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id="toc-Open_questions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Open_questions"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Open questions</span> </div> </a> <ul id="toc-Open_questions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Diagonalization_in_broader_context" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Diagonalization_in_broader_context"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Diagonalization in broader context</span> </div> </a> <ul id="toc-Diagonalization_in_broader_context-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Version_for_Quine's_New_Foundations" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Version_for_Quine's_New_Foundations"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Version for Quine's New Foundations</span> </div> </a> <ul id="toc-Version_for_Quine's_New_Foundations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Notes</span> </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Cantor's diagonal argument</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 34 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-34" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">34 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%AD%D8%AC%D8%A9_%D9%83%D8%A7%D9%86%D8%AA%D9%88%D8%B1_%D8%A7%D9%84%D9%82%D8%B7%D8%B1%D9%8A%D8%A9" title="حجة كانتور القطرية – Arabic" lang="ar" hreflang="ar" data-title="حجة كانتور القطرية" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Diagonalitzaci%C3%B3_de_Cantor" title="Diagonalització de Cantor – Catalan" lang="ca" hreflang="ca" data-title="Diagonalització de Cantor" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Cantorova_diagon%C3%A1ln%C3%AD_metoda" title="Cantorova diagonální metoda – Czech" lang="cs" hreflang="cs" data-title="Cantorova diagonální metoda" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Cantors_diagonalbevis" title="Cantors diagonalbevis – Danish" lang="da" hreflang="da" data-title="Cantors diagonalbevis" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Cantors_zweites_Diagonalargument" title="Cantors zweites Diagonalargument – German" lang="de" hreflang="de" data-title="Cantors zweites Diagonalargument" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Cantori_diagonaalt%C3%B5estus" title="Cantori diagonaaltõestus – Estonian" lang="et" hreflang="et" data-title="Cantori diagonaaltõestus" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Argumento_de_la_diagonal_de_Cantor" title="Argumento de la diagonal de Cantor – Spanish" lang="es" hreflang="es" data-title="Argumento de la diagonal de Cantor" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Diagonala_argumento_de_Cantor" title="Diagonala argumento de Cantor – Esperanto" lang="eo" hreflang="eo" data-title="Diagonala argumento de Cantor" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Cantorren_Argudio_Diagonala" title="Cantorren Argudio Diagonala – Basque" lang="eu" hreflang="eu" data-title="Cantorren Argudio Diagonala" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%A7%D8%B3%D8%AA%D8%AF%D9%84%D8%A7%D9%84_%D9%82%D8%B7%D8%B1%DB%8C_%DA%A9%D8%A7%D9%86%D8%AA%D9%88%D8%B1" title="استدلال قطری کانتور – Persian" lang="fa" hreflang="fa" data-title="استدلال قطری کانتور" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Argument_de_la_diagonale_de_Cantor" title="Argument de la diagonale de Cantor – French" lang="fr" hreflang="fr" data-title="Argument de la diagonale de Cantor" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%8C%80%EA%B0%81%EC%84%A0_%EB%85%BC%EB%B2%95" title="대각선 논법 – Korean" lang="ko" hreflang="ko" data-title="대각선 논법" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%BF%D5%A1%D5%B6%D5%BF%D5%B8%D6%80%D5%AB_%D5%A1%D5%B6%D5%AF%D5%B5%D5%B8%D6%82%D5%B6%D5%A1%D5%A3%D5%AE%D5%A1%D5%B5%D5%AB%D5%B6_%D6%83%D5%A1%D5%BD%D5%BF%D5%A1%D6%80%D5%AF" title="Կանտորի անկյունագծային փաստարկ – Armenian" lang="hy" hreflang="hy" data-title="Կանտորի անկյունագծային փաստարկ" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Argomento_diagonale_di_Cantor" title="Argomento diagonale di Cantor – Italian" lang="it" hreflang="it" data-title="Argomento diagonale di Cantor" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%94%D7%90%D7%9C%D7%9B%D7%A1%D7%95%D7%9F_%D7%A9%D7%9C_%D7%A7%D7%A0%D7%98%D7%95%D7%A8" title="האלכסון של קנטור – Hebrew" lang="he" hreflang="he" data-title="האלכסון של קנטור" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%99%E1%83%90%E1%83%9C%E1%83%A2%E1%83%9D%E1%83%A0%E1%83%98%E1%83%A1_%E1%83%93%E1%83%98%E1%83%90%E1%83%92%E1%83%9D%E1%83%9C%E1%83%90%E1%83%9A%E1%83%A3%E1%83%A0%E1%83%98_%E1%83%90%E1%83%A0%E1%83%92%E1%83%A3%E1%83%9B%E1%83%94%E1%83%9C%E1%83%A2%E1%83%98" title="კანტორის დიაგონალური არგუმენტი – Georgian" lang="ka" hreflang="ka" data-title="კანტორის დიაგონალური არგუმენტი" data-language-autonym="ქართული" data-language-local-name="Georgian" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-lmo mw-list-item"><a href="https://lmo.wikipedia.org/wiki/Argument_de_la_diagunala_de_Cantor" title="Argument de la diagunala de Cantor – Lombard" lang="lmo" hreflang="lmo" data-title="Argument de la diagunala de Cantor" data-language-autonym="Lombard" data-language-local-name="Lombard" class="interlanguage-link-target"><span>Lombard</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/%C3%81tl%C3%B3s_elj%C3%A1r%C3%A1s" title="Átlós eljárás – Hungarian" lang="hu" hreflang="hu" data-title="Átlós eljárás" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%95%E0%B4%BE%E0%B4%A8%E0%B5%8D%E0%B4%B1%E0%B4%B1%E0%B5%81%E0%B4%9F%E0%B5%86_%E0%B4%A1%E0%B4%AF%E0%B4%97%E0%B4%A3%E0%B5%BD_%E0%B4%86%E0%B5%BC%E0%B4%97%E0%B5%8D%E0%B4%AF%E0%B5%81%E0%B4%AE%E0%B5%86%E0%B4%A8%E0%B5%8D%E0%B4%B1%E0%B5%8D" title="കാന്ററുടെ ഡയഗണൽ ആർഗ്യുമെന്റ് – Malayalam" lang="ml" hreflang="ml" data-title="കാന്ററുടെ ഡയഗണൽ ആർഗ്യുമെന്റ്" data-language-autonym="മലയാളം" data-language-local-name="Malayalam" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Diagonaalbewijs_van_Cantor" title="Diagonaalbewijs van Cantor – Dutch" lang="nl" hreflang="nl" data-title="Diagonaalbewijs van Cantor" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%82%AB%E3%83%B3%E3%83%88%E3%83%BC%E3%83%AB%E3%81%AE%E5%AF%BE%E8%A7%92%E7%B7%9A%E8%AB%96%E6%B3%95" title="カントールの対角線論法 – Japanese" lang="ja" hreflang="ja" data-title="カントールの対角線論法" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Metoda_przek%C4%85tniowa" title="Metoda przekątniowa – Polish" lang="pl" hreflang="pl" data-title="Metoda przekątniowa" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Argumento_de_diagonaliza%C3%A7%C3%A3o_de_Cantor" title="Argumento de diagonalização de Cantor – Portuguese" lang="pt" hreflang="pt" data-title="Argumento de diagonalização de Cantor" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%94%D0%B8%D0%B0%D0%B3%D0%BE%D0%BD%D0%B0%D0%BB%D1%8C%D0%BD%D1%8B%D0%B9_%D0%B0%D1%80%D0%B3%D1%83%D0%BC%D0%B5%D0%BD%D1%82" title="Диагональный аргумент – Russian" lang="ru" hreflang="ru" data-title="Диагональный аргумент" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Cantor%27s_diagonal_argument" title="Cantor's diagonal argument – Simple English" lang="en-simple" hreflang="en-simple" data-title="Cantor's diagonal argument" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Cantorova_diagon%C3%A1lna_met%C3%B3da" title="Cantorova diagonálna metóda – Slovak" lang="sk" hreflang="sk" data-title="Cantorova diagonálna metóda" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Cantorjev_diagonalni_dokaz" title="Cantorjev diagonalni dokaz – Slovenian" lang="sl" hreflang="sl" data-title="Cantorjev diagonalni dokaz" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Cantorin_diagonaaliargumentti" title="Cantorin diagonaaliargumentti – Finnish" lang="fi" hreflang="fi" data-title="Cantorin diagonaaliargumentti" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%95%E0%AF%87%E0%AE%A3%E0%AF%8D%E0%AE%9F%E0%AE%B0%E0%AE%BF%E0%AE%A9%E0%AF%8D_%E0%AE%95%E0%AF%8B%E0%AE%A3%E0%AE%B2%E0%AF%8D%E0%AE%95%E0%AF%8B%E0%AE%9F%E0%AF%81_%E0%AE%A8%E0%AE%BF%E0%AE%B1%E0%AF%81%E0%AE%B5%E0%AE%B2%E0%AF%8D%E0%AE%AE%E0%AF%81%E0%AE%B1%E0%AF%88" title="கேண்டரின் கோணல்கோடு நிறுவல்முறை – Tamil" lang="ta" hreflang="ta" data-title="கேண்டரின் கோணல்கோடு நிறுவல்முறை" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%81%E0%B8%B2%E0%B8%A3%E0%B8%AD%E0%B9%89%E0%B8%B2%E0%B8%87%E0%B9%80%E0%B8%AB%E0%B8%95%E0%B8%B8%E0%B8%9C%E0%B8%A5%E0%B9%81%E0%B8%99%E0%B8%A7%E0%B8%97%E0%B9%81%E0%B8%A2%E0%B8%87%E0%B8%82%E0%B8%AD%E0%B8%87%E0%B8%84%E0%B8%B1%E0%B8%99%E0%B8%97%E0%B8%AD%E0%B8%A3%E0%B9%8C" title="การอ้างเหตุผลแนวทแยงของคันทอร์ – Thai" lang="th" hreflang="th" data-title="การอ้างเหตุผลแนวทแยงของคันทอร์" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Cantor%27un_k%C3%B6%C5%9Fegen_y%C3%B6ntemi" title="Cantor'un köşegen yöntemi – Turkish" lang="tr" hreflang="tr" data-title="Cantor'un köşegen yöntemi" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a 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src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b7/Diagonal_argument_01_svg.svg/250px-Diagonal_argument_01_svg.svg.png" decoding="async" width="250" height="325" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b7/Diagonal_argument_01_svg.svg/375px-Diagonal_argument_01_svg.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b7/Diagonal_argument_01_svg.svg/500px-Diagonal_argument_01_svg.svg.png 2x" data-file-width="177" data-file-height="230" /></a><figcaption>An illustration of Cantor's diagonal argument (in base 2) for the existence of <a href="/wiki/Uncountable_set" title="Uncountable set">uncountable sets</a>. The sequence at the bottom cannot occur anywhere in the enumeration of sequences above.</figcaption></figure> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Aplicaci%C3%B3n_2_inyectiva_sobreyectiva02.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/69/Aplicaci%C3%B3n_2_inyectiva_sobreyectiva02.svg/250px-Aplicaci%C3%B3n_2_inyectiva_sobreyectiva02.svg.png" decoding="async" width="250" height="200" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/69/Aplicaci%C3%B3n_2_inyectiva_sobreyectiva02.svg/375px-Aplicaci%C3%B3n_2_inyectiva_sobreyectiva02.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/69/Aplicaci%C3%B3n_2_inyectiva_sobreyectiva02.svg/500px-Aplicaci%C3%B3n_2_inyectiva_sobreyectiva02.svg.png 2x" data-file-width="250" data-file-height="200" /></a><figcaption>An <a href="/wiki/Infinite_set" title="Infinite set">infinite set</a> may have the same <a href="/wiki/Cardinality" title="Cardinality">cardinality</a> as a proper <a href="/wiki/Subset" title="Subset">subset</a> of itself, as the depicted <a href="/wiki/Bijection" title="Bijection">bijection</a> <i>f</i>(<i>x</i>)=2<i>x</i> from the <a href="/wiki/Natural_numbers" class="mw-redirect" title="Natural numbers">natural</a> to the <a href="/wiki/Even_numbers" class="mw-redirect" title="Even numbers">even numbers</a> demonstrates. Nevertheless, infinite sets of different cardinalities exist, as Cantor's diagonal argument shows.</figcaption></figure> <p><b>Cantor's diagonal argument</b> (among various similar names<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>note 1<span class="cite-bracket">]</span></a></sup>) is a <a href="/wiki/Mathematical_proof" title="Mathematical proof">mathematical proof</a> that there are <a href="/wiki/Infinite_set" title="Infinite set">infinite sets</a> which cannot be put into <a href="/wiki/Bijection" title="Bijection">one-to-one correspondence</a> with the infinite set of <a href="/wiki/Natural_number" title="Natural number">natural numbers</a> – informally, that there are <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">sets</a> which in some sense contain more elements than there are positive integers. Such sets are now called <a href="/wiki/Uncountable_set" title="Uncountable set">uncountable sets</a>, and the size of infinite sets is treated by the theory of <a href="/wiki/Cardinal_number" title="Cardinal number">cardinal numbers</a>, which Cantor began. </p><p><a href="/wiki/Georg_Cantor" title="Georg Cantor">Georg Cantor</a> published this proof in 1891,<sup id="cite_ref-Cantor.1891_2-0" class="reference"><a href="#cite_note-Cantor.1891-2"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Simmons1993_3-0" class="reference"><a href="#cite_note-Simmons1993-3"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 20–">: 20– </span></sup><sup id="cite_ref-Rubin1976_4-0" class="reference"><a href="#cite_note-Rubin1976-4"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> but it was not <a href="/wiki/Cantor%27s_first_uncountability_proof" class="mw-redirect" title="Cantor's first uncountability proof">his first proof</a> of the uncountability of the <a href="/wiki/Real_number" title="Real number">real numbers</a>, which appeared in 1874.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Bloch2011_6-0" class="reference"><a href="#cite_note-Bloch2011-6"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> However, it demonstrates a general technique that has since been used in a wide range of proofs,<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> including the first of <a href="/wiki/G%C3%B6del%27s_incompleteness_theorems" title="Gödel's incompleteness theorems">Gödel's incompleteness theorems</a><sup id="cite_ref-Simmons1993_3-1" class="reference"><a href="#cite_note-Simmons1993-3"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> and Turing's answer to the <i><a href="/wiki/Entscheidungsproblem" title="Entscheidungsproblem">Entscheidungsproblem</a></i>. Diagonalization arguments are often also the source of contradictions like <a href="/wiki/Russell%27s_paradox" title="Russell's paradox">Russell's paradox</a><sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> and <a href="/wiki/Richard%27s_paradox" title="Richard's paradox">Richard's paradox</a>.<sup id="cite_ref-Simmons1993_3-2" class="reference"><a href="#cite_note-Simmons1993-3"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 27">: 27 </span></sup> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Uncountable_set">Uncountable set</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Cantor%27s_diagonal_argument&action=edit&section=1" title="Edit section: Uncountable set"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Cantor considered the set <i>T</i> of all infinite <a href="/wiki/Sequences" class="mw-redirect" title="Sequences">sequences</a> of <a href="/wiki/Binary_digits" class="mw-redirect" title="Binary digits">binary digits</a> (i.e. each digit is zero or one).<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>note 2<span class="cite-bracket">]</span></a></sup> He begins with a <a href="/wiki/Constructive_proof" title="Constructive proof">constructive proof</a> of the following <a href="/wiki/Lemma_(mathematics)" title="Lemma (mathematics)">lemma</a>: </p> <dl><dd>If <i>s</i><sub>1</sub>, <i>s</i><sub>2</sub>, ... , <i>s</i><sub><i>n</i></sub>, ... is any enumeration of elements from <i>T</i>,<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>note 3<span class="cite-bracket">]</span></a></sup> then an element <i>s</i> of <i>T</i> can be constructed that doesn't correspond to any <i>s</i><sub><i>n</i></sub> in the enumeration.</dd></dl> <p>The proof starts with an enumeration of elements from <i>T</i>, for example </p> <dl><dd><table> <tbody><tr> <td><i>s</i><sub>1</sub> =</td> <td>(0,</td> <td>0,</td> <td>0,</td> <td>0,</td> <td>0,</td> <td>0,</td> <td>0,</td> <td>...) </td></tr> <tr> <td><i>s</i><sub>2</sub> =</td> <td>(1,</td> <td>1,</td> <td>1,</td> <td>1,</td> <td>1,</td> <td>1,</td> <td>1,</td> <td>...) </td></tr> <tr> <td><i>s</i><sub>3</sub> =</td> <td>(0,</td> <td>1,</td> <td>0,</td> <td>1,</td> <td>0,</td> <td>1,</td> <td>0,</td> <td>...) </td></tr> <tr> <td><i>s</i><sub>4</sub> =</td> <td>(1,</td> <td>0,</td> <td>1,</td> <td>0,</td> <td>1,</td> <td>0,</td> <td>1,</td> <td>...) </td></tr> <tr> <td><i>s</i><sub>5</sub> =</td> <td>(1,</td> <td>1,</td> <td>0,</td> <td>1,</td> <td>0,</td> <td>1,</td> <td>1,</td> <td>...) </td></tr> <tr> <td><i>s</i><sub>6</sub> =</td> <td>(0,</td> <td>0,</td> <td>1,</td> <td>1,</td> <td>0,</td> <td>1,</td> <td>1,</td> <td>...) </td></tr> <tr> <td><i>s</i><sub>7</sub> =</td> <td>(1,</td> <td>0,</td> <td>0,</td> <td>0,</td> <td>1,</td> <td>0,</td> <td>0,</td> <td>...) </td></tr> <tr> <td>... </td></tr></tbody></table></dd></dl> <p>Next, a sequence <i>s</i> is constructed by choosing the 1st digit as <a href="/wiki/Ones%27_complement" title="Ones' complement">complementary</a> to the 1st digit of <i>s</i><sub><i>1</i></sub> (swapping <b>0</b>s for <b>1</b>s and vice versa), the 2nd digit as complementary to the 2nd digit of <i>s</i><sub><i>2</i></sub>, the 3rd digit as complementary to the 3rd digit of <i>s</i><sub><i>3</i></sub>, and generally for every <i>n</i>, the <i>n</i><sup>th</sup> digit as complementary to the <i>n</i><sup>th</sup> digit of <i>s</i><sub><i>n</i></sub>. For the example above, this yields </p> <dl><dd><table> <tbody><tr> <td><i>s</i><sub>1</sub></td> <td>=</td> <td>(<u><b>0</b></u>,</td> <td>0,</td> <td>0,</td> <td>0,</td> <td>0,</td> <td>0,</td> <td>0,</td> <td>...) </td></tr> <tr> <td><i>s</i><sub>2</sub></td> <td>=</td> <td>(1,</td> <td><u><b>1</b></u>,</td> <td>1,</td> <td>1,</td> <td>1,</td> <td>1,</td> <td>1,</td> <td>...) </td></tr> <tr> <td><i>s</i><sub>3</sub></td> <td>=</td> <td>(0,</td> <td>1,</td> <td><u><b>0</b></u>,</td> <td>1,</td> <td>0,</td> <td>1,</td> <td>0,</td> <td>...) </td></tr> <tr> <td><i>s</i><sub>4</sub></td> <td>=</td> <td>(1,</td> <td>0,</td> <td>1,</td> <td><u><b>0</b></u>,</td> <td>1,</td> <td>0,</td> <td>1,</td> <td>...) </td></tr> <tr> <td><i>s</i><sub>5</sub></td> <td>=</td> <td>(1,</td> <td>1,</td> <td>0,</td> <td>1,</td> <td><u><b>0</b></u>,</td> <td>1,</td> <td>1,</td> <td>...) </td></tr> <tr> <td><i>s</i><sub>6</sub></td> <td>=</td> <td>(0,</td> <td>0,</td> <td>1,</td> <td>1,</td> <td>0,</td> <td><u><b>1</b></u>,</td> <td>1,</td> <td>...) </td></tr> <tr> <td><i>s</i><sub>7</sub></td> <td>=</td> <td>(1,</td> <td>0,</td> <td>0,</td> <td>0,</td> <td>1,</td> <td>0,</td> <td><u><b>0</b></u>,</td> <td>...) </td></tr> <tr> <td>... </td></tr> <tr> <td> </td></tr> <tr> <td><i>s</i></td> <td>=</td> <td>(<u><b>1</b></u>,</td> <td><u><b>0</b></u>,</td> <td><u><b>1</b></u>,</td> <td><u><b>1</b></u>,</td> <td><u><b>1</b></u>,</td> <td><u><b>0</b></u>,</td> <td><u><b>1</b></u>,</td> <td>...) </td></tr></tbody></table></dd></dl> <p>By construction, <i>s</i> is a member of <i>T</i> that differs from each <i>s</i><sub><i>n</i></sub>, since their <i>n</i><sup>th</sup> digits differ (highlighted in the example). Hence, <i>s</i> cannot occur in the enumeration. </p><p>Based on this lemma, Cantor then uses a <a href="/wiki/Proof_by_contradiction" title="Proof by contradiction">proof by contradiction</a> to show that: </p> <dl><dd>The set <i>T</i> is uncountable.</dd></dl> <p>The proof starts by assuming that <i>T</i> is <a href="/wiki/Countable_set#Definition" title="Countable set">countable</a>. Then all its elements can be written in an enumeration <i>s</i><sub>1</sub>, <i>s</i><sub>2</sub>, ... , <i>s</i><sub><i>n</i></sub>, ... . Applying the previous lemma to this enumeration produces a sequence <i>s</i> that is a member of <i>T</i>, but is not in the enumeration. However, if <i>T</i> is enumerated, then every member of <i>T</i>, including this <i>s</i>, is in the enumeration. This contradiction implies that the original assumption is false. Therefore, <i>T</i> is uncountable.<sup id="cite_ref-Cantor.1891_2-1" class="reference"><a href="#cite_note-Cantor.1891-2"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Real_numbers">Real numbers</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Cantor%27s_diagonal_argument&action=edit&section=2" title="Edit section: Real numbers"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The uncountability of the <a href="/wiki/Real_number" title="Real number">real numbers</a> was already established by <a href="/wiki/Cantor%27s_first_uncountability_proof" class="mw-redirect" title="Cantor's first uncountability proof">Cantor's first uncountability proof</a>, but it also follows from the above result. To prove this, an <a href="/wiki/Injective_function" title="Injective function">injection</a> will be constructed from the set <i>T</i> of infinite binary strings to the set <b>R</b> of real numbers. Since <i>T</i> is uncountable, the <a href="/wiki/Image_(mathematics)" title="Image (mathematics)">image</a> of this function, which is a subset of <b>R</b>, is uncountable. Therefore, <b>R</b> is uncountable. Also, by using a method of construction devised by Cantor, a <a href="/wiki/Bijection" title="Bijection">bijection</a> will be constructed between <i>T</i> and <b>R</b>. Therefore, <i>T</i> and <b>R</b> have the same cardinality, which is called the "<a href="/wiki/Cardinality_of_the_continuum" title="Cardinality of the continuum">cardinality of the continuum</a>" and is usually denoted by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {c}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">c</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {c}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b21924b960341255be18e538e51404718f29cbc0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.905ex; height:1.676ex;" alt="{\displaystyle {\mathfrak {c}}}"></span> or <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{\aleph _{0}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="normal">ℵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{\aleph _{0}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/779da5db4ed54fa334dd92089cdf1c284e45febb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.231ex; height:2.676ex;" alt="{\displaystyle 2^{\aleph _{0}}}"></span>. </p><p>An injection from <i>T</i> to <b>R</b> is given by mapping binary strings in <i>T</i> to <a href="/wiki/Decimal_fractions" class="mw-redirect" title="Decimal fractions">decimal fractions</a>, such as mapping <i>t</i> = 0111... to the decimal 0.0111.... This function, defined by <span class="nowrap"><i>f</i><span class="nowrap"> </span>(<i>t</i>) = 0.<i>t</i></span>, is an injection because it maps different strings to different numbers.<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>note 4<span class="cite-bracket">]</span></a></sup> </p><p>Constructing a bijection between <i>T</i> and <b>R</b> is slightly more complicated. Instead of mapping 0111... to the decimal 0.0111..., it can be mapped to the <a href="/wiki/Radix" title="Radix">base</a> <i>b</i> number: 0.0111...<sub><i>b</i></sub>. This leads to the family of functions: <span class="nowrap"><i>f</i><sub><i>b</i></sub><span class="nowrap"> </span>(<i>t</i>) = 0.<i>t</i><sub><i>b</i></sub></span>. The functions <span class="nowrap"><i>f</i><span class="nowrap"> </span><sub><i>b</i></sub>(<i>t</i>)</span> are injections, except for <span class="nowrap"><i>f</i><span class="nowrap"> </span><sub>2</sub>(<i>t</i>)</span>. This function will be modified to produce a bijection between <i>T</i> and <b>R</b>. </p> <table class="wikitable collapsible collapsed"> <tbody><tr> <th>Construction of a bijection between <i>T</i> and <b>R</b> </th></tr> <tr style="text-align: left; vertical-align: top"> <td><style data-mw-deduplicate="TemplateStyles:r1237032888/mw-parser-output/.tmulti">.mw-parser-output .tmulti .multiimageinner{display:flex;flex-direction:column}.mw-parser-output .tmulti .trow{display:flex;flex-direction:row;clear:left;flex-wrap:wrap;width:100%;box-sizing:border-box}.mw-parser-output .tmulti .tsingle{margin:1px;float:left}.mw-parser-output .tmulti .theader{clear:both;font-weight:bold;text-align:center;align-self:center;background-color:transparent;width:100%}.mw-parser-output .tmulti .thumbcaption{background-color:transparent}.mw-parser-output .tmulti .text-align-left{text-align:left}.mw-parser-output .tmulti .text-align-right{text-align:right}.mw-parser-output .tmulti .text-align-center{text-align:center}@media all and (max-width:720px){.mw-parser-output .tmulti .thumbinner{width:100%!important;box-sizing:border-box;max-width:none!important;align-items:center}.mw-parser-output .tmulti .trow{justify-content:center}.mw-parser-output .tmulti .tsingle{float:none!important;max-width:100%!important;box-sizing:border-box;text-align:center}.mw-parser-output .tmulti .tsingle .thumbcaption{text-align:left}.mw-parser-output .tmulti .trow>.thumbcaption{text-align:center}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .tmulti .multiimageinner img{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .tmulti .multiimageinner img{background-color:white}}</style><div class="thumb tmulti tright"><div class="thumbinner multiimageinner" style="width:192px;max-width:192px"><div class="trow"><div class="tsingle" style="width:100px;max-width:100px"><div class="thumbimage" style="height:147px;overflow:hidden"><span typeof="mw:File"><a href="/wiki/File:Linear_transformation_svg.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/8/82/Linear_transformation_svg.svg/98px-Linear_transformation_svg.svg.png" decoding="async" width="98" height="147" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/82/Linear_transformation_svg.svg/147px-Linear_transformation_svg.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/82/Linear_transformation_svg.svg/196px-Linear_transformation_svg.svg.png 2x" data-file-width="106" data-file-height="159" /></a></span></div><div class="thumbcaption">The function <i>h</i>: (0,1) → (−π/2,π/2)</div></div><div class="tsingle" style="width:88px;max-width:88px"><div class="thumbimage" style="height:147px;overflow:hidden"><span typeof="mw:File"><a href="/wiki/File:Tangent_one_period.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a3/Tangent_one_period.svg/86px-Tangent_one_period.svg.png" decoding="async" width="86" height="148" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a3/Tangent_one_period.svg/129px-Tangent_one_period.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a3/Tangent_one_period.svg/172px-Tangent_one_period.svg.png 2x" data-file-width="338" data-file-height="580" /></a></span></div><div class="thumbcaption">The function tan: (−π/2,π/2) → <b>R</b></div></div></div></div></div> <p>This construction uses a method devised by Cantor that was published in 1878. He used it to construct a bijection between the <a href="/wiki/Closed_interval" class="mw-redirect" title="Closed interval">closed interval</a> [0, 1] and the <a href="/wiki/Irrational_number" title="Irrational number">irrationals</a> in the <a href="/wiki/Open_interval" class="mw-redirect" title="Open interval">open interval</a> (0, 1). He first removed a <a href="/wiki/Countably_infinite" class="mw-redirect" title="Countably infinite">countably infinite</a> subset from each of these sets so that there is a bijection between the remaining uncountable sets. Since there is a bijection between the countably infinite subsets that have been removed, combining the two bijections produces a bijection between the original sets.<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> </p><p>Cantor's method can be used to modify the function <span class="nowrap"><i>f</i><span class="nowrap"> </span><sub>2</sub>(<i>t</i>) = 0.<i>t</i><sub>2</sub></span> to produce a bijection from <i>T</i> to (0, 1). Because some numbers have two binary expansions, <span class="nowrap"><i>f</i><span class="nowrap"> </span><sub>2</sub>(<i>t</i>)</span> is not even <a href="/wiki/Injective_function" title="Injective function">injective</a>. For example, <span class="nowrap"><i>f</i><span class="nowrap"> </span><sub>2</sub>(1000...) =</span> 0.1000...<sub>2</sub> = 1/2 and <span class="nowrap"><i>f</i><span class="nowrap"> </span><sub>2</sub>(0111...) =</span> 0.0111...<sub>2</sub> = <span class="nowrap"><a href="/wiki/Infinite_series" class="mw-redirect" title="Infinite series">1/4 + 1/8 + 1/16 + ...</a> =</span> 1/2, so both 1000... and 0111... map to the same number, 1/2. </p><p>To modify <span class="nowrap"><i>f</i><sub>2</sub><span class="nowrap"> </span>(<i>t</i>)</span>, observe that it is a bijection except for a countably infinite subset of (0, 1) and a countably infinite subset of <i>T</i>. It is not a bijection for the numbers in (0, 1) that have two <a href="/wiki/Binary_expansion" class="mw-redirect" title="Binary expansion">binary expansions</a>. These are called <a href="/wiki/Dyadic_rational" title="Dyadic rational">dyadic</a> numbers and have the form <span class="nowrap"><i>m</i><span class="nowrap"> </span>/<span class="nowrap"> </span>2<sup><i>n</i></sup></span> where <i>m</i> is an odd integer and <i>n</i> is a natural number. Put these numbers in the sequence: <i>r</i> = (1/2, 1/4, 3/4, 1/8, 3/8, 5/8, 7/8, ...). Also, <span class="nowrap"><i>f</i><sub>2</sub><span class="nowrap"> </span>(<i>t</i>)</span> is not a bijection to (0, 1) for the strings in <i>T</i> appearing after the <a href="/wiki/Binary_point" class="mw-redirect" title="Binary point">binary point</a> in the binary expansions of 0, 1, and the numbers in sequence <i>r</i>. Put these eventually-constant strings in the sequence: <i>s</i> = (<style data-mw-deduplicate="TemplateStyles:r1239334494">@media screen{html.skin-theme-clientpref-night .mw-parser-output div:not(.notheme)>.tmp-color,html.skin-theme-clientpref-night .mw-parser-output p>.tmp-color,html.skin-theme-clientpref-night .mw-parser-output table:not(.notheme) .tmp-color{color:inherit!important}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output div:not(.notheme)>.tmp-color,html.skin-theme-clientpref-os .mw-parser-output p>.tmp-color,html.skin-theme-clientpref-os .mw-parser-output table:not(.notheme) .tmp-color{color:inherit!important}}</style><span class="tmp-color" style="color:#808080">000</span>..., <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:#808080">111</span>..., 1<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:#808080">000</span>..., 0<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:#808080">111</span>..., 01<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:#808080">000</span>..., 00<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:#808080">111</span>..., 11<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:#808080">000</span>..., 10<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:#808080">111</span>..., ...). Define the bijection <i>g</i>(<i>t</i>) from <i>T</i> to (0, 1): If <i>t</i> is the <i>n</i><sup>th</sup> string in sequence <i>s</i>, let <i>g</i>(<i>t</i>) be the <i>n</i><sup>th</sup> number in sequence <i>r</i><span class="nowrap"> </span>; otherwise, <i>g</i>(<i>t</i>) = 0.<i>t</i><sub>2</sub>. </p><p>To construct a bijection from <i>T</i> to <b>R</b>, start with the <a href="/wiki/Trigonometric_functions" title="Trigonometric functions">tangent function</a> tan(<i>x</i>), which is a bijection from (−π/2, π/2) to <b>R</b> (see the figure shown on the right). Next observe that the <a href="/wiki/Linear_function" title="Linear function">linear function</a> <i>h</i>(<i>x</i>) = <span class="nowrap">π<i>x</i> – π/2</span> is a bijection from (0, 1) to (−π/2, π/2) (see the figure shown on the left). The <a href="/wiki/Function_composition" title="Function composition">composite function</a> tan(<i>h</i>(<i>x</i>)) = <span class="nowrap">tan(π<i>x</i> – π/2)</span> is a bijection from (0, 1) to <b>R</b>. Composing this function with <i>g</i>(<i>t</i>) produces the function tan(<i>h</i>(<i>g</i>(<i>t</i>))) = <span class="nowrap">tan(π<i>g</i>(<i>t</i>) – π/2)</span>, which is a bijection from <i>T</i> to <b>R</b>. </p> </td></tr></tbody></table> <div class="mw-heading mw-heading3"><h3 id="General_sets">General sets</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Cantor%27s_diagonal_argument&action=edit&section=3" title="Edit section: General sets"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Diagonal_argument_powerset_svg.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c7/Diagonal_argument_powerset_svg.svg/250px-Diagonal_argument_powerset_svg.svg.png" decoding="async" width="250" height="216" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c7/Diagonal_argument_powerset_svg.svg/375px-Diagonal_argument_powerset_svg.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c7/Diagonal_argument_powerset_svg.svg/500px-Diagonal_argument_powerset_svg.svg.png 2x" data-file-width="266" data-file-height="230" /></a><figcaption>Illustration of the generalized diagonal argument: The set <i>T</i> = {<i>n</i>∈<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {N} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {N} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fdf9a96b565ea202d0f4322e9195613fb26a9bed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {N} }"></span>: <i>n</i>∉<i>f</i>(<i>n</i>)} at the bottom cannot occur anywhere in the range of <i>f</i>:<a href="/wiki/Natural_number" title="Natural number"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {N} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {N} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fdf9a96b565ea202d0f4322e9195613fb26a9bed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {N} }"></span></a>→<a href="/wiki/Power_set" title="Power set"><i><b>P</b></i></a>(<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {N} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {N} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fdf9a96b565ea202d0f4322e9195613fb26a9bed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {N} }"></span>). The example mapping <i>f</i> happens to correspond to the example enumeration <i>s</i> in the <a href="#Lead">above</a> picture.</figcaption></figure> <p>A generalized form of the diagonal argument was used by Cantor to prove <a href="/wiki/Cantor%27s_theorem" title="Cantor's theorem">Cantor's theorem</a>: for every <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">set</a> <i>S</i>, the <a href="/wiki/Power_set" title="Power set">power set</a> of <i>S</i>—that is, the set of all <a href="/wiki/Subset" title="Subset">subsets</a> of <i>S</i> (here written as <i><b>P</b></i>(<i>S</i>))—cannot be in bijection with <i>S</i> itself. This proof proceeds as follows: </p><p>Let <i>f</i> be any <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">function</a> from <i>S</i> to <i><b>P</b></i>(<i>S</i>). It suffices to prove <i>f</i> cannot be <a href="/wiki/Surjective" class="mw-redirect" title="Surjective">surjective</a>. That means that some member <i>T</i> of <i><b>P</b></i>(<i>S</i>), i.e. some subset of <i>S</i>, is not in the <a href="/wiki/Image_(mathematics)" title="Image (mathematics)">image</a> of <i>f</i>. As a candidate consider the set: </p> <dl><dd><i>T</i> = { <i>s</i> ∈ <i>S</i>: <i>s</i> ∉ <i>f</i>(<i>s</i>) }.</dd></dl> <p>For every <i>s</i> in <i>S</i>, either <i>s</i> is in <i>T</i> or not. If <i>s</i> is in <i>T</i>, then by definition of <i>T</i>, <i>s</i> is not in <i>f</i>(<i>s</i>), so <i>T</i> is not equal to <i>f</i>(<i>s</i>). On the other hand, if <i>s</i> is not in <i>T</i>, then by definition of <i>T</i>, <i>s</i> is in <i>f</i>(<i>s</i>), so again <i>T</i> is not equal to <i>f</i>(<i>s</i>); cf. picture. For a more complete account of this proof, see <a href="/wiki/Cantor%27s_theorem" title="Cantor's theorem">Cantor's theorem</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Consequences">Consequences</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Cantor%27s_diagonal_argument&action=edit&section=4" title="Edit section: Consequences"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Ordering_of_cardinals">Ordering of cardinals</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Cantor%27s_diagonal_argument&action=edit&section=5" title="Edit section: Ordering of cardinals"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>With equality defined as the existence of a bijection between their underlying sets, Cantor also defines binary predicate of cardinalities <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |S|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |S|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/28d901e98a035ff4c0e37fe6dd8e750ece6c1f0b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.793ex; height:2.843ex;" alt="{\displaystyle |S|}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |T|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |T|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/073a0b1ce9d4c91f30329289adbb91aae47f87ea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.93ex; height:2.843ex;" alt="{\displaystyle |T|}"></span> in terms of the <a href="/wiki/Cardinality#Comparing_sets" title="Cardinality">existence of injections</a> between <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"></span>. It has the properties of a <a href="/wiki/Preorder" title="Preorder">preorder</a> and is here written "<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \leq }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>≤</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \leq }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/440568a09c3bfdf0e1278bfa79eb137c04e94035" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \leq }"></span>". One can embed the naturals into the binary sequences, thus proving various <i>injection existence</i> statements explicitly, so that in this sense <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |{\mathbb {N} }|\leq |2^{\mathbb {N} }|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |{\mathbb {N} }|\leq |2^{\mathbb {N} }|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43f7da221aa622f211f62ee110fd2cc3c24f2c1d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.945ex; height:3.176ex;" alt="{\displaystyle |{\mathbb {N} }|\leq |2^{\mathbb {N} }|}"></span>, where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{\mathbb {N} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{\mathbb {N} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7869cc2aba37b268552ae3f98559e646c417eb2b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.581ex; height:2.676ex;" alt="{\displaystyle 2^{\mathbb {N} }}"></span> denotes the function space <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathbb {N} }\to \{0,1\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mrow> <mo stretchy="false">→</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathbb {N} }\to \{0,1\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef05d2a6293661c84cec3475b4c042c30773fc20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.976ex; height:2.843ex;" alt="{\displaystyle {\mathbb {N} }\to \{0,1\}}"></span>. But following from the argument in the previous sections, there is <i>no surjection</i> and so also no bijection, i.e. the set is uncountable. For this one may write <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |{\mathbb {N} }|<|2^{\mathbb {N} }|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo><</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |{\mathbb {N} }|<|2^{\mathbb {N} }|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6a391e3ea4956ce929f5b2eb84450f42dde2c288" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.945ex; height:3.176ex;" alt="{\displaystyle |{\mathbb {N} }|<|2^{\mathbb {N} }|}"></span>, where "<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle <}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo><</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle <}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33737c89a17785dacc8638b4d66db3d5c8670de1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:1.843ex;" alt="{\displaystyle <}"></span>" is understood to mean the existence of an injection together with the proven absence of a bijection (as opposed to alternatives such as the negation of Cantor's preorder, or a definition in terms of <a href="/wiki/Von_Neumann_cardinal_assignment" title="Von Neumann cardinal assignment">assigned</a> <a href="/wiki/Ordinal_numbers" class="mw-redirect" title="Ordinal numbers">ordinals</a>). Also <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |S|<|{\mathcal {P}}(S)|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo><</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">P</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>S</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |S|<|{\mathcal {P}}(S)|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da9dc0c26421c09fd69504c54fcc911611b00a45" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.197ex; height:2.843ex;" alt="{\displaystyle |S|<|{\mathcal {P}}(S)|}"></span> in this sense, as has been shown, and at the same time it is the case that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \neg (|{\mathcal {P}}(S)|\leq |S|)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">¬</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">P</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>S</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \neg (|{\mathcal {P}}(S)|\leq |S|)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e60701251b9bbf3c173c2a42a56b6e789d5bbd8a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.557ex; height:2.843ex;" alt="{\displaystyle \neg (|{\mathcal {P}}(S)|\leq |S|)}"></span>, for all sets <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"></span>. </p><p>Assuming the <a href="/wiki/Law_of_excluded_middle" title="Law of excluded middle">law of excluded middle</a>, <a href="/wiki/Characteristic_functions" class="mw-redirect" title="Characteristic functions">characteristic functions</a> surject onto powersets, and then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |2^{S}|=|{\mathcal {P}}(S)|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">P</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>S</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |2^{S}|=|{\mathcal {P}}(S)|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d617669671e72a25e0ed83f1e0f8024f9558349" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.153ex; height:3.176ex;" alt="{\displaystyle |2^{S}|=|{\mathcal {P}}(S)|}"></span>. So the uncountable <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{\mathbb {N} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{\mathbb {N} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7869cc2aba37b268552ae3f98559e646c417eb2b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.581ex; height:2.676ex;" alt="{\displaystyle 2^{\mathbb {N} }}"></span> is also not enumerable and it can also be mapped onto <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathbb {N} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathbb {N} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9bbdf4a20a0a0a50daaa524863051d5d22c9fb23" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle {\mathbb {N} }}"></span>. Classically, the <a href="/wiki/Schr%C3%B6der%E2%80%93Bernstein_theorem" title="Schröder–Bernstein theorem">Schröder–Bernstein theorem</a> is valid and says that any two sets which are in the injective image of one another are in bijection as well. Here, every unbounded subset of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathbb {N} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathbb {N} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9bbdf4a20a0a0a50daaa524863051d5d22c9fb23" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle {\mathbb {N} }}"></span> is then in bijection with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathbb {N} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathbb {N} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9bbdf4a20a0a0a50daaa524863051d5d22c9fb23" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle {\mathbb {N} }}"></span> itself, and every <a href="/wiki/Subcountable" class="mw-redirect" title="Subcountable">subcountable</a> set (a property in terms of surjections) is then already countable, i.e. in the surjective image of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathbb {N} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathbb {N} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9bbdf4a20a0a0a50daaa524863051d5d22c9fb23" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle {\mathbb {N} }}"></span>. In this context the possibilities are then exhausted, making "<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \leq }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>≤</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \leq }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/440568a09c3bfdf0e1278bfa79eb137c04e94035" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \leq }"></span>" a <a href="/wiki/Partial_order" class="mw-redirect" title="Partial order">non-strict partial order</a>, or even a <a href="/wiki/Total_order" title="Total order">total order</a> when assuming <a href="/wiki/Axiom_of_choice" title="Axiom of choice">choice</a>. The diagonal argument thus establishes that, although both sets under consideration are infinite, there are actually <i>more</i> infinite sequences of ones and zeros than there are natural numbers. Cantor's result then also implies that the notion of the <a href="/wiki/Set_of_all_sets" class="mw-redirect" title="Set of all sets">set of all sets</a> is inconsistent: If <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"></span> were the set of all sets, then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {P}}(S)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">P</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>S</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {P}}(S)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b51bfe21fd15a7b5531fd9635a87a3863be6025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.012ex; height:2.843ex;" alt="{\displaystyle {\mathcal {P}}(S)}"></span> would at the same time be bigger than <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"></span> and a subset of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"></span>. </p> <div class="mw-heading mw-heading4"><h4 id="In_the_absence_of_excluded_middle">In the absence of excluded middle</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Cantor%27s_diagonal_argument&action=edit&section=6" title="Edit section: In the absence of excluded middle"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Also in <a href="/wiki/Constructivism_(mathematics)" class="mw-redirect" title="Constructivism (mathematics)">constructive mathematics</a>, there is no surjection from the full domain <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathbb {N} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathbb {N} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9bbdf4a20a0a0a50daaa524863051d5d22c9fb23" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle {\mathbb {N} }}"></span> onto the space of functions <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathbb {N} }^{\mathbb {N} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathbb {N} }^{\mathbb {N} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e6699687943087822616a37468d23367daaef9fa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.097ex; height:2.676ex;" alt="{\displaystyle {\mathbb {N} }^{\mathbb {N} }}"></span> or onto the collection of subsets <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {P}}({\mathbb {N} })}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">P</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {P}}({\mathbb {N} })}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac642434f7b2266edc6d6fa6d6085cd563dae455" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.191ex; height:2.843ex;" alt="{\displaystyle {\mathcal {P}}({\mathbb {N} })}"></span>, which is to say these two collections are uncountable. Again using "<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle <}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo><</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle <}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33737c89a17785dacc8638b4d66db3d5c8670de1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:1.843ex;" alt="{\displaystyle <}"></span>" for proven injection existence in conjunction with bijection absence, one has <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathbb {N} }<2^{\mathbb {N} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mrow> <mo><</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathbb {N} }<2^{\mathbb {N} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/055426b59af1004b7a115c382a0e2c88877f8ac4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.358ex; height:2.676ex;" alt="{\displaystyle {\mathbb {N} }<2^{\mathbb {N} }}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S<{\mathcal {P}}(S)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo><</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">P</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>S</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S<{\mathcal {P}}(S)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1c93812e192b76eaf5b3f7f21bf812acc3fd2e43" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.61ex; height:2.843ex;" alt="{\displaystyle S<{\mathcal {P}}(S)}"></span>. Further, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \neg ({\mathcal {P}}(S)\leq S)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">¬</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">P</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>S</mi> <mo stretchy="false">)</mo> <mo>≤</mo> <mi>S</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \neg ({\mathcal {P}}(S)\leq S)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b53bb5659be02da36324f4c7911f2c6355cbf4c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.969ex; height:2.843ex;" alt="{\displaystyle \neg ({\mathcal {P}}(S)\leq S)}"></span>, as previously noted. Likewise, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{\mathbb {N} }\leq {\mathbb {N} }^{\mathbb {N} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mrow> </msup> <mo>≤</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{\mathbb {N} }\leq {\mathbb {N} }^{\mathbb {N} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b5ce4bec43a6285c69e7f6015a4a342be41a2114" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:8.777ex; height:2.843ex;" alt="{\displaystyle 2^{\mathbb {N} }\leq {\mathbb {N} }^{\mathbb {N} }}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{S}\leq {\mathcal {P}}(S)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msup> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">P</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>S</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{S}\leq {\mathcal {P}}(S)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/77df68ccc9a3f4b1ac8c336df0b4168204617cd1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.565ex; height:3.176ex;" alt="{\displaystyle 2^{S}\leq {\mathcal {P}}(S)}"></span> and of course <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S\leq S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>≤</mo> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S\leq S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/60c87611d958399e753ca1c6fa21669b6fc732a9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.097ex; height:2.343ex;" alt="{\displaystyle S\leq S}"></span>, also in <a href="/wiki/Constructive_set_theory" title="Constructive set theory">constructive set theory</a>. </p><p>It is however harder or impossible to order ordinals and also cardinals, constructively. For example, the Schröder–Bernstein theorem requires the law of excluded middle.<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> In fact, the standard ordering on the reals, extending the ordering of the rational numbers, is not necessarily decidable either. Neither are most properties of interesting classes of functions decidable, by <a href="/wiki/Rice%27s_theorem" title="Rice's theorem">Rice's theorem</a>, i.e. the set of counting numbers for the subcountable sets may not be <a href="/wiki/Recursive_set" class="mw-redirect" title="Recursive set">recursive</a> and can thus fail to be countable. The elaborate collection of subsets of a set is constructively not exchangeable with the collection of its characteristic functions. In an otherwise constructive context (in which the law of excluded middle is not taken as axiom), it is consistent to adopt non-classical axioms that contradict consequences of the law of excluded middle. Uncountable sets such as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{\mathbb {N} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{\mathbb {N} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7869cc2aba37b268552ae3f98559e646c417eb2b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.581ex; height:2.676ex;" alt="{\displaystyle 2^{\mathbb {N} }}"></span> or <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathbb {N} }^{\mathbb {N} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathbb {N} }^{\mathbb {N} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e6699687943087822616a37468d23367daaef9fa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.097ex; height:2.676ex;" alt="{\displaystyle {\mathbb {N} }^{\mathbb {N} }}"></span> may be asserted to be <a href="/wiki/Subcountability" title="Subcountability">subcountable</a>.<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> This is a notion of size that is redundant in the classical context, but otherwise need not imply countability. The existence of injections from the uncountable <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{\mathbb {N} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{\mathbb {N} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7869cc2aba37b268552ae3f98559e646c417eb2b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.581ex; height:2.676ex;" alt="{\displaystyle 2^{\mathbb {N} }}"></span> or <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathbb {N} }^{\mathbb {N} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathbb {N} }^{\mathbb {N} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e6699687943087822616a37468d23367daaef9fa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.097ex; height:2.676ex;" alt="{\displaystyle {\mathbb {N} }^{\mathbb {N} }}"></span> into <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathbb {N} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathbb {N} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9bbdf4a20a0a0a50daaa524863051d5d22c9fb23" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle {\mathbb {N} }}"></span> is here possible as well.<sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> So the cardinal relation fails to be <a href="/wiki/Antisymmetric_relation" title="Antisymmetric relation">antisymmetric</a>. Consequently, also in the presence of function space sets that are even classically uncountable, <a href="/wiki/Intuitionist" class="mw-redirect" title="Intuitionist">intuitionists</a> do not accept this relation to constitute a hierarchy of transfinite sizes.<sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> When the <a href="/wiki/Axiom_of_powerset" class="mw-redirect" title="Axiom of powerset">axiom of powerset</a> is not adopted, in a constructive framework even the subcountability of all sets is then consistent. That all said, in common set theories, the non-existence of a set of all sets also already follows from <a href="/wiki/Axiom_schema_of_predicative_separation" title="Axiom schema of predicative separation">Predicative Separation</a>. </p><p>In a set theory, theories of mathematics are <a href="/wiki/Model_theory" title="Model theory">modeled</a>. Weaker logical axioms mean fewer constraints and so allow for a richer class of models. A set may be identified as a <a href="/wiki/Construction_of_the_real_numbers" title="Construction of the real numbers">model of the field of real numbers</a> when it fulfills some <a href="/wiki/Tarski%27s_axiomatization_of_the_reals" title="Tarski's axiomatization of the reals">axioms of real numbers</a> or a <a href="/wiki/Constructive_analysis" title="Constructive analysis">constructive rephrasing</a> thereof. Various models have been studied, such as the <a href="/wiki/Construction_of_the_real_numbers#Construction_from_Cauchy_sequences" title="Construction of the real numbers">Cauchy reals</a> or the <a href="/wiki/Dedekind_cut" title="Dedekind cut">Dedekind reals</a>, among others. The former relate to quotients of sequences while the later are well-behaved cuts taken from a powerset, if they exist. In the presence of excluded middle, those are all isomorphic and uncountable. Otherwise, <a href="/wiki/Effective_topos#Realizability_topoi" title="Effective topos">variants</a> of the Dedekind reals can be countable<sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> or inject into the naturals, but not jointly. When assuming <a href="/wiki/Countable_choice" class="mw-redirect" title="Countable choice">countable choice</a>, constructive Cauchy reals even without an explicit <a href="/wiki/Modulus_of_convergence" title="Modulus of convergence">modulus of convergence</a> are then <a href="/wiki/Cauchy_sequence#Completeness" title="Cauchy sequence">Cauchy-complete</a><sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> and Dedekind reals simplify so as to become isomorphic to them. Indeed, here choice also aids diagonal constructions and when assuming it, Cauchy-complete models of the reals are uncountable. </p> <div class="mw-heading mw-heading3"><h3 id="Open_questions">Open questions</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Cantor%27s_diagonal_argument&action=edit&section=7" title="Edit section: Open questions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Motivated by the insight that the set of real numbers is "bigger" than the set of natural numbers, one is led to ask if there is a set whose <a href="/wiki/Cardinality" title="Cardinality">cardinality</a> is "between" that of the integers and that of the reals. This question leads to the famous <a href="/wiki/Continuum_hypothesis" title="Continuum hypothesis">continuum hypothesis</a>. Similarly, the question of whether there exists a set whose cardinality is between |<i>S</i>| and |<i><b>P</b></i>(<i>S</i>)| for some infinite <i>S</i> leads to the <a href="/wiki/Generalized_continuum_hypothesis" class="mw-redirect" title="Generalized continuum hypothesis">generalized continuum hypothesis</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Diagonalization_in_broader_context">Diagonalization in broader context</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Cantor%27s_diagonal_argument&action=edit&section=8" title="Edit section: Diagonalization in broader context"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Russell%27s_paradox" title="Russell's paradox">Russell's paradox</a> has shown that set theory that includes an <a href="/wiki/Unrestricted_comprehension" class="mw-redirect" title="Unrestricted comprehension">unrestricted comprehension</a> scheme is contradictory. Note that there is a similarity between the construction of <i>T</i> and the set in Russell's paradox. Therefore, depending on how we modify the axiom scheme of comprehension in order to avoid Russell's paradox, arguments such as the non-existence of a set of all sets may or may not remain valid. </p><p>Analogues of the diagonal argument are widely used in mathematics to prove the existence or nonexistence of certain objects. For example, the conventional proof of the unsolvability of the <a href="/wiki/Halting_problem" title="Halting problem">halting problem</a> is essentially a diagonal argument. Also, diagonalization was originally used to show the existence of arbitrarily hard <a href="/wiki/Complexity_classes" class="mw-redirect" title="Complexity classes">complexity classes</a> and played a key role in early attempts to prove <a href="/wiki/P_versus_NP" class="mw-redirect" title="P versus NP">P does not equal NP</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Version_for_Quine's_New_Foundations"><span id="Version_for_Quine.27s_New_Foundations"></span>Version for Quine's New Foundations</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Cantor%27s_diagonal_argument&action=edit&section=9" title="Edit section: Version for Quine's New Foundations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The above proof fails for <a href="/wiki/W._V._Quine" class="mw-redirect" title="W. V. Quine">W. V. Quine</a>'s "<a href="/wiki/New_Foundations" title="New Foundations">New Foundations</a>" set theory (NF). In NF, the <a href="/wiki/Unrestricted_comprehension" class="mw-redirect" title="Unrestricted comprehension">naive axiom scheme of comprehension</a> is modified to avoid the paradoxes by introducing a kind of "local" <a href="/wiki/Type_theory" title="Type theory">type theory</a>. In this axiom scheme, </p> <dl><dd>{ <i>s</i> ∈ <i>S</i>: <i>s</i> ∉ <i>f</i>(<i>s</i>) }</dd></dl> <p>is <i>not</i> a set — i.e., does not satisfy the axiom scheme. On the other hand, we might try to create a modified diagonal argument by noticing that </p> <dl><dd>{ <i>s</i> ∈ <i>S</i>: <i>s</i> ∉ <i>f</i>({<i>s</i>}) }</dd></dl> <p><i>is</i> a set in NF. In which case, if <i><b>P</b></i><sub>1</sub>(<i>S</i>) is the set of one-element subsets of <i>S</i> and <i>f</i> is a proposed bijection from <i><b>P</b></i><sub>1</sub>(<i>S</i>) to <i><b>P</b></i>(<i>S</i>), one is able to use <a href="/wiki/Proof_by_contradiction" title="Proof by contradiction">proof by contradiction</a> to prove that |<i><b>P</b></i><sub>1</sub>(<i>S</i>)| < |<i><b>P</b></i>(<i>S</i>)|. </p><p>The proof follows by the fact that if <i>f</i> were indeed a map <i>onto</i> <i><b>P</b></i>(<i>S</i>), then we could find <i>r</i> in <i>S</i>, such that <i>f</i>({<i>r</i>}) coincides with the modified diagonal set, above. We would conclude that if <i>r</i> is not in <i>f</i>({<i>r</i>}), then <i>r</i> is in <i>f</i>({<i>r</i>}) and vice versa. </p><p>It is <i>not</i> possible to put <i><b>P</b></i><sub>1</sub>(<i>S</i>) in a one-to-one relation with <i>S</i>, as the two have different types, and so any function so defined would violate the typing rules for the comprehension scheme. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Cantor%27s_diagonal_argument&action=edit&section=10" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Cantor%27s_first_uncountability_proof" class="mw-redirect" title="Cantor's first uncountability proof">Cantor's first uncountability proof</a></li> <li><a href="/wiki/Controversy_over_Cantor%27s_theory" title="Controversy over Cantor's theory">Controversy over Cantor's theory</a></li> <li><a href="/wiki/Diagonal_lemma" title="Diagonal lemma">Diagonal lemma</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Cantor%27s_diagonal_argument&action=edit&section=11" title="Edit section: Notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text">the <b>diagonalisation argument</b>, the <b>diagonal slash argument</b>, the <b>anti-diagonal argument</b>, the <b>diagonal method</b>, and <b>Cantor's diagonalization proof</b></span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text">Cantor used "<i>m</i> and "<i>w</i>" instead of "0" and "1", "<i>M</i>" instead of "<i>T</i>", and "<i>E</i><sub><i>i</i></sub>" instead of "<i>s</i><sub><i>i</i></sub>".</span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text">Cantor does not assume that every element of <i>T</i> is in this enumeration.</span> </li> <li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text">While 0.0111... and 0.1000... would be equal if interpreted as binary fractions (destroying injectivity), they are different when interpreted as decimal fractions, as is done by <i>f</i>. On the other hand, since <i>t</i> is a binary string, the equality 0.0999... = 0.1000... of decimal fractions is not relevant here.</span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Cantor%27s_diagonal_argument&action=edit&section=12" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-Cantor.1891-2"><span class="mw-cite-backlink">^ <a href="#cite_ref-Cantor.1891_2-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Cantor.1891_2-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFGeorg_Cantor1891" class="citation journal cs1">Georg Cantor (1891). <a rel="nofollow" class="external text" href="https://www.digizeitschriften.de/dms/img/?PID=GDZPPN002113910&physid=phys84#navi">"Ueber eine elementare Frage der Mannigfaltigkeitslehre"</a>. <i><a href="/wiki/Jahresbericht_der_Deutschen_Mathematiker-Vereinigung" class="mw-redirect" title="Jahresbericht der Deutschen Mathematiker-Vereinigung">Jahresbericht der Deutschen Mathematiker-Vereinigung</a></i>. <b>1</b>: 75–78.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Jahresbericht+der+Deutschen+Mathematiker-Vereinigung&rft.atitle=Ueber+eine+elementare+Frage+der+Mannigfaltigkeitslehre&rft.volume=1&rft.pages=75-78&rft.date=1891&rft.au=Georg+Cantor&rft_id=https%3A%2F%2Fwww.digizeitschriften.de%2Fdms%2Fimg%2F%3FPID%3DGDZPPN002113910%26physid%3Dphys84%23navi&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACantor%27s+diagonal+argument" class="Z3988"></span> English translation: <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEwald1996" class="citation book cs1">Ewald, William B., ed. (1996). <i>From Immanuel Kant to David Hilbert: A Source Book in the Foundations of Mathematics, Volume 2</i>. Oxford University Press. pp. 920–922. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-19-850536-1" title="Special:BookSources/0-19-850536-1"><bdi>0-19-850536-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=From+Immanuel+Kant+to+David+Hilbert%3A+A+Source+Book+in+the+Foundations+of+Mathematics%2C+Volume+2&rft.pages=920-922&rft.pub=Oxford+University+Press&rft.date=1996&rft.isbn=0-19-850536-1&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACantor%27s+diagonal+argument" class="Z3988"></span></span> </li> <li id="cite_note-Simmons1993-3"><span class="mw-cite-backlink">^ <a href="#cite_ref-Simmons1993_3-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Simmons1993_3-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-Simmons1993_3-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKeith_Simmons1993" class="citation book cs1"><a href="/wiki/Keith_Simmons_(philosopher)" title="Keith Simmons (philosopher)">Keith Simmons</a> (30 July 1993). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=wEj3Spept0AC&pg=PA20"><i>Universality and the Liar: An Essay on Truth and the Diagonal Argument</i></a>. Cambridge University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-43069-2" title="Special:BookSources/978-0-521-43069-2"><bdi>978-0-521-43069-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Universality+and+the+Liar%3A+An+Essay+on+Truth+and+the+Diagonal+Argument&rft.pub=Cambridge+University+Press&rft.date=1993-07-30&rft.isbn=978-0-521-43069-2&rft.au=Keith+Simmons&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DwEj3Spept0AC%26pg%3DPA20&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACantor%27s+diagonal+argument" class="Z3988"></span></span> </li> <li id="cite_note-Rubin1976-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-Rubin1976_4-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRudin1976" class="citation book cs1">Rudin, Walter (1976). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/principlesofmath00rudi/page/30"><i>Principles of Mathematical Analysis</i></a></span> (3rd ed.). New York: McGraw-Hill. p. <a rel="nofollow" class="external text" href="https://archive.org/details/principlesofmath00rudi/page/30">30</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0070856133" title="Special:BookSources/0070856133"><bdi>0070856133</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Principles+of+Mathematical+Analysis&rft.place=New+York&rft.pages=30&rft.edition=3rd&rft.pub=McGraw-Hill&rft.date=1976&rft.isbn=0070856133&rft.aulast=Rudin&rft.aufirst=Walter&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fprinciplesofmath00rudi%2Fpage%2F30&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACantor%27s+diagonal+argument" class="Z3988"></span></span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGray1994" class="citation cs2">Gray, Robert (1994), <a rel="nofollow" class="external text" href="http://www.maa.org/sites/default/files/pdf/upload_library/22/Ford/Gray819-832.pdf">"Georg Cantor and Transcendental Numbers"</a> <span class="cs1-format">(PDF)</span>, <i><a href="/wiki/American_Mathematical_Monthly" class="mw-redirect" title="American Mathematical Monthly">American Mathematical Monthly</a></i>, <b>101</b> (9): 819–832, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2975129">10.2307/2975129</a>, <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2975129">2975129</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=American+Mathematical+Monthly&rft.atitle=Georg+Cantor+and+Transcendental+Numbers&rft.volume=101&rft.issue=9&rft.pages=819-832&rft.date=1994&rft_id=info%3Adoi%2F10.2307%2F2975129&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2975129%23id-name%3DJSTOR&rft.aulast=Gray&rft.aufirst=Robert&rft_id=http%3A%2F%2Fwww.maa.org%2Fsites%2Fdefault%2Ffiles%2Fpdf%2Fupload_library%2F22%2FFord%2FGray819-832.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACantor%27s+diagonal+argument" class="Z3988"></span></span> </li> <li id="cite_note-Bloch2011-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-Bloch2011_6-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBloch2011" class="citation book cs1">Bloch, Ethan D. (2011). <span class="id-lock-limited" title="Free access subject to limited trial, subscription normally required"><a rel="nofollow" class="external text" href="https://archive.org/details/realnumbersreala00edbl"><i>The Real Numbers and Real Analysis</i></a></span>. New York: Springer. p. <a rel="nofollow" class="external text" href="https://archive.org/details/realnumbersreala00edbl/page/n458">429</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-72176-7" title="Special:BookSources/978-0-387-72176-7"><bdi>978-0-387-72176-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Real+Numbers+and+Real+Analysis&rft.place=New+York&rft.pages=429&rft.pub=Springer&rft.date=2011&rft.isbn=978-0-387-72176-7&rft.aulast=Bloch&rft.aufirst=Ethan+D.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Frealnumbersreala00edbl&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACantor%27s+diagonal+argument" class="Z3988"></span></span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSheppard2014" class="citation book cs1">Sheppard, Barnaby (2014). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=RXzsAwAAQBAJ"><i>The Logic of Infinity</i></a> (illustrated ed.). Cambridge University Press. p. 73. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-107-05831-6" title="Special:BookSources/978-1-107-05831-6"><bdi>978-1-107-05831-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Logic+of+Infinity&rft.pages=73&rft.edition=illustrated&rft.pub=Cambridge+University+Press&rft.date=2014&rft.isbn=978-1-107-05831-6&rft.aulast=Sheppard&rft.aufirst=Barnaby&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DRXzsAwAAQBAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACantor%27s+diagonal+argument" class="Z3988"></span> <a rel="nofollow" class="external text" href="https://books.google.com/books?id=RXzsAwAAQBAJ&pg=PA73">Extract of page 73</a></span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation book cs1"><a rel="nofollow" class="external text" href="http://plato.stanford.edu/entries/russell-paradox"><i>Russell's paradox</i></a>. Stanford encyclopedia of philosophy. 2021.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Russell%27s+paradox&rft.pub=Stanford+encyclopedia+of+philosophy&rft.date=2021&rft_id=http%3A%2F%2Fplato.stanford.edu%2Fentries%2Frussell-paradox&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACantor%27s+diagonal+argument" class="Z3988"></span></span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBertrand_Russell1931" class="citation book cs1">Bertrand Russell (1931). <i>Principles of mathematics</i>. Norton. pp. 363–366.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Principles+of+mathematics&rft.pages=363-366&rft.pub=Norton&rft.date=1931&rft.au=Bertrand+Russell&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACantor%27s+diagonal+argument" class="Z3988"></span></span> </li> <li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text">See page 254 of <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGeorg_Cantor1878" class="citation cs2">Georg Cantor (1878), <a rel="nofollow" class="external text" href="http://www.digizeitschriften.de/dms/img/?PID=GDZPPN002156806">"Ein Beitrag zur Mannigfaltigkeitslehre"</a>, <i>Journal für die Reine und Angewandte Mathematik</i>, <b>84</b>: 242–258</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+f%C3%BCr+die+Reine+und+Angewandte+Mathematik&rft.atitle=Ein+Beitrag+zur+Mannigfaltigkeitslehre&rft.volume=84&rft.pages=242-258&rft.date=1878&rft.au=Georg+Cantor&rft_id=http%3A%2F%2Fwww.digizeitschriften.de%2Fdms%2Fimg%2F%3FPID%3DGDZPPN002156806&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACantor%27s+diagonal+argument" class="Z3988"></span>. This proof is discussed in <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJoseph_Dauben1979" class="citation cs2">Joseph Dauben (1979), <i>Georg Cantor: His Mathematics and Philosophy of the Infinite</i>, Harvard University Press, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-674-34871-0" title="Special:BookSources/0-674-34871-0"><bdi>0-674-34871-0</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Georg+Cantor%3A+His+Mathematics+and+Philosophy+of+the+Infinite&rft.pub=Harvard+University+Press&rft.date=1979&rft.isbn=0-674-34871-0&rft.au=Joseph+Dauben&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACantor%27s+diagonal+argument" class="Z3988"></span>, pp. 61–62, 65. On page 65, Dauben proves a result that is stronger than Cantor's. He lets "<i>φ<sub>ν</sub></i> denote any sequence of rationals in [0, 1]." Cantor lets <i>φ<sub>ν</sub></i> denote a sequence <a href="/wiki/Enumeration" title="Enumeration">enumerating</a> the rationals in [0, 1], which is the kind of sequence needed for his construction of a bijection between [0, 1] and the irrationals in (0, 1).</span> </li> <li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPradicBrown2019" class="citation arxiv cs1">Pradic, Pierre; Brown, Chad E. (2019). "Cantor-Bernstein implies Excluded Middle". <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1904.09193">1904.09193</a></span> [<a rel="nofollow" class="external text" href="https://arxiv.org/archive/math.LO">math.LO</a>].</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=preprint&rft.jtitle=arXiv&rft.atitle=Cantor-Bernstein+implies+Excluded+Middle&rft.date=2019&rft_id=info%3Aarxiv%2F1904.09193&rft.aulast=Pradic&rft.aufirst=Pierre&rft.au=Brown%2C+Chad+E.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACantor%27s+diagonal+argument" class="Z3988"></span></span> </li> <li id="cite_note-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-15">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBell2004" class="citation cs2"><a href="/wiki/John_Lane_Bell" title="John Lane Bell">Bell, John L.</a> (2004), <a rel="nofollow" class="external text" href="https://publish.uwo.ca/~jbell/russ.pdf">"Russell's paradox and diagonalization in a constructive context"</a> <span class="cs1-format">(PDF)</span>, in Link, Godehard (ed.), <i>One hundred years of Russell's paradox</i>, De Gruyter Series in Logic and its Applications, vol. 6, de Gruyter, Berlin, pp. 221–225, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=2104745">2104745</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Russell%27s+paradox+and+diagonalization+in+a+constructive+context&rft.btitle=One+hundred+years+of+Russell%27s+paradox&rft.series=De+Gruyter+Series+in+Logic+and+its+Applications&rft.pages=221-225&rft.pub=de+Gruyter%2C+Berlin&rft.date=2004&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D2104745%23id-name%3DMR&rft.aulast=Bell&rft.aufirst=John+L.&rft_id=https%3A%2F%2Fpublish.uwo.ca%2F~jbell%2Fruss.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACantor%27s+diagonal+argument" class="Z3988"></span></span> </li> <li id="cite_note-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-16">^</a></b></span> <span class="reference-text">Rathjen, M. "<a rel="nofollow" class="external text" href="http://www1.maths.leeds.ac.uk/~rathjen/acend.pdf">Choice principles in constructive and classical set theories</a>", Proceedings of the Logic Colloquium, 2002</span> </li> <li id="cite_note-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-17">^</a></b></span> <span class="reference-text">Bauer, A. "<a rel="nofollow" class="external text" href="http://math.andrej.com/wp-content/uploads/2011/06/injection.pdf">An injection from N^N to N</a>", 2011</span> </li> <li id="cite_note-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-18">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEttore_Carruccio2006" class="citation book cs1">Ettore Carruccio (2006). <i>Mathematics and Logic in History and in Contemporary Thought</i>. Transaction Publishers. p. 354. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-202-30850-0" title="Special:BookSources/978-0-202-30850-0"><bdi>978-0-202-30850-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mathematics+and+Logic+in+History+and+in+Contemporary+Thought&rft.pages=354&rft.pub=Transaction+Publishers&rft.date=2006&rft.isbn=978-0-202-30850-0&rft.au=Ettore+Carruccio&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACantor%27s+diagonal+argument" class="Z3988"></span></span> </li> <li id="cite_note-19"><span class="mw-cite-backlink"><b><a href="#cite_ref-19">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBauerHanson2024" class="citation arxiv cs1">Bauer; Hanson (2024). "The Countable Reals". <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/2404.01256">2404.01256</a></span> [<a rel="nofollow" class="external text" href="https://arxiv.org/archive/math.LO">math.LO</a>].</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=preprint&rft.jtitle=arXiv&rft.atitle=The+Countable+Reals&rft.date=2024&rft_id=info%3Aarxiv%2F2404.01256&rft.au=Bauer&rft.au=Hanson&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACantor%27s+diagonal+argument" class="Z3988"></span></span> </li> <li id="cite_note-20"><span class="mw-cite-backlink"><b><a href="#cite_ref-20">^</a></b></span> <span class="reference-text">Robert S. Lubarsky, <a rel="nofollow" class="external text" href="https://arxiv.org/pdf/1510.00639.pdf"><i>On the Cauchy Completeness of the Constructive Cauchy Reals</i></a>, July 2015</span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Cantor%27s_diagonal_argument&action=edit&section=13" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="http://www.mathpages.com/home/kmath371.htm">Cantor's Diagonal Proof</a> at MathPages</li> <li><span class="citation mathworld" id="Reference-Mathworld-Cantor_Diagonal_Method"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeisstein" class="citation web cs1"><a href="/wiki/Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a> <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/CantorDiagonalMethod.html">"Cantor Diagonal Method"</a>. <i><a href="/wiki/MathWorld" title="MathWorld">MathWorld</a></i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=MathWorld&rft.atitle=Cantor+Diagonal+Method&rft.au=Weisstein%2C+Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FCantorDiagonalMethod.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACantor%27s+diagonal+argument" class="Z3988"></span></span></li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline 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href="/wiki/Tuple" title="Tuple">tuple</a></li></ul></li> <li><a href="/wiki/Family_of_sets" title="Family of sets">Family</a></li> <li><a href="/wiki/Forcing_(mathematics)" title="Forcing (mathematics)">Forcing</a></li> <li><a href="/wiki/Bijection" title="Bijection">One-to-one correspondence</a></li> <li><a href="/wiki/Ordinal_number" title="Ordinal number">Ordinal number</a></li> <li><a href="/wiki/Set-builder_notation" title="Set-builder notation">Set-builder notation</a></li> <li><a href="/wiki/Transfinite_induction" title="Transfinite induction">Transfinite induction</a></li> <li><a href="/wiki/Venn_diagram" title="Venn diagram">Venn diagram</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Set_(mathematics)" title="Set (mathematics)">Set</a> types</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Amorphous_set" title="Amorphous set">Amorphous</a></li> <li><a href="/wiki/Countable_set" title="Countable set">Countable</a></li> <li><a href="/wiki/Empty_set" title="Empty set">Empty</a></li> <li><a href="/wiki/Finite_set" title="Finite set">Finite</a> (<a href="/wiki/Hereditarily_finite_set" title="Hereditarily finite set">hereditarily</a>)</li> <li><a href="/wiki/Filter_(set_theory)" title="Filter (set theory)">Filter</a> <ul><li><a href="/wiki/Filter_(set_theory)" title="Filter (set theory)">base</a></li> <li><a href="/wiki/Filter_(set_theory)#Filters_and_prefilters" title="Filter (set theory)">subbase</a></li> <li><a href="/wiki/Ultrafilter_on_a_set" title="Ultrafilter on a set">Ultrafilter</a></li></ul></li> <li><a href="/wiki/Fuzzy_set" title="Fuzzy set">Fuzzy</a></li> <li><a href="/wiki/Infinite_set" title="Infinite set">Infinite</a> (<a href="/wiki/Dedekind-infinite_set" title="Dedekind-infinite set">Dedekind-infinite</a>)</li> <li><a href="/wiki/Computable_set" title="Computable set">Recursive</a></li> <li><a href="/wiki/Singleton_(mathematics)" title="Singleton (mathematics)">Singleton</a></li> <li><a href="/wiki/Subset" title="Subset">Subset <b>·</b> Superset</a></li> <li><a href="/wiki/Transitive_set" title="Transitive set">Transitive</a></li> <li><a href="/wiki/Uncountable_set" title="Uncountable set">Uncountable</a></li> <li><a href="/wiki/Universal_set" title="Universal set">Universal</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Theories</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Alternative_set_theory" class="mw-redirect" title="Alternative set theory">Alternative</a></li> <li><a href="/wiki/Set_theory#Formalized_set_theory" title="Set theory">Axiomatic</a></li> <li><a href="/wiki/Naive_set_theory" title="Naive set theory">Naive</a></li> <li><a href="/wiki/Cantor%27s_theorem" title="Cantor's theorem">Cantor's theorem</a></li></ul> <ul><li><a href="/wiki/Zermelo_set_theory" title="Zermelo set theory">Zermelo</a> <ul><li><a href="/wiki/General_set_theory" title="General set theory">General</a></li></ul></li> <li><i><a href="/wiki/Principia_Mathematica" title="Principia Mathematica">Principia Mathematica</a></i> <ul><li><a href="/wiki/New_Foundations" title="New Foundations">New Foundations</a></li></ul></li> <li><a href="/wiki/Zermelo%E2%80%93Fraenkel_set_theory" title="Zermelo–Fraenkel set theory">Zermelo–Fraenkel </a> <ul><li><a href="/wiki/Von_Neumann%E2%80%93Bernays%E2%80%93G%C3%B6del_set_theory" title="Von Neumann–Bernays–Gödel set theory">von Neumann–Bernays–Gödel </a> <ul><li><a href="/wiki/Morse%E2%80%93Kelley_set_theory" title="Morse–Kelley set theory">Morse–Kelley</a></li></ul></li> <li><a href="/wiki/Kripke%E2%80%93Platek_set_theory" title="Kripke–Platek set theory">Kripke–Platek</a></li> <li><a href="/wiki/Tarski%E2%80%93Grothendieck_set_theory" title="Tarski–Grothendieck set theory">Tarski–Grothendieck</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><div class="hlist"><ul><li><a href="/wiki/Paradoxes_of_set_theory" title="Paradoxes of set theory">Paradoxes</a></li><li>Problems</li></ul></div></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Russell%27s_paradox" title="Russell's paradox">Russell's paradox</a></li> <li><a href="/wiki/Suslin%27s_problem" title="Suslin's problem">Suslin's problem</a></li> <li><a href="/wiki/Burali-Forti_paradox" title="Burali-Forti paradox">Burali-Forti paradox</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Category:Set_theorists" title="Category:Set theorists">Set theorists</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Paul_Bernays" title="Paul Bernays">Paul Bernays</a></li> <li><a href="/wiki/Georg_Cantor" title="Georg Cantor">Georg Cantor</a></li> <li><a href="/wiki/Paul_Cohen" title="Paul Cohen">Paul Cohen</a></li> <li><a href="/wiki/Richard_Dedekind" title="Richard Dedekind">Richard Dedekind</a></li> <li><a href="/wiki/Abraham_Fraenkel" title="Abraham Fraenkel">Abraham Fraenkel</a></li> <li><a href="/wiki/Kurt_G%C3%B6del" title="Kurt Gödel">Kurt Gödel</a></li> <li><a href="/wiki/Thomas_Jech" title="Thomas Jech">Thomas Jech</a></li> <li><a href="/wiki/John_von_Neumann" title="John von Neumann">John von Neumann</a></li> <li><a href="/wiki/Willard_Van_Orman_Quine" title="Willard Van Orman Quine">Willard Quine</a></li> <li><a href="/wiki/Bertrand_Russell" title="Bertrand Russell">Bertrand Russell</a></li> <li><a href="/wiki/Thoralf_Skolem" title="Thoralf Skolem">Thoralf Skolem</a></li> <li><a href="/wiki/Ernst_Zermelo" title="Ernst Zermelo">Ernst Zermelo</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"></div><div role="navigation" class="navbox" aria-labelledby="Mathematical_logic" style="padding:3px"><table class="nowraplinks mw-collapsible mw-collapsed navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Mathematical_logic" title="Template:Mathematical logic"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Mathematical_logic" title="Template talk:Mathematical logic"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Mathematical_logic" title="Special:EditPage/Template:Mathematical logic"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Mathematical_logic" style="font-size:114%;margin:0 4em"><a href="/wiki/Mathematical_logic" title="Mathematical logic">Mathematical logic</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">General</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Axiom" title="Axiom">Axiom</a> <ul><li><a href="/wiki/List_of_axioms" title="List of axioms">list</a></li></ul></li> <li><a href="/wiki/Cardinality" title="Cardinality">Cardinality</a></li> <li><a href="/wiki/First-order_logic" title="First-order logic">First-order logic</a></li> <li><a href="/wiki/Formal_proof" title="Formal proof">Formal proof</a></li> <li><a href="/wiki/Formal_semantics_(logic)" class="mw-redirect" title="Formal semantics (logic)">Formal semantics</a></li> <li><a href="/wiki/Foundations_of_mathematics" title="Foundations of mathematics">Foundations of mathematics</a></li> <li><a href="/wiki/Information_theory" title="Information theory">Information theory</a></li> <li><a href="/wiki/Lemma_(mathematics)" title="Lemma (mathematics)">Lemma</a></li> <li><a href="/wiki/Logical_consequence" title="Logical consequence">Logical consequence</a></li> <li><a href="/wiki/Structure_(mathematical_logic)" title="Structure (mathematical logic)">Model</a></li> <li><a href="/wiki/Theorem" title="Theorem">Theorem</a></li> <li><a href="/wiki/Theory_(mathematical_logic)" title="Theory (mathematical logic)">Theory</a></li> <li><a href="/wiki/Type_theory" title="Type theory">Type theory</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Theorems (<a href="/wiki/Category:Theorems_in_the_foundations_of_mathematics" title="Category:Theorems in the foundations of mathematics">list</a>)<br /> and <a href="/wiki/Paradoxes_of_set_theory" title="Paradoxes of set theory">paradoxes</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/G%C3%B6del%27s_completeness_theorem" title="Gödel's completeness theorem">Gödel's completeness</a> and <a href="/wiki/G%C3%B6del%27s_incompleteness_theorems" title="Gödel's incompleteness theorems">incompleteness theorems</a></li> <li><a href="/wiki/Tarski%27s_undefinability_theorem" title="Tarski's undefinability theorem">Tarski's undefinability</a></li> <li><a href="/wiki/Banach%E2%80%93Tarski_paradox" title="Banach–Tarski paradox">Banach–Tarski paradox</a></li> <li>Cantor's <a href="/wiki/Cantor%27s_theorem" title="Cantor's theorem">theorem,</a> <a href="/wiki/Cantor%27s_paradox" title="Cantor's paradox">paradox</a> and <a class="mw-selflink selflink">diagonal argument</a></li> <li><a href="/wiki/Compactness_theorem" title="Compactness theorem">Compactness</a></li> <li><a href="/wiki/Halting_problem" title="Halting problem">Halting problem</a></li> <li><a href="/wiki/Lindstr%C3%B6m%27s_theorem" title="Lindström's theorem">Lindström's</a></li> <li><a href="/wiki/L%C3%B6wenheim%E2%80%93Skolem_theorem" title="Löwenheim–Skolem theorem">Löwenheim–Skolem</a></li> <li><a href="/wiki/Russell%27s_paradox" title="Russell's paradox">Russell's paradox</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Logic" title="Logic">Logics</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th id="Traditional" scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Term_logic" title="Term logic">Traditional</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Classical_logic" title="Classical logic">Classical logic</a></li> <li><a href="/wiki/Logical_truth" title="Logical truth">Logical truth</a></li> <li><a href="/wiki/Tautology_(logic)" title="Tautology (logic)">Tautology</a></li> <li><a href="/wiki/Proposition" title="Proposition">Proposition</a></li> <li><a href="/wiki/Inference" title="Inference">Inference</a></li> <li><a href="/wiki/Logical_equivalence" title="Logical equivalence">Logical equivalence</a></li> <li><a href="/wiki/Consistency" title="Consistency">Consistency</a> <ul><li><a href="/wiki/Equiconsistency" title="Equiconsistency">Equiconsistency</a></li></ul></li> <li><a href="/wiki/Argument" title="Argument">Argument</a></li> <li><a href="/wiki/Soundness" title="Soundness">Soundness</a></li> <li><a href="/wiki/Validity_(logic)" title="Validity (logic)">Validity</a></li> <li><a href="/wiki/Syllogism" title="Syllogism">Syllogism</a></li> <li><a href="/wiki/Square_of_opposition" title="Square of opposition">Square of opposition</a></li> <li><a href="/wiki/Venn_diagram" title="Venn diagram">Venn diagram</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Propositional_calculus" title="Propositional calculus">Propositional</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Boolean_algebra" title="Boolean algebra">Boolean algebra</a></li> <li><a href="/wiki/Boolean_function" title="Boolean function">Boolean functions</a></li> <li><a href="/wiki/Logical_connective" title="Logical connective">Logical connectives</a></li> <li><a href="/wiki/Propositional_calculus" title="Propositional calculus">Propositional calculus</a></li> <li><a href="/wiki/Propositional_formula" title="Propositional formula">Propositional formula</a></li> <li><a href="/wiki/Truth_table" title="Truth table">Truth tables</a></li> <li><a href="/wiki/Many-valued_logic" title="Many-valued logic">Many-valued logic</a> <ul><li><a href="/wiki/Three-valued_logic" title="Three-valued logic">3</a></li> <li><a href="/wiki/Finite-valued_logic" title="Finite-valued logic">finite</a></li> <li><a href="/wiki/Infinite-valued_logic" title="Infinite-valued logic">∞</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Predicate_logic" class="mw-redirect" title="Predicate logic">Predicate</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/First-order_logic" title="First-order logic">First-order</a> <ul><li><a href="/wiki/List_of_first-order_theories" title="List of first-order theories"><span style="font-size:85%;">list</span></a></li></ul></li> <li><a href="/wiki/Second-order_logic" title="Second-order logic">Second-order</a> <ul><li><a href="/wiki/Monadic_second-order_logic" title="Monadic second-order logic">Monadic</a></li></ul></li> <li><a href="/wiki/Higher-order_logic" title="Higher-order logic">Higher-order</a></li> <li><a href="/wiki/Fixed-point_logic" title="Fixed-point logic">Fixed-point</a></li> <li><a href="/wiki/Free_logic" title="Free logic">Free</a></li> <li><a href="/wiki/Quantifier_(logic)" title="Quantifier (logic)">Quantifiers</a></li> <li><a href="/wiki/Predicate_(mathematical_logic)" title="Predicate (mathematical logic)">Predicate</a></li> <li><a href="/wiki/Monadic_predicate_calculus" title="Monadic predicate calculus">Monadic predicate calculus</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Set_theory" title="Set theory">Set theory</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><td colspan="2" class="navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Zermelo%E2%80%93Fraenkel_set_theory" title="Zermelo–Fraenkel set theory">Set</a> <ul><li><a href="/wiki/Hereditary_set" title="Hereditary set">hereditary</a></li></ul></li> <li><a href="/wiki/Class_(set_theory)" title="Class (set theory)">Class</a></li> <li>(<a href="/wiki/Urelement" title="Urelement">Ur-</a>)<a href="/wiki/Element_(mathematics)" title="Element (mathematics)">Element</a></li> <li><a href="/wiki/Ordinal_number" title="Ordinal number">Ordinal number</a></li> <li><a href="/wiki/Extensionality" title="Extensionality">Extensionality</a></li> <li><a href="/wiki/Forcing_(mathematics)" title="Forcing (mathematics)">Forcing</a></li> <li><a href="/wiki/Relation_(mathematics)" title="Relation (mathematics)">Relation</a> <ul><li><a href="/wiki/Equivalence_relation" title="Equivalence relation">equivalence</a></li> <li><a href="/wiki/Partition_of_a_set" title="Partition of a set">partition</a></li></ul></li> <li>Set operations: <ul><li><a href="/wiki/Intersection_(set_theory)" title="Intersection (set theory)">intersection</a></li> <li><a href="/wiki/Union_(set_theory)" title="Union (set theory)">union</a></li> <li><a href="/wiki/Complement_(set_theory)" title="Complement (set theory)">complement</a></li> <li><a href="/wiki/Cartesian_product" title="Cartesian product">Cartesian product</a></li> <li><a href="/wiki/Power_set" title="Power set">power set</a></li> <li><a href="/wiki/List_of_set_identities_and_relations" title="List of set identities and relations">identities</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Types of <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">sets</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Countable_set" title="Countable set">Countable</a></li> <li><a href="/wiki/Uncountable_set" title="Uncountable set">Uncountable</a></li> <li><a href="/wiki/Empty_set" title="Empty set">Empty</a></li> <li><a href="/wiki/Inhabited_set" title="Inhabited set">Inhabited</a></li> <li><a href="/wiki/Singleton_(mathematics)" title="Singleton (mathematics)">Singleton</a></li> <li><a href="/wiki/Finite_set" title="Finite set">Finite</a></li> <li><a href="/wiki/Infinite_set" title="Infinite set">Infinite</a></li> <li><a href="/wiki/Transitive_set" title="Transitive set">Transitive</a></li> <li><a href="/wiki/Ultrafilter_(set_theory)" class="mw-redirect" title="Ultrafilter (set theory)">Ultrafilter</a></li> <li><a href="/wiki/Recursive_set" class="mw-redirect" title="Recursive set">Recursive</a></li> <li><a href="/wiki/Fuzzy_set" title="Fuzzy set">Fuzzy</a></li> <li><a href="/wiki/Universal_set" title="Universal set">Universal</a></li> <li><a href="/wiki/Universe_(mathematics)" title="Universe (mathematics)">Universe</a> <ul><li><a href="/wiki/Constructible_universe" title="Constructible universe">constructible</a></li> <li><a href="/wiki/Grothendieck_universe" title="Grothendieck universe">Grothendieck</a></li> <li><a href="/wiki/Von_Neumann_universe" title="Von Neumann universe">Von Neumann</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Map_(mathematics)" title="Map (mathematics)">Maps</a> and <a href="/wiki/Cardinality" title="Cardinality">cardinality</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Function_(mathematics)" title="Function (mathematics)">Function</a>/<a href="/wiki/Map_(mathematics)" title="Map (mathematics)">Map</a> <ul><li><a href="/wiki/Domain_of_a_function" title="Domain of a function">domain</a></li> <li><a href="/wiki/Codomain" title="Codomain">codomain</a></li> <li><a href="/wiki/Image_(mathematics)" title="Image (mathematics)">image</a></li></ul></li> <li><a href="/wiki/Injective_function" title="Injective function">In</a>/<a href="/wiki/Surjective_function" title="Surjective function">Sur</a>/<a href="/wiki/Bijection" title="Bijection">Bi</a>-jection</li> <li><a href="/wiki/Schr%C3%B6der%E2%80%93Bernstein_theorem" title="Schröder–Bernstein theorem">Schröder–Bernstein theorem</a></li> <li><a href="/wiki/Isomorphism" title="Isomorphism">Isomorphism</a></li> <li><a href="/wiki/G%C3%B6del_numbering" title="Gödel numbering">Gödel numbering</a></li> <li><a href="/wiki/Enumeration" title="Enumeration">Enumeration</a></li> <li><a href="/wiki/Large_cardinal" title="Large cardinal">Large cardinal</a> <ul><li><a href="/wiki/Inaccessible_cardinal" title="Inaccessible cardinal">inaccessible</a></li></ul></li> <li><a href="/wiki/Aleph_number" title="Aleph number">Aleph number</a></li> <li><a href="/wiki/Operation_(mathematics)" title="Operation (mathematics)">Operation</a> <ul><li><a href="/wiki/Binary_operation" title="Binary operation">binary</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Set theories</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Zermelo%E2%80%93Fraenkel_set_theory" title="Zermelo–Fraenkel set theory">Zermelo–Fraenkel</a> <ul><li><a href="/wiki/Axiom_of_choice" title="Axiom of choice">axiom of choice</a></li> <li><a href="/wiki/Continuum_hypothesis" title="Continuum hypothesis">continuum hypothesis</a></li></ul></li> <li><a href="/wiki/General_set_theory" title="General set theory">General</a></li> <li><a href="/wiki/Kripke%E2%80%93Platek_set_theory" title="Kripke–Platek set theory">Kripke–Platek</a></li> <li><a href="/wiki/Morse%E2%80%93Kelley_set_theory" title="Morse–Kelley set theory">Morse–Kelley</a></li> <li><a href="/wiki/Naive_set_theory" title="Naive set theory">Naive</a></li> <li><a href="/wiki/New_Foundations" title="New Foundations">New Foundations</a></li> <li><a href="/wiki/Tarski%E2%80%93Grothendieck_set_theory" title="Tarski–Grothendieck set theory">Tarski–Grothendieck</a></li> <li><a href="/wiki/Von_Neumann%E2%80%93Bernays%E2%80%93G%C3%B6del_set_theory" title="Von Neumann–Bernays–Gödel set theory">Von Neumann–Bernays–Gödel</a></li> <li><a href="/wiki/Ackermann_set_theory" title="Ackermann set theory">Ackermann</a></li> <li><a href="/wiki/Constructive_set_theory" title="Constructive set theory">Constructive</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Formal_system" title="Formal system">Formal systems</a> (<a href="/wiki/List_of_formal_systems" title="List of formal systems"><span style="font-size:85%;">list</span></a>),<br /><a href="/wiki/Formal_language" title="Formal language">language</a> and <a href="/wiki/Syntax_(logic)" title="Syntax (logic)">syntax</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><td colspan="2" class="navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Alphabet_(formal_languages)" title="Alphabet (formal languages)">Alphabet</a></li> <li><a href="/wiki/Arity" title="Arity">Arity</a></li> <li><a href="/wiki/Automata_theory" title="Automata theory">Automata</a></li> <li><a href="/wiki/Axiom_schema" title="Axiom schema">Axiom schema</a></li> <li><a href="/wiki/Expression_(mathematics)" title="Expression (mathematics)">Expression</a> <ul><li><a href="/wiki/Ground_expression" title="Ground expression">ground</a></li></ul></li> <li><a href="/wiki/Extension_by_new_constant_and_function_names" title="Extension by new constant and function names">Extension</a> <ul><li><a href="/wiki/Extension_by_definitions" title="Extension by definitions">by definition</a></li> <li><a href="/wiki/Conservative_extension" title="Conservative extension">conservative</a></li></ul></li> <li><a href="/wiki/Finitary_relation" title="Finitary relation">Relation</a></li> <li><a href="/wiki/Formation_rule" title="Formation rule">Formation rule</a></li> <li><a href="/wiki/Formal_grammar" title="Formal grammar">Grammar</a></li> <li><a href="/wiki/Well-formed_formula" title="Well-formed formula">Formula</a> <ul><li><a href="/wiki/Atomic_formula" title="Atomic formula">atomic</a></li> <li><a href="/wiki/Sentence_(mathematical_logic)" title="Sentence (mathematical logic)">closed</a></li> <li><a href="/wiki/Ground_formula" class="mw-redirect" title="Ground formula">ground</a></li> <li><a href="/wiki/Open_formula" title="Open formula">open</a></li></ul></li> <li><a href="/wiki/Free_variables_and_bound_variables" title="Free variables and bound variables">Free/bound variable</a></li> <li><a href="/wiki/Formal_language" title="Formal language">Language</a></li> <li><a href="/wiki/Metalanguage" title="Metalanguage">Metalanguage</a></li> <li><a href="/wiki/Logical_connective" title="Logical connective">Logical connective</a> <ul><li><a href="/wiki/Negation" title="Negation">¬</a></li> <li><a href="/wiki/Logical_disjunction" title="Logical disjunction">∨</a></li> <li><a href="/wiki/Logical_conjunction" title="Logical conjunction">∧</a></li> <li><a href="/wiki/Material_conditional" title="Material conditional">→</a></li> <li><a href="/wiki/Logical_biconditional" title="Logical biconditional">↔</a></li> <li><a href="/wiki/Logical_equality" title="Logical equality">=</a></li></ul></li> <li><a href="/wiki/Predicate_(mathematical_logic)" title="Predicate (mathematical logic)">Predicate</a> <ul><li><a href="/wiki/Functional_predicate" title="Functional predicate">functional</a></li> <li><a href="/wiki/Predicate_variable" title="Predicate variable">variable</a></li> <li><a href="/wiki/Propositional_variable" title="Propositional variable">propositional variable</a></li></ul></li> <li><a href="/wiki/Formal_proof" title="Formal proof">Proof</a></li> <li><a href="/wiki/Quantifier_(logic)" title="Quantifier (logic)">Quantifier</a> <ul><li><a href="/wiki/Existential_quantification" title="Existential quantification">∃</a></li> <li><a href="/wiki/Uniqueness_quantification" title="Uniqueness quantification">!</a></li> <li><a href="/wiki/Universal_quantification" title="Universal quantification">∀</a></li> <li><a href="/wiki/Quantifier_rank" title="Quantifier rank">rank</a></li></ul></li> <li><a href="/wiki/Sentence_(mathematical_logic)" title="Sentence (mathematical logic)">Sentence</a> <ul><li><a href="/wiki/Atomic_sentence" title="Atomic sentence">atomic</a></li> <li><a href="/wiki/Spectrum_of_a_sentence" title="Spectrum of a sentence">spectrum</a></li></ul></li> <li><a href="/wiki/Signature_(logic)" title="Signature (logic)">Signature</a></li> <li><a href="/wiki/String_(formal_languages)" class="mw-redirect" title="String (formal languages)">String</a></li> <li><a href="/wiki/Substitution_(logic)" title="Substitution (logic)">Substitution</a></li> <li><a href="/wiki/Symbol_(formal)" title="Symbol (formal)">Symbol</a> <ul><li><a href="/wiki/Uninterpreted_function" title="Uninterpreted function">function</a></li> <li><a href="/wiki/Logical_constant" title="Logical constant">logical/constant</a></li> <li><a href="/wiki/Non-logical_symbol" title="Non-logical symbol">non-logical</a></li> <li><a href="/wiki/Variable_(mathematics)" title="Variable (mathematics)">variable</a></li></ul></li> <li><a href="/wiki/Term_(logic)" title="Term (logic)">Term</a></li> <li><a href="/wiki/Theory_(mathematical_logic)" title="Theory (mathematical logic)">Theory</a> <ul><li><a href="/wiki/List_of_mathematical_theories" title="List of mathematical theories"><span style="font-size:85%;">list</span></a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><span class="nowrap">Example <a href="/wiki/Axiomatic_system" title="Axiomatic system">axiomatic<br />systems</a> <span style="font-size:85%;">(<a href="/wiki/List_of_first-order_theories" title="List of first-order theories">list</a>)</span></span></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li>of <a href="/wiki/True_arithmetic" title="True arithmetic">arithmetic</a>: <ul><li><a href="/wiki/Peano_axioms" title="Peano axioms">Peano</a></li> <li><a href="/wiki/Second-order_arithmetic" title="Second-order arithmetic">second-order</a></li> <li><a href="/wiki/Elementary_function_arithmetic" title="Elementary function arithmetic">elementary function</a></li> <li><a href="/wiki/Primitive_recursive_arithmetic" title="Primitive recursive arithmetic">primitive recursive</a></li> <li><a href="/wiki/Robinson_arithmetic" title="Robinson arithmetic">Robinson</a></li> <li><a href="/wiki/Skolem_arithmetic" title="Skolem arithmetic">Skolem</a></li></ul></li> <li>of the <a href="/wiki/Construction_of_the_real_numbers" title="Construction of the real numbers">real numbers</a> <ul><li><a href="/wiki/Tarski%27s_axiomatization_of_the_reals" title="Tarski's axiomatization of the reals">Tarski's axiomatization</a></li></ul></li> <li>of <a href="/wiki/Axiomatization_of_Boolean_algebras" class="mw-redirect" title="Axiomatization of Boolean algebras">Boolean algebras</a> <ul><li><a href="/wiki/Boolean_algebras_canonically_defined" title="Boolean algebras canonically defined">canonical</a></li> <li><a href="/wiki/Minimal_axioms_for_Boolean_algebra" title="Minimal axioms for Boolean algebra">minimal axioms</a></li></ul></li> <li>of <a href="/wiki/Foundations_of_geometry" title="Foundations of geometry">geometry</a>: <ul><li><a href="/wiki/Euclidean_geometry" title="Euclidean geometry">Euclidean</a>: <ul><li><a href="/wiki/Euclid%27s_Elements" title="Euclid's Elements"><i>Elements</i></a></li> <li><a href="/wiki/Hilbert%27s_axioms" title="Hilbert's axioms">Hilbert's</a></li> <li><a href="/wiki/Tarski%27s_axioms" title="Tarski's axioms">Tarski's</a></li></ul></li> <li><a href="/wiki/Non-Euclidean_geometry" title="Non-Euclidean geometry">non-Euclidean</a></li></ul></li></ul> <ul><li><i><a href="/wiki/Principia_Mathematica" title="Principia Mathematica">Principia Mathematica</a></i></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Proof_theory" title="Proof theory">Proof theory</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Formal_proof" title="Formal proof">Formal proof</a></li> <li><a href="/wiki/Natural_deduction" title="Natural deduction">Natural deduction</a></li> <li><a href="/wiki/Logical_consequence" title="Logical consequence">Logical consequence</a></li> <li><a href="/wiki/Rule_of_inference" title="Rule of inference">Rule of inference</a></li> <li><a href="/wiki/Sequent_calculus" title="Sequent calculus">Sequent calculus</a></li> <li><a href="/wiki/Theorem" title="Theorem">Theorem</a></li> <li><a href="/wiki/Formal_system" title="Formal system">Systems</a> <ul><li><a href="/wiki/Axiomatic_system" title="Axiomatic system">axiomatic</a></li> <li><a href="/wiki/Deductive_system" class="mw-redirect" title="Deductive system">deductive</a></li> <li><a href="/wiki/Hilbert_system" title="Hilbert system">Hilbert</a> <ul><li><a href="/wiki/List_of_Hilbert_systems" class="mw-redirect" title="List of Hilbert systems">list</a></li></ul></li></ul></li> <li><a href="/wiki/Complete_theory" title="Complete theory">Complete theory</a></li> <li><a href="/wiki/Independence_(mathematical_logic)" title="Independence (mathematical logic)">Independence</a> (<a href="/wiki/List_of_statements_independent_of_ZFC" title="List of statements independent of ZFC">from ZFC</a>)</li> <li><a href="/wiki/Proof_of_impossibility" title="Proof of impossibility">Proof of impossibility</a></li> <li><a href="/wiki/Ordinal_analysis" title="Ordinal analysis">Ordinal analysis</a></li> <li><a href="/wiki/Reverse_mathematics" title="Reverse mathematics">Reverse mathematics</a></li> <li><a href="/wiki/Self-verifying_theories" title="Self-verifying theories">Self-verifying theories</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Model_theory" title="Model theory">Model theory</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Interpretation_(logic)" title="Interpretation (logic)">Interpretation</a> <ul><li><a href="/wiki/Interpretation_function" class="mw-redirect" title="Interpretation function">function</a></li> <li><a href="/wiki/Interpretation_(model_theory)" title="Interpretation (model theory)">of models</a></li></ul></li> <li><a href="/wiki/Structure_(mathematical_logic)" title="Structure (mathematical logic)">Model</a> <ul><li><a href="/wiki/Elementary_equivalence" title="Elementary equivalence">equivalence</a></li> <li><a href="/wiki/Finite_model_theory" title="Finite model theory">finite</a></li> <li><a href="/wiki/Saturated_model" title="Saturated model">saturated</a></li> <li><a href="/wiki/Spectrum_of_a_theory" title="Spectrum of a theory">spectrum</a></li> <li><a href="/wiki/Substructure_(mathematics)" title="Substructure (mathematics)">submodel</a></li></ul></li> <li><a href="/wiki/Non-standard_model" title="Non-standard model">Non-standard model</a> <ul><li><a href="/wiki/Non-standard_model_of_arithmetic" title="Non-standard model of arithmetic">of arithmetic</a></li></ul></li> <li><a href="/wiki/Diagram_(mathematical_logic)" title="Diagram (mathematical logic)">Diagram</a> <ul><li><a href="/wiki/Elementary_diagram" title="Elementary diagram">elementary</a></li></ul></li> <li><a href="/wiki/Categorical_theory" title="Categorical theory">Categorical theory</a></li> <li><a href="/wiki/Model_complete_theory" title="Model complete theory">Model complete theory</a></li> <li><a href="/wiki/Satisfiability" title="Satisfiability">Satisfiability</a></li> <li><a href="/wiki/Semantics_of_logic" title="Semantics of logic">Semantics of logic</a></li> <li><a href="/wiki/Strength_(mathematical_logic)" title="Strength (mathematical logic)">Strength</a></li> <li><a href="/wiki/Theories_of_truth" class="mw-redirect" title="Theories of truth">Theories of truth</a> <ul><li><a href="/wiki/Semantic_theory_of_truth" title="Semantic theory of truth">semantic</a></li> <li><a href="/wiki/Tarski%27s_theory_of_truth" class="mw-redirect" title="Tarski's theory of truth">Tarski's</a></li> <li><a href="/wiki/Kripke%27s_theory_of_truth" class="mw-redirect" title="Kripke's theory of truth">Kripke's</a></li></ul></li> <li><a href="/wiki/T-schema" title="T-schema">T-schema</a></li> <li><a href="/wiki/Transfer_principle" title="Transfer principle">Transfer principle</a></li> <li><a href="/wiki/Truth_predicate" title="Truth predicate">Truth predicate</a></li> <li><a href="/wiki/Truth_value" title="Truth value">Truth value</a></li> <li><a href="/wiki/Type_(model_theory)" title="Type (model theory)">Type</a></li> <li><a href="/wiki/Ultraproduct" title="Ultraproduct">Ultraproduct</a></li> <li><a href="/wiki/Validity_(logic)" title="Validity (logic)">Validity</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Computability_theory" title="Computability theory">Computability theory</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Church_encoding" title="Church encoding">Church encoding</a></li> <li><a href="/wiki/Church%E2%80%93Turing_thesis" title="Church–Turing thesis">Church–Turing thesis</a></li> <li><a href="/wiki/Computably_enumerable_set" title="Computably enumerable set">Computably enumerable</a></li> <li><a href="/wiki/Computable_function" title="Computable function">Computable function</a></li> <li><a href="/wiki/Computable_set" title="Computable set">Computable set</a></li> <li><a href="/wiki/Decision_problem" title="Decision problem">Decision problem</a> <ul><li><a href="/wiki/Decidability_(logic)" title="Decidability (logic)">decidable</a></li> <li><a href="/wiki/Undecidable_problem" title="Undecidable problem">undecidable</a></li> <li><a href="/wiki/P_(complexity)" title="P (complexity)">P</a></li> <li><a href="/wiki/NP_(complexity)" title="NP (complexity)">NP</a></li> <li><a href="/wiki/P_versus_NP_problem" title="P versus NP problem">P versus NP problem</a></li></ul></li> <li><a href="/wiki/Kolmogorov_complexity" title="Kolmogorov complexity">Kolmogorov complexity</a></li> <li><a href="/wiki/Lambda_calculus" title="Lambda calculus">Lambda calculus</a></li> <li><a href="/wiki/Primitive_recursive_function" title="Primitive recursive function">Primitive recursive function</a></li> <li><a href="/wiki/Recursion" title="Recursion">Recursion</a></li> <li><a href="/wiki/Recursive_set" class="mw-redirect" title="Recursive set">Recursive set</a></li> <li><a href="/wiki/Turing_machine" title="Turing machine">Turing machine</a></li> <li><a href="/wiki/Type_theory" title="Type theory">Type theory</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Abstract_logic" title="Abstract logic">Abstract logic</a></li> <li><a href="/wiki/Algebraic_logic" title="Algebraic logic">Algebraic logic</a></li> <li><a href="/wiki/Automated_theorem_proving" title="Automated theorem proving">Automated theorem proving</a></li> <li><a href="/wiki/Category_theory" title="Category theory">Category theory</a></li> <li><a href="/wiki/Concrete_category" title="Concrete category">Concrete</a>/<a href="/wiki/Category_(mathematics)" title="Category (mathematics)">Abstract category</a></li> <li><a href="/wiki/Category_of_sets" title="Category of sets">Category of sets</a></li> <li><a href="/wiki/History_of_logic" title="History of logic">History of logic</a></li> <li><a href="/wiki/History_of_mathematical_logic" class="mw-redirect" title="History of mathematical logic">History of mathematical logic</a> <ul><li><a href="/wiki/Timeline_of_mathematical_logic" title="Timeline of mathematical logic">timeline</a></li></ul></li> <li><a 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