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vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#History"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>History</span> </div> </a> <button aria-controls="toc-History-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 History subsection</span> </button> <ul id="toc-History-sublist" class="vector-toc-list"> <li id="toc-Circa_1930–1952" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Circa_1930–1952"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Circa 1930–1952</span> </div> </a> <ul id="toc-Circa_1930–1952-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Later_developments" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Later_developments"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Later developments</span> </div> </a> <ul id="toc-Later_developments-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-The_thesis_as_a_definition" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#The_thesis_as_a_definition"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>The thesis as a definition</span> </div> </a> <ul id="toc-The_thesis_as_a_definition-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Success_of_the_thesis" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Success_of_the_thesis"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Success of the thesis</span> </div> </a> <ul id="toc-Success_of_the_thesis-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Informal_usage_in_proofs" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Informal_usage_in_proofs"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Informal usage in proofs</span> </div> </a> <ul id="toc-Informal_usage_in_proofs-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Variations" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Variations"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Variations</span> </div> </a> <ul id="toc-Variations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Philosophical_implications" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Philosophical_implications"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Philosophical implications</span> </div> </a> <ul id="toc-Philosophical_implications-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Non-computable_functions" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Non-computable_functions"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Non-computable functions</span> </div> </a> <ul id="toc-Non-computable_functions-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">8</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Footnotes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Footnotes"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Footnotes</span> </div> </a> <ul id="toc-Footnotes-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">10</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">11</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 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class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 35 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-35" 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">35 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%A3%D8%B7%D8%B1%D9%88%D8%AD%D8%A9_%D8%AA%D8%B4%D8%B1%D8%B4-%D8%AA%D9%88%D8%B1%D9%8A%D9%86%D8%BA" title="أطروحة تشرش-تورينغ – Arabic" lang="ar" hreflang="ar" data-title="أطروحة تشرش-تورينغ" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%A2%D0%B5%D0%B7%D0%B8%D1%81_%D0%BD%D0%B0_%D0%A7%D1%8A%D1%80%D1%87" title="Тезис на Чърч – Bulgarian" lang="bg" hreflang="bg" data-title="Тезис на Чърч" 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/Tesi_de_Church-Turing" title="Tesi de Church-Turing – Catalan" lang="ca" hreflang="ca" data-title="Tesi de Church-Turing" 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/Churchova%E2%80%93Turingova_teze" title="Churchova–Turingova teze – Czech" lang="cs" hreflang="cs" data-title="Churchova–Turingova teze" 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/Church-Turing-tesen" title="Church-Turing-tesen – Danish" lang="da" hreflang="da" data-title="Church-Turing-tesen" 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/Church-Turing-These" title="Church-Turing-These – German" lang="de" hreflang="de" data-title="Church-Turing-These" 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/Churchi_tees" title="Churchi tees – Estonian" lang="et" hreflang="et" data-title="Churchi tees" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Tesis_de_Church-Turing" title="Tesis de Church-Turing – Spanish" lang="es" hreflang="es" data-title="Tesis de Church-Turing" 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/%C4%88ur%C4%89a_tezo" title="Ĉurĉa tezo – Esperanto" lang="eo" hreflang="eo" data-title="Ĉurĉa tezo" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%AA%D8%B2_%DA%86%D8%B1%DA%86-%D8%AA%D9%88%D8%B1%DB%8C%D9%86%DA%AF" title="تز چرچ-تورینگ – Persian" lang="fa" hreflang="fa" data-title="تز چرچ-تورینگ" 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/Th%C3%A8se_de_Church" title="Thèse de Church – French" lang="fr" hreflang="fr" data-title="Thèse de Church" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%B2%98%EC%B9%98-%ED%8A%9C%EB%A7%81_%EB%85%BC%EC%A0%9C" title="처치-튜링 논제 – Korean" lang="ko" hreflang="ko" data-title="처치-튜링 논제" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Church-Turingova_teza" title="Church-Turingova teza – Croatian" lang="hr" hreflang="hr" data-title="Church-Turingova teza" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Tesis_Church_Turing" title="Tesis Church Turing – Indonesian" lang="id" hreflang="id" data-title="Tesis Church Turing" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Tesi_di_Church-Turing" title="Tesi di Church-Turing – Italian" lang="it" hreflang="it" data-title="Tesi di Church-Turing" 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%AA%D7%96%D7%AA_%D7%A6%27%D7%A8%D7%A5%27-%D7%98%D7%99%D7%95%D7%A8%D7%99%D7%A0%D7%92" title="תזת צ'רץ'-טיורינג – Hebrew" lang="he" hreflang="he" data-title="תזת צ'רץ'-טיורינג" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Church%E2%80%93Turing-t%C3%A9zis" title="Church–Turing-tézis – Hungarian" lang="hu" hreflang="hu" data-title="Church–Turing-tézis" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%9A%E0%B5%BC%E0%B4%9A%E0%B5%8D%E0%B4%9A%E0%B5%8D-%E0%B4%9F%E0%B5%8D%E0%B4%AF%E0%B5%82%E0%B4%B1%E0%B4%BF%E0%B4%99%E0%B5%8D%E0%B4%99%E0%B5%8D_%E0%B4%B8%E0%B4%BF%E0%B4%A6%E0%B5%8D%E0%B4%A7%E0%B4%BE%E0%B4%A8%E0%B5%8D%E0%B4%A4%E0%B4%82" title="ചർച്ച്-ട്യൂറിങ്ങ് സിദ്ധാന്തം – Malayalam" lang="ml" hreflang="ml" data-title="ചർച്ച്-ട്യൂറിങ്ങ് സിദ്ധാന്തം" 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/Church-Turing-hypothese" title="Church-Turing-hypothese – Dutch" lang="nl" hreflang="nl" data-title="Church-Turing-hypothese" 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%81%E3%83%A3%E3%83%BC%E3%83%81%EF%BC%9D%E3%83%81%E3%83%A5%E3%83%BC%E3%83%AA%E3%83%B3%E3%82%B0%E3%81%AE%E3%83%86%E3%83%BC%E3%82%BC" title="チャーチ＝チューリングのテーゼ – Japanese" lang="ja" hreflang="ja" data-title="チャーチ＝チューリングのテーゼ" 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-pms mw-list-item"><a href="https://pms.wikipedia.org/wiki/Tesi_%C3%ABd_Church" title="Tesi ëd Church – Piedmontese" lang="pms" hreflang="pms" data-title="Tesi ëd Church" 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/Hipoteza_Churcha-Turinga" title="Hipoteza Churcha-Turinga – Polish" lang="pl" hreflang="pl" data-title="Hipoteza Churcha-Turinga" 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/Tese_de_Church-Turing" title="Tese de Church-Turing – Portuguese" lang="pt" hreflang="pt" data-title="Tese de Church-Turing" 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/Teza_Church-Turing" title="Teza Church-Turing – Romanian" lang="ro" hreflang="ro" data-title="Teza Church-Turing" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A2%D0%B5%D0%B7%D0%B8%D1%81_%D0%A7%D1%91%D1%80%D1%87%D0%B0_%E2%80%94_%D0%A2%D1%8C%D1%8E%D1%80%D0%B8%D0%BD%D0%B3%D0%B0" title="Тезис Чёрча — Тьюринга – Russian" lang="ru" hreflang="ru" data-title="Тезис Чёрча — Тьюринга" 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/Church%E2%80%93Turing_thesis" title="Church–Turing thesis – Simple English" lang="en-simple" hreflang="en-simple" data-title="Church–Turing thesis" 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-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%A7%D0%B5%D1%80%D1%87-%D0%A2%D1%98%D1%83%D1%80%D0%B8%D0%BD%D0%B3%D0%BE%D0%B2%D0%B0_%D1%82%D0%B5%D0%B7%D0%B0" title="Черч-Тјурингова теза – Serbian" lang="sr" hreflang="sr" data-title="Черч-Тјурингова теза" 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/Church-Turingova_teza" title="Church-Turingova teza – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Church-Turingova teza" 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/Churchin%E2%80%93Turingin_teesi" title="Churchin–Turingin teesi – Finnish" lang="fi" hreflang="fi" data-title="Churchin–Turingin teesi" 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/Church-Turings_hypotes" title="Church-Turings hypotes – Swedish" lang="sv" hreflang="sv" data-title="Church-Turings hypotes" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Tesis_na_Church-Turing" title="Tesis na Church-Turing – Tagalog" lang="tl" hreflang="tl" data-title="Tesis na Church-Turing" data-language-autonym="Tagalog" data-language-local-name="Tagalog" class="interlanguage-link-target"><span>Tagalog</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Church-Turing_tezi" title="Church-Turing tezi – Turkish" lang="tr" hreflang="tr" data-title="Church-Turing tezi" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%A2%D0%B5%D0%B7%D0%B0_%D0%A7%D0%B5%D1%80%D1%87%D0%B0_%E2%80%94_%D0%A2%D1%8E%D1%80%D1%96%D0%BD%D0%B3%D0%B0" title="Теза Черча — Тюрінга – Ukrainian" lang="uk" hreflang="uk" data-title="Теза Черча — Тюрінга" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E5%9C%96%E9%9D%88%E8%AB%96%E9%A1%8C" title="圖靈論題 – Cantonese" lang="yue" hreflang="yue" data-title="圖靈論題" data-language-autonym="粵語" data-language-local-name="Cantonese" class="interlanguage-link-target"><span>粵語</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a 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searchaux" style="display:none">Thesis on the nature of computability</div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">"Church's thesis" redirects here. For the axiom CT in constructive mathematics, see <a href="/wiki/Church%27s_thesis_(constructive_mathematics)" title="Church's thesis (constructive mathematics)">Church's thesis (constructive mathematics)</a>.</div> <p class="mw-empty-elt"> </p><p>In <a href="/wiki/Computability_theory_(computation)" class="mw-redirect" title="Computability theory (computation)">computability theory</a>, the <b>Church–Turing thesis</b> (also known as <b>computability thesis</b>,<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> the <b>Turing–Church thesis</b>,<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> the <b>Church–Turing conjecture</b>, <b>Church's thesis</b>, <b>Church's conjecture</b>, and <b>Turing's thesis</b>) is a <a href="https://en.wiktionary.org/wiki/thesis" class="extiw" title="wiktionary:thesis">thesis</a> about the nature of <a href="/wiki/Computable_function" title="Computable function">computable functions</a>. It states that a <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">function</a> on the <a href="/wiki/Natural_numbers" class="mw-redirect" title="Natural numbers">natural numbers</a> can be calculated by an <a href="/wiki/Effective_method" title="Effective method">effective method</a> if and only if it is computable by a <a href="/wiki/Turing_machine" title="Turing machine">Turing machine</a>. The thesis is named after American mathematician <a href="/wiki/Alonzo_Church" title="Alonzo Church">Alonzo Church</a> and the British mathematician <a href="/wiki/Alan_Turing" title="Alan Turing">Alan Turing</a>. Before the precise definition of computable function, mathematicians often used the informal term <a href="/wiki/Effectively_calculable" class="mw-redirect" title="Effectively calculable">effectively calculable</a> to describe functions that are computable by paper-and-pencil methods. In the 1930s, several independent attempts were made to <a href="/wiki/Formal_system" title="Formal system">formalize</a> the notion of <a href="/wiki/Computability" title="Computability">computability</a>: </p> <ul><li>In 1933, <a href="/wiki/Kurt_G%C3%B6del" title="Kurt Gödel">Kurt Gödel</a>, with <a href="/wiki/Jacques_Herbrand" title="Jacques Herbrand">Jacques Herbrand</a>, formalized the definition of the class of <a href="/wiki/General_recursive_function" title="General recursive function">general recursive functions</a>: the smallest class of functions (with arbitrarily many arguments) that is closed under <a href="/wiki/Function_composition" title="Function composition">composition</a>, <a href="/wiki/Recursion" title="Recursion">recursion</a>, and <a href="/wiki/%CE%9C_operator" title="Μ operator">minimization</a>, and includes <a href="/wiki/Zero_function" class="mw-redirect" title="Zero function">zero</a>, <a href="/wiki/Successor_function" title="Successor function">successor</a>, and all <a href="/wiki/Projection_function" class="mw-redirect" title="Projection function">projections</a>.</li> <li>In 1936, <a href="/wiki/Alonzo_Church" title="Alonzo Church">Alonzo Church</a> created a method for defining functions called the <a href="/wiki/Lambda_calculus" title="Lambda calculus">λ-calculus</a>. Within λ-calculus, he defined an encoding of the natural numbers called the <a href="/wiki/Church_numerals" class="mw-redirect" title="Church numerals">Church numerals</a>. A function on the natural numbers is called <a href="/wiki/Lambda-recursive_function" class="mw-redirect" title="Lambda-recursive function">λ-computable</a> if the corresponding function on the Church numerals can be represented by a term of the λ-calculus.</li> <li>Also in 1936, before learning of Church's work,<sup id="cite_ref-TuringLearn_6-0" class="reference"><a href="#cite_note-TuringLearn-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Alan_Turing" title="Alan Turing">Alan Turing</a> created a theoretical model for machines, now called Turing machines, that could carry out calculations from inputs by manipulating symbols on a tape. Given a suitable encoding of the natural numbers as sequences of symbols, a function on the natural numbers is called <a href="/wiki/Computable_function" title="Computable function">Turing computable</a> if some Turing machine computes the corresponding function on encoded natural numbers.</li></ul> <p>Church,<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> <a href="/wiki/Stephen_Cole_Kleene" title="Stephen Cole Kleene">Kleene</a>,<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> and Turing<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup><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> proved that these three formally defined classes of computable functions coincide: a function is λ-computable if and only if it is Turing computable, and if and only if it is <i>general recursive</i>. This has led mathematicians and computer scientists to believe that the concept of computability is accurately characterized by these three equivalent processes. Other formal attempts to characterize computability have subsequently strengthened this belief (see <a href="#Success_of_the_thesis">below</a>). </p><p>On the other hand, the Church–Turing thesis states that the above three formally-defined classes of computable functions coincide with the <i>informal</i> notion of an effectively calculable function. Although the thesis has near-universal acceptance, it cannot be formally proven, as the concept of effective calculability is only informally defined. </p><p>Since its inception, variations on the original thesis have arisen, including statements about what can physically be realized by a computer in our universe (<a href="/wiki/Physical_Church-Turing_thesis" class="mw-redirect" title="Physical Church-Turing thesis">physical Church-Turing thesis</a>) and what can be efficiently computed (<a href="#complexity-theoretic_Church–Turing_thesis">Church–Turing thesis (complexity theory)</a>). These variations are not due to Church or Turing, but arise from later work in <a href="/wiki/Computational_complexity_theory" title="Computational complexity theory">complexity theory</a> and <a href="/wiki/Digital_physics" title="Digital physics">digital physics</a>. The thesis also has implications for the <a href="/wiki/Philosophy_of_mind" title="Philosophy of mind">philosophy of mind</a> (see <a href="#Philosophical_implications">below</a>). </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Statement_in_Church's_and_Turing's_words"><span id="Statement_in_Church.27s_and_Turing.27s_words"></span>Statement in Church's and Turing's words</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Church%E2%80%93Turing_thesis&action=edit&section=1" title="Edit section: Statement in Church's and Turing's words"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Effective_method" title="Effective method">Effective method</a></div> <p><a href="/wiki/J._Barkley_Rosser" title="J. Barkley Rosser">J. B. Rosser</a> (<a href="#CITEREFRosser1939">1939</a>) addresses the notion of "effective computability" as follows: "Clearly the existence of CC and RC (Church's and Rosser's proofs) presupposes a precise definition of 'effective'. 'Effective method' is here used in the rather special sense of a method each step of which is precisely predetermined and which is certain to produce the answer in a finite number of steps".<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> Thus the adverb-adjective "effective" is used in a sense of "1a: producing a decided, decisive, or desired effect", and "capable of producing a result".<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><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> </p><p>In the following, the words "effectively calculable" will mean "produced by any intuitively 'effective' means whatsoever" and "effectively computable" will mean "produced by a Turing-machine or equivalent mechanical device". Turing's "definitions" given in a footnote in his 1938 Ph.D. thesis <i><a href="/wiki/Systems_of_Logic_Based_on_Ordinals" title="Systems of Logic Based on Ordinals">Systems of Logic Based on Ordinals</a></i>, supervised by Church, are virtually the same: </p> <blockquote><p><sup>†</sup> We shall use the expression "computable function" to mean a function calculable by a machine, and let "effectively calculable" refer to the intuitive idea without particular identification with any one of these definitions.<sup id="cite_ref-Turing_1938_thesis_p8_15-0" class="reference"><a href="#cite_note-Turing_1938_thesis_p8-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup></p></blockquote> <p>The thesis can be stated as: <i>Every effectively calculable function is a computable function</i>.<sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> Church also stated that "No computational procedure will be considered as an algorithm unless it can be represented as a Turing Machine".<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (November 2019)">citation needed</span></a></i>]</sup> Turing stated it this way: </p> <blockquote><p>It was stated ... that "a function is effectively calculable if its values can be found by some purely mechanical process". We may take this literally, understanding that by a purely mechanical process one which could be carried out by a machine. The development ... leads to ... an identification of computability<sup>†</sup> with effective calculability. [<sup>†</sup> is the footnote quoted above.]<sup id="cite_ref-Turing_1938_thesis_p8_15-1" class="reference"><a href="#cite_note-Turing_1938_thesis_p8-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup></p></blockquote> <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=Church%E2%80%93Turing_thesis&action=edit&section=2" title="Edit section: History"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/History_of_the_Church%E2%80%93Turing_thesis" title="History of the Church–Turing thesis">History of the Church–Turing thesis</a></div> <p>One of the important problems for logicians in the 1930s was the <a href="/wiki/Entscheidungsproblem" title="Entscheidungsproblem">Entscheidungsproblem</a> of <a href="/wiki/David_Hilbert" title="David Hilbert">David Hilbert</a> and <a href="/wiki/Wilhelm_Ackermann" title="Wilhelm Ackermann">Wilhelm Ackermann</a>,<sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> which asked whether there was a mechanical procedure for separating mathematical truths from mathematical falsehoods. This quest required that the notion of "algorithm" or "effective calculability" be pinned down, at least well enough for the quest to begin.<sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> But from the very outset <a href="/wiki/Alonzo_Church" title="Alonzo Church">Alonzo Church</a>'s attempts began with a debate that continues to this day.<sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> <span class="clarify-content" style="padding-left:0.1em; padding-right:0.1em; color:var(--color-subtle, #54595d); border:1px solid var(--border-color-subtle, #c8ccd1);">Was</span><sup class="noprint Inline-Template Template-Clarify" style="margin-left:0.1em; white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Please_clarify" title="Wikipedia:Please clarify"><span title="It should be made clear who asked the question (Peter Smith, mentioned in the previous footnote?). (revised March 2024) (March 2019)">clarify</span></a></i>]</sup> the notion of "effective calculability" to be (i) an "axiom or axioms" in an axiomatic system, (ii) merely a <i>definition</i> that "identified" two or more propositions, (iii) an <i>empirical hypothesis</i> to be verified by observation of natural events, or (iv) just <i>a proposal</i> for the sake of argument (i.e. a "thesis")? </p> <div class="mw-heading mw-heading3"><h3 id="Circa_1930–1952"><span id="Circa_1930.E2.80.931952"></span>Circa 1930–1952</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Church%E2%80%93Turing_thesis&action=edit&section=3" title="Edit section: Circa 1930–1952"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In the course of studying the problem, Church and his student <a href="/wiki/Stephen_Cole_Kleene" title="Stephen Cole Kleene">Stephen Kleene</a> introduced the notion of <a href="/wiki/Lambda_calculus" title="Lambda calculus">λ-definable functions</a>, and they were able to prove that several large classes of functions frequently encountered in number theory were λ-definable.<sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> The debate began when Church proposed to Gödel that one should define the "effectively computable" functions as the λ-definable functions. Gödel, however, was not convinced and called the proposal "thoroughly unsatisfactory".<sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> Rather, in correspondence with Church (c. 1934–1935), Gödel proposed <i>axiomatizing</i> the notion of "effective calculability"; indeed, in a 1935 letter to Kleene, Church reported that: </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>His [Gödel's] only idea at the time was that it might be possible, in terms of effective calculability as an undefined notion, to state a set of axioms which would embody the generally accepted properties of this notion, and to do something on that basis.<sup id="cite_ref-sieg160_22-0" class="reference"><a href="#cite_note-sieg160-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup></p></blockquote> <p>But Gödel offered no further guidance. Eventually, he would suggest his recursion, modified by Herbrand's suggestion, that Gödel had detailed in his 1934 lectures in Princeton NJ (Kleene and <a href="/wiki/J._B._Rosser" class="mw-redirect" title="J. B. Rosser">Rosser</a> transcribed the notes). But he did not think that the two ideas could be satisfactorily identified "except heuristically".<sup id="cite_ref-23" class="reference"><a href="#cite_note-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup> </p><p>Next, it was necessary to identify and prove the equivalence of two notions of effective calculability. Equipped with the λ-calculus and "general" recursion, Kleene with help of Church and <a href="/wiki/J._Barkley_Rosser" title="J. Barkley Rosser">J. Barkley Rosser</a> produced proofs (1933, 1935) to show that the two calculi are equivalent. Church subsequently modified his methods to include use of Herbrand–Gödel recursion and then proved (1936) that the <a href="/wiki/Entscheidungsproblem" title="Entscheidungsproblem">Entscheidungsproblem</a> is unsolvable: there is no algorithm that can determine whether a <a href="/wiki/Well_formed_formula" class="mw-redirect" title="Well formed formula">well formed formula</a> has a <a href="/wiki/Beta_normal_form" title="Beta normal form">beta normal form</a>.<sup id="cite_ref-24" class="reference"><a href="#cite_note-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup> </p><p>Many years later in a letter to Davis (c. 1965), Gödel said that "he was, at the time of these [1934] lectures, not at all convinced that his concept of recursion comprised all possible recursions".<sup id="cite_ref-25" class="reference"><a href="#cite_note-25"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup> By 1963–1964 Gödel would disavow Herbrand–Gödel recursion and the λ-calculus in favor of the Turing machine as the definition of "algorithm" or "mechanical procedure" or "formal system".<sup id="cite_ref-26" class="reference"><a href="#cite_note-26"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup> </p><p><b>A hypothesis leading to a natural law?</b>: In late 1936 <a href="/wiki/Alan_Turing" title="Alan Turing">Alan Turing</a>'s paper (also proving that the <a href="/wiki/Entscheidungsproblem" title="Entscheidungsproblem">Entscheidungsproblem</a> is unsolvable) was delivered orally, but had not yet appeared in print.<sup id="cite_ref-On_Computable_27-0" class="reference"><a href="#cite_note-On_Computable-27"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup> On the other hand, <a href="/wiki/Emil_Post" class="mw-redirect" title="Emil Post">Emil Post</a>'s 1936 paper had appeared and was certified independent of Turing's work.<sup id="cite_ref-28" class="reference"><a href="#cite_note-28"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup> Post strongly disagreed with Church's "identification" of effective computability with the λ-calculus and recursion, stating: </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1244412712"><blockquote class="templatequote"><p>Actually the work already done by Church and others carries this identification considerably beyond the working hypothesis stage. But to mask this identification under a definition… blinds us to the need of its continual verification.<sup id="cite_ref-29" class="reference"><a href="#cite_note-29"><span class="cite-bracket">[</span>29<span class="cite-bracket">]</span></a></sup></p></blockquote> <p>Rather, he regarded the notion of "effective calculability" as merely a "working hypothesis" that might lead by <a href="/wiki/Inductive_reasoning" title="Inductive reasoning">inductive reasoning</a> to a "<a href="/wiki/Natural_law" title="Natural law">natural law</a>" rather than by "a definition or an axiom".<sup id="cite_ref-30" class="reference"><a href="#cite_note-30"><span class="cite-bracket">[</span>30<span class="cite-bracket">]</span></a></sup> This idea was "sharply" criticized by Church.<sup id="cite_ref-31" class="reference"><a href="#cite_note-31"><span class="cite-bracket">[</span>31<span class="cite-bracket">]</span></a></sup> </p><p>Thus Post in his 1936 paper was also discounting <a href="/wiki/Kurt_G%C3%B6del" title="Kurt Gödel">Kurt Gödel</a>'s suggestion to Church in 1934–1935 that the thesis might be expressed as an axiom or set of axioms.<sup id="cite_ref-sieg160_22-1" class="reference"><a href="#cite_note-sieg160-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> </p><p><b>Turing adds another definition, Rosser equates all three</b>: Within just a short time, Turing's 1936–1937 paper "On Computable Numbers, with an Application to the Entscheidungsproblem"<sup id="cite_ref-On_Computable_27-1" class="reference"><a href="#cite_note-On_Computable-27"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup> appeared. In it he stated another notion of "effective computability" with the introduction of his a-machines (now known as the <a href="/wiki/Turing_machine" title="Turing machine">Turing machine</a> abstract computational model). And in a proof-sketch added as an "Appendix" to his 1936–1937 paper, Turing showed that the classes of functions defined by λ-calculus and Turing machines coincided.<sup id="cite_ref-32" class="reference"><a href="#cite_note-32"><span class="cite-bracket">[</span>32<span class="cite-bracket">]</span></a></sup> Church was quick to recognise how compelling Turing's analysis was. In his review of Turing's paper he made clear that Turing's notion made "the identification with effectiveness in the ordinary (not explicitly defined) sense evident immediately".<sup id="cite_ref-33" class="reference"><a href="#cite_note-33"><span class="cite-bracket">[</span>33<span class="cite-bracket">]</span></a></sup> </p><p>In a few years (1939) Turing would propose, like Church and Kleene before him, that <i>his</i> formal definition of mechanical computing agent was the correct one.<sup id="cite_ref-34" class="reference"><a href="#cite_note-34"><span class="cite-bracket">[</span>34<span class="cite-bracket">]</span></a></sup> Thus, by 1939, both Church (1934) and Turing (1939) had individually proposed that their "formal systems" should be <i>definitions</i> of "effective calculability";<sup id="cite_ref-35" class="reference"><a href="#cite_note-35"><span class="cite-bracket">[</span>35<span class="cite-bracket">]</span></a></sup> neither framed their statements as <i>theses</i>. </p><p>Rosser (1939) formally identified the three notions-as-definitions: </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1244412712"><blockquote class="templatequote"><p>All three <i>definitions</i> are equivalent, so it does not matter which one is used.<sup id="cite_ref-36" class="reference"><a href="#cite_note-36"><span class="cite-bracket">[</span>36<span class="cite-bracket">]</span></a></sup></p></blockquote> <p><b>Kleene proposes <i>Thesis I</i></b>: This left the overt expression of a "thesis" to Kleene. In 1943 Kleene proposed his "Thesis I":<sup id="cite_ref-Davis274_37-0" class="reference"><a href="#cite_note-Davis274-37"><span class="cite-bracket">[</span>37<span class="cite-bracket">]</span></a></sup> </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1244412712"><blockquote class="templatequote"><p>This heuristic fact [general recursive functions are effectively calculable] ... led Church to state the following thesis. The same thesis is implicit in Turing's description of computing machines. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1244412712"></p><blockquote class="templatequote" style="font-size:inherit;quotes:none;"><p>Thesis I. <i>Every effectively calculable function (effectively decidable predicate) is general recursive</i> [Kleene's italics]</p></blockquote><p> Since a precise mathematical definition of the term effectively calculable (effectively decidable) has been wanting, we can take this thesis ... as a definition of it ...</p></blockquote> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1244412712"><blockquote class="templatequote"><p>...the thesis has the character of an hypothesis—a point emphasized by Post and by Church. If we consider the thesis and its converse as definition, then the hypothesis is an hypothesis about the application of the mathematical theory developed from the definition. For the acceptance of the hypothesis, there are, as we have suggested, quite compelling grounds.</p></blockquote> <p><b>The Church–Turing Thesis</b>: Stephen Kleene, in <i>Introduction To Metamathematics</i>, finally goes on to formally name "Church's Thesis" and "Turing's Thesis", using his theory of recursive realizability. Kleene having switched from presenting his work in the terminology of Church-Kleene lambda definability, to that of Gödel-Kleene recursiveness (partial recursive functions). In this transition, Kleene modified Gödel's general recursive functions to allow for proofs of the unsolvability of problems in the Intuitionism of E. J. Brouwer. In his graduate textbook on logic, "Church's thesis" is introduced and basic mathematical results are demonstrated to be unrealizable. Next, Kleene proceeds to present "Turing's thesis", where results are shown to be uncomputable, using his simplified derivation of a Turing machine based on the work of Emil Post. Both theses are proven equivalent by use of "Theorem XXX". </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1244412712"><blockquote class="templatequote" style="font-size:inherit;quotes:none"><p>Thesis I. <i>Every effectively calculable function (effectively decidable predicate) is general recursive</i>.<sup id="cite_ref-38" class="reference"><a href="#cite_note-38"><span class="cite-bracket">[</span>38<span class="cite-bracket">]</span></a></sup></p></blockquote> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1244412712"><blockquote class="templatequote" style="font-size:inherit;quotes:none"><p>Theorem XXX: The following classes of partial functions are coextensive, i.e. have the same members: (a) the partial recursive functions, (b) the computable functions ...<sup id="cite_ref-Kleene_1952_p376_39-0" class="reference"><a href="#cite_note-Kleene_1952_p376-39"><span class="cite-bracket">[</span>39<span class="cite-bracket">]</span></a></sup></p></blockquote> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1244412712"><blockquote class="templatequote" style="font-size:inherit;quotes:none"><p>Turing's thesis: Turing's thesis that every function which would naturally be regarded as computable is computable under his definition, i.e. by one of his machines, is equivalent to Church's thesis by Theorem XXX.<sup id="cite_ref-Kleene_1952_p376_39-1" class="reference"><a href="#cite_note-Kleene_1952_p376-39"><span class="cite-bracket">[</span>39<span class="cite-bracket">]</span></a></sup></p></blockquote> <p>Kleene, finally, uses for the first time the term the "Church-Turing thesis" in a section in which he helps to give clarifications to concepts in Alan Turing's paper "The Word Problem in Semi-Groups with Cancellation", as demanded in a critique from William Boone.<sup id="cite_ref-40" class="reference"><a href="#cite_note-40"><span class="cite-bracket">[</span>40<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Later_developments">Later developments</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Church%E2%80%93Turing_thesis&action=edit&section=4" title="Edit section: Later developments"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>An attempt to better understand the notion of "effective computability" led <a href="/wiki/Robin_Gandy" title="Robin Gandy">Robin Gandy</a> (Turing's student and friend) in 1980 to analyze <i>machine</i> computation (as opposed to human-computation acted out by a Turing machine). Gandy's curiosity about, and analysis of, <a href="/wiki/Cellular_automata" class="mw-redirect" title="Cellular automata">cellular automata</a> (including <a href="/wiki/Conway%27s_game_of_life" class="mw-redirect" title="Conway's game of life">Conway's game of life</a>), parallelism, and crystalline automata, led him to propose four "principles (or constraints) ... which it is argued, any machine must satisfy".<sup id="cite_ref-41" class="reference"><a href="#cite_note-41"><span class="cite-bracket">[</span>41<span class="cite-bracket">]</span></a></sup> His most-important fourth, "the principle of causality" is based on the "finite velocity of propagation of effects and signals; contemporary physics rejects the possibility of instantaneous action at a distance".<sup id="cite_ref-42" class="reference"><a href="#cite_note-42"><span class="cite-bracket">[</span>42<span class="cite-bracket">]</span></a></sup> From these principles and some additional constraints—(1a) a lower bound on the linear dimensions of any of the parts, (1b) an upper bound on speed of propagation (the velocity of light), (2) discrete progress of the machine, and (3) deterministic behavior—he produces a theorem that "What can be calculated by a device satisfying principles I–IV is computable."<sup id="cite_ref-43" class="reference"><a href="#cite_note-43"><span class="cite-bracket">[</span>43<span class="cite-bracket">]</span></a></sup> </p><p>In the late 1990s <a href="/w/index.php?title=Wilfried_Sieg&action=edit&redlink=1" class="new" title="Wilfried Sieg (page does not exist)">Wilfried Sieg</a> analyzed Turing's and Gandy's notions of "effective calculability" with the intent of "sharpening the informal notion, formulating its general features axiomatically, and investigating the axiomatic framework".<sup id="cite_ref-44" class="reference"><a href="#cite_note-44"><span class="cite-bracket">[</span>44<span class="cite-bracket">]</span></a></sup> In his 1997 and 2002 work Sieg presents a series of constraints on the behavior of a <i>computor</i>—"a human computing agent who proceeds mechanically". These constraints reduce to: </p> <ul><li>"(B.1) (Boundedness) There is a fixed bound on the number of symbolic configurations a computor can immediately recognize.</li> <li>"(B.2) (Boundedness) There is a fixed bound on the number of internal states a computor can be in.</li> <li>"(L.1) (Locality) A computor can change only elements of an observed symbolic configuration.</li> <li>"(L.2) (Locality) A computor can shift attention from one symbolic configuration to another one, but the new observed configurations must be within a bounded distance of the immediately previously observed configuration.</li> <li>"(D) (Determinacy) The immediately recognizable (sub-)configuration determines uniquely the next computation step (and id [instantaneous description])"; stated another way: "A computor's internal state together with the observed configuration fixes uniquely the next computation step and the next internal state."<sup id="cite_ref-45" class="reference"><a href="#cite_note-45"><span class="cite-bracket">[</span>45<span class="cite-bracket">]</span></a></sup></li></ul> <p>The matter remains in active discussion within the academic community.<sup id="cite_ref-46" class="reference"><a href="#cite_note-46"><span class="cite-bracket">[</span>46<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-47" class="reference"><a href="#cite_note-47"><span class="cite-bracket">[</span>47<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="The_thesis_as_a_definition">The thesis as a definition</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Church%E2%80%93Turing_thesis&action=edit&section=5" title="Edit section: The thesis as a definition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The thesis can be viewed as nothing but an ordinary mathematical definition. Comments by Gödel on the subject suggest this view, e.g. "the correct definition of mechanical computability was established beyond any doubt by Turing".<sup id="cite_ref-48" class="reference"><a href="#cite_note-48"><span class="cite-bracket">[</span>48<span class="cite-bracket">]</span></a></sup> The case for viewing the thesis as nothing more than a definition is made explicitly by <a href="/wiki/Robert_I._Soare" title="Robert I. Soare">Robert I. Soare</a>,<sup id="cite_ref-Soare_5-1" class="reference"><a href="#cite_note-Soare-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> where it is also argued that Turing's definition of computability is no less likely to be correct than the epsilon-delta definition of a <a href="/wiki/Continuous_function" title="Continuous function">continuous function</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Success_of_the_thesis">Success of the thesis</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Church%E2%80%93Turing_thesis&action=edit&section=6" title="Edit section: Success of the thesis"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Other formalisms (besides recursion, the <a href="/wiki/Lambda_calculus" title="Lambda calculus">λ-calculus</a>, and the <a href="/wiki/Turing_machine" title="Turing machine">Turing machine</a>) have been proposed for describing effective calculability/computability. Kleene (1952) adds to the list the functions "<i>reckonable</i> in the system S<sub>1</sub>" of <a href="/wiki/Kurt_G%C3%B6del" title="Kurt Gödel">Kurt Gödel</a> 1936, and <a href="/wiki/Emil_Post" class="mw-redirect" title="Emil Post">Emil Post</a>'s (1943, 1946) "<i>canonical</i> [also called <i>normal</i>] <i>systems</i>".<sup id="cite_ref-49" class="reference"><a href="#cite_note-49"><span class="cite-bracket">[</span>49<span class="cite-bracket">]</span></a></sup> In the 1950s <a href="/wiki/Hao_Wang_(academic)" title="Hao Wang (academic)">Hao Wang</a> and <a href="/wiki/Martin_Davis_(mathematician)" title="Martin Davis (mathematician)">Martin Davis</a> greatly simplified the one-tape Turing-machine model (see <a href="/wiki/Post%E2%80%93Turing_machine" title="Post–Turing machine">Post–Turing machine</a>). <a href="/wiki/Marvin_Minsky" title="Marvin Minsky">Marvin Minsky</a> expanded the model to two or more tapes and greatly simplified the tapes into "up-down counters", which Melzak and <a href="/wiki/Joachim_Lambek" title="Joachim Lambek">Lambek</a> further evolved into what is now known as the <a href="/wiki/Counter_machine" title="Counter machine">counter machine</a> model. In the late 1960s and early 1970s researchers expanded the counter machine model into the <a href="/wiki/Register_machine" title="Register machine">register machine</a>, a close cousin to the modern notion of the <a href="/wiki/Computer" title="Computer">computer</a>. Other models include <a href="/wiki/Combinatory_logic" title="Combinatory logic">combinatory logic</a> and <a href="/wiki/Markov_algorithm" title="Markov algorithm">Markov algorithms</a>. Gurevich adds the <a href="/wiki/Pointer_machine" title="Pointer machine">pointer machine</a> model of Kolmogorov and Uspensky (1953, 1958): "... they just wanted to ... convince themselves that there is no way to extend the notion of computable function."<sup id="cite_ref-50" class="reference"><a href="#cite_note-50"><span class="cite-bracket">[</span>50<span class="cite-bracket">]</span></a></sup> </p><p>All these contributions involve proofs that the models are computationally equivalent to the Turing machine; such models are said to be <a href="/wiki/Turing_complete" class="mw-redirect" title="Turing complete">Turing complete</a>. Because all these different attempts at formalizing the concept of "effective calculability/computability" have yielded equivalent results, it is now generally assumed that the Church–Turing thesis is correct. In fact, Gödel (1936) proposed something stronger than this; he observed that there was something "absolute" about the concept of "reckonable in S<sub>1</sub>": </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1244412712"><blockquote class="templatequote"><p>It may also be shown that a function which is computable ['reckonable'] in one of the systems S<sub>i</sub>, or even in a system of transfinite type, is already computable [reckonable] in S<sub>1</sub>. Thus the concept 'computable' ['reckonable'] is in a certain definite sense 'absolute', while practically all other familiar metamathematical concepts (e.g. provable, definable, etc.) depend quite essentially on the system to which they are defined ...<sup id="cite_ref-51" class="reference"><a href="#cite_note-51"><span class="cite-bracket">[</span>51<span class="cite-bracket">]</span></a></sup></p></blockquote> <div class="mw-heading mw-heading2"><h2 id="Informal_usage_in_proofs">Informal usage in proofs</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Church%E2%80%93Turing_thesis&action=edit&section=7" title="Edit section: Informal usage in proofs"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Proofs in computability theory often invoke the Church–Turing thesis in an informal way to establish the computability of functions while avoiding the (often very long) details which would be involved in a rigorous, formal proof.<sup id="cite_ref-52" class="reference"><a href="#cite_note-52"><span class="cite-bracket">[</span>52<span class="cite-bracket">]</span></a></sup> To establish that a function is computable by Turing machine, it is usually considered sufficient to give an informal English description of how the function can be effectively computed, and then conclude "by the Church–Turing thesis" that the function is Turing computable (equivalently, partial recursive). </p><p>Dirk van Dalen gives the following example for the sake of illustrating this informal use of the Church–Turing thesis:<sup id="cite_ref-53" class="reference"><a href="#cite_note-53"><span class="cite-bracket">[</span>53<span class="cite-bracket">]</span></a></sup> </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1244412712"><blockquote class="templatequote"><p>Example: Each infinite <a href="/wiki/Recursively_enumerable" class="mw-redirect" title="Recursively enumerable">recursively enumerable</a> (RE) set contains an infinite <a href="/wiki/Recursive_set" class="mw-redirect" title="Recursive set">recursive set</a>. </p><p>Proof: Let A be infinite RE. We list the elements of A effectively, n<sub>0</sub>, n<sub>1</sub>, n<sub>2</sub>, n<sub>3</sub>, ... </p><p>From this list we extract an increasing sublist: put m<sub>0</sub> = n<sub>0</sub>, after finitely many steps we find an n<sub>k</sub> such that n<sub>k</sub> > m<sub>0</sub>, put m<sub>1</sub> = n<sub>k</sub>. We repeat this procedure to find m<sub>2</sub> > m<sub>1</sub>, etc. this yields an effective listing of the subset B={m<sub>0</sub>, m<sub>1</sub>, m<sub>2</sub>,...} of A, with the property m<sub>i</sub> < m<sub>i+1</sub>. </p> <p><i>Claim</i>. B is decidable. For, in order to test k in B we must check if k = m<sub>i</sub> for some i. Since the sequence of m<sub>i</sub>'s is increasing we have to produce at most k+1 elements of the list and compare them with k. If none of them is equal to k, then k not in B. Since this test is effective, B is decidable and, <b>by Church's thesis</b>, recursive.</p></blockquote> <p>In order to make the above example completely rigorous, one would have to carefully construct a Turing machine, or λ-function, or carefully invoke recursion axioms, or at best, cleverly invoke various theorems of computability theory. But because the computability theorist believes that Turing computability correctly captures what can be computed effectively, and because an effective procedure is spelled out in English for deciding the set B, the computability theorist accepts this as proof that the set is indeed recursive. </p> <div class="mw-heading mw-heading2"><h2 id="Variations">Variations</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Church%E2%80%93Turing_thesis&action=edit&section=8" title="Edit section: Variations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The success of the Church–Turing thesis prompted variations of the thesis to be proposed. For example, the <b>physical Church–Turing thesis</b> states: "All physically computable functions are Turing-computable."<sup id="cite_ref-54" class="reference"><a href="#cite_note-54"><span class="cite-bracket">[</span>54<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page: 101">: 101 </span></sup> </p><p><span class="anchor" id="complexity-theoretic_Church–Turing_thesis"></span>The Church–Turing thesis says nothing about the efficiency with which one model of computation can simulate another. It has been proved for instance that a (multi-tape) <a href="/wiki/Universal_Turing_machine" title="Universal Turing machine">universal Turing machine</a> only suffers a logarithmic slowdown factor in simulating any Turing machine.<sup id="cite_ref-55" class="reference"><a href="#cite_note-55"><span class="cite-bracket">[</span>55<span class="cite-bracket">]</span></a></sup> </p><p>A variation of the Church–Turing thesis addresses whether an arbitrary but "reasonable" model of computation can be efficiently simulated. This is called the <b>feasibility thesis</b>,<sup id="cite_ref-56" class="reference"><a href="#cite_note-56"><span class="cite-bracket">[</span>56<span class="cite-bracket">]</span></a></sup> also known as the (<b>classical</b>) <b>complexity-theoretic Church–Turing thesis</b> or the <b>extended Church–Turing thesis</b>, which is not due to Church or Turing, but rather was realized gradually in the development of <a href="/wiki/Computational_complexity_theory" title="Computational complexity theory">complexity theory</a>. It states:<sup id="cite_ref-kaye_57-0" class="reference"><a href="#cite_note-kaye-57"><span class="cite-bracket">[</span>57<span class="cite-bracket">]</span></a></sup> "A <a href="/wiki/Probabilistic_Turing_machine" title="Probabilistic Turing machine">probabilistic Turing machine</a> can efficiently simulate any realistic model of computation." The word 'efficiently' here means up to <a href="/wiki/Polynomial-time_reduction" title="Polynomial-time reduction">polynomial-time reductions</a>. This thesis was originally called <b>computational complexity-theoretic Church–Turing thesis</b> by Ethan Bernstein and <a href="/wiki/Umesh_Vazirani" title="Umesh Vazirani">Umesh Vazirani</a> (1997). The complexity-theoretic Church–Turing thesis, then, posits that all 'reasonable' models of computation yield the same class of problems that can be computed in polynomial time. Assuming the conjecture that probabilistic polynomial time (<a href="/wiki/Bounded-error_probabilistic_polynomial" class="mw-redirect" title="Bounded-error probabilistic polynomial">BPP</a>) equals deterministic polynomial time (<a href="/wiki/P_(complexity)" title="P (complexity)">P</a>), the word 'probabilistic' is optional in the complexity-theoretic Church–Turing thesis. A similar thesis, called the <b>invariance thesis</b>, was introduced by Cees F. Slot and Peter van Emde Boas. It states: <span style="padding-right:.15em;">"</span>'Reasonable' machines can simulate each other within a polynomially bounded overhead in time and a constant-factor overhead in space."<sup id="cite_ref-58" class="reference"><a href="#cite_note-58"><span class="cite-bracket">[</span>58<span class="cite-bracket">]</span></a></sup> The thesis originally appeared in a paper at <a href="/wiki/Symposium_on_Theory_of_Computing" title="Symposium on Theory of Computing">STOC</a>'84, which was the first paper to show that polynomial-time overhead and constant-space overhead could be <i>simultaneously</i> achieved for a simulation of a <a href="/wiki/Random_Access_Machine" class="mw-redirect" title="Random Access Machine">Random Access Machine</a> on a Turing machine.<sup id="cite_ref-59" class="reference"><a href="#cite_note-59"><span class="cite-bracket">[</span>59<span class="cite-bracket">]</span></a></sup> </p><p>If <a href="/wiki/BQP" title="BQP">BQP</a> is shown to be a strict superset of <a href="/wiki/Bounded-error_probabilistic_polynomial" class="mw-redirect" title="Bounded-error probabilistic polynomial">BPP</a>, it would invalidate the complexity-theoretic Church–Turing thesis. In other words, there would be efficient <a href="/wiki/Quantum_algorithms" class="mw-redirect" title="Quantum algorithms">quantum algorithms</a> that perform tasks that do not have efficient <a href="/wiki/Probabilistic_algorithms" class="mw-redirect" title="Probabilistic algorithms">probabilistic algorithms</a>. This would not however invalidate the original Church–Turing thesis, since a quantum computer can always be simulated by a Turing machine, but it would invalidate the classical complexity-theoretic Church–Turing thesis for efficiency reasons. Consequently, the <b>quantum complexity-theoretic Church–Turing thesis</b> states:<sup id="cite_ref-kaye_57-1" class="reference"><a href="#cite_note-kaye-57"><span class="cite-bracket">[</span>57<span class="cite-bracket">]</span></a></sup> "A <a href="/wiki/Quantum_Turing_machine" title="Quantum Turing machine">quantum Turing machine</a> can efficiently simulate any realistic model of computation." </p><p>Eugene Eberbach and Peter Wegner claim that the Church–Turing thesis is sometimes interpreted too broadly, stating "Though [...] Turing machines express the behavior of algorithms, the broader assertion that algorithms precisely capture what can be computed is invalid".<sup id="cite_ref-60" class="reference"><a href="#cite_note-60"><span class="cite-bracket">[</span>60<span class="cite-bracket">]</span></a></sup> They claim that forms of computation not captured by the thesis are relevant today, terms which they call <a href="/wiki/Hypercomputation" title="Hypercomputation">super-Turing computation</a>. </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=Church%E2%80%93Turing_thesis&action=edit&section=9" title="Edit section: Philosophical implications"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Philosophers have interpreted the Church–Turing thesis as having implications for the <a href="/wiki/Philosophy_of_mind" title="Philosophy of mind">philosophy of mind</a>.<sup id="cite_ref-61" class="reference"><a href="#cite_note-61"><span class="cite-bracket">[</span>61<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-62" class="reference"><a href="#cite_note-62"><span class="cite-bracket">[</span>62<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-63" class="reference"><a href="#cite_note-63"><span class="cite-bracket">[</span>63<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Jack_Copeland" title="Jack Copeland">B. Jack Copeland</a> states that it is an open empirical question whether there are actual deterministic physical processes that, in the long run, elude simulation by a Turing machine; furthermore, he states that it is an open empirical question whether any such processes are involved in the working of the human brain.<sup id="cite_ref-64" class="reference"><a href="#cite_note-64"><span class="cite-bracket">[</span>64<span class="cite-bracket">]</span></a></sup> There are also some important open questions which cover the relationship between the Church–Turing thesis and physics, and the possibility of <a href="/wiki/Hypercomputation" title="Hypercomputation">hypercomputation</a>. When applied to physics, the thesis has several possible meanings: </p> <ol><li>The universe is equivalent to a Turing machine; thus, computing <a href="/wiki/Computable_function" title="Computable function">non-recursive functions</a> is physically impossible. This has been termed the strong Church–Turing thesis, or <a href="/wiki/Church%E2%80%93Turing%E2%80%93Deutsch_principle" title="Church–Turing–Deutsch principle">Church–Turing–Deutsch principle</a>, and is a foundation of <a href="/wiki/Digital_physics" title="Digital physics">digital physics</a>.</li> <li>The universe is not equivalent to a Turing machine (i.e., the laws of physics are not Turing-computable), but incomputable physical events are not "harnessable" for the construction of a <a href="/wiki/Hypercomputation" title="Hypercomputation">hypercomputer</a>. For example, a universe in which physics involves random <a href="/wiki/Real_numbers" class="mw-redirect" title="Real numbers">real numbers</a>, as opposed to <a href="/wiki/Computable_number" title="Computable number">computable reals</a>, would fall into this category.</li> <li>The universe is a <a href="/wiki/Hypercomputation" title="Hypercomputation">hypercomputer</a>, and it is possible to build physical devices to harness this property and calculate non-recursive functions. For example, it is an open question whether all <a href="/wiki/Quantum_mechanics" title="Quantum mechanics">quantum mechanical</a> events are Turing-computable, although it is known that rigorous models such as <a href="/wiki/Quantum_Turing_machine" title="Quantum Turing machine">quantum Turing machines</a> are equivalent to deterministic Turing machines. (They are not necessarily efficiently equivalent; see above.) <a href="/wiki/John_Lucas_(philosopher)" title="John Lucas (philosopher)">John Lucas</a> and <a href="/wiki/Roger_Penrose" title="Roger Penrose">Roger Penrose</a> have suggested that the human mind might be the result of some kind of quantum-mechanically enhanced, "non-algorithmic" computation.<sup id="cite_ref-65" class="reference"><a href="#cite_note-65"><span class="cite-bracket">[</span>65<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-66" class="reference"><a href="#cite_note-66"><span class="cite-bracket">[</span>66<span class="cite-bracket">]</span></a></sup></li></ol> <p>There are many other technical possibilities which fall outside or between these three categories, but these serve to illustrate the range of the concept. </p><p>Philosophical aspects of the thesis, regarding both physical and biological computers, are also discussed in Odifreddi's 1989 textbook on recursion theory.<sup id="cite_ref-67" class="reference"><a href="#cite_note-67"><span class="cite-bracket">[</span>67<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Pages: 101–123">: 101–123 </span></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Non-computable_functions">Non-computable functions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Church%E2%80%93Turing_thesis&action=edit&section=10" title="Edit section: Non-computable functions"><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 solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output .ambox.mbox-small-left{margin:4px 1em 4px 0;overflow:hidden;width:238px;border-collapse:collapse;font-size:88%;line-height:1.25em}.mw-parser-output .ambox-speedy{border-left:10px solid #b32424;background-color:#fee7e6}.mw-parser-output .ambox-delete{border-left:10px solid #b32424}.mw-parser-output .ambox-content{border-left:10px solid #f28500}.mw-parser-output .ambox-style{border-left:10px solid #fc3}.mw-parser-output .ambox-move{border-left:10px solid #9932cc}.mw-parser-output .ambox-protection{border-left:10px solid #a2a9b1}.mw-parser-output .ambox .mbox-text{border:none;padding:0.25em 0.5em;width:100%}.mw-parser-output .ambox .mbox-image{border:none;padding:2px 0 2px 0.5em;text-align:center}.mw-parser-output .ambox .mbox-imageright{border:none;padding:2px 0.5em 2px 0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .mbox-image-div{width:52px}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}@media print{body.ns-0 .mw-parser-output .ambox{display:none!important}}</style><table class="box-One_source plainlinks metadata ambox ambox-content ambox-one_source" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><span><img 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" /></span></span></div></td><td class="mbox-text"><div class="mbox-text-span">This section <b>relies largely or entirely upon a <a href="/wiki/Wikipedia:Identifying_reliable_sources" class="mw-redirect" title="Wikipedia:Identifying reliable sources">single source</a></b>.<span class="hide-when-compact"> Relevant discussion may be found on the <a href="/wiki/Talk:Church%E2%80%93Turing_thesis##" title="Talk:Church–Turing thesis">talk page</a>. Please help <a class="external text" href="https://en.wikipedia.org/w/index.php?title=Church%E2%80%93Turing_thesis&action=edit">improve this article</a> by introducing <a href="/wiki/Wikipedia:Citing_sources" title="Wikipedia:Citing sources">citations</a> to additional sources at this section.</span> <span class="date-container"><i>(<span class="date">November 2017</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> <p>One can formally define functions that are not computable. A well-known example of such a function is the <a href="/wiki/Busy_Beaver" class="mw-redirect" title="Busy Beaver">Busy Beaver</a> function. This function takes an input <i>n</i> and returns the largest number of symbols that a <a href="/wiki/Turing_machine" title="Turing machine">Turing machine</a> with <i>n</i> states can print before halting, when run with no input. Finding an upper bound on the busy beaver function is equivalent to solving the <a href="/wiki/Halting_problem" title="Halting problem">halting problem</a>, a problem known to be unsolvable by Turing machines. Since the busy beaver function cannot be computed by Turing machines, the Church–Turing thesis states that this function cannot be effectively computed by any method. </p><p>Several computational models allow for the computation of (Church-Turing) non-computable functions. These are known as <a href="/wiki/Hypercomputation" title="Hypercomputation">hypercomputers</a>. Mark Burgin argues that <a href="/wiki/Super-recursive_algorithm" title="Super-recursive algorithm">super-recursive algorithms</a> such as inductive Turing machines disprove the Church–Turing thesis.<sup id="cite_ref-68" class="reference"><a href="#cite_note-68"><span class="cite-bracket">[</span>68<span class="cite-bracket">]</span></a></sup><sup class="noprint Inline-Template" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citing_sources" title="Wikipedia:Citing sources"><span title="This citation requires a reference to the specific page or range of pages in which the material appears. (November 2017)">page needed</span></a></i>]</sup> His argument relies on a definition of algorithm broader than the ordinary one, so that non-computable functions obtained from some inductive Turing machines are called computable. This interpretation of the Church–Turing thesis differs from the interpretation commonly accepted in computability theory, discussed above. The argument that super-recursive algorithms are indeed algorithms in the sense of the Church–Turing thesis has not found broad acceptance within the computability research community.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (May 2020)">citation needed</span></a></i>]</sup> </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Church%E2%80%93Turing_thesis&action=edit&section=11" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Abstract_machine" title="Abstract machine">Abstract machine</a></li> <li><a href="/wiki/Church%27s_thesis_(constructive_mathematics)" title="Church's thesis (constructive mathematics)">Church's thesis in constructive mathematics</a></li> <li><a href="/wiki/Church%E2%80%93Turing%E2%80%93Deutsch_principle" title="Church–Turing–Deutsch principle">Church–Turing–Deutsch principle</a>, which states that every <a href="/wiki/Physical_process" class="mw-redirect" title="Physical process">physical process</a> can be simulated by a universal computing device</li> <li><a href="/wiki/Computability_logic" title="Computability logic">Computability logic</a></li> <li><a href="/wiki/Computability_theory_(computation)" class="mw-redirect" title="Computability theory (computation)">Computability theory</a></li> <li><a href="/wiki/Decidability_(logic)" title="Decidability (logic)">Decidability</a></li> <li><a href="/wiki/Hypercomputation" title="Hypercomputation">Hypercomputation</a></li> <li><a href="/wiki/Model_of_computation" title="Model of computation">Model of computation</a></li> <li><a href="/wiki/Oracle_(computer_science)" class="mw-redirect" title="Oracle (computer science)">Oracle (computer science)</a></li> <li><a href="/wiki/Super-recursive_algorithm" title="Super-recursive algorithm">Super-recursive algorithm</a></li> <li><a href="/wiki/Turing_completeness" title="Turing completeness">Turing completeness</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Footnotes">Footnotes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Church%E2%80%93Turing_thesis&action=edit&section=12" title="Edit section: Footnotes"><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"> <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"><a href="/wiki/Robert_Soare" class="mw-redirect" title="Robert Soare">Robert Soare</a>, <a rel="nofollow" class="external text" href="http://www.people.cs.uchicago.edu/~soare/History/turing.pdf">"Turing Oracle Machines, Online Computing, and Three Displacements in Computability Theory"</a></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"><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="CITEREFRabin2012" class="citation audio-visual cs1"><a href="/wiki/Michael_O._Rabin" title="Michael O. Rabin">Rabin, Michael O.</a> (June 2012). <a rel="nofollow" class="external text" href="http://videolectures.net/turing100_rabin_turing_church_goedel/"><i>Turing, Church, Gödel, Computability, Complexity and Randomization: A Personal View</i></a>. Event occurs at 9:36<span class="reference-accessdate">. Retrieved <span class="nowrap">2021-12-05</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Turing%2C+Church%2C+G%C3%B6del%2C+Computability%2C+Complexity+and+Randomization%3A+A+Personal+View&rft.date=2012-06&rft.aulast=Rabin&rft.aufirst=Michael+O.&rft_id=http%3A%2F%2Fvideolectures.net%2Fturing100_rabin_turing_church_goedel%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AChurch%E2%80%93Turing+thesis" class="Z3988"></span></span> </li> <li id="cite_note-TuringNewman-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-TuringNewman_3-0">^</a></b></span> <span class="reference-text">Correspondence between <a href="/wiki/Max_Newman" title="Max Newman">Max Newman</a> and Church in <a rel="nofollow" class="external text" href="https://findingaids.princeton.edu/collections/C0948/c00385">Alonzo Church papers</a></span> </li> <li id="cite_note-EssentialTuring-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-EssentialTuring_4-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTuring2004" class="citation book cs1">Turing, Alan (2004). <a rel="nofollow" class="external text" href="http://www.cse.chalmers.se/~aikmitr/papers/Turing.pdf"><i>The essential Turing : seminal writings in computing, logic, philosophy, artificial intelligence, and artificial life, plus the secrets of Enigma</i></a> <span class="cs1-format">(PDF)</span>. Oxford: Clarendon Press. p. 44. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780198250791" title="Special:BookSources/9780198250791"><bdi>9780198250791</bdi></a><span class="reference-accessdate">. Retrieved <span class="nowrap">2021-12-06</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+essential+Turing+%3A+seminal+writings+in+computing%2C+logic%2C+philosophy%2C+artificial+intelligence%2C+and+artificial+life%2C+plus+the+secrets+of+Enigma&rft.place=Oxford&rft.pages=44&rft.pub=Clarendon+Press&rft.date=2004&rft.isbn=9780198250791&rft.aulast=Turing&rft.aufirst=Alan&rft_id=http%3A%2F%2Fwww.cse.chalmers.se%2F~aikmitr%2Fpapers%2FTuring.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AChurch%E2%80%93Turing+thesis" class="Z3988"></span></span> </li> <li id="cite_note-Soare-5"><span class="mw-cite-backlink">^ <a href="#cite_ref-Soare_5-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Soare_5-1"><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="CITEREFSoare1996" class="citation journal cs1"><a href="/wiki/Robert_I._Soare" title="Robert I. Soare">Soare, Robert I.</a> (September 1996). "Computability and Recursion". <i>Bulletin of Symbolic Logic</i>. <b>2</b> (3): 284–321. <a href="/wiki/CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.35.5803">10.1.1.35.5803</a></span>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F420992">10.2307/420992</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/420992">420992</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:5894394">5894394</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Bulletin+of+Symbolic+Logic&rft.atitle=Computability+and+Recursion&rft.volume=2&rft.issue=3&rft.pages=284-321&rft.date=1996-09&rft_id=https%3A%2F%2Fciteseerx.ist.psu.edu%2Fviewdoc%2Fsummary%3Fdoi%3D10.1.1.35.5803%23id-name%3DCiteSeerX&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A5894394%23id-name%3DS2CID&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F420992%23id-name%3DJSTOR&rft_id=info%3Adoi%2F10.2307%2F420992&rft.aulast=Soare&rft.aufirst=Robert+I.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AChurch%E2%80%93Turing+thesis" class="Z3988"></span></span> </li> <li id="cite_note-TuringLearn-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-TuringLearn_6-0">^</a></b></span> <span class="reference-text">Church's paper was presented to the American Mathematical Society on 19 April 1935 and published on 15 April 1936. Turing, who had made substantial progress in writing up his own results, was disappointed to learn of Church's proof upon its publication.<sup id="cite_ref-TuringNewman_3-0" class="reference"><a href="#cite_note-TuringNewman-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-EssentialTuring_4-0" class="reference"><a href="#cite_note-EssentialTuring-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> Turing quickly completed his paper and rushed it to publication; it was received by the <i>Proceedings of the London Mathematical Society</i> on 28 May 1936, read on 12 November 1936, and published in series 2, volume 42 (1936–1937); it appeared in two sections: in Part 3 (pages 230–240), issued on 30 November 1936 and in Part 4 (pages 241–265), issued on 23 December 1936; Turing added corrections in volume 43 (1937), pp. 544–546.<sup id="cite_ref-Soare_5-0" class="reference"><a href="#cite_note-Soare-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page: 45">: 45 </span></sup></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"><a href="#CITEREFChurch1936a">Church 1936a</a></span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><a href="#CITEREFKleene1936">Kleene 1936</a></span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><a href="#CITEREFTuring1937a">Turing 1937a</a></span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text"><a href="#CITEREFKleene1936">Kleene 1936</a></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"><a href="#CITEREFTuring1937b">Turing 1937b</a>. Proof outline on p. 153: <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 \lambda {\mbox{-definable}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>-definable</mtext> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda {\mbox{-definable}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d6c738af9eca0d38afefb0086eeda0af4e847dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.24ex; height:2.176ex;" alt="{\displaystyle \lambda {\mbox{-definable}}}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\stackrel {triv}{\implies }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-REL"> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mspace width="thickmathspace" /> <mo stretchy="false">⟹</mo> <mspace width="thickmathspace" /> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mi>r</mi> <mi>i</mi> <mi>v</mi> </mrow> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\stackrel {triv}{\implies }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b351515c182febf6ac8e0776971a5d59bacd3eb6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.096ex; height:3.676ex;" alt="{\displaystyle {\stackrel {triv}{\implies }}}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda {\mbox{-}}K{\mbox{-definable}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>-</mtext> </mstyle> </mrow> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>-definable</mtext> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda {\mbox{-}}K{\mbox{-definable}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb207a7d1eda3e4427dad70b3ba764ca15f5acd3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:14.081ex; height:2.176ex;" alt="{\displaystyle \lambda {\mbox{-}}K{\mbox{-definable}}}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\stackrel {160}{\implies }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-REL"> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mspace width="thickmathspace" /> <mo stretchy="false">⟹</mo> <mspace width="thickmathspace" /> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>160</mn> </mrow> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\stackrel {160}{\implies }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/70829859183c6c699fa6ec613d9311e293b816ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.096ex; height:3.676ex;" alt="{\displaystyle {\stackrel {160}{\implies }}}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mbox{Turing computable}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>Turing computable</mtext> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mbox{Turing computable}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3084223de52f82cad93553f677a206c2957b06f6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:19.319ex; height:2.509ex;" alt="{\displaystyle {\mbox{Turing computable}}}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\stackrel {161}{\implies }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-REL"> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mspace width="thickmathspace" /> <mo stretchy="false">⟹</mo> <mspace width="thickmathspace" /> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>161</mn> </mrow> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\stackrel {161}{\implies }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2873e188be2ba5438ac119acbe19f3f216db723d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.096ex; height:3.676ex;" alt="{\displaystyle {\stackrel {161}{\implies }}}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu {\mbox{-recursive}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>-recursive</mtext> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu {\mbox{-recursive}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c2208f29f466cfa33360598507960c4cedd32c7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.18ex; height:2.676ex;" alt="{\displaystyle \mu {\mbox{-recursive}}}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\stackrel {Kleene}{\implies }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-REL"> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mspace width="thickmathspace" /> <mo stretchy="false">⟹</mo> <mspace width="thickmathspace" /> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>K</mi> <mi>l</mi> <mi>e</mi> <mi>e</mi> <mi>n</mi> <mi>e</mi> </mrow> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\stackrel {Kleene}{\implies }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffc36b8d9792097493bd8b7fb37cf21e82dceaed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.236ex; height:3.676ex;" alt="{\displaystyle {\stackrel {Kleene}{\implies }}}"></span><sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> <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 \lambda {\mbox{-definable}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>-definable</mtext> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda {\mbox{-definable}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d6c738af9eca0d38afefb0086eeda0af4e847dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.24ex; height:2.176ex;" alt="{\displaystyle \lambda {\mbox{-definable}}}"></span></span> </li> <li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text"><a href="#CITEREFRosser1939">Rosser 1939</a> in <a href="#CITEREFDavis1965">Davis 1965</a>:225.</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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation encyclopaedia cs1">"effective". <i>Merriam Webster's New Collegiate Dictionary</i> (9th ed.).</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=effective&rft.btitle=Merriam+Webster%27s+New+Collegiate+Dictionary&rft.edition=9th&rfr_id=info%3Asid%2Fen.wikipedia.org%3AChurch%E2%80%93Turing+thesis" class="Z3988"></span></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">See also <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation encyclopaedia cs1">"effective". <a rel="nofollow" class="external text" href="http://www.merriam-webster.com/dictionary/effective"><i>Merriam-Webster's Online Dictionary</i></a> (11th ed.)<span class="reference-accessdate">. Retrieved <span class="nowrap">2014-07-26</span></span>,</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=effective&rft.btitle=Merriam-Webster%27s+Online+Dictionary&rft.edition=11th&rft_id=http%3A%2F%2Fwww.merriam-webster.com%2Fdictionary%2Feffective&rfr_id=info%3Asid%2Fen.wikipedia.org%3AChurch%E2%80%93Turing+thesis" class="Z3988"></span> which also gives these definitions for "effective" – the first ["producing a decided, decisive, or desired effect"] as the definition for sense "1a" of the word "effective", and the second ["capable of producing a result"] as part of the "Synonym Discussion of EFFECTIVE" there, (in the introductory part, where it summarizes the similarities between the meanings of the words "effective", "effectual", "efficient", and "efficacious").</span> </li> <li id="cite_note-Turing_1938_thesis_p8-15"><span class="mw-cite-backlink">^ <a href="#cite_ref-Turing_1938_thesis_p8_15-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Turing_1938_thesis_p8_15-1"><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="CITEREFTuring1938" class="citation thesis cs1">Turing, A. M. (1938). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20121023103503/https://webspace.princeton.edu/users/jedwards/Turing%20Centennial%202012/Mudd%20Archive%20files/12285_AC100_Turing_1938.pdf"><i>Systems of Logic Based on Ordinals</i></a> <span class="cs1-format">(PDF)</span> (PhD). Princeton University. p. 8. Archived from <a rel="nofollow" class="external text" href="https://webspace.princeton.edu/users/jedwards/Turing%20Centennial%202012/Mudd%20Archive%20files/12285_AC100_Turing_1938.pdf">the original</a> <span class="cs1-format">(PDF)</span> on 2012-10-23<span class="reference-accessdate">. Retrieved <span class="nowrap">2012-06-23</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Adissertation&rft.title=Systems+of+Logic+Based+on+Ordinals&rft.inst=Princeton+University&rft.date=1938&rft.aulast=Turing&rft.aufirst=A.+M.&rft_id=https%3A%2F%2Fwebspace.princeton.edu%2Fusers%2Fjedwards%2FTuring%2520Centennial%25202012%2FMudd%2520Archive%2520files%2F12285_AC100_Turing_1938.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AChurch%E2%80%93Turing+thesis" class="Z3988"></span></span> </li> <li id="cite_note-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-16">^</a></b></span> <span class="reference-text"><a href="#CITEREFGandy1980">Gandy (1980</a>:123) states it this way: <i>What is effectively calculable is computable.</i> He calls this "Church's Thesis".</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">David Hilbert and Wilhelm Ackermann: Grundzüge der theoretischen Logik, Berlin, Germany, Springer, 1st ed. 1928. (6th ed. 1972, <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/3-540-05843-5" title="Special:BookSources/3-540-05843-5">3-540-05843-5</a>) English Translation: David Hilbert and Wilhelm Ackermann: Principles of Mathematical Logic, AMS Chelsea Publishing, Providence, Rhode Island, USA, 1950.</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">Davis's commentary before Church 1936 <i>An Unsolvable Problem of Elementary Number Theory</i> in Davis 1965:88. Church uses the words "effective calculability" on page 100ff.</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">In his review of <i>Church's Thesis after 70 Years</i> edited by Adam Olszewski et al. 2006, Peter Smith's criticism of a paper by Muraswski and Wolenski suggests 4 "lines" re the status of the Church–Turing Thesis: (1) empirical hypothesis (2) axiom or theorem, (3) definition, (4) explication. But Smith opines that (4) is indistinguishable from (3), cf. Smith (2007-07-11) <i>Church's Thesis after 70 Years</i> at <a rel="nofollow" class="external free" href="http://www.logicmatters.net/resources/pdfs/CTT.pdf">http://www.logicmatters.net/resources/pdfs/CTT.pdf</a></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. footnote 3 in <a href="#CITEREFChurch1936a">Church 1936a</a> <i>An Unsolvable Problem of Elementary Number Theory</i> in <a href="#CITEREFDavis1965">Davis 1965</a>:89.</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"><a href="#CITEREFDawson1997">Dawson 1997</a>:99.</span> </li> <li id="cite_note-sieg160-22"><span class="mw-cite-backlink">^ <a href="#cite_ref-sieg160_22-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-sieg160_22-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">Sieg 1997:160.</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">Sieg 1997:160 quoting from the 1935 letter written by Church to Kleene, cf. Footnote 3 in Gödel 1934 in <a href="#CITEREFDavis1965">Davis 1965</a>:44.</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">cf. Church 1936 in <a href="#CITEREFDavis1965">Davis 1965</a>:105ff..</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">Davis's commentary before Gödel 1934 in <a href="#CITEREFDavis1965">Davis 1965</a>:40.</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">For a detailed discussion of Gödel's adoption of Turing's machines as models of computation, see <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFShagrir2006" class="citation book cs1"><a href="/wiki/Oron_Shagrir" title="Oron Shagrir">Shagrir, Oron</a> (2006-06-15). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20151217145831/http://moon.cc.huji.ac.il/oron-shagrir/papers/Goedel_on_Turing_on_Computability.pdf">"Gödel on Turing on Computability"</a> <span class="cs1-format">(PDF)</span>. <i>Church's Thesis After 70 Years</i>. De Gruyter. pp. 393–419. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1515%2F9783110325461.393">10.1515/9783110325461.393</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-11-032494-5" title="Special:BookSources/978-3-11-032494-5"><bdi>978-3-11-032494-5</bdi></a>. Archived from <a rel="nofollow" class="external text" href="http://moon.cc.huji.ac.il/oron-shagrir/papers/Goedel_on_Turing_on_Computability.pdf">the original</a> <span class="cs1-format">(PDF)</span> on 2015-12-17<span class="reference-accessdate">. Retrieved <span class="nowrap">2016-02-08</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=G%C3%B6del+on+Turing+on+Computability&rft.btitle=Church%27s+Thesis+After+70+Years&rft.pages=393-419&rft.pub=De+Gruyter&rft.date=2006-06-15&rft_id=info%3Adoi%2F10.1515%2F9783110325461.393&rft.isbn=978-3-11-032494-5&rft.aulast=Shagrir&rft.aufirst=Oron&rft_id=http%3A%2F%2Fmoon.cc.huji.ac.il%2Foron-shagrir%2Fpapers%2FGoedel_on_Turing_on_Computability.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AChurch%E2%80%93Turing+thesis" class="Z3988"></span></span> </li> <li id="cite_note-On_Computable-27"><span class="mw-cite-backlink">^ <a href="#cite_ref-On_Computable_27-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-On_Computable_27-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><a href="#CITEREFTuring1937a">Turing 1937a</a>.</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">cf. Editor's footnote to Post 1936 <i>Finite Combinatory Process. Formulation I.</i> at <a href="#CITEREFDavis1965">Davis 1965</a>:289.</span> </li> <li id="cite_note-29"><span class="mw-cite-backlink"><b><a href="#cite_ref-29">^</a></b></span> <span class="reference-text">Post 1936 in <a href="#CITEREFDavis1965">Davis 1965</a>:291, footnote 8.</span> </li> <li id="cite_note-30"><span class="mw-cite-backlink"><b><a href="#cite_ref-30">^</a></b></span> <span class="reference-text">Post 1936 in Davis 1952:291.</span> </li> <li id="cite_note-31"><span class="mw-cite-backlink"><b><a href="#cite_ref-31">^</a></b></span> <span class="reference-text">Sieg 1997:171 and 176–177.</span> </li> <li id="cite_note-32"><span class="mw-cite-backlink"><b><a href="#cite_ref-32">^</a></b></span> <span class="reference-text">Turing 1936–1937 in <a href="#CITEREFDavis1965">Davis 1965</a>:263ff..</span> </li> <li id="cite_note-33"><span class="mw-cite-backlink"><b><a href="#cite_ref-33">^</a></b></span> <span class="reference-text"><a href="#CITEREFChurch1937">Church 1937</a>.</span> </li> <li id="cite_note-34"><span class="mw-cite-backlink"><b><a href="#cite_ref-34">^</a></b></span> <span class="reference-text">Turing 1939 in Davis:160.</span> </li> <li id="cite_note-35"><span class="mw-cite-backlink"><b><a href="#cite_ref-35">^</a></b></span> <span class="reference-text">cf. Church 1934 in <a href="#CITEREFDavis1965">Davis 1965</a>:100, also Turing 1939 in <a href="#CITEREFDavis1965">Davis 1965</a>:160.</span> </li> <li id="cite_note-36"><span class="mw-cite-backlink"><b><a href="#cite_ref-36">^</a></b></span> <span class="reference-text">italics added, <a href="#CITEREFRosser1939">Rosser 1939</a> in <a href="#CITEREFDavis1965">Davis 1965</a>:226.</span> </li> <li id="cite_note-Davis274-37"><span class="mw-cite-backlink"><b><a href="#cite_ref-Davis274_37-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFKleene1943">Kleene 1943</a>, p. 60 in <a href="#CITEREFDavis1965">Davis 1965</a>:274. Footnotes omitted.</span> </li> <li id="cite_note-38"><span class="mw-cite-backlink"><b><a href="#cite_ref-38">^</a></b></span> <span class="reference-text"><a href="#CITEREFKleene1952">Kleene 1952</a>:300.</span> </li> <li id="cite_note-Kleene_1952_p376-39"><span class="mw-cite-backlink">^ <a href="#cite_ref-Kleene_1952_p376_39-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Kleene_1952_p376_39-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><a href="#CITEREFKleene1952">Kleene 1952</a>:376.</span> </li> <li id="cite_note-40"><span class="mw-cite-backlink"><b><a href="#cite_ref-40">^</a></b></span> <span class="reference-text"><a href="#CITEREFKleene1952">Kleene 1952</a>:382, 536</span> </li> <li id="cite_note-41"><span class="mw-cite-backlink"><b><a href="#cite_ref-41">^</a></b></span> <span class="reference-text"><a href="#CITEREFGandy1980">Gandy 1980</a>:123ff.</span> </li> <li id="cite_note-42"><span class="mw-cite-backlink"><b><a href="#cite_ref-42">^</a></b></span> <span class="reference-text"><a href="#CITEREFGandy1980">Gandy 1980</a>:135</span> </li> <li id="cite_note-43"><span class="mw-cite-backlink"><b><a href="#cite_ref-43">^</a></b></span> <span class="reference-text"><a href="#CITEREFGandy1980">Gandy 1980</a>:126</span> </li> <li id="cite_note-44"><span class="mw-cite-backlink"><b><a href="#cite_ref-44">^</a></b></span> <span class="reference-text">Sieg 1998–1999 in <a href="#CITEREFSiegSommerTalcott2002">Sieg, Sommer & Talcott 2002</a>:390ff.; also Sieg 1997:154ff.</span> </li> <li id="cite_note-45"><span class="mw-cite-backlink"><b><a href="#cite_ref-45">^</a></b></span> <span class="reference-text">In a footnote Sieg breaks Post's 1936 (B) into (B.1) and (B.2) and (L) into (L.1) and (L.2) and describes (D) differently. With respect to his proposed <a href="/w/index.php?title=Gandy_machine&action=edit&redlink=1" class="new" title="Gandy machine (page does not exist)">Gandy machine</a> he later adds LC.1, LC.2, GA.1 and GA.2. These are complicated; see Sieg 1998–1999 in <a href="#CITEREFSiegSommerTalcott2002">Sieg, Sommer & Talcott 2002</a>:390ff..</span> </li> <li id="cite_note-46"><span class="mw-cite-backlink"><b><a href="#cite_ref-46">^</a></b></span> <span class="reference-text">A collection of papers can be found in <a href="#CITEREFOlszewskiWoleńskiJanusz2006">Olszewski, Woleński & Janusz (2006)</a>. Also a review of this collection: <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSmith2007" class="citation web cs1">Smith, Peter (2007-07-11). <a rel="nofollow" class="external text" href="http://www.logicmatters.net/resources/pdfs/CTT.pdf">"Church's Thesis after 70 Years"</a> <span class="cs1-format">(PDF)</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Church%27s+Thesis+after+70+Years&rft.date=2007-07-11&rft.aulast=Smith&rft.aufirst=Peter&rft_id=http%3A%2F%2Fwww.logicmatters.net%2Fresources%2Fpdfs%2FCTT.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AChurch%E2%80%93Turing+thesis" class="Z3988"></span></span> </li> <li id="cite_note-47"><span class="mw-cite-backlink"><b><a href="#cite_ref-47">^</a></b></span> <span class="reference-text">See also <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHodges2005" class="citation web cs1">Hodges, Andrew (2005). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20160304032827/http://www.turing.org.uk/publications/ct70.pdf">"Did Church and Turing Have a Thesis about Machines?"</a> <span class="cs1-format">(PDF)</span>. Archived from <a rel="nofollow" class="external text" href="http://www.turing.org.uk/publications/ct70.pdf">the original</a> <span class="cs1-format">(PDF)</span> on 2016-03-04<span class="reference-accessdate">. Retrieved <span class="nowrap">2014-07-27</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Did+Church+and+Turing+Have+a+Thesis+about+Machines%3F&rft.date=2005&rft.aulast=Hodges&rft.aufirst=Andrew&rft_id=http%3A%2F%2Fwww.turing.org.uk%2Fpublications%2Fct70.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AChurch%E2%80%93Turing+thesis" class="Z3988"></span></span> </li> <li id="cite_note-48"><span class="mw-cite-backlink"><b><a href="#cite_ref-48">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGödel1995" class="citation book cs1"><a href="/wiki/Kurt_G%C3%B6del" title="Kurt Gödel">Gödel, Kurt</a> (1995) [193?]. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=gDzbuUwma5MC&pg=PA164">"Undecidable Diophantine Propositions"</a>. In <a href="/wiki/Solomon_Feferman" title="Solomon Feferman">Feferman, Solomon</a> (ed.). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=gDzbuUwma5MC"><i>Collected Works</i></a>. Vol. 3. New York: <a href="/wiki/Oxford_University_Press" title="Oxford University Press">Oxford University Press</a>. p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=gDzbuUwma5MC&pg=PA168">168</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-507255-6" title="Special:BookSources/978-0-19-507255-6"><bdi>978-0-19-507255-6</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/928791907">928791907</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Undecidable+Diophantine+Propositions&rft.btitle=Collected+Works&rft.place=New+York&rft.pages=168&rft.pub=Oxford+University+Press&rft.date=1995&rft_id=info%3Aoclcnum%2F928791907&rft.isbn=978-0-19-507255-6&rft.aulast=G%C3%B6del&rft.aufirst=Kurt&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DgDzbuUwma5MC%26pg%3DPA164&rfr_id=info%3Asid%2Fen.wikipedia.org%3AChurch%E2%80%93Turing+thesis" class="Z3988"></span></span> </li> <li id="cite_note-49"><span class="mw-cite-backlink"><b><a href="#cite_ref-49">^</a></b></span> <span class="reference-text">Kleene 1952:320</span> </li> <li id="cite_note-50"><span class="mw-cite-backlink"><b><a href="#cite_ref-50">^</a></b></span> <span class="reference-text">Gurevich 1988:2</span> </li> <li id="cite_note-51"><span class="mw-cite-backlink"><b><a href="#cite_ref-51">^</a></b></span> <span class="reference-text">Translation of Gödel (1936) by Davis in <i>The Undecidable</i> p. 83, differing in the use of the word 'reckonable' in the translation in Kleene (1952) p. 321</span> </li> <li id="cite_note-52"><span class="mw-cite-backlink"><b><a href="#cite_ref-52">^</a></b></span> <span class="reference-text">Horsten in <a href="#CITEREFOlszewskiWoleńskiJanusz2006">Olszewski, Woleński & Janusz 2006</a>:256.</span> </li> <li id="cite_note-53"><span class="mw-cite-backlink"><b><a href="#cite_ref-53">^</a></b></span> <span class="reference-text"><a href="#CITEREFGabbay2001">Gabbay 2001</a>:284</span> </li> <li id="cite_note-54"><span class="mw-cite-backlink"><b><a href="#cite_ref-54">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPiccinini2007" class="citation journal cs1"><a href="/wiki/Gualtiero_Piccinini" title="Gualtiero Piccinini">Piccinini, Gualtiero</a> (January 2007). <a rel="nofollow" class="external text" href="http://www.umsl.edu/~piccininig/Computationalism_Church-Turing_Thesis_Church-Turing_Fallacy.pdf">"Computationalism, the Church–Turing Thesis, and the Church–Turing Fallacy"</a> <span class="cs1-format">(PDF)</span>. <i>Synthese</i>. <b>154</b> (1): 97–120. <a href="/wiki/CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.360.9796">10.1.1.360.9796</a></span>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs11229-005-0194-z">10.1007/s11229-005-0194-z</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:494161">494161</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20080424151259/http://www.umsl.edu/~piccininig/Computationalism_Church-Turing_Thesis_Church-Turing_Fallacy.pdf">Archived</a> <span class="cs1-format">(PDF)</span> from the original on 2008-04-24.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Synthese&rft.atitle=Computationalism%2C+the+Church%E2%80%93Turing+Thesis%2C+and+the+Church%E2%80%93Turing+Fallacy&rft.volume=154&rft.issue=1&rft.pages=97-120&rft.date=2007-01&rft_id=https%3A%2F%2Fciteseerx.ist.psu.edu%2Fviewdoc%2Fsummary%3Fdoi%3D10.1.1.360.9796%23id-name%3DCiteSeerX&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A494161%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1007%2Fs11229-005-0194-z&rft.aulast=Piccinini&rft.aufirst=Gualtiero&rft_id=http%3A%2F%2Fwww.umsl.edu%2F~piccininig%2FComputationalism_Church-Turing_Thesis_Church-Turing_Fallacy.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AChurch%E2%80%93Turing+thesis" class="Z3988"></span></span> </li> <li id="cite_note-55"><span class="mw-cite-backlink"><b><a href="#cite_ref-55">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAroraBarak2009" class="citation book cs1">Arora, Sanjeev; Barak, Boaz (2009). <a rel="nofollow" class="external text" href="http://www.cs.princeton.edu/theory/complexity/"><i>Complexity Theory: A Modern Approach</i></a>. <a href="/wiki/Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-42426-4" title="Special:BookSources/978-0-521-42426-4"><bdi>978-0-521-42426-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Complexity+Theory%3A+A+Modern+Approach&rft.pub=Cambridge+University+Press&rft.date=2009&rft.isbn=978-0-521-42426-4&rft.aulast=Arora&rft.aufirst=Sanjeev&rft.au=Barak%2C+Boaz&rft_id=http%3A%2F%2Fwww.cs.princeton.edu%2Ftheory%2Fcomplexity%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AChurch%E2%80%93Turing+thesis" class="Z3988"></span> Sections 1.4, "Machines as strings and the universal Turing machine" and 1.7, "Proof of theorem 1.9".</span> </li> <li id="cite_note-56"><span class="mw-cite-backlink"><b><a href="#cite_ref-56">^</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://web.archive.org/web/20051124084833/http://www.claymath.org/millennium/P_vs_NP/Official_Problem_Description.pdf">"Official Problem Description"</a> <span class="cs1-format">(PDF)</span>. Archived from <a rel="nofollow" class="external text" href="http://www.claymath.org/millennium/P_vs_NP/Official_Problem_Description.pdf">the original</a> <span class="cs1-format">(PDF)</span> on 2005-11-24.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Official+Problem+Description&rft_id=http%3A%2F%2Fwww.claymath.org%2Fmillennium%2FP_vs_NP%2FOfficial_Problem_Description.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AChurch%E2%80%93Turing+thesis" class="Z3988"></span></span> </li> <li id="cite_note-kaye-57"><span class="mw-cite-backlink">^ <a href="#cite_ref-kaye_57-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-kaye_57-1"><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="CITEREFKayeLaflammeMosca2007" class="citation book cs1">Kaye, Phillip; Laflamme, Raymond; Mosca, Michele (2007). <i>An introduction to quantum computing</i>. Oxford University Press. pp. 5–6. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-857049-3" title="Special:BookSources/978-0-19-857049-3"><bdi>978-0-19-857049-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=An+introduction+to+quantum+computing&rft.pages=5-6&rft.pub=Oxford+University+Press&rft.date=2007&rft.isbn=978-0-19-857049-3&rft.aulast=Kaye&rft.aufirst=Phillip&rft.au=Laflamme%2C+Raymond&rft.au=Mosca%2C+Michele&rfr_id=info%3Asid%2Fen.wikipedia.org%3AChurch%E2%80%93Turing+thesis" class="Z3988"></span></span> </li> <li id="cite_note-58"><span class="mw-cite-backlink"><b><a href="#cite_ref-58">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFvan_Emde_Boas1990" class="citation book cs1">van Emde Boas, Peter (1990). "Machine Models and Simulations". <i>Handbook of Theoretical Computer Science A</i>. <a href="/wiki/Elsevier" title="Elsevier">Elsevier</a>. p. 5.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Machine+Models+and+Simulations&rft.btitle=Handbook+of+Theoretical+Computer+Science+A&rft.pages=5&rft.pub=Elsevier&rft.date=1990&rft.aulast=van+Emde+Boas&rft.aufirst=Peter&rfr_id=info%3Asid%2Fen.wikipedia.org%3AChurch%E2%80%93Turing+thesis" class="Z3988"></span></span> </li> <li id="cite_note-59"><span class="mw-cite-backlink"><b><a href="#cite_ref-59">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSlotvan_Emde_Boas1984" class="citation conference cs1">Slot, C.; van Emde Boas, P. (December 1984). <i>On tape versus core: an application of space efficient perfect hash functions to the invariance of space</i>. <a href="/wiki/Symposium_on_Theory_of_Computing" title="Symposium on Theory of Computing">STOC</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=conference&rft.btitle=On+tape+versus+core%3A+an+application+of+space+efficient+perfect+hash+functions+to+the+invariance+of+space&rft.date=1984-12&rft.aulast=Slot&rft.aufirst=C.&rft.au=van+Emde+Boas%2C+P.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AChurch%E2%80%93Turing+thesis" class="Z3988"></span></span> </li> <li id="cite_note-60"><span class="mw-cite-backlink"><b><a href="#cite_ref-60">^</a></b></span> <span class="reference-text"><a href="#CITEREFEberbachWegner2003">Eberbach & Wegner 2003</a>, p. 287.</span> </li> <li id="cite_note-61"><span class="mw-cite-backlink"><b><a href="#cite_ref-61">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAbramson2011" class="citation journal cs1">Abramson, Darren (2011). <a rel="nofollow" class="external text" href="https://dl.acm.org/doi/abs/10.1007/s11023-011-9236-0">"Philosophy of mind is (in part) philosophy of computer science"</a>. <i>Minds and Machines</i>. <b>21</b> (2): 203–219. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs11023-011-9236-0">10.1007/s11023-011-9236-0</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:32116031">32116031</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Minds+and+Machines&rft.atitle=Philosophy+of+mind+is+%28in+part%29+philosophy+of+computer+science.&rft.volume=21&rft.issue=2&rft.pages=203-219&rft.date=2011&rft_id=info%3Adoi%2F10.1007%2Fs11023-011-9236-0&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A32116031%23id-name%3DS2CID&rft.aulast=Abramson&rft.aufirst=Darren&rft_id=https%3A%2F%2Fdl.acm.org%2Fdoi%2Fabs%2F10.1007%2Fs11023-011-9236-0&rfr_id=info%3Asid%2Fen.wikipedia.org%3AChurch%E2%80%93Turing+thesis" class="Z3988"></span></span> </li> <li id="cite_note-62"><span class="mw-cite-backlink"><b><a href="#cite_ref-62">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCopeland2017" class="citation encyclopaedia cs1"><a href="/wiki/Jack_Copeland" title="Jack Copeland">Copeland, B. Jack</a> (2017-11-10). <a rel="nofollow" class="external text" href="https://plato.stanford.edu/entries/church-turing/">"The Church-Turing Thesis"</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=The+Church-Turing+Thesis&rft.btitle=Stanford+Encyclopedia+of+Philosophy&rft.date=2017-11-10&rft.aulast=Copeland&rft.aufirst=B.+Jack&rft_id=https%3A%2F%2Fplato.stanford.edu%2Fentries%2Fchurch-turing%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AChurch%E2%80%93Turing+thesis" class="Z3988"></span></span> </li> <li id="cite_note-63"><span class="mw-cite-backlink"><b><a href="#cite_ref-63">^</a></b></span> <span class="reference-text">For a good place to encounter original papers see <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFChalmers2002" class="citation book cs1"><a href="/wiki/David_Chalmers" title="David Chalmers">Chalmers, David J.</a>, ed. (2002). <i>Philosophy of Mind: Classical and Contemporary Readings</i>. New York: Oxford University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-514581-6" title="Special:BookSources/978-0-19-514581-6"><bdi>978-0-19-514581-6</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/610918145">610918145</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Philosophy+of+Mind%3A+Classical+and+Contemporary+Readings&rft.place=New+York&rft.pub=Oxford+University+Press&rft.date=2002&rft_id=info%3Aoclcnum%2F610918145&rft.isbn=978-0-19-514581-6&rfr_id=info%3Asid%2Fen.wikipedia.org%3AChurch%E2%80%93Turing+thesis" class="Z3988"></span></span> </li> <li id="cite_note-64"><span class="mw-cite-backlink"><b><a href="#cite_ref-64">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCopeland2004" class="citation book cs1"><a href="/wiki/Jack_Copeland" title="Jack Copeland">Copeland, B. Jack</a> (2004). "Computation". In <a href="/wiki/Luciano_Floridi" title="Luciano Floridi">Floridi, Luciano</a> (ed.). <i>The Blackwell guide to the philosophy of computing and information</i>. Wiley-Blackwell. p. 15. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-631-22919-3" title="Special:BookSources/978-0-631-22919-3"><bdi>978-0-631-22919-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Computation&rft.btitle=The+Blackwell+guide+to+the+philosophy+of+computing+and+information&rft.pages=15&rft.pub=Wiley-Blackwell&rft.date=2004&rft.isbn=978-0-631-22919-3&rft.aulast=Copeland&rft.aufirst=B.+Jack&rfr_id=info%3Asid%2Fen.wikipedia.org%3AChurch%E2%80%93Turing+thesis" class="Z3988"></span></span> </li> <li id="cite_note-65"><span class="mw-cite-backlink"><b><a href="#cite_ref-65">^</a></b></span> <span class="reference-text">cf. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPenrose1990" class="citation book cs1"><a href="/wiki/Roger_Penrose" title="Roger Penrose">Penrose, Roger</a> (1990). "Algorithms and Turing machines". <a href="/wiki/The_Emperor%27s_New_Mind" title="The Emperor's New Mind"><i>The Emperor's New Mind: Concerning Computers, Minds, and the Laws of Physics</i></a>. Oxford: Oxford University Press. pp. 47–49. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-851973-7" title="Special:BookSources/978-0-19-851973-7"><bdi>978-0-19-851973-7</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/456785846">456785846</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Algorithms+and+Turing+machines&rft.btitle=The+Emperor%27s+New+Mind%3A+Concerning+Computers%2C+Minds%2C+and+the+Laws+of+Physics&rft.place=Oxford&rft.pages=47-49&rft.pub=Oxford+University+Press&rft.date=1990&rft_id=info%3Aoclcnum%2F456785846&rft.isbn=978-0-19-851973-7&rft.aulast=Penrose&rft.aufirst=Roger&rfr_id=info%3Asid%2Fen.wikipedia.org%3AChurch%E2%80%93Turing+thesis" class="Z3988"></span></span> </li> <li id="cite_note-66"><span class="mw-cite-backlink"><b><a href="#cite_ref-66">^</a></b></span> <span class="reference-text">Also the description of "the non-algorithmic nature of mathematical insight", <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPenrose1990" class="citation book cs1"><a href="/wiki/Roger_Penrose" title="Roger Penrose">Penrose, Roger</a> (1990). "Where lies the physics of mind?". <a href="/wiki/The_Emperor%27s_New_Mind" title="The Emperor's New Mind"><i>The Emperor's New Mind: Concerning Computers, Minds, and the Laws of Physics</i></a>. Oxford: Oxford University Press. pp. 416–418. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-851973-7" title="Special:BookSources/978-0-19-851973-7"><bdi>978-0-19-851973-7</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/456785846">456785846</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Where+lies+the+physics+of+mind%3F&rft.btitle=The+Emperor%27s+New+Mind%3A+Concerning+Computers%2C+Minds%2C+and+the+Laws+of+Physics&rft.place=Oxford&rft.pages=416-418&rft.pub=Oxford+University+Press&rft.date=1990&rft_id=info%3Aoclcnum%2F456785846&rft.isbn=978-0-19-851973-7&rft.aulast=Penrose&rft.aufirst=Roger&rfr_id=info%3Asid%2Fen.wikipedia.org%3AChurch%E2%80%93Turing+thesis" class="Z3988"></span></span> </li> <li id="cite_note-67"><span class="mw-cite-backlink"><b><a href="#cite_ref-67">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPiergiorgio_Odifreddi1989" class="citation book cs1"><a href="/wiki/Piergiorgio_Odifreddi" title="Piergiorgio Odifreddi">Piergiorgio Odifreddi</a> (1989). <i>Classical Recursion Theory</i>. Studies in Logic and the Foundations of Mathematics. Vol. 125. Amsterdam, Netherlands: North Holland.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Classical+Recursion+Theory&rft.place=Amsterdam%2C+Netherlands&rft.series=Studies+in+Logic+and+the+Foundations+of+Mathematics&rft.pub=North+Holland&rft.date=1989&rft.au=Piergiorgio+Odifreddi&rfr_id=info%3Asid%2Fen.wikipedia.org%3AChurch%E2%80%93Turing+thesis" class="Z3988"></span></span> </li> <li id="cite_note-68"><span class="mw-cite-backlink"><b><a href="#cite_ref-68">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBurgin2005" class="citation book cs1">Burgin, Mark (2005). <i>Super-Recursive Algorithms</i>. Monographs in Computer Science. New York: Springer. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-95569-8" title="Special:BookSources/978-0-387-95569-8"><bdi>978-0-387-95569-8</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/990755791">990755791</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Super-Recursive+Algorithms&rft.place=New+York&rft.series=Monographs+in+Computer+Science&rft.pub=Springer&rft.date=2005&rft_id=info%3Aoclcnum%2F990755791&rft.isbn=978-0-387-95569-8&rft.aulast=Burgin&rft.aufirst=Mark&rfr_id=info%3Asid%2Fen.wikipedia.org%3AChurch%E2%80%93Turing+thesis" class="Z3988"></span></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=Church%E2%80%93Turing_thesis&action=edit&section=13" title="Edit section: References"><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="CITEREFBarwiseKeislerKunen1980" class="citation book cs1"><a href="/wiki/Jon_Barwise" title="Jon Barwise">Barwise, Jon</a>; <a href="/wiki/H._J._Keisler" class="mw-redirect" title="H. J. Keisler">Keisler, H.J.</a>; <a href="/wiki/Kenneth_Kunen" title="Kenneth Kunen">Kunen, Kenneth</a>, eds. (1980). <i>The Kleene Symposium</i>. Amsterdam: North-Holland Publishing Company. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-444-85345-5" title="Special:BookSources/978-0-444-85345-5"><bdi>978-0-444-85345-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Kleene+Symposium&rft.place=Amsterdam&rft.pub=North-Holland+Publishing+Company&rft.date=1980&rft.isbn=978-0-444-85345-5&rfr_id=info%3Asid%2Fen.wikipedia.org%3AChurch%E2%80%93Turing+thesis" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBen-Amram2005" class="citation journal cs1">Ben-Amram, A. M. (2005). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20170706063735/http://www2.mta.ac.il/~amirben/downloadable/church-turing.ps">"The Church-Turing Thesis and its Look-Alikes"</a>. <i><a href="/wiki/SIGACT_News" class="mw-redirect" title="SIGACT News">SIGACT News</a></i>. <b>36</b> (3): 113–116. <a href="/wiki/CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.74.7308">10.1.1.74.7308</a></span>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1145%2F1086649.1086651">10.1145/1086649.1086651</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:13566703">13566703</a>. Archived from <a rel="nofollow" class="external text" href="http://www2.mta.ac.il/~amirben/downloadable/church-turing.ps">the original</a> <span class="cs1-format">(PS)</span> on 2017-07-06<span class="reference-accessdate">. Retrieved <span class="nowrap">2017-10-24</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=SIGACT+News&rft.atitle=The+Church-Turing+Thesis+and+its+Look-Alikes&rft.volume=36&rft.issue=3&rft.pages=113-116&rft.date=2005&rft_id=https%3A%2F%2Fciteseerx.ist.psu.edu%2Fviewdoc%2Fsummary%3Fdoi%3D10.1.1.74.7308%23id-name%3DCiteSeerX&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A13566703%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1145%2F1086649.1086651&rft.aulast=Ben-Amram&rft.aufirst=A.+M.&rft_id=http%3A%2F%2Fwww2.mta.ac.il%2F~amirben%2Fdownloadable%2Fchurch-turing.ps&rfr_id=info%3Asid%2Fen.wikipedia.org%3AChurch%E2%80%93Turing+thesis" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBernsteinVazirani1997" class="citation journal cs1">Bernstein, E.; Vazirani, U. (1997). 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Turing, On Computable Numbers, with an Application to the Entscheidungsproblem". <i>Journal of Symbolic Logic</i>. <b>2</b> (1): 42–43. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2268810">10.2307/2268810</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2268810">2268810</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Symbolic+Logic&rft.atitle=Review%3A+A.+M.+Turing%2C+On+Computable+Numbers%2C+with+an+Application+to+the+Entscheidungsproblem&rft.volume=2&rft.issue=1&rft.pages=42-43&rft.date=1937-03&rft_id=info%3Adoi%2F10.2307%2F2268810&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2268810%23id-name%3DJSTOR&rft.aulast=Church&rft.aufirst=Alonzo&rfr_id=info%3Asid%2Fen.wikipedia.org%3AChurch%E2%80%93Turing+thesis" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFChurch1941" class="citation book cs1">Church, Alonzo (1941). <i>The Calculi of Lambda-Conversion</i>. Princeton: Princeton University Press.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Calculi+of+Lambda-Conversion&rft.place=Princeton&rft.pub=Princeton+University+Press&rft.date=1941&rft.aulast=Church&rft.aufirst=Alonzo&rfr_id=info%3Asid%2Fen.wikipedia.org%3AChurch%E2%80%93Turing+thesis" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCooperOdifreddi,_P.2003" class="citation book cs1">Cooper, S. B.; Odifreddi, P. (2003). "Incomputability in Nature". In S. B. Cooper; S. S. Goncharov (eds.). <i>Computability and Models: Perspectives East and West</i>. Kluwer Academic/Plenum Publishers. pp. 137–160.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Incomputability+in+Nature&rft.btitle=Computability+and+Models%3A+Perspectives+East+and+West&rft.pages=137-160&rft.pub=Kluwer+Academic%2FPlenum+Publishers&rft.date=2003&rft.aulast=Cooper&rft.aufirst=S.+B.&rft.au=Odifreddi%2C+P.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AChurch%E2%80%93Turing+thesis" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDavis1965" class="citation book cs1"><a href="/wiki/Martin_Davis_(mathematician)" title="Martin Davis (mathematician)">Davis, Martin</a>, ed. (1965). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/undecidablebasic0000davi"><i>The Undecidable, Basic Papers on Undecidable Propositions, Unsolvable Problems And Computable Functions</i></a></span>. New York: Raven Press.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Undecidable%2C+Basic+Papers+on+Undecidable+Propositions%2C+Unsolvable+Problems+And+Computable+Functions&rft.place=New+York&rft.pub=Raven+Press&rft.date=1965&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fundecidablebasic0000davi&rfr_id=info%3Asid%2Fen.wikipedia.org%3AChurch%E2%80%93Turing+thesis" class="Z3988"></span> Includes original papers by Gödel, Church, Turing, Rosser, Kleene, and Post mentioned in this section.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDawson1997" class="citation book cs1">Dawson, John W. Jr. (1997). <i>Logical Dilemmas: The Life and Work of Kurt Gödel</i>. Wellesley, Massachusetts, US: <a href="/wiki/A._K._Peters" class="mw-redirect" title="A. K. Peters">A. K. 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North-Holland Publishing Company. pp. 123–148.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Church%27s+Thesis+and+the+Principles+for+Mechanisms&rft.btitle=The+Kleene+Symposium&rft.pages=123-148&rft.pub=North-Holland+Publishing+Company&rft.date=1980&rft.aulast=Gandy&rft.aufirst=Robin&rfr_id=info%3Asid%2Fen.wikipedia.org%3AChurch%E2%80%93Turing+thesis" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGandy1994" class="citation book cs1">Gandy, Robin (1994). Herken, Rolf (ed.). <i>The universal Turing Machine: A Half-Century Survey</i>. New York: Wien Springer–Verlag. pp. 51ff. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-211-82637-9" title="Special:BookSources/978-3-211-82637-9"><bdi>978-3-211-82637-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+universal+Turing+Machine%3A+A+Half-Century+Survey&rft.place=New+York&rft.pages=51ff&rft.pub=Wien+Springer%E2%80%93Verlag&rft.date=1994&rft.isbn=978-3-211-82637-9&rft.aulast=Gandy&rft.aufirst=Robin&rfr_id=info%3Asid%2Fen.wikipedia.org%3AChurch%E2%80%93Turing+thesis" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGödel1965" class="citation book cs1">Gödel, Kurt (1965) [1934]. "On Undecidable Propositions of Formal Mathematical Systems". In Davis, Martin (ed.). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/undecidablebasic0000davi"><i>The Undecidable</i></a></span>. Kleene and Rosser (lecture note-takers); Institute for Advanced Study (lecture sponsor). New York: Raven Press.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=On+Undecidable+Propositions+of+Formal+Mathematical+Systems&rft.btitle=The+Undecidable&rft.place=New+York&rft.pub=Raven+Press&rft.date=1965&rft.aulast=G%C3%B6del&rft.aufirst=Kurt&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fundecidablebasic0000davi&rfr_id=info%3Asid%2Fen.wikipedia.org%3AChurch%E2%80%93Turing+thesis" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGödel1936" class="citation journal cs1 cs1-prop-foreign-lang-source">Gödel, Kurt (1936). "Über die Lāange von Beweisen" [On The Length of Proofs]. <i>Ergenbnisse Eines Mathematishen Kolloquiums</i> (in German) (7). Heft: 23–24.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Ergenbnisse+Eines+Mathematishen+Kolloquiums&rft.atitle=%C3%9Cber+die+L%C4%81ange+von+Beweisen&rft.issue=7&rft.pages=23-24&rft.date=1936&rft.aulast=G%C3%B6del&rft.aufirst=Kurt&rfr_id=info%3Asid%2Fen.wikipedia.org%3AChurch%E2%80%93Turing+thesis" class="Z3988"></span> Cited by <a href="#CITEREFKleene1952">Kleene (1952)</a>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGurevich1988" class="citation journal cs1"><a href="/wiki/Yuri_Gurevich" title="Yuri Gurevich">Gurevich, Yuri</a> (June 1988). "On Kolmogorov Machines and Related Issues". <i>Bulletin of European Association for Theoretical Computer Science</i> (35): 71–82.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Bulletin+of+European+Association+for+Theoretical+Computer+Science&rft.atitle=On+Kolmogorov+Machines+and+Related+Issues&rft.issue=35&rft.pages=71-82&rft.date=1988-06&rft.aulast=Gurevich&rft.aufirst=Yuri&rfr_id=info%3Asid%2Fen.wikipedia.org%3AChurch%E2%80%93Turing+thesis" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGurevich2000" class="citation journal cs1">Gurevich, Yuri (July 2000). <a rel="nofollow" class="external text" href="http://research.microsoft.com/~gurevich/Opera/141.pdf">"Sequential Abstract State Machines Capture Sequential Algorithms"</a> <span class="cs1-format">(PDF)</span>. <i>ACM Transactions on Computational Logic</i>. <b>1</b> (1): 77–111. <a href="/wiki/CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.146.3017">10.1.1.146.3017</a></span>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1145%2F343369.343384">10.1145/343369.343384</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:2031696">2031696</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20031016105643/http://research.microsoft.com/~gurevich/Opera/141.pdf">Archived</a> <span class="cs1-format">(PDF)</span> from the original on 2003-10-16.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=ACM+Transactions+on+Computational+Logic&rft.atitle=Sequential+Abstract+State+Machines+Capture+Sequential+Algorithms&rft.volume=1&rft.issue=1&rft.pages=77-111&rft.date=2000-07&rft_id=https%3A%2F%2Fciteseerx.ist.psu.edu%2Fviewdoc%2Fsummary%3Fdoi%3D10.1.1.146.3017%23id-name%3DCiteSeerX&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A2031696%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1145%2F343369.343384&rft.aulast=Gurevich&rft.aufirst=Yuri&rft_id=http%3A%2F%2Fresearch.microsoft.com%2F~gurevich%2FOpera%2F141.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AChurch%E2%80%93Turing+thesis" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHerbrand1932" class="citation journal cs1 cs1-prop-foreign-lang-source"><a href="/wiki/Jacques_Herbrand" title="Jacques Herbrand">Herbrand, Jacques</a> (1932). "Sur la non-contradiction de l'arithmétique". <i>Journal für die Reine und Angewandte Mathematik</i> (in French). <b>166</b>: 1–8. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1515%2Fcrll.1932.166.1">10.1515/crll.1932.166.1</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:116636410">116636410</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+f%C3%BCr+die+Reine+und+Angewandte+Mathematik&rft.atitle=Sur+la+non-contradiction+de+l%27arithm%C3%A9tique&rft.volume=166&rft.pages=1-8&rft.date=1932&rft_id=info%3Adoi%2F10.1515%2Fcrll.1932.166.1&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A116636410%23id-name%3DS2CID&rft.aulast=Herbrand&rft.aufirst=Jacques&rfr_id=info%3Asid%2Fen.wikipedia.org%3AChurch%E2%80%93Turing+thesis" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHofstadter1999" class="citation book cs1"><a href="/wiki/Douglas_Hofstadter" title="Douglas Hofstadter">Hofstadter, Douglas R.</a> (1999-02-05). "Chapter XVII: Church, Turing, Tarski, and Others". <a href="/wiki/G%C3%B6del,_Escher,_Bach:_an_Eternal_Golden_Braid" class="mw-redirect" title="Gödel, Escher, Bach: an Eternal Golden Braid"><i>Gödel, Escher, Bach: an Eternal Golden Braid</i></a> (Twentieth-anniversary ed.). <a href="/wiki/Basic_Books" title="Basic Books">Basic Books</a>. pp. 559–585. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-465-02656-7" title="Special:BookSources/0-465-02656-7"><bdi>0-465-02656-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Chapter+XVII%3A+Church%2C+Turing%2C+Tarski%2C+and+Others&rft.btitle=G%C3%B6del%2C+Escher%2C+Bach%3A+an+Eternal+Golden+Braid&rft.pages=559-585&rft.edition=Twentieth-anniversary&rft.pub=Basic+Books&rft.date=1999-02-05&rft.isbn=0-465-02656-7&rft.aulast=Hofstadter&rft.aufirst=Douglas+R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AChurch%E2%80%93Turing+thesis" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKleene1935" class="citation journal cs1"><a href="/wiki/Stephen_Cole_Kleene" title="Stephen Cole Kleene">Kleene, Stephen Cole</a> (January 1935). "A Theory of Positive Integers in Formal Logic". <i>American Journal of Mathematics</i>. <b>57</b> (1): 153–173 & 219–244. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2372027">10.2307/2372027</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2372027">2372027</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=American+Journal+of+Mathematics&rft.atitle=A+Theory+of+Positive+Integers+in+Formal+Logic&rft.volume=57&rft.issue=1&rft.pages=153-173+%26+219-244&rft.date=1935-01&rft_id=info%3Adoi%2F10.2307%2F2372027&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2372027%23id-name%3DJSTOR&rft.aulast=Kleene&rft.aufirst=Stephen+Cole&rfr_id=info%3Asid%2Fen.wikipedia.org%3AChurch%E2%80%93Turing+thesis" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKleene1936" class="citation journal cs1">Kleene, Stephen Cole (1936). "Lambda-Definability and Recursiveness". <i><a href="/wiki/Duke_Mathematical_Journal" title="Duke Mathematical Journal">Duke Mathematical Journal</a></i>. <b>2</b> (2): 340–353. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1215%2Fs0012-7094-36-00227-2">10.1215/s0012-7094-36-00227-2</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Duke+Mathematical+Journal&rft.atitle=Lambda-Definability+and+Recursiveness&rft.volume=2&rft.issue=2&rft.pages=340-353&rft.date=1936&rft_id=info%3Adoi%2F10.1215%2Fs0012-7094-36-00227-2&rft.aulast=Kleene&rft.aufirst=Stephen+Cole&rfr_id=info%3Asid%2Fen.wikipedia.org%3AChurch%E2%80%93Turing+thesis" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKleene1943" class="citation journal cs1">Kleene, Stephen Cole (1943). <a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F1990131">"Recursive Predicates and Quantifiers"</a>. <i>Transactions of the American Mathematical Society</i>. <b>53</b> (1): 41–73. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F1990131">10.2307/1990131</a></span>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/1990131">1990131</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Transactions+of+the+American+Mathematical+Society&rft.atitle=Recursive+Predicates+and+Quantifiers&rft.volume=53&rft.issue=1&rft.pages=41-73&rft.date=1943&rft_id=info%3Adoi%2F10.2307%2F1990131&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F1990131%23id-name%3DJSTOR&rft.aulast=Kleene&rft.aufirst=Stephen+Cole&rft_id=https%3A%2F%2Fdoi.org%2F10.2307%252F1990131&rfr_id=info%3Asid%2Fen.wikipedia.org%3AChurch%E2%80%93Turing+thesis" class="Z3988"></span> Reprinted in <a href="#CITEREFDavis1965"><i>The Undecidable</i></a>, p. 255ff. Kleene refined his definition of "general recursion" and proceeded in his chapter "12. Algorithmic theories" to posit "Thesis I" (p. 274); he would later repeat this thesis (in <a href="#CITEREFKleene1952">Kleene 1952</a>:300) and name it "Church's Thesis" (<a href="#CITEREFKleene1952">Kleene 1952</a>:317) (i.e., the <a href="/wiki/Church_thesis" class="mw-redirect" title="Church thesis">Church thesis</a>).</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKleene1952" class="citation book cs1">Kleene, Stephen Cole (1952). <i>Introduction to Metamathematics</i>. North-Holland. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/523942">523942</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Metamathematics&rft.pub=North-Holland&rft.date=1952&rft_id=info%3Aoclcnum%2F523942&rft.aulast=Kleene&rft.aufirst=Stephen+Cole&rfr_id=info%3Asid%2Fen.wikipedia.org%3AChurch%E2%80%93Turing+thesis" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKnuth1973" class="citation book cs1 cs1-prop-long-vol"><a href="/wiki/Donald_Knuth" title="Donald Knuth">Knuth, Donald</a> (1973). <i>The Art of Computer Programming</i>. Vol. 1/Fundamental Algorithms (2nd ed.). Addison–Wesley.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Art+of+Computer+Programming&rft.edition=2nd&rft.pub=Addison%E2%80%93Wesley&rft.date=1973&rft.aulast=Knuth&rft.aufirst=Donald&rfr_id=info%3Asid%2Fen.wikipedia.org%3AChurch%E2%80%93Turing+thesis" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKugel2005" class="citation journal cs1">Kugel, Peter (November 2005). "It's time to think outside the computational box". <i>Communications of the ACM</i>. <b>48</b> (11): 32–37. <a href="/wiki/CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.137.6939">10.1.1.137.6939</a></span>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1145%2F1096000.1096001">10.1145/1096000.1096001</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:29843806">29843806</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Communications+of+the+ACM&rft.atitle=It%27s+time+to+think+outside+the+computational+box&rft.volume=48&rft.issue=11&rft.pages=32-37&rft.date=2005-11&rft_id=https%3A%2F%2Fciteseerx.ist.psu.edu%2Fviewdoc%2Fsummary%3Fdoi%3D10.1.1.137.6939%23id-name%3DCiteSeerX&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A29843806%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1145%2F1096000.1096001&rft.aulast=Kugel&rft.aufirst=Peter&rfr_id=info%3Asid%2Fen.wikipedia.org%3AChurch%E2%80%93Turing+thesis" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLewis,_H.R.Papadimitriou,_C.H.1998" class="citation book cs1"><a href="/wiki/Harry_R._Lewis" title="Harry R. Lewis">Lewis, H.R.</a>; <a href="/wiki/Christos_H._Papadimitriou" class="mw-redirect" title="Christos H. Papadimitriou">Papadimitriou, C.H.</a> (1998). <i>Elements of the Theory of Computation</i>. Upper Saddle River, New Jersey, US: Prentice-Hall.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Elements+of+the+Theory+of+Computation&rft.place=Upper+Saddle+River%2C+New+Jersey%2C+US&rft.pub=Prentice-Hall&rft.date=1998&rft.au=Lewis%2C+H.R.&rft.au=Papadimitriou%2C+C.H.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AChurch%E2%80%93Turing+thesis" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFManna2003" class="citation book cs1"><a href="/wiki/Zohar_Manna" title="Zohar Manna">Manna, Zohar</a> (2003) [1974]. <i>Mathematical Theory of Computation</i>. Dover. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-43238-0" title="Special:BookSources/978-0-486-43238-0"><bdi>978-0-486-43238-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mathematical+Theory+of+Computation&rft.place=Dover&rft.date=2003&rft.isbn=978-0-486-43238-0&rft.aulast=Manna&rft.aufirst=Zohar&rfr_id=info%3Asid%2Fen.wikipedia.org%3AChurch%E2%80%93Turing+thesis" class="Z3988"></span><span class="cs1-maint citation-comment"><code class="cs1-code">{{<a href="/wiki/Template:Cite_book" title="Template:Cite book">cite book</a>}}</code>: CS1 maint: location missing publisher (<a href="/wiki/Category:CS1_maint:_location_missing_publisher" title="Category:CS1 maint: location missing publisher">link</a>)</span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMarkov1960" class="citation journal cs1"><a href="/wiki/Andrey_Markov_Jr." title="Andrey Markov Jr.">Markov, A. A.</a> (1960) [1954]. "The Theory of Algorithms". <i>American Mathematical Society Translations</i>. <b>2</b> (15): 1–14.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=American+Mathematical+Society+Translations&rft.atitle=The+Theory+of+Algorithms&rft.volume=2&rft.issue=15&rft.pages=1-14&rft.date=1960&rft.aulast=Markov&rft.aufirst=A.+A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AChurch%E2%80%93Turing+thesis" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFOlszewskiWoleńskiJanusz2006" class="citation book cs1">Olszewski, Adam; Woleński, Jan; Janusz, Robert, eds. (2006). <i>Church's Thesis After 70 Years</i>. Frankfurt: Ontos. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-938793-09-1" title="Special:BookSources/978-3-938793-09-1"><bdi>978-3-938793-09-1</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/909679288">909679288</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Church%27s+Thesis+After+70+Years&rft.place=Frankfurt&rft.pub=Ontos&rft.date=2006&rft_id=info%3Aoclcnum%2F909679288&rft.isbn=978-3-938793-09-1&rfr_id=info%3Asid%2Fen.wikipedia.org%3AChurch%E2%80%93Turing+thesis" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPour-El,_M._B.Richards,_J.I.1989" class="citation book cs1"><a href="/wiki/Marian_Pour-El" title="Marian Pour-El">Pour-El, M. B.</a>; Richards, J.I. (1989). <a href="/wiki/Computability_in_Analysis_and_Physics" title="Computability in Analysis and Physics"><i>Computability in Analysis and Physics</i></a>. <a href="/wiki/Springer_Verlag" class="mw-redirect" title="Springer Verlag">Springer Verlag</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Computability+in+Analysis+and+Physics&rft.pub=Springer+Verlag&rft.date=1989&rft.au=Pour-El%2C+M.+B.&rft.au=Richards%2C+J.I.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AChurch%E2%80%93Turing+thesis" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRosser1939" class="citation journal cs1"><a href="/wiki/J._B._Rosser" class="mw-redirect" title="J. B. Rosser">Rosser, J. B.</a> (1939). "An Informal Exposition of Proofs of Godel's Theorem and Church's Theorem". <i>The Journal of Symbolic Logic</i>. <b>4</b> (2): 53–60. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2269059">10.2307/2269059</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2269059">2269059</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:39499392">39499392</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Journal+of+Symbolic+Logic&rft.atitle=An+Informal+Exposition+of+Proofs+of+Godel%27s+Theorem+and+Church%27s+Theorem&rft.volume=4&rft.issue=2&rft.pages=53-60&rft.date=1939&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A39499392%23id-name%3DS2CID&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2269059%23id-name%3DJSTOR&rft_id=info%3Adoi%2F10.2307%2F2269059&rft.aulast=Rosser&rft.aufirst=J.+B.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AChurch%E2%80%93Turing+thesis" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSiegSommerTalcott2002" class="citation book cs1">Sieg, Wilfried; Sommer, Richard; Talcott, Carolyn, eds. (2002). <i>Reflections on the Foundations of Mathematics: Essays in Honor of Solomon Feferman</i>. Lecture Notes in Logic. Vol. 15. <a href="/wiki/A._K._Peters,_Ltd." class="mw-redirect" title="A. K. Peters, Ltd.">A. K. Peters, Ltd.</a> <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-56881-169-7" title="Special:BookSources/978-1-56881-169-7"><bdi>978-1-56881-169-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Reflections+on+the+Foundations+of+Mathematics%3A+Essays+in+Honor+of+Solomon+Feferman&rft.series=Lecture+Notes+in+Logic&rft.pub=A.+K.+Peters%2C+Ltd.&rft.date=2002&rft.isbn=978-1-56881-169-7&rfr_id=info%3Asid%2Fen.wikipedia.org%3AChurch%E2%80%93Turing+thesis" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSyropoulos2008" class="citation book cs1">Syropoulos, Apostolos (2008). <i>Hypercomputation: Computing Beyond the Church–Turing Barrier</i>. Springer. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-30886-9" title="Special:BookSources/978-0-387-30886-9"><bdi>978-0-387-30886-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Hypercomputation%3A+Computing+Beyond+the+Church%E2%80%93Turing+Barrier&rft.pub=Springer&rft.date=2008&rft.isbn=978-0-387-30886-9&rft.aulast=Syropoulos&rft.aufirst=Apostolos&rfr_id=info%3Asid%2Fen.wikipedia.org%3AChurch%E2%80%93Turing+thesis" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTuring1937a" class="citation cs2"><a href="/wiki/Alan_Turing" title="Alan Turing">Turing, A. M.</a> (1937a) [Delivered to the Society November 1936], <a rel="nofollow" class="external text" href="http://www.comlab.ox.ac.uk/activities/ieg/e-library/sources/tp2-ie.pdf">"On Computable Numbers, with an Application to the Entscheidungsproblem"</a> <span class="cs1-format">(PDF)</span>, <i>Proceedings of the London Mathematical Society</i>, 2, vol. 42, pp. 230–265, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1112%2Fplms%2Fs2-42.1.230">10.1112/plms/s2-42.1.230</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:73712">73712</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Proceedings+of+the+London+Mathematical+Society&rft.atitle=On+Computable+Numbers%2C+with+an+Application+to+the+Entscheidungsproblem&rft.volume=42&rft.pages=230-265&rft.date=1937&rft_id=info%3Adoi%2F10.1112%2Fplms%2Fs2-42.1.230&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A73712%23id-name%3DS2CID&rft.aulast=Turing&rft.aufirst=A.+M.&rft_id=http%3A%2F%2Fwww.comlab.ox.ac.uk%2Factivities%2Fieg%2Fe-library%2Fsources%2Ftp2-ie.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AChurch%E2%80%93Turing+thesis" class="Z3988"></span> and <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTuring1938" class="citation news cs1">Turing, A. M. (1938). "On Computable Numbers, with an Application to the Entscheidungsproblem: A correction". <i>Proceedings of the London Mathematical Society</i>. 2. Vol. 43 (published 1937). pp. 544–546. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1112%2Fplms%2Fs2-43.6.544">10.1112/plms/s2-43.6.544</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Proceedings+of+the+London+Mathematical+Society&rft.atitle=On+Computable+Numbers%2C+with+an+Application+to+the+Entscheidungsproblem%3A+A+correction&rft.volume=43&rft.pages=544-546&rft.date=1938&rft_id=info%3Adoi%2F10.1112%2Fplms%2Fs2-43.6.544&rft.aulast=Turing&rft.aufirst=A.+M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AChurch%E2%80%93Turing+thesis" class="Z3988"></span> (See also: <a href="#CITEREFDavis1965">Davis 1965</a>:115ff.)</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTuring1937b" class="citation journal cs1">Turing, Alan Mathison (December 1937b). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20200809215242/http://pdfs.semanticscholar.org/ee8c/779e7823814a5f1746d883ca77b26671b617.pdf">"Computability and λ-Definability"</a> <span class="cs1-format">(PDF)</span>. <i>Journal of Symbolic Logic</i>. <b>2</b> (4): 153–163. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2268280">10.2307/2268280</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2268280">2268280</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:2317046">2317046</a>. Archived from <a rel="nofollow" class="external text" href="http://pdfs.semanticscholar.org/ee8c/779e7823814a5f1746d883ca77b26671b617.pdf">the original</a> <span class="cs1-format">(PDF)</span> on 2020-08-09.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Symbolic+Logic&rft.atitle=Computability+and+%CE%BB-Definability&rft.volume=2&rft.issue=4&rft.pages=153-163&rft.date=1937-12&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A2317046%23id-name%3DS2CID&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2268280%23id-name%3DJSTOR&rft_id=info%3Adoi%2F10.2307%2F2268280&rft.aulast=Turing&rft.aufirst=Alan+Mathison&rft_id=http%3A%2F%2Fpdfs.semanticscholar.org%2Fee8c%2F779e7823814a5f1746d883ca77b26671b617.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AChurch%E2%80%93Turing+thesis" 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=Church%E2%80%93Turing_thesis&action=edit&section=14" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="https://plato.stanford.edu/entries/church-turing/">"The Church–Turing Thesis"</a> entry by <a href="/wiki/Jack_Copeland" title="Jack Copeland">B. Jack Copeland</a> in the <i><a href="/wiki/Stanford_Encyclopedia_of_Philosophy" title="Stanford Encyclopedia of Philosophy">Stanford Encyclopedia of Philosophy</a></i>.</li> <li><a rel="nofollow" class="external text" href="https://plato.stanford.edu/entries/computation-physicalsystems/">"Computation in Physical Systems"</a> entry by <a href="/wiki/Gualtiero_Piccinini" title="Gualtiero Piccinini">Gualtiero Piccinini</a> in the <i><a href="/wiki/Stanford_Encyclopedia_of_Philosophy" title="Stanford Encyclopedia of Philosophy">Stanford Encyclopedia of Philosophy</a></i>—a comprehensive philosophical treatment of relevant issues.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKaznatcheev2014" class="citation journal cs1">Kaznatcheev, Artem (2014-09-11). <a rel="nofollow" class="external text" href="https://egtheory.wordpress.com/2014/09/11/transcendental-idealism-and-posts-variant-of-the-church-turing-thesis/">"Transcendental idealism and Post's variant of the Church-Turing thesis"</a>. <i>Journal of Symbolic Logic</i>. <b>1</b> (3): 103–105.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Symbolic+Logic&rft.atitle=Transcendental+idealism+and+Post%27s+variant+of+the+Church-Turing+thesis&rft.volume=1&rft.issue=3&rft.pages=103-105&rft.date=2014-09-11&rft.aulast=Kaznatcheev&rft.aufirst=Artem&rft_id=https%3A%2F%2Fegtheory.wordpress.com%2F2014%2F09%2F11%2Ftranscendental-idealism-and-posts-variant-of-the-church-turing-thesis%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AChurch%E2%80%93Turing+thesis" class="Z3988"></span></li> <li>A <a rel="nofollow" class="external text" href="https://projecteuclid.org/euclid.ndjfl/1093637642">special issue</a> (Vol. 28, No. 4, 1987) of the <i><a href="/wiki/Notre_Dame_Journal_of_Formal_Logic" title="Notre Dame Journal of Formal Logic">Notre Dame Journal of Formal Logic</a></i> was devoted to the Church–Turing thesis.</li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist .mw-empty-li{display:none}.mw-parser-output .hlist dt::after{content:": "}.mw-parser-output .hlist dd::after,.mw-parser-output .hlist li::after{content:" · ";font-weight:bold}.mw-parser-output .hlist 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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/Lambda_calculus" title="Lambda calculus">Lambda calculus</a> <ul><li><a href="/wiki/Simply_typed_lambda_calculus" title="Simply typed lambda calculus">Simply typed lambda calculus</a></li> <li><a class="mw-selflink selflink">Church–Turing thesis</a></li> <li><a href="/wiki/Church_encoding" title="Church encoding">Church encoding</a></li></ul></li> <li><a href="/wiki/Frege%E2%80%93Church_ontology" title="Frege–Church ontology">Frege–Church ontology</a></li> <li><a href="/wiki/Church%E2%80%93Rosser_theorem" title="Church–Rosser theorem">Church–Rosser theorem</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Students</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/Alan_Turing" title="Alan Turing">Alan Turing</a></li> <li><a href="/wiki/C._Anthony_Anderson" title="C. Anthony Anderson">C. Anthony Anderson</a></li> <li><a href="/wiki/Peter_B._Andrews" title="Peter B. Andrews">Peter Andrews</a></li> <li><a href="/wiki/George_Alfred_Barnard" title="George Alfred Barnard">George Alfred Barnard</a></li> <li><a href="/wiki/William_Boone_(mathematician)" title="William Boone (mathematician)">William Boone</a></li> <li><a href="/wiki/Martin_Davis_(mathematician)" title="Martin Davis (mathematician)">Martin Davis</a></li> <li><a href="/wiki/William_Bigelow_Easton" title="William Bigelow Easton">William Easton</a></li> <li><a href="/wiki/Alfred_Foster_(mathematician)" title="Alfred Foster (mathematician)">Alfred Foster</a></li> <li><a href="/wiki/Leon_Henkin" title="Leon Henkin">Leon Henkin</a></li> <li><a href="/wiki/John_George_Kemeny" class="mw-redirect" title="John George Kemeny">John George Kemeny</a></li> <li><a href="/wiki/Stephen_Cole_Kleene" title="Stephen Cole Kleene">Stephen Cole Kleene</a></li> <li><a href="/wiki/Simon_B._Kochen" title="Simon B. Kochen">Simon B. Kochen</a></li> <li><a href="/wiki/Maurice_L%27Abb%C3%A9" title="Maurice L'Abbé">Maurice L'Abbé</a></li> <li><a href="/wiki/Gary_R._Mar" title="Gary R. Mar">Gary R. Mar</a></li> <li><a href="/wiki/Michael_O._Rabin" title="Michael O. Rabin">Michael O. Rabin</a></li> <li><a href="/wiki/Nicholas_Rescher" title="Nicholas Rescher">Nicholas Rescher</a></li> <li><a href="/wiki/Hartley_Rogers,_Jr" class="mw-redirect" title="Hartley Rogers, Jr">Hartley Rogers, Jr</a></li> <li><a href="/wiki/J._Barkley_Rosser" title="J. Barkley Rosser">J. Barkley Rosser</a></li> <li><a href="/wiki/Dana_Scott" title="Dana Scott">Dana Scott</a></li> <li><a href="/wiki/Norman_Shapiro" title="Norman Shapiro">Norman Shapiro</a></li> <li><a href="/wiki/Raymond_Smullyan" title="Raymond Smullyan">Raymond Smullyan</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Institutions</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/Princeton_University" title="Princeton University">Princeton University</a></li> <li><a href="/wiki/University_of_California,_Los_Angeles" title="University of California, Los Angeles">University of California, Los Angeles</a></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-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Alonzo_Church_(college_president)" title="Alonzo Church (college president)">Alonzo Church (college president)</a></li> <li><a href="/wiki/A._C._Croom" title="A. C. Croom">A. C. Croom</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"></div><div role="navigation" class="navbox" aria-labelledby="Mathematical_logic" style="padding:3px"><table class="nowraplinks mw-collapsible mw-collapsed navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Mathematical_logic" title="Template:Mathematical logic"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Mathematical_logic" title="Template talk:Mathematical logic"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Mathematical_logic" title="Special:EditPage/Template:Mathematical logic"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Mathematical_logic" style="font-size:114%;margin:0 4em"><a href="/wiki/Mathematical_logic" title="Mathematical logic">Mathematical logic</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">General</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Axiom" title="Axiom">Axiom</a> <ul><li><a href="/wiki/List_of_axioms" title="List of axioms">list</a></li></ul></li> <li><a href="/wiki/Cardinality" title="Cardinality">Cardinality</a></li> <li><a href="/wiki/First-order_logic" title="First-order logic">First-order logic</a></li> <li><a href="/wiki/Formal_proof" title="Formal proof">Formal proof</a></li> <li><a href="/wiki/Formal_semantics_(logic)" class="mw-redirect" title="Formal semantics (logic)">Formal semantics</a></li> <li><a href="/wiki/Foundations_of_mathematics" title="Foundations of mathematics">Foundations of mathematics</a></li> <li><a href="/wiki/Information_theory" title="Information theory">Information theory</a></li> <li><a href="/wiki/Lemma_(mathematics)" title="Lemma (mathematics)">Lemma</a></li> <li><a href="/wiki/Logical_consequence" title="Logical consequence">Logical consequence</a></li> <li><a href="/wiki/Structure_(mathematical_logic)" title="Structure (mathematical logic)">Model</a></li> <li><a href="/wiki/Theorem" title="Theorem">Theorem</a></li> <li><a href="/wiki/Theory_(mathematical_logic)" title="Theory (mathematical logic)">Theory</a></li> <li><a href="/wiki/Type_theory" title="Type theory">Type theory</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Theorems (<a href="/wiki/Category:Theorems_in_the_foundations_of_mathematics" title="Category:Theorems in the foundations of mathematics">list</a>)<br /> and <a href="/wiki/Paradoxes_of_set_theory" title="Paradoxes of set theory">paradoxes</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/G%C3%B6del%27s_completeness_theorem" title="Gödel's completeness theorem">Gödel's completeness</a> and <a href="/wiki/G%C3%B6del%27s_incompleteness_theorems" title="Gödel's incompleteness theorems">incompleteness theorems</a></li> <li><a href="/wiki/Tarski%27s_undefinability_theorem" title="Tarski's undefinability theorem">Tarski's undefinability</a></li> <li><a href="/wiki/Banach%E2%80%93Tarski_paradox" title="Banach–Tarski paradox">Banach–Tarski paradox</a></li> <li>Cantor's <a href="/wiki/Cantor%27s_theorem" title="Cantor's theorem">theorem,</a> <a href="/wiki/Cantor%27s_paradox" title="Cantor's paradox">paradox</a> and <a 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 href="/wiki/Russell%27s_paradox" title="Russell's paradox">Russell's paradox</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Logic" title="Logic">Logics</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th id="Traditional" scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Term_logic" title="Term logic">Traditional</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Classical_logic" title="Classical logic">Classical logic</a></li> <li><a href="/wiki/Logical_truth" title="Logical truth">Logical truth</a></li> <li><a href="/wiki/Tautology_(logic)" title="Tautology (logic)">Tautology</a></li> <li><a href="/wiki/Proposition" title="Proposition">Proposition</a></li> <li><a href="/wiki/Inference" title="Inference">Inference</a></li> <li><a href="/wiki/Logical_equivalence" title="Logical equivalence">Logical equivalence</a></li> <li><a href="/wiki/Consistency" title="Consistency">Consistency</a> <ul><li><a href="/wiki/Equiconsistency" title="Equiconsistency">Equiconsistency</a></li></ul></li> <li><a href="/wiki/Argument" title="Argument">Argument</a></li> <li><a href="/wiki/Soundness" title="Soundness">Soundness</a></li> <li><a href="/wiki/Validity_(logic)" title="Validity (logic)">Validity</a></li> <li><a href="/wiki/Syllogism" title="Syllogism">Syllogism</a></li> <li><a href="/wiki/Square_of_opposition" title="Square of opposition">Square of opposition</a></li> <li><a href="/wiki/Venn_diagram" title="Venn diagram">Venn diagram</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Propositional_calculus" title="Propositional calculus">Propositional</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Boolean_algebra" title="Boolean algebra">Boolean algebra</a></li> <li><a href="/wiki/Boolean_function" title="Boolean function">Boolean functions</a></li> <li><a href="/wiki/Logical_connective" title="Logical connective">Logical connectives</a></li> <li><a href="/wiki/Propositional_calculus" title="Propositional calculus">Propositional calculus</a></li> <li><a href="/wiki/Propositional_formula" title="Propositional formula">Propositional formula</a></li> <li><a href="/wiki/Truth_table" title="Truth table">Truth tables</a></li> <li><a href="/wiki/Many-valued_logic" title="Many-valued logic">Many-valued logic</a> <ul><li><a href="/wiki/Three-valued_logic" title="Three-valued logic">3</a></li> <li><a href="/wiki/Finite-valued_logic" title="Finite-valued logic">finite</a></li> <li><a href="/wiki/Infinite-valued_logic" title="Infinite-valued logic">∞</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Predicate_logic" class="mw-redirect" title="Predicate logic">Predicate</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/First-order_logic" title="First-order logic">First-order</a> <ul><li><a href="/wiki/List_of_first-order_theories" title="List of first-order theories"><span style="font-size:85%;">list</span></a></li></ul></li> <li><a href="/wiki/Second-order_logic" title="Second-order logic">Second-order</a> <ul><li><a href="/wiki/Monadic_second-order_logic" title="Monadic second-order logic">Monadic</a></li></ul></li> <li><a href="/wiki/Higher-order_logic" title="Higher-order logic">Higher-order</a></li> <li><a href="/wiki/Fixed-point_logic" title="Fixed-point logic">Fixed-point</a></li> <li><a href="/wiki/Free_logic" title="Free logic">Free</a></li> <li><a href="/wiki/Quantifier_(logic)" title="Quantifier (logic)">Quantifiers</a></li> <li><a href="/wiki/Predicate_(mathematical_logic)" title="Predicate (mathematical logic)">Predicate</a></li> <li><a href="/wiki/Monadic_predicate_calculus" title="Monadic predicate calculus">Monadic predicate calculus</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Set_theory" title="Set theory">Set theory</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><td colspan="2" class="navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Zermelo%E2%80%93Fraenkel_set_theory" title="Zermelo–Fraenkel set theory">Set</a> <ul><li><a href="/wiki/Hereditary_set" title="Hereditary set">hereditary</a></li></ul></li> <li><a href="/wiki/Class_(set_theory)" title="Class (set theory)">Class</a></li> <li>(<a href="/wiki/Urelement" title="Urelement">Ur-</a>)<a href="/wiki/Element_(mathematics)" title="Element (mathematics)">Element</a></li> <li><a href="/wiki/Ordinal_number" title="Ordinal number">Ordinal number</a></li> <li><a href="/wiki/Extensionality" title="Extensionality">Extensionality</a></li> <li><a href="/wiki/Forcing_(mathematics)" title="Forcing (mathematics)">Forcing</a></li> <li><a href="/wiki/Relation_(mathematics)" title="Relation (mathematics)">Relation</a> <ul><li><a href="/wiki/Equivalence_relation" title="Equivalence relation">equivalence</a></li> <li><a href="/wiki/Partition_of_a_set" title="Partition of a set">partition</a></li></ul></li> <li>Set operations: <ul><li><a href="/wiki/Intersection_(set_theory)" title="Intersection (set theory)">intersection</a></li> <li><a href="/wiki/Union_(set_theory)" title="Union (set theory)">union</a></li> <li><a href="/wiki/Complement_(set_theory)" title="Complement (set theory)">complement</a></li> <li><a href="/wiki/Cartesian_product" title="Cartesian product">Cartesian product</a></li> <li><a href="/wiki/Power_set" title="Power set">power set</a></li> <li><a href="/wiki/List_of_set_identities_and_relations" title="List of set identities and relations">identities</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Types of <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">sets</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Countable_set" title="Countable set">Countable</a></li> <li><a href="/wiki/Uncountable_set" title="Uncountable set">Uncountable</a></li> <li><a href="/wiki/Empty_set" title="Empty set">Empty</a></li> <li><a href="/wiki/Inhabited_set" title="Inhabited set">Inhabited</a></li> <li><a href="/wiki/Singleton_(mathematics)" title="Singleton (mathematics)">Singleton</a></li> <li><a href="/wiki/Finite_set" title="Finite set">Finite</a></li> <li><a href="/wiki/Infinite_set" title="Infinite set">Infinite</a></li> <li><a href="/wiki/Transitive_set" title="Transitive set">Transitive</a></li> <li><a href="/wiki/Ultrafilter_(set_theory)" class="mw-redirect" title="Ultrafilter (set theory)">Ultrafilter</a></li> <li><a href="/wiki/Recursive_set" class="mw-redirect" title="Recursive set">Recursive</a></li> <li><a href="/wiki/Fuzzy_set" title="Fuzzy set">Fuzzy</a></li> <li><a href="/wiki/Universal_set" title="Universal set">Universal</a></li> <li><a href="/wiki/Universe_(mathematics)" title="Universe (mathematics)">Universe</a> <ul><li><a href="/wiki/Constructible_universe" title="Constructible universe">constructible</a></li> <li><a href="/wiki/Grothendieck_universe" title="Grothendieck universe">Grothendieck</a></li> <li><a href="/wiki/Von_Neumann_universe" title="Von Neumann universe">Von Neumann</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Map_(mathematics)" title="Map (mathematics)">Maps</a> and <a href="/wiki/Cardinality" title="Cardinality">cardinality</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Function_(mathematics)" title="Function (mathematics)">Function</a>/<a href="/wiki/Map_(mathematics)" title="Map (mathematics)">Map</a> <ul><li><a href="/wiki/Domain_of_a_function" title="Domain of a function">domain</a></li> <li><a href="/wiki/Codomain" title="Codomain">codomain</a></li> <li><a href="/wiki/Image_(mathematics)" title="Image (mathematics)">image</a></li></ul></li> <li><a href="/wiki/Injective_function" title="Injective function">In</a>/<a href="/wiki/Surjective_function" title="Surjective function">Sur</a>/<a href="/wiki/Bijection" title="Bijection">Bi</a>-jection</li> <li><a href="/wiki/Schr%C3%B6der%E2%80%93Bernstein_theorem" title="Schröder–Bernstein theorem">Schröder–Bernstein theorem</a></li> <li><a href="/wiki/Isomorphism" title="Isomorphism">Isomorphism</a></li> <li><a href="/wiki/G%C3%B6del_numbering" title="Gödel numbering">Gödel numbering</a></li> <li><a href="/wiki/Enumeration" title="Enumeration">Enumeration</a></li> <li><a href="/wiki/Large_cardinal" title="Large cardinal">Large cardinal</a> <ul><li><a href="/wiki/Inaccessible_cardinal" title="Inaccessible cardinal">inaccessible</a></li></ul></li> <li><a href="/wiki/Aleph_number" title="Aleph number">Aleph number</a></li> <li><a href="/wiki/Operation_(mathematics)" title="Operation (mathematics)">Operation</a> <ul><li><a href="/wiki/Binary_operation" title="Binary operation">binary</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Set theories</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Zermelo%E2%80%93Fraenkel_set_theory" title="Zermelo–Fraenkel set theory">Zermelo–Fraenkel</a> <ul><li><a href="/wiki/Axiom_of_choice" title="Axiom of choice">axiom of choice</a></li> <li><a href="/wiki/Continuum_hypothesis" title="Continuum hypothesis">continuum hypothesis</a></li></ul></li> <li><a href="/wiki/General_set_theory" title="General set theory">General</a></li> <li><a href="/wiki/Kripke%E2%80%93Platek_set_theory" title="Kripke–Platek set theory">Kripke–Platek</a></li> <li><a href="/wiki/Morse%E2%80%93Kelley_set_theory" title="Morse–Kelley set theory">Morse–Kelley</a></li> <li><a href="/wiki/Naive_set_theory" title="Naive set theory">Naive</a></li> <li><a href="/wiki/New_Foundations" title="New Foundations">New Foundations</a></li> <li><a href="/wiki/Tarski%E2%80%93Grothendieck_set_theory" title="Tarski–Grothendieck set theory">Tarski–Grothendieck</a></li> <li><a href="/wiki/Von_Neumann%E2%80%93Bernays%E2%80%93G%C3%B6del_set_theory" title="Von Neumann–Bernays–Gödel set theory">Von Neumann–Bernays–Gödel</a></li> <li><a href="/wiki/Ackermann_set_theory" title="Ackermann set theory">Ackermann</a></li> <li><a href="/wiki/Constructive_set_theory" title="Constructive set theory">Constructive</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Formal_system" title="Formal system">Formal systems</a> (<a href="/wiki/List_of_formal_systems" title="List of formal systems"><span style="font-size:85%;">list</span></a>),<br /><a href="/wiki/Formal_language" title="Formal language">language</a> and <a href="/wiki/Syntax_(logic)" title="Syntax (logic)">syntax</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><td colspan="2" class="navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Alphabet_(formal_languages)" title="Alphabet (formal languages)">Alphabet</a></li> <li><a href="/wiki/Arity" title="Arity">Arity</a></li> <li><a href="/wiki/Automata_theory" title="Automata theory">Automata</a></li> <li><a href="/wiki/Axiom_schema" title="Axiom schema">Axiom schema</a></li> <li><a href="/wiki/Expression_(mathematics)" title="Expression (mathematics)">Expression</a> <ul><li><a href="/wiki/Ground_expression" title="Ground expression">ground</a></li></ul></li> <li><a href="/wiki/Extension_by_new_constant_and_function_names" title="Extension by new constant and function names">Extension</a> <ul><li><a href="/wiki/Extension_by_definitions" title="Extension by definitions">by definition</a></li> <li><a href="/wiki/Conservative_extension" title="Conservative extension">conservative</a></li></ul></li> <li><a href="/wiki/Finitary_relation" title="Finitary relation">Relation</a></li> <li><a href="/wiki/Formation_rule" title="Formation rule">Formation rule</a></li> <li><a href="/wiki/Formal_grammar" title="Formal grammar">Grammar</a></li> <li><a href="/wiki/Well-formed_formula" title="Well-formed formula">Formula</a> <ul><li><a href="/wiki/Atomic_formula" title="Atomic formula">atomic</a></li> <li><a href="/wiki/Sentence_(mathematical_logic)" title="Sentence (mathematical logic)">closed</a></li> <li><a href="/wiki/Ground_formula" class="mw-redirect" title="Ground formula">ground</a></li> <li><a href="/wiki/Open_formula" title="Open formula">open</a></li></ul></li> <li><a href="/wiki/Free_variables_and_bound_variables" title="Free variables and bound variables">Free/bound variable</a></li> <li><a href="/wiki/Formal_language" title="Formal language">Language</a></li> <li><a href="/wiki/Metalanguage" title="Metalanguage">Metalanguage</a></li> <li><a href="/wiki/Logical_connective" title="Logical connective">Logical connective</a> <ul><li><a href="/wiki/Negation" title="Negation">¬</a></li> <li><a href="/wiki/Logical_disjunction" title="Logical disjunction">∨</a></li> <li><a href="/wiki/Logical_conjunction" title="Logical conjunction">∧</a></li> <li><a href="/wiki/Material_conditional" title="Material conditional">→</a></li> <li><a href="/wiki/Logical_biconditional" title="Logical biconditional">↔</a></li> <li><a href="/wiki/Logical_equality" title="Logical equality">=</a></li></ul></li> <li><a href="/wiki/Predicate_(mathematical_logic)" title="Predicate (mathematical logic)">Predicate</a> <ul><li><a href="/wiki/Functional_predicate" title="Functional predicate">functional</a></li> <li><a href="/wiki/Predicate_variable" title="Predicate variable">variable</a></li> <li><a href="/wiki/Propositional_variable" title="Propositional variable">propositional variable</a></li></ul></li> <li><a href="/wiki/Formal_proof" title="Formal proof">Proof</a></li> <li><a href="/wiki/Quantifier_(logic)" title="Quantifier (logic)">Quantifier</a> <ul><li><a href="/wiki/Existential_quantification" title="Existential quantification">∃</a></li> <li><a href="/wiki/Uniqueness_quantification" title="Uniqueness quantification">!</a></li> <li><a href="/wiki/Universal_quantification" title="Universal quantification">∀</a></li> <li><a href="/wiki/Quantifier_rank" title="Quantifier rank">rank</a></li></ul></li> <li><a href="/wiki/Sentence_(mathematical_logic)" title="Sentence (mathematical logic)">Sentence</a> <ul><li><a href="/wiki/Atomic_sentence" title="Atomic sentence">atomic</a></li> <li><a href="/wiki/Spectrum_of_a_sentence" title="Spectrum of a sentence">spectrum</a></li></ul></li> <li><a href="/wiki/Signature_(logic)" title="Signature (logic)">Signature</a></li> <li><a href="/wiki/String_(formal_languages)" class="mw-redirect" title="String (formal languages)">String</a></li> <li><a href="/wiki/Substitution_(logic)" title="Substitution (logic)">Substitution</a></li> <li><a href="/wiki/Symbol_(formal)" title="Symbol (formal)">Symbol</a> <ul><li><a href="/wiki/Uninterpreted_function" title="Uninterpreted function">function</a></li> <li><a href="/wiki/Logical_constant" title="Logical constant">logical/constant</a></li> <li><a href="/wiki/Non-logical_symbol" title="Non-logical symbol">non-logical</a></li> <li><a href="/wiki/Variable_(mathematics)" title="Variable (mathematics)">variable</a></li></ul></li> <li><a href="/wiki/Term_(logic)" title="Term (logic)">Term</a></li> <li><a href="/wiki/Theory_(mathematical_logic)" title="Theory (mathematical logic)">Theory</a> <ul><li><a href="/wiki/List_of_mathematical_theories" title="List of mathematical theories"><span style="font-size:85%;">list</span></a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><span class="nowrap">Example <a href="/wiki/Axiomatic_system" title="Axiomatic system">axiomatic<br />systems</a> <span style="font-size:85%;">(<a href="/wiki/List_of_first-order_theories" title="List of first-order theories">list</a>)</span></span></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li>of <a href="/wiki/True_arithmetic" title="True arithmetic">arithmetic</a>: <ul><li><a href="/wiki/Peano_axioms" title="Peano axioms">Peano</a></li> <li><a href="/wiki/Second-order_arithmetic" title="Second-order arithmetic">second-order</a></li> <li><a href="/wiki/Elementary_function_arithmetic" title="Elementary function arithmetic">elementary function</a></li> <li><a href="/wiki/Primitive_recursive_arithmetic" title="Primitive recursive arithmetic">primitive recursive</a></li> <li><a href="/wiki/Robinson_arithmetic" title="Robinson arithmetic">Robinson</a></li> <li><a href="/wiki/Skolem_arithmetic" title="Skolem arithmetic">Skolem</a></li></ul></li> <li>of the <a href="/wiki/Construction_of_the_real_numbers" title="Construction of the real numbers">real numbers</a> <ul><li><a href="/wiki/Tarski%27s_axiomatization_of_the_reals" title="Tarski's axiomatization of the reals">Tarski's axiomatization</a></li></ul></li> <li>of <a href="/wiki/Axiomatization_of_Boolean_algebras" class="mw-redirect" title="Axiomatization of Boolean algebras">Boolean algebras</a> <ul><li><a href="/wiki/Boolean_algebras_canonically_defined" title="Boolean algebras canonically defined">canonical</a></li> <li><a href="/wiki/Minimal_axioms_for_Boolean_algebra" title="Minimal axioms for Boolean algebra">minimal axioms</a></li></ul></li> <li>of <a href="/wiki/Foundations_of_geometry" title="Foundations of geometry">geometry</a>: <ul><li><a href="/wiki/Euclidean_geometry" title="Euclidean geometry">Euclidean</a>: <ul><li><a href="/wiki/Euclid%27s_Elements" title="Euclid's Elements"><i>Elements</i></a></li> <li><a href="/wiki/Hilbert%27s_axioms" title="Hilbert's axioms">Hilbert's</a></li> <li><a href="/wiki/Tarski%27s_axioms" title="Tarski's axioms">Tarski's</a></li></ul></li> <li><a href="/wiki/Non-Euclidean_geometry" title="Non-Euclidean geometry">non-Euclidean</a></li></ul></li></ul> <ul><li><i><a href="/wiki/Principia_Mathematica" title="Principia Mathematica">Principia Mathematica</a></i></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Proof_theory" title="Proof theory">Proof theory</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Formal_proof" title="Formal proof">Formal proof</a></li> <li><a href="/wiki/Natural_deduction" title="Natural deduction">Natural deduction</a></li> <li><a href="/wiki/Logical_consequence" title="Logical consequence">Logical consequence</a></li> <li><a href="/wiki/Rule_of_inference" title="Rule of inference">Rule of inference</a></li> <li><a href="/wiki/Sequent_calculus" title="Sequent calculus">Sequent calculus</a></li> <li><a href="/wiki/Theorem" title="Theorem">Theorem</a></li> <li><a href="/wiki/Formal_system" title="Formal system">Systems</a> <ul><li><a href="/wiki/Axiomatic_system" title="Axiomatic system">axiomatic</a></li> <li><a href="/wiki/Deductive_system" class="mw-redirect" title="Deductive system">deductive</a></li> <li><a href="/wiki/Hilbert_system" title="Hilbert system">Hilbert</a> <ul><li><a href="/wiki/List_of_Hilbert_systems" class="mw-redirect" title="List of Hilbert systems">list</a></li></ul></li></ul></li> <li><a href="/wiki/Complete_theory" title="Complete theory">Complete theory</a></li> <li><a href="/wiki/Independence_(mathematical_logic)" title="Independence (mathematical logic)">Independence</a> (<a href="/wiki/List_of_statements_independent_of_ZFC" title="List of statements independent of ZFC">from ZFC</a>)</li> <li><a href="/wiki/Proof_of_impossibility" title="Proof of impossibility">Proof of impossibility</a></li> <li><a href="/wiki/Ordinal_analysis" title="Ordinal analysis">Ordinal analysis</a></li> <li><a href="/wiki/Reverse_mathematics" title="Reverse mathematics">Reverse mathematics</a></li> <li><a href="/wiki/Self-verifying_theories" title="Self-verifying theories">Self-verifying theories</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Model_theory" title="Model theory">Model theory</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Interpretation_(logic)" title="Interpretation (logic)">Interpretation</a> <ul><li><a href="/wiki/Interpretation_function" class="mw-redirect" title="Interpretation function">function</a></li> <li><a href="/wiki/Interpretation_(model_theory)" title="Interpretation (model theory)">of models</a></li></ul></li> <li><a href="/wiki/Structure_(mathematical_logic)" title="Structure (mathematical logic)">Model</a> <ul><li><a href="/wiki/Elementary_equivalence" title="Elementary equivalence">equivalence</a></li> <li><a href="/wiki/Finite_model_theory" title="Finite model theory">finite</a></li> <li><a href="/wiki/Saturated_model" title="Saturated model">saturated</a></li> <li><a href="/wiki/Spectrum_of_a_theory" title="Spectrum of a theory">spectrum</a></li> <li><a href="/wiki/Substructure_(mathematics)" title="Substructure (mathematics)">submodel</a></li></ul></li> <li><a href="/wiki/Non-standard_model" title="Non-standard model">Non-standard model</a> <ul><li><a href="/wiki/Non-standard_model_of_arithmetic" title="Non-standard model of arithmetic">of arithmetic</a></li></ul></li> <li><a href="/wiki/Diagram_(mathematical_logic)" title="Diagram (mathematical logic)">Diagram</a> <ul><li><a href="/wiki/Elementary_diagram" title="Elementary diagram">elementary</a></li></ul></li> <li><a href="/wiki/Categorical_theory" title="Categorical theory">Categorical theory</a></li> <li><a href="/wiki/Model_complete_theory" title="Model complete theory">Model complete theory</a></li> <li><a href="/wiki/Satisfiability" title="Satisfiability">Satisfiability</a></li> <li><a href="/wiki/Semantics_of_logic" title="Semantics of logic">Semantics of logic</a></li> <li><a href="/wiki/Strength_(mathematical_logic)" title="Strength (mathematical logic)">Strength</a></li> <li><a href="/wiki/Theories_of_truth" class="mw-redirect" title="Theories of truth">Theories of truth</a> <ul><li><a href="/wiki/Semantic_theory_of_truth" title="Semantic theory of truth">semantic</a></li> <li><a href="/wiki/Tarski%27s_theory_of_truth" class="mw-redirect" title="Tarski's theory of truth">Tarski's</a></li> <li><a href="/wiki/Kripke%27s_theory_of_truth" class="mw-redirect" title="Kripke's theory of truth">Kripke's</a></li></ul></li> <li><a href="/wiki/T-schema" title="T-schema">T-schema</a></li> <li><a href="/wiki/Transfer_principle" title="Transfer principle">Transfer principle</a></li> <li><a href="/wiki/Truth_predicate" title="Truth predicate">Truth predicate</a></li> <li><a href="/wiki/Truth_value" title="Truth value">Truth value</a></li> <li><a href="/wiki/Type_(model_theory)" title="Type (model theory)">Type</a></li> <li><a href="/wiki/Ultraproduct" title="Ultraproduct">Ultraproduct</a></li> <li><a href="/wiki/Validity_(logic)" title="Validity (logic)">Validity</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Computability_theory" title="Computability theory">Computability theory</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Church_encoding" title="Church encoding">Church encoding</a></li> <li><a class="mw-selflink selflink">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> 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