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vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#History"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>History</span> </div> </a> <ul id="toc-History-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Applied_versions" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Applied_versions"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Applied versions</span> </div> </a> <ul id="toc-Applied_versions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Applications_and_related_topics" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Applications_and_related_topics"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Applications and related topics</span> </div> </a> <button aria-controls="toc-Applications_and_related_topics-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Applications and related topics subsection</span> </button> <ul id="toc-Applications_and_related_topics-sublist" class="vector-toc-list"> <li id="toc-Russell-like_paradoxes" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Russell-like_paradoxes"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.1</span> <span>Russell-like paradoxes</span> </div> </a> <ul id="toc-Russell-like_paradoxes-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Related_paradoxes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Related_paradoxes"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Related paradoxes</span> </div> </a> <ul id="toc-Related_paradoxes-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">9</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">10</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">11</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Sources" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Sources"> <div class="vector-toc-text"> <span class="vector-toc-numb">12</span> <span>Sources</span> </div> </a> <ul id="toc-Sources-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">13</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" title="Table of Contents" > <input type="checkbox" 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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 47 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-47" 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">47 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/%D9%85%D9%81%D8%A7%D8%B1%D9%82%D8%A9_%D8%B1%D8%A7%D8%B3%D9%84" title="مفارقة راسل – Arabic" lang="ar" hreflang="ar" data-title="مفارقة راسل" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Paradoxa_de_Russell" title="Paradoxa de Russell – Asturian" lang="ast" hreflang="ast" data-title="Paradoxa de Russell" data-language-autonym="Asturianu" data-language-local-name="Asturian" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%9F%D0%B0%D1%80%D0%B0%D0%B4%D0%BE%D0%BA%D1%81_%D0%BD%D0%B0_%D0%A0%D1%8A%D1%81%D0%B5%D0%BB" title="Парадокс на Ръсел – Bulgarian" lang="bg" hreflang="bg" data-title="Парадокс на Ръсел" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Paradoxa_de_Russell" title="Paradoxa de Russell – Catalan" lang="ca" hreflang="ca" data-title="Paradoxa de Russell" 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/Russell%C5%AFv_paradox" title="Russellův paradox – Czech" lang="cs" hreflang="cs" data-title="Russellův paradox" 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/Russells_paradoks" title="Russells paradoks – Danish" lang="da" hreflang="da" data-title="Russells paradoks" 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/Russellsche_Antinomie" title="Russellsche Antinomie – German" lang="de" hreflang="de" data-title="Russellsche Antinomie" 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/Russelli_paradoks" title="Russelli paradoks – Estonian" lang="et" hreflang="et" data-title="Russelli paradoks" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%A0%CE%B1%CF%81%CE%AC%CE%B4%CE%BF%CE%BE%CE%BF_%CF%84%CE%BF%CF%85_%CE%A1%CE%AC%CF%83%CE%B5%CE%BB" title="Παράδοξο του Ράσελ – Greek" lang="el" hreflang="el" data-title="Παράδοξο του Ράσελ" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Paradoja_de_Russell" title="Paradoja de Russell – Spanish" lang="es" hreflang="es" data-title="Paradoja de Russell" 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/Rusela_paradokso" title="Rusela paradokso – Esperanto" lang="eo" hreflang="eo" data-title="Rusela paradokso" 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/Russellen_paradoxa" title="Russellen paradoxa – Basque" lang="eu" hreflang="eu" data-title="Russellen paradoxa" 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/%D9%BE%D8%A7%D8%B1%D8%A7%D8%AF%D9%88%DA%A9%D8%B3_%D8%B1%D8%A7%D8%B3%D9%84" title="پارادوکس راسل – Persian" lang="fa" hreflang="fa" data-title="پارادوکس راسل" 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/Paradoxe_de_Russell" title="Paradoxe de Russell – French" lang="fr" hreflang="fr" data-title="Paradoxe de Russell" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Paradoxo_de_Russell" title="Paradoxo de Russell – Galician" lang="gl" hreflang="gl" data-title="Paradoxo de Russell" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%9F%AC%EC%85%80%EC%9D%98_%EC%97%AD%EC%84%A4" title="러셀의 역설 – Korean" lang="ko" hreflang="ko" data-title="러셀의 역설" 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/%D5%8C%D5%A1%D5%BD%D5%A5%D5%AC%D5%AB_%D5%BA%D5%A1%D6%80%D5%A1%D5%A4%D5%B8%D6%84%D5%BD" title="Ռասելի պարադոքս – Armenian" lang="hy" hreflang="hy" data-title="Ռասելի պարադոքս" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Russellov_paradoks" title="Russellov paradoks – Croatian" lang="hr" hreflang="hr" data-title="Russellov paradoks" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Russell-%C3%BEvers%C3%B6gn" title="Russell-þversögn – Icelandic" lang="is" hreflang="is" data-title="Russell-þversögn" data-language-autonym="Íslenska" data-language-local-name="Icelandic" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Paradosso_di_Russell" title="Paradosso di Russell – Italian" lang="it" hreflang="it" data-title="Paradosso di Russell" 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%A4%D7%A8%D7%93%D7%95%D7%A7%D7%A1_%D7%A9%D7%9C_%D7%A8%D7%90%D7%A1%D7%9C" title="הפרדוקס של ראסל – Hebrew" lang="he" hreflang="he" data-title="הפרדוקס של ראסל" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-lmo mw-list-item"><a href="https://lmo.wikipedia.org/wiki/Paradoss_da_Russell" title="Paradoss da Russell – Lombard" lang="lmo" hreflang="lmo" data-title="Paradoss da Russell" 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/Russell-paradoxon" title="Russell-paradoxon – Hungarian" lang="hu" hreflang="hu" data-title="Russell-paradoxon" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%A0%D0%B0%D1%81%D0%B5%D0%BB%D0%BE%D0%B2_%D0%BF%D0%B0%D1%80%D0%B0%D0%B4%D0%BE%D0%BA%D1%81" title="Раселов парадокс – Macedonian" lang="mk" hreflang="mk" data-title="Раселов парадокс" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%B1%E0%B4%B8%E0%B5%8D%E0%B4%B8%E0%B4%B2%E0%B4%BF%E0%B4%A8%E0%B5%8D%E0%B4%B1%E0%B5%86_%E0%B4%B5%E0%B4%BF%E0%B4%B0%E0%B5%8B%E0%B4%A7%E0%B4%BE%E0%B4%AD%E0%B4%BE%E0%B4%B8%E0%B4%82" title="റസ്സലിന്റെ വിരോധാഭാസം – Malayalam" lang="ml" hreflang="ml" data-title="റസ്സലിന്റെ വിരോധാഭാസം" 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/Russellparadox" title="Russellparadox – Dutch" lang="nl" hreflang="nl" data-title="Russellparadox" 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%83%A9%E3%83%83%E3%82%BB%E3%83%AB%E3%81%AE%E3%83%91%E3%83%A9%E3%83%89%E3%83%83%E3%82%AF%E3%82%B9" title="ラッセルのパラドックス – Japanese" lang="ja" hreflang="ja" data-title="ラッセルのパラドックス" 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-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Russells_paradoks" title="Russells paradoks – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Russells paradoks" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Russells_paradoks" title="Russells paradoks – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Russells paradoks" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-pms mw-list-item"><a href="https://pms.wikipedia.org/wiki/Parad%C3%B2ss_%C3%ABd_Russell" title="Paradòss ëd Russell – Piedmontese" lang="pms" hreflang="pms" data-title="Paradòss ëd Russell" data-language-autonym="Piemontèis" data-language-local-name="Piedmontese" class="interlanguage-link-target"><span>Piemontèis</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Antynomia_Russella" title="Antynomia Russella – Polish" lang="pl" hreflang="pl" data-title="Antynomia Russella" 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/Paradoxo_de_Russell" title="Paradoxo de Russell – Portuguese" lang="pt" hreflang="pt" data-title="Paradoxo de Russell" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Paradoxul_lui_Russell" title="Paradoxul lui Russell – Romanian" lang="ro" hreflang="ro" data-title="Paradoxul lui Russell" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru badge-Q17437796 badge-featuredarticle mw-list-item" title="featured article badge"><a href="https://ru.wikipedia.org/wiki/%D0%9F%D0%B0%D1%80%D0%B0%D0%B4%D0%BE%D0%BA%D1%81_%D0%A0%D0%B0%D1%81%D1%81%D0%B5%D0%BB%D0%B0" title="Парадокс Рассела – Russian" lang="ru" hreflang="ru" data-title="Парадокс Рассела" 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/Russell%27s_paradox" title="Russell's paradox – Simple English" lang="en-simple" hreflang="en-simple" data-title="Russell's paradox" 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/Russellov_paradox" title="Russellov paradox – Slovak" lang="sk" hreflang="sk" data-title="Russellov paradox" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%A0%D0%B0%D1%81%D0%B5%D0%BB%D0%BE%D0%B2_%D0%BF%D0%B0%D1%80%D0%B0%D0%B4%D0%BE%D0%BA%D1%81" title="Раселов парадокс – Serbian" lang="sr" hreflang="sr" data-title="Раселов парадокс" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Raselov_paradoks" title="Raselov paradoks – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Raselov paradoks" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbo-Croatian" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Russellin_paradoksi" title="Russellin paradoksi – Finnish" lang="fi" hreflang="fi" data-title="Russellin paradoksi" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Russells_paradox" title="Russells paradox – Swedish" lang="sv" hreflang="sv" data-title="Russells paradox" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%9B%E0%B8%8F%E0%B8%B4%E0%B8%97%E0%B8%A3%E0%B8%A3%E0%B8%A8%E0%B8%99%E0%B9%8C%E0%B8%82%E0%B8%AD%E0%B8%87%E0%B8%A3%E0%B8%B1%E0%B8%AA%E0%B9%80%E0%B8%8B%E0%B8%B4%E0%B8%A5%E0%B8%A5%E0%B9%8C" title="ปฏิทรรศน์ของรัสเซิลล์ – Thai" lang="th" hreflang="th" 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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><table class="sidebar nomobile nowraplinks sidebar-person vcard hlist" style="border-color: black"><tbody><tr><th class="sidebar-title"><table><tbody><tr> <td class="sidebar-person-title-image" style="background-color:blue;color:inherit;"><span class="mw-image-border" typeof="mw:File"><a href="/wiki/File:Bertrand_Russell_transparent_bg.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a6/Bertrand_Russell_transparent_bg.png/75px-Bertrand_Russell_transparent_bg.png" decoding="async" width="75" height="111" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a6/Bertrand_Russell_transparent_bg.png/113px-Bertrand_Russell_transparent_bg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a6/Bertrand_Russell_transparent_bg.png/150px-Bertrand_Russell_transparent_bg.png 2x" data-file-width="1255" data-file-height="1859" /></a></span></td> <td class="sidebar-person-title" style="background-color:blue;color:white;"><div><span class="tmp-color" style="color:white">This article is part of <br />a series about</span></div><span class="vcard"><span class="fn"><a href="/wiki/Bertrand_Russell" title="Bertrand Russell"><span class="tmpl-colored-link" style="color: white; text-decoration: inherit;">Bertrand Russell</span></a></span></span></td> </tr></tbody></table></th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Philosophical_views_of_Bertrand_Russell" title="Philosophical views of Bertrand Russell">Views on philosophy</a></li> <li><a href="/wiki/Political_views_of_Bertrand_Russell" title="Political views of Bertrand Russell">Views on society</a> (<a href="/wiki/Bertrand_Russell_Peace_Foundation" title="Bertrand Russell Peace Foundation">Peace Foundation</a>)</li> <li><a href="/wiki/Bertrand_Russell_Professorship_of_Philosophy" title="Bertrand Russell Professorship of Philosophy">Professorship of Philosophy</a></li> <li><a href="/wiki/The_Bertrand_Russell_Case" title="The Bertrand Russell Case">Appointment court case</a></li> <li><a class="mw-selflink selflink">Russell's paradox</a></li> <li><a href="/wiki/Peano%E2%80%93Russell_notation" title="Peano–Russell notation">Peano–Russell notation</a></li> <li><a href="/wiki/Copleston%E2%80%93Russell_debate" title="Copleston–Russell debate">Copleston–Russell debate</a></li> <li><a href="/wiki/Russell%E2%80%93Einstein_Manifesto" title="Russell–Einstein Manifesto">Russell–Einstein Manifesto</a></li> <li><a href="/wiki/Russell_Tribunal" title="Russell Tribunal">Russell Tribunal</a></li> <li><a href="/wiki/Russell%27s_teapot" title="Russell's teapot">Russell's teapot</a></li> <li><a href="/wiki/Theory_of_descriptions" title="Theory of descriptions">Theory of descriptions</a></li> <li><a href="/wiki/Logical_atomism" title="Logical atomism">Logical atomism</a></li></ul> <hr /></td> </tr><tr><td class="sidebar-navbar"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374" /><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Bertrand_Russell_sidebar" title="Template:Bertrand Russell sidebar"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Bertrand_Russell_sidebar" title="Template talk:Bertrand Russell sidebar"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Bertrand_Russell_sidebar" title="Special:EditPage/Template:Bertrand Russell sidebar"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <p>In <a href="/wiki/Mathematical_logic" title="Mathematical logic">mathematical logic</a>, <b>Russell's paradox</b> (also known as <b>Russell's antinomy</b>) is a <a href="/wiki/Paradoxes_of_set_theory" title="Paradoxes of set theory">set-theoretic paradox</a> published by the <a href="/wiki/United_Kingdom" title="United Kingdom">British</a> <a href="/wiki/Philosopher" class="mw-redirect" title="Philosopher">philosopher</a> and <a href="/wiki/Mathematician" title="Mathematician">mathematician</a>, <a href="/wiki/Bertrand_Russell" title="Bertrand Russell">Bertrand Russell</a>, in 1901.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> Russell's paradox shows that every <a href="/wiki/Set_theory" title="Set theory">set theory</a> that contains an <a href="/wiki/Unrestricted_comprehension_principle" class="mw-redirect" title="Unrestricted comprehension principle">unrestricted comprehension principle</a> leads to contradictions.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> According to the unrestricted comprehension principle, for any sufficiently well-defined <a href="/wiki/Property_(mathematics)" title="Property (mathematics)">property</a>, there is the <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">set</a> of all and only the objects that have that property. Let <i>R</i> be the set of all sets that are not members of themselves. (This set is sometimes called "the Russell set".) If <i>R</i> is not a member of itself, then its definition entails that it is a member of itself; yet, if it is a member of itself, then it is not a member of itself, since it is the set of all sets that are not members of themselves. The resulting contradiction is Russell's paradox. In symbols: </p> <dl><dd><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 {\text{Let }}R=\{x\mid x\not \in x\}{\text{, then }}R\in R\iff R\not \in R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>Let </mtext> </mrow> <mi>R</mi> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mo>∣</mo> <mi>x</mi> <mo>∉</mo> <mi>x</mi> <mo fence="false" stretchy="false">}</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>, then </mtext> </mrow> <mi>R</mi> <mo>∈</mo> <mi>R</mi> <mspace width="thickmathspace"></mspace> <mo stretchy="false">⟺</mo> <mspace width="thickmathspace"></mspace> <mi>R</mi> <mo>∉</mo> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{Let }}R=\{x\mid x\not \in x\}{\text{, then }}R\in R\iff R\not \in R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1083e5691d2b959d103e2a6c3a9585a1b25b0438" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:45.889ex; height:2.843ex;" alt="{\displaystyle {\text{Let }}R=\{x\mid x\not \in x\}{\text{, then }}R\in R\iff R\not \in R}" /></span></dd></dl> <p>Russell also showed that a version of the paradox could be derived in the <a href="/wiki/Axiomatic_system" title="Axiomatic system">axiomatic system</a> constructed by the German philosopher and mathematician <a href="/wiki/Gottlob_Frege" title="Gottlob Frege">Gottlob Frege</a>, hence undermining Frege's attempt to reduce mathematics to logic and calling into question the <a href="/wiki/Logicism" title="Logicism">logicist programme</a>. Two influential ways of avoiding the paradox were both proposed in 1908: Russell's own <a href="/wiki/Type_theory" title="Type theory">type theory</a> and the <a href="/wiki/Zermelo_set_theory" title="Zermelo set theory">Zermelo set theory</a>. In particular, Zermelo's axioms restricted the unlimited comprehension principle. With the additional contributions of <a href="/wiki/Abraham_Fraenkel" title="Abraham Fraenkel">Abraham Fraenkel</a>, Zermelo set theory developed into the now-standard <a href="/wiki/Zermelo%E2%80%93Fraenkel_set_theory" title="Zermelo–Fraenkel set theory">Zermelo–Fraenkel set theory</a> (commonly known as ZFC when including the <a href="/wiki/Axiom_of_choice" title="Axiom of choice">axiom of choice</a>). The main difference between Russell's and Zermelo's solution to the paradox is that Zermelo modified the axioms of set theory while maintaining a standard logical language, while Russell modified the logical language itself. The language of ZFC, with the <a href="/wiki/Axiom_schema_of_replacement#History" title="Axiom schema of replacement">help of Thoralf Skolem</a>, turned out to be that of <a href="/wiki/First-order_logic" title="First-order logic">first-order logic</a>.<sup id="cite_ref-FraenkelBar-Hillel1973_4-0" class="reference"><a href="#cite_note-FraenkelBar-Hillel1973-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p><p>The paradox had already been discovered independently in 1899 by the German mathematician <a href="/wiki/Ernst_Zermelo" title="Ernst Zermelo">Ernst Zermelo</a>.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> However, Zermelo did not publish the idea, which remained known only to <a href="/wiki/David_Hilbert" title="David Hilbert">David Hilbert</a>, <a href="/wiki/Edmund_Husserl" title="Edmund Husserl">Edmund Husserl</a>, and other academics at the <a href="/wiki/University_of_G%C3%B6ttingen" title="University of Göttingen">University of Göttingen</a>. At the end of the 1890s, <a href="/wiki/Georg_Cantor" title="Georg Cantor">Georg Cantor</a> – considered the founder of modern set theory – had already realized that his theory would lead to a contradiction, as he told Hilbert and <a href="/wiki/Richard_Dedekind" title="Richard Dedekind">Richard Dedekind</a> by letter.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Informal_presentation">Informal presentation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Russell%27s_paradox&action=edit&section=1" title="Edit section: Informal presentation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Most sets commonly encountered are not members of themselves. Let us call a set "normal" if it is not a member of itself, and "abnormal" if it is a member of itself. Clearly every set must be either normal or abnormal. For example, consider the set of all <a href="/wiki/Square" title="Square">squares</a> in a <a href="/wiki/Plane_(geometry)" class="mw-redirect" title="Plane (geometry)">plane</a>. This set is not itself a square in the plane, thus it is not a member of itself and is therefore normal. In contrast, the complementary set that contains everything which is <b>not</b> a square in the plane is itself not a square in the plane, and so it is one of its own members and is therefore abnormal. </p><p>Now we consider the set of all normal sets, <i>R</i>, and try to determine whether <i>R</i> is normal or abnormal. If <i>R</i> were normal, it would be contained in the set of all normal sets (itself), and therefore be abnormal; on the other hand if <i>R</i> were abnormal, it would not be contained in the set of all normal sets (itself), and therefore be normal. This leads to the conclusion that <i>R</i> is neither normal nor abnormal: Russell's paradox. </p> <div class="mw-heading mw-heading2"><h2 id="Formal_presentation">Formal presentation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Russell%27s_paradox&action=edit&section=2" title="Edit section: Formal presentation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The term "<a href="/wiki/Naive_set_theory" title="Naive set theory">naive set theory</a>" is used in various ways. In one usage, naive set theory is a formal theory, that is formulated in a <a href="/wiki/Predicate_logic" class="mw-redirect" title="Predicate logic">first-order language</a> with a binary non-logical <a href="/wiki/Predicate_(mathematical_logic)" title="Predicate (mathematical logic)">predicate</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 \in }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∈</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \in }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6fe4d5b0a594c1da89b5e78e7dfbeed90bdcc32f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:1.843ex;" alt="{\displaystyle \in }" /></span>, and that includes the <a href="/wiki/Axiom_of_extensionality" title="Axiom of extensionality">axiom of extensionality</a>: </p> <dl><dd><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 \forall x\,\forall y\,(\forall z\,(z\in x\iff z\in y)\implies x=y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mspace width="thinmathspace"></mspace> <mi mathvariant="normal">∀</mi> <mi>y</mi> <mspace width="thinmathspace"></mspace> <mo stretchy="false">(</mo> <mi mathvariant="normal">∀</mi> <mi>z</mi> <mspace width="thinmathspace"></mspace> <mo stretchy="false">(</mo> <mi>z</mi> <mo>∈</mo> <mi>x</mi> <mspace width="thickmathspace"></mspace> <mo stretchy="false">⟺</mo> <mspace width="thickmathspace"></mspace> <mi>z</mi> <mo>∈</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mspace width="thickmathspace"></mspace> <mo stretchy="false">⟹</mo> <mspace width="thickmathspace"></mspace> <mi>x</mi> <mo>=</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall x\,\forall y\,(\forall z\,(z\in x\iff z\in y)\implies x=y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b9d8a133bb6f92e82b05505f5454bae8180e36ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:41.441ex; height:2.843ex;" alt="{\displaystyle \forall x\,\forall y\,(\forall z\,(z\in x\iff z\in y)\implies x=y)}" /></span></dd></dl> <p>and the axiom schema of <a href="/wiki/Unrestricted_comprehension" class="mw-redirect" title="Unrestricted comprehension">unrestricted comprehension</a>: </p> <dl><dd><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 \exists y\forall x(x\in y\iff \varphi (x))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∃</mi> <mi>y</mi> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>∈</mo> <mi>y</mi> <mspace width="thickmathspace"></mspace> <mo stretchy="false">⟺</mo> <mspace width="thickmathspace"></mspace> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \exists y\forall x(x\in y\iff \varphi (x))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c622da728ddd9727f6bba25fb95e7d308634bfc7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.762ex; height:2.843ex;" alt="{\displaystyle \exists y\forall x(x\in y\iff \varphi (x))}" /></span></dd></dl> <p>for any predicate <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 \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }" /></span> with <span class="texhtml mvar" style="font-style:italic;">x</span> as a free variable inside <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 \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }" /></span>. Substitute <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 x\notin x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∉</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\notin x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/221b2b8a0e97121c78fd1588eb7e9faaff3674ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.5ex; height:2.676ex;" alt="{\displaystyle x\notin x}" /></span> for <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 \varphi (x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi (x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c4046f1f2de7df04bde418ba2bc4d3898ac2385" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.659ex; height:2.843ex;" alt="{\displaystyle \varphi (x)}" /></span> to get </p> <dl><dd><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 \exists y\forall x(x\in y\iff x\notin x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∃</mi> <mi>y</mi> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>∈</mo> <mi>y</mi> <mspace width="thickmathspace"></mspace> <mo stretchy="false">⟺</mo> <mspace width="thickmathspace"></mspace> <mi>x</mi> <mo>∉</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \exists y\forall x(x\in y\iff x\notin x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/464db9afa31466235c9febcbac405f42a33dac2e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.603ex; height:2.843ex;" alt="{\displaystyle \exists y\forall x(x\in y\iff x\notin x)}" /></span></dd></dl> <p>Then by <a href="/wiki/Existential_instantiation" title="Existential instantiation">existential instantiation</a> (reusing the symbol <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 y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}" /></span>) and <a href="/wiki/Universal_instantiation" title="Universal instantiation">universal instantiation</a> we have </p> <dl><dd><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 y\in y\iff y\notin y,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>∈</mo> <mi>y</mi> <mspace width="thickmathspace"></mspace> <mo stretchy="false">⟺</mo> <mspace width="thickmathspace"></mspace> <mi>y</mi> <mo>∉</mo> <mi>y</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y\in y\iff y\notin y,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d88b5ba50d04b85d4ad639bb25f59e953365953" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.847ex; height:2.676ex;" alt="{\displaystyle y\in y\iff y\notin y,}" /></span></dd></dl> <p>a contradiction. Therefore, this naive set theory is <a href="/wiki/Consistency" title="Consistency">inconsistent</a>.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Philosophical_implications">Philosophical implications</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Russell%27s_paradox&action=edit&section=3" title="Edit section: Philosophical implications"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Prior to Russell's paradox (and to other similar paradoxes discovered around the time, such as the <a href="/wiki/Burali-Forti_paradox" title="Burali-Forti paradox">Burali-Forti paradox</a>), a common conception of the idea of set was the "extensional concept of set", as recounted by von Neumann and Morgenstern:<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </p> <style data-mw-deduplicate="TemplateStyles:r1244412712">.mw-parser-output .templatequote{overflow:hidden;margin:1em 0;padding:0 32px}.mw-parser-output .templatequotecite{line-height:1.5em;text-align:left;margin-top:0}@media(min-width:500px){.mw-parser-output .templatequotecite{padding-left:1.6em}}</style><blockquote class="templatequote"><p>A set is an arbitrary collection of objects, absolutely no restriction being placed on the nature and number of these objects, the elements of the set in question. The elements constitute and determine the set as such, without any ordering or relationship of any kind between them.</p></blockquote> <p>In particular, there was no distinction between sets and proper classes as collections of objects. Additionally, the existence of each of the elements of a collection was seen as sufficient for the existence of the set of said elements. However, paradoxes such as Russell's and Burali-Forti's showed the impossibility of this conception of set, by examples of collections of objects that do not form sets, despite all said objects being existent. </p> <div class="mw-heading mw-heading2"><h2 id="Set-theoretic_responses">Set-theoretic responses</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Russell%27s_paradox&action=edit&section=4" title="Edit section: Set-theoretic responses"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>From the <a href="/wiki/Principle_of_explosion" title="Principle of explosion">principle of explosion</a> of <a href="/wiki/Classical_logic" title="Classical logic">classical logic</a>, <i>any</i> proposition can be proved from a <a href="/wiki/Contradiction" title="Contradiction">contradiction</a>. Therefore, the presence of contradictions like Russell's paradox in an axiomatic set theory is disastrous; since if any formula can be proved true it destroys the conventional meaning of truth and falsity. Further, since set theory was seen as the basis for an axiomatic development of all other branches of mathematics, Russell's paradox threatened the foundations of mathematics as a whole. This motivated a great deal of research around the turn of the 20th century to develop a consistent (contradiction-free) set theory. </p><p>In 1908, <a href="/wiki/Ernst_Zermelo" title="Ernst Zermelo">Ernst Zermelo</a> proposed an <a href="/wiki/Axiomatic_system" title="Axiomatic system">axiomatization</a> of set theory that avoided the paradoxes of naive set theory by replacing arbitrary set comprehension with weaker existence axioms, such as his <a href="/wiki/Axiom_of_separation" class="mw-redirect" title="Axiom of separation">axiom of separation</a> (<i>Aussonderung</i>). (Avoiding paradox was not Zermelo's original intention, but instead to document which assumptions he used in proving the <a href="/wiki/Well-ordering_theorem" title="Well-ordering theorem">well-ordering theorem</a>.)<sup id="cite_ref-Maddy_9-0" class="reference"><a href="#cite_note-Maddy-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> Modifications to this axiomatic theory proposed in the 1920s by <a href="/wiki/Abraham_Fraenkel" title="Abraham Fraenkel">Abraham Fraenkel</a>, <a href="/wiki/Thoralf_Skolem" title="Thoralf Skolem">Thoralf Skolem</a>, and by Zermelo himself resulted in the axiomatic set theory called <a href="/wiki/ZFC" class="mw-redirect" title="ZFC">ZFC</a>. This theory became widely accepted once Zermelo's <a href="/wiki/Axiom_of_choice" title="Axiom of choice">axiom of choice</a> ceased to be controversial, and ZFC has remained the canonical <a href="/wiki/Axiomatic_set_theory" class="mw-redirect" title="Axiomatic set theory">axiomatic set theory</a> down to the present day. </p><p>ZFC does not assume that, for every property, there is a set of all things satisfying that property. Rather, it asserts that given any set <i>X</i>, any subset of <i>X</i> definable using <a href="/wiki/First-order_logic" title="First-order logic">first-order logic</a> exists. The object <i>R</i> defined by Russell's paradox above cannot be constructed as a subset of any set <i>X</i>, and is therefore not a set in ZFC. In some extensions of ZFC, notably in <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 set theory</a>, objects like <i>R</i> are called <a href="/wiki/Proper_class" class="mw-redirect" title="Proper class">proper classes</a>. </p><p>ZFC is silent about types, although the <a href="/wiki/Von_Neumann_universe" title="Von Neumann universe">cumulative hierarchy</a> has a notion of layers that resemble types. Zermelo himself never accepted Skolem's formulation of ZFC using the language of first-order logic. As José Ferreirós notes, Zermelo insisted instead that "propositional functions (conditions or predicates) used for separating off subsets, as well as the replacement functions, can be 'entirely <i>arbitrary</i><span class="nowrap" style="padding-left:0.1em;">'</span> [ganz <i>beliebig</i>]"; the modern interpretation given to this statement is that Zermelo wanted to include <a href="/wiki/Higher-order_logic" title="Higher-order logic">higher-order quantification</a> in order to avoid <a href="/wiki/Skolem%27s_paradox" title="Skolem's paradox">Skolem's paradox</a>. Around 1930, Zermelo also introduced (apparently independently of von Neumann), the <a href="/wiki/Axiom_of_foundation" class="mw-redirect" title="Axiom of foundation">axiom of foundation</a>, thus—as Ferreirós observes—"by forbidding 'circular' and 'ungrounded' sets, it [ZFC] incorporated one of the crucial motivations of TT [type theory]—the principle of the types of arguments". This 2nd order ZFC preferred by Zermelo, including axiom of foundation, allowed a rich cumulative hierarchy. Ferreirós writes that "Zermelo's 'layers' are essentially the same as the types in the contemporary versions of simple TT [type theory] offered by Gödel and Tarski. One can describe the cumulative hierarchy into which Zermelo developed his models as the universe of a cumulative TT in which transfinite types are allowed. (Once we have adopted an impredicative standpoint, abandoning the idea that classes are constructed, it is not unnatural to accept transfinite types.) Thus, simple TT and ZFC could now be regarded as systems that 'talk' essentially about the same intended objects. The main difference is that TT relies on a strong higher-order logic, while Zermelo employed second-order logic, and ZFC can also be given a first-order formulation. The first-order 'description' of the cumulative hierarchy is much weaker, as is shown by the existence of countable models (Skolem's paradox), but it enjoys some important advantages."<sup id="cite_ref-Ferreirós2008_10-0" class="reference"><a href="#cite_note-Ferreirós2008-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> </p><p>In ZFC, given a set <i>A</i>, it is possible to define a set <i>B</i> that consists of exactly the sets in <i>A</i> that are not members of themselves. <i>B</i> cannot be in <i>A</i> by the same reasoning in Russell's Paradox. This variation of Russell's paradox shows that no set contains everything. </p><p>Through the work of Zermelo and others, especially <a href="/wiki/John_von_Neumann" title="John von Neumann">John von Neumann</a>, the structure of what some see as the "natural" objects described by ZFC eventually became clear: they are the elements of the <a href="/wiki/Von_Neumann_universe" title="Von Neumann universe">von Neumann universe</a>, <i>V</i>, built up from the <a href="/wiki/Empty_set" title="Empty set">empty set</a> by <a href="/wiki/Transfinite_recursion" class="mw-redirect" title="Transfinite recursion">transfinitely iterating</a> the <a href="/wiki/Power_set" title="Power set">power set</a> operation. It is thus now possible again to reason about sets in a non-axiomatic fashion without running afoul of Russell's paradox, namely by reasoning about the elements of <i>V</i>. Whether it is <i>appropriate</i> to think of sets in this way is a point of contention among the rival points of view on the <a href="/wiki/Philosophy_of_mathematics" title="Philosophy of mathematics">philosophy of mathematics</a>. </p><p>Other solutions to Russell's paradox, with an underlying strategy closer to that of <a href="/wiki/Type_theory" title="Type theory">type theory</a>, include <a href="/wiki/Willard_van_Orman_Quine" class="mw-redirect" title="Willard van Orman Quine">Quine</a>'s <a href="/wiki/New_Foundations" title="New Foundations">New Foundations</a> and <a href="/wiki/Scott%E2%80%93Potter_set_theory" title="Scott–Potter set theory">Scott–Potter set theory</a>. Yet another approach is to define multiple membership relation with appropriately modified comprehension scheme, as in the <a href="/wiki/Double_extension_set_theory" title="Double extension set theory">Double extension set theory</a>. </p> <div class="mw-heading mw-heading2"><h2 id="History">History</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Russell%27s_paradox&action=edit&section=5" title="Edit section: History"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Russell discovered the paradox in May<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> or June 1901.<sup id="cite_ref-auto_12-0" class="reference"><a href="#cite_note-auto-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> By his own account in his 1919 <i>Introduction to Mathematical Philosophy</i>, he "attempted to discover some flaw in Cantor's proof that there is no greatest cardinal".<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> In a 1902 letter,<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> he announced the discovery to <a href="/wiki/Gottlob_Frege" title="Gottlob Frege">Gottlob Frege</a> of the paradox in Frege's 1879 <i><a href="/wiki/Begriffsschrift" title="Begriffsschrift">Begriffsschrift</a></i> and framed the problem in terms of both logic and set theory, and in particular in terms of Frege's definition of <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">function</a>:<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>a<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>b<span class="cite-bracket">]</span></a></sup> </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1244412712" /><blockquote class="templatequote"><p>There is just one point where I have encountered a difficulty. You state (p. 17 [p. 23 above]) that a function too, can act as the indeterminate element. This I formerly believed, but now this view seems doubtful to me because of the following contradiction. Let <b>w</b> be the predicate: to be a predicate that cannot be predicated of itself. Can <b>w</b> be predicated of itself? From each answer its opposite follows. Therefore we must conclude that <b>w</b> is not a predicate. Likewise there is no class (as a totality) of those classes which, each taken as a totality, do not belong to themselves. From this I conclude that under certain circumstances a definable collection [Menge] does not form a totality.</p></blockquote> <p>Russell would go on to cover it at length in his 1903 <i><a href="/wiki/The_Principles_of_Mathematics" title="The Principles of Mathematics">The Principles of Mathematics</a></i>, where he repeated his first encounter with the paradox:<sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1244412712" /><blockquote class="templatequote"><p>Before taking leave of fundamental questions, it is necessary to examine more in detail the singular contradiction, already mentioned, with regard to predicates not predicable of themselves. ... I may mention that I was led to it in the endeavour to reconcile Cantor's proof....</p></blockquote> <p>Russell wrote to Frege about the paradox just as Frege was preparing the second volume of his <i>Grundgesetze der Arithmetik</i>.<sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> Frege responded to Russell very quickly; his letter dated 22 June 1902 appeared, with van Heijenoort's commentary in Heijenoort 1967:126–127. Frege then wrote an appendix admitting to the paradox,<sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> and proposed a solution that Russell would endorse in his <i>Principles of Mathematics</i>,<sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> but was later considered by some to be unsatisfactory.<sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> For his part, Russell had his work at the printers and he added an appendix on the <a href="/wiki/Type_theory" title="Type theory">doctrine of types</a>.<sup id="cite_ref-22" class="reference"><a href="#cite_note-22"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> </p><p>Ernst Zermelo in his (1908) <i>A new proof of the possibility of a well-ordering</i> (published at the same time he published "the first axiomatic set theory")<sup id="cite_ref-23" class="reference"><a href="#cite_note-23"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> laid claim to prior discovery of the <a href="/wiki/Antinomy" title="Antinomy">antinomy</a> in Cantor's naive set theory. He states: "And yet, even the elementary form that Russell<sup>9</sup> gave to the set-theoretic antinomies could have persuaded them [J. König, Jourdain, F. Bernstein] that the solution of these difficulties is not to be sought in the surrender of well-ordering but only in a suitable restriction of the notion of set".<sup id="cite_ref-24" class="reference"><a href="#cite_note-24"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> Footnote 9 is where he stakes his claim: </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1244412712" /><blockquote class="templatequote"><p><sup>9</sup><i>1903</i>, pp. 366–368. I had, however, discovered this antinomy myself, independently of Russell, and had communicated it prior to 1903 to Professor Hilbert among others.<sup id="cite_ref-25" class="reference"><a href="#cite_note-25"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup></p></blockquote> <p>Frege sent a copy of his <i>Grundgesetze der Arithmetik</i> to Hilbert; as noted above, Frege's last volume mentioned the paradox that Russell had communicated to Frege. After receiving Frege's last volume, on 7 November 1903, Hilbert wrote a letter to Frege in which he said, referring to Russell's paradox, "I believe Dr. Zermelo discovered it three or four years ago". A written account of Zermelo's actual argument was discovered in the <i>Nachlass</i> of <a href="/wiki/Edmund_Husserl" title="Edmund Husserl">Edmund Husserl</a>.<sup id="cite_ref-26" class="reference"><a href="#cite_note-26"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup> </p><p>In 1923, <a href="/wiki/Ludwig_Wittgenstein" title="Ludwig Wittgenstein">Ludwig Wittgenstein</a> proposed to "dispose" of Russell's paradox as follows: </p> <blockquote> <p>The reason why a function cannot be its own argument is that the sign for a function already contains the prototype of its argument, and it cannot contain itself. For let us suppose that the function F(fx) could be its own argument: in that case there would be a proposition <b>F(F(fx))</b>, in which the outer function <b>F</b> and the inner function <b>F</b> must have different meanings, since the inner one has the form <b>O(fx)</b> and the outer one has the form <b>Y(O(fx))</b>. Only the letter 'F' is common to the two functions, but the letter by itself signifies nothing. This immediately becomes clear if instead of <b>F(Fu)</b> we write <b>(do) : F(Ou) . Ou = Fu</b>. That disposes of Russell's paradox. (<i><a href="/wiki/Tractatus_Logico-Philosophicus" title="Tractatus Logico-Philosophicus">Tractatus Logico-Philosophicus</a></i>, 3.333) </p> </blockquote> <p>Russell and <a href="/wiki/Alfred_North_Whitehead" title="Alfred North Whitehead">Alfred North Whitehead</a> wrote their three-volume <i><a href="/wiki/Principia_Mathematica" title="Principia Mathematica">Principia Mathematica</a></i> hoping to achieve what Frege had been unable to do. They sought to banish the paradoxes of <a href="/wiki/Naive_set_theory" title="Naive set theory">naive set theory</a> by employing a theory of types they devised for this purpose. While they succeeded in grounding arithmetic in a fashion, it is not at all evident that they did so by purely logical means. While <i>Principia Mathematica</i> avoided the known paradoxes and allows the derivation of a great deal of mathematics, its system gave rise to new problems. </p><p>In any event, <a href="/wiki/Kurt_G%C3%B6del" title="Kurt Gödel">Kurt Gödel</a> in 1930–31 proved that while the logic of much of <i>Principia Mathematica</i>, now known as first-order logic, is <a href="/wiki/G%C3%B6del%27s_completeness_theorem" title="Gödel's completeness theorem">complete</a>, <a href="/wiki/Peano_axioms" title="Peano axioms">Peano arithmetic</a> is necessarily incomplete if it is <a href="/wiki/Consistent" class="mw-redirect" title="Consistent">consistent</a>. This is very widely—though not universally—regarded as having shown the <a href="/wiki/Logicist" class="mw-redirect" title="Logicist">logicist</a> program of Frege to be impossible to complete. </p><p>In 2001, A Centenary International Conference celebrating the first hundred years of Russell's paradox was held in Munich and its proceedings have been published.<sup id="cite_ref-auto_12-1" class="reference"><a href="#cite_note-auto-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Applied_versions">Applied versions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Russell%27s_paradox&action=edit&section=6" title="Edit section: Applied versions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1251242444">.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px 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class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><a href="/wiki/File:Question_book-new.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/50px-Question_book-new.svg.png" decoding="async" width="50" height="39" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/75px-Question_book-new.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/100px-Question_book-new.svg.png 2x" data-file-width="512" data-file-height="399" /></a></span></div></td><td class="mbox-text"><div class="mbox-text-span">This section <b>does not <a href="/wiki/Wikipedia:Citing_sources" title="Wikipedia:Citing sources">cite</a> any <a href="/wiki/Wikipedia:Verifiability" title="Wikipedia:Verifiability">sources</a></b>.<span class="hide-when-compact"> Please help <a href="/wiki/Special:EditPage/Russell%27s_paradox" title="Special:EditPage/Russell's paradox">improve this section</a> by <a href="/wiki/Help:Referencing_for_beginners" title="Help:Referencing for beginners">adding citations to reliable sources</a>. Unsourced material may be challenged and <a href="/wiki/Wikipedia:Verifiability#Burden_of_evidence" title="Wikipedia:Verifiability">removed</a>.</span> <span class="date-container"><i>(<span class="date">March 2021</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1251242444" /><table class="box-Original_research plainlinks metadata ambox ambox-content ambox-Original_research" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/b/b4/Ambox_important.svg/40px-Ambox_important.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/b/b4/Ambox_important.svg/60px-Ambox_important.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/b/b4/Ambox_important.svg/80px-Ambox_important.svg.png 2x" data-file-width="40" data-file-height="40" /></span></span></div></td><td class="mbox-text"><div class="mbox-text-span">This section <b>possibly contains <a href="/wiki/Wikipedia:No_original_research" title="Wikipedia:No original research">original research</a></b>.<span class="hide-when-compact"> Please <a class="external text" href="https://en.wikipedia.org/w/index.php?title=Russell%27s_paradox&action=edit">improve it</a> by <a href="/wiki/Wikipedia:Verifiability" title="Wikipedia:Verifiability">verifying</a> the claims made and adding <a href="/wiki/Wikipedia:Citing_sources#Inline_citations" title="Wikipedia:Citing sources">inline citations</a>. Statements consisting only of original research should be removed.</span> <span class="date-container"><i>(<span class="date">March 2021</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table><sup class="noprint Inline-Template" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Accuracy_dispute#Disputed_statement" title="Wikipedia:Accuracy dispute"><span title="The material near this tag is possibly inaccurate or nonfactual. (March 2021)">dubious</span></a> – <a href="/wiki/Talk:Russell%27s_paradox#Dubious" title="Talk:Russell's paradox">discuss</a></i>]</sup> </div> </div><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <p>There are some versions of this paradox that are closer to real-life situations and may be easier to understand for non-logicians. For example, the <a href="/wiki/Barber_paradox" title="Barber paradox">barber paradox</a> supposes a barber who shaves all men who do not shave themselves and only men who do not shave themselves. When one thinks about whether the barber should shave himself or not, a similar paradox begins to emerge.<sup id="cite_ref-27" class="reference"><a href="#cite_note-27"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup> </p><p>An easy refutation of the "layman's versions" such as the barber paradox seems to be that no such barber exists, or that the barber is not a man, and so can exist without paradox. The whole point of Russell's paradox is that the answer "such a set does not exist" means the definition of the notion of set within a given theory is unsatisfactory. Note the difference between the statements "such a set does not exist" and "it is an <a href="/wiki/Empty_set" title="Empty set">empty set</a>". It is like the difference between saying "There is no bucket" and saying "The bucket is empty". </p><p>A notable exception to the above may be the <a href="/wiki/Grelling%E2%80%93Nelson_paradox" title="Grelling–Nelson paradox">Grelling–Nelson paradox</a>, in which words and meaning are the elements of the scenario rather than people and hair-cutting. Though it is easy to refute the barber's paradox by saying that such a barber does not (and <i>cannot</i>) exist, it is impossible to say something similar about a meaningfully defined word. </p><p>One way that the paradox has been dramatised is as follows: Suppose that every public library has to compile a catalogue of all its books. Since the catalogue is itself one of the library's books, some librarians include it in the catalogue for completeness; while others leave it out as it being one of the library's books is self evident. Now imagine that all these catalogues are sent to the national library. Some of them include themselves in their listings, others do not. The national librarian compiles two master catalogues—one of all the catalogues that list themselves, and one of all those that do not. </p><p>The question is: should these master catalogues list themselves? The 'catalogue of all catalogues that list themselves' is no problem. If the librarian does not include it in its own listing, it remains a true catalogue of those catalogues that do include themselves. If he does include it, it remains a true catalogue of those that list themselves. However, just as the librarian cannot go wrong with the first master catalogue, he is doomed to fail with the second. When it comes to the 'catalogue of all catalogues that do not list themselves', the librarian cannot include it in its own listing, because then it would include itself, and so belong in the other catalogue, that of catalogues that do include themselves. However, if the librarian leaves it out, the catalogue is incomplete. Either way, it can never be a true master catalogue of catalogues that do not list themselves. </p> <div class="mw-heading mw-heading2"><h2 id="Applications_and_related_topics">Applications and related topics</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Russell%27s_paradox&action=edit&section=7" title="Edit section: Applications and related topics"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Russell-like_paradoxes">Russell-like paradoxes</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Russell%27s_paradox&action=edit&section=8" title="Edit section: Russell-like paradoxes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>As illustrated above for the barber paradox, Russell's paradox is not hard to extend. Take: </p> <ul><li>A <a href="/wiki/Transitive_verb" title="Transitive verb">transitive verb</a> <span class="nowrap">⟨V⟩</span>, that can be applied to its <a href="/wiki/Substantive" class="mw-redirect" title="Substantive">substantive</a> form.</li></ul> <p>Form the sentence: </p> <dl><dd>The <span class="nowrap">⟨V⟩</span>er that <span class="nowrap">⟨V⟩</span>s all (and only those) who do not <span class="nowrap">⟨V⟩</span> themselves,</dd></dl> <p>Sometimes the "all" is replaced by "all <span class="nowrap">⟨V⟩</span>ers". </p><p>An example would be "paint": </p> <dl><dd>The <i>paint</i>er that <i>paint</i>s all (and only those) that do not <i>paint</i> themselves.</dd></dl> <p>or "elect" </p> <dl><dd>The <i>elect</i>or (<a href="/wiki/Group_representation" title="Group representation">representative</a>), that <i>elect</i>s all that do not <i>elect</i> themselves.</dd></dl> <p>In the <a href="/wiki/The_Big_Bang_Theory_(season_8)#Episodes" class="mw-redirect" title="The Big Bang Theory (season 8)">Season 8</a> episode of <i><a href="/wiki/The_Big_Bang_Theory" title="The Big Bang Theory">The Big Bang Theory</a></i>, "The Skywalker Intrusion", <a href="/wiki/Sheldon_Cooper" title="Sheldon Cooper">Sheldon Cooper</a> analyzes the song "<a href="/wiki/Play_That_Funky_Music" title="Play That Funky Music">Play That Funky Music</a>", concluding that the lyrics present a musical example of Russell's Paradox.<sup id="cite_ref-28" class="reference"><a href="#cite_note-28"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup> </p><p>Paradoxes that fall in this scheme include: </p> <ul><li><a href="/wiki/Barber_paradox" title="Barber paradox">The barber with "shave"</a>.</li> <li>The original Russell's paradox with "contain": The container (Set) that contains all (containers) that do not contain themselves.</li> <li>The <a href="/wiki/Grelling%E2%80%93Nelson_paradox" title="Grelling–Nelson paradox">Grelling–Nelson paradox</a> with "describer": The describer (word) that describes all words, that do not describe themselves.</li> <li><a href="/wiki/Richard%27s_paradox" title="Richard's paradox">Richard's paradox</a> with "denote": The denoter (number) that denotes all denoters (numbers) that do not denote themselves. (In this paradox, all descriptions of numbers get an assigned number. The term "that denotes all denoters (numbers) that do not denote themselves" is here called <i>Richardian</i>.)</li> <li>"I am lying.", namely the <a href="/wiki/Liar_paradox" title="Liar paradox">liar paradox</a> and <a href="/wiki/Epimenides_paradox" title="Epimenides paradox">Epimenides paradox</a>, whose origins are ancient</li> <li><a href="/wiki/Russell%E2%80%93Myhill_paradox" class="mw-redirect" title="Russell–Myhill paradox">Russell–Myhill paradox</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Related_paradoxes">Related paradoxes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Russell%27s_paradox&action=edit&section=9" title="Edit section: Related paradoxes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>The <a href="/wiki/Burali-Forti_paradox" title="Burali-Forti paradox">Burali-Forti paradox</a>, about the <a href="/wiki/Order_type" title="Order type">order type</a> of all <a href="/wiki/Well-ordering" class="mw-redirect" title="Well-ordering">well-orderings</a></li> <li>The <a href="/wiki/Kleene%E2%80%93Rosser_paradox" title="Kleene–Rosser paradox">Kleene–Rosser paradox</a>, showing that the original <a href="/wiki/Lambda_calculus" title="Lambda calculus">lambda calculus</a> is inconsistent, by means of a self-negating statement</li> <li><a href="/wiki/Curry%27s_paradox" title="Curry's paradox">Curry's paradox</a> (named after <a href="/wiki/Haskell_Curry" title="Haskell Curry">Haskell Curry</a>), which does not require <a href="/wiki/Negation" title="Negation">negation</a></li> <li>The <a href="/wiki/Interesting_number_paradox" title="Interesting number paradox">smallest uninteresting integer</a> paradox</li> <li><a href="/wiki/Girard%27s_paradox" class="mw-redirect" title="Girard's paradox">Girard's paradox</a> in <a href="/wiki/Type_theory" title="Type theory">type theory</a></li></ul> <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=Russell%27s_paradox&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/Basic_Law_V" class="mw-redirect" title="Basic Law V">Basic Law V</a></li> <li><a href="/wiki/Cantor%27s_diagonal_argument" title="Cantor's diagonal argument">Cantor's diagonal argument</a> – Proof in set theory</li> <li><a href="/wiki/G%C3%B6del%27s_incompleteness_theorems" title="Gödel's incompleteness theorems">Gödel's incompleteness theorems</a> – Limitative results in mathematical logic</li> <li><a href="/wiki/Hilbert%27s_first_problem" class="mw-redirect" title="Hilbert's first problem">Hilbert's first problem</a> – Proposition in mathematical logic<span style="display:none" class="category-annotation-with-redirected-description">Pages displaying short descriptions of redirect targets</span></li> <li>"<a href="/wiki/On_Denoting" title="On Denoting">On Denoting</a>"</li> <li><a href="/wiki/Paradoxes_of_set_theory" title="Paradoxes of set theory">Paradoxes of set theory</a></li> <li><a href="/wiki/Quine%27s_paradox" title="Quine's paradox">Quine's paradox</a></li> <li><a href="/wiki/Self-reference" title="Self-reference">Self-reference</a></li> <li><a href="/wiki/List_of_self%E2%80%93referential_paradoxes" class="mw-redirect" title="List of self–referential paradoxes">List of self–referential paradoxes</a></li> <li><a href="/wiki/Strange_loop" title="Strange loop">Strange loop</a> – Cyclic structure that goes through several levels in a hierarchical system</li> <li><a href="/wiki/Universal_set" title="Universal set">Universal set</a> – Mathematical set containing all objects</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=Russell%27s_paradox&action=edit&section=11" title="Edit section: Notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist reflist-lower-alpha"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-15">^</a></b></span> <span class="reference-text">In the following, p. 17 refers to a page in the original <i>Begriffsschrift</i>, and page 23 refers to the same page in van Heijenoort 1967</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">Remarkably, this letter was unpublished until van Heijenoort 1967—it appears with van Heijenoort's commentary at van Heijenoort 1967:124–125.</span> </li> </ol></div></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=Russell%27s_paradox&action=edit&section=12" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626" /><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><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">Russell, Bertrand, "Correspondence with Frege}. In Gottlob Frege <i>Philosophical and Mathematical Correspondence</i>. Translated by Hans Kaal., University of Chicago Press, Chicago, 1980.</span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text">Russell, Bertrand. <i><a href="/wiki/The_Principles_of_Mathematics" title="The Principles of Mathematics">The Principles of Mathematics</a></i>. 2d. ed. Reprint, New York: W. W. Norton & Company, 1996. (First published in 1903.)</span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text">Irvine, A. D., H. Deutsch (2021). "Russell's Paradox". Stanford Encyclopedia of Philosophy (Spring 2021 Edition), E. N. Zalta (ed.), <a rel="nofollow" class="external autonumber" href="https://plato.stanford.edu/entries/russell-paradox/">[1]</a></span> </li> <li id="cite_note-FraenkelBar-Hillel1973-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-FraenkelBar-Hillel1973_4-0">^</a></b></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="CITEREFA.A._FraenkelY._Bar-HillelA._Levy1973" class="citation book cs1">A.A. Fraenkel; Y. Bar-Hillel; A. Levy (1973). <i>Foundations of Set Theory</i>. Elsevier. pp. <span class="nowrap">156–</span>157. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-08-088705-0" title="Special:BookSources/978-0-08-088705-0"><bdi>978-0-08-088705-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Foundations+of+Set+Theory&rft.pages=%3Cspan+class%3D%22nowrap%22%3E156-%3C%2Fspan%3E157&rft.pub=Elsevier&rft.date=1973&rft.isbn=978-0-08-088705-0&rft.au=A.A.+Fraenkel&rft.au=Y.+Bar-Hillel&rft.au=A.+Levy&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARussell%27s+paradox" 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">Bernhard Rang, Wolfgang Thomas: Zermelo's Discovery of the "Russell Paradox", Historia Mathematica 8.</span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><a href="/wiki/Walter_Purkert" title="Walter Purkert">Walter Purkert</a>, Hans J. Ilgauds: <i>Vita Mathematica - Georg Cantor</i>, Birkhäuser, 1986, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/3-764-31770-1" title="Special:BookSources/3-764-31770-1">3-764-31770-1</a></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="CITEREFIrvineDeutsch2014" class="citation encyclopaedia cs1">Irvine, Andrew David; Deutsch, Harry (2014). <a rel="nofollow" class="external text" href="http://plato.stanford.edu/archives/win2014/entries/russell-paradox/">"Russell's Paradox"</a>. In Zalta, Edward N. (ed.). <i>The Stanford Encyclopedia of Philosophy</i>.</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&rft.btitle=The+Stanford+Encyclopedia+of+Philosophy&rft.date=2014&rft.aulast=Irvine&rft.aufirst=Andrew+David&rft.au=Deutsch%2C+Harry&rft_id=http%3A%2F%2Fplato.stanford.edu%2Farchives%2Fwin2014%2Fentries%2Frussell-paradox%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARussell%27s+paradox" class="Z3988"></span></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">R. Bunn, <a rel="nofollow" class="external text" href="https://open.library.ubc.ca/media/stream/pdf/831/1.0100043/1">Infinite Sets and Numbers</a> (1967), pp.176–178. Ph.D dissertation, University of British Columbia</span> </li> <li id="cite_note-Maddy-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-Maddy_9-0">^</a></b></span> <span class="reference-text">P. Maddy, "<a rel="nofollow" class="external text" href="https://www.cs.umd.edu/~gasarch/BLOGPAPERS/belaxioms1.pdf">Believing the Axioms I</a>" (1988). Association for Symbolic Logic.</span> </li> <li id="cite_note-Ferreirós2008-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-Ferreirós2008_10-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFJosé_Ferreirós2008" class="citation book cs1">José Ferreirós (2008). <i>Labyrinth of Thought: A History of Set Theory and Its Role in Modern Mathematics</i> (2nd ed.). Springer. § Zermelo's cumulative hierarchy pp. 374-378. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-7643-8350-3" title="Special:BookSources/978-3-7643-8350-3"><bdi>978-3-7643-8350-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Labyrinth+of+Thought%3A+A+History+of+Set+Theory+and+Its+Role+in+Modern+Mathematics&rft.pages=%C2%A7+Zermelo%27s+cumulative+hierarchy+pp.+374-378&rft.edition=2nd&rft.pub=Springer&rft.date=2008&rft.isbn=978-3-7643-8350-3&rft.au=Jos%C3%A9+Ferreir%C3%B3s&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARussell%27s+paradox" class="Z3988"></span></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"><i>The Autobiography of Bertrand Russell</i>, George Allen and Unwin Ltd., 1971, page 147: "At the end of the Lent Term [1901], I went back to Fernhurst, where I set to work to write out the logical deduction of mathematics which afterwards became <i>Principia Mathematica</i>. I thought the work was nearly finished but <i>in the month of May</i> [emphasis added] I had an intellectual set-back […]. Cantor had a proof that there is no greatest number, and it seemed to me that the number of all the things in the world ought to be the greatest possible. Accordingly, I examined his proof with some minuteness, and endeavoured to apply it to the class of all the things there are. This led me to consider those classes which are not members of themselves, and to ask whether the class of such classes is or is not a member of itself. I found that either answer implies its contradictory".</span> </li> <li id="cite_note-auto-12"><span class="mw-cite-backlink">^ <a href="#cite_ref-auto_12-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-auto_12-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFGodehard_Link2004" class="citation cs2">Godehard Link (2004), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=Xg6QpedPpcsC&pg=PA350"><i>One hundred years of Russell's paradox</i></a>, Walter de Gruyter, p. 350, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-11-017438-0" title="Special:BookSources/978-3-11-017438-0"><bdi>978-3-11-017438-0</bdi></a><span class="reference-accessdate">, retrieved <span class="nowrap">2016-02-22</span></span></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=One+hundred+years+of+Russell%27s+paradox&rft.pages=350&rft.pub=Walter+de+Gruyter&rft.date=2004&rft.isbn=978-3-11-017438-0&rft.au=Godehard+Link&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DXg6QpedPpcsC%26pg%3DPA350&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARussell%27s+paradox" 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">Russell 1920:136</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="CITEREFGottlob_Frege,_Michael_Beaney1997" class="citation cs2">Gottlob Frege, Michael Beaney (1997), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=4ktC0UrG4V8C&pg=PA253"><i>The Frege reader</i></a>, Wiley, p. 253, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-631-19445-3" title="Special:BookSources/978-0-631-19445-3"><bdi>978-0-631-19445-3</bdi></a><span class="reference-accessdate">, retrieved <span class="nowrap">2016-02-22</span></span></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Frege+reader&rft.pages=253&rft.pub=Wiley&rft.date=1997&rft.isbn=978-0-631-19445-3&rft.au=Gottlob+Frege%2C+Michael+Beaney&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D4ktC0UrG4V8C%26pg%3DPA253&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARussell%27s+paradox" class="Z3988"></span>. Also van Heijenoort 1967:124–125</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">Russell 1903:101</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">cf van Heijenoort's commentary before Frege's <i>Letter to Russell</i> in van Heijenoort 1964:126.</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">van Heijenoort's commentary, cf van Heijenoort 1967:126; Frege starts his analysis by this exceptionally honest comment : "Hardly anything more unfortunate can befall a scientific writer than to have one of the foundations of his edifice shaken after the work is finished. This was the position I was placed in by a letter of Mr Bertrand Russell, just when the printing of this volume was nearing its completion" (Appendix of <i>Grundgesetze der Arithmetik, vol. II</i>, in <i>The Frege Reader</i>, p. 279, translation by Michael Beaney</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">cf van Heijenoort's commentary, cf van Heijenoort 1967:126. The added text reads as follows: "<i>Note</i>. The second volume of Gg., which appeared too late to be noticed in the Appendix, contains an interesting discussion of the contradiction (pp. 253–265), suggesting that the solution is to be found by denying that two <a href="/wiki/Propositional_function" title="Propositional function">propositional functions</a> that determine equal classes must be equivalent. As it seems very likely that this is the true solution, the reader is strongly recommended to examine Frege's argument on the point" (Russell 1903:522); The abbreviation Gg. stands for Frege's <i>Grundgezetze der Arithmetik</i>. Begriffsschriftlich abgeleitet. Vol. I. Jena, 1893. Vol. II. 1903.</span> </li> <li id="cite_note-21"><span class="mw-cite-backlink"><b><a href="#cite_ref-21">^</a></b></span> <span class="reference-text">Livio states that "While Frege did make some desperate attempts to remedy his axiom system, he was unsuccessful. The conclusion appeared to be disastrous ..." Livio 2009:188. But van Heijenoort in his commentary before Frege's (1902) <i>Letter to Russell</i> describes Frege's proposed "way out" in some detail—the matter has to do with the " 'transformation of the generalization of an equality into an equality of courses-of-values. For Frege a function is something incomplete, 'unsaturated<span style="padding-right:.15em;">'</span>"; this seems to contradict the contemporary notion of a "function in extension"; see Frege's wording at page 128: "Incidentally, it seems to me that the expression 'a predicate is predicated of itself' is not exact. ...Therefore I would prefer to say that 'a concept is predicated of its own extension' [etc]". But he waffles at the end of his suggestion that a function-as-concept-in-extension can be written as predicated of its function. van Heijenoort cites Quine: "For a late and thorough study of Frege's "way out", see <i>Quine 1956</i>": "On Frege's way out", <i>Mind 64</i>, 145–159; reprinted in <i>Quine 1955b</i>: <i>Appendix. Completeness of quantification theory. Loewenheim's theorem</i>, enclosed as a pamphlet with part of the third printing (1955) of <i>Quine 1950</i> and incorporated in the revised edition (1959), 253—260" (cf REFERENCES in van Heijenoort 1967:649)</span> </li> <li id="cite_note-22"><span class="mw-cite-backlink"><b><a href="#cite_ref-22">^</a></b></span> <span class="reference-text">Russell mentions this fact to Frege, cf van Heijenoort's commentary before Frege's (1902) <i>Letter to Russell</i> in van Heijenoort 1967:126</span> </li> <li id="cite_note-23"><span class="mw-cite-backlink"><b><a href="#cite_ref-23">^</a></b></span> <span class="reference-text">van Heijenoort's commentary before Zermelo (1908a) <i>Investigations in the foundations of set theory</i> I in van Heijenoort 1967:199</span> </li> <li id="cite_note-24"><span class="mw-cite-backlink"><b><a href="#cite_ref-24">^</a></b></span> <span class="reference-text">van Heijenoort 1967:190–191. In the section before this he objects strenuously to the notion of <a href="/wiki/Impredicativity" title="Impredicativity">impredicativity</a> as defined by Poincaré (and soon to be taken by Russell, too, in his 1908 <i>Mathematical logic as based on the theory of types</i> cf van Heijenoort 1967:150–182).</span> </li> <li id="cite_note-25"><span class="mw-cite-backlink"><b><a href="#cite_ref-25">^</a></b></span> <span class="reference-text">Ernst Zermelo (1908) <i>A new proof of the possibility of a well-ordering</i> in van Heijenoort 1967:183–198. Livio 2009:191 reports that Zermelo "discovered Russell's paradox independently as early as 1900"; Livio in turn cites Ewald 1996 and van Heijenoort 1967 (cf Livio 2009:268).</span> </li> <li id="cite_note-26"><span class="mw-cite-backlink"><b><a href="#cite_ref-26">^</a></b></span> <span class="reference-text">B. Rang and W. Thomas, "Zermelo's discovery of the 'Russell Paradox'", <i>Historia Mathematica</i>, v. 8 n. 1, 1981, pp. 15–22. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0315-0860%2881%2990002-1">10.1016/0315-0860(81)90002-1</a></span> </li> <li id="cite_note-27"><span class="mw-cite-backlink"><b><a href="#cite_ref-27">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.oxfordreference.com/display/10.1093/oi/authority.20110803095446216">"barber paradox"</a>. <i>Oxford Reference</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2024-02-04</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Oxford+Reference&rft.atitle=barber+paradox&rft_id=https%3A%2F%2Fwww.oxfordreference.com%2Fdisplay%2F10.1093%2Foi%2Fauthority.20110803095446216&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARussell%27s+paradox" class="Z3988"></span></span> </li> <li id="cite_note-28"><span class="mw-cite-backlink"><b><a href="#cite_ref-28">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.mprnews.org/story/2016/09/27/play-that-funky-music-no-1-40-years-ago">"Play That Funky Music Was No. 1 40 Years Ago"</a>. <i><a href="/wiki/Minnesota_Public_Radio" title="Minnesota Public Radio">Minnesota Public Radio</a></i>. September 27, 2016<span class="reference-accessdate">. Retrieved <span class="nowrap">January 30,</span> 2022</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Minnesota+Public+Radio&rft.atitle=Play+That+Funky+Music+Was+No.+1+40+Years+Ago&rft.date=2016-09-27&rft_id=https%3A%2F%2Fwww.mprnews.org%2Fstory%2F2016%2F09%2F27%2Fplay-that-funky-music-no-1-40-years-ago&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARussell%27s+paradox" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Sources">Sources</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Russell%27s_paradox&action=edit&section=13" title="Edit section: Sources"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239549316">.mw-parser-output .refbegin{margin-bottom:0.5em}.mw-parser-output .refbegin-hanging-indents>ul{margin-left:0}.mw-parser-output .refbegin-hanging-indents>ul>li{margin-left:0;padding-left:3.2em;text-indent:-3.2em}.mw-parser-output .refbegin-hanging-indents ul,.mw-parser-output .refbegin-hanging-indents ul li{list-style:none}@media(max-width:720px){.mw-parser-output .refbegin-hanging-indents>ul>li{padding-left:1.6em;text-indent:-1.6em}}.mw-parser-output .refbegin-columns{margin-top:0.3em}.mw-parser-output .refbegin-columns ul{margin-top:0}.mw-parser-output .refbegin-columns li{page-break-inside:avoid;break-inside:avoid-column}@media screen{.mw-parser-output .refbegin{font-size:90%}}</style><div class="refbegin" style=""> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFPotter2004" class="citation cs2">Potter, Michael (15 January 2004), <i>Set Theory and its Philosophy</i>, <a href="/wiki/Clarendon_Press" class="mw-redirect" title="Clarendon Press">Clarendon Press</a> (<a href="/wiki/Oxford_University_Press" title="Oxford University Press">Oxford University Press</a>), <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-926973-0" title="Special:BookSources/978-0-19-926973-0"><bdi>978-0-19-926973-0</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Set+Theory+and+its+Philosophy&rft.pub=Clarendon+Press+%28Oxford+University+Press%29&rft.date=2004-01-15&rft.isbn=978-0-19-926973-0&rft.aulast=Potter&rft.aufirst=Michael&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARussell%27s+paradox" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFvan_Heijenoort1967" class="citation cs2"><a href="/wiki/Jean_van_Heijenoort" title="Jean van Heijenoort">van Heijenoort, Jean</a> (1967), <i>From Frege to Gödel: A Source Book in Mathematical Logic, 1879–1931, (third printing 1976)</i>, Cambridge, Massachusetts: <a href="/wiki/Harvard_University_Press" title="Harvard University Press">Harvard University Press</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-674-32449-8" title="Special:BookSources/0-674-32449-8"><bdi>0-674-32449-8</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+Frege+to+G%C3%B6del%3A+A+Source+Book+in+Mathematical+Logic%2C+1879%E2%80%931931%2C+%28third+printing+1976%29&rft.place=Cambridge%2C+Massachusetts&rft.pub=Harvard+University+Press&rft.date=1967&rft.isbn=0-674-32449-8&rft.aulast=van+Heijenoort&rft.aufirst=Jean&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARussell%27s+paradox" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFLivio2009" class="citation cs2"><a href="/wiki/Mario_Livio" title="Mario Livio">Livio, Mario</a> (6 January 2009), <i>Is God a Mathematician?</i>, New York: <a href="/wiki/Simon_%26_Schuster" title="Simon & Schuster">Simon & Schuster</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-7432-9405-8" title="Special:BookSources/978-0-7432-9405-8"><bdi>978-0-7432-9405-8</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Is+God+a+Mathematician%3F&rft.place=New+York&rft.pub=Simon+%26+Schuster&rft.date=2009-01-06&rft.isbn=978-0-7432-9405-8&rft.aulast=Livio&rft.aufirst=Mario&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARussell%27s+paradox" class="Z3988"></span></li></ul> </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=Russell%27s_paradox&action=edit&section=14" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFKaplan2022" class="citation web cs1">Kaplan, Jeffrey (2022). <a rel="nofollow" class="external text" href="https://www.youtube.com/watch?v=ymGt7I4Yn3k">"Russell's Paradox - a simple explanation of a profound problem"</a>. <i>YouTube</i><span class="reference-accessdate">. Retrieved <span class="nowrap">25 November</span> 2023</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=YouTube&rft.atitle=Russell%27s+Paradox+-+a+simple+explanation+of+a+profound+problem&rft.date=2022&rft.aulast=Kaplan&rft.aufirst=Jeffrey&rft_id=https%3A%2F%2Fwww.youtube.com%2Fwatch%3Fv%3DymGt7I4Yn3k&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARussell%27s+paradox" class="Z3988"></span></li></ul> <style data-mw-deduplicate="TemplateStyles:r1235681985">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:var(--background-color-interactive-subtle,#f8f9fa);display:flow-root}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output .side-box-image{padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .side-box-imageright{padding:2px 0.9em 2px 0;text-align:center}@media(min-width:500px){.mw-parser-output .side-box-flex{display:flex;align-items:center}.mw-parser-output .side-box-text{flex:1;min-width:0}}@media(min-width:720px){.mw-parser-output .side-box{width:238px}.mw-parser-output .side-box-right{clear:right;float:right;margin-left:1em}.mw-parser-output .side-box-left{margin-right:1em}}</style><style data-mw-deduplicate="TemplateStyles:r1237033735">@media print{body.ns-0 .mw-parser-output .sistersitebox{display:none!important}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}</style><div class="side-box side-box-right plainlinks sistersitebox"><style data-mw-deduplicate="TemplateStyles:r1126788409">.mw-parser-output .plainlist ol,.mw-parser-output .plainlist ul{line-height:inherit;list-style:none;margin:0;padding:0}.mw-parser-output .plainlist ol li,.mw-parser-output .plainlist ul li{margin-bottom:0}</style> <div class="side-box-flex"> <div class="side-box-image"><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Wikiversity_logo_2017.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Wikiversity_logo_2017.svg/40px-Wikiversity_logo_2017.svg.png" decoding="async" width="40" height="33" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Wikiversity_logo_2017.svg/60px-Wikiversity_logo_2017.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Wikiversity_logo_2017.svg/80px-Wikiversity_logo_2017.svg.png 2x" data-file-width="626" data-file-height="512" /></a></span></div> <div class="side-box-text plainlist">Wikiversity has learning resources about <i><b><a href="https://en.wikiversity.org/wiki/Russell%27s_paradox" class="extiw" title="v:Russell's paradox">Russell's paradox</a></b></i></div></div> </div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite class="citation encyclopaedia cs1"><a rel="nofollow" class="external text" href="http://www.iep.utm.edu/par-russ">"Russell's Paradox"</a>. <i><a href="/wiki/Internet_Encyclopedia_of_Philosophy" title="Internet Encyclopedia of Philosophy">Internet Encyclopedia of Philosophy</a></i>.</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&rft.btitle=Internet+Encyclopedia+of+Philosophy&rft_id=http%3A%2F%2Fwww.iep.utm.edu%2Fpar-russ&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARussell%27s+paradox" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFIrvine2016" class="citation encyclopaedia cs1">Irvine, Andrew David (2016). <a rel="nofollow" class="external text" href="https://plato.stanford.edu/entries/russell-paradox/">"Russell's Paradox"</a>. In <a href="/wiki/Edward_N._Zalta" title="Edward N. Zalta">Zalta, Edward N.</a> (ed.). <i><a href="/wiki/Stanford_Encyclopedia_of_Philosophy" title="Stanford Encyclopedia of Philosophy">Stanford Encyclopedia of Philosophy</a></i>.</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&rft.btitle=Stanford+Encyclopedia+of+Philosophy&rft.date=2016&rft.aulast=Irvine&rft.aufirst=Andrew+David&rft_id=https%3A%2F%2Fplato.stanford.edu%2Fentries%2Frussell-paradox%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARussell%27s+paradox" class="Z3988"></span></li> <li><span class="citation mathworld" id="Reference-Mathworld-Russell's_Antinomy"><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/RussellsAntinomy.html">"Russell's Antinomy"</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=Russell%27s+Antinomy&rft.au=Weisstein%2C+Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FRussellsAntinomy.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARussell%27s+paradox" class="Z3988"></span></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.cut-the-knot.org/selfreference/russell.shtml">"Russell's Paradox"</a>. <i><a href="/wiki/Cut-the-Knot" class="mw-redirect" title="Cut-the-Knot">Cut-the-Knot</a></i><span class="reference-accessdate">. Retrieved <span class="nowrap">25 November</span> 2023</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Cut-the-Knot&rft.atitle=Russell%27s+Paradox&rft_id=https%3A%2F%2Fwww.cut-the-knot.org%2Fselfreference%2Frussell.shtml&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARussell%27s+paradox" class="Z3988"></span></li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1236075235">.mw-parser-output .navbox{box-sizing:border-box;border:1px solid #a2a9b1;width:100%;clear:both;font-size:88%;text-align:center;padding:1px;margin:1em auto 0}.mw-parser-output .navbox .navbox{margin-top:0}.mw-parser-output .navbox+.navbox,.mw-parser-output .navbox+.navbox-styles+.navbox{margin-top:-1px}.mw-parser-output .navbox-inner,.mw-parser-output .navbox-subgroup{width:100%}.mw-parser-output .navbox-group,.mw-parser-output .navbox-title,.mw-parser-output .navbox-abovebelow{padding:0.25em 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navbox-subgroup" style="border-spacing:0"><tbody><tr><th id="Views_on_philosophy62" scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Bertrand_Russell%27s_philosophical_views" class="mw-redirect" title="Bertrand Russell's philosophical views">Views on philosophy</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/Copleston%E2%80%93Russell_debate" title="Copleston–Russell debate">Copleston–Russell debate</a></li> <li><a href="/wiki/Logical_atomism" title="Logical atomism">Logical atomism</a></li> <li><a href="/wiki/Russell%27s_teapot" title="Russell's teapot">Russell's teapot</a></li> <li><a href="/wiki/Theory_of_descriptions" title="Theory of descriptions">Theory of descriptions</a></li></ul> </div></td></tr></tbody></table><div> </div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th id="Views_on_society55" scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Bertrand_Russell%27s_political_views" class="mw-redirect" title="Bertrand Russell's political views">Views on society</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/Russell%E2%80%93Einstein_Manifesto" title="Russell–Einstein Manifesto">Russell–Einstein Manifesto</a></li> <li><a href="/wiki/Russell_Tribunal" title="Russell Tribunal">Russell Tribunal</a></li></ul> </div></td></tr></tbody></table><div> </div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th id="Mathematics11" scope="row" class="navbox-group" style="width:1%">Mathematics</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/Peano%E2%80%93Russell_notation" title="Peano–Russell notation">Peano–Russell notation</a></li> <li><a class="mw-selflink selflink">Russell's paradox</a></li></ul> </div></td></tr></tbody></table><div></div></td><td class="noviewer navbox-image" rowspan="4" style="width:1px;padding:0 0 0 2px"><div><span typeof="mw:File"><a href="/wiki/File:Bertrand_Russell_transparent_bg.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a6/Bertrand_Russell_transparent_bg.png/105px-Bertrand_Russell_transparent_bg.png" decoding="async" width="105" height="156" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a6/Bertrand_Russell_transparent_bg.png/158px-Bertrand_Russell_transparent_bg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a6/Bertrand_Russell_transparent_bg.png/210px-Bertrand_Russell_transparent_bg.png 2x" data-file-width="1255" data-file-height="1859" /></a></span></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Works</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><i><a href="/wiki/The_Principles_of_Mathematics" title="The Principles of Mathematics">The Principles of Mathematics</a></i> (1903)</li> <li><i><a href="/wiki/On_Denoting" title="On Denoting">On Denoting</a></i> (1905)</li> <li><i><a href="/wiki/Principia_Mathematica" title="Principia Mathematica">Principia Mathematica</a></i> (1910–1913)</li> <li><i><a href="/wiki/The_Problems_of_Philosophy" title="The Problems of Philosophy">The Problems of Philosophy</a></i> (1912)</li> <li><i><a href="/wiki/Why_Men_Fight_(book)" title="Why Men Fight (book)">Why Men Fight</a></i> (1916)</li> <li><i><a href="/wiki/Introduction_to_Mathematical_Philosophy" title="Introduction to Mathematical Philosophy">Introduction to Mathematical Philosophy</a></i> (1919)</li> <li><i><a href="/wiki/Free_Thought_and_Official_Propaganda" title="Free Thought and Official Propaganda">Free Thought and Official Propaganda</a></i> (1922)</li> <li><i><a href="/wiki/Why_I_Am_Not_a_Christian" title="Why I Am Not a Christian">Why I Am Not a Christian</a></i> (1927)</li> <li><i><a href="/wiki/Marriage_and_Morals" title="Marriage and Morals">Marriage and Morals</a></i> (1929)</li> <li><i><a href="/wiki/In_Praise_of_Idleness_and_Other_Essays" title="In Praise of Idleness and Other Essays">In Praise of Idleness and Other Essays</a></i> (1935)</li> <li><i><a href="/wiki/Power:_A_New_Social_Analysis" title="Power: A New Social Analysis">Power: A New Social Analysis</a></i> (1938)</li> <li><i><a href="/wiki/A_History_of_Western_Philosophy" title="A History of Western Philosophy">A History of Western Philosophy</a></i> (1945)</li> <li><i><a href="/wiki/My_Philosophical_Development" title="My Philosophical Development">My Philosophical Development</a></i> (1959)</li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Family</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/Alys_Pearsall_Smith" title="Alys Pearsall Smith">Alys Pearsall Smith</a> (wife, 1894–1921)</li> <li><a href="/wiki/Dora_Russell" title="Dora Russell">Dora Russell</a> (wife, 1921–35)</li> <li><a href="/wiki/Patricia_Russell" title="Patricia Russell">Patricia Russell</a> (wife, 1936–51)</li> <li><a href="/wiki/Edith_Finch_Russell" title="Edith Finch Russell">Edith Finch Russell</a> (wife, 1952–70)</li> <li><a href="/wiki/John_Russell,_4th_Earl_Russell" title="John Russell, 4th Earl Russell">John Russell, 4th Earl Russell</a> (son)</li> <li><a href="/wiki/Conrad_Russell,_5th_Earl_Russell" title="Conrad Russell, 5th Earl Russell">Conrad Russell, 5th Earl Russell</a> (son)</li> <li><a href="/wiki/Frank_Russell,_2nd_Earl_Russell" title="Frank Russell, 2nd Earl Russell">Frank Russell, 2nd Earl Russell</a> (brother)</li> <li><a href="/wiki/John_Russell,_Viscount_Amberley" title="John Russell, Viscount Amberley">John Russell, Viscount Amberley</a> (father)</li> <li><a href="/wiki/Katharine_Russell,_Viscountess_Amberley" title="Katharine Russell, Viscountess Amberley">Katharine Russell, Viscountess Amberley</a> (mother)</li> <li><a href="/wiki/John_Stuart_Mill" title="John Stuart Mill">John Stuart Mill</a> (godfather)</li> <li><a href="/wiki/John_Russell,_1st_Earl_Russell" title="John Russell, 1st Earl Russell">John Russell, 1st Earl Russell</a> (paternal grandfather)</li> <li><a href="/wiki/Henrietta_Stanley,_Baroness_Stanley_of_Alderley" title="Henrietta Stanley, Baroness Stanley of Alderley">Henrietta Stanley, Baroness Stanley of Alderley</a> (maternal grandmother)</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-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/The_Bertrand_Russell_Case" title="The Bertrand Russell Case">Appointment court case</a></li> <li><a href="/wiki/Earl_Russell" title="Earl Russell">Earl Russell</a></li> <li><a href="/wiki/Bertrand_Russell_Peace_Foundation" title="Bertrand Russell Peace Foundation">Peace Foundation</a></li> <li><a href="/wiki/Bertrand_Russell_Professorship_of_Philosophy" title="Bertrand Russell Professorship of Philosophy">Professorship of Philosophy</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="3"><div><b><span class="noviewer" typeof="mw:File"><span title="Category"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/16px-Symbol_category_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/23px-Symbol_category_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/31px-Symbol_category_class.svg.png 2x" data-file-width="180" data-file-height="185" /></span></span> <a href="/wiki/Category:Works_by_Bertrand_Russell" title="Category:Works by Bertrand Russell">Category: Works by Bertrand Russell</a></b></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="Notable_paradoxes289" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse 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:Paradoxes" title="Template:Paradoxes"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Paradoxes" title="Template talk:Paradoxes"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Paradoxes" title="Special:EditPage/Template:Paradoxes"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Notable_paradoxes289" style="font-size:114%;margin:0 4em">Notable <a href="/wiki/Paradox" title="Paradox">paradoxes</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Philosophical</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/Paradox_of_analysis" title="Paradox of analysis">Analysis</a></li> <li><a href="/wiki/Buridan%27s_bridge" title="Buridan's bridge">Buridan's bridge</a></li> <li><a href="/wiki/Dream_argument" title="Dream argument">Dream argument</a></li> <li><a href="/wiki/Epicurean_paradox" title="Epicurean paradox">Epicurean</a></li> <li><a href="/wiki/Paradox_of_fiction" title="Paradox of fiction">Fiction</a></li> <li><a href="/wiki/Fitch%27s_paradox_of_knowability" title="Fitch's paradox of knowability">Fitch's knowability</a></li> <li><a href="/wiki/Argument_from_free_will" title="Argument from free will">Free will</a></li> <li><a href="/wiki/New_riddle_of_induction" title="New riddle of induction">Goodman's</a></li> <li><a href="/wiki/Paradox_of_hedonism" title="Paradox of hedonism">Hedonism</a></li> <li><a href="/wiki/Liberal_paradox" title="Liberal paradox">Liberal</a></li> <li><a href="/wiki/Meno" title="Meno">Meno's</a></li> <li><a href="/wiki/Mere_addition_paradox" title="Mere addition paradox">Mere addition</a></li> <li><a href="/wiki/Moore%27s_paradox" title="Moore's paradox">Moore's</a></li> <li><a href="/wiki/Newcomb%27s_paradox" title="Newcomb's paradox">Newcomb's</a></li> <li><a href="/wiki/Paradox_of_nihilism" title="Paradox of nihilism">Nihilism</a></li> <li><a href="/wiki/Omnipotence_paradox" title="Omnipotence paradox">Omnipotence</a></li> <li><a href="/wiki/Preface_paradox" title="Preface paradox">Preface</a></li> <li><a href="/wiki/Wittgenstein_on_Rules_and_Private_Language" title="Wittgenstein on Rules and Private Language">Rule-following</a></li> <li><a href="/wiki/Sorites_paradox" title="Sorites paradox">Sorites</a></li> <li><a href="/wiki/Ship_of_Theseus" title="Ship of Theseus">Theseus' ship</a></li> <li><a href="/wiki/White_Horse_Dialogue" title="White Horse Dialogue">White horse</a></li> <li><a href="/wiki/Zeno%27s_paradoxes" title="Zeno's paradoxes">Zeno's</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Logical</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/Barber_paradox" title="Barber paradox">Barber</a></li> <li><a href="/wiki/Berry_paradox" title="Berry paradox">Berry</a></li> <li><a href="/wiki/Bhartrhari%27s_paradox" class="mw-redirect" title="Bhartrhari's paradox">Bhartrhari's</a></li> <li><a href="/wiki/Burali-Forti_paradox" title="Burali-Forti paradox">Burali-Forti</a></li> <li><a href="/wiki/Paradox_of_the_Court" title="Paradox of the Court">Court</a></li> <li><a href="/wiki/Crocodile_dilemma" title="Crocodile dilemma">Crocodile</a></li> <li><a href="/wiki/Curry%27s_paradox" title="Curry's paradox">Curry's</a></li> <li><a href="/wiki/Epimenides_paradox" title="Epimenides paradox">Epimenides</a></li> <li><a href="/wiki/Free_choice_inference" title="Free choice inference">Free choice paradox</a></li> <li><a href="/wiki/Grelling%E2%80%93Nelson_paradox" title="Grelling–Nelson paradox">Grelling–Nelson</a></li> <li><a href="/wiki/Kleene%E2%80%93Rosser_paradox" title="Kleene–Rosser paradox">Kleene–Rosser</a></li> <li><a href="/wiki/Liar_paradox" title="Liar paradox">Liar</a> <ul><li><a href="/wiki/Card_paradox" title="Card paradox">Card</a></li> <li><a href="/wiki/No%E2%80%93no_paradox" title="No–no paradox">No-no</a></li> <li><a href="/wiki/Pinocchio_paradox" title="Pinocchio paradox">Pinocchio</a></li> <li><a href="/wiki/Quine%27s_paradox" title="Quine's paradox">Quine's</a></li> <li><a href="/wiki/Yablo%27s_paradox" class="mw-redirect" title="Yablo's paradox">Yablo's</a></li></ul></li> <li><a href="/wiki/Opposite_Day" title="Opposite Day">Opposite Day</a></li> <li><a href="/wiki/Paradoxes_of_set_theory" title="Paradoxes of set theory">Paradoxes of set theory</a></li> <li><a href="/wiki/Richard%27s_paradox" title="Richard's paradox">Richard's</a></li> <li><a class="mw-selflink selflink">Russell's</a></li> <li><a href="/wiki/I_know_that_I_know_nothing" title="I know that I know nothing">Socratic</a></li> <li><a href="/wiki/Hilbert%27s_paradox_of_the_Grand_Hotel" title="Hilbert's paradox of the Grand Hotel">Hilbert's Hotel</a></li> <li><a href="/wiki/Temperature_paradox" title="Temperature paradox">Temperature paradox</a></li> <li><a href="/wiki/Barbershop_paradox" title="Barbershop paradox">Barbershop</a></li> <li><a href="/wiki/Catch-22_(logic)" title="Catch-22 (logic)">Catch-22</a></li> <li><a href="/wiki/Chicken_or_the_egg" title="Chicken or the egg">Chicken or the egg</a></li> <li><a href="/wiki/Drinker_paradox" title="Drinker paradox">Drinker</a></li> <li><a href="/wiki/Paradoxes_of_material_implication" title="Paradoxes of material implication">Entailment</a></li> <li><a href="/wiki/Lottery_paradox" title="Lottery paradox">Lottery</a></li> <li><a href="/wiki/Plato%27s_beard" title="Plato's beard">Plato's beard</a></li> <li><a href="/wiki/Raven_paradox" title="Raven paradox">Raven</a></li> <li><a href="/wiki/Imperative_logic#Ross's_paradox" title="Imperative logic">Ross's</a></li> <li><a href="/wiki/Unexpected_hanging_paradox" title="Unexpected hanging paradox">Unexpected hanging</a></li> <li>"<a href="/wiki/What_the_Tortoise_Said_to_Achilles" title="What the Tortoise Said to Achilles">What the Tortoise Said to Achilles</a>"</li> <li><a href="/wiki/Heat_death_paradox" title="Heat death paradox">Heat death paradox</a></li> <li><a href="/wiki/Olbers%27s_paradox" title="Olbers's paradox">Olbers's paradox</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Economic</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/Allais_paradox" title="Allais paradox">Allais</a></li> <li><a href="/wiki/The_Antitrust_Paradox" title="The Antitrust Paradox">Antitrust</a></li> <li><a href="/wiki/Arrow_information_paradox" title="Arrow information paradox">Arrow information</a></li> <li><a href="/wiki/Bertrand_paradox_(economics)" title="Bertrand paradox (economics)">Bertrand</a></li> <li><a href="/wiki/Braess%27s_paradox" title="Braess's paradox">Braess's</a></li> <li><a href="/wiki/Paradox_of_competition" title="Paradox of competition">Competition</a></li> <li><a href="/wiki/Income_and_fertility" title="Income and fertility">Income and fertility</a></li> <li><a href="/wiki/Downs%E2%80%93Thomson_paradox" title="Downs–Thomson paradox">Downs–Thomson</a></li> <li><a href="/wiki/Easterlin_paradox" title="Easterlin paradox">Easterlin</a></li> <li><a href="/wiki/Edgeworth_paradox" title="Edgeworth paradox">Edgeworth</a></li> <li><a href="/wiki/Ellsberg_paradox" title="Ellsberg paradox">Ellsberg</a></li> <li><a href="/wiki/European_paradox" title="European paradox">European</a></li> <li><a href="/wiki/Gibson%27s_paradox" title="Gibson's paradox">Gibson's</a></li> <li><a href="/wiki/Giffen_good" title="Giffen good">Giffen good</a></li> <li><a href="/wiki/Icarus_paradox" title="Icarus paradox">Icarus</a></li> <li><a href="/wiki/Jevons_paradox" title="Jevons paradox">Jevons</a></li> <li><a href="/wiki/Leontief_paradox" title="Leontief paradox">Leontief</a></li> <li><a href="/wiki/Lerner_paradox" title="Lerner paradox">Lerner</a></li> <li><a href="/wiki/Lucas_paradox" title="Lucas paradox">Lucas</a></li> <li><a href="/wiki/Mandeville%27s_paradox" title="Mandeville's paradox">Mandeville's</a></li> <li><a href="/wiki/Mayfield%27s_paradox" title="Mayfield's paradox">Mayfield's</a></li> <li><a href="/wiki/Metzler_paradox" title="Metzler paradox">Metzler</a></li> <li><a href="/wiki/Resource_curse" title="Resource curse">Plenty</a></li> <li><a href="/wiki/Productivity_paradox" title="Productivity paradox">Productivity</a></li> <li><a href="/wiki/Paradox_of_prosperity" title="Paradox of prosperity">Prosperity</a></li> <li><a href="/wiki/Scitovsky_paradox" title="Scitovsky paradox">Scitovsky</a></li> <li><a href="/wiki/Service_recovery_paradox" title="Service recovery paradox">Service recovery</a></li> <li><a href="/wiki/St._Petersburg_paradox" title="St. Petersburg paradox">St. Petersburg</a></li> <li><a href="/wiki/Paradox_of_thrift" title="Paradox of thrift">Thrift</a></li> <li><a href="/wiki/Paradox_of_toil" title="Paradox of toil">Toil</a></li> <li><a href="/wiki/Tullock_paradox" class="mw-redirect" title="Tullock paradox">Tullock</a></li> <li><a href="/wiki/Paradox_of_value" title="Paradox of value">Value</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Decision theory</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/Abilene_paradox" title="Abilene paradox">Abilene</a></li> <li><a href="/wiki/Apportionment_paradox" title="Apportionment paradox">Apportionment</a> <ul><li><a href="/wiki/House_monotonicity" title="House monotonicity">Alabama</a></li> <li><a href="/wiki/Coherence_(fairness)" title="Coherence (fairness)">New states</a></li> <li><a href="/wiki/State-population_monotonicity" class="mw-redirect" title="State-population monotonicity">Population</a></li></ul></li> <li><a href="/wiki/Arrow%27s_impossibility_theorem" title="Arrow's impossibility theorem">Arrow's</a></li> <li><a href="/wiki/Buridan%27s_ass" title="Buridan's ass">Buridan's ass</a></li> <li><a href="/wiki/Chainstore_paradox" title="Chainstore paradox">Chainstore</a></li> <li><a href="/wiki/Condorcet_paradox" title="Condorcet paradox">Condorcet's</a></li> <li><a href="/wiki/Decision-making_paradox" title="Decision-making paradox">Decision-making</a></li> <li><a href="/wiki/Paradox_of_voting" title="Paradox of voting">Downs</a></li> <li><a href="/wiki/Ellsberg_paradox" title="Ellsberg paradox">Ellsberg</a></li> <li><a href="/wiki/Fenno%27s_paradox" title="Fenno's paradox">Fenno's</a></li> <li><a href="/wiki/Fredkin%27s_paradox" title="Fredkin's paradox">Fredkin's</a></li> <li><a href="/wiki/The_Green_Paradox" title="The Green Paradox">Green</a></li> <li><a href="/wiki/Hedgehog%27s_dilemma" title="Hedgehog's dilemma">Hedgehog's</a></li> <li><a href="/wiki/Inventor%27s_paradox" title="Inventor's paradox">Inventor's</a></li> <li><a href="/wiki/Kavka%27s_toxin_puzzle" title="Kavka's toxin puzzle">Kavka's toxin puzzle</a></li> <li><a href="/wiki/Morton%27s_fork" title="Morton's fork">Morton's fork</a></li> <li><a href="/wiki/Navigation_paradox" title="Navigation paradox">Navigation</a></li> <li><a href="/wiki/Newcomb%27s_paradox" title="Newcomb's paradox">Newcomb's</a></li> <li><a href="/wiki/Parrondo%27s_paradox" title="Parrondo's paradox">Parrondo's</a></li> <li><a href="/wiki/Preparedness_paradox" title="Preparedness paradox">Preparedness</a></li> <li><a href="/wiki/Prevention_paradox" title="Prevention paradox">Prevention</a></li> <li><a href="/wiki/Prisoner%27s_dilemma" title="Prisoner's dilemma">Prisoner's dilemma</a></li> <li><a href="/wiki/Paradox_of_tolerance" title="Paradox of tolerance">Tolerance</a></li> <li><a href="/wiki/Willpower_paradox" title="Willpower paradox">Willpower</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div> <ul><li><span class="noviewer" typeof="mw:File"><span title="List-Class article"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/d/db/Symbol_list_class.svg/16px-Symbol_list_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/d/db/Symbol_list_class.svg/23px-Symbol_list_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/d/db/Symbol_list_class.svg/31px-Symbol_list_class.svg.png 2x" data-file-width="180" data-file-height="185" /></span></span> <a href="/wiki/List_of_paradoxes" title="List of paradoxes">List</a></li> <li><span class="noviewer" typeof="mw:File"><span title="Category"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/16px-Symbol_category_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/23px-Symbol_category_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/31px-Symbol_category_class.svg.png 2x" data-file-width="180" data-file-height="185" /></span></span> <a href="/wiki/Category:Paradoxes" title="Category:Paradoxes">Category</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" /><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374" /><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374" /></div><div role="navigation" class="navbox" aria-labelledby="Set_theory409" 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="3"><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:Set_theory" title="Template:Set theory"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Set_theory" title="Template talk:Set theory"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Set_theory" title="Special:EditPage/Template:Set theory"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Set_theory409" style="font-size:114%;margin:0 4em"><a href="/wiki/Set_theory" title="Set theory">Set theory</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Overview</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/Set_(mathematics)" title="Set (mathematics)">Set (mathematics)</a></li></ul> </div></td><td class="noviewer navbox-image" rowspan="8" style="width:1px;padding:0 0 0 2px"><div><span typeof="mw:File"><a href="/wiki/Venn_diagram" title="Venn diagram"><img alt="Venn diagram of set intersection" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/6d/Venn_A_intersect_B.svg/100px-Venn_A_intersect_B.svg.png" decoding="async" width="100" height="71" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/6d/Venn_A_intersect_B.svg/150px-Venn_A_intersect_B.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/6d/Venn_A_intersect_B.svg/200px-Venn_A_intersect_B.svg.png 2x" data-file-width="350" data-file-height="250" /></a></span></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Axiom" title="Axiom">Axioms</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/Axiom_of_adjunction" title="Axiom of adjunction">Adjunction</a></li> <li><a href="/wiki/Axiom_of_choice" title="Axiom of choice">Choice</a> <ul><li><a href="/wiki/Axiom_of_countable_choice" title="Axiom of countable choice">countable</a></li> <li><a href="/wiki/Axiom_of_dependent_choice" title="Axiom of dependent choice">dependent</a></li> <li><a href="/wiki/Axiom_of_global_choice" title="Axiom of global choice">global</a></li></ul></li> <li><a href="/wiki/Axiom_of_constructibility" title="Axiom of constructibility">Constructibility (V=L)</a></li> <li><a href="/wiki/Axiom_of_determinacy" title="Axiom of determinacy">Determinacy</a> <ul><li><a href="/wiki/Axiom_of_projective_determinacy" title="Axiom of projective determinacy">projective</a></li></ul></li> <li><a href="/wiki/Axiom_of_extensionality" title="Axiom of extensionality">Extensionality</a></li> <li><a href="/wiki/Axiom_of_infinity" title="Axiom of infinity">Infinity</a></li> <li><a href="/wiki/Axiom_of_limitation_of_size" title="Axiom of limitation of size">Limitation of size</a></li> <li><a href="/wiki/Axiom_of_pairing" title="Axiom of pairing">Pairing</a></li> <li><a href="/wiki/Axiom_of_power_set" title="Axiom of power set">Power set</a></li> <li><a href="/wiki/Axiom_of_regularity" title="Axiom of regularity">Regularity</a></li> <li><a href="/wiki/Axiom_of_union" title="Axiom of union">Union</a></li> <li><a href="/wiki/Martin%27s_axiom" title="Martin's axiom">Martin's axiom</a></li></ul> <ul><li><a href="/wiki/Axiom_schema" title="Axiom schema">Axiom schema</a> <ul><li><a href="/wiki/Axiom_schema_of_replacement" title="Axiom schema of replacement">replacement</a></li> <li><a href="/wiki/Axiom_schema_of_specification" title="Axiom schema of specification">specification</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Set_(mathematics)#Basic_operations" title="Set (mathematics)">Operations</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/Cartesian_product" title="Cartesian product">Cartesian product</a></li> <li><a href="/wiki/Complement_(set_theory)" title="Complement (set theory)">Complement</a> (i.e. set difference)</li> <li><a href="/wiki/De_Morgan%27s_laws" title="De Morgan's laws">De Morgan's laws</a></li> <li><a href="/wiki/Disjoint_union" title="Disjoint union">Disjoint union</a></li> <li><a href="/wiki/List_of_set_identities_and_relations" title="List of set identities and relations">Identities</a></li> <li><a href="/wiki/Intersection_(set_theory)" title="Intersection (set theory)">Intersection</a></li> <li><a href="/wiki/Power_set" title="Power set">Power set</a></li> <li><a href="/wiki/Symmetric_difference" title="Symmetric difference">Symmetric difference</a></li> <li><a href="/wiki/Union_(set_theory)" title="Union (set theory)">Union</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><div class="hlist"><ul><li>Concepts</li><li>Methods</li></ul></div></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/Almost" title="Almost">Almost</a></li> <li><a href="/wiki/Cardinality" title="Cardinality">Cardinality</a></li> <li><a href="/wiki/Cardinal_number" title="Cardinal number">Cardinal number</a> (<a href="/wiki/Large_cardinal" title="Large cardinal">large</a>)</li> <li><a href="/wiki/Class_(set_theory)" title="Class (set theory)">Class</a></li> <li><a href="/wiki/Constructible_universe" title="Constructible universe">Constructible universe</a></li> <li><a href="/wiki/Continuum_hypothesis" title="Continuum hypothesis">Continuum hypothesis</a></li> <li><a href="/wiki/Cantor%27s_diagonal_argument" title="Cantor's diagonal argument">Diagonal argument</a></li> <li><a href="/wiki/Element_(mathematics)" title="Element (mathematics)">Element</a> <ul><li><a href="/wiki/Ordered_pair" title="Ordered pair">ordered pair</a></li> <li><a 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 class="mw-selflink selflink">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_logic326" 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_logic326" 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 href="/wiki/Cantor%27s_diagonal_argument" title="Cantor's diagonal argument">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 class="mw-selflink selflink">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="Traditional95" 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 href="/wiki/Logicism" title="Logicism">Logicism</a></li> <li><a href="/wiki/Mathematical_object" title="Mathematical object">Mathematical object</a></li> <li><a href="/wiki/Philosophy_of_mathematics" title="Philosophy of mathematics">Philosophy of mathematics</a></li> <li><a href="/wiki/Supertask" title="Supertask">Supertask</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div><b><span class="nowrap"><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Nuvola_apps_edu_mathematics_blue-p.svg" class="mw-file-description"><img alt="icon" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/16px-Nuvola_apps_edu_mathematics_blue-p.svg.png" decoding="async" width="16" height="16" class="mw-file-element" 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