarrow{Q}_*ʻ\text{max}_Qʻ\text{ᗡ}ʻR \cap \text{ᗡ}ʻR = \iota ʻ\text{max}_Qʻ\text{ᗡ}ʻR.\\ +[*37·26.*53·301] &\supset . Rʻʻ(\overleftarrow{{Q}_{*}}ʻ\text{max}_Qʻ\text{ᗡ}ʻR) = \overrightarrow{R}ʻ\text{max}_Qʻ\text{ᗡ}ʻR &\qquad \text{(2)}\\ +\vdash . (1). (2) . \supset \vdash \colon\ldotp \text{Hp}.& \supset : \overrightarrow{R}ʻ \text{max}_Qʻ\text{ᗡ}ʻR \subset \alpha . \supset . \text{max}_Qʻ\text{ᗡ}ʻR \in (R\overline{Q} _{\text{cn}}\alpha ).\\ +[*230·11] &\supset . RQ_{\text{cn}} \alpha &\qquad \text{(3)}\\ +\vdash .*205·36. \supset \vdash \colon\ldotp \text{Hp}. &\supset : y\in CʻQ \cap \text{ᗡ}ʻR . Rʻʻ\overleftarrow{{Q}_{*}}ʻy + \subset \alpha . \supset . Rʻʻ\overleftarrow{Q}_* ʻ\text{max}_Qʻ\text{ᗡ}ʻR \subset \alpha :\\ +[*230·11] \supset : RQ_{\text{cn}}\alpha . &\supset . Rʻʻ\overleftarrow{Q}_*ʻ\text{max}_Qʻ\text{ᗡ}ʻR \subset \alpha .\\ +[(2)] &\supset . \overrightarrow{R}ʻ\text{max}_Qʻ\text{ᗡ}ʻR \subset \alpha &\qquad \text{(4)}\\ +\vdash . (3). (4). \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*230·54.</b> \(\vdash \colon\ldotp Q \in \text{trans} \cap \text{connex} . \text{E}! \text{max}_Qʻ\text{ᗡ}ʻR . \supset : \kappa \subset \overleftarrow{{Q}_{\text{cn}}}ʻR + . \equiv . P ʻ\kappa \in \overleftarrow{Q}{\text{cn}}ʻR\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *230·53. \supset \vdash \colon\colon \text{Hp}. \supset \colon\ldotp \kappa \subset \overleftarrow{{Q}_{\text{cn}}}ʻR + . &\equiv : \alpha \in \kappa . \supset _a . \overrightarrow{R}ʻ\text{max}_Qʻ\text{ᗡ}ʻR \subset \alpha :\\ +[*40·15] &\equiv : \overrightarrow{R} ʻ \text{max}_Q ʻ\text{ᗡ}ʻR \subset pʻ\kappa :\\ +[*230·53] &\equiv : pʻ\kappa \in \overleftarrow{Q}_{\text{cn}}ʻR \colon\colon\supset \vdash . \text{Prop} +\end{array} +\]</p> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_727">[Pg 727]</span></p> +<h2 class="nobreak" id="*231"> +*231. LIMITING SECTIONS AND ULTIMATE OSCILLATION +OF A FUNCTION.</h2> +</div> + + +<p class="nind"><i>Summary of</i> *231.</p> + +<p>In the present number we are concerned with the limiting section +defined in a series \(P\), to which the values of a function +\(R\) belong, as the arguments to the function increase in the +argument-series \(Q\). That is, we are concerned with the section +consisting of those terms \(x\) of \(CʻP\) which are such that, +however great the argument to \(R\) becomes, there are still values +at least as great as \(x\). Such terms as \(x\) may be said to be +\(P\)-persistent; \(x\) is \(P\)-persistent if the function does not +ultimately become and remain less than \(x\). The class of persistent +terms is called the <i>limiting section</i>. The limiting section +may be defined as follows. If \(\alpha\) is any class into which +\(R\) is \(Q\)-convergent, then the section \(P_{*}ʻʻ\alpha\) is such +that the values of the function are ultimately contained in it. The +product of such terms as \(P_{*}ʻʻ\alpha\) is the smallest section +having this property. Hence if \(x\) be any member of this section, +then ultimately (<i>i.e.</i> for arguments far enough along the \(Q\) +series) the values of the function \(R\) do not persistently remain +less than \(x\) in the \(P\) series. Thus the product of such terms as +\(P_{*}ʻʻ\alpha\) is the limiting section, and we may therefore put +\[ +P\overline{R} _{\text{sc}}Q=pʻP_{*}ʻʻʻ\overleftarrow{Q}_{\text{cn}}ʻR\cap CʻP \quad\text{Df}, +\] +where the letters "\(\text{sc}\)" are intended to suggest "section." +(The factor \(CʻP\) on the right is superfluous except when +\(\overleftarrow{Q}_{\text{cn}}ʻR=\Lambda\), <i>i.e.</i> when \(CʻQ\cap\text{ᗡ}ʻR=\Lambda\).)</p> + +<p>We will call the limiting section of \(\breve{P}\), <i>i.e.</i> +\(\breve{P} \overline{R} _{\text{sc}}Q\), the "limiting upper +section." It will be seen that if \(x\) is a member of +\(\breve{P}\overline{R} _{\text{sc}}Q\), then the function does not ultimately +become and remain, as far as some of its arguments are concerned, +greater than \(x\), that is, however great we make the argument, we +still find values not greater than \(x\). Hence if \(x\) belongs to +both \(P\overline{R} _{\text{sc}}Q\) and \(\breve{P} \overline{R}_{\text{sc}}Q\), +we find values not less than \(x\) and values not greater than \(x\) +however great we make the argument. This class, \(P\overline{R}_{\text{sc}}Q\cap \breve{P} \overline{R} _{\text{sc}}Q\), +may therefore be regarded as the class of ultimate values of the +function. We will call it the "ultimate oscillation" of the function, +since, as the argument approaches \(\infty\), the value of the function +ultimately oscillates in this stretch of \(P\), and no smaller stretch +has the same property. We will denote this class by "\(P\overline{R}_{\text{os}}Q\)," +where "\(\text{os}\)" is intended to suggest "oscillation." +\(P\overline{R} _{\text{os}}Q\) is a stretch in \(CʻP\), because it is +the product of two sections. Hence we shall also call it the "limiting<span class="pagenum" id="Page_728">[Pg 728]</span> +stretch." When the function has a definite limit as the argument +approaches \(\infty\), the limiting stretch must not contain more than +one term.</p> + +<p>Limits of functions for arguments \(x\) in the middle of \(CʻQ\cap\text{ᗡ}ʻR\), +which will be considered later, are derived from the +limits considered in the present number by limiting the field of \(Q\) +to predecessors of \(x\).</p> + +<p>In this number we prove the following propositions among others:</p> + +<p class="nind"><b>*231·103.</b> \(\vdash .P\overline{R} _{\text{os}}Q=P_{\text{po}}\overline{R} _{\text{os}}Q=P_{*}\overline{R} _{\text{os}}Q\)</p> + +<p class="nind"><b>*231·12.</b> \(\vdash .P\overline{R} _{\text{sc}}Q=pʻP_{*}ʻʻʻRʻʻʻ\overleftarrow{Q}_{*}ʻʻ(CʻQ\cap \text{ᗡ}ʻR)\cap CʻP\)</p> + +<p class="nind"><b>*231·13.</b> \(\vdash .P\overline{R} _{\text{sc}}Q\in \text{sect}ʻP\)</p> + +<p class="nind"><b>*231·141.</b> \(\vdash :Q_{*}\in \text{connex} .RQ_{\text{cn}}(\overleftarrow{P}_{*}ʻx).\supset .x\in P\overline{R} _{\text{sc}}Q\)</p> + +<p class="nind"><b>*231·191.</b> \[\begin{align}\vdash :P_{\text{po}}\in \text{connex} .&\exists !P\overline{R} _{\text{os}}Q.\supset .\\ +&P\overline{R} _{\text{sc}}Q=P_{*}ʻʻ(P\overline{R} _{\text{os}}Q).Pʻʻ(P\overline{R} _{\text{sc}}Q)=P_{\text{po}}ʻʻ(P\overline{R} + _{\text{os}}Q)\end{align}\]</p> + +<p class="nind"><b>*231·192.</b> \[\begin{align}\vdash \colon\ldotp P_{\text{po}}\in &\text{connex} .\exists !P\overline{R} _{\text{os}}Q.\exists !P\overline{R} _{\text{os}}Q'.\supset + :\\ +&P\overline{R} _{\text{os}}Q=P\overline{R} _{\text{os}}Q'.\equiv .P\overline{R} _{\text{sc}}Q=P\overline{R} _{\text{sc}}Q'.\breve{P} \overline{R} _{\text{sc}}Q=\breve{P} + \overline{R} _{\text{sc}}Q'\end{align}\]</p> + +<p class="nind"><b>*231·193.</b> \[\begin{align}\vdash :P_{\text{po}}\in \text{Ser}.P\overline{R} _{\text{os}}Q&\in 1.\supset .\\ +&P\overline{R} _{\text{os}}Q=\iota ʻ\text{max}_{P}ʻ(P\overline{R} _{\text{sc}}Q)=\iota ʻ\text{min}_{P}ʻ(\breve{P} + \overline{R} _{\text{sc}}Q)\end{align}\]</p> + +<p>This proposition is frequently used in the present section.</p> + +<p>In all ordinary circumstances, we shall have \(CʻP=P\overline{R}_{\text{sc}}Q\cup \breve{P} \overline{R} _{\text{sc}}Q\), +so that if the upper and lower limiting sections do not have more than +one term in common (<i>i.e.</i> if \(P\overline{R} _{\text{os}}Q\in +1)\), they define a Dedekind cut in \(P\). The following propositions +are concerned with this fact:</p> + +<p class="nind"><b>*231·202.</b> \(\vdash :P_{*},Q_{*}\in \text{connex} .\exists !P\overline{R} _{\text{sc}}Q.\supset .CʻP-(P\overline{R} _{\text{sc}}Q)\subset + \breve{P} \overline{R} _{\text{sc}}Q\)</p> + +<p class="nind"><b>*231·21.</b> \[\begin{align}\vdash :P_{*},Q_{*}\in \text{connex} .CʻQ\cap \text{ᗡ}ʻR\subset Q_{*}ʻʻ\breve{R} ʻʻ&CʻP.\supset .\\ +&CʻP=P\overline{R} _{\text{sc}}Q\cup \breve{P} \overline{R} _{\text{sc}}Q\end{align}\]</p> + +<p class="nind"><b>*231·22.</b> \(\vdash :P_{*},Q_{*}\in \text{connex} .RʻʻCʻQ\subset CʻP.\supset .CʻP=P\overline{R} _{\text{sc}}Q\cup \breve{P} \overline{R} _{\text{sc}}Q\)</p> + +<p>Note that "\(RʻʻCʻQ\subset CʻP\)" is the hypothesis that for arguments belonging +to \(CʻQ\), the values belong to \(CʻP\).</p> + +<p class="nind"><b>*231·24.</b> \(\vdash :P_{*}\in \text{connex} .RʻʻCʻQ\subset CʻP.{\sim}\{RQ_{\text{cn}}(\overrightarrow{P}_{*}ʻx)\}.\supset .\overrightarrow{P}_{*}ʻx\subset + P\overline{R} _{\text{sc}}Q\)</p> + +<hr class="tb"> + +<p class="nind"><b>*231·01.</b> \(P\overline{R} _{\text{sc}}Q=pʻP_{*}ʻʻʻ\overleftarrow{Q}_{\text{cn}}ʻR\cap CʻP \quad\text{Df}\)</p> + +<p class="nind"><b>*231·02.</b> \(P\overline{R} _{\text{os}}Q=P\overline{R} _{\text{sc}}Q\cap \breve{P} \overline{R} _{\text{sc}}Q \quad\text{Df}\)</p> + +<p><span class="pagenum" id="Page_729">[Pg 729]</span></p> + +<p class="nind"><b>*231·1.</b> \(\vdash .P\overline{R} _{\text{sc}}Q=pʻP_{*}ʻʻʻ\overleftarrow{Q}_{\text{cn}}ʻR\cap CʻP \quad[(*231·01)]\)</p> + +<p class="nind"><b>*231·101.</b> \(\vdash .P\overline{R} _{\text{os}}Q=P\overline{R} _{\text{sc}}Q\cap \breve{P} \overline{R} _{\text{sc}}Q \quad[(*231·02)]\)</p> + +<p class="nind"><b>*231·102.</b> \(\vdash .P\overline{R} _{\text{sc}}Q=P_{\text{po}}\overline{R} _{\text{sc}}Q=P_{*}\overline{R} _{\text{sc}}Q + \quad[*231·1.*91·602.*90·4]\)</p> + +<p class="nind"><b>*231·103.</b> \(\vdash .P\overline{R} _{\text{os}}Q=P_{\text{po}}\overline{R} _{\text{os}}Q=P{*}\overline{R} _{\text{os}}Q \quad[*231·102·101]\)</p> + +<p class="nind"><b>*231·11.</b> \(\vdash \colon\ldotp x\in P\overline{R} _{\text{sc}}Q.\equiv :RQ_{\text{cn}}\alpha .\supset _{\alpha }.x\in P_{*}ʻʻ\alpha + :x\in CʻP \quad[*231·1]\)</p> + +<p class="nind"><b>*231·111.</b> \[\begin{align}&\vdash \colon\ldotp x\in P\overline{R} _{\text{sc}}Q.\equiv :y\in CʻQ\cap \text{ᗡ}ʻR.Rʻʻ\overleftarrow{Q}_{*}ʻy\subset + \alpha .\supset _{y,\alpha }.x\in P_{*}ʻʻ\alpha :x\in CʻP\\ +&[*231·11.*230·11]\end{align}\]</p> + +<p class="nind"><b>*231·112.</b> \(\vdash \colon\ldotp x\in P\overline{R} _{\text{sc}}Q.\equiv :y\in CʻQ\cap \text{ᗡ}ʻR.\supset _{y}.x\in P_{*}ʻʻRʻʻ\overleftarrow{Q}_{*}ʻy:x\in + CʻP\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*231·111.*22·42.\supset \\ +\vdash \colon\ldotp x\in P\overline{R} _{\text{sc}}Q.&\supset :y\in CʻQ\cap \text{ᗡ}ʻR.\supset _{y}.x\in P_{*}ʻʻRʻʻ\overleftarrow{Q}_{*}yʻ:x\in + CʻP &\qquad \text{(1)}\\ +\vdash .*37·2.\supset \\ +\vdash \colon\ldotp y\in CʻQ\cap \text{ᗡ}ʻ&R.\supset _{y}.x\in P_{*}ʻʻRʻʻ\overleftarrow{Q}_{*}ʻy:x\in CʻP:\supset :\\ +&y\in CʻQ\cap \text{ᗡ}ʻR.Rʻʻ\overleftarrow{Q}_{*}ʻy\subset \alpha .\supset _{y,\alpha }.x\in P_{*}ʻʻ\alpha :x\in CʻP:\\ +[*231·111] &\supset :x\in P\overline{R} _{\text{sc}}Q &\qquad \text{(2)}\\ +\vdash .(1).(2).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*231·113.</b> \[\begin{align}&\vdash \colon\ldotp x\in P\overline{R} _{\text{sc}}Q.\equiv :y\in CʻQ\cap \text{ᗡ}ʻR.\supset _{y}.x(P_{*}\mid R\mid \breve{Q} _{*})y:x\in + CʻP\\ +&[*231·112.*37·3]\end{align}\]</p> + +<p>If \(R\) is a one-valued function (<i>i.e.</i> a one-many relation), +and if we write \(x\leq x'\) for \(xP_{*}x'\), and \(y\leq y'\) for +\(yQ_{*}y'\), we have +\[ +x\in P\overline{R} _{\text{sc}}Q.\equiv :y\in CʻQ\cap \text{ᗡ}ʻR.\supset _{y}.(\exists y').y\leq y'.x\leq Rʻy':x\in CʻP. +\] +That is, \(x\) belongs to \(P\overline{R} _{\text{sc}}Q\) if, for any +argument \(y\) in \(CʻQ\), we can find an argument \(y'\) greater than +or equal to \(y\), for which the value is greater than or equal to +\(x\).</p> + +<p class="nind"><b><a id="*231·12">*231·12</a>.</b> \(\vdash .P\overline{R} _{\text{sc}}Q=pʻP_{*}ʻʻʻRʻʻʻ\overleftarrow{Q}_{*}ʻʻ(CʻQ\cap \text{ᗡ}ʻR)\cap CʻP \quad[*231·112]\)</p> + +<p>This is usually the most convenient formula for \(P\overline{R}_{\text{sc}}Q\).</p> + +<p class="nind"><b>*231·121.</b> \[\begin{align}\vdash :\exists !&CʻQ\cap \text{ᗡ}ʻR.\supset .\\ +&P\overline{R} _{\text{sc}}Q=pʻP_{*}ʻʻʻ\overleftarrow{Q}_{\text{cn}}ʻR=pʻP_{*}ʻʻʻRʻʻʻ\overleftarrow{Q}_{*}ʻʻ(CʻQ\cap \text{ᗡ}ʻR)\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*230·253.\supset \vdash :\text{Hp}.&\supset .\exists !\overleftarrow{Q}_{\text{cn}}ʻR.\\ +[*40·23.*37·47] &\supset .pʻP_{*}ʻʻʻ\overleftarrow{Q}_{\text{cn}}ʻR\subset sʻP_{*}ʻʻʻ\overleftarrow{Q}_{\text{cn}}ʻR.\\ +[*40·38.*37·16] &\supset .pʻP_{*}ʻʻʻ\overleftarrow{Q}_{\text{cn}}ʻR\subset CʻP &\qquad \text{(1)}\\ +\vdash .*40·23.\supset \vdash :\text{Hp}.\supset .pʻP_{*}ʻʻʻRʻʻʻ\overleftarrow{Q}_{*}ʻʻ(CʻQ\cap \text{ᗡ}ʻR)&\subset sʻP_{*}ʻʻʻRʻʻʻ\overleftarrow{Q}_{*}ʻʻ(CʻQ\cap + \text{ᗡ}ʻR)\\ +[*40·38.*37·16] &\subset CʻP &\qquad \text{(2)}\\ +\vdash .(1).(2).*231·1·12.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_730">[Pg 730]</span></p> + +<p class="nind"><b>*231·13.</b> \(\vdash .P\overline{R}_{\text{sc}}Q\in \text{sect}ʻP \quad[*211·631·13.*231·12]\)</p> + +<p class="nind"><b>*231·131.</b> \(\vdash .P\overline{R}_{\text{sc}}Q \subset CʻP \quad[*231·1]\)</p> + +<p class="nind"><b>*231·132.</b> \(\vdash :\exists !CʻQ\cap \text{ᗡ}ʻR.\supset .P\overline{R}_{\text{sc}}Q\subset P_{*}ʻʻRʻʻ\breve{Q} _{*}ʻʻ\text{ᗡ}ʻR\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*40·23.*231·121.\supset \vdash :\text{Hp}.\supset .P\overline{R}_{\text{sc}}Q&\subset sʻP_{*}ʻʻʻRʻʻʻ\overleftarrow{Q}_{*}ʻʻ(CʻQ\cap \text{ᗡ}ʻR)\\ +[*40·38] &\subset P_{*}ʻʻRʻʻsʻ\overleftarrow{Q}_{*}ʻʻ(CʻQ\cap \text{ᗡ}ʻR)\\ +[*40·52.*37·265]&\subset P_{*}ʻʻRʻʻ\breve{Q} _{*}ʻʻ\text{ᗡ}ʻR:\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*231·133.</b> \(\vdash :CʻP\cap \text{ᗡ}ʻR=\Lambda .\supset .P\overline{R}_{\text{sc}}Q=CʻP \quad[*231·12.*37·29.*40·2]\)</p> + +<p class="nind"><b>*231·134.</b> \(\vdash .P_{\text{po}}ʻʻ(P\overline{R}_{\text{sc}}Q)=Pʻʻ(P\overline{R}_{\text{sc}}Q) \quad[*211·131.*231·13]\)</p> + +<p class="nind"><b>*231·14.</b> \[\begin{align}&\vdash \colon\colon R\in 1\rightarrow \text{Cls}.\supset \colon\ldotp x\in P\overline{R}_{\text{sc}}Q.\equiv :\\ +&y\in CʻQ\cap \text{ᗡ}ʻR.\supset _{y}.(\exists z).yQ_{*}z.xP_{*}(Rʻz):x\in CʻP \quad[*71·7.*231·113]\end{align}\]</p> + +<p class="nind"><b>*231·141.</b> \(\vdash :Q_{*}\in \text{connex} .RQ_{\text{cn}}(\overleftarrow{P}_{*}ʻx).\supset .x\in P\overline{R}_{\text{sc}}Q\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*230·4.\supset \vdash :y\in R\overline{Q}_{\text{cn}}(\overleftarrow{P}_{*}ʻx).z\in \text{ᗡ}ʻR.yQ_{*}z.&\supset + .z\in R\overline{Q}_{\text{cn}}(\overleftarrow{P}_{*}ʻx).\\ +[*230·171] &\supset .x\in P_{*}ʻʻRʻʻ\overleftarrow{Q}_{*}ʻz &\qquad \text{(1)}\\ +\vdash .*230·171.*96·3.\supset\\ +\vdash :y\in R\overline{Q}_{\text{cn}}(\overleftarrow{P}_{*}ʻx).zQ_{*}y.&\supset .x\in P_{*}ʻʻRʻʻ\overleftarrow{Q}_{*}ʻy.\overleftarrow{Q}_{*}ʻy\subset + \overleftarrow{Q}_{*}ʻz.\\ +[*37·2] &\supset .x\in P_{*}ʻʻRʻʻ\overleftarrow{Q}_{*}ʻz &\qquad \text{(2)}\\ +\vdash .(1).(2).&\supset \vdash :\text{Hp}.y\in P\overline{Q}_{\text{cn}}(\overleftarrow{P}_{*}ʻx).z\in CʻQ\cap \text{ᗡ}ʻR.\supset .x\in P_{*}ʻʻRʻʻ\overleftarrow{Q}_{*}ʻz + &\qquad \text{(3)}\\ +\vdash .(3).*230·11.&\supset \vdash \colon\ldotp \text{Hp}.\supset :z\in CʻQ\cap \text{ᗡ}ʻR.\supset _{z}.x\in P_{*}ʻʻRʻʻ\overleftarrow{Q}_{*}ʻz &\qquad \text{(4)}\\ +\vdash .*230·151. &\supset \vdash :\text{Hp}.\supset .x\in CʻP &\qquad \text{(5)}\\ +\vdash .(4).(5).*231·112.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*231·142.</b> \(\vdash :\alpha \in \text{sect}ʻP.RQ_{\text{cn}}\alpha .\supset .P\overline{R}_{\text{sc}}Q\subset \alpha\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*231·1.*40·12.\supset \vdash :\text{Hp}.&\supset .P\overline{R}_{\text{sc}}Q\subset P_{*}ʻʻ\alpha .\\ +[*211·13] &\supset .P\overline{R}_{\text{sc}}Q\subset \alpha :\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*231·143.</b> \(\vdash :RQ_{\text{cn}}(\overrightarrow{P}_{*}ʻx).\supset .P\overline{R}_{\text{sc}}Q\subset \overrightarrow{P}_{*}ʻx + \quad[*231·142.*211·13]\)</p> + +<p class="nind"><b>*231·144.</b> \(\vdash :RQ_{\text{cn}}(\overrightarrow{P}_{\text{po}}ʻx).\supset .P\overline{R}_{\text{sc}}Q\subset \overrightarrow{P}_{\text{po}}ʻx + \quad[*231·142.*211·16]\)</p> + +<p class="nind"><b>*231·15.</b> \(\vdash :RʻʻCʻQ\subset CʻP.\supset .CʻP\cap pʻ\overrightarrow{P}_{*}ʻʻCʻP\subset P\overline{R}_{\text{sc}}Q\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*37·2.\supset \vdash \colon\ldotp \text{Hp}.&\supset :Rʻʻ\overleftarrow{Q}_{*}ʻy\subset CʻP:\\ +[*40·16] &\supset :pʻ\overrightarrow{P}_{*}ʻʻCʻP\subset pʻ\overrightarrow{P}_{*}ʻʻRʻʻ\overleftarrow{Q}_{*}ʻy:\\ +[*40·23] &\supset :y\in CʻQ\cap \text{ᗡ}ʻR.\supset _{y}.pʻ\overrightarrow{P}_{*}ʻʻCʻP\subset P_{*}ʻʻRʻʻ\overleftarrow{Q}_{*}ʻy:\\ +[*231·12] &\supset :CʻP\cap pʻ\overrightarrow{P}_{*}ʻʻCʻP\subset P\overline{R}_{\text{sc}}Q\colon\ldotp \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_731">[Pg 731]</span></p> + +<p class="nind"><b>*231·151.</b> \[\begin{align}&\vdash :P_{*}\in \text{connex} .CʻP\cap pʻ\overrightarrow{P}_{*}ʻʻCʻP\subset P\overline{R} _{\text{sc}}Q.\supset + .\overrightarrow{B}ʻP\subset P\overline{R} _{\text{sc}}Q\\ +&[*202·521]\end{align}\]</p> + +<p class="nind"><b>*231·152.</b> \[\begin{align}&\vdash :P_{*}\in \text{connex} .CʻP\cap pʻ\overrightarrow{P}_{*}ʻʻCʻP\subset P\overline{R} _{\text{sc}}Q.\exists + !\overrightarrow{B}ʻP.\supset .BʻP\in P\overline{R} _{\text{sc}}Q\\ +&[*231·151.*202·523]\end{align}\]</p> + +<p>The hypothesis \(CʻP\cap pʻ\overrightarrow{P}_{*}ʻʻCʻP\subset P\overline{R} _{\text{sc}}Q\) is verified not only when +\(RʻʻCʻQ\subset CʻP\), but also under certain more general hypotheses. Two such +hypotheses, namely +\[ +\begin{aligned} +&CʻQ\cap \text{ᗡ}ʻR\subset \breve{R} ʻʻCʻP\\ +\text{and}\quad &CʻQ\cap \text{ᗡ}ʻR\subset Q_{*}ʻʻ\breve{R} ʻʻCʻP, +\end{aligned} +\] +are considered in the following propositions.</p> + +<p class="nind"><b>*231·153.</b> \(\vdash :CʻQ\cap \text{ᗡ}ʻR\subset Q_{*}ʻʻ\breve{R} ʻʻCʻP.\supset .CʻP\cap pʻ\overrightarrow{P}_{*}ʻʻCʻP\subset P\overline{R} _{\text{sc}}Q\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*37·1.\supset \vdash \colon\colon&\text{Hp}.\supset \colon\ldotp y\in CʻQ\cap \text{ᗡ}ʻR.\supset _{y}:(\exists z).z\in CʻP.z(R\mid \breve{Q} _{*})y:\\ +[*40·51] \supset _{y}:x \in pʻ\overrightarrow{P}_{*}ʻʻCʻP.&\supset _{x}.(\exists z).z\in CʻP.z(R\mid \breve{Q} _{*})y.xP_{*}z.\\ +[*34·1] &\supset _{x}.x(P_{*}\mid R\mid \breve{Q} _{*})y &\qquad \text{(1)}\\ +\vdash .(1).\text{Comm}.\supset \\ +\vdash \colon\ldotp \text{Hp}.x\in CʻP\cap pʻ\overrightarrow{P}_{*}ʻʻCʻP.\supset :y\in CʻQ\cap \text{ᗡ}ʻR.&\supset _{y}.x(P_{*}\mid R\mid \breve{Q} _{*})y:x\in + CʻP:\\ +[*231·113] &\supset :x\in P\overline{R} _{\text{sc}}Q\colon\ldotp \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*231·154.</b> \(\vdash :RʻʻCʻQ\subset CʻP.\supset .CʻQ\cap \text{ᗡ}ʻR\subset \breve{R} ʻʻCʻP\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*37·2.\supset \vdash :\text{Hp}.&\supset .\breve{R} ʻʻRʻʻCʻQ\subset \breve{R} ʻʻCʻP.\\ +[*37·501] &\supset .CʻQ\cap \text{ᗡ}ʻR\subset \breve{R} ʻʻCʻP:\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*231·155.</b> \(\vdash :CʻQ\cap \text{ᗡ}ʻR\subset \breve{R} ʻʻCʻP.\supset .CʻQ\cap \text{ᗡ}ʻR\subset Q_{*}ʻʻ\breve{R} ʻʻCʻP\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*22·43·45.\supset \vdash :\text{Hp}.\supset .CʻQ\cap \text{ᗡ}ʻR&\subset CʻQ\cap \breve{R} ʻʻCʻP\\ +[*90·33] &\subset Q_{*}ʻʻ\breve{R} ʻʻCʻP:\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*231·156.</b> \[\begin{align}\vdash \colon\ldotp CʻQ\cap \text{ᗡ}ʻR\subset Q_{*}ʻʻ\breve{R} ʻʻCʻP.&\equiv :\Lambda {\sim}\in P_{*}ʻʻʻRʻʻʻ\overleftarrow{Q}_{*}ʻʻ(CʻQ\cap + \text{ᗡ}ʻR):\\ +&\equiv :z\in CʻQ\cap \text{ᗡ}ʻR.\supset _{z}.\exists !CʻP\cap Rʻʻ\overleftarrow{Q}_{*}ʻz\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*37·1.\supset \vdash \colon\ldotp &CʻQ\cap \text{ᗡ}ʻR\subset Q_{*}ʻʻ\breve{R} ʻʻCʻP.\equiv :\\ +&z\in CʻQ\cap \text{ᗡ}ʻR.\supset _{z}.(\exists x).x\in CʻP.z(Q_{*}\mid \breve{R} )x:\\ +[*37·3] &\equiv :z\in CʻQ\cap \text{ᗡ}ʻR.\supset _{z}.(\exists x).x\in CʻP.x\in Rʻʻ\overleftarrow{Q}_{*}ʻz:\\ +[*22·33] &\equiv :z\in CʻQ\cap \text{ᗡ}ʻR.\supset _{z}.\exists !CʻP\cap Rʻʻ\overleftarrow{Q}_{*}ʻz: &\qquad \text{(1)}\\ +[*37·265·43] &\equiv :z\in CʻQ\cap \text{ᗡ}ʻR.\supset _{z}.\exists !P_{*}ʻʻRʻʻ\overleftarrow{Q}_{*}ʻz &\qquad \text{(2)}\\ +\vdash .(1).(2).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_732">[Pg 732]</span></p> + +<p class="nind"><b>*231·16.</b> \[\begin{align}\vdash :P_{*}\in \text{connex} .\exists !\overrightarrow{B}ʻP.&CʻQ\cap \text{ᗡ}ʻR\subset Q_{*}ʻʻ\breve{R} ʻʻCʻP.\supset .\\ +&\exists !P\overline{R} _{\text{sc}}Q.BʻP\in P\overline{R} _{\text{sc}}Q \quad[*231·152·153]\end{align}\]</p> + +<p class="nind"><b>*231·161.</b> \[\begin{align}&\vdash :P_{*}\in \text{connex} .\exists !\overrightarrow{B}ʻP.RʻʻCʻQ\subset CʻP.\supset .\exists !P\overline{R} _{\text{sc}}Q.BʻP\in + P\overline{R} _{\text{sc}}Q\\ +&[*231·154·155·16]\end{align}\]</p> + +<p class="nind"><b>*231·17.</b> \(\vdash :RʻʻCʻQ\subset CʻP.\supset .R\overline{Q} _{\text{cn}}\alpha \subset R\overline{Q} _{\text{cn}}(P_{*}ʻʻ\alpha )\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*90·13.\supset \vdash \colon\ldotp \text{Hp}.&\supset :y\in CʻQ.\supset .Rʻʻ\overleftarrow{Q}_{*}ʻy\subset CʻP:\\ +[*230·1] \supset :y\in R\overline{Q} _{\text{cn}}\alpha .&\supset .y\in CʻQ\cap \text{ᗡ}ʻR.Rʻʻ\overleftarrow{Q}_{*}ʻy\subset \alpha \cap CʻP.\\ +[*90·33] &\supset .y\in CʻQ\cap \text{ᗡ}ʻR.Rʻʻ\overleftarrow{Q}_{*}ʻy\subset P_{*}ʻʻ\alpha .\\ +[*230·1] &\supset .y\in R\overline{Q} _{\text{cn}}(P_{*}ʻʻ\alpha )\colon\ldotp \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*231·171.</b> \(\vdash :RʻʻCʻQ\subset CʻP.R\overline{Q} _{\text{cn}}\alpha .\supset .R\overline{Q} _{\text{cn}}(P_{*}ʻʻ\alpha ) \quad[*231·17.*230·11]\)</p> + +<p class="nind"><b>*231·18.</b> \(\vdash :RʻʻCʻQ\subset CʻP.\supset .P\overline{R} _{\text{sc}}Q=pʻ(\text{sect}ʻP\cap \overleftarrow{Q}_{\text{cn}}ʻR)\cap CʻP\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*231·11.*211·13.\supset \\ +\vdash \colon\ldotp x\in P\overline{R} _{\text{sc}}Q.&\supset :\beta Q_{\text{cn}}R.\beta \in \text{sect}ʻP.\supset _{\beta }.x\in \beta :x\in CʻP &\qquad \text{(1)}\\ +\vdash .*231·171.\supset \\ +\vdash \colon\colon\text{Hp}.&\supset \colon\ldotp RQ_{\text{cn}}(P_{*}ʻʻ\alpha ).\supset _{\alpha }.x\in P_{*}ʻʻ\alpha :\supset :RQ_{\text{cn}}\alpha + .\supset _{\alpha }.x\in P_{*}ʻʻ\alpha \colon\ldotp \\ +[*13·195.*231·11] +&\supset \colon\ldotp (\exists \alpha ).\beta =P_{*}ʻʻ\alpha .RQ_{\text{cn}}\beta .\supset _{\beta }.x\in \beta :x\in CʻP:\supset :x\in P\overline{R} _{\text{sc}}Q\colon\ldotp + \\ +[*211·13]&\supset \colon\ldotp \beta \in \text{sect}ʻP.R\overline{Q} _{\text{cn}}\beta .\supset _{\beta }.x\in \beta :x\in CʻP:\supset :x\in P\overline{R} _{\text{sc}}Q + &\qquad \text{(2)}\\ +\vdash .(1).(2).\supset \\ +\vdash \colon\colon\text{Hp}.&\supset \colon\ldotp x\in P\overline{R} _{\text{sc}}Q.\equiv :\beta \in \text{sect}ʻP.RQ_{\text{cn}}\beta .\supset _{\beta }.x\in + \beta :x\in CʻP\colon\colon\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*231·181.</b> \(\vdash :P\in \text{Ser}.RʻʻCʻQ\subset CʻP.\supset .P\overline{R} _{\text{sc}}Q=CʻP\cap pʻ(\overrightarrow{P}ʻʻCʻP\cap \overleftarrow{Q}_{\text{cn}}ʻR)\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*231·18.*211·302.*40·16.\supset \\ +\vdash :\text{Hp}.&\supset .P\overline{R} _{\text{sc}}Q\subset CʻP\cap pʻ(\overrightarrow{P}ʻʻCʻP\cap \overleftarrow{Q}_{\text{cn}}ʻR) &\qquad \text{(1)}\\ +\vdash .*40·55.*230·211.\supset \vdash \colon\ldotp \alpha \in \text{sect}ʻ&P\cap \overleftarrow{Q}_{\text{cn}}ʻR.z\in CʻP\cap pʻ\overleftarrow{P}ʻʻ\alpha .\supset :\\ +&\overrightarrow{P}ʻz\in \overleftarrow{Q}_{\text{cn}}ʻR:\\ +[*40·12] &\supset :x\in pʻ(\overrightarrow{P}ʻʻCʻP\cap \overleftarrow{Q}_{\text{cn}}ʻR).\supset .x\in \overrightarrow{P}ʻz &\qquad \text{(2)}\\ +\vdash .(2).\text{Comm}.\supset \vdash \colon\ldotp x\in CʻP\cap &pʻ(\overrightarrow{P}ʻʻCʻP\cap \overleftarrow{Q}_{\text{cn}}ʻR).\alpha + \in \text{sect}ʻP\cap \overleftarrow{Q}_{\text{cn}}ʻR.\supset :\\ +&x\in CʻP:z\in CʻP\cap pʻ\overleftarrow{P}ʻʻ\alpha .\supset _{z}.x\in \overrightarrow{P}ʻz:\\ +[*40·41] &\supset :x\in CʻP\cap pʻ\overrightarrow{P}ʻʻ(CʻP\cap pʻ\overleftarrow{P}ʻʻ\alpha ) &\qquad \text{(3)}\\ +\vdash .*211·711.\supset \\ +\vdash :\text{Hp}.\alpha \in \text{sect}ʻP.\supset .CʻP\cap pʻ\overrightarrow{P}ʻʻ(CʻP\cap pʻ\overleftarrow{P}ʻʻ\alpha )&=CʻP\cap pʻ\overrightarrow{P}ʻʻ(CʻP-\alpha )\\ +[*211·7·711] &=CʻP-(CʻP-\alpha ) &\qquad \text{(4)}\\ +\vdash .(3).(4).\supset\\ +\vdash \colon\ldotp \text{Hp}.x\in CʻP\cap pʻ(\overrightarrow{P}ʻʻCʻP\cap \overleftarrow{Q}_{\text{cn}}ʻR).&\supset :\alpha \in \text{sect}ʻP\cap \overleftarrow{Q}_{\text{cn}}ʻR.\supset + _{\alpha }.x\in \alpha :\\ +[*231·18] &\supset :x\in P\overline{R} _{\text{sc}}Q &\qquad \text{(5)}\\ +\vdash .(1).(5).\supset \vdash .\text{Prop} +\end{array}\]</p> + +<p><span class="pagenum" id="Page_733">[Pg 733]</span></p> + +<p class="nind"><b>*231·182.</b> \[\begin{align}\vdash :P\in \text{Ser}.Rʻʻ&CʻQ\subset CʻP.\exists !CʻP-(P\overline{R} _{\text{sc}}Q).\supset .\\ +&P\overline{R} _{\text{sc}}Q=pʻ(\overrightarrow{P}ʻʻCʻP\cap \overleftarrow{Q}_{\text{cn}}ʻR).\exists + !(\overrightarrow{P} ʻʻCʻP \cap \overleftarrow{Q}_{\text{cn}}ʻR)\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*231·181.\supset\\ +\vdash :\text{Hp}.\exists !(\overrightarrow{P}ʻʻCʻP\cap \overleftarrow{Q}_{\text{cn}}ʻR).&\supset .P\overline{R} _{\text{sc}}Q=pʻ(\overrightarrow{P}ʻʻCʻP\cap + \overleftarrow{Q}_{\text{cn}}ʻR) &\qquad \text{(1)}\\ +\vdash .*230·11.&\supset \vdash \colon\ldotp \text{Hp}.\overrightarrow{P} ʻʻCʻP \cap \overleftarrow{Q}_{\text{cn}}ʻR=\Lambda .\supset :\\ +x\in CʻP.y\in CʻQ\cap \text{ᗡ}ʻR.&\supset _{x,y}.{\sim}(Rʻʻ\overleftarrow{Q}_{*}ʻy\subset \overrightarrow{P}ʻx).\\ +[*90·33.\text{Hp}] &\supset _{x,y}.{\sim}(P_{*}ʻʻRʻʻ\overleftarrow{Q}_{*}ʻy\subset \overrightarrow{P}ʻx).\\ +[*211·56] &\supset _{x,y}.\overrightarrow{P}_{*}ʻx\subset P_{*}ʻʻRʻʻ\overleftarrow{Q}_{*}ʻy.\\ +[*90·13] &\supset _{x,y}.x\in P_{*}ʻʻRʻʻ\overleftarrow{Q}_{*}ʻy &\qquad \text{(2)}\\ +\vdash .(2).*231·12.&\supset \vdash :\text{Hp}.\overrightarrow{P}ʻʻCʻP\cap \overleftarrow{Q}_{\text{cn}}ʻR=\Lambda .\supset .CʻP\subset P\overline{R} _{\text{sc}}Q + &\qquad \text{(3)}\\ +\vdash .(3).\text{Transp}.&\supset \vdash .\text{Hp}.\supset .\exists !\overrightarrow{P}ʻʻCʻP\cap \overleftarrow{Q}_{\text{cn}}ʻR &\qquad \text{(4)}\\ +\vdash .(1).(4).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*231·19.</b> \[\begin{align}\vdash :P\in \text{trans}.Q_{*}\in \text{connex} .Rʻʻ&CʻQ\subset CʻP.\supset .\\ +&P\overline{R} _{\text{os}}Q=pʻ(\text{str}ʻP\cap \overleftarrow{Q}_{\text{cn}}ʻR)\cap CʻP\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*231·18·101.\supset \vdash \colon\colon\text{Hp}.\supset \colon\ldotp \\ +x\in P\overline{R} _{\text{os}}Q.&\equiv :\alpha \in \text{sect}ʻP.\beta \in \text{sect}ʻ\breve{P} .\alpha ,\beta \in \overleftarrow{Q}_{\text{cn}}ʻR.\supset _{\alpha ,\beta }.x\in + \alpha \cap \beta :x\in CʻP:\\ +[*13·191.*11·35]&\equiv :(\exists \alpha ,\beta ).\alpha \in \text{sect}ʻP.\beta \in \text{sect}ʻ\breve{P} .\alpha ,\beta \in \overleftarrow{Q}_{\text{cn}}ʻR.\gamma + =\alpha \cap \beta .\supset _{\gamma }.\\ +&x\in \gamma :x\in CʻP &\qquad \text{(1)}\\ +\vdash .*230·42.\supset \vdash \colon\ldotp \text{Hp}.\supset :\alpha ,\beta \in \overleftarrow{Q}_{\text{cn}}ʻR.&\equiv .\alpha \cap \beta \in \overleftarrow{Q}_{\text{cn}}ʻR + &\qquad \text{(2)}\\ +\vdash .*215·16.\supset \\ +\vdash \colon\ldotp \text{Hp}.\supset :(\exists \alpha ,\beta ).\alpha \in \text{sect}ʻP.\beta \in \text{sect}ʻ\breve{P} .\gamma =\alpha \cap \beta .&\equiv .\gamma \in \text{str}ʻP &\qquad \text{(3)}\\ +\vdash .(1).(2).(3).\supset \\ +\vdash \colon\colon\text{Hp}.\supset \colon\ldotp x\in P\overline{R} _{\text{os}}Q.&\equiv :\gamma \in \text{str}ʻP.RQ_{\text{cn}}\gamma .\supset _{\gamma }.x\in + \gamma \colon\colon\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_734">[Pg 734]</span></p> + +<p class="nind"><b>*231·191.</b> \[\begin{align}\vdash :P_{\text{po}}\in \text{connex} .&\exists !P\overline{R} _{\text{os}}Q.\supset .\\ +&P\overline{R} _{\text{sc}}Q=P_{*}ʻʻ(P\overline{R} _{\text{os}}Q).Pʻʻ(P\overline{R} _{\text{sc}}Q)=P_{\text{po}}ʻʻ(P\overline{R} _{\text{os}}Q)\\ +&[*215·165.*231·13·101]\end{align}\]</p> + +<p class="nind"><b>*231·192.</b> \[\begin{align}\vdash \colon\ldotp P_{\text{po}}\in &\text{connex} .\exists !P\overline{R} _{\text{os}}Q.\exists !P\overline{R} _{\text{os}}Qʻ.\supset + :\\ +&P\overline{R} _{\text{os}}Q=P\overline{R} _{\text{os}}Qʻ.\equiv .P\overline{R} _{\text{sc}}Q=P\overline{R} _{\text{sc}}Qʻ.\breve{P} \overline{R} _{\text{sc}}Q=\breve{P} + \overline{R} _{\text{sc}}Qʻ\\ +&[*231·191·101]\end{align}\]</p> + +<p class="nind"><b>*231·193.</b> \[\begin{align}\vdash :P_{\text{po}}\in \text{Ser}.P\overline{R} _{\text{os}}Q\in &1.\supset .\\ +&P\overline{R} _{\text{os}}Q=\iota ʻ\text{max}_{P}ʻ(P\overline{R} _{\text{sc}}Q)=\iota ʻ\text{min}_{P}ʻ(\breve{P} \overline{R} _{\text{sc}}Q)\\ +&[*215·166.*231·13·101]\end{align}\]</p> + +<p>This proposition is of fundamental importance.</p> + +<p class="nind"><b>*231·2.</b> \[\begin{align}\vdash :P_{*},Q_{*}\in \text{connex} .CʻQ\cap \text{ᗡ}ʻR\subset Q_{*}ʻʻ\breve{R} ʻʻ&CʻP.\supset .\\ +&CʻP-(P\overline{R} _{\text{sc}}Q)\subset \breve{P} \overline{R} _{\text{sc}}Q\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*231·112.\supset \\ +\vdash :x\in CʻP-(P\overline{R} _{\text{sc}}Q).&\supset .(\exists y).y\in CʻQ\cap \text{ᗡ}ʻR.x\in CʻP-P_{*}ʻʻRʻʻ\overleftarrow{Q}_{*}ʻy &\qquad \text{(1)}\\ +\vdash .*202·501.*90·33.\supset \\ +\vdash \colon\ldotp \text{Hp}.&x\in CʻP-P_{*}ʻʻRʻʻ\overleftarrow{Q}_{*}ʻy.\supset :x\in pʻ\overleftarrow{P}_{*}ʻʻ(Rʻʻ\overleftarrow{Q}_{*}ʻy\cap CʻP):\\ +[*96·3] &\supset :yQ_{*}z.\supset _{z}.x\in pʻ\overleftarrow{P}_{*}ʻʻ(Rʻʻ\overleftarrow{Q}_{*}ʻz\cap CʻP):\\ +[*40·61] \supset :yQ_{*}z.z\in \text{ᗡ}ʻR.&\supset _{z}.x\in \breve{P} _{*}ʻʻ(Rʻʻ\overleftarrow{Q}_{*}ʻz\cap CʻP).\\ +[*37·265] & \supset _{z}.x\in \breve{P} _{*}ʻʻRʻʻ\overleftarrow{Q}_{*}ʻz: &\qquad \text{(2)}\\ +[*90·12] &\supset :y\in CʻQ\cap \text{ᗡ}ʻR.\supset .x\in \breve{P} _{*}ʻʻRʻʻ\overleftarrow{Q}_{*}ʻy:\\ +[*96·3.*37·2]&\supset :y\in CʻQ\cap \text{ᗡ}ʻR.zQ_{*}y.\supset _{z}.x\in \breve{P} _{*}ʻʻRʻʻ\overleftarrow{Q}_{*}ʻz &\qquad \text{(3)}\\ +\vdash .(2).(3).*202·137.\supset \vdash \colon\ldotp \text{Hp}.&y\in CʻQ\cap \text{ᗡ}ʻR.x\in CʻP-P_{*}ʻʻRʻʻ\overleftarrow{Q}_{*}ʻy.\supset :\\ +&z\in CʻQ\cap \text{ᗡ}ʻR.\supset _{z}.x\in \breve{P} _{*}ʻʻRʻʻ\overleftarrow{Q}_{*}.ʻz:x\in CʻP:\\ +[*231·112] &\supset :x\in \breve{P} \overline{R} _{\text{sc}}Q &\qquad \text{(4)}\\ +\vdash .(1).(4).&\supset \vdash \colon\ldotp \text{Hp}.\supset :x\in CʻP-(P\overline{R} _{\text{sc}}Q).\supset .x\in \breve{P} \overline{R} _{\text{sc}}Q\colon\ldotp + \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p>This proposition is fundamental in the theory of limiting segments.</p> + +<p class="nind"><b>*231·201.</b> \[\begin{align}&\vdash :P_{*},Q_{*}\in \text{connex} .RʻʻCʻQ\subset CʻP.\supset .CʻP-(P\overline{R} _{\text{sc}}Q)\subset + \breve{P} \overline{R} _{\text{sc}}Q\\ +&[*231·2·154·155]\end{align}\]</p> + +<p class="nind"><b>*231·202.</b> \(\vdash :P_{*},Q_{*}\in \text{connex} .\exists !P\overline{R} _{\text{sc}}Q.\supset .CʻP-(P\overline{R} _{\text{sc}}Q)\subset + \breve{P} \overline{R} _{\text{sc}}Q\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*40·22.\text{Transp}.*231·12.\supset \\ +\vdash :\text{Hp}.&\supset .\Lambda {\sim}\in P_{*}ʻʻʻRʻʻʻ\overleftarrow{Q}_{*}ʻʻ(CʻQ\cap \text{ᗡ}ʻR).\\ +[*231·156] &\supset .CʻQ\cap \text{ᗡ}ʻR\subset Q_{*}ʻʻ\breve{R} ʻʻCʻP.\\ +[*231·2] &\supset .CʻP-(P\overline{R} _{\text{sc}}Q)\subset \breve{P} \overline{R} _{\text{sc}}Q:\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_735">[Pg 735]</span></p> + +<p class="nind"><b>*231·21.</b> \[\begin{align}\vdash :P_{*},Q_{*}\in \text{connex} .&CʻQ\cap \text{ᗡ}ʻR\subset Q_{*}ʻʻ\breve{R} ʻʻCʻP.\supset .\\ +&CʻP=P\overline{R} _{\text{sc}}Q\cup \breve{P} \overline{R} _{\text{sc}}Q\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*231·13.&\supset \vdash :P\overline{R} _{\text{sc}}Q\subset CʻP.\breve{P} \overline{R} _{\text{sc}}Q\subset CʻP &\qquad \text{(1)}\\ +\vdash .*231·2. &\supset \vdash :\text{Hp}.\supset .CʻP\subset P\overline{R} _{\text{sc}}Q\cup \breve{P} \overline{R} _{\text{sc}}Q &\qquad \text{(2)}\\ +\vdash .(1).(2).&\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*231·22">*231·22</a>.</b> \[\begin{align}&\vdash :P_{*},Q_{*}\in \text{connex} .RʻʻCʻQ\subset CʻP.\supset .CʻP=P\overline{R} _{\text{sc}}Q\cup + \breve{P} \overline{R} _{\text{sc}}Q\\ +&[*231·201·13]\end{align}\]</p> + +<p class="nind"><b>*231·23.</b> \[\begin{align}\vdash \colon\ldotp P_{*}\in \text{connex} .Rʻʻ&CʻQ\subset CʻP.\supset :\\ +&\exists !Rʻʻ\overleftarrow{Q}_{*}ʻy-\overrightarrow{P}_{*}ʻx.\supset .\overrightarrow{P}_{*}ʻx\subset + P_{\text{po}}ʻʻRʻʻ\overleftarrow{Q}_{*}ʻy\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*90·14. &\supset \vdash :\text{Hp}.\supset .Rʻʻ\overleftarrow{Q}_{*}ʻy\subset CʻP &\qquad \text{(1)}\\ +\vdash .(1)*90·21.&\supset \vdash :\text{Hp}.\exists !Rʻʻ\overleftarrow{Q}_{*}ʻy-\overrightarrow{P}_{*}ʻx.\supset + .\exists !P_{*}ʻʻRʻʻ\overleftarrow{Q}_{*}ʻy-\overrightarrow{P}_{*}ʻx.\\ +[*211·56.*202·13] &\supset .\overrightarrow{P}_{*}ʻx\subset P_{\text{po}}ʻʻRʻʻ\overleftarrow{Q}_{*}ʻy:\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*231·24.</b> \(\vdash :P_{*}\in \text{connex} .RʻʻCʻQ\subset CʻP.{\sim}{RQ_{\text{cn}}(\overrightarrow{P}_{*}ʻx)}.\supset .\overrightarrow{P}_{*}x\subset + P\overline{R} _{\text{sc}}Q\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*230·11.\supset \vdash \colon\ldotp \text{Hp}.\supset :y\in CʻQ\cap \text{ᗡ}ʻR.&\supset _{y}.\exists !Rʻʻ\overleftarrow{Q}_{*}ʻy-\overrightarrow{P}_{*}ʻx.\\ +[*231·23] &\supset _{y}.\overrightarrow{P}_{*}ʻx\subset P_{\text{po}}ʻʻRʻʻ\overleftarrow{Q}_{*}ʻy:\\ +[*91·54.*40·44] &\supset :\overrightarrow{P}_{*}ʻx\subset pʻP_{*}ʻʻʻRʻʻʻ\overleftarrow{Q}_{*}ʻʻ(CʻQ\cap \text{ᗡ}ʻR) &\qquad \text{(1)}\\ +\vdash .(1).*231·12.*90·14.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*231·25.</b> \[\begin{align}\vdash \colon\ldotp &P\in \text{Ser}.Q_{*}\in \text{connex} .RʻʻCʻQ\subset CʻP.P\overline{R} _{\text{os}}Q=\Lambda .\\ +&\text{E}!\text{limax}_Pʻ(P\overline{R} _{\text{sc}}Q).\supset :\text{limax}_Pʻ(P\overline{R} _{\text{sc}}Q=\text{limin}_Pʻ(\breve{P} \overline{R} _{\text{sc}}Q).\lor.\\ +&\text{limax}_Pʻ(P\overline{R} _{\text{sc}}Q)P_{1}\text{limin}_Pʻ(\breve{P} \overline{R} _{\text{sc}}Q)\\ +&[*215·54·541.*231·13·22]\end{align}\]</p> + +<p class="nind"><b>*231·251.</b> \[\begin{align}\vdash :\text{Hp}*231·25.&\text{limax}_Pʻ(P\overline{R} _{\text{sc}}Q){\sim}\in \text{D}ʻP_{1}.\supset .\\ +&\text{limax}_Pʻ(P\overline{R} _{\text{sc}}Q)=\text{limin}_Pʻ(\breve{P} \overline{R} _{\text{sc}}Q) \quad[*231·25]\end{align}\]</p> + +<p class="nind"><b>*231·252.</b> \[\begin{align}\vdash :&P\in \text{Ser}.Q_{*}\in \text{connex} .RʻʻCʻQ\subset CʻP.P\overline{R} _{\text{os}}\in 0\cup 1.\\ +&\text{E}!\text{limax}_Pʻ(P\overline{R} _{\text{sc}}Q).\text{limax}_Pʻ(P\overline{R} _{\text{sc}}Q){\sim}e\in \text{D}ʻP_{1}.\supset .\\ +&\text{limax}_Pʻ(P\overline{R} _{\text{sc}}Q)=\text{limin}_Pʻ(\breve{P} \overline{R} _{\text{sc}}Q)\\ +&[*215·543.*231·13·22]\end{align}\]</p> + +<p><span class="pagenum" id="Page_736">[Pg 736]</span></p> + +<p class="nind"><b>*231·4.</b> \(\vdash :Q\in \text{trans}\cap \text{connex} .\text{E}!\text{max}_{Q}ʻ\text{ᗡ}ʻR.\supset + .P\overline{R} _{\text{sc}}Q=P_{*}ʻʻ\overrightarrow{R}ʻ\text{max}_{Q}ʻ\text{ᗡ}ʻR\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*230·53.*231·121.\supset \\ +\vdash \colon\colon\text{Hp}.\supset \colon\ldotp x\in P\overline{R} _{\text{sc}}Q.&\equiv :\overrightarrow{R}ʻ\text{max}_{Q}ʻ\text{ᗡ}ʻR\subset \alpha .\supset _{\alpha }.x\in + P_{*}ʻʻ\alpha :\\ +[*37·2.*22·42] &\equiv :x\in P_{*}ʻʻ\overrightarrow{R}ʻ\text{max}_{Q}ʻ\text{ᗡ}ʻR\colon\colon\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*231·41.</b> \(\vdash :Q\in \text{trans}\cap \text{connex} .\text{E}!Rʻ\text{max}_{Q}ʻ\text{ᗡ}ʻR.\supset + .P\overline{R} _{\text{sc}}Q=\overrightarrow{P}_{*}ʻRʻ\text{max}_{Q}ʻ\text{ᗡ}ʻR\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*30·5.*231·4.*53·31.\supset \\ +\vdash :\text{Hp}.\supset .P\overline{R} _{\text{sc}}Q&=P_{*}ʻʻ\iota ʻRʻ\text{max}_{Q}ʻ\text{ᗡ}ʻR\\ +[*53·301] &=\overrightarrow{P}_{*}ʻRʻ\text{max}_{Q}ʻ\text{ᗡ}ʻR:\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_737">[Pg 737]</span></p> +<h2 class="nobreak" id="*232"> +*232. ON THE OSCILLATION OF A FUNCTION AS THE ARGUMENT +APPROACHES A GIVEN LIMIT.</h2> +</div> + + +<p class="nind"><i>Summary of</i> *232.</p> + +<p>In the preceding number, we considered the ultimate oscillation +of a function when the argument grows without limit. If, in the +propositions of the last number, we confine the field of \(Q\) to +\(\overrightarrow{Q}ʻx\), where \(x\in \text{ᗡ}ʻQ\), the ultimate +oscillation becomes the ultimate oscillation as the argument +approaches \(x\) from below. If the ultimate oscillation consists +of a single term, this is the limit of the function as the argument +approaches \(x\) from below. If, instead of confining the argument +to \(\overrightarrow{Q}ʻx\), we confine it to any other class whose +limit is \(x\), we shall, under a very usual hypothesis, obtain +the same value for the ultimate oscillation as if we confined it +to \(\overrightarrow{Q}ʻx\). And more generally, under a similar +hypothesis, if \(\alpha\) and \(\beta\) are two classes of arguments +which define the same section (<i>i.e.</i> such that \(Q_{*}ʻʻ\alpha=Q_{*}ʻʻ\beta\)), +then, whether or not this section has a limit, the ultimate sections +and the ultimate oscillation are the same for \(\alpha\) as they +are for \(\beta\). Hence we are led to consider first the result +of confining the field of \(Q\), not to \(\overrightarrow{Q}ʻx\), +but to any class \(\alpha\). In order not to have to exclude +explicitly the case in which \(\alpha \in 1\), we deal with +\(Q_{*}\unicode{x0294f}\alpha\), not \(Q\unicode{x0294f}\alpha\). Hence +we are led to the following definitions:</p> + +<p class="nind"><b>*232·01.</b> \((P\overline{R} Q)_{\text{sc}}ʻ\alpha =P\overline{R} _{\text{sc}}(Q_{*}\unicode{x0294f}\alpha ) \quad\text{Df}\)</p> + +<p class="nind"><b>*232·02.</b> \((P\overline{R} Q)_{\text{os}}ʻ\alpha =P\overline{R} _{os}(Q_{*}\unicode{x0294f}\alpha ) \quad\text{Df}\)</p> + +<p>Most of the propositions of the present number are immediate +consequences of corresponding propositions in <a href="#*231">*231</a>. The most important +application of the propositions of the present number is to the case +where \(\alpha\) is of the form \(\overrightarrow{Q}ʻx\), \(x\) being +a member of \(\delta _{Q}ʻ\text{ᗡ}ʻR\). We may, in this case, take +in place of \(\overrightarrow{Q}ʻx\) any other class of arguments +(<i>e.g.</i> a progression of arguments \(x_{1}\), \(x_{2}\), ... +\(x_{\nu }\), ...) having \(x\) for its limit, without altering the +limiting sections or the ultimate oscillation. Hence the limit of the +function for a given argument (if it exists) may be determined by +choosing any selection of arguments having the given argument as their +limit (cf. <a href="#*233·142">*233·142</a>, below).</p> + +<p><span class="pagenum" id="Page_738">[Pg 738]</span></p> + +<p>From the definition of \((P\overline{R} Q)_{\text{sc}}ʻ\alpha\) we +obtain immediately</p> + +<p class="nind"><b>*232·11.</b> \[\begin{align}\vdash \colon\ldotp x\in (P\overline{R} Q)_{\text{sc}}ʻ&\alpha .\equiv :\\ +&y\in \alpha \cap CʻQ\cap \text{ᗡ}ʻR.\supset _{y}.x\in P_{*}ʻʻRʻʻ(\alpha \cap \overleftarrow{Q}_{*}ʻy):x\in CʻP\end{align}\]</p> + +<p>We prove that \((P\overline{R} Q)_{\text{sc}}ʻ\alpha = (P\overline{R}Q)_{\text{sc}}ʻ(\alpha \cap CʻQ\cap \text{ᗡ}ʻR)\) +(<a href="#*232·131">*232·131</a>), and that if \(\alpha \cap CʻQ\cap \text{ᗡ}ʻR=\Lambda\), the +two limiting sections and the ultimate oscillation are all equal to +\(CʻP\) (<a href="#*232·15">*232·15</a>). Also we have</p> + +<p class="nind"><b>*232·14.</b> \(\vdash :Q\in \text{trans}\cap \text{connex} .\alpha \cap CʻQ{\sim}\in 1.\supset .(P\overline{R} Q)_{\text{sc}}ʻ\alpha =P\overline{R} _{\text{sc}}(Q\unicode{x0294f}\alpha + )\)</p> + +<p>Thus the substitution of \(Q_{*}\) for \(Q\) in our definitions has +the effect of making them applicable to unit classes, and of enabling +us to substitute the hypothesis \(Q_{*}\in \text{connex}\) for \(Q\in\text{trans}\cap \text{connex}\). +But when \(Q\) is transitive and connected (and therefore when \(Q\) is +a series), the substitution of \(Q_{*}\) for \(Q\) in the definitions +makes no difference unless \(\alpha\) is a unit class. This case is +trivial, since the only interest of our definitions is when a has no +maximum in \(Q\).</p> + +<p>From <a href="#*231·22">*231·22</a> we obtain</p> + +<p class="nind"><b>*232·22.</b> \[\begin{align}\vdash :P_{*},Q_{*}\unicode{x0294f}\alpha \in \text{connex} .Rʻʻ(\alpha \cap CʻQ)&\subset CʻP.\supset .\\ +CʻP=(P\overline{R} Q)_{\text{sc}}ʻ\alpha \cup (\breve{P} \overline{R} Q)_{\text{sc}}ʻ\alpha \end{align}\]</p> + +<p>We have next a set of propositions concerned in discovering +circumstances under which two classes \(\alpha\) and \(\beta\) which +determine the same section in \(Q\) (and therefore have the same limit, +if any) give the same values for the two limiting sections. For this +purpose, it is only necessary to discover circumstances under which +we may substitute \(Q_{*}ʻʻ(\alpha \cap \text{ᗡ}ʻR)\) for \(\alpha\). +When this can be done, the ultimate oscillation of the function as the +argument approaches the limit of a can be determined by taking any set +of arguments having this limit. We have</p> + +<p class="nind"><b>*232·301.</b> \(\vdash .(P\overline{R} Q)_{\text{sc}}ʻ\alpha \subset (P\overline{R} Q)_{\text{sc}}ʻQ_{*}ʻʻ(\alpha \cap \text{ᗡ}ʻR)\)</p> + +<p class="nind"><b>*232·32.</b> \(\vdash :(P\overline{R} Q)_{\text{os}}ʻQ_{*}ʻʻ(\alpha \cap \text{ᗡ}ʻR)\in 0\cup 1.\supset .(P\overline{R} Q)_{\text{os}}ʻ\alpha \in 0\cup 1\)</p> + +<p>Thus if the function has a limit as the argument approaches the limit +of \(Q_{*}ʻʻ(\alpha \cap \text{ᗡ}ʻR)\), it also has a limit as the +argument approaches the limit of \(\alpha\).</p> + +<p class="nind"><b>*232·33.</b> \[\begin{align}&\vdash :P_{*},Q_{*}\unicode{x0294f}\alpha \in \text{connex} .Rʻʻ(\alpha \cap CʻQ)\subset CʻP.\supset .\\ +&(P\overline{R} Q)_{\text{sc}}ʻ\alpha \cup (\breve{P} \overline{R}Q)_{\text{sc}}ʻ\alpha =(P\overline{R} Q)_{\text{sc}}ʻQ_{*}ʻʻ(\alpha + \cap \text{ᗡ}ʻR)\cup (\breve{P} \overline{R} Q)_{\text{sc}}ʻQ_{*}ʻʻ(\alpha \cap \text{ᗡ}ʻR)=CʻP\end{align}\]</p> + +<p class="nind">whence</p> + +<p class="nind"><b>*232·34.</b> \[\begin{align}&\vdash :\text{Hp} *232·33.(P\overline{R} Q)_{\text{os}}ʻQ_{*}ʻʻ(\alpha \cap \text{ᗡ}ʻR)=\Lambda .\supset .\\ +&(P\overline{R} Q)_{\text{sc}}ʻ\alpha =(P\overline{R} Q)_{\text{sc}}ʻQ_{*}ʻʻ(\alpha + \cap \text{ᗡ}ʻR).(\breve{P} \overline{R} Qʻ\alpha =(\breve{P} \overline{R}Q)_{\text{sc}}ʻQ*ʻʻ(\alpha \cap \text{ᗡ}ʻR)\end{align}\]</p> + +<p>We have also</p> + +<p class="nind"><b>*232·341.</b> \[\begin{align}&\vdash :P_{*}\in \text{connex} .\exists !(P\overline{R} Q)_{\text{os}}ʻ\alpha .(P\overline{R} .Q)_{\text{os}}ʻQ_{*}ʻʻ(\alpha + \cap \text{ᗡ}ʻR)\in 1.\supset .\\ +&(P\overline{R} Q)_{\text{sc}}ʻ\alpha =(P\overline{R} Q)_{\text{sc}}ʻQ_{*}ʻʻ(\alpha + \cap \text{ᗡ}ʻR).(\breve{P} \overline{R} Q)_{\text{sc}}ʻ\alpha =(\breve{P} \overline{R} Q)_{\text{sc}}ʻQ_{*}ʻʻ(\alpha \cap \text{ᗡ}ʻR)\end{align}\]</p> + +<p><span class="pagenum" id="Page_739">[Pg 739]</span></p> + +<p>Hence we arrive at the conclusion that, if \(P_{\text{po}}\) +is a series, and \(x\) is the limit of the function for the +class a \(Q_{*}ʻʻ(\alpha \cap \text{ᗡ}ʻR)\), if \(x\) is a +member of \((P\overline{R}Q)_{\text{sc}}ʻ\alpha\), it is +its maximum (<a href="#*232·352">*232·352</a>), while if \(x\) is not a member of +\((P\overline{R}Q)_{\text{sc}}ʻ\alpha\), it is its sequent (<a href="#*232·356">*232·356</a>), +assuming \((P\overline{R}Q)_{\text{sc}}ʻ\alpha \cup (\breve{P}\overline{R}Q)_{\text{sc}}ʻ\alpha = CʻP\), +which, as we saw (<a href="#*233·22">*233·22</a>), is generally the case, and +assuming also \(P \in \text{Ser}\). On the other hand, if +\((P\overline{R}Q)_{\text{sc}}ʻ\alpha\) has no maximum, \(x\) is +the minimum of \((P\overline{R}Q)_{\text{sc}}ʻ\alpha\); and if +\((P\overline{R}Q)_{\text{sc}}ʻ\alpha\) has a maximum other than \(x\), +this is \(P_1ʻx\) (<a href="#*232·357">*232·357</a><a href="#*232·358">·358</a>). This latter case is impossible unless +\(x\) has an immediate predecessor. Hence we arrive at the following +proposition:</p> + +<p class="nind"><b>*232·38.</b> \[\begin{align}\vdash : P \in \text{Ser} . Q_{*} \unicode{x0294f} &\alpha \in \text{connex} . Rʻʻ(\alpha \cap CʻQ) \subset CʻP .\\ +&(P\overline{R}Q)_{\text{os}}ʻQ_{*}ʻʻ(\alpha \cap \text{ᗡ}ʻR) \in 0 \cup (1 - \text{Cl}ʻCʻP_1) . \supset .\\ +&\overrightarrow{\text{limax}} _pʻ(P\overline{R}Q)_{\text{sc}}ʻ\alpha = \overrightarrow{\text{limax}} _pʻ(P\overline{R}Q)_{\text{sc}}ʻQ_{*}ʻʻ(\alpha + \cap \text{ᗡ}ʻR) .\\ +&\overrightarrow{\text{limin}} _pʻ(\breve{P} \overline{R}Q)_{\text{sc}}ʻ\alpha = \overrightarrow{\text{limin}} _pʻ(\breve{P} \overline{R}Q)_{\text{sc}}ʻQ_{*}ʻʻ(\alpha + \cap \text{ᗡ}ʻR)\end{align}\]</p> + +<p>Applying this to a series having Dedekindian continuity, we know +that \(P_1 = \dot{\Lambda}\), and that the \(\text{limax}\) and +\(\text{limin}\) always exist. Hence</p> + +<p class="nind"><b>*232·39.</b> \[\begin{align}\vdash \colon\ldotp P \in \text{Ser} \cap &\text{Ded} . P^{2} = P . Q_{*} \in \text{connex} . RʻʻCʻQ \subset CʻP . \supset :\\ +&(P\overline{R}Q)_{\text{os}}ʻQ_{*}ʻʻ(\alpha \cap \text{ᗡ}ʻR) \in 0 \cup 1 . \supset _\alpha .\\ +&\text{limax}_pʻ(P\overline{R}Q)_{\text{sc}}ʻ\alpha = \text{limax}_pʻ(P\overline{R}Q)_{\text{sc}}ʻQ_{*}ʻʻ(\alpha \cap \text{ᗡ}ʻR) =\\ +&\text{limin}_pʻ(\breve{P} \overline{R}Q)_{\text{sc}}ʻ\alpha = \text{limin}_pʻ(\breve{P} \overline{R}Q)_{\text{sc}}ʻQ_{*}ʻʻ(\alpha \cap \text{ᗡ}ʻR)\end{align}\]</p> + +<p>That is to say, if the value-series \(P\) has Dedekindian continuity, +and contains all values for arguments in \(CʻQ\), then, provided the +function has a definite limit for the class \(Q_{*}ʻʻ(\alpha \cap\text{ᗡ}ʻR)\), +this is its limit also for the class \(\alpha\); that is +to say, any collection of arguments having the same limit or maximum as +a given section will give the same limit for the function.</p> + +<hr class="tb"> + +<p class="nind"><b>*232·01.</b> \((P\overline{R}Q)_{\text{sc}}ʻ\alpha = P\overline{R}_{\text{sc}}(Q_{*} \unicode{x0294f} \alpha ) \quad\text{Df}\)</p> + +<p class="nind"><b>*232·02.</b> \((P\overline{R}Q)_{\text{os}}ʻ\alpha = P\overline{R}_{\text{os}}(Q_{*} \unicode{x0294f} \alpha ) \quad\text{Df}\)</p> + +<p class="nind"><b>*232·1.</b> \(\vdash . (P\overline{R}Q)_{\text{sc}}ʻ\alpha = P\overline{R}_{\text{sc}}(Q_{*} \unicode{x0294f} \alpha ) \quad[(*232·01)]\)</p> + +<p class="nind"><b>*232·101.</b> \(\vdash . (P\overline{R}Q)_{\text{os}}ʻ\alpha = P\overline{R}_{\text{os}}(Q_{*} \unicode{x0294f} \alpha ) = (P\overline{R}Q)_{\text{sc}}ʻ\alpha + \cap (\breve{P} \overline{R}Q)_{\text{sc}}ʻ\alpha \quad[(*232·02)]\)</p> + +<p class="nind"><b>*232·11.</b> \[\begin{align}\vdash \colon\ldotp x \in (P\overline{R}Q)_{\text{sc}}ʻ&\alpha . \equiv :\\ +&y \in \alpha \cap CʻQ \cap \text{ᗡ}ʻR . \supset _y . x \in P_{*}ʻʻRʻʻ(\alpha \cap \overleftarrow{Q}_*ʻy) : x \in CʻP\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *90·41·42 . *231·112 . \supset\\ +\vdash \colon\ldotp x \in (P\overline{R}Q)_{\text{sc}}ʻ\alpha . \equiv : y \in \alpha \cap CʻQ \cap \text{ᗡ}ʻR . \supset _y . x \in P_{*}ʻʻRʻʻ(\overleftarrow{{Q}_{*}} + \unicode{x0294f} \alpha ])ʻy : x \in CʻP :\\ +[*35·102] \equiv : y \in \alpha \cap CʻQ \cap \text{ᗡ}ʻR . \supset _y . x \in P_{*}ʻʻRʻʻ(\alpha \cap \overleftarrow{Q}_*ʻy) : x \in CʻP \colon\ldotp \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_740">[Pg 740]</span></p> + +<p class="nind"><b>*232·12.</b> \[\begin{align}&\vdash .(P\overline{R} Q)_{\text{sc}}ʻ\alpha =pʻP_{*}ʻʻʻRʻʻʻ(\alpha \cap )ʻʻ\overleftarrow{Q}_{*}ʻʻ(\alpha + \cap CʻQ\cap \text{ᗡ}ʻR)\cap CʻP\\ +&[*232·11]\end{align}\]</p> + +<p class="nind"><b>*232·121.</b> \(\vdash :\gamma =\alpha \cap CʻQ\cap \text{ᗡ}ʻR.\supset .(P\overline{R} Q)_{\text{sc}}ʻ\alpha =pʻP_{*}ʻʻʻRʻʻʻ(\gamma \cap )ʻʻ\overleftarrow{Q}_{*}ʻʻ\gamma + \cap CʻP\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*90·13. &\supset \vdash .\alpha \cap \overleftarrow{Q}_{*}ʻy=\alpha \cap CʻQ\cap \overleftarrow{Q}_{*}ʻy.\\ +[*37·26] &\supset \vdash .Rʻʻ(\alpha \cap \overleftarrow{Q}_{*}ʻy)=Rʻʻ(\alpha \cap CʻQ\cap \text{ᗡ}ʻR\cap \overleftarrow{Q}_{*}ʻy) &\qquad \text{(1)}\\ +\vdash .(1).*232·11.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*232·13.</b> \[\begin{align}&\vdash :\alpha \cap CʻQ\cap \text{ᗡ}ʻR=\beta \cap CʻQ\cap \text{ᗡ}ʻR.\supset .(P\overline{R} Q)_{\text{sc}}ʻ\alpha + =(P\overline{R} Q)_{\text{sc}}ʻ\beta\\ +&[*232·121]\end{align}\]</p> + +<p class="nind"><b><a id="*232·131">*232·131</a>.</b> \(\vdash .(P\overline{R} Q)_{\text{sc}}ʻ\alpha =(P\overline{R} Q)_{\text{sc}}ʻ(\alpha \cap CʻQ\cap \text{ᗡ}ʻR) \quad[*232·13]\)</p> + +<p>From the above propositions it follows that the values of +\((P\overline{R} Q)_{\text{sc}}ʻ\alpha\), \((\breve{P} \overline{R}Q)_{\text{sc}}ʻ\alpha\), +and \((P\overline{R} Q)_{\text{os}}ʻ\alpha\) depend only upon +\(\alpha\cap CʻQ\cap \text{ᗡ}ʻR\); thus if \(\alpha\) is not contained +in \(CʻQ\cap \text{ᗡ}ʻR\), the part not contained in \(CʻQ\cap\text{ᗡ}ʻR\) +is irrelevant.</p> + +<p class="nind"><b><a id="*232·14">*232·14</a>.</b> \[\begin{align}&\vdash :Q\in \text{trans} \cap\text{connex} .\alpha \cap CʻQ{\sim}\in 1.\supset .(P\overline{R} Q)_{\text{sc}}ʻ\alpha =P\overline{R} _{\text{sc}}(Q\unicode{x0294f}\alpha + )\\ +&[*232·1.*202·54·541]\end{align}\]</p> + +<p class="nind"><b><a id="*232·15">*232·15</a>.</b> \[\begin{align}&\vdash :\alpha \cap CʻQ\cap \text{ᗡ}ʻR=\Lambda .\supset .(P\overline{R} Q)_{\text{sc}}ʻ\alpha =(\breve{P} \overline{R} Q)_{\text{sc}}ʻ\alpha + =(P\overline{R} Q)_{\text{os}}ʻ\alpha =CʻP\\ +&[*232·12·101.*37·29.*40·2]\end{align}\]</p> + +<p class="nind"><b>*232·151.</b> \[\begin{align}&\vdash :\dot{\exists} !P.(P\overline{R} Q)_{\text{os}}ʻ\alpha =\Lambda .\supset .\exists !\alpha \cap CʻQ\cap \text{ᗡ}ʻR\\ +&[*232·15.\text{Transp}.*33·24]\end{align}\]</p> + +<p class="nind"><b>*232·2.</b> \(\vdash :CʻQ\cap \text{ᗡ}ʻR\subset \alpha .\supset .(P\overline{R} Q)_{\text{sc}}ʻ\alpha =P\overline{R} _{\text{sc}}Q\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*22·621.*232·11.\supset \\ +\vdash \colon\colon\text{Hp}.\supset \colon\ldotp x\in (P\overline{R} Q)_{\text{sc}}ʻ\alpha .&\equiv :y\in CʻQ\cap \text{ᗡ}ʻR.\supset _{y}.x\in + P_{*}ʻʻRʻʻ\overleftarrow{Q}_{*}ʻy:\\ +[*231·112] &\equiv :x\in P\overline{R} _{\text{sc}}Q\colon\colon\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*232·21.</b> \[\begin{align}&\vdash :P_{*},Q_{*}\unicode{x0294f}\alpha \in \text{connex} .\alpha \cap CʻQ\cap \text{ᗡ}ʻR\subset Q_{*}ʻʻ(\alpha + \cap \breve{R} ʻʻCʻP).\supset .\\ +&CʻP=(P\overline{R} Q)_{\text{sc}}ʻ\alpha \cup (\breve{P} \overline{R} Q)_{\text{sc}}ʻ\alpha \\ +&\left[*231·21 \frac{Q_{*}\unicode{x0294f}\alpha}{Q}\right]\end{align}\]</p> + +<p class="nind"><b>*232·22.</b> \[\begin{align}\vdash :P_{*},Q_{*}\unicode{x0294f}\alpha \in \text{connex} .Rʻʻ&(\alpha \cap CʻQ)\subset CʻP.\supset .\\ +&CʻP=(P\overline{R} Q)_{\text{sc}}ʻ\alpha \cup (\breve{P} \overline{R} Q)_{\text{sc}}ʻ\alpha \quad[*231·22]\end{align}\]</p> + +<p class="nind"><b>*232·23.</b> \(\vdash :y\in CʻQ\cap \text{ᗡ}ʻR.\supset .(P\overline{R} Q)_{\text{sc}}ʻ\iota ʻy=P_{*}ʻʻ\overrightarrow{R}ʻy\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*232·11.*13·191.\supset \\ +\vdash \colon\ldotp \text{Hp}.\supset :x\in (P\overline{R} Q)_{\text{sc}}ʻ\iota ʻy.\equiv .x\in P_{*}ʻʻRʻʻ(\iota ʻy\cap \overleftarrow{Q}]_{*}ʻy)\colon\ldotp + \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_741">[Pg 741]</span></p> + +<p class="nind"><b>*232·24.</b> \[\begin{align}\vdash :Q\in \text{trans}\cap \text{connex} .\text{E}!\text{max}_{Q}ʻ&(\alpha \cap \text{ᗡ}ʻR).\supset .\\ +&(P\overline{R} Q)_{\text{sc}}ʻ\alpha =P_{*}ʻʻ\overrightarrow{R}ʻ\text{max}_{Q}ʻ(\alpha \cap \text{ᗡ}ʻR)\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*232·14.&\supset \vdash :\text{Hp}.\alpha \cap CʻQ\cap \text{ᗡ}ʻR{\sim}\in 1.\supset .\\ +(P\overline{R} Q)_{\text{sc}}ʻ\alpha &=P\overline{R} _{\text{sc}}\{Q\unicode{x0294f}(\alpha \cap \text{ᗡ}ʻR)\}\\ +[*231·4.*205·9]&=P_{*}ʻʻ\overrightarrow{R}ʻ\text{max}_{Q}ʻ(\alpha \cap \text{ᗡ}ʻR) &\qquad \text{(1)}\\ +\vdash .*205·17.*232·23·131.\supset \\ +\vdash :\alpha \cap CʻQ\cap \text{ᗡ}ʻR\in 1.\supset .(P\overline{R} Q)_{\text{sc}}ʻ\alpha &=P_{*}ʻʻ\overrightarrow{R}ʻ\text{max}_{Q}ʻ(\alpha + \cap \text{ᗡ}ʻR) &\qquad \text{(2)}\\ +\vdash .(1).(2).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*232·3.</b> \(\vdash :\alpha \subset \text{ᗡ}ʻR.\supset .(P\overline{R} Q)_{\text{sc}}ʻ\alpha \subset (P\overline{R} Q)_{\text{sc}}ʻQ_{*}ʻʻ\alpha\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*96·3.&\supset \vdash :y\in Q_{*}ʻʻ\alpha .\supset .(\exists z).z\in \alpha \cap CʻQ.\overleftarrow{Q}_{*}ʻz\subset \overleftarrow{Q}_{*}ʻy.\\ +[\text{Fact}.*37·2]&\supset .(\exists z).z\in \alpha \cap CʻQ.P_{*}ʻʻRʻʻ(\alpha \cap \overleftarrow{Q}_{*}ʻz)\subset P_{*}ʻʻRʻʻ(\alpha \cap \overleftarrow{Q}_{*}ʻy) + &\qquad \text{(1)}\\ +\vdash .(1).*232·11.\supset \vdash \colon\ldotp \text{Hp}.x\in (P\overline{R} Q)_{\text{sc}}ʻ\alpha .\supset :y\in Q_{*}ʻʻ\alpha .&\supset .x\in P_{*}ʻʻRʻʻ(\alpha + \cap \overleftarrow{Q}_{*}ʻy).\\ +[*90·33] &\supset .x\in P_{*}ʻʻRʻʻ(Q_{*}ʻʻ\alpha \cap \overleftarrow{Q}_{*}ʻy):\\ +[*232·11] &\supset :x\in (P\overline{R} Q)_{\text{sc}}ʻQ_{*}ʻʻ\alpha \colon\ldotp \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*232·301.</b> \(\vdash .(P\overline{R} Q)_{\text{sc}}ʻ\alpha \subset (P\overline{R} Q)_{\text{sc}}ʻQ_{*}ʻʻ(\alpha \cap \text{ᗡ}ʻR)\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*232·13.\supset \vdash .(P\overline{R} Q)_{\text{sc}}ʻ\alpha &=(P\overline{R} Q)_{\text{sc}}ʻ(\alpha \cap \text{ᗡ}ʻR)\\ +[*232·3] & \subset (P\overline{R} Q)_{\text{sc}}ʻQ_{*}ʻʻ(\alpha \cap \text{ᗡ}ʻR).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*232·31.</b> \(\vdash .(P\overline{R} Q)_{\text{os}}ʻ\alpha \subset (P\overline{R} Q)_{\text{os}}ʻQ_{*}ʻʻ(\alpha \cap \text{ᗡ}ʻR)\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*232·301 \frac{\breve{P}}{P}.\supset \vdash .(\breve{P} \overline{R} Q)_{\text{sc}}ʻ\alpha \subset (\breve{P} \overline{R} Q)_{\text{sc}}ʻQ_{*}ʻʻ(\alpha + \cap \text{ᗡ}ʻR) &\qquad \text{(1)}\\ +\vdash .*232·301.(1).*232·101.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*232·32.</b> \(\vdash :(P\overline{R} Q)_{\text{os}}ʻQ_{*}ʻʻ(\alpha \cap \text{ᗡ}ʻR)\in 0\cup 1.\supset .(P\overline{R} Q)_{\text{os}}ʻ\alpha + \in 0\cup 1 \quad[*232·31]\)</p> + +<p class="nind"><b>*232·33.</b> \[\begin{align}&\vdash :P_{*},Q_{*}\unicode{x0294f}\alpha \in \text{connex} .Rʻʻ(\alpha \cap CʻQ)\subset CʻP.\supset .\\ +&(P\overline{R} Q)_{\text{sc}}ʻ\alpha \cup (\breve{P} \overline{R} Q)_{\text{sc}}ʻ\alpha =(P\overline{R} Q)_{\text{sc}}ʻQ_{*}ʻʻ(\alpha + \cap \text{ᗡ}ʻR)\cup (\breve{P} \overline{R} Q)_{\text{sc}}ʻQ_{*}ʻʻ(\alpha \cap \text{ᗡ}ʻR)=CʻP\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*232·22·301.\supset \\ +\vdash :\text{Hp}.\supset .CʻP\subset (P\overline{R} Q)_{\text{sc}}ʻQ_{*}ʻʻ(\alpha \cap \text{ᗡ}ʻR)\cup (\breve{P} \overline{R} Q)_{\text{sc}}ʻQ_{*}ʻʻ(\alpha + \cap \text{ᗡ}ʻR) &\qquad \text{(1)}\\ +\vdash .(1).*231·131.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*232·34">*232·34</a>.</b> \[\begin{align}&\vdash :\text{Hp}*232·33.(P\overline{R} Q)_{\text{os}}ʻQ_{*}ʻʻ(\alpha \cap \text{ᗡ}ʻR)=\Lambda .\supset .\\ +&(P\overline{R} Q)_{\text{sc}}ʻ\alpha =(P\overline{R} Q)_{\text{sc}}ʻQ_{*}ʻʻ(\alpha \cap \text{ᗡ}ʻR).(\breve{P} \overline{R} Q)_{\text{sc}}ʻ\alpha + =(\breve{P} \overline{R} Q)_{\text{sc}}ʻQ_{*}ʻʻ(\alpha \cap \text{ᗡ}ʻR)\\ +&[*232·33·301.*24·482]\end{align}\]</p> + +<p><span class="pagenum" id="Page_742">[Pg 742]</span></p> + +<p class="nind"><b>*232·341.</b> \[\begin{align}&\vdash : P_{*} \in \text{connex} . \exists ! (P\overline{R} Q)_{\text{os}}ʻ\alpha .(P\overline{R} Q)_{\text{os}}ʻQ_{*}ʻʻ(\alpha + \cap \text{ᗡ}ʻR) \in 1 . \supset .\\ +&(P\overline{R} Q)_{\text{sc}}ʻ\alpha = (P\overline{R} Q)_{\text{sc}}ʻQ_{*}ʻʻ(\alpha \cap \text{ᗡ}ʻR) . (\breve{P} RQ)_{\text{sc}}ʻ\alpha + = . (\breve{P} RQ)_{\text{sc}}ʻQ_{*}ʻʻ(\alpha \cap \text{ᗡ}ʻR)\\ +&[*231·192 . *232·31. *60·38]\end{align}\]</p> + +<p class="nind"><b>*232·35.</b> \[\begin{align}&\vdash : P_{*} \in \text{connex} . (P\overline{R} Q)_{\text{os}}ʻQ_{*}ʻʻ(\alpha \cap \text{ᗡ}ʻR) = {℩}ʻx . \supset .\\ +&(P\overline{R} Q)_{\text{sc}}ʻ\alpha \subset \overrightarrow{P}_{*}ʻx . (\breve{P} \overline{R} Q)_{\text{sc}}ʻ\alpha \subset \overleftarrow{P}_{*}ʻx\\ +&[*232·301.*231·191]\end{align}\]</p> + +<p class="nind"><b>*232·351.</b> \(\vdash : \text{Hp} *232·35 . x \in (P\overline{R} Q)_{\text{sc}}ʻ\alpha . \supset . (P\overline{R} Q)_{\text{sc}}ʻ\alpha + = \overrightarrow{P} _{*}ʻx\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *231*13. \supset \vdash : \text{Hp} . \supset . \overrightarrow{P}_{*}ʻx \supset (P\overline{R} Q)_{\text{sc}}ʻ\alpha &\qquad \text{(1)}\\ +\vdash . (1). *232·35. \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*232·352">*232·352</a>.</b> \[\begin{align}&\vdash : \text{Hp} *232·351 . P_{\text{po}} \,\unicode{x2abd}\, J . \supset . x = \text{max}_{P}ʻ(P\overline{R} Q)_{\text{sc}}ʻ\alpha \\ +&[*211·8 . *205·197 . *232·351]\end{align}\]</p> + +<p class="nind"><b>*232·353.</b> \[\begin{align}\vdash : \text{Hp} *232·35. (P\overline{R} Q)_{\text{sc}}ʻ\alpha \cup (\breve{P} \overline{R} Q)_{\text{sc}}ʻ\alpha + = CʻP . &x{\sim}\in (P\overline{R} Q)_{\text{sc}}ʻ\alpha . \supset .\\ +&(\breve{P}\overline{R} Q)_{\text{sc}}ʻ\alpha = \overleftarrow{P}_{*}ʻx\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *231·13. \supset \vdash : \text{Hp} . &\supset . x \in (\breve{P} \overline{R} Q)_{\text{sc}}ʻ\alpha .\\ +[*232·351] &\supset . (\breve{P} \overline{R} Q)_{\text{sc}}ʻ\alpha = \overleftarrow{P}_{*}ʻx : \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*232·354.</b> \(\vdash : \text{Hp} *232·353 . P_{\text{po}} \,\unicode{x2abd}\, J. \supset . x = \text{min}_{P}ʻ(\breve{P} \overline{R} Q)_{\text{sc}}ʻ\alpha + \quad\left[*232·352 \frac{\breve{P}}{P}\right]\)</p> + +<p class="nind"><b>*232·355.</b> \(\vdash : \text{Hp} *232·353. \supset . (P\overline{R} Q)_{\text{sc}}ʻ\alpha = \overrightarrow{P}_{\text{po}}ʻx\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *232·35. \supset \vdash : \text{Hp} . \supset . (P\overline{R} Q)_{\text{sc}}ʻ\alpha &\subset \overrightarrow{P}_{*}ʻx - \iota ʻx\\ +[*91·542] &\subset \overrightarrow{P}_{\text{po}}ʻx &\qquad \text{(1)}\\ +\vdash . *232·353. \supset \vdash : \text{Hp}. &\supset . CʻP -\overleftarrow{P}_{*}ʻx \subset CʻP - (\breve{P} \overline{R} Q)_{\text{sc}} ʻ\alpha .\\ +[*202·101.\text{Hp}] &\supset . \overrightarrow{P}_{\text{po}}ʻx \subset (P\overline{R} Q)_{\text{sc}}ʻ\alpha &\qquad \text{(2)}\\ +\vdash . (1). (2). \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*232·356">*232·356</a>.</b> \[\begin{align}&\vdash \colon\ldotp P \in \text{Ser}. (P\overline{R} Q)_{\text{os}}ʻQ_{*}ʻʻ(\alpha \cap \text{ᗡ}ʻR) = \iota ʻx.\\ +&(P\overline{R} Q)_{\text{sc}}ʻ\alpha \cup (\breve{P}\overline{R} Q)_{\text{sc}}ʻ\alpha =CʻP . \supset : x{\sim}\in (P\overline{R} Q)_{\text{sc}}ʻ\alpha + . \supset . x = \text{seq}_{P}ʻ(P\overline{R} Q)_{\text{sc}}ʻ\alpha \\ +&[*206·172 . *231·13 . *232·355]\end{align}\]</p> + +<p class="nind"><b><a id="*232·357">*232·357</a>.</b> \(\vdash : \text{Hp}*232·35 . P_{\text{po}} \,\unicode{x2abd}\, J. {\sim}\text{E}!\text{max}_{P}ʻ(P\overline{R} Q)_{\text{sc}}ʻ\alpha + . \supset .x= \text{min}_{P}ʻ(\breve{P} \overline{R} Q)_{\text{sc}}ʻ\alpha\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *232·352. \text{Transp} . \supset \vdash : \text{Hp}. &\supset . x{\sim}\in (P\overline{R} Q)_{\text{sc}}ʻ\alpha .\\ +[*232·354] &\supset . x = \text{min}_{P}ʻ(\breve{P} \overline{R} Q)_{\text{sc}}\alpha : \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_743">[Pg 743]</span></p> + +<p class="nind"><b><a id="*232·358">*232·358</a>.</b> \[\begin{align}&\vdash :\text{Hp}*232·35.P_{\text{po}}\,\unicode{x2abd}\, J.(P\overline{R} Q)_{\text{sc}}ʻ\alpha \cup (\breve{P} \overline{R} Q)_{\text{sc}}ʻ\alpha + =CʻP.\\ +&\text{E}!\text{max}_{P}ʻ(P\overline{R} Q)_{\text{sc}}ʻ\alpha .\text{max}_{P}ʻ(P\overline{R} Q)_{\text{sc}}ʻ\alpha \neq x.\supset .\text{max}_{P}ʻ(P\overline{R} + Q)_{\text{sc}}ʻ\alpha P_{1}x\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*232·352.\text{Transp}.\supset \vdash :\text{Hp}.&\supset .x{\sim}\in (P\overline{R} Q)_{\text{sc}}ʻ\alpha .\\ +[*232·356] &\supset.x=\text{seq}_{P}ʻ(P\overline{R} Q)_{\text{sc}}ʻ\alpha .\\ +[*206·5] & \supset .\text{max}_{P}ʻ(P\overline{R} Q)_{\text{sc}}ʻ\alpha P_{1}x:\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*232·36.</b> \[\begin{align}&\vdash \colon\ldotp P\in \text{Ser}.(P\overline{R} Q)_{\text{os}}ʻQ_{*}ʻʻ(\alpha \cap \text{ᗡ}ʻR)=\iota ʻx.\\ +&(P\overline{R} Q)_{\text{sc}}ʻ\alpha \cup (\breve{P} \overline{R} Q)_{\text{sc}}ʻ\alpha =CʻP.\supset :\\ +&x\in (P\overline{R} Q)_{os}ʻ\alpha .\supset .x=\text{max}_{P}ʻ(P\overline{R} Q)_{\text{sc}}ʻ\alpha =\text{min}_{P}ʻ(\breve{P} \overline{R} Q)_{\text{sc}}ʻ\alpha + :\\ +&x\in (P\overline{R} Q)_{\text{sc}}ʻ\alpha -(\breve{P} \overline{R} Q)_{\text{sc}}ʻ\alpha .\supset .x=\text{max}_{P}ʻ(\breve{P} \overline{R} Q)_{\text{sc}}ʻ\alpha + =\text{prec}_{P}ʻ(P\overline{R} Q)_{\text{sc}}ʻ\alpha :\\ +&x\in (\breve{P} \overline{R} Q)_{\text{sc}}ʻ\alpha -(P\overline{R} Q)_{\text{sc}}ʻ\alpha .\supset .x=\text{seq}_{P}ʻ(P\overline{R} Q)_{\text{sc}}ʻ\alpha + =\text{min}_{P}ʻ(\breve{P} \overline{R} Q)_{\text{sc}}ʻ\alpha \\ +&[*232·352·354·356]\end{align}\]</p> + +<p class="nind"><b>*232·361.</b> \(\vdash :\text{Hp}*232·36.x{\sim}\in \text{ᗡ}ʻP_{1}.\supset .x=\text{limax}_Pʻ(P\overline{R} Q)_{\text{sc}}ʻ\alpha\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*232·358.\text{Transp}.\supset \\ +\vdash \colon\ldotp \text{Hp}.\supset :\text{E}!\text{max}_{P}ʻ(P\overline{R} Q)_{\text{sc}}ʻ\alpha .&\supset .\text{max}_{P}ʻ(P\overline{R} Q)_{\text{sc}}ʻ\alpha + =x &\qquad \text{(1)}\\ +\vdash .*232·352.\text{Transp}.\supset \\ +\vdash \colon\ldotp \text{Hp}.\supset :{\sim}\text{E}!\text{max}_{P}ʻ(P\overline{R} Q)_{\text{sc}}ʻ\alpha .&\supset .x{\sim}\in (P\overline{R} Q)_{\text{sc}}ʻ\alpha .\\ +[*232·356] &\supset .x=\text{seq}_{P}ʻ(P\overline{R} Q)ʻ\alpha &\qquad \text{(2)}\\ +\vdash .(1).(2).*207·46.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*232·37.</b> \[\begin{align}&\vdash :P\in \text{Ser}.(P\overline{R} Q)_{\text{os}}ʻQ_{*}ʻʻ(\alpha \cap \text{ᗡ}ʻR)\in 1-\text{Cl}ʻ\text{ᗡ}ʻP_{1}.\\ +&(P\overline{R} Q)_{\text{sc}}ʻ\alpha \cup (\breve{P} \overline{R} Q)_{\text{sc}}ʻ\alpha =CʻP.\supset .\\ +&\text{limax}_Pʻ(P\overline{R} Q)_{\text{sc}}ʻ\alpha =\text{max}_{P}ʻ(P\overline{R} Q)_{\text{sc}}ʻQ_{*}ʻʻ(\alpha \cap \text{ᗡ}ʻR)\\ +&= \breve{\iota} ʻ(P\overline{R} Q)_{\text{os}}ʻQ_{*}ʻʻ(\alpha \cap \text{ᗡ}ʻR)\\ +&[*232·361.*231·193]\end{align}\]</p> + +<p class="nind"><b>*232·38.</b> \[\begin{align}&\vdash :P\in \text{Ser}.Q_{*}\unicode{x0294f}\alpha \in \text{connex} .Rʻʻ(\alpha \cap CʻQ)\subset CʻP.\\ +&(P\overline{R} Q)_{\text{os}}ʻQ_{*}ʻʻ(\alpha \cap \text{ᗡ}ʻR)\in 0\cup (1-\text{Cl}ʻCʻP_{1}).\supset .\\ +&\overrightarrow{\text{limax}} _{P}ʻ(P\overline{R} Q)_{\text{sc}}ʻ\alpha =\overrightarrow{\text{limax}} _{P}ʻ(P\overline{R} Q)_{\text{sc}}ʻQ_{*}ʻʻ(\alpha + \cap \text{ᗡ}ʻR).\\ +&\overrightarrow{\text{limin}} _{P}ʻ(\breve{P} \overline{R} Q)_{\text{sc}}ʻ\alpha =\overrightarrow{\text{limin}} _{P}ʻ(\breve{P} + \overline{R} Q)_{\text{sc}}ʻQ_{*}ʻʻ(\alpha \cap \text{ᗡ}ʻR)\\ +&[*232·33·34·37]\end{align}\]</p> + +<p><span class="pagenum" id="Page_744">[Pg 744]</span></p> + +<p class="nind"><b><a id="*232·39">*232·39</a>.</b> \[\begin{align}\vdash \colon\ldotp P\in \text{Ser}\cap &\text{Ded}.P^{2}=P.Q_{*}\in \text{connex} .RʻʻCʻQ\subset CʻP.\supset :\\ +&(P\overline{R} Q)_{\text{os}}ʻQ_{*}ʻʻ(\alpha \cap \text{ᗡ}ʻR)\in 0\cup 1.\supset _{\alpha }.\\ +&\text{limax}_Pʻ(P\overline{R} Q)_{\text{sc}}ʻ\alpha =\text{limax}_Pʻ(P\overline{R} Q)_{\text{sc}}ʻQ_{*}ʻʻ(\alpha \cap \text{ᗡ}ʻR)\\ +&=\text{limin}_Pʻ(\breve{P} \overline{R} Q)_{\text{sc}}ʻ\alpha =\text{limin}_Pʻ(\breve{P} \overline{R} Q)_{\text{sc}}ʻQ_{*}ʻʻ(\alpha \cap \text{ᗡ}ʻR)\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*201·63.*232·38.&\supset \vdash :\text{Hp}.(P\overline{R} Q)_{\text{os}}ʻQ_{*}ʻʻ(\alpha \cap \text{ᗡ}ʻR)\in 0\cup 1.\supset .\\ +&\text{limax}_Pʻ(P\overline{R} Q)_{\text{sc}}ʻ\alpha =\text{limax}_Pʻ(P\overline{R} Q)_{\text{sc}}ʻQ_{*}ʻʻ(\alpha \cap \text{ᗡ}ʻR).\\ +&\text{limin}_Pʻ(\breve{P} \overline{R} Q)_{\text{sc}}ʻ\alpha =\text{limin}_Pʻ(\breve{P} \overline{R} Q)_{\text{sc}}ʻQ_{*}ʻʻ(\alpha + \cap \text{ᗡ}ʻR) &\qquad \text{(1)}\\ +\vdash .*231·193.&\supset \vdash :\text{Hp}(1).(P\overline{R} Q)_{\text{os}}ʻQ_{*}ʻʻ(\alpha \cap \text{ᗡ}ʻR)\in 1.\supset .\\ +&\text{limax}_Pʻ(P\overline{R} Q)_{\text{sc}}ʻQ_{*}ʻʻ(\alpha \cap \text{ᗡ}ʻR)=\text{limin}_Pʻ(\breve{P} \overline{R} Q)_{\text{sc}}ʻQ_{*}ʻʻ(\alpha + \cap \text{ᗡ}ʻR) &\qquad \text{(2)}\\ +\vdash .*214·42.*232·33.&\supset \vdash :\text{Hp}(1).(P\overline{R} Q)_{\text{os}}ʻQ_{*}ʻʻ(\alpha \cap \text{ᗡ}ʻR)=\Lambda .\supset .\\ +&\text{limax}_Pʻ(P\overline{R} Q)_{\text{sc}}ʻQ_{*}ʻʻ(\alpha \cap \text{ᗡ}ʻR)=\text{limin}_Pʻ(\breve{P} \overline{R} Q)_{\text{sc}}ʻQ_{*}ʻʻ(\alpha + \cap \text{ᗡ}ʻR) &\qquad \text{(3)}\\ +\vdash .(1).(2).(3).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*232·5.</b> \[\begin{align}&\vdash .(P\overline{R} Q)_{\text{sc}}ʻ\overrightarrow{Q}ʻx=CʻP\cap \hat{y} \{z\in \overrightarrow{Q}ʻx\cap \text{ᗡ}ʻR.\supset _{z}.y\in P_{*}ʻʻRʻʻ(\overrightarrow{Q}ʻx\cap + \overleftarrow{Q}_{*}ʻz)\}\\ +&[*232·11]\end{align}\]</p> + +<p class="nind"><b>*232·51.</b> \[\begin{align}\vdash :Q\in \text{trans}\cap &\text{connex} .\text{E}!\text{max}_{Q}ʻ(\overrightarrow{Q}ʻx\cap \text{ᗡ}ʻR).\supset .\\ +&(P\overline{R} Q)_{\text{sc}}ʻ\overrightarrow{Q}ʻx=P_{*}ʻʻ\overrightarrow{R}ʻ\text{max}_{Q}ʻ(\overrightarrow{Q}ʻx\cap \text{ᗡ}ʻR) \quad[*232·24]\end{align}\]</p> + +<p class="nind"><b>*232·511.</b> \[\begin{align}\vdash :Q\in \text{trans}\cap &\text{connex} .\text{E}!Rʻ\text{max}_{Q}ʻ(\overrightarrow{Q}ʻx\cap \text{ᗡ}ʻR).\supset .\\ +&(P\overline{R} Q)_{\text{sc}}ʻ\overrightarrow{Q}ʻx=\overrightarrow{P}_{*}ʻRʻ\text{max}_{Q}ʻ(\overrightarrow{Q}ʻx\cap \text{ᗡ}ʻR) \quad[*232·51]\end{align}\]</p> + +<p class="nind"><b>*232·52.</b> \[\begin{align}\vdash :Q\in \text{connex} .yQx.\overrightarrow{Q}ʻx\cap &(\overleftarrow{Q}ʻy\cup {℩}ʻy)\cap \text{ᗡ}ʻR=\Lambda .\supset .\\ +&(P\overline{R} Q)_{\text{sc}}ʻ\overrightarrow{Q}ʻx=(P\overline{R} Q)_{\text{sc}}ʻ\overrightarrow{Q}ʻy \quad[*232·13]\end{align}\]</p> + +<p class="nind"><b>*232·53.</b> \(\vdash :Q\in \text{connex} .z\in \overrightarrow{Q}ʻx\cap \text{ᗡ}ʻR.\supset .(P\overline{R} Q)_{\text{sc}}ʻ\overrightarrow{Q}ʻx=(P\overline{R} Q)_{\text{sc}}ʻ(\overrightarrow{Q}ʻx\cap + \overleftarrow{Q}_{*}ʻz)\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*232·5.*96·3.&\supset \vdash \colon\ldotp \text{Hp}.y\in (P\overline{R} Q)_{\text{sc}}ʻ\overrightarrow{Q}ʻx.\supset :\\ +&u\in \overrightarrow{Q}ʻx\cap \overleftarrow{Q}_{*}ʻz\cap \text{ᗡ}ʻR.\supset _{u}.y\in P_{*}ʻʻRʻʻ(\overrightarrow{Q}ʻx\cap + \overleftarrow{Q}_{*}ʻu).\overleftarrow{Q}_{*}ʻu\subset \overleftarrow{Q}_{*}ʻz:\\ +[*22·621.*232·11]&\supset :y\in (P\overline{R} Q)_{\text{sc}}ʻ(\overrightarrow{Q}ʻx\cap \overleftarrow{Q}_{*}ʻz) &\qquad \text{(1)}\\ +\vdash .*232·11.*37·2.&\supset \vdash \colon\ldotp \text{Hp}.y\in (P\overline{R} Q)_{\text{sc}}ʻ(\overrightarrow{Q}ʻx\cap \overleftarrow{Q}_{*}ʻz).\supset :\\ +&u\in \overrightarrow{Q}ʻx\cap \overleftarrow{Q}_{*}ʻz\cap \text{ᗡ}ʻR.\supset _{u}.y\in P_{*}ʻʻRʻʻ(\overrightarrow{Q}ʻx\cap \overleftarrow{Q}_{*}ʻu) + &\qquad \text{(2)}\\ +[*96·3]&\supset :u\in \overrightarrow{Q}ʻx\cap \overrightarrow{Q}_{*}ʻz\cap \text{ᗡ}ʻR.\supset _{u}.y\in P_{*}ʻʻRʻʻ(\overrightarrow{Q}ʻx\cap \overleftarrow{Q}_{*}ʻu) + &\qquad \text{(3)}\\ +\vdash .(2).(3).\supset \vdash \colon\ldotp \text{Hp}(2).&\supset :u\in \overrightarrow{Q}ʻx\cap \text{ᗡ}ʻR.\supset _{u}.y\in P_{*}ʻʻRʻʻ(\overrightarrow{Q}ʻx\cap + \overleftarrow{Q}_{*}ʻu):\\ +[*232·5] &\supset :y\in (P\overline{R} Q)_{\text{sc}}ʻ\overrightarrow{Q}ʻx &\qquad \text{(4)}\\ +\vdash .(1).(4).\supset \vdash .\text{Prop} +\end{array} +\]</p> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_745">[Pg 745]</span></p> +<h2 class="nobreak" id="*233"> +*233. ON THE LIMITS OF FUNCTIONS.</h2> +</div> + + +<p class="nind"><i>Summary of</i> *233.</p> + +<p>There are four limits of a function as the argument approaches some +term \(a\) in the argument-series, namely the upper and lower limits +of the ultimate oscillation for approaches from below and above +respectively. If the ultimate oscillation for approaches to \(a\) +from below reduces to a single term, <i>i.e.</i> if \((P\overline{R}Q)_{\text{os}}ʻ\overrightarrow{Q}ʻa \in 1\), +that one term is <i>the</i> limit of the function for approaches to +\(a\) from below. If this one term is also the ultimate oscillation +for approaches from above, we may call it simply the limit of the +function for the argument \(a\). This may or may not (when it exists) +be equal to the value for the argument \(a\). It is characteristic +of <i>continuous</i> functions that <i>the</i> limit exists for +every argument, and is always equal to the value for that argument. +Continuous functions will be considered in <a href="#*234">*234</a>.</p> + +<p>The upper limit or maximum of the ultimate oscillation as the argument +approaches \(a\) is the upper limit or maximum of the ultimate section. +Hence if we put +\[ +R(PQ)ʻa =\text{limax}_Pʻ(P\overline{R} Q)_{\text{sc}}ʻ\overrightarrow{Q}ʻa \quad\text{Df}, +\] +the four limits of the function as the argument approaches \(a\) will be +\[ +R(PQ)ʻa,\quad R(\breve{P} Q)ʻa,\quad R(P\breve{Q} )ʻa,\quad R(\breve{P} \breve{Q} )ʻa . +\] +It will be seen that \(R(PQ)ʻa\) is a function of +\(\overrightarrow{Q}ʻa\). It may happen that, if we put \(\alpha\) in +place of \(\overrightarrow{Q}ʻa\), the function will have a definite +limit as the argument increases in \(\alpha\), although \(\alpha\) has +no limit or maximum. Thus if, for example, \(Q\) consists of the series +of rationals, and \(P\) of the series of real numbers, if \(\alpha\) +is a class of rationals not having a rational limit, we may regard +the limit of the function (if it exists), as the argument increases +in \(\alpha\), as the value of the function for the irrational limit +of \(\alpha\). In this way we can extend the domain of definition of a +function. In order to be able to deal with the cases in which +\(\alpha\) has no limit, we put +\[ +(P\overline{R} Q)_{\text{lmx}}ʻ\alpha =\text{limax}_Pʻ(P\overline{R} Q)_{\text{sc}}ʻ\alpha \quad\text{Df}. +\] +If \(P\) is a Dedekindian series, \((P\overline{R}Q)_{\text{lmx}}ʻ\alpha\) +always exists. If we take \(\alpha\) to be any segment of \(Q\), we +thus get a new function, derived from \(R\), but having segments of +\(Q\) instead of members of \(CʻQ\) as its arguments. Thus if \(R\) +had<span class="pagenum" id="Page_746">[Pg 746]</span> rationale for its arguments, this new function will have real +numbers for its arguments. (Real numbers may be regarded as segments of +the series of rationals.)</p> + +<p>The function \(R(PQ)ʻ\alpha\) is a particular case of the above; thus +we take as our definition +\[ +R(PQ)ʻ\alpha =(P\overline{R} Q)_{\text{lmx}}ʻ\overrightarrow{Q}ʻ\alpha \quad\text{Df}, +\] +or, what comes to the same thing, +\[ +R(PQ)=(P\overline{R} Q)_{\text{lmx}}\mid \overrightarrow{Q} \quad\text{Df}. +\]</p> + +<p>The following propositions of this number are important:</p> + +<p class="nind"><b>*233·15.</b> \[\begin{align}\vdash \colon\ldotp P\in &\text{Ser}\cap \text{Ded}.(P\overline{R} Q)_{\text{sc}}ʻ\alpha \cup (\breve{P} \overline{R} Q)_{\text{sc}}ʻ\alpha + =CʻP.(P\overline{R} Q)_{\text{os}}ʻ\alpha =\Lambda .\supset :\\ +&(P\overline{R} Q)_{\text{lmx}}ʻ\alpha =(\breve{P} \overline{R} Q)_{\text{lmx}}ʻ\alpha .\lor.\{(P\overline{R} Q)_{\text{lmx}}ʻ\alpha \}P_{1}\{(\breve{P} + \overline{R} Q)_{\text{lmx}}ʻ\alpha\}\end{align}\]</p> + +<p class="nind"><b>*233·16.</b> \[\begin{align}\vdash \colon\ldotp P\in \text{Ser}\cap \text{Ded}.&P^{2}=P.Q_{*}\in \text{connex} .RʻʻCʻQ\subset CʻP.\supset :\\ +&(P\overline{R} Q)_{\text{os}}ʻ\alpha \in 0\cup 1.\supset _{\alpha }.(P\overline{R} Q)_{\text{lmx}}ʻ\alpha + =(\breve{P} \overline{R} Q)_{\text{lmx}}ʻ\alpha\end{align}\]</p> + +<p><a href="#*233·2">*233·2</a>—<a href="#*233·25">·25</a> are applications of the more important of the propositions +<a href="#*232·34">*232·34</a>—<a href="#*232·39">·39</a>, showing circumstances under which the limit of the +function for the class \(\alpha\) is the same as for the class +\(Q_{*}ʻʻ(\alpha \cap \text{ᗡ}ʻR)\).</p> + +<p><a href="#*233·4">*233·4</a> and following propositions apply the earlier propositions +of <a href="#*233">*233</a> to the case where \(\alpha\) is replaced by +\(\overrightarrow{Q}ʻ\alpha\), and therefore \((P\overline{R}Q)_{\text{lmx}}ʻ\alpha\) +is replaced by \(R(PQ)ʻ\alpha\). We have</p> + +<p class="nind"><b>*233·43.</b> \[\begin{align}\vdash :P_{\text{po}}\in \text{Ser}.(P\overline{R} Q)_{\text{os}}ʻ\overrightarrow{Q}ʻa &\in 1.\supset .\\ +&R(PQ)ʻa =R(\breve{P} Q)ʻa =\breve{{℩}} ʻ(P\overline{R} Q)_{\text{os}}ʻ\overrightarrow{Q}ʻa\end{align}\]</p> + +<p class="nind"><b>*233·433.</b> \[\begin{align}\vdash \colon\ldotp P\in \text{Ser}.&Q_{*}\unicode{x0294f}\overrightarrow{Q}ʻa +\in \text{connex} .Rʻʻ\overrightarrow{Q}ʻa \subset CʻP.(P\overline{R} Q)_{\text{os}}ʻ\overrightarrow{Q}ʻa=\Lambda .\\ +&\text{E}!R(PQ)ʻa.\text{E}!R(\breve{P} Q)ʻa.\supset :\\ +&R(PQ)ʻa=R(\breve{P} Q)ʻa.\lor.\{R(PQ)ʻa \}P_{1}\{R(\breve{P} Q)ʻa\}\end{align}\]</p> + +<p class="nind"><b>*233·45.</b> \[\begin{align}\vdash \colon\ldotp P\in \text{Ser}\cap \text{Ded}.&P^{2}=P.Q_{*}\in \text{connex} .RʻʻCʻQ\subset CʻP.\supset :\\ +&R(PQ)ʻa =R(\breve{P} Q)ʻa.\equiv _{a}.(P\overline{R} Q)_{\text{os}}ʻ\overrightarrow{Q}ʻa \in 0\cup 1\end{align}\]</p> + +<p><i>I.e.</i> in a series having Dedekindian continuity, the necessary +and sufficient condition that the two limits of the function as the +argument approaches \(\alpha\) from below should be equal is that the +ultimate oscillation should not have more than one term.</p> + +<p><span class="pagenum" id="Page_747">[Pg 747]</span></p> + +<p>We have next a set of propositions (<a href="#*233·5">*233·5</a>—<a href="#*233·53">·53</a>) on the possibility of +replacing \(\overrightarrow{Q}ʻa\) by a class \(\alpha\) having \(a\) +for its limit, without altering the limits of the function. We have to +begin with</p> + +<p class="nind"><b>*233·5.</b> \(\vdash :Q\in \text{Ser}.\alpha =\text{lt}_{Q}ʻ(\alpha \cap \text{ᗡ}ʻR).\supset .\overrightarrow{Q}ʻ\alpha =Q_{*}ʻʻ(\alpha \cap \text{ᗡ}ʻR)\)</p> + +<p class="nind">in virtue of <a href="#*207·291">*207·291</a>. Thence by earlier propositions of this number,</p> + +<p class="nind"><b>*233·512.</b> \[\begin{align}\vdash \colon\ldotp &\text{Hp}*233·5.P\in \text{Ser}.Rʻʻ(\alpha \cap CʻQ)\subset CʻP.(P\overline{R} Q)_{\text{os}}ʻ\overrightarrow{Q}ʻa + =\iota ʻx.\supset :\\ +&x=R(PQ)ʻa =R(\breve{P} Q)ʻa:x=(P\overline{R} Q)_{\text{lmx}}ʻ\alpha .\lor.(P\overline{R} Q)_{\text{lmx}}ʻ\alpha P_{1}x\end{align}\]</p> + +<p class="nind">whence we obtain</p> + +<p class="nind"><b>*233·514.</b> \(\vdash :\text{Hp}*233·512.x{\sim}\in CʻP_{1}.\supset .x=(P\overline{R} Q)_{\text{lmx}}ʻ\alpha =(\breve{P} \overline{R} Q)_{\text{lmx}}ʻ\alpha\)</p> + +<p>Thus if \(P\), \(Q\) are series, and \(x\) is the limit of the function +for the argument \(\alpha\) (\(x\) being a term which has no immediate +successor or predecessor), \(x\) is the limit of the function for any +class of arguments whose limit is \(\alpha\). Hence we arrive at the +proposition</p> + +<p class="nind"><b>*233·53.</b> \[\begin{align}\vdash :Q\in \text{Ser}.&P\in \text{Ser}\cap \text{Ded}.P^{2}=P.RʻʻCʻQ\subset CʻP.\alpha \subset \text{ᗡ}ʻR.\text{E}!\text{lt}_{Q}ʻ\alpha .\\ +&(P\overline{R} Q)_{\text{os}}ʻQ_{*}ʻʻ\alpha \in 0\cup 1.\supset .\\ +&(P\overline{R} Q)_{\text{lmx}}ʻ\alpha =(\breve{P} \overline{R} Q)_{\text{lmx}}ʻ\alpha =R(PQ)ʻ\text{lt}_{Q}ʻ\alpha + =R(\breve{P} Q)ʻ\text{lt}_{Q}ʻ\alpha\end{align}\]</p> + +<p>Thus if \(P\) has Dedekindian continuity, and \(\alpha\) is a class +of arguments having a limit, and if the ultimate oscillation as the +argument approaches this limit has not more than one term, the limit of +the function for the class \(\alpha\) exists, and is equal to the limit +of the function for the argument \(\text{lt}_{Q}ʻ\alpha\).</p> + +<hr class="tb"> + +<p class="nind"><b>*233·01.</b> \((P\overline{R} Q)_{\text{lmx}}=\text{limax}_P\mid (P\overline{R} Q)_{\text{sc}} \quad\text{Df}\)</p> + +<p class="nind"><b>*233·02.</b> \(R(PQ)=(P\overline{R} Q)_{\text{lmx}}\mid \overrightarrow{Q} \quad\text{Df}\)</p> + +<p class="nind"><b>*233·1.</b> \(\vdash :y\{(P\overline{R} Q)_{\text{lmx}}\}\alpha .\equiv .y(\text{limax}_P)\{(P\overline{R} Q)_{\text{sc}}ʻ\alpha\} \quad[(*233·01)]\)</p> + +<p class="nind"><b>*233·101.</b> \(\vdash :y=(P\overline{R} Q)_{\text{lmx}}ʻ\alpha .\equiv .y=\text{limax}_Pʻ(P\overline{R} Q)_{\text{sc}}ʻ\alpha \quad[*233·1]\)</p> + +<p class="nind"><b>*233·102.</b> \[\begin{align}\vdash :\text{E}!\text{limax}_Pʻ(P\overline{R} Q)_{\text{sc}}ʻ\alpha .&\equiv .(P\overline{R} Q)_{\text{lmx}}ʻ\alpha + =\text{limax}_Pʻ(P\overline{R} Q)_{\text{sc}}ʻ\alpha\\. +&\equiv .\text{E}!(P\overline{R} Q)_{\text{lmx}}ʻ\alpha \quad[*233·101.*14·28]\end{align}\]</p> + +<p class="nind"><b>*233·103.</b> \(\vdash :P\in \text{connex} .\supset .(P\overline{R} Q)_{\text{lmx}}\in 1\rightarrow \text{Cls} \quad[*207·41.*233·1]\)</p> + +<p class="nind"><b>*233·11.</b> \[\begin{align}&\vdash \colon\ldotp P\in \text{Ser}.\supset :y=(P\overline{R} Q)_{\text{lmx}}ʻ\alpha + .\equiv .y\in CʻP.\overrightarrow{P}ʻy=Pʻʻ(P\overline{R} Q)_{\text{sc}}ʻ\alpha \\ +&[*207·51.*233·101]\end{align}\]</p> + +<p class="nind"><b>*233·111.</b> \[\begin{align}\vdash \colon\ldotp &P\in \text{Ser}.\exists !Pʻʻ(P\overline{R} Q)_{\text{sc}}ʻ\alpha .\supset :\\ +&y=(P\overline{R} Q)_{\text{lmx}}ʻ\alpha .\equiv .\overrightarrow{P}ʻy=Pʻʻ(P\overline{R} Q)_{\text{sc}}ʻ\alpha \quad[*207·52.*233·101]\end{align}\]</p> + +<p class="nind"><b>*233·12.</b> \[\begin{align}\vdash \colon\ldotp P\in \text{Ser}.{\sim}\text{E}!&\text{max}_{P}ʻ(P\overline{R} Q)_{\text{sc}}ʻ\alpha .\supset :\\ +&y=(P\overline{R} Q)_{\text{lmx}}ʻ\alpha .\equiv .y\in CʻP.\overrightarrow{P}ʻy=(P\overline{R} Q)_{\text{sc}}ʻ\alpha\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*231·13.*211·41.\supset \vdash :\text{Hp}.\supset .(P\overline{R} Q)_{\text{sc}}ʻ\alpha =Pʻʻ(P\overline{R} Q)_{\text{sc}}ʻ\alpha &\qquad \text{(1)}\\ +\vdash .(1).*233·11.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_748">[Pg 748]</span></p> + +<p class="nind"><b>*233·13.</b> \[\begin{align}\vdash :P\in \text{connex} \cap &\text{Ded}.\supset .\\ +&\text{E}!(P\overline{R} Q)_{\text{lmx}}ʻ\alpha .(P\overline{R} Q)_{\text{lmx}}ʻ\alpha=\text{limax}_Pʻ(P\overline{R} Q)_{\text{sc}}ʻ\alpha\\ +&[*233·102·103.*214·11]\end{align}\]</p> + +<p class="nind"><b>*233·14.</b> \[\begin{align}&\vdash :P\in \text{Ser}.(P\overline{R} Q)_{\text{os}}ʻ\alpha \in 1.\supset .(P\overline{R} Q)_{\text{lmx}}ʻ\alpha + =(\breve{P} \overline{R} Q)_{\text{lmx}}ʻ\alpha =\breve{\iota} ʻ(P\overline{R} Q)_{\text{os}}ʻ\alpha \\ +&[*231·193.*233·102]\end{align}\]</p> + +<p class="nind"><b>*233·141.</b> \[\begin{align}\vdash \colon\ldotp P\in \text{Ser}.&(P\overline{R} Q)_{\text{sc}}ʻ\alpha \cup (\breve{P} \overline{R} Q)_{\text{sc}}ʻ\alpha + =CʻP.(P\overline{R} Q)_{\text{os}}ʻ\alpha =\Lambda .\supset :\\ +&\text{E}!(P\overline{R} Q)_{\text{lmx}}ʻ\alpha .\equiv .\text{E}!(\breve{P} \overline{R} Q)_{\text{lmx}}ʻ\alpha \\ +&[*211·727.*233·102.*231·13]\end{align}\]</p> + +<p class="nind"><b><a id="*233·142">*233·142</a>.</b> \[\begin{align}&\vdash :P\in \text{Ser}.Q_{*}\unicode{x0294f}\alpha \in \text{connex} .\\ +&Rʻʻ(\alpha \cap CʻQ)\subset CʻP.(P\overline{R} Q)_{\text{os}}ʻQ_{*}ʻʻ(\alpha \cap \text{ᗡ}ʻR)\in 0\cup 1.\\ +&\text{E}!(P\overline{R} Q)_{\text{lmx}}ʻQ_{*}ʻʻ(\alpha \cap \text{ᗡ}ʻR).(P\overline{R} Q)_{\text{lmx}}ʻQ_{*}ʻʻ(\alpha \cap \text{ᗡ}ʻR){\sim}\in CʻP_{1}.\supset + .\\ +(P\overline{R} Q)_{\text{lmx}}ʻ\alpha &=(\breve{P} \overline{R} Q)_{\text{lmx}}ʻ\alpha =(P\overline{R} Q)_{\text{lmx}}ʻQ_{*}ʻʻ(\alpha \cap \text{ᗡ}ʻR)\\ +&= (\breve{P} \overline{R} Q)_{\text{lmx}}ʻQ_{*}ʻʻ(\alpha \cap \text{ᗡ}ʻR)\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*231·252.\supset \vdash :\text{Hp}.&\supset .(P\overline{R} Q)_{\text{lmx}}ʻQ_{*}ʻʻ(\alpha \cap \text{ᗡ}ʻR)=(\breve{P} \overline{R} Q)_{\text{lmx}}ʻQ_{*}ʻʻ(\alpha + \cap \text{ᗡ}ʻR) &\qquad \text{(1)}\\ +\vdash .*232·37.*233·14.&\supset \vdash :\text{Hp}.(P\overline{R} Q)_{\text{os}}ʻQ_{*}ʻʻ(\alpha \cap \text{ᗡ}ʻR)\in 1.\supset .\\ +(P\overline{R} Q)_{\text{lmx}}ʻ\alpha &=(\breve{P} \overline{R} Q)_{\text{lmx}}ʻ\alpha =(P\overline{R} Q)_{\text{lmx}}ʻQ_{*}ʻʻ(\alpha \cap \text{ᗡ}ʻR)\\ +&=(\breve{P} \overline{R} Q)_{\text{lmx}}ʻQ_{*}ʻʻ(\alpha \cap \text{ᗡ}ʻR) &\qquad \text{(2)}\\ +\vdash .(1).*232·34.\supset \vdash :\text{Hp}.&(P\overline{R} Q)_{\text{os}}ʻQ_{*}ʻʻ(\alpha \cap \text{ᗡ}ʻR)=\Lambda .\supset .\\ +(P\overline{R} Q)_{\text{lmx}}ʻ\alpha &=(P\overline{R} Q)_{\text{lmx}}ʻQ_{*}ʻʻ(\alpha \cap \text{ᗡ}ʻR)=(\breve{P} \overline{R} Q)_{\text{lmx}}ʻQ_{*}ʻʻ(\alpha + \cap \text{ᗡ}ʻR)\\ +&=(\breve{P} \overline{R} Q)_{\text{lmx}}ʻ\alpha &\qquad \text{(3)}\\ +\vdash .(2).(3).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*233·15.</b> \[\begin{align}\vdash \colon\ldotp P\in &\text{Ser}\cap \text{Ded}.(P\overline{R} Q)_{\text{sc}}ʻ\alpha \cup (\breve{P} \overline{R} Q)_{\text{sc}}ʻ\alpha + =CʻP.(P\overline{R} Q)_{\text{os}}ʻ\alpha =\Lambda .\supset :\\ +&(P\overline{R} Q)_{\text{lmx}}ʻ\alpha =(\breve{P} \overline{R} Q)_{\text{lmx}}ʻ\alpha .\lor.\{(P\overline{R} Q)_{\text{lmx}}ʻ\alpha \}P_{1}\{(\breve{P} + \overline{R} Q)_{\text{lmx}}ʻ\alpha\}\\ +&[*214·43.*233·13.*231·13]\end{align}\]</p> + +<p class="nind"><b>*233·16.</b> \[\begin{align}\vdash \colon\ldotp P\in \text{Ser}\cap &\text{Ded}.P^{2}=P.Q_{*}\in \text{connex} .RʻʻCʻQ\subset CʻP.\supset :\\ +&(P\overline{R} Q)_{\text{os}}ʻ\alpha \in 0\cup 1.\supset _{\alpha }.(P\overline{R} Q)_{\text{lmx}}ʻ\alpha + =(\breve{P} \overline{R} Q)_{\text{lmx}}ʻ\alpha\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*232·22.&\supset \vdash :\text{Hp}.\supset .CʻP=(P\overline{R} Q)_{\text{sc}}ʻ\alpha \cup (\breve{P} \overline{R} Q)_{\text{sc}}ʻ\alpha &\qquad \text{(1)}\\ +\vdash .*201·65.&\supset \vdash :\text{Hp}.\supset .P_{1}=\dot{\Lambda} &\qquad \text{(2)}\\ +\vdash .(1).(2).*233·14·15.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_749">[Pg 749]</span></p> + +<p class="nind"><b>*233·17.</b> \(\vdash \colon\ldotp \alpha \cap CʻQ\cap \text{ᗡ}ʻR=\Lambda .\supset :y=(P\overline{R} Q)_{\text{lmx}}ʻ\alpha .\equiv .y=Bʻ\breve{P}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*232·15.*233·101.\supset \\ +\vdash \colon\ldotp \text{Hp}.\supset :y=(P\overline{R} Q)_{\text{lmx}}ʻ\alpha .&\equiv .y=\text{limax}_PʻCʻP.\\ +[*206·2.*93·117] &\equiv .y=Bʻ\breve{P} \colon\ldotp \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*233·171.</b> \(\vdash :\alpha \cap CʻQ\cap \text{ᗡ}ʻR=\Lambda .\supset .{\sim}\{(P\overline{R} Q)_{\text{lmx}}ʻ\alpha =(\breve{P} \overline{R} Q)_{\text{lmx}}ʻ\alpha\}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*93·102.\supset \vdash .{\sim}(BʻP=Bʻ\breve{P} ) &\qquad \text{(1)}\\ +\vdash .(1).*233·17.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*233·172.</b> \[\begin{align}\vdash :\alpha \cap CʻQ\cap \text{ᗡ}ʻR=\Lambda .\text{E}!(P\overline{R} Q)_{\text{lmx}}ʻ\alpha .\text{E}!(\breve{P} &\overline{R} Q)_{\text{lmx}}ʻ\alpha + .\supset .\\ +&(P\overline{R} Q)_{\text{os}}ʻ\alpha {\sim}\in 0\cup 1\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*233·171.*232·15.\supset \\ +\vdash :\text{Hp}.&\supset .(P\overline{R} Q)_{\text{lmx}}ʻ\alpha ,(\breve{P} \overline{R} Q)_{\text{lmx}}ʻ\alpha \in (P\overline{R} Q)_{\text{os}}ʻ\alpha + .(P\overline{R} Q)_{\text{lmx}}ʻ\alpha \neq (\breve{P} \overline{R} Q)_{\text{lmx}}ʻ\alpha .\\ +[*52·41]&\supset .(P\overline{R} Q)_{\text{os}}ʻ\alpha {\sim}\in 0\cup 1:\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*233·173.</b> \[\begin{align}\vdash :(P\overline{R} Q)_{\text{os}}ʻ\alpha \in 0\cup 1.\text{E}!&(P\overline{R} Q)_{\text{lmx}}ʻ\alpha + .\text{E}!(\breve{P} \overline{R} Q)_{\text{lmx}}ʻ\alpha .\supset .\\ +&\exists !\alpha \cap CʻQ\cap \text{ᗡ}ʻR \quad[*233·172.\text{Transp}]\end{align}\]</p> + +<p class="nind"><b>*233·174.</b> \(\vdash :P \,\unicode{x2abd}\, J.(P\overline{R} Q)_{\text{os}}ʻ\alpha \in 1.\supset .\exists !\alpha \cap CʻQ\cap \text{ᗡ}ʻR\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*200·12.\supset \vdash :\text{Hp}.&\supset .{\sim}\{CʻP\subset (P\overline{R} Q)_{\text{os}}ʻ\alpha\}.\\ +[*232·15] &\supset .\exists !\alpha \cap CʻQ\cap \text{ᗡ}ʻR:\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*233·2">*233·2</a>.</b> \[\begin{align}&\vdash :Q_{*}\unicode{x0294f}\alpha \in \text{connex} .P\in \text{Ser}.Rʻʻ(\alpha \cap CʻQ)\subset CʻP.\\ +&(P\overline{R} Q)_{\text{os}}ʻQ_{*}ʻʻ(\alpha \cap \text{ᗡ}ʻR)=\Lambda .\text{E}!(P\overline{R} Q)_{\text{lmx}}ʻQ_{*}ʻʻ(\alpha \cap \text{ᗡ}ʻR).\supset .\\ +&(P\overline{R} Q)_{\text{lmx}}ʻ\alpha =(P\overline{R} Q)_{\text{lmx}}ʻQ_{*}ʻʻ(\alpha \cap \text{ᗡ}ʻR).(\breve{P} R\overline{R} Q)_{\text{lmx}}ʻ\alpha + =(\breve{P} R\overline{R} Q)_{\text{lmx}}ʻQ_{*}ʻʻ(\alpha \cap \text{ᗡ}ʻR)\\ +&[*232·34.*211·727.*233·102]\end{align}\]</p> + +<p class="nind"><b>*233·21.</b> \[\begin{align}\vdash :&P_{\text{po}}\in \text{Ser}.\exists !(P\overline{R} Q)_{\text{os}}ʻ\alpha .(P\overline{R} Q)_{\text{os}}ʻQ_{*}ʻʻ(\alpha + \cap \text{ᗡ}ʻR)\in 1.\supset .\\ +&(P\overline{R} Q)_{\text{lmx}}ʻ\alpha =(\breve{P} \overline{R} Q)_{\text{lmx}}ʻ\alpha =(P\overline{R} Q)_{\text{lmx}}ʻQ_{*}ʻʻ(\alpha \cap \text{ᗡ}ʻR)\\ +&=(\breve{P} \overline{R} Q)_{\text{lmx}}ʻQ_{*}ʻʻ(\alpha \cap \text{ᗡ}ʻR)=\breve{\iota} ʻ(P\overline{R} Q)_{\text{os}}ʻQ_{*}ʻʻ(\alpha \cap \text{ᗡ}ʻR)\\ +&[*232·341.*231·193]\end{align}\]</p> + +<p class="nind"><b><a id="*233·22">*233·22</a>.</b> \[\begin{align}&\vdash \colon\ldotp P\in \text{Ser}.(P\overline{R} Q)_{\text{os}}ʻQ_{*}ʻʻ(\alpha \cap \text{ᗡ}ʻR)=\iota ʻx.\\ +&(P\overline{R} Q)_{\text{sc}}ʻ\alpha \cup (\breve{P} \overline{R} Q)_{\text{sc}}ʻ\alpha =CʻP.\supset :\\ +&x=(P\overline{R} Q)_{\text{lmx}}ʻ\alpha .\lor.(P\overline{R} Q)_{\text{lmx}}ʻ\alpha P_{1}x.(P\overline{R} Q)_{\text{lmx}}ʻ\alpha =\text{max}_{P}ʻ(P\overline{R} + Q)_{\text{sc}}ʻ\alpha\end{align}\]</p> + +<p><span class="pagenum" id="Page_750">[Pg 750]</span></p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*232·352.&\supset \vdash :\text{Hp}.x\in (P\overline{R} Q)_{\text{sc}}ʻ\alpha .\supset .x=(P\overline{R} Q)_{\text{lmx}}ʻ\alpha + &\qquad \text{(1)}\\ +\vdash .*232·356.&\supset \vdash :\text{Hp}.x{\sim}\in (P\overline{R} Q)_{\text{sc}}ʻ\alpha .{\sim}\text{E}!\text{max}_{P}ʻ(P\overline{R} Q)_{\text{sc}}ʻ\alpha + .\supset .\\ +&x=(P\overline{R} Q)_{\text{lmx}}ʻ\alpha &\qquad \text{(2)}\\ +\vdash .*232·358.*207·42.&\supset \vdash :\text{Hp}.x{\sim}\in (P\overline{R} Q)_{\text{sc}}ʻ\alpha .\text{E}!\text{max}_{P}ʻ(P\overline{R} Q)_{\text{sc}}ʻ\alpha + .\supset .\\ +&\text{max}_{P}ʻ(P\overline{R} Q)ʻ\alpha P_{1}x.(P\overline{R} Q)_{\text{lmx}}ʻ\alpha =\text{max}_{P}ʻ(P\overline{R} Q)_{\text{sc}}ʻ\alpha + &\qquad \text{(3)}\\ +\vdash .(1).(2).(3).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*233·23.</b> \[\begin{align}&\vdash :\text{Hp}*233·22.\supset .\\ +&x=(P\breve{R} Q)_{\text{lmx}}ʻQ_{*}ʻʻ(\alpha \cap \text{ᗡ}ʻR)=(\breve{P} \overline{R} Q)_{\text{lmx}}ʻQ_{*}ʻʻ(\alpha + \cap \text{ᗡ}ʻR) \quad[*231·193]\end{align}\]</p> + +<p class="nind"><b>*233·24.</b> \(\vdash :\text{Hp}*233·22.x{\sim}\in \text{ᗡ}ʻP_{1}.\supset .x=(P\overline{R} Q)_{\text{lmx}}ʻ\alpha \quad[*233·22]\)</p> + +<p class="nind"><b>*233·241.</b> \[\begin{align}&\vdash :\text{Hp}*233·22.x{\sim}\in CʻP_{1}.\supset .x=(P\overline{R} Q)_{\text{lmx}}ʻ\alpha =(\breve{P}\overline{R} Q)_{\text{lmx}}ʻ\alpha + \\ +&\left[*233·24 \frac{\breve{P}}{P}.*233·24\right]\end{align}\]</p> + +<p class="nind"><b><a id="*233·25">*233·25</a>.</b> \[\begin{align}&\vdash \colon\ldotp P\in \text{Ser}\cap \text{Ded}.P^{2}=P.Q_{*}\in \text{connex} .RʻʻCʻQ\subset CʻP.\supset :\\ +&(P\overline{R} Q)_{\text{os}}ʻQ_{*}ʻʻ(\alpha \cap \text{ᗡ}ʻR)\in 0\cup 1.\supset .\\ +&(P\overline{R} Q)_{\text{lmx}}ʻ\alpha =(P\overline{R} Q)_{\text{lmx}}ʻQ_{*}ʻʻ(\alpha \cap \text{ᗡ}ʻR)=(\breve{P} \overline{R} Q)_{\text{lmx}}ʻ\alpha + =(\breve{P} \overline{R} Q)_{\text{lmx}}ʻQ_{*}ʻʻ(\alpha \cap \text{ᗡ}ʻR)\\ +&[*232·39]\end{align}\]</p> + +<p class="nind"><b><a id="*233·4">*233·4</a>.</b> \(\vdash :y{R(PQ)}\alpha .\equiv .y\{(P\overline{R} Q)_{\text{lmx}}\}\overrightarrow{Q}ʻ\alpha \quad[(*233·02)]\)</p> + +<p class="nind"><b>*233·401.</b> \(\vdash :y=R(PQ)ʻ\alpha .\equiv .y=(P\overline{R} Q)_{\text{lmx}}ʻ\overrightarrow{Q}ʻ\alpha \quad[*233·4]\)</p> + +<p class="nind"><b>*233·402.</b> \(\vdash :P\in \text{connex} .\supset .R(PQ)\in 1\rightarrow \text{Cls} \quad[*207·41]\)</p> + +<p class="nind"><b>*233·41.</b> \(\vdash :y=R(PQ)ʻ\alpha .\equiv .y=(P\overline{R} Q)_{\text{lmx}}ʻ(\overrightarrow{Q}ʻ\alpha \cap \text{ᗡ}ʻR)\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*232·13.\supset \vdash .(P\overline{R} Q)_{\text{sc}}ʻ\overrightarrow{Q}ʻ\alpha =(P\overline{R} Q)_{\text{sc}}ʻ(\overrightarrow{Q}ʻ\alpha + \cap \text{ᗡ}ʻR) &\qquad \text{(1)}\\ +\vdash .(1).*233·401·101.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*233·42.</b> \[\begin{align}\vdash :.Q\in \text{trans}&\cap \text{connex} .\text{E}!\text{max}_{Q}ʻ(\overrightarrow{Q}ʻ\alpha \cap \text{ᗡ}ʻR).\supset :\\ +&y=R(PQ)ʻ\alpha .\equiv .y=\text{limax}_PʻP_{*}ʻʻ\overrightarrow{R}ʻ \text{max}_{Q}ʻ(\overrightarrow{Q}ʻ\alpha \cap \text{ᗡ}ʻR)\\ +&[*232·24.*233·401·101]\end{align}\]</p> + +<p class="nind"><b>*233·421.</b> \[\begin{align}&\vdash :P\in \text{ᗡ}ʻʻʻJ\cap \text{trans}.Q\in \text{trans}\cap \text{connex} .Rʻ\text{max}_{Q}ʻ(\alpha \cap \text{ᗡ}ʻR)\in C^{P}.\supset .\\ +&R(PQ)ʻ\alpha =Rʻ\text{max}_{Q}ʻ(\overrightarrow{Q}ʻ\alpha \cap \text{ᗡ}ʻR)\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*233·42.\supset \vdash \colon\ldotp \text{Hp}.\supset :y=R(PQ)ʻ\alpha .&\equiv .y=\text{limax}_Pʻ\overrightarrow{P}_{Q}ʻRʻ\text{max}_{Q}ʻ(\overrightarrow{Q}ʻ\alpha + \cap \text{ᗡ}ʻR).\\ +[*205·197] &\equiv .y=Rʻ\text{max}_{Q}ʻ(\overrightarrow{Q}ʻ\alpha \cap \text{ᗡ}ʻR)\colon\ldotp \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*233·422.</b> \(\vdash \colon\ldotp \overrightarrow{Q}ʻ\alpha \cap \text{ᗡ}ʻR=\Lambda .\supset :y=R(PQ)ʻ\alpha .\equiv .y=Bʻ\breve{P} \quad[*233·17]\)</p> + +<p><span class="pagenum" id="Page_751">[Pg 751]</span></p> + +<p class="nind"><b><a id="*233·423">*233·423</a>.</b> \(\vdash :\overrightarrow{Q}ʻa\cap \text{ᗡ}ʻR=\Lambda .\supset .{\sim}\{R(PQ)ʻa=R(\breve{P} Q)ʻa\} \quad[*233·171]\)</p> + +<p class="nind"><b>*233·424.</b> \[\begin{align}&\vdash :\overrightarrow{Q}ʻa\cap \text{ᗡ}ʻR=\Lambda .\text{E}!R(PQ)ʻa.\text{E}!R(\breve{P} Q)ʻa.\supset .(P\overline{R}Q)_{\text{os}}ʻ\overrightarrow{Q}ʻa{\sim}\in + 0\cup 1\\ +&[*233·172]\end{align}\]</p> + +<p class="nind"><b>*233·425.</b> \[\begin{align}&\vdash :(P\overline{R}Q)_{\text{os}}ʻ\overrightarrow{Q}ʻa\in + 0\cup 1.\text{E}!R(PQ)ʻa.\text{E}!P(\breve{P} Q)ʻa.\supset .\exists !\overrightarrow{Q}ʻa\cap \text{ᗡ}ʻR\\ +&[*233·424.\text{Transp}]\end{align}\]</p> + +<p class="nind"><b>*233·426.</b> \(\vdash :P\,\unicode{x2abd}\, J.(P\overline{R}Q)_{\text{os}}ʻ\overrightarrow{Q}ʻa\in 1.\supset .\exists !\overrightarrow{Q}ʻa\cap \text{ᗡ}ʻR \quad[*233·174]\)</p> + +<p class="nind"><b>*233·43.</b> \[\begin{align}\vdash :P_{\text{po}}\in \text{Ser}.&(P\overline{R}Q)_{\text{os}}ʻ\overrightarrow{Q}ʻa\in 1.\supset.\\ +&R(PQ)ʻa = R(\breve{P} Q)ʻa = \breve{\iota} ʻ(P\overline{R}Q)_{\text{os}}ʻ\overrightarrow{Q}ʻ\alpha \quad[*231·193]\end{align}\]</p> + +<p class="nind"><b>*233·431.</b> \[\begin{align}&\vdash :P\in \text{trans}\cap \text{connex} .(P\overline{R}Q)_{\text{os}}ʻ\overrightarrow{Q}ʻa{\sim}\in 0\cup 1.\\ +&\text{E}!R(PQ)ʻa.\text{E}!R(\breve{P} Q)ʻa.\supset .\{R(\breve{P} Q)ʻa\}P\{R(PQ)ʻa\}\\ +&[*215·52.*231·13·101]\end{align}\]</p> + +<p class="nind"><b>*233·432.</b> \[\begin{align}&\vdash :P\in \text{trans}\cap \text{connex} .(P\overline{R}Q)_{\text{os}}ʻ\overrightarrow{Q}ʻa=\Lambda .\\ +&\text{E}!R(PQ)ʻa.\text{E}!R(\breve{P} Q)ʻa.\supset .\{R(PQ)ʻa\}P_{*}\{R(\breve{P} Q)ʻa\} \quad[*215·53]\end{align}\]</p> + +<p class="nind"><b>*233·433.</b> \[\begin{align}&\vdash \colon\ldotp P\in \text{Ser}.Q_{*}\unicode{x0294f}\overrightarrow{Q}ʻa\in \text{connex} .Rʻʻ\overrightarrow{Q}ʻa\subset\\ +&CʻP.(P\overline{R}Q)_{\text{os}}ʻ\overrightarrow{Q}ʻa=\Lambda . +\text{E}!R(PQ)ʻa.\text{E}!R(\breve{P} Q)ʻa.\supset :R(PQ)ʻa=R(\breve{P} Q)ʻa.\lor.\{R(PQ)ʻa\}P_1\{R(\breve{P} Q)ʻa\}\\ +&[*215·54.*232·22]\end{align}\]</p> + +<p class="nind"><b>*233·434.</b> \[\begin{align}&\vdash :P\in \text{Ser}.Q_{*}\unicode{x0294f}\overrightarrow{Q}ʻa\in \text{connex} .Rʻʻ\overrightarrow{Q}ʻa\subset CʻP.\text{E}!R(PQ)ʻa.\\ +&\text{E}!R(\breve{P} Q)ʻa.\supset .\{R(PQ)ʻa\}(P_1\unicode{x228d} \breve{P} _*)\{R(\breve{P} Q)ʻa\} \quad[*233·43·431·433]\end{align}\]</p> + +<p class="nind"><b>*233·435.</b> \[\begin{align}&\vdash :P\in \text{Ser}.R(PQ)ʻa=R(\breve{P} Q)ʻa.\supset .(P\overline{R}Q)_{\text{os}}ʻ\overrightarrow{Q}ʻa\in 0\cup 1\\ +&[*233·431.\text{Transp}]\end{align}\]</p> + +<p class="nind"><b>*233·44.</b> \[\begin{align}&\vdash \colon\ldotp P\in \text{Ser}.Q_{*}\unicode{x0294f}\overrightarrow{Q}ʻa\in \text{connex} .Rʻʻ\overrightarrow{Q}ʻa\subset CʻP.\text{E}!R(PQ)ʻa.\\ +&\text{E}!R(\breve{P} Q)ʻa.{\sim}{R(PQ)ʻa\in \text{D}ʻP_1.R(\breve{P} Q)ʻa\in \text{ᗡ}ʻP_1}.\supset :\\ +&R(PQ)ʻa=R(\breve{P} Q)ʻa.\equiv .(P\overline{R}Q)_{\text{os}}ʻ\overrightarrow{Q}ʻa\in 0\cup 1 \quad[*233·426·43·433·435]\end{align}\]</p> + +<p class="nind"><b>*233·45.</b> \[\begin{align}\vdash \colon\ldotp P\in \text{Ser}\cap \text{Ded}.P^{2}&=P.Q_{*}\in \text{connex} .RʻʻCʻQ\subset CʻP.\supset :\\ +&R(PQ)ʻa=R(\breve{P} Q)ʻa.\equiv _a.(P\overline{R}Q)_{\text{os}}ʻ\overrightarrow{Q}ʻa\in 0\cup 1\\ +&[*233·13.*201·65.*233·44]\end{align}\]</p> + +<p><span class="pagenum" id="Page_752">[Pg 752]</span></p> + +<p class="nind"><b><a id="*233·5">*233·5</a>.</b> \(\vdash :Q\in \text{Ser}.a=\text{lt}_Qʻ(\alpha \cap \text{ᗡ}ʻR).\supset .\overrightarrow{Q}ʻa=Q_{*}ʻʻ(\alpha \cap \text{ᗡ}ʻR) \quad[*207·291]\)</p> + +<p class="nind"><b>*233·501.</b> \(\vdash \colon\ldotp Q\in \text{Ser}.\alpha=\text{lt}_{Q}ʻ(\alpha + \cap \text{ᗡ}ʻR).\supset :\exists !\overrightarrow{Q}ʻa\cap \text{ᗡ}ʻR.\equiv .\exists !\alpha \cap CʻQ\cap \text{ᗡ}ʻR\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*233·5.\supset \vdash \colon\ldotp \text{Hp}.\supset :\exists !\overrightarrow{Q}ʻa\cap \text{ᗡ}ʻR.&\equiv .\exists !Q_{*}ʻʻ(\alpha + \cap \text{ᗡ}ʻR)\cap \text{ᗡ}ʻR. &\qquad \text{(1)}\\ +[*37·29·265] &\supset .\exists !\alpha \cap \text{ᗡ}ʻR\cap CʻQ &\qquad \text{(2)}\\ +\vdash .*90·33.*22·43.&\supset \vdash :x\in \alpha \cap CʻQ\cap \text{ᗡ}ʻR.\supset .x\in Q_{*}ʻʻ(\alpha \cap \text{ᗡ}ʻR).x\in \text{ᗡ}ʻR &\qquad \text{(3)}\\ +\vdash .(3).*10·28. &\supset \vdash :\exists !\alpha \cap CʻQ\cap \text{ᗡ}ʻR.\supset .\exists !Q_{*}ʻʻ(\alpha \cap \text{ᗡ}ʻR)\cap \text{ᗡ}ʻR &\qquad \text{(4)}\\ +\vdash .(1).(2).(4).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*233·51.</b> \[\begin{align}\vdash :\text{Hp}*233·5.&P\in \text{Ser}.Rʻʻ(\alpha \cap CʻQ)\subset CʻP.(P\overline{R} Q)_{\text{os}}ʻ\overrightarrow{Q}ʻa=\Lambda .\\ +&\text{E}!R(PQ)ʻ\alpha.\supset .(P\overline{R} Q)_{\text{lmx}}ʻ\alpha =R(PQ)ʻ\alpha \quad[*233·2·5]\end{align}\]</p> + +<p class="nind"><b>*233·511.</b> \[\begin{align}&\vdash :\text{Hp}*233·5.P\in \text{Ser}.\exists !(P\overline{R} Q)_{\text{os}}ʻ\alpha .(P\overline{R} Q)_{\text{os}}ʻ\overrightarrow{Q}ʻ\alpha\in + 1.\supset .\\ +&(P\overline{R} Q)_{\text{lmx}}ʻ\alpha =(\breve{P} \overline{R} Q)_{\text{lmx}}ʻ\alpha + =R(PQ)ʻa=R(\breve{P} Q)ʻa=\breve{\iota} ʻ(P\overline{R} Q)_{\text{os}}ʻ\overrightarrow{Q}ʻ\alpha\\ +&[*233·501·5·21]\end{align}\]</p> + +<p class="nind"><b>*233·512.</b> \[\begin{align}&\vdash \colon\ldotp \text{Hp}*233·5.P\in \text{Ser}.Rʻʻ(\alpha \cap CʻQ)\subset CʻP.(P\overline{R} Q)_{\text{os}}ʻ\overrightarrow{Q}ʻ\alpha=\iota + ʻx.\supset :\\ +&x=R(PQ)ʻ\alpha=R(\breve{P} Q)ʻ\alpha:x=(P\overline{R} Q)_{\text{lmx}}ʻ\alpha .\lor.(P\overline{R} Q)_{\text{lmx}}ʻ\alpha P_{1}x\\ +&[*233·22·23.*232·22]\end{align}\]</p> + +<p class="nind"><b>*233·513.</b> \(\vdash :\text{Hp}*233·512.x{\sim}\in \text{ᗡ}ʻP_{1}.\supset .x=(P\overline{R} Q)_{\text{lmx}}ʻ\alpha \quad[*233·512]\)</p> + +<p class="nind"><b>*233·514.</b> \[\begin{align}&\vdash :\text{Hp}*233·512.x{\sim}\in CʻP_{1}.\supset .x=(P\overline{R} Q)_{\text{lmx}}ʻ\alpha =(\breve{P} \overline{R} Q)_{\text{lmx}}ʻ\alpha + \\ +&\left[*233·513\frac{\breve{P}}{P}.*233·513\right]\end{align}\]</p> + +<p class="nind"><b>*233·515.</b> \[\begin{align}\vdash :P,Q&\in \text{Ser}.a=\text{lt}_{Q}ʻ(\alpha \cap \text{ᗡ}ʻR).Rʻʻ(CʻQ\cap \alpha )\subset CʻP.\\ +&(P\overline{R} Q)_{\text{os}}ʻ\overrightarrow{Q}ʻ\alpha\in 0\cup 1.\text{E}!R(PQ)ʻ\alpha.R(PQ)ʻ\alpha{\sim}\in CʻP_{1}.\supset .\\ +&(P\overline{R} Q)_{\text{lmx}}ʻ\alpha =(\breve{P} \overline{R} Q)_{\text{lmx}}ʻ\alpha =R(PQ)ʻ\alpha=R(\breve{P} Q)ʻ\alpha\\ +&[*233·142·5]\end{align}\]</p> + +<p class="nind"><b>*233·516.</b> \[\begin{align}\vdash :P,Q&\in \text{Ser}.\text{E}!R(PQ)ʻ\text{lt}_{Q}ʻ\alpha .R(PQ)ʻ\text{lt}_{Q}ʻ\alpha {\sim}\in CʻP_{1}.\\ +&Rʻʻ(CʻQ\cap \alpha )\subset CʻP.(P\overline{R} Q)_{\text{os}}ʻ\overrightarrow{Q}ʻ\text{lt}_{Q}ʻ\alpha \in 0\cup 1.\supset .\\ +&(P\overline{R} Q)_{\text{lmx}}ʻ\alpha = (\breve{P} \overline{R} Q)_{\text{lmx}}ʻ\alpha = R(PQ)ʻ\text{lt}_{Q}ʻ\alpha = R(\breve{P} Q)ʻ\text{lt}_{Q}ʻ\alpha \\ +&[*233·515]\end{align}\]</p> + +<p class="nind"><b>*233·52.</b> \[\begin{align}\vdash \colon\ldotp &\text{Hp}*233·5.P\in \text{Ser}\cap \text{Ded}.P^{2}=P.RʻʻCʻQ\subset CʻP.\supset :\\ +&(P\overline{R} Q)_{\text{os}}ʻ\overrightarrow{Q}ʻa\in 0\cup 1.\supset .\\ +&(P\overline{R} Q)_{\text{lmx}}ʻ\alpha =(\breve{P} \overline{R} Q)_{\text{lmx}}ʻ\alpha =R(PQ)ʻ\alpha=R(\breve{P} Q)ʻa \quad[*233·25]\end{align}\]</p> + +<p class="nind"><b><a id="*233·53">*233·53</a>.</b> \[\begin{align}\vdash :Q\in \text{Ser}.P\in &\text{Ser}\cap \text{Ded}.P^{2}=P.RʻʻCʻQ\subset CʻP.\alpha \subset \text{ᗡ}ʻR.\text{E}!\text{lt}_{Q}ʻ\alpha .\\ +&(P\overline{R} Q)_{\text{os}}ʻQ_{*}ʻʻ\alpha \in 0\cup 1.\supset .\\ +&(P\overline{R} Q)_{\text{lmx}}ʻ\alpha =(\breve{P} \overline{R} Q)_{\text{lmx}}ʻ\alpha =R(PQ)ʻ\text{lt}_{Q}ʻ\alpha =R(\breve{P} Q)ʻ\text{lt}_{Q}ʻ\alpha\\ +&[*233·52]\end{align}\]</p> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_753">[Pg 753]</span></p> +<h2 class="nobreak" id="*234"> +*234. CONTINUITY OF FUNCTIONS.</h2> +</div> + + +<p class="nind"><i>Summary of</i> *234.</p> + +<p>In the present number we are concerned with the definition and analysis +of the continuity of functions. The following definition of continuity +is given by Dini<a id="FNanchor_17" href="#Footnote_17" class="fnanchor">[17]</a>:</p> + +<div class="blockquot"> + +<p>"We call it [the function] <i>continuous</i> for \(x = a\), or in the +point \(a\), in which it has the value \(f(a)\), if, for every positive +number \(\sigma\), different from 0 but as small as we please, there +exists a positive number \({\in}\), different from 0, such that, for +all values of \(\delta\) which are numerically less than \({\in}\), +the difference \(f(a + \delta ) - f(a)\) is numerically less than +\(\sigma\). In other words, \(f(x)\) is continuous in the point \(x =a\) +where it has the value \(f(a)\), if the limit of its values to the +right and left of \(a\) is the same and equal to \(f(a)\)...."</p> +</div> + +<p>By the second form of the above definition, the function \(R\) of +previous numbers is to be called continuous at the point \(a\) if +\[ +R(PQ)ʻa = R(\breve{P} Q)ʻa = R(P\breve{Q} )ʻa = R(\breve{P} \breve{Q} )ʻa = Rʻa. +\] +The first form of the definition can also be so stated as to be free +from any reference to number, and derivable from the ideas dealt with +in the previous numbers of the present section. For this purpose, +instead of "a positive number \(\sigma\)" we take an interval in which +\(Rʻa\) is contained, say \(P(z - w)\). Similarly the "values of +\(\delta\) which are numerically less than \({\in}\)" are replaced by +arguments in a certain interval containing \(a\).</p> + +<p>By <a href="#*233·423">*233·423</a>, if the limits of the function as the argument approaches +\(a\) are to be all equal, \(a\) must not be the maximum or minimum +of \(\text{ᗡ}ʻR\). We therefore take the interval containing \(a\) to +be an interval in which the end-points are not included, say \(Q(y -y')\). +Thus our definition becomes +\[ +\begin{aligned} +(\text{A})\quad Rʻa\in &P(z-w).\supset _{z, w}.\\ +&(\exists y,y').y,y'\in \text{ᗡ}ʻR.a\in Q(y-y').RʻʻQ(y\vdash\dashv y')\subset P(z-w) +\end{aligned} +\]</p> + +<p>We require further, what is tacitly assumed in Dini's definition, that +\(Rʻa\) is a member of \(CʻP\) which has no immediate predecessor or +successor, <i>i.e.</i> +\[ +Rʻa\in CʻP-CʻP_{1}. +\]</p> + +<p><span class="pagenum" id="Page_754">[Pg 754]</span></p> + +<p>In order to deal more easily with the above definition, we analyse it +into the product of four factors, which concern respectively \(P\) and +\(Q\), \(\breve{P}\) and \(Q\), \(P\) and \(\breve{Q}\), \(\breve{P}\) +and \(\breve{Q}\). In the first place, it is obvious that (A) is the +product of +\[ +(\text{B})\quad Rʻa \in P(z-w). \supset _{z,w} . (\exists y) . y \in \text{ᗡ}ʻR . y \in \overrightarrow{Q}ʻa . RʻʻQ(y \unicode{x27dd} a) \subset P(z-w) +\] +and a factor obtained by substituting \(\breve{Q}\) for +\(Q\) in (\(\text{B}\)). If \(Q_{*} \in \text{connex}\), and +\(P_{\text{po}}\in\text{Ser}\), (\(\text{B}\)) is the product of +\[ +(\text{C}) Rʻa \in \overrightarrow{P}_{\text{po}}ʻw + . \supset _w . (\exists y) . y \in \text{ᗡ}ʻR . y \in \overrightarrow{Q}ʻa . RʻʻQ (y \unicode{x27dd} a) \subset \overrightarrow{P}_{*}ʻw +\] +and a factor obtained by writing \(\breve{P}\) for \(P\) and \(z\) for +\(w\) in (\(\text{C}\)); and in virtue of \(Rʻa {\sim}\in CʻP_{1}\), +(\(\text{C}\)) becomes +\[ +Rʻa \in \overrightarrow{P}_{\text{po}}ʻw . \supset _w . (\exists y) . y \in \text{ᗡ}ʻR . y \in \overrightarrow{Q}ʻa . RʻʻQ (y \unicode{x27dd} a) \subset \overrightarrow{P}_{\text{po}}ʻw. +\] +<i>i.e.</i> if \(Q\) is transitive, +\[ +(\text{D}) Rʻa \in \overrightarrow{P}_{\text{po}}ʻw . \supset _w . R(Q_{*}\unicode{x0294f}\overrightarrow{Q}ʻa)_{\text{cn}} (\overrightarrow{P}_{\text{po}}ʻw) +\]</p> + +<p>Hence the function is continuous for the argument a if a satisfies +(\(\text{D}\)) and the three other hypotheses resulting from replacing +\(P\) by \(\breve{P}\), or \(Q\) by \(\breve{Q}\), or \(P\) and +\(Q\) by \(\breve{P}\) and \(\breve{Q}\). If we substitute \(x\) for +\(Rʻa\), and \(Q\) for \(Q_{*} \unicode{x0294f} \overrightarrow{Q}ʻa\), +(\(\text{D}\)) becomes +\[ +(\text{E}) \overrightarrow{P}_{\text{po}}ʻʻ\overleftarrow{P}_{\text{po}}ʻx \subset \overleftarrow{Q}_{\text{cn}}ʻR +\]</p> + +<p>Hence continuity can be studied by studying the hypothesis +(\(\text{E}\)), and replacing \(x\) by \(Rʻa\) and \(Q\) by +\(Q_{*}\unicode{x0294f} \overrightarrow{Q}ʻa\).</p> + +<p>The hypothesis (\(\text{E}\)) is an interesting one on its own account. +We put +\[ +\text{sc}(P,Q)ʻR = CʻP \cap \hat{x} (\overrightarrow{P}_{\text{po}}ʻʻ\overleftarrow{P}_{\text{po}}ʻx \subset \overleftarrow{Q}_{\text{cn}}ʻR) \quad\text{Df}. +\] +Thus "\(x \in \text{sc}(P,Q)ʻR\)" means that \(x\) is a member of the +value-series such that, if \(y\) is any later member, the function +ultimately becomes less than \(y\). If we put further +\[ +\text{os}(P,Q)ʻR = \text{sc}(P,Q)ʻR \cap \text{sc}(\breve{P} ,Q)ʻR \quad\text{Df}, +\] +then, if \(x\) is a member of \(\text{os}(P,Q)ʻR\), the function +ultimately becomes less than any later member of \(CʻP\), and greater +than any earlier member. Hence \(x\) is the limit of the function as +the argument increases indefinitely. Hence, if we substitute +\(Q_{*}\unicode{x0294f} \overrightarrow{Q}ʻa\) for \(Q\), and if +\(x\in \text{os}(P,Q_{*} \unicode{x0294f} \overrightarrow{Q}ʻa)ʻR\), +\(x\) is the limit of the function as the argument approaches a from +below, <i>i.e.</i> +\[ +R(PQ)ʻa = R(\breve{P} Q)ʻa = x. +\] +(This is proved in <a href="#*234·462">*234·462</a>.) Hence, putting \(Rʻa\) in place of \(x\), +the function is continuous from below at the point \(a\) if +\[ +Rʻa \in \text{os}(P,Q_{*} \unicode{x0294f} \overrightarrow{Q}ʻa)ʻR, +\]<span class="pagenum" id="Page_755">[Pg 755]</span> +and is continuous from above if +\[ +Rʻa\in \text{os}(P,\breve{Q} _{*}\unicode{x0294f}\overleftarrow{Q}ʻa)ʻR. +\] +These results, and various others connected with them, are proved +below. The equivalence of Dini's two definitions is proved in <a href="#*234·63">*234·63</a>. +It will be observed that practically nothing in the theory of +continuous functions requires the use of numbers.</p> + +<p>We use the symbol "\(\text{ct}(PQ)ʻR\)" for the class of arguments +\(a\) for which the limit of the function for approaches to a from +below is \(Rʻa\). Thus, in virtue of what was said above, we may put +\[ +\text{ct}(PQ)ʻR=\hat{a}\{Rʻa\in \text{os}(P,Q_{*}\unicode{x0294f}\overrightarrow{Q}_{\text{po}}ʻa)ʻR-CʻP_{1}\} \quad\text{Df}. +\] +Then a function is continuous at the point \(a\) if a belongs to the +two classes \(\text{ct}(PQ)ʻR\) and \(\text{ct}(P\breve{Q} )ʻR\). Hence +we put +\[ +\text{contin}(PQ)ʻR=\text{ct}(PQ)ʻR\cap \text{ct}(P\breve{Q} )ʻR \quad\text{Df}. +\] +The function \(R\) is continuous with respect to \(P\) and \(Q\) if it +is continuous for all arguments in \(CʻQ\). Thus we put +\[ +P\overline{\text{contin}} Q=\hat{R}\{\exists !CʻQ\cap \text{ᗡ}ʻR.CʻQ\cap \text{ᗡ}ʻR\subset \text{contin}(PQ)ʻR\} \quad\text{Df}. +\]</p> + +<p>Our propositions in this number begin with the properties of +\(\text{sc}(P,Q)ʻR\) and \(\text{os}(P,Q)ʻR\). We have</p> + +<p class="nind"><b>*234·103.</b> \(\vdash :P_{\text{po}}\in \text{Ser}.\exists !\text{os}(P,Q)ʻR.\supset .P\overline{R} _{\text{os}}Q\in 0\cup 1\)</p> + +<p>Thus the hypothesis \(\exists !\text{os}(P,Q)ʻR\) enables us to use +propositions of previous numbers having the hypothesis \(P\overline{R}_{\text{os}}Q\in 0\cup 1\).</p> + +<p>The identification of our definitions with the usual definitions of +continuity of functions proceeds by means of the proposition</p> + +<p class="nind"><b>*234·12.</b> \[\begin{align}\vdash \colon\colon Q_{*}\in &\text{connex} .\supset \colon\ldotp x\in \text{os}(P,Q)ʻR\cap \text{D}ʻP\cap \text{ᗡ}ʻP.\equiv :\\ +&x\in \text{D}ʻP\cap \text{ᗡ}ʻP:x\in P(z-w).\supset _{z,w}.RQ_{\text{cn}}\{P(z-w)\}\end{align}\]</p> + +<p>We have a collection of propositions dealing with the relations of +\(\text{sc}(P,Q)ʻR\) to \(P\overline{R} _{\text{sc}}Q\) and +\(\breve{P}\overline{R}_{\text{sc}}Q\). \(\text{sc}(P,Q)ʻR\) is +an upper section of \(P\) (<a href="#*234·131">*234·131</a>); \(\text{sc}(P,Q)ʻR\) is the +complement of \(Pʻʻ(P\overline{R} _{\text{sc}}Q)\), <i>i.e.</i> of +\(P\overline{R} _{\text{sc}}Q\) without its maximum (if any). This is +expressed in the following proposition:</p> + +<p class="nind"><b>*234·174.</b> \[\begin{align}\vdash :P_{\text{po}}\in \text{Ser}.Q_{*}\in \text{connex} .Rʻʻ&CʻQ\subset CʻP.\supset .\\ +&CʻP\cap pʻ\overrightarrow{P}_{\text{po}}ʻʻ\text{sc}(P,Q)ʻR=Pʻʻ(P\overline{R} _{\text{sc}}Q)=CʻP-\text{sc}(P,Q)ʻR\end{align}\]</p> + +<p>We thus arrive at</p> + +<p class="nind"><b>*234·182.</b> \[\begin{align}\vdash :P\in \text{Ser}.Q_{*}\in \text{connex} .Rʻʻ&CʻQ\subset CʻP.\supset .\\ +&\overrightarrow{\text{limax}} _{P}ʻ(P\overline{R} _{\text{sc}}Q) = \overrightarrow{\text{min}}_{P}ʻ\text{sc}(P, + Q)ʻR\end{align}\]</p> + +<p><span class="pagenum" id="Page_756">[Pg 756]</span></p> + +<p>Thus \(\text{os}(P,Q)ʻR\) is contained in +\(\overrightarrow{\text{max}}_{P}ʻ(P\overline{R} _{\text{sc}}Q)\cup\overrightarrow{\text{min}}_{P}ʻ(\breve{P} \overline{R}_{\text{sc}}Q)\) +(*234·201), and therefore has not more than two terms (<a href="#*234·202">*234·202</a>). If +\(P\overline{R}_{\text{os}}Q\) has one term, this is the only member +of \(\text{os}(P,Q)ʻR\) (<a href="#*234·203">*234·203</a>). If \(\text{os}(P,Q)ʻR\) has two +terms, they have the relation \(P_{1}\) (<a href="#*234·242">*234·242</a>); hence if \(P\) +is a compact series, and \(\text{os}(P,Q)ʻR\) is not null, its only +member is both \(\text{limax}_Pʻ(P\overline{R}_{\text{sc}}Q)\) and +\(\text{limin}_Pʻ(\breve{P} \overline{R}_{\text{sc}}Q)\) (<a href="#*234·25">*234·25</a>), +while conversely, if \(\text{limax}_Pʻ(P\overline{R}_{\text{sc}}Q)\) +and \(\text{limin}_Pʻ(\breve{P} \overline{R}_{\text{sc}}Q)\) are +equal, each is the only member of \(\text{os}(P,Q)ʻR\) (<a href="#*234·251">*234·251</a>).</p> + +<p>We now apply the above results to the limits of a function as its +argument approaches the limit of a class \(\alpha\). This is done, as +before, by substituting \(Q_{*}\unicode{x0294f}\alpha\) for \(Q\). +We arrive at the proposition (<a href="#*234·33">*234·33</a>) that if \(P\) has Dedekindian +continuity, and \(\text{os}(P,Q_{*}\unicode{x0294f}\alpha )ʻR\) is not +null, its only member is both \((P\overline{R} Q)_{\text{lmx}}ʻ\alpha\) +and \((\breve{P} \overline{R} Q)_{\text{lmx}}ʻ\alpha\), <i>i.e.</i> is +the limit of the function as the argument increases in \(\alpha\).</p> + +<p>We then take for \(\alpha\) the particular value +\(\overrightarrow{Q}_{\text{po}}ʻa\), so that we become concerned +with what happens when the argument approaches a from below. For the +comparison of our definition of continuity with such definitions as the +one quoted from Dini above, we have</p> + +<p class="nind"><b>*234·41.</b> \[\begin{align}\vdash \colon\colon Q\in \text{trans}.Q_{*}\unicode{x0294f}&\overrightarrow{Q}ʻa\in \text{connex} .\supset \colon\ldotp\\ +&x\in \text{os}(P,Q_{*}\unicode{x0294f}\overrightarrow{Q}ʻa)ʻR\cap \text{D}ʻP\cap \text{ᗡ}ʻP.\equiv :\\ +&x\in \text{D}ʻP\cap \text{ᗡ}ʻP:x\in P(z-w).\supset _{z,w}.\\ +&(\exists y).y\in \overrightarrow{Q}ʻa\cap \text{ᗡ}ʻR.RʻʻQ(y\unicode{x27dd} a)\subset P(z-w)\end{align}\]</p> + +<p><i>I.e.</i> if \(x\) is neither the first nor the +last member of the \(P\)-series, \(x\) belongs to +\(\text{os}(P,Q_{*}\unicode{x0294f}\overrightarrow{Q}ʻa)ʻR\) when, and +only when, given any interval \(P(z-w)\), however small, in which \(x\) +is contained, there is an argument \(y\) earlier than \(a\), such that +the value of the function for all arguments earlier than a but not +earlier than \(y\) lies in the interval \(P(z-w)\).</p> + +<p>We deduce from previous propositions that, with the usual hypothesis as +to \(Q\), if \(P\) is a Dedekindian series, +\[ +R(PQ)ʻa=\text{limin}_Pʻ\text{sc}(P,Q_{*}\unicode{x0294f}\overrightarrow{Q}ʻa)ʻR \quad(*234·422), +\] +and if \(P\) is a series and +\(\text{os}(P,Q_{*}\unicode{x0294f}\overrightarrow{Q}ʻa)\) is a unit +class, its only member is both \(R(PQ)ʻa\) and \(R(\breve{P} Q)ʻa\), +<i>i.e.</i> is the limit of the function for approaches to \(a\) from +below (<a href="#*234·43">*234·43</a>). The following proposition sums up our results:</p> + +<p class="nind"><b>*234·45.</b> \[\begin{align}\vdash \colon\ldotp P\in &\text{Ser}.Q\in \text{trans}.Q_{*}\unicode{x0294f}\overrightarrow{Q}ʻa\in + \text{connex} .Rʻʻ\overrightarrow{Q}ʻa\subset CʻP.P^{2}=P.\supset :\\ +\exists !\text{os}(P,Q_{*}\unicode{x0294f}\overrightarrow{Q}ʻa)ʻR.&\equiv .\text{os}(P,Q_{*}\unicode{x0294f}\overrightarrow{Q}ʻa)ʻR=\iota ʻR(PQ)ʻa.\\ +&\equiv .\text{os}(P,Q_{*}\unicode{x0294f}\overrightarrow{Q}ʻa)ʻR=\iota ʻR(\breve{P} Q)ʻa.\\ +&\equiv .R(PQ)ʻa=R(\breve{P} Q)ʻa\end{align}\]</p> + +<p><span class="pagenum" id="Page_757">[Pg 757]</span></p> + +<p>Thus \(\exists!\text{os}(P,Q_{*}\unicode{x0294f}\overrightarrow{Q}ʻa)ʻR\) is, +in a compact series, the necessary and sufficient condition for +the existence of a definite limit of the function as the argument +approaches \(a\) from below.</p> + +<p>Without assuming \(P^{2} = P\), if \(x\) is a member of +\(\text{os}(P,Q_{*}\unicode{x0294f}\overrightarrow{Q}ʻa)ʻR\), and +if \(x\) has no immediate predecessor or successor, so that in the +neighbourhood of \(x\) the series is compact, we still have +\(x =R(PQ)ʻa = R(\breve{P} Q)ʻa\) (<a href="#*234·462">*234·462</a>).</p> + +<p>We next consider \(\text{ct}(PQ)ʻR\). By the definition we have</p> + +<p class="nind"><b>*234·5.</b> \(\vdash :a\in \text{ct}(PQ)ʻR.\equiv .Rʻa\in \text{os}(P,Q_{*}\unicode{x0294f}\overrightarrow{Q}_{\text{po}}ʻa)ʻR-CʻP_{1}\)</p> + +<p>Thus \(a\) is an argument for which the function has a single value +which has no immediate predecessor or successor in \(P\), and which, +in virtue of <a href="#*234·462">*234·462</a>, is the limit of the function as the argument +approaches \(a\) from below (<a href="#*234·52">*234·52</a>). The cases when \(Rʻa = BʻP\) or +\(Rʻa = Bʻ\breve{P}\) require special attention; excluding these cases, +we arrive at</p> + +<p class="nind"><b>*234·51.</b> \[\begin{align}\vdash &\colon\colon Q\in \text{trans}.Q_{*}\unicode{x0294f}\overrightarrow{Q}ʻa\in + \text{connex} .Rʻa\in \text{D}ʻP\cap \text{ᗡ}ʻP.\supset \colon\ldotp \\ +&a\in \text{ct}(PQ)ʻR.\equiv :Rʻa{\sim}\in CʻP_{1}:Rʻa\in P(z-w).\supset _{z,w}.\\ +&(\exists y).y\in \overrightarrow{Q}ʻa\cap \text{ᗡ}ʻR.RʻʻQ(y\vdash\dashv a)\subset P(z-w)\end{align}\]</p> + +<p>This proposition is analogous to <a href="#*234·41">*234·41</a>.</p> + +<p>We prove (<a href="#*234·562">*234·562</a>) that if \(P\), \(Q\) are series, and \(\alpha\) +is any class of arguments for which all the values belong to \(CʻP\), +and if \(\alpha\) has a limit at which the function is continuous from +below, then the limit of the function, as the argument increases in +\(\alpha\), is the value of the function at the limit of \(\alpha\).</p> + +<p>We next consider \(\text{contin}(PQ)ʻR\), which is defined as +\(\text{ct}(PQ)ʻR\cap \text{ct}(P\breve{Q} )ʻR\). We show that if \(P\) +is a series whose field contains \(Rʻʻ(\overleftrightarrow{Q}ʻa\) +and \(Q\) is transitive, and \(Q_{*} \unicode{x0294f}\overleftrightarrow{Q}ʻa\) +is connected, and \(Rʻa\) is neither \(BʻP\) nor \(Bʻ\breve{P}\), +then if \(a\) belongs to the class \(\text{contin}(PQ)ʻR\), \(Rʻa\) is the +limit of the function for the argument \(a\) for approaches either +from below or from above (<a href="#*234·62">*234·62</a>). If \(P\) is compact, the converse +also holds (<a href="#*234·63">*234·63</a>). Our definition of a point of continuity is thus +identified with the second form of Dini's definition quoted above. It +is identified with the first form by the following proposition: In the +circumstances of *234·62, if \(Rʻa\in \text{D}ʻP\cap \text{ᗡ}ʻP\), we +have (<a href="#*234·64">*234·64</a>) +\[ +\begin{aligned} +a\in \text{contin}(PQ)ʻ&R.\equiv :Rʻa\in CʻP-CʻP_{1}:Rʻa\in P(z-w).\supset _{z,w}.\\ +&(\exists y,y').y,y'\in \text{ᗡ}ʻR.a\in Q(y-y').RʻʻQ(y\vdash\dashv y')\subset P(z-w), +\end{aligned} +\] +<i>i.e.</i> \(a\) is a point of continuity when, and only when, the +value \(Rʻa\) for the argument \(a\) is a member of the \(P\)-series +having no immediate predecessor or successor, and if \(Rʻa\) is +contained in the interval \(P(z-w)\), then,<span class="pagenum" id="Page_758">[Pg 758]</span> however small this +interval may be, two arguments \(y\), \(y'\) can be found such that +a lies between them, and the values for all arguments from \(y\) to +\(y'\) (both included) lie in the interval \(P(z-w)\).</p> + +<p>We end with a few propositions on continuous functions. The last of +these (<a href="#*234·73">*234·73</a>) states that, if \(P\) is a compact series and \(Q\) +is transitive and connected, then \(R\) is continuous with respect to +\(P\) and \(Q\) when, and only when, it has arguments in \(CʻQ\), and +for all such arguments \(a\) we have +\[ +R(PQ)ʻa=R(\breve{P} Q)ʻa=R(P\breve{Q} )ʻa=R(\breve{P} \breve{Q} )ʻa=Rʻa, +\] +<i>i.e.</i> the value for every argument is the limit for that argument +for approaches either from above or from below.</p> + +<hr class="tb"> + +<p class="nind"><b>*234·01.</b> \(\text{sc}(P,Q)ʻR=CʻP\cap \hat{x} (\overrightarrow{P}_{\text{po}}ʻʻ\overleftarrow{P}_{\text{po}}ʻx\subset \overleftarrow{Q}_{\text{cn}}ʻR) + \quad\text{Df}\)</p> + +<p class="nind"><b>*234·02.</b> \(\text{os}(P,Q)ʻR=\text{sc}(P,Q)ʻR\cap \text{sc}(\breve{P},Q)ʻR \quad\text{Df}\)</p> + +<p class="nind"><b>*234·03.</b> \(\text{ct}(PQ)ʻR=\hat{a} \{Rʻa\in \text{os}(P,Q_{*}\unicode{x0294f}\overrightarrow{Q}_{\text{po}}ʻa)ʻR-CʻP_{1}\} \quad\text{Df}\)</p> + +<p class="nind"><b>*234·04.</b> \(\text{contin}(PQ)ʻR=\text{ct}(PQ)ʻR\cap \text{ct}(P\breve{Q})ʻR \quad\text{Df}\)</p> + +<p class="nind"><b>*234·05.</b> \(P\overline{\text{contin}} Q=\hat{R} \{\exists !CʻQ\cap \text{ᗡ}ʻR.CʻQ\cap \text{ᗡ}ʻR\subset \text{contin}(PQ)ʻR\} \quad\text{Df}\)</p> + +<p class="nind"><b>*234·1.</b> \[\begin{align}\vdash \colon\ldotp &x\in \text{sc}(P,Q)ʻR.\equiv :x\in CʻP:xP_{\text{po}}w.\supset _{w}.RQ_{\text{cn}}(\overrightarrow{P} + _{\text{pο}}ʻw):\\ +&\equiv :x\in CʻP:xP_{\text{pο}}w.\supset _{w}.(\exists y).y\in CʻQ\cap \text{ᗡ}ʻR.Rʻʻ\overleftarrow{Q}_{*}ʻy\subset \overrightarrow{P}_{\text{po}}ʻw\\ +&[*230·11.(*234·01)]\end{align}\]</p> + +<p class="nind"><b><a id="*234·101">*234·101</a>.</b> \(\vdash :P_{\text{po}}\in \text{Ser}.x\in \text{sc}(P,Q)ʻR.\supset .P\overline{R} _{\text{sc}}Q\subset \overrightarrow{P}_{*}ʻx\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*40·16.(*234·01).\supset \\ +\vdash :\text{Hp}.\supset .x\in CʻP.pʻP_{*}ʻʻʻ\overleftarrow{Q}_{\text{cn}}ʻR\cap + CʻP&\subset pʻP _{*}ʻʻʻ\overrightarrow{P}_{\text{po}}ʻʻ\overleftarrow{P}_{\text{po}}ʻx\cap CʻP\\ +[*91·574] & \subset pʻ\overrightarrow{P}_{\text{po}}ʻʻ\overleftarrow{P}_{\text{po}}ʻx\cap CʻP\\ +[*204·65.*91·602] &\subset \overrightarrow{P}_{*}ʻx &\qquad \text{(1)}\\ +\vdash .(1).*231·1.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*234·102.</b> \(\vdash :P_{\text{po}}\in \text{Ser}.x\in \text{os}(P,Q)ʻR.\supset .P\overline{R} _{\text{os}}Q\subset \iota ʻx\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*234·1·101.(*234·02)\supset \vdash :\text{Hp}.&\supset .x\in CʻP.P\overline{R} _{\text{os}}Q\subset \overrightarrow{P}_{*}ʻx\cap \overleftarrow{P}_{*}ʻx.\\ +[*200·39] &\supset .P\overline{R} _{\text{os}}Q\subset \iota ʻx:\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*234·103.</b> \(\vdash :P_{\text{po}}\in \text{Ser}.\exists !\text{os}(P,Q)ʻR.\supset .P\overline{R} _{\text{os}}Q\in 0\cup 1\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*234·102.\supset \vdash :\text{Hp}.&\supset .(\exists x).P\overline{R} _{\text{os}}Q\subset \iota ʻx\\ +[*51·401] &\supset .P\overline{R} _{\text{os}}Q\in 0\cup 1:\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_759">[Pg 759]</span></p> + +<p class="nind"><b>*234·104.</b> \(\vdash :RQ_{\text{cn}}(\overrightarrow{P}_{*}ʻx).\supset .x\in \text{sc}(P,Q)ʻR\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*91·52. &\supset \vdash :xP_{\text{po}}z.\supset .\overrightarrow{P}_{*}ʻx\subset \overrightarrow{P}_{\text{po}}ʻz &\qquad \text{(1)}\\ +\vdash .(1).*230·211·151.&\supset \vdash \colon\ldotp \text{Hp}.\supset :xP_{\text{po}}z.\supset _{z}.RQ_{\text{cn}}(\overrightarrow{P}_{\text{po}}ʻz):x\in CʻP:\\ +[*234·1] &\supset :x\in \text{sc}(P,Q)ʻR\colon\ldotp \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*234·105.</b> \(\vdash :P_{\text{po}}\in \text{Ser}.x\in \text{sc}(P,Q)ʻR\cap \text{D}ʻP_{1}.\supset .RQ_{\text{cn}}(\overrightarrow{P}_{*}ʻx)\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*201·63.*121·254.\supset \vdash \colon\colon\text{Hp}.xP_{1}z.\supset \colon\ldotp yP_{\text{po}}z.&\supset :{\sim}(xP_{\text{po}}y):\\ +[*202·103] &\supset :yP_{\text{po}}x.\lor.y=x &\qquad \text{(1)}\\ +\vdash .(1).*91·54. \supset \vdash \colon\ldotp \text{Hp}.xP_{1}z.&\supset :\overrightarrow{P}_{\text{po}}ʻz\subset \overrightarrow{P}_{*}ʻx:\\ +[*230·211] &\supset :RQ_{\text{cn}}(\overrightarrow{P}_{\text{po}}ʻz).\supset .RQ_{\text{cn}}(\overrightarrow{P}_{*}ʻx) &\qquad \text{(2)}\\ +\vdash .*234·1. &\supset \vdash :\text{Hp}.\supset .(\exists z).xP_{1}z.RQ_{\text{cn}}(\overrightarrow{P}_{\text{po}}ʻz) &\qquad \text{(3)}\\ +\vdash .(2)·(3).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p>When \(x{\sim}\in \text{D}ʻP_{1}\), the above proposition is not +necessarily true: it may fail if \(x=\text{min}_{P}ʻ\text{sc}(P,Q)ʻR\).</p> + +<p>It is to be observed that \(\text{sc}(P,Q)ʻR\) and \(\text{os}(P,Q)ʻR\) +are functions of \(P_{\text{po}}\), so that they are unchanged when +\(P_{\text{po}}\) is substituted for \(P\). Hence the hypothesis +\(P_{\text{po}}\in \text{Ser}\) is as effective, with regard to them, +as the hypothesis \(P\in \text{Ser}\). This is stated in the following +proposition.</p> + +<p class="nind"><b>*234·106.</b> \(\vdash .\text{sc}(P,Q)ʻR=\text{sc}(P_{\text{po}},Q)ʻR.\text{os}(P,Q)ʻR=\text{os}(P_{\text{po}},Q)ʻR \quad[*234·1]\)</p> + +<p class="nind"><b>*234·107.</b> \(\vdash \colon\ldotp x\in CʻP-\text{D}ʻP_{1}.\supset :x\in \text{sc}(P,Q)ʻR.\equiv .\overrightarrow{P}_{*}ʻʻ\overleftarrow{P}_{\text{po}}ʻx\subset + \overleftarrow{Q}_{\text{cn}}ʻR\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*121·254.\supset \vdash \colon\ldotp \text{Hp}.&\supset :x{\sim}\in \text{D}ʻ(P_{\text{po}})_{1}:\\ +[*201·61] & \supset :x{\sim}\in \text{D}ʻ{P_{\text{po}}\dot{-} P_{\text{po}}^{2}}:\\ +[*10·51] \supset :xP_{\text{po}}y.&\supset .xP_{\text{po}}^{2}y.\\ +[*91·574] &\supset .(\exists z).xP_{\text{po}}z.\overrightarrow{P}_{*}ʻz\subset \overrightarrow{P}_{\text{po}}ʻy &\qquad \text{(1)}\\ +\vdash .(1).*230·211.\supset \\ +\vdash \colon\ldotp \text{Hp}:xP_{\text{po}}y.&\supset _{y}.RQ_{\text{cn}}\overrightarrow{P}_{*}ʻy:\supset :xP_{\text{po}}y.\supset + _{y}.RQ_{\text{cn}}\overrightarrow{P}_{\text{po}}ʻy &\qquad \text{(2)}\\ +\vdash .*91·54.*230·211.\supset \\ +\vdash \colon\ldotp xP_{\text{po}}y.&\supset _{y}.RQ_{\text{cn}}\overrightarrow{P}_{\text{po}}ʻy:\supset :xP_{\text{po}}y.\supset + _{y}.RQ_{\text{cn}}\overrightarrow{P}_{*}ʻy &\qquad \text{(3)}\\ +\vdash .(2).(3).\supset \vdash \colon\ldotp \text{Hp}.&\supset :\overrightarrow{P}_{*}ʻʻ\overleftarrow{P}_{\text{po}}ʻx\subset \overleftarrow{Q}_{\text{cn}}ʻR.\equiv + .\overrightarrow{P}_{\text{po}}ʻʻ\overleftarrow{P}_{\text{po}}ʻx\subset \overleftarrow{Q}_{\text{cn}}ʻR &\qquad \text{(4)}\\ +\vdash .(4).*234·1.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*234·11.</b> \[\begin{align}\vdash \colon\ldotp &x\in \text{D}ʻP\cap \text{ᗡ}ʻP:x\in P(z-w).\supset _{z,w}.RQ_{\text{cn}}{P(z-w)}:\equiv :\\ +&x\in \text{D}ʻP\cap \text{ᗡ}ʻP:x\in P(z-w).\supset _{z,w}.\\ +&(\exists y).y\in CʻQ\cap \text{ᗡ}ʻR.Rʻʻ\overleftarrow{Q}_{*}ʻy\subset P(z-w) \quad[*230·11]\end{align}\]</p> + +<p><span class="pagenum" id="Page_760">[Pg 760]</span></p> + +<p class="nind"><b>*234·111.</b> \[\begin{align}\vdash \colon\ldotp x\in \text{D}ʻP\cap \text{ᗡ}ʻP:x\in P(z-w).\supset _{z,w}.&RQ_{\text{cn}}\{P(z-w)\}:\supset .\\ +&x\in \text{os}(P,Q)ʻR\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*230·211.\supset \\ +\vdash \colon\colon\text{Hp}.\supset \colon\ldotp &x\in \text{D}ʻP\cap \text{ᗡ}ʻP\colon\ldotp xP_{\text{po}}w:(\exists z).zP_{\text{po}}x:\supset + _{w}.RQ_{\text{cn}}\overrightarrow{P}_{\text{po}}ʻw\colon\ldotp \\ +[*91·504] &\supset \colon\ldotp x\in \text{D}ʻP:xP_{\text{po}}w.\supset _{w}.RQ_{\text{cn}}\overrightarrow{P}_{\text{po}}ʻw\colon\ldotp \\ +[*234·1] &\supset \colon\ldotp x\in \text{sc}(P,Q)ʻR &\qquad \text{(1)}\\ +\text{Similarly}\quad &\vdash :\text{Hp}.\supset .\text{sc}(\breve{P} ,Q)ʻR &\qquad \text{(2)}\\ +\vdash .(1).(2).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*234·12.</b> \[\begin{align}\vdash \colon\colon Q_{*}\in \text{connex} .&\supset \colon\ldotp x\in \text{os}(P,Q)ʻR\cap \text{D}ʻP\cap \text{ᗡ}ʻP.\equiv :\\ +&x\in \text{D}ʻP\cap \text{ᗡ}ʻP:x\in P(z-w).\supset _{z,w}.RQ_{\text{cn}}\{P(z-w)\}\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*234·1.&\supset \vdash \colon\ldotp x\in \text{os}(P,Q)ʻR\cap \text{D}ʻP\cap \text{ᗡ}ʻP.\equiv :\\ +&x\in \text{D}ʻP\cap \text{ᗡ}ʻP:xP_{\text{po}}w.\supset _{w}.RQ_{\text{cn}}(\overrightarrow{P}_{\text{po}}ʻw):zP_{\text{po}}x.\supset +_{z}.RQ_{\text{cn}}(\overleftarrow{P}_{\text{po}}ʻz):\\ +[*11·71]&\equiv :x\in \text{D}ʻP\cap \text{ᗡ}ʻP:zP_{\text{po}}x.xP_{\text{po}}w.\supset +_{z,w}.RQ_{\text{cn}}(\overrightarrow{P}_{\text{po}}ʻw).RQ_{\text{cn}}(\overleftarrow{P}_{\text{po}}ʻz) &\qquad \text{(1)}\\ +\vdash .*230·42.\supset \vdash \colon\ldotp \text{Hp}.\supset :&RQ_{\text{cn}}\overrightarrow{P}_{\text{po}}ʻw.RQ_{\text{cn}}(\overleftarrow{P}_{\text{po}}ʻz).\equiv.\\ +\end{array} +\]</p> + +<p class="nind"><b>*234·121.</b> \(\vdash .\overrightarrow{B}ʻ\breve{P} \subset \text{sc}(P,Q)ʻR \quad[*93·104.(*234·01)]\)</p> + +<p class="nind"><b>*234·122.</b> \[\begin{align}\vdash \colon\ldotp P_{\text{po}}\in &\text{connex} .x=BʻP.\supset :\\ +&x\in \text{os}(P,Q)ʻR.\equiv .x\in \text{sc}(P,Q)ʻR.\equiv .\overrightarrow{P}_{\text{po}}ʻʻ\text{ᗡ}ʻP\subset \overleftarrow{Q}_{\text{cn}}ʻR\\ +&[*234·121.(*234·02).*234·1.*205·253]\end{align}\]</p> + +<p class="nind"><b>*234·13.</b> \(\vdash :x\in \text{sc}(P,Q)ʻR.\supset .\overleftarrow{P}_{*}ʻx\subset \text{sc}(P,Q)ʻR\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*96·3.*91·74.*90·13.\supset \vdash :xP_{*}z.&\supset .\overleftarrow{P}_{\text{po}}ʻz\subset \overleftarrow{P}_{\text{po}}ʻx.z\in CʻP.\\ +[*37·2] &\supset .\overrightarrow{P}_{\text{po}}ʻʻ\overleftarrow{P}_{\text{po}}ʻz\subset \overleftarrow{P}_{\text{po}}ʻʻ\overleftarrow{P}_{\text{po}}ʻx.z\in + CʻP &\qquad \text{(1)}\\ +\vdash .(1).(*234·01).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*234·131">*234·131</a>.</b> \(\vdash .\text{sc}(P,Q)ʻR=\breve{P} _{*}ʻʻ\text{sc}(P,Q)ʻR.\text{sc}(P,Q)ʻR\in \text{sect}ʻ\breve{P}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*90·21.*234·1.&\supset \vdash .\text{sc}(P,Q)ʻR\subset \breve{P} _{*}ʻʻ\text{sc}(P,Q)ʻR &\qquad \text{(1)}\\ +\vdash .*234·13. &\supset \vdash .\breve{P} _{*}ʻʻ\text{sc}(P,Q)ʻR\subset \text{sc}(P,Q)ʻR &\qquad \text{(2)}\\ +\vdash .(1).(2).*211·13.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_761">[Pg 761]</span></p> + +<p class="nind"><b>*234·14.</b> \(\vdash :Q_{*}\in \text{connex} .x\in \text{sc}(P,Q)ʻR.\supset .x\in CʻP.\overleftarrow{P}_{\text{po}}ʻx\subset \breve{P} \overline{R} _{\text{sc}}Q\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*234·1.\supset \vdash \colon\ldotp \text{Hp}.\supset :x\in CʻP:xP_{\text{po}}z.&\supset _{z}.RQ_{\text{cn}}(\overrightarrow{P}_{\text{po}}ʻz).\\ +[*230·211] &\supset _{z}.RQ_{\text{cn}}(\overrightarrow{P}_{*}ʻz).\\ +[*231·141] &\supset _{z}.z\in \breve{P} \overline{R} _{\text{sc}}Q\colon\ldotp \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*234·141.</b> \(\vdash :Q_{*}\in \text{connex} .\exists !\text{sc}(P,Q)ʻR.\supset .\exists !\breve{P} \overline{R} _{\text{sc}}Q \quad[*234·14]\)</p> + +<p class="nind"><b>*234·142.</b> \(\vdash :\exists !\text{sc}(P,Q)ʻR\cap \text{D}ʻP.\supset .\exists !CʻQ\cap \text{ᗡ}ʻR\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*234·1.\supset \\ +\vdash \colon\ldotp x\in \text{sc}(P,Q)ʻR\cap \text{D}ʻP.&\supset :x\in \text{D}ʻP:(\exists w).xP_{\text{po}}w.\supset .\exists !CʻQ\cap \text{ᗡ}ʻR:\\ +[*91·504] & \supset :\exists !CʻQ\cap \text{ᗡ}ʻR\colon\ldotp \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*234·15.</b> \(\vdash :P_{*},Q_{*}\in \text{connex} .\exists !\text{sc}(P,Q)ʻR.\supset .P\overline{R} _{\text{sc}}Q\cup + \breve{P} \overline{R} _{\text{sc}}Q=CʻP\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*231·202.*234·141.&\supset \vdash :\text{Hp}.\supset .CʻP-P\overline{R} _{\text{sc}}Q\subset \breve{P} \overline{R} _{\text{sc}}Q &\qquad \text{(1)}\\ +\vdash .*231·1. &\supset \vdash .P\overline{R} _{\text{sc}}Q\cup \breve{P} \overline{R} _{\text{sc}}Q\subset CʻP &\qquad \text{(2)}\\ +\vdash .(1).(2).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*234·16.</b> \[\begin{align}&\vdash :P_{\text{po}}\in \text{Ser}.Q_{*}\in \text{connex} .\supset .\\ +&P\overline{R} _{\text{sc}}Q\subset pʻ\overrightarrow{P}_{*}ʻʻ\text{sc}(P,Q)ʻR.\breve{P} _{\text{po}}ʻʻ\text{sc}(P,Q)ʻR\subset \breve{P} \overline{R} _{\text{sc}}Q + \quad[*234·101·14]\end{align}\]</p> + +<p class="nind"><b>*234·161.</b> \[\begin{align}\vdash \colon\ldotp P_{\text{po}}\in \text{Ser}.RʻʻCʻQ\subset CʻP.P\overline{R} _{\text{sc}}Q&\subset \overrightarrow{P}_{*}ʻx.\supset + :\\ +&P\overline{R} _{\text{sc}}Q=\overrightarrow{P}_{*}ʻx.\lor.RQ_{\text{cn}}(\overrightarrow{P}_{*}ʻx)\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*231·24.\supset \vdash :\text{Hp}.{\sim}\{RQ_{\text{cn}}(\overrightarrow{P}_{*}ʻx)\}.&\supset .\overrightarrow{P}_{*}ʻx\subset + P\overline{R} _{\text{sc}}Q.\\ +[\text{Hp}.*22·41] &\supset .\overrightarrow{P}_{*}ʻx=P\overline{R} _{\text{sc}}Q:\in \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*234·162.</b> \[\begin{align}\vdash :P_{\text{po}}\in \text{Ser}.RʻʻCʻQ\subset &CʻP.\overrightarrow{P}_{*}ʻx=P\overline{R} _{\text{sc}}Q.x\in CʻP.\supset .\\ +&x\in \text{sc}(P,Q)ʻR\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*202·5.\supset \vdash \colon\ldotp \text{Hp}.xP_{\text{po}}z.&\supset :z{\sim}\in P\overline{R} _{\text{sc}}Q:\\ +[*231·12] &\supset :(\exists y).y\in CʻQ\cap \text{ᗡ}ʻR.z{\sim}\in P_{*}ʻʻRʻʻ\overleftarrow{Q}_{*}y:\\ +[*211·56] & \supset :(\exists y).y\in CʻQ\cap \text{ᗡ}ʻR.P_{*}ʻʻRʻʻ\overleftarrow{Q}_{*}ʻy\subset \overrightarrow{P}_{\text{po}}ʻz:\\ +[*90·33] &\supset :(\exists y).y\in CʻQ\cap \text{ᗡ}ʻR.Rʻʻ\overleftarrow{Q}_{*}ʻy\subset \overrightarrow{P}_{\text{po}}ʻz:\\ +[*230·11] &\supset :RQ_{\text{cn}}(\overrightarrow{P}_{\text{po}}ʻz) &\qquad \text{(1)}\\ +\vdash .(1).*234·1.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_762">[Pg 762]</span></p> + +<p class="nind"><b>*234·17.</b> \[\begin{align}\vdash \colon\ldotp P_{\text{po}}\in \text{Ser}.RʻʻCʻQ\subset &CʻP.\supset :\\ +&x\in \text{sc}(P,Q)ʻR.\equiv .x\in CʻP.P\overline{R} _{\text{sc}}Q\subset \overrightarrow{P}_{*}ʻx\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*234·1·101.\supset \vdash \colon\ldotp \text{Hp}.&\supset :x\in \text{sc}(P,Q)ʻR.\supset .x\in CʻP.P\overline{R} _{\text{sc}}Q\subset \overrightarrow{P}_{*}ʻx + &\qquad \text{(1)}\\ +\vdash .*234·161·162·104.\supset \vdash \colon\ldotp \text{Hp}.&\supset :x\in CʻP.P\overline{R} _{\text{sc}}Q\subset \overrightarrow{P}_{*}ʻx.\supset .\\ +&x\in \text{sc}(P,Q)ʻR &\qquad \text{(2)}\\ +\vdash .(1).(2).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*234·171.</b> \[\begin{align}\vdash :P_{\text{po}}\in \text{Ser}.RʻʻCʻQ\subset CʻP.x\in CʻP-&\text{sc}(P,Q)ʻR.\supset .\\ +&\overrightarrow{P}_{*}ʻx\subset Pʻʻ(P\overline{R} _{\text{sc}}Q)\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*234·17.\supset \vdash :\text{Hp}.\supset .\exists !P\overline{R} _{\text{sc}}Q-\overrightarrow{P}_{*}ʻx &\qquad \text{(1)}\\ +\vdash .(1).*211·56.*231·13.\supset \vdash :\text{Hp}.\supset .\overrightarrow{P}_{*}ʻx\subset P_{\text{po}}ʻʻ(P\overline{R} _{\text{sc}}Q) &\qquad \text{(2)}\\ +\vdash .(2).*231·134.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*234·172.</b> \(\vdash :P_{\text{po}}\in \text{Ser}.\supset .CʻP-\text{sc}(P,Q)ʻR=CʻP\cap pʻ\overrightarrow{P}_{\text{po}}ʻʻ\text{sc}(P,Q)ʻR\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*200·5.\supset \vdash :\text{Hp}.&\supset .CʻP\cap pʻ\overrightarrow{P}_{\text{po}}ʻʻ\text{sc}(P,Q)ʻR\subset CʻP-\text{sc}(P,Q)ʻR &\qquad \text{(1)}\\ +\vdash .*234·131.\supset\\ +\vdash :x\in \text{sc}(P,Q)ʻR.y\in CʻP-\text{sc}(P,Q)ʻR.&\supset .{\sim}(xP_{*}y).x,y\in CʻP.\\ +[*202·103] &\supset .yP_{\text{po}}x &\qquad \text{(2)}\\ +\vdash .(1).(2).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*234·173.</b> \[\begin{align}&\vdash :P_{\text{po}}\in + \text{Ser}.\exists !\text{sc}(P,Q)ʻR.\supset .CʻP-\text{sc}(P,Q)ʻR=pʻ\overrightarrow{P}_{\text{po}}ʻʻ\text{sc}(P,Q)ʻR\\ +&[*234·172.*40·61.*37·15]\end{align}\]</p> + +<p class="nind"><b>*234·174.</b> \[\begin{align}\vdash :P_{\text{po}}\in &\text{Ser}.Q_{*}\in \text{connex} .RʻʻCʻQ\subset CʻP.\supset .\\ +&CʻP\cap pʻ\overrightarrow{P}_{\text{po}}ʻʻ\text{sc}(P,Q)ʻR=P ʻʻ(P \overline{R} _{\text{sc}}Q)=CʻP-\text{sc}(P,Q)ʻR\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*234·171·172.\supset \vdash :\text{Hp}.&\supset .P_{*}ʻʻ{CʻP\cap pʻ\overrightarrow{P}_{\text{po}}ʻʻ\text{sc}(P,Q)ʻR}\subset Pʻʻ(P \overline{R} _{\text{sc}}Q).\\ +[*90·21] &\supset .CʻP\cap pʻ\overrightarrow{P}_{\text{po}}ʻʻ\text{sc}(P,Q)ʻR\subset Pʻʻ(P \overline{R} _{\text{sc}}Q) &\qquad \text{(1)}\\ +\vdash .*234·16.*37·2.\supset \vdash :\text{Hp}.\supset .Pʻʻ(P \overline{R} _{\text{sc}}Q)&\subset Pʻʻpʻ\overrightarrow{P}_{*}ʻʻ\text{sc}(P,Q)ʻR\\ +[*40·37.*91·52] & \subset pʻ\overrightarrow{P}_{\text{po}}ʻʻ\text{sc}(P,Q)ʻR &\qquad \text{(2)}\\ +\vdash .*37·15. &\supset \vdash .Pʻʻ(P\overline{R} _{\text{sc}}Q)\subset \text{D}ʻP. &\qquad \text{(3)}\\ +\vdash .(1).(2).(3).*234·172.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*234·175.</b> \[\begin{align}&\vdash :\text{Hp}*234·174.\exists !\text{sc}(P,Q)ʻR.\supset .pʻ\overrightarrow{P}_{\text{po}}ʻʻ\text{sc}(P,Q)ʻR=Pʻʻ(P + \overline{R} _{\text{sc}}Q)\\ +&[*234·174.*40·61.*37·15]\end{align}\]</p> + +<p><span class="pagenum" id="Page_763">[Pg 763]</span></p> + +<p class="nind"><b>*234·18.</b> \[\begin{align}\vdash :&P_{\text{po}}\in \text{Ser}.Q_{*}\in \text{connex} .RʻʻCʻQ\subset CʻP.\supset .\\ +&CʻP=\text{sc}(P,Q)ʻR\cup Pʻʻ(P\overline{R} _{\text{sc}}Q).\text{sc}(P,Q)ʻR\cap Pʻʻ(P\overline{R} _{\text{sc}}Q)=\Lambda .\\ +&\text{sc}(P,Q)ʻR=CʻP-Pʻʻ(P\overline{R} _{\text{sc}}Q)\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*234·174.*24·411.&\supset \vdash :\text{Hp}.\supset .CʻP=\text{sc}(P,Q)ʻR\cup Pʻʻ(P\overline{R} _{\text{sc}}Q) &\qquad \text{(1)}\\ +\vdash .*234·174. \supset \vdash :\text{Hp}.&\supset .Pʻʻ(P\overline{R} _{\text{sc}}Q)\subset pʻ\overrightarrow{P}_{\text{po}}ʻʻ\text{sc}(P,Q)ʻR.\\ +[*200·5] &\supset .\text{sc}(P,Q)ʻR\cap Pʻʻ(P\overline{R} _{\text{sc}}Q)=\Lambda &\qquad \text{(2)}\\ +\vdash .*24·492.*234·174.&\supset \vdash :\text{Hp}.\supset .\text{sc}(P,Q)ʻR=CʻP-Pʻʻ(P\overline{R} _{\text{sc}}Q) &\qquad \text{(3)}\\ +\vdash .(1).(2).(3).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p>In virtue of this proposition, \(Pʻʻ(P\overline{R} _{\text{sc}}Q)\) and +\(\text{sc}(P,Q)ʻR\) are complementary sections of \(P\), <i>i.e.</i> +they constitute a Dedekind cut in \(P\).</p> + +<p class="nind"><b>*234·181.</b> \[\begin{align}\vdash :P_{\text{po}}\in \text{Ser}.Q_{*}\in &\text{connex} .RʻʻCʻQ\subset CʻP.\supset .\\ +&P\overline{R} _{\text{sc}}Q\cap \text{sc}(P,Q)ʻR=\overrightarrow{\text{max}}_{P}ʻ(P\overline{R} _{\text{sc}}Q).\\ +&\text{sc}(P,Q)ʻR=(CʻP-P\overline{R} _{\text{sc}}Q)\cup \overrightarrow{\text{max}}_{P}ʻ(P\overline{R} _{\text{sc}}Q)\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*234·18.\supset \vdash :\text{Hp}.\supset .P\overline{R} _{\text{sc}}Q\cap \text{sc}(P,Q)ʻR&=P\overline{R} _{\text{sc}}Q-Pʻʻ(P\overline{R} _{\text{sc}}Q)\\ +[*205·111] &=\overrightarrow{\text{max}}_{P}ʻ(P\overline{R} _{\text{sc}}Q) &\qquad \text{(1)}\\ +\vdash .*24·412.*231·13.\supset \\ +\vdash :\text{Hp}.\supset .CʻP-Pʻʻ(P\overline{R} _{\text{sc}}Q)&=\{CʻP-(P\overline{R} _{\text{sc}}Q)\}\cup \{(P\overline{R} _{\text{sc}}Q)-Pʻʻ(P\overline{R} + _{\text{sc}}Q)\}.\\ +[*234·18.*205·111] \supset .\text{sc}(P,Q)ʻR&=(CʻP-P\overline{R} _{\text{sc}}Q)\cup \overrightarrow{\text{max}}_{P}ʻ(P\overline{R} _{\text{sc}}Q) + &\qquad \text{(2)}\\ +\vdash .(1).(2).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*234·182.</b> \[\begin{align}\vdash :P\in \text{Ser}.Q_{*}\in \text{connex} .RʻʻCʻQ&\subset CʻP.\supset .\\ +&\overrightarrow{\text{limax}} _{P}ʻ(P\overline{R} _{\text{sc}}Q)=\overrightarrow{\text{min}}_{P}ʻ\text{sc}(P,Q)ʻR\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*207·51.\supset \vdash \colon\ldotp \text{Hp}.\supset :x&=\text{limax}_Pʻ(P\overline{R} _{\text{sc}}Q).\equiv + .x\in CʻP.\overrightarrow{P}ʻx=Pʻʻ(P\overline{R} _{\text{sc}}Q).\\ +[*234·174] &\equiv .x\in CʻP.\overrightarrow{P}ʻx=CʻP\cap pʻ\overrightarrow{P}ʻʻ\text{sc}(P,Q)ʻR &\qquad \text{(1)}\\ +\vdash .*200·52.\supset \\ +\vdash :\text{Hp}.x\in CʻP.\overrightarrow{P}ʻx&=CʻP\cap pʻ\overrightarrow{P}ʻʻ\text{sc}(P,Q)ʻR.\supset .CʻP\neq CʻP\cap pʻ\overrightarrow{P}ʻʻ\text{sc}(P,Q)ʻR.\\ +[*40·2.\text{Transp}]\supset .\exists !\text{sc}(P,Q)ʻR.\\ +[*40·62] \supset .CʻP\cap pʻ\overrightarrow{P}ʻʻ\text{sc}(P,Q)ʻR&=pʻ\overrightarrow{P}ʻʻ\text{sc}(P,Q)ʻR.\\ +[*13·12] \supset .\overrightarrow{P}ʻx&=pʻ\overrightarrow{P}ʻʻ\text{sc}(P,Q)ʻR &\qquad \text{(2)}\\ +\vdash .*22·621.\supset \\ +\vdash :\overrightarrow{P}ʻx&=pʻ\overrightarrow{P}ʻʻ\text{sc}(P,Q)ʻR.\supset .\overrightarrow{P}ʻx=CʻP\cap pʻ\overrightarrow{P}ʻʻ\text{sc}(P,Q)ʻR &\qquad \text{(3)}\\ +\vdash .(2).(3).\supset \vdash \colon\ldotp \text{Hp}.x\in CʻP.\supset :\\ +\overrightarrow{P}ʻx&=CʻP\cap pʻ\overrightarrow{P}ʻʻ\text{sc}(P,Q)ʻR.\equiv .\overrightarrow{P}ʻx=pʻ\overrightarrow{P}ʻʻ\text{sc}(P,Q)ʻR &\qquad \text{(4)}\\ +\vdash .(1).(4).\supset \\ +\vdash \colon\ldotp \text{Hp}.\supset :x&=\text{limax}_Pʻ(P\overline{R} _{\text{sc}}Q).\equiv .x\in CʻP.\overrightarrow{P}ʻx=pʻ\overrightarrow{P}ʻʻ\text{sc}(P,Q)ʻR.\\ +[*205·67] &\equiv .x=\text{min}_{P}ʻ\text{sc}(P,Q)ʻR\colon\ldotp \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_764">[Pg 764]</span></p> + +<p class="nind"><b>*234·183.</b> \[\begin{align}&\vdash :\text{Hp}*234·18.\text{sc}(P,Q)ʻR=\Lambda .\supset .P\overline{R} _{\text{sc}}Q=CʻP.{\sim}\text{E}!Bʻ\breve{P}\\ +&[*234·181·121]\end{align}\]</p> + +<p class="nind"><b>*234·2.</b> \[\begin{align}&\vdash :P_{\text{po}}\in \text{Ser}.RʻʻCʻQ\subset CʻP.Q_{*}\in \text{connex} .\supset .\\ +\quad\quad\quad\text{os}(P,Q)ʻR=&\{\overrightarrow{\text{min}}_{P}ʻ(\breve{P} \overline{R} _{\text{sc}}Q)-P\overline{R} _{\text{sc}}Q\}\cup \{\overrightarrow{\text{max}}_{P}ʻ(P\overline{R} + _{\text{sc}}Q)-\breve{P} \overline{R} _{\text{sc}}Q\}\cup \\ +&\{\overrightarrow{\text{max}}_{P}ʻ(P\overline{R} _{\text{sc}}Q)\cap \overrightarrow{\text{min}}_{P}ʻ(\breve{P} \overline{R} _{\text{sc}}Q)\}\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{align}&\vdash :P_{\text{po}}\in \text{Ser}.RʻʻCʻQ\subset CʻP.Q_{*}\in \text{connex} .\supset .\\ +\quad\quad\quad\text{os}(P,Q)ʻR=&\{\overrightarrow{\text{min}}_{P}ʻ(\breve{P} \overline{R} _{\text{sc}}Q)-P\overline{R} _{\text{sc}}Q\}\cup \{\overrightarrow{\text{max}}_{P}ʻ(P\overline{R} + _{\text{sc}}Q)-\breve{P} \overline{R} _{\text{sc}}Q\}\cup \\ +&\{\overrightarrow{\text{max}}_{P}ʻ(P\overline{R} _{\text{sc}}Q)\cap \overrightarrow{\text{min}}_{P}ʻ(\breve{P} \overline{R} _{\text{sc}}Q)\}\end{align} +\]</p> + +<p class="nind"><b>*234·201.</b> \[\begin{align}&\vdash :\text{Hp}*234·2.\supset .\text{os}(P,Q)ʻR\subset \overrightarrow{\text{max}}_{P}ʻ(P\overline{R} _{\text{sc}}Q)\cup + \overrightarrow{\text{min}}_{P}ʻ(\breve{P} \overline{R} _{\text{sc}}Q)\\ +&[*234·2]\end{align}\]</p> + +<p class="nind"><b><a id="*234·202">*234·202</a>.</b> \[\begin{align}&\vdash :\text{Hp}*234·2.\supset .\text{os}(P,Q)ʻR\in 0\cup 1\cup 2\\ +&[*234·201.*205·681.*60·391]\end{align}\]</p> + +<p class="nind"><b><a id="*234·203">*234·203</a>.</b> \[\begin{align}&\vdash :\text{Hp}*234·2.P\overline{R} _{\text{os}}Q\in 1.\supset .\\ +&\text{os}(P,Q)ʻP\in 1.\text{os}(P,Q)ʻR = \iota ʻ\text{max}_{P}ʻ(P\overline{R} _{\text{sc}}Q)=\iota ʻ\text{min}_{P}ʻ(\breve{P} \overline{R} _{\text{sc}}Q)=P\overline{R} + _{\text{os}}Q\\ +&[*231·193·103.*205·68.*234·2]\end{align}\]</p> + +<p class="nind"><b>*234·204.</b> \(\vdash :P_{\text{po}}\in \text{Ser}.P\overline{R} _{\text{os}}Q{\sim}\in 0\cup 1.\supset .\text{os}(P,Q)ʻR=\Lambda \quad[*234·103]\)</p> + +<p class="nind"><b>*234·21.</b> \[\begin{align}\vdash :\text{Hp}*234·2.P\overline{R} _{\text{os}}Q = &\Lambda .\supset .\\ +&\text{os}(P,Q)ʻR=\overrightarrow{\text{max}}_{P}ʻ(P\overline{R} _{\text{sc}}Q)\cup \overrightarrow{\text{min}}_{P}ʻ(\breve{P} + \overline{R} _{\text{sc}}Q)\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*205·11·111. \supset\\ +\vdash :\text{Hp}.\supset .\overrightarrow{\text{max}}_{P}ʻ(P\overline{R} _{\text{sc}}Q)\subset -(\breve{P} \overline{R} _{\text{sc}}Q).\overrightarrow{\text{min}}_{P}ʻ(\breve{P} + \overline{R} _{\text{sc}}Q)\subset -(P\overline{R} _{\text{sc}}Q) &\qquad \text{(1)}\\ +\vdash .(1).*234·2.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*234·23.</b> \[\begin{align}&\vdash \colon\ldotp \text{Hp}*234·2.P\overline{R} _{\text{os}}Q{\sim}\in 1.\text{os}(P,Q)ʻR\in 1.\supset :\\ +P\overline{R} _{\text{os}}Q=\Lambda :&\text{os}(P,Q)ʻR=\iota ʻ\text{max}_{P}ʻ(P\overline{R} _{\text{sc}}Q).{\sim}\text{E}!\text{min}_{P}ʻ(\breve{P} + \overline{R} _{\text{sc}}Q).\lor.\\ +&\text{os}(P,Q)ʻR = \iota ʻ\text{min}_{P}ʻ(\breve{P} \overline{R} _{\text{sc}}Q).{\sim}\text{E}!\text{max}_{P}ʻ(P\overline{R} _{\text{sc}}Q)\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*234·103.\supset \vdash :\text{Hp}.&\supset .P\overline{R} _{\text{sc}}Q=\Lambda &\qquad \text{(1)}\\ +[*234·21] &\supset .\text{os}(P,Q)ʻR=\overrightarrow{\text{max}}_{P}ʻ(P\overline{R} _{\text{sc}}Q)\cup \overrightarrow{\text{min}}_{P}ʻ(\breve{P} + \overline{R} _{\text{sc}}Q) &\qquad \text{(2)}\\ +\vdash .*52·41.\supset \vdash :P\overline{R} _{\text{os}}Q=\Lambda .&\text{E}!\text{max}_{P}ʻ(P\overline{R} _{\text{sc}}Q).\text{E}!\text{min}_{P}ʻ(\breve{P} + \overline{R} _{\text{sc}}Q).\supset .\\ +&\{\overrightarrow{\text{max}}_{P}ʻ(P\overline{R} _{\text{sc}}Q)\cup \overrightarrow{\text{min}}_{P}ʻ(\breve{P} \overline{R} _{\text{sc}}Q)\}{\sim}\in + 1 &\qquad \text{(3)}\\ +\vdash.(1).(2).(3).\text{Transp}.\supset \\ +\vdash \colon\ldotp \text{Hp}.\supset :{\sim}&\text{E}!\text{max}_{P}ʻ(P\overline{R} _{\text{sc}}Q).\lor.{\sim}\text{E}!\text{min}_{P}ʻ(\breve{P} \overline{R} _{\text{sc}}Q) + &\qquad \text{(4)}\\ +\vdash .(2).*205·681.\supset \vdash \colon\ldotp \text{Hp}.\supset :&\text{E}!\text{max}_{P}ʻ(P\overline{R} _{\text{sc}}Q).\lor.\text{E}!\text{min}_{P}ʻ(\breve{P} + \overline{R} _{\text{sc}}Q) &\qquad \text{(5)}\\ +\vdash .(1).(2).(4).(5).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_765">[Pg 765]</span></p> + +<p class="nind"><b>*234·24.</b> \[\begin{align}&\vdash \colon\ldotp P \in \text{Ser}. Q_{*} \in \text{connex} . RʻʻCʻQ \subset CʻP . \supset :\\ +&\text{os}(P,Q)ʻR \in 1 . \supset . \text{os}(P,Q)ʻR = \iota ʻ\text{limax}_Pʻ(P\overline{R} _{\text{sc}}Q) + = \iota ʻ\text{limin}_Pʻ(\breve{P} \overline{R} _{\text{sc}}Q)\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *234·203 . *207·42 . \supset \\ +\vdash : \text{Hp} . P\overline{R} _{\text{os}}Q \in 1 . &\supset . \text{os}(P,Q)ʻR = \iota ʻ\text{limax}_Pʻ(P\overline{R} _{\text{sc}}Q) + = \iota ʻ\text{limin}_Pʻ(\breve{P} \overline{R} _{\text{sc}}Q) &\qquad \text{(1)}\\ +\vdash . *234·23 . *211·728 . *207·42 . \supset \\ +\vdash : \text{Hp}.P\overline{R} _{\text{os}}Q{\sim}\in 1 . &\supset .\text{os}(P,Q)ʻR = \iota ʻ\text{limax}_Pʻ(P\overline{R} _{\text{sc}}Q) + = \iota ʻ\text{limin}_Pʻ(\breve{P} \overline{R} _{\text{sc}}Q) &\qquad \text{(2)}\\ +\vdash . (1). (2). \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*234·241.</b> \(\vdash : \text{Hp} *234·2 . \text{os}(P,Q)ʻR \in 2. \supset . P\overline{R} _{\text{os}}Q = \Lambda\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *234·103. &\supset \vdash : \text{Hp}.\supset . P\overline{R} _{\text{os}}Q \in 0\cup 1 &\qquad \text{(1)}\\ +\vdash . *234·203. \text{Transp}. &\supset \vdash : \text{Hp}. \supset . P\overline{R} _{\text{os}}Q{\sim}\in 1 &\qquad \text{(2)}\\ +\vdash . (1). (2). \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*234·242">*234·242</a>.</b> \[\begin{align}&\vdash : \text{Hp} *234·2. \text{os}(P,Q)ʻR \in 2. \supset .\\ +&\text{os}(P,Q)ʻR = \iota ʻ\text{max}_{P}ʻ(P\overline{R} _{\text{sc}}Q)\cup \iota ʻ\text{min}_{P}ʻ(\breve{P} \overline{R} _{\text{sc}}Q) . \text{max}_{P}ʻ(P\overline{R} + _{\text{sc}}Q)P_{1} \text{min}_{P}ʻ(\breve{P} \overline{R} _{\text{sc}}Q)\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *234·201. *205·3. \supset \vdash : \text{Hp}. \supset . \text{E}! &\text{max}_{P}ʻ(P\overline{R} _{\text{sc}}Q). \text{E}! \text{min}_{P}ʻ(\breve{P} + \overline{R} _{\text{sc}}Q).\\ +&\text{max}_{P}ʻ(P\overline{R} _{\text{sc}}Q) \neq \text{min}_{P}ʻ(\breve{P}\overline{R} _{\text{sc}}Q) &\qquad \text{(1)}\\ +\vdash . *234·241·15. \supset \vdash : \text{Hp}. &\supset . \breve{P} \overline{R} _{\text{sc}}Q = CʻP - P\overline{R} _{\text{sc}}Q.\\ +[*211·8.(1)] &\supset . \text{max}_{P}ʻ(P\overline{R} _{\text{sc}}Q) = \text{max}(P_{\text{po}})ʻ(P\overline{R} _{\text{sc}}Q).\\ +&\text{min}_{P}ʻ(\breve{P} \overline{R} _{\text{sc}}Q) =\text{seq}(P_{\text{po}})ʻ(P\overline{R} _{\text{sc}}Q).\\ +[*206·5.*201·63] &\supset . \{\text{max}_{P}ʻ(P\overline{R} _{\text{sc}}Q)\} (P_{\text{po}})_{1} \{\text{min}_{P}ʻ(\breve{P} + \overline{R} _{\text{sc}}Q)\}.\\ +[*121·254] &\supset . \{\text{max}_{P}ʻ(P\overline{R} _{\text{sc}}Q)\} P_{1} \{\text{min}_{P}ʻ(\breve{P} \overline{R} _{\text{sc}}Q)\} + &\qquad \text{(2)}\\ +\vdash . (1). (2). *234·201. \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*234·243.</b> \[\begin{align}\vdash : \text{Hp} *234·24 . \exists ! &\text{os}(P,Q)ʻR. \supset .\\ +&\text{E}! \text{limax}_Pʻ(P\overline{R} _{\text{sc}}Q). \text{E}! \text{limin}_Pʻ(\breve{P} \overline{R} _{\text{sc}}Q)\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *234·202. \supset \vdash :\text{Hp}. \supset . \text{os}(P,Q)ʻP \in 1 \cup 2 &\qquad \text{(1)}\\ +\vdash . (1). *234·24·242. \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*234·244.</b> \(\vdash : \text{Hp} *234·2 . P^{2} = P. \supset . \text{os}(P,Q)ʻR \in 0 \cup 1\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . *234·242·202. \supset \vdash : \text{Hp} *234·2 . \text{os}(P,Q)ʻR {\sim} \in 0 \cup 1 . \supset . \dot{\exists} ! P_{1} &\qquad \text{(1)}\\ +\vdash .(1).\text{Transp}. *201·65. \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*234·25">*234·25</a>.</b> \[\begin{align}\vdash : \text{Hp} *234·2. P^{2} &= P. \exists ! \text{os}(P,Q)ʻR . \supset .\\ +&\text{os}(P,Q)ʻR = \iota ʻ\text{limax}_Pʻ(P\overline{R} _{\text{sc}}Q) = \iota ʻ\text{limin}_Pʻ(\breve{P} \overline{R} _{\text{sc}}Q) +&[*234·244·24]\end{align}\]</p> + +<p><span class="pagenum" id="Page_766">[Pg 766]</span></p> + +<p class="nind"><b><a id="*234·251">*234·251</a>.</b> \[\begin{align}&\vdash :\text{Hp}*234·24.\text{limax}_Pʻ(P\overline{R} _{\text{sc}}Q)=\text{limin}_Pʻ(\breve{P} \overline{R} _{\text{sc}}Q).\supset .\\ +&\text{os}(P,Q)ʻR=\iota ʻ\text{limax}_Pʻ(P\overline{R} _{\text{sc}}Q)=\iota ʻ\text{min}_{P}ʻ\text{sc}(P,Q)ʻR=\iota ʻ\text{max}_{P}ʻ\text{sc}(\breve{P} + ,Q)ʻR\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*234·18.*207·51.\supset \\ +\vdash :\text{Hp}.\supset .\text{sc}(P,Q)ʻR&=CʻP-\overrightarrow{P}ʻ\text{limax}_Pʻ(P\overline{R} _{\text{sc}}Q).\\ +\text{sc}(\breve{P} ,Q)ʻR&=CʻP-\overleftarrow{P}ʻ\text{limin}_Pʻ(\breve{P} \overline{R} _{\text{sc}}Q).\\ +[\text{Hp}.*202·101] \supset .\text{os}(P,Q)ʻR&=CʻP\cap \iota ʻ\text{limax}_Pʻ(P\overline{R} _{\text{sc}}Q).\\ +[*51·31] &=\iota ʻ\text{limax}_Pʻ(P\overline{R} _{\text{sc}}Q) &\qquad \text{(1)}\\ +[*234·182] &=\iota ʻ\text{min}_{P}ʻ\text{sc}(P,Q)ʻR &\qquad \text{(2)}\\ +\left[(2) \frac{\breve{P}}{P}\right] & =\iota ʻ\text{max}_{P}ʻ\text{sc}(\breve{P},Q)ʻR &\qquad \text{(3)}\\ +\vdash .(1).(2).(3).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*234·26.</b> \[\begin{align}\vdash \colon\ldotp \text{Hp}*234·2.P^{2}&=P.\supset :\\ +\exists !\text{os}(P,Q)ʻR.&\equiv .\text{os}(P,Q)ʻR=\iota ʻ\text{limax}_Pʻ(P\overline{R} _{\text{sc}}Q).\\ +&\equiv .\text{os}(P,Q)ʻR=\iota ʻ\text{limin}_Pʻ(\breve{P} \overline{R} _{\text{sc}}Q).\\ +&\equiv .\text{os}(P,Q)ʻR=\iota ʻ\text{min}_{P}ʻ\text{sc}(PQ)ʻR.\\ +&\equiv .\text{os}(P,Q)ʻR=\iota ʻ\text{max}_{P}ʻ\text{sc}(\breve{P} Q)ʻR.\\ +&\equiv .\text{limax}_Pʻ(P\overline{R} _{\text{sc}}Q)=\text{limin}_Pʻ(\breve{P} \overline{R} _{\text{sc}}Q)\\ +&[*234·25·251·182.*51·161]\end{align}\]</p> + +<p class="nind"><b>*234·27.</b> \(\vdash :\text{Hp}*234·24.x\in \text{os}(P,Q)ʻR-\text{ᗡ}ʻP_{1}.\supset .x=\text{limax}_Pʻ(P\overline{R} _{\text{sc}}Q)\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*234·24. &\supset \vdash :\text{Hp}.\text{os}(P,Q)ʻR\in 1.\supset .x=\text{limax}_Pʻ(P\overline{R} _{\text{sc}}Q) &\qquad \text{(1)}\\ +\vdash .*234·242. &\supset \vdash :\text{Hp}.\text{os}(P,Q)ʻR\in 2.\supset .x=\text{limax}_Pʻ(P\overline{R} _{\text{sc}}Q) &\qquad \text{(2)}\\ +\vdash .*234·202. &\supset \vdash :\text{Hp}.\supset .\text{os}(P,Q)ʻR\in 1\cup 2 &\qquad \text{(3)}\\ +\vdash .(1).(2).(3).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*234·271.</b> \[\begin{align}&\vdash :\text{Hp}.234·24.x\in \text{os}(P,Q)ʻR-\text{D}ʻP_{1}.\supset .x=\text{limin}_Pʻ(\breve{P} \overline{R} _{\text{sc}}Q)\\ +&\left[*234·27 \frac{\breve{P}}{P}\right]\end{align}\]</p> + +<p class="nind"><b>*234·272.</b> \[\begin{align}\vdash :\text{Hp}.*234·24.&x\in \text{os}(P,Q)ʻR-CʻP_{1}.\supset .\\ +&x=\text{limax}_Pʻ(P\overline{R} _{\text{sc}}Q)=\text{limin}_Pʻ(\breve{P} \overline{R} _{\text{sc}}Q) \quad[*234·27·271]\end{align}\]</p> + +<p><span class="pagenum" id="Page_767">[Pg 767]</span></p> + +<p>The remaining propositions of the present number are for the most +part immediate consequences of those already proved. In order to +obtain, from propositions already proved, propositions concerning +the limit of a function as the argument approaches the limit of +some class of arguments \(\alpha\), we only have to substitute +\(Q_{*}\unicode{x0294f}\alpha\) for \(Q\). In order to obtain the limit +of a function as the argument approaches a given term \(a\), we take +\(Q_{*}\unicode{x0294f}\overrightarrow{Q}ʻa\) in place of \(Q\).</p> + +<p class="nind"><b>*234·3.</b> \[\begin{align}&\vdash \colon\ldotp x\in \text{sc}(P,Q_{*}\unicode{x0294f}\alpha )ʻR.\equiv :\\ +&x\in CʻP:xP_{\text{po}}w.\supset _{w}.(\exists y).y\in \alpha \cap CʻQ\cap \text{ᗡ}ʻR.Rʻʻ(\alpha \cap \overrightarrow{Q}_{*}ʻy)\subset + \overrightarrow{P}_{\text{po}}ʻw\\ +&[*234·1]\end{align}\]</p> + +<p class="nind"><b>*234·301.</b> \[\begin{align}\vdash \colon\colon Q_{*}\unicode{x0294f}\alpha \in &\text{connex} .\supset \colon\ldotp x\in \text{os}(P,Q_{*}\unicode{x0294f}\alpha + )ʻR\cap \text{D}ʻP\cap \text{ᗡ}ʻP.\equiv :\\ +&x\in \text{D}ʻP\cap \text{ᗡ}ʻP:x\in P(z-w).\supset _{z,w}.\\ +&(\exists y).y\in \alpha \cap CʻQ\cap \text{ᗡ}ʻR.Rʻʻ(\alpha \cap \overleftarrow{Q}_{*}ʻy)\subset P(z-w)\\ +[*234·12]\end{align}\]</p> + +<p class="nind"><b>*234·31.</b> \[\begin{align}&\vdash :P_{\text{po}}\in \text{Ser}.Q_{*}\unicode{x0294f}\alpha \in \text{connex} .Rʻʻ(\alpha \cap CʻQ)\subset CʻP.\supset .\\ +&CʻP-\text{sc}(P,Q_{*}\unicode{x0294f}\alpha )ʻR=CʻP\cap pʻ\overrightarrow{P}_{\text{po}}ʻʻ\text{sc}(P,Q_{*}\unicode{x0294f}\alpha )ʻR=Pʻʻ(P\overline{R} Q)_{\text{sc}}ʻ\alpha + \\ +&[*234·174]\end{align}\]</p> + +<p class="nind"><b>*234·311.</b> \[\begin{align}\vdash :\text{Hp}*234·31.\supset .CʻP=&\text{sc}(P,Q_{*}\unicode{x0294f}\alpha )ʻR\cup Pʻʻ(P\overline{R} Q)_{\text{sc}}ʻ\alpha .\\ +&\text{sc}(P,Q_{*}\unicode{x0294f}\alpha )ʻR\cap Pʻʻ(P\overline{R} Q)_{\text{sc}}ʻ\alpha =\Lambda .\\ +&\text{sc}(P,Q_{*}\unicode{x0294f}\alpha )ʻR=CʻP-Pʻʻ(P\overline{R} Q)_{\text{sc}}ʻ\alpha\\ +&[*234·18]\end{align}\]</p> + +<p class="nind"><b>*234·312.</b> \[\begin{align}\vdash \colon\ldotp P\in \text{Ser}.Q_{*}&\unicode{x0294f}\alpha \in \text{connex} .Rʻʻ(\alpha \cap CʻQ)\subset CʻP.\supset :\\ +&\text{E}!(P\overline{R} Q)_{\text{lmx}}ʻ\alpha .\equiv .\text{E}!\text{min}_{P}ʻ\text{sc}(P,Q_{*}\unicode{x0294f}\alpha )ʻR.\\ +&\equiv .(P\overline{R} Q)_{\text{lmx}}ʻ\alpha =\text{min}_{P}ʻ\text{sc}(P,Q_{*}\unicode{x0294f}\alpha )ʻR\\ +&[*234·182]\end{align}\]</p> + +<p class="nind"><b>*234·32.</b> \[\begin{align}&\vdash \colon\ldotp P_{\text{po}}\in \text{Ser}.Q_{*}\unicode{x0294f}\alpha + \in \text{connex}.Rʻʻ(\alpha\cap CʻQ)\subset CʻP.\supset:(P\overline{R} Q)_{\text{os}}ʻ\alpha \in 1.\supset .\\ +&\text{os}(P,Q_{*}\unicode{x0294f}\alpha )ʻR=(P\overline{R} Q)_{\text{os}}ʻ\alpha =\iota ʻ\text{max}_{P}ʻ(P\overline{R} Q)_{\text{sc}}ʻ\alpha + =\iota ʻ\text{min}_{P}ʻ(\breve{P} \overline{R} Q)_{\text{sc}}ʻ\alpha\\ +&[*234·203]\end{align}\]</p> + +<p class="nind"><b>*234·321.</b> \[\begin{align}&\vdash \colon\colon\text{Hp}*234·32.\text{os}(P,Q_{*}\unicode{x0294f}\alpha )ʻR\in 1.\supset \colon\ldotp (P\overline{R} Q)_{\text{os}}ʻ\alpha + {\sim}\in 1.\supset :\\ +(P\overline{R} Q)_{\text{os}}ʻ\alpha =&\Lambda :\text{os}(P,Q_{*}\unicode{x0294f}\alpha )ʻR=\iota ʻ\text{max}_{P}ʻ(P\overline{R} Q)_{\text{sc}}ʻ\alpha + .{\sim}\text{E}!\text{min}_{P}ʻ(\breve{P} \overline{R} Q)_{\text{sc}}ʻ\alpha .\\ +&\lor.\text{os}(P,Q_{*}\unicode{x0294f}\alpha )ʻR=\iota ʻ\text{min}_{P}ʻ(\breve{P} \overline{R} Q)_{\text{sc}}ʻ\alpha .{\sim}\text{E}!\text{max}_{P}ʻ(P\overline{R} + Q)_{\text{sc}}ʻ\alpha\\ +&[*234·23]\end{align}\]</p> + +<p class="nind"><b>*234·322.</b> \[\begin{align}&\vdash :\text{Hp}.*234·312.\text{os}(P,Q_{*}\unicode{x0294f}\alpha )ʻR\in 1.\supset .\\ +&\text{os}(P,Q_{*}\unicode{x0294f}\alpha )ʻR=\iota ʻ(P\overline{R} Q)_{\text{lmx}}ʻ\alpha =\iota ʻ(\breve{P} \overline{R} Q)_{\text{lmx}}ʻ\alpha +\quad[*234·24]\end{align}\]</p> + +<p class="nind"><b>*234·329.</b> \[\begin{align}&\vdash :\text{Hp}*234·32.\text{os}(P,Q_{*}\unicode{x0294f}\alpha )ʻR\in 2.\supset .\\ +&\text{os}(P,Q_{*}\unicode{x0294f}\alpha )ʻR=\iota ʻ\text{max}_{P}ʻ(P\overline{R} Q)_{\text{sc}}ʻ\alpha \cup \iota ʻ\text{min}_{P}ʻ(\breve{P}\overline{R} Q)_{\text{sc}}ʻ\alpha .\\ +&\{\text{max}_{P}ʻ(P\overline{R} Q)_{\text{sc}\}ʻ\alpha }P_{1}\{\text{min}_{P}ʻ(\breve{P} \overline{R} Q)_{\text{sc}}ʻ\alpha\}\\ +&[*234·242]\end{align}\]</p> + +<p class="nind"><b><a id="*234·33">*234·33</a>.</b> \[\begin{align}\vdash :\text{Hp}&*234·32.P^{2}=P.\exists !\text{os}(P,Q_{*}\unicode{x0294f}\alpha )ʻR.\supset .\\ +&\text{os}(P,Q_{*}\unicode{x0294f}\alpha )ʻR=\iota ʻ(P\overline{R} Q)_{\text{lmx}}ʻ\alpha =\iota ʻ(\breve{P} \overline{R} Q)_{\text{lmx}}ʻ\alpha + \quad[*234·25]\end{align}\]</p> + +<p class="nind"><b>*234·331.</b> \[\begin{align}&\vdash : \text{Hp}*234·312.(P\overline{R} Q)_{\text{lmx}}ʻ\alpha =(\breve{P} \overline{R} Q)_{\text{lmx}}ʻ\alpha .\supset .\\ +\text{os}(P,Q_{*}\unicode{x0294f}\alpha )ʻR&=\iota ʻ(P\overline{R} Q)_{\text{lmx}}ʻ\alpha =\iota ʻ(\breve{P} \overline{R} Q)_{\text{lmx}}ʻ\alpha \\ +&=\iota ʻ\text{min}_{P}ʻ\text{sc}(P,Q_{*}\unicode{x0294f}\alpha )ʻR=\iota ʻ\text{max}_{P}ʻ\text{sc}(\breve{P} ,Q_{*}\unicode{x0294f}\alpha )ʻR\\ +&[*234·251]\end{align}\]</p> + +<p><span class="pagenum" id="Page_768">[Pg 768]</span></p> + +<p class="nind"><b>*234·34.</b> \[\begin{align}&\vdash \colon\ldotp \text{Hp}*234·32.P^{2}=P.\supset :\\ +\exists !\text{os}(P,Q_{*}\unicode{x0294f}\alpha )ʻR.&\equiv .\text{os}(P,Q_{*}\unicode{x0294f}\alpha )ʻR=\iota ʻ(P\overline{R} Q)_{\text{lmx}}ʻ\alpha .\\ +&\equiv .\text{os}(P,Q_{*}\unicode{x0294f}\alpha )ʻR=\iota ʻ(\breve{P} \overline{R} Q)_{\text{lmx}}ʻ\alpha .\\ +&\equiv .(P\overline{R} Q)_{\text{lmx}}ʻ\alpha =(\breve{P} \overline{R} Q)_{\text{lmx}}ʻ\alpha \\ +&[*234·26]\end{align}\]</p> + +<p class="nind"><b>*234·35.</b> \[\begin{align}&\vdash :\text{Hp}*234·312.x\in \text{os}(P,Q_{*}\unicode{x0294f}\alpha )ʻR-\text{ᗡ}ʻP_{1}.\supset + .x=(P\overline{R} Q)_{\text{lmx}}ʻ\alpha\\ +&[*234·27]\end{align}\]</p> + +<p class="nind"><b>*234·351.</b> \[\begin{align}&\vdash :\text{Hp}*234·312.x\in \text{os}(P,Q_{*}\unicode{x0294f}\alpha )ʻR-\text{D}ʻP_{1}.\supset.x=(\breve{P} \overline{R} Q)_{\text{lmx}}ʻ\alpha + \\ +&\left[*234·35 \frac{\breve{P}}{P}\right]\end{align}\]</p> + +<p class="nind"><b>*234·352.</b> \[\begin{align}\vdash :\text{Hp}*234·312.x\in &\text{os}(P,Q_{*}\unicode{x0294f}\alpha )ʻR-CʻP_{1}.\supset .\\ +&x=(P\overline{R} Q)_{\text{lmx}}ʻ\alpha =(\breve{P} \overline{R} Q)_{\text{lmx}}ʻ\alpha \quad[*234·35·351]\end{align}\]</p> + +<p class="nind"><b>*234·4.</b> \[\begin{align}&\vdash \colon\ldotp x\in \text{sc}(P,Q_{*}\unicode{x0294f}\overrightarrow{Q}_{\text{po}}ʻa)ʻR.\equiv :\\ +&x\in CʻP:xP_{\text{po}}w.\supset _{w}.(\exists y).y\in \overrightarrow{Q}_{\text{po}}ʻa\cap \text{ᗡ}ʻR.RʻʻQ(y\unicode{x27dd} a)\subset \overrightarrow{P}_{\text{po}}ʻw\\ +&[*234·3.(*121·012)]\end{align}\]</p> + +<p class="nind"><b><a id="*234·41">*234·41</a>.</b> \[\begin{align}&\vdash \colon\colon Q\in \text{trans}.Q_{*}\unicode{x0294f}\overrightarrow{Q}ʻa\in \text{connex} .\supset \colon\ldotp \\ +&x\in \text{os}(P,Q_{*}\unicode{x0294f}\overrightarrow{Q}ʻa)ʻR\cap \text{D}ʻP\cap \text{ᗡ}ʻP.\equiv :x\in \text{D}ʻP\cap \text{ᗡ}ʻP:\\ +&x\in P(z-w).\supset _{z,w}.(\exists y).y\in \overrightarrow{Q}ʻa\cap \text{ᗡ}ʻR.RʻʻQ(y\unicode{x27dd} a)\subset P(z-w)\\ +&[*234·301.(*121·012).*201·18]\end{align}\]</p> + +<p class="nind"><b>*234·42.</b> \[\begin{align}\vdash \colon\ldotp P\in \text{Ser}.Q\in &\text{trans}.Q_{*}\unicode{x0294f}\overrightarrow{Q}ʻa\in + \text{connex} .Rʻʻ\overrightarrow{Q}ʻa\subset CʻP.\supset :\\ +&\overrightarrow{R(PQ)}ʻa=\overrightarrow{\text{min}}_{P}ʻ\text{sc}(P,Q_{*}\unicode{x0294f}\overrightarrow{Q}ʻa)ʻR \quad[*234·182]\end{align}\]</p> + +<p class="nind"><b>*234·421.</b> \[\begin{align}\vdash \colon\ldotp P_{\text{po}}\in &\text{Ser}.Q\in \text{trans}.Q_{*}\unicode{x0294f}\overrightarrow{Q}ʻa\in + \text{connex} .Rʻʻ\overrightarrow{Q}ʻa\subset CʻP.\supset :\\ +&\text{sc}(P,Q_{*}\unicode{x0294f}\overrightarrow{Q}ʻa)ʻR=\Lambda .\supset .\overrightarrow{R(PQ)}ʻa=\overrightarrow{B}ʻ\breve{P} \quad[*234·183]\end{align}\]</p> + +<p class="nind"><b>*234·422.</b> \[\begin{align}&\vdash :\text{Hp}*234·42.P\in \text{Ded}.\supset .R(PQ)ʻa=\text{limin}_Pʻ\text{sc}(P,Q_{*}\unicode{x0294f}\overrightarrow{Q}ʻa)ʻR\\ +&[*233·13.*234·42]\end{align}\]</p> + +<p class="nind"><b><a id="*234·43">*234·43</a>.</b> \[\begin{align}\vdash :\text{Hp}*234·42.&\text{os}(P,Q_{*}\unicode{x0294f}\overrightarrow{Q}ʻa)ʻR\in 1.\supset .\\ +&\text{os}(P,Q_{*}\unicode{x0294f}\overrightarrow{Q}ʻa)=\iota ʻR(PQ)ʻa= \iota ʻR(\breve{P} Q)ʻa \quad[*234·322]\end{align}\]</p> + +<p class="nind"><b>*234·439.</b> \[\begin{align}\vdash :\text{Hp}&*234·421.\text{os}(P,Q_{*}\unicode{x0294f}\overrightarrow{Q}ʻa)ʻR\in 2.\supset .\\ +&\text{os}(P,Q_{*}\unicode{x0294f}\overrightarrow{Q}ʻa)ʻR=\iota ʻR(PQ)ʻa\cup \iota ʻR(\breve{P} Q)ʻa.\\ +&\{R(PQ)ʻa\}P_{1}\{R(\breve{P} Q)ʻa\} \quad[*234·329]\end{align}\]</p> + +<p class="nind"><b>*234·44.</b> \[\begin{align}\vdash :\text{Hp}&*234·421.P^{2}=P.\exists !\text{os}(P,Q_{*}\unicode{x0294f}\overrightarrow{Q}ʻa)ʻR.\supset .\\ +&\text{os}(P,Q_{*}\unicode{x0294f}\overrightarrow{Q}ʻa)ʻR=\iota ʻR(PQ)ʻa=\iota ʻR(\breve{P} Q)ʻa \quad[*234·33]\end{align}\]</p> + +<p><span class="pagenum" id="Page_769">[Pg 769]</span></p> + +<p class="nind"><b>*234·441.</b> \[\begin{align}\vdash :\text{Hp}*234·42.&R(PQ)ʻa=R(\breve{P} Q)ʻa.\supset .\\ +&\text{os}(P,Q_{*}\unicode{x0294f}\overrightarrow{Q}ʻa)=\iota ʻR(PQ)ʻa=\iota ʻR(\breve{P} Q)ʻa \quad[*234·331]\end{align}\]</p> + +<p class="nind"><b>*234·45.</b> \[\begin{align}&\vdash \colon\ldotp P\in \text{Ser}.Q\in \text{trans}.Q_{*}\unicode{x0294f}\overrightarrow{Q}ʻa\in + \text{connex} .Rʻʻ\overrightarrow{Q}ʻa\subset CʻP.P^{2}=P.\supset :\\ +\exists !\text{os}(P,Q_{*}\unicode{x0294f}\overrightarrow{Q}ʻa)ʻR.&\equiv .\text{os}(P,Q_{*}\unicode{x0294f}\overrightarrow{Q}ʻa)ʻR=\iota ʻR(PQ)ʻa.\\ +&\equiv .\text{os}(P,Q_{*}\unicode{x0294f}\overrightarrow{Q}ʻa)ʻR=\iota ʻR(\breve{P} Q)ʻa.\\ +&\equiv .R(PQ)ʻa=R(\breve{P} Q)ʻa \quad[*234·34]\end{align}\]</p> + +<p class="nind"><b>*234·46.</b> \[\begin{align}&\vdash :\text{Hp}*234·42.x\in \text{os}(P,Q_{*}\unicode{x0294f}\overrightarrow{Q}ʻa)ʻR-\text{ᗡ}ʻP_1.\supset .x=R(PQ)ʻa\\ +&[*234·35]\end{align}\]</p> + +<p class="nind"><b>*234·461.</b> \[\begin{align}&\vdash :\text{Hp}*234·42.x\in \text{os}(P,Q_{*}\unicode{x0294f}\overrightarrow{Q}ʻa)ʻR-\text{D}ʻP_1.\supset .x=R(\breve{P} Q)ʻa\\ +&\left[*234·46 \frac{\breve{P}}{P}\right]\end{align}\]</p> + +<p class="nind"><b><a id="*234·462">*234·462</a>.</b> \[\begin{align}\vdash :\text{Hp}*234·42.x\in &\text{os}(P,Q_{*}\unicode{x0294f}\overrightarrow{Q}ʻa)ʻR-CʻP_1.\supset .\\ +&x=R(PQ)ʻa=R(\breve{P} Q)ʻa \quad[*234·46·461]\end{align}\]</p> + +<p class="nind"><b>*234·5.</b> \(\vdash :a\in \text{ct}(PQ)ʻR.\equiv .Rʻa\in \text{os}(P,Q_{*}\unicode{x0294f}\overrightarrow{Q}_{\text{po}}ʻa)ʻR-CʻP_1 \quad[(*234·03)]\)</p> + +<p class="nind"><b>*234·51.</b> \[\begin{align}&\vdash \colon\colon Q\in \text{trans}.Q_{*}\unicode{x0294f}\overrightarrow{Q}ʻa\in + \text{connex} .Rʻa\in \text{D}ʻP\cap \text{ᗡ}ʻP.\supset \colon\ldotp \\ +&a\in \text{ct}(PQ)ʻR.\equiv :Rʻa{\sim}\in CʻP_1:Rʻa\in P(z-w).\supset _{z,w}.\\ +&(\exists y).y\in \overrightarrow{Q}ʻa\cap \text{ᗡ}ʻR.RʻʻQ(y\vdash\dashv a)\subset P(z-w)\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*234·5·4.*53·31.\supset \\ +\vdash \colon\colon\text{Hp}.\supset \colon\ldotp a\in \text{ct}(PQ)ʻR.\equiv :Rʻa\in \text{D}ʻP\cap \text{ᗡ}ʻP-CʻP_1:Rʻa\in P(z-w).\supset _{z,w}.\\ +(\exists y).y\in \overrightarrow{Q}ʻa\cap \text{ᗡ}ʻR.RʻʻQ(y\unicode{x27dd} a)\subset P(z-w).Rʻʻ\iota ʻa\subset P(z-w) &\qquad \text{(1)}\\ +\vdash .(1).*121·242.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*234·52">*234·52</a>.</b> \[\begin{align}\vdash \colon\ldotp P\in &\text{Ser}.Q\in \text{trans}.Q_{*}\unicode{x0294f}\overrightarrow{Q}ʻa\in + \text{connex} .Rʻʻ\overrightarrow{Q}ʻa\subset CʻP.\supset :\\ +&a\in \text{ct}(PQ)ʻR.\supset .R(PQ)ʻa=R(\breve{P} Q)ʻa=Rʻa \quad[*234·462·5]\end{align}\]</p> + +<p class="nind"><b>*234·521.</b> \[\begin{align}&\vdash :\text{Hp}*234·52.a\in \text{ct}(PQ)ʻR.\supset .\text{os}(P,Q_{*}\unicode{x0294f}\overrightarrow{Q}ʻa)=\iota ʻRʻa\\ +&[*234·441·52]\end{align}\]</p> + +<p class="nind"><b>*234·522.</b> \[\begin{align}\vdash \colon\ldotp \text{Hp}*234·52.P^{2}=&P.\supset :\\ +&a\in \text{ct}(PQ)ʻR.\equiv .R(PQ)ʻa=R(\breve{P} Q)ʻa=Rʻa\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*234·45.\supset \\ +\vdash \colon\ldotp \text{Hp}.\supset :R(PQ)ʻa=R(\breve{P} Q)ʻa=Rʻa.&\supset .\text{os}(P,Q_{*}\unicode{x0294f}\overrightarrow{Q}ʻa)ʻR=\iota ʻRʻa.\\ +[*234·5.*201·65] &\supset .a\in \text{ct}(PQ)ʻR &\qquad \text{(1)}\\ +\vdash .(1).*234·52.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_770">[Pg 770]</span></p> + +<p class="nind"><b>*234·53.</b> \[\begin{align}\vdash \colon\colon P_{\text{po}}&\in \text{connex} .Q\in \text{trans}.Rʻa=BʻP.\supset \colon\ldotp \\ +a\in \text{ct}(PQ)ʻR.&\equiv :BʻP{\sim}\in \text{D}ʻP_{1}:w\in \text{ᗡ}ʻP.\supset _{w}.\\ +&(\exists y).y\in \overrightarrow{Q}ʻa\cap \text{ᗡ}ʻR.RʻʻQ(y\vdash\dashv a)\subset \overrightarrow{P}_{\text{po}}ʻw\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*234·122.*53·31.*234·5.\supset \\ +\vdash \colon\colon\text{Hp}.\supset \colon\ldotp a\in \text{ct}(PQ)ʻR.&\equiv :BʻP{\sim}\in \text{D}ʻP_{1}:(BʻP)P_{\text{po}}w.\supset _{w}.\\ +&(\exists y).y\in \overrightarrow{Q}ʻa\cap \text{ᗡ}ʻR.Rʻʻ(\overleftarrow{Q}_{*}ʻy\cap \overrightarrow{Q}ʻa)\subset \overrightarrow{P}_{\text{po}}ʻw.Rʻʻ\iota + ʻa\subset \overrightarrow{P}_{\text{po}}ʻw:\\ +[*202·522.*205·253.*201·18]&\equiv :BʻP{\sim}\in \text{D}ʻP_{1}:\\ +&w\in \text{ᗡ}ʻP.\supset _{w}.(\exists y).y\in \overrightarrow{Q}ʻa\cap \text{ᗡ}ʻR.RʻʻQ(y\vdash\dashv a)\subset \overrightarrow{P}_{\text{po}}ʻw\colon\colon\supset + \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*234·54.</b> \(\vdash :a\in \text{ct}(PQ)ʻR.\supset .a\in \text{ᗡ}ʻR\cap \breve{Q} _{\text{po}}ʻʻ\text{ᗡ}ʻR.Rʻa\in CʻP\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*234·5·1.(*234·02).&\supset \vdash :\text{Hp}.\supset .Rʻa\in CʻP &\qquad \text{(1)}\\ +\vdash .(1).*234·5.(*234·02).\supset \\ +\vdash \colon\ldotp \text{Hp}.&\supset :\exists !\text{sc}(P,Q_{*}\unicode{x0294f}\overrightarrow{Q}_{\text{po}}ʻa)ʻR\cap + \text{D}ʻP.\lor.\exists !\text{sc}(\breve{P} ,Q_{*}\unicode{x0294f}\overrightarrow{Q}_{\text{po}}ʻa)ʻR\cap \text{ᗡ}ʻP:\\ +[*234·142]&\supset :\exists !\overrightarrow{Q}_{\text{po}}ʻa\cap \text{ᗡ}ʻR:\\ +[*37·46] &\supset :a\in \breve{Q} _{\text{po}}ʻʻ\text{ᗡ}ʻR &\qquad \text{(2)}\\ +\vdash .(1).*14·21.*33·43.&\supset \vdash :\text{Hp}.\supset .a\in \text{ᗡ}ʻR &\qquad \text{(3)}\\ +\vdash .(1).(2).(3).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*234·55.</b> \(\vdash .{\sim}\{\text{min}(Q_{\text{po}})ʻ\text{ᗡ}ʻR\in \text{ct}(PQ)ʻR\} \quad[*234·54.\text{Transp}]\)</p> + +<p class="nind"><b>*234·56.</b> \[\begin{align}&\vdash :\text{Hp}*234·52.a\in \text{ct}(PQ)ʻR.\supset .\\ +&(P\overline{R} Q)_{\text{os}}ʻ\overrightarrow{Q}ʻa\in 0\cup 1.\text{E}!R(PQ)ʻa.R(PQ)ʻa{\sim}\in CʻP_{1}.Rʻa=R(PQ)ʻa\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*234·5. \supset \vdash :\text{Hp}.&\supset .\exists !\text{os}(P,Q_{*}\unicode{x0294f}\overrightarrow{Q}ʻa)ʻR.Rʻa{\sim}\in CʻP_{1} .\\ +[*234·103] &\supset .(P\overline{R} Q)_{\text{os}}ʻ\overrightarrow{Q}ʻa\in 0\cup 1.Rʻa{\sim}\in CʻP_{1} &\qquad \text{(1)}\\ +\vdash .*234·52.\supset \vdash :\text{Hp}.&\supset .Rʻa=R(PQ)ʻa.\text{E}!R(PQ)ʻa &\qquad \text{(2)}\\ +\vdash .(1).(2).\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*234·561.</b> \[\begin{align}\vdash :P,Q\in \text{Ser}.&a\in \text{ct}(PQ)ʻR.a=\text{lt}_{Q}ʻ(\alpha \cap \text{ᗡ}ʻR).Rʻʻ\overrightarrow{Q}ʻa\subset CʻP.\supset .\\ +&(P\overline{R}Q)_{\text{lmx}}ʻ\alpha =Rʻa=(\breve{P} \overline{R} Q)_{\text{lmx}}ʻ\alpha \quad[*233·515.*234·56]\end{align}\]</p> + +<p class="nind"><b><a id="*234·562">*234·562</a>.</b> \[\begin{align}\vdash :P,Q\in &\text{Ser}.\text{lt}_{Q}ʻ(\alpha \cap \text{ᗡ}ʻR)\in \text{ct}(PQ)ʻR.Rʻʻ(\alpha \cap CʻQ)\subset CʻP.\supset .\\ +&(P\overline{R} Q)_{\text{lmx}}ʻ\alpha =(\breve{P} \overline{R} Q)_{\text{lmx}}ʻ\alpha =Rʻ\text{lt}_{Q}ʻ\alpha \quad[*233·516.*234·56]\end{align}\]</p> + +<p><span class="pagenum" id="Page_771">[Pg 771]</span></p> + +<p>That is, if \(\alpha\) is any class of arguments having a limit at +which the function is continuous, then the limit of the function, as +the argument approaches the limit of the set of arguments, is the value +of the function for that limit.</p> + +<p class="nind"><b>*234·6.</b> \(\vdash :a\in \text{contin}(PQ)ʻR.\equiv .a\in \text{ct}(PQ)ʻR\cap \text{ct}(P\breve{Q} )ʻR \quad[(*234·04)]\)</p> + +<p class="nind"><b>*234·61.</b> \[\begin{align}&\vdash \colon\colon P_{\text{po}}\in \text{Ser}.Q\in \text{trans}.Q_{*}\unicode{x0294f}\overleftrightarrow{Q}ʻa\in + \text{connex} .Rʻa\in \text{D}ʻP\cap \text{ᗡ}ʻP.\supset \colon\ldotp \\ +&a\in \text{contin}(PQ)ʻR.\equiv :Rʻa{\sim}\in CʻP_{1}:Rʻa\in P(z-w).\supset _{z,w}.\\ +&(\exists y,y').a\in Q(y-y').y,y'\in \text{ᗡ}ʻR.RʻʻQ(y\vdash\dashv y')\subset P(z-w)\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*234·51.\supset \vdash \colon\colon\text{Hp}.\supset \colon\ldotp a\in &\text{contin}(PQ)ʻR.\equiv :\\ +&Rʻa\in \text{D}ʻP\cap \text{ᗡ}ʻP-CʻP_{1}:Rʻa\in P(z-w).\supset _{z,w}.\\ +&(\exists y,y').y\in \overrightarrow{Q}ʻa\cap \text{ᗡ}ʻR.y'\in \overleftarrow{Q}ʻa\cap \text{ᗡ}ʻR.\\ +&RʻʻQ(y\vdash\dashv a)\cup RʻʻQ(a\vdash\dashv y')\subset P(z-w) &\qquad \text{(1)}\\ +\vdash .(1).*201·19.*202·17.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*234·62">*234·62</a>.</b> \[\begin{align}\vdash \colon\ldotp \text{Hp}*234·61.&P\in \text{trans}.Rʻʻ\overleftrightarrow{Q}ʻa\subset CʻP.\supset :a\in \text{contin}(PQ)ʻR.\supset .\\ +&R(PQ)ʻa=R(\breve{P} Q)ʻa=R(P\breve{Q} )ʻa=R(\breve{P} \breve{Q} )ʻa=Rʻa\\ +[*234·52·6]\end{align}\]</p> + +<p class="nind"><b><a id="*234·63">*234·63</a>.</b> \[\begin{align}\vdash \colon\ldotp \text{Hp}*234·62.&P^{2}=P.\supset :a\in \text{contin}(PQ)ʻR.\equiv .\\ +&R(PQ)ʻa = R(\breve{P} Q)ʻa = R(P\breve{Q} )ʻa = R(\breve{P} \breve{Q} )ʻa = Rʻa\\ +[*234·522·6]\end{align}\]</p> + +<p class="nind"><b><a id="*234·64">*234·64</a>.</b> \[\begin{align}&\vdash \colon\colon\text{Hp}*234·62.Rʻa\in \text{D}ʻP\cap \text{ᗡ}ʻP.\supset \colon\ldotp a\in \text{contin}(P,Q)ʻR.\equiv :\\ +Rʻa\in CʻP-&CʻP_{1}:Rʻa\in P(z-w).\supset _{z,w}.\\ +&(\exists y,y').y,y'\in \text{ᗡ}ʻR.a\in Q(y-y').RʻʻQ(y\vdash\dashv y')\subset P(z-w)\\ +&[*234·51·6]\end{align}\]</p> + +<p class="nind"><b>*234·7.</b> \[\begin{align}&\vdash :R\in P\,\overline{\text{contin}}\,Q.\equiv .\exists !CʻQ\cap \text{ᗡ}ʻR.CʻQ\cap \text{ᗡ}ʻR\subset \text{contin}(PQ)ʻR\\ +&[(*234·04)]\end{align}\]</p> + +<p class="nind"><b>*234·71.</b> \(\vdash :R\in P\,\overline{\text{contin}}\,Q.\supset .R\upharpoonright CʻQ\in 1\rightarrow \text{Cls}.RʻʻCʻQ\subset CʻP\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*234·7·6·5.\supset \\ +\vdash \colon\ldotp \text{Hp}.\supset :a\in CʻQ\cap \text{ᗡ}ʻR.&\supset .Rʻa\in \text{os}(P,Q_{*}\unicode{x0294f}\overrightarrow{Q}_{\text{po}}ʻa)ʻR.\\ +[*234·1] &\supset .Rʻa\in CʻP. &\qquad \text{(1)}\\ +[*14·21] &\supset .\text{E}!Rʻa &\qquad \text{(2)}\\ +\vdash .(2).*71·572. \supset \vdash :\text{Hp}.&\supset .R\upharpoonright CʻQ\in 1\rightarrow \text{Cls} &\qquad \text{(3)}\\ +\vdash .(1).(2).*37·61.&\supset \vdash :\text{Hp}.\supset .Rʻʻ(CʻQ\cap \text{ᗡ}ʻR)\subset CʻP &\qquad \text{(4)}\\ +\vdash .(3).(4).*37·26.&\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*234·72.</b> \[\begin{align}&\vdash \colon\ldotp P\in \text{Ser}.Q\in \text{trans}\cap \text{connex} .R\in P\,\overline{\text{contin}}\,Q.\supset :\\ +&a\in CʻQ\cap \text{ᗡ}ʻR.\supset _{a}.R(PQ)ʻa=R(\breve{P} Q)ʻa=R(P\breve{Q})ʻa=R(\breve{P} \breve{Q} )ʻa=Rʻa\\ +&[*234·62·7]\end{align}\]</p> + +<p><span class="pagenum" id="Page_772">[Pg 772]</span></p> + +<p class="nind"><b><a id="*234·73">*234·73</a>.</b> \[\begin{align}\vdash \colon\colon P\in \text{Ser}.&P^{2}=P.Q\in \text{trans}\cap \text{connex} .\supset \colon\ldotp \\ +&R\in P\,\overline{\text{contin}}\,Q.\equiv :\exists !CʻQ\cap \text{ᗡ}ʻR:a\in CʻQ\cap \text{ᗡ}ʻR.\supset _{a} .\\ +&R(PQ)ʻa=R(\breve{P} Q)ʻa=R(P\breve{Q} )ʻa=R(\breve{P} \breve{Q} )ʻa=Rʻa\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .*234·7·71.\supset \vdash \colon\colon\text{Hp}.\supset \colon\ldotp R\in P&\overline{\text{contin}} Q.\equiv :\exists !CʻQ\cap \text{ᗡ}ʻR.RʻʻCʻQ\subset CʻP:\\ +&a\in CʻQ\cap \text{ᗡ}ʻR.\supset _{a}.a\in \text{contin}(PQ)ʻR:\\ +[*234·63]\equiv :\exists !&CʻQ\cap \text{ᗡ}ʻR.RʻʻCʻQ\subset CʻP:a\in CʻQ\cap \text{ᗡ}ʻR.\supset _{a}.\\ +&R(PQ)ʻa=R(\breve{P} Q)ʻa=R(P\breve{Q} )ʻa=R(\breve{P} \breve{Q} )ʻa=Rʻa &\qquad \text{(1)}\\ +\vdash .*233·401·101.\supset \\ +\vdash \colon\ldotp a\in CʻQ\cap \text{ᗡ}ʻR.\supset _{a}.R(PQ)ʻa=Rʻa:&\supset :a\in CʻQ\cap \text{ᗡ}ʻR.\supset _{a}.Rʻa\in CʻP:\\ +[*37·61·26] &\supset :RʻʻCʻQ\subset CʻP &\qquad \text{(2)}\\ +\vdash .(1).(2).\supset \vdash .\text{Prop} +\end{array} +\]</p> + + + +<div class="footnotes"><h3>FOOTNOTES:</h3> + +<div class="footnote"> + +<p class="nind"> +<a id="Footnote_17" href="#FNanchor_17" class="label">[17]</a> +<i>Theorie der Functionen einer veränderlichen reellen +Grösse</i>, Chap. <span class="allsmcap">IV</span>. § 30, p. 50.</p> + +</div> +</div> + + + +<hr class="tb"> + + +<p class="nindc space-above2 space-below2"> +CAMBRIDGE: PRINTED BY JOHN CLAY, M.A. AT THE UNIVERSITY PRESS</p> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<div class="transnote spa1"> +<p class="nindc"><b>TRANSCRIBER’S NOTES</b></p> + + +<p>All items in the Errata, from all three volumes, have been +added and corrected accordingly. +</p> + +<p>The author's notation as *102·72·73 is an abbreviation for *102·72 +and *102·73 respectively.</p> + +<p>The lemmas *113·01 (page 302): *120·450 (page 209); *122·436 (page +268) *122·473 and *126·122 (page xxxiii); *124·62 (page 279); *151·45 +(page 315); *165·372 (page 387) although they were mentioned by the +authors, they have not been described in the corresponding sections.</p> + +<p>From Section C, the authors use the lower case “a” as a +limit and the Greek letter \(\alpha\) as a class.</p> + +<p>The alt texts for the illustrations in this book have been created +by the Post-Processor.</p> + +</div></div> + +</body> +</html> diff --git a/78255-src/README-math.txt b/78255-src/README-math.txt new file mode 100644 index 0000000..adabf5a --- /dev/null +++ b/78255-src/README-math.txt @@ -0,0 +1,96 @@ +MathJax HTML source file instructions +================================== +This project is a math heavy eBook. The source is a preliminary HTML file that +uses MathJax to define mathematical expressions, which is processed to generate +a final HTML file with SVG images. + +This source file is kept for the purpose of applying errata fixes. Although the +MathJax takes some learning, it is clearer than the generated final. This also +allows the SVG images to be regenerated with changes. + + +Tools +===== +See the ppmath GitHub repository: + https://github.com/DistributedProofreaders/ppmath. +Follow the instructions to install m2svg. + +Command line: + m2svg -i input.htm -o output.htm + +- The SVG files will be placed in a subdirectory of the working directory + called "images". + +- In the converted file, the maths expressions, delimited by the tags `\[` + and `\]` for *display* expressions or `\(` and `\)` for *inline* + expressions, are replaced by `<img>` links. + +- The "data-tex" attribute will contain the original maths expression. + + +Inline code example +=================== +For the expression \(\mathrm{AB}^{2} = \mathrm{AG} \times \mathrm{BD}\), the +input `\(\mathrm{AB}^{2} = \mathrm{AG} \times \mathrm{BD}\)` + +becomes + `<span class="nowrap"><img style="vertical-align: -0.186ex; width: 16.872ex; + height: 2.253ex;" src="images/4.svg" alt="" data-tex="\mathrm{AB}^{2} + = \mathrm{AG} \times \mathrm{BD}">,</span>` + +The file images/4.svg displays the desired expression. + + +Source files structure +====================== +(eBook 75107 is used as an example) + +- 75107/ + - README-math.txt (this file) + - 75107-h/ + - 75107-h.htm (final HTML file) + - images/ + - 75107-src/ + - 75107-src.htm (source HTML file with MathJax) + + +SVG fixup for ebookmaker +======================== +Now, the SVG files contain a "data-variant" attribute that causes errors. +It needs to be removed by downloading and running this utility: +https://github.com/user-attachments/files/25548572/remove_data_variant_attribute.py + +Command line: + python remove_data_variant_attribute.py images + +Hopefully, this step will be removed in the future. + + +Submission process +================== +- Generated final HTML and images should be submitted as normal. +- In addition, the source HTML will be included, and needs to be renamed + to #####-src.htm by the whitewasher or the Workflow app. +- This readme will need to be added by the whitewasher or the Workflow app. + - Having it with the eBook makes it obvious, and avoids issues with + procedures changing in the future. + + +Errata process +============== +(eBook 75107 is used as an example) + +1. Download the project files using Errata Workbench, and unzip. +2. Install m2svg if not already done. +3. Make desired changes to 75107-src.htm. +4. Execute command line `m2svg -i 75107-src.htm -o 75107-h.htm` + - The image files will be placed in a subdirectory of the working directory + called images. +5. Move 75107-h.htm to the 75107-h directory. +6. Move the contents of the images directory to the 75107-h/images directory. + - Rename the existing 75107-h/images directory to images-old. + - Move the new images directory to 75107-h. + - Check images-old, move any non-generated images (JPG, PNG, etc.). to + images. + - Remove images-old and any other temporary files. +7. Zip the project directory and upload to Errata Workbench. diff --git a/LICENSE.txt b/LICENSE.txt index 6c72794..b5dba15 100644 --- a/LICENSE.txt +++ b/LICENSE.txt @@ -7,5 +7,5 @@ the "Copyright How-To" at https://www.gutenberg.org. No investigation has been made concerning possible copyrights in jurisdictions other than the United States. Anyone seeking to utilize -this eBook outside of the United States should confirm copyright +this book outside of the United States should confirm copyright status under the laws that apply to them. @@ -1,2 +1,2 @@ -Project Gutenberg (https://www.gutenberg.org) public repository for eBook #78255 -(https://www.gutenberg.org/ebooks/78255) +Project Gutenberg (https://www.gutenberg.org) public repository for +book #78255 (https://www.gutenberg.org/ebooks/78255) |