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class="vector-toc-list"> </ul> </li> <li id="toc-Consistency" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Consistency"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.3</span> <span>Consistency</span> </div> </a> <ul id="toc-Consistency-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Systems_which_contain_arithmetic" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Systems_which_contain_arithmetic"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.4</span> <span>Systems which contain arithmetic</span> </div> </a> <ul id="toc-Systems_which_contain_arithmetic-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Conflicting_goals" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Conflicting_goals"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.5</span> <span>Conflicting goals</span> </div> </a> <ul id="toc-Conflicting_goals-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-First_incompleteness_theorem" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#First_incompleteness_theorem"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>First incompleteness theorem</span> </div> </a> <button aria-controls="toc-First_incompleteness_theorem-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 First incompleteness theorem subsection</span> </button> <ul id="toc-First_incompleteness_theorem-sublist" class="vector-toc-list"> <li id="toc-Syntactic_form_of_the_Gödel_sentence" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Syntactic_form_of_the_Gödel_sentence"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Syntactic form of the Gödel sentence</span> </div> </a> <ul id="toc-Syntactic_form_of_the_Gödel_sentence-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Truth_of_the_Gödel_sentence" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Truth_of_the_Gödel_sentence"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Truth of the Gödel sentence</span> </div> </a> <ul id="toc-Truth_of_the_Gödel_sentence-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Relationship_with_the_liar_paradox" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Relationship_with_the_liar_paradox"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Relationship with the liar paradox</span> </div> </a> <ul id="toc-Relationship_with_the_liar_paradox-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Extensions_of_Gödel's_original_result" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Extensions_of_Gödel's_original_result"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.4</span> <span>Extensions of Gödel's original result</span> </div> </a> <ul id="toc-Extensions_of_Gödel's_original_result-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Second_incompleteness_theorem" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Second_incompleteness_theorem"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Second incompleteness theorem</span> </div> </a> <button aria-controls="toc-Second_incompleteness_theorem-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 Second incompleteness theorem subsection</span> </button> <ul id="toc-Second_incompleteness_theorem-sublist" class="vector-toc-list"> <li id="toc-Expressing_consistency" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Expressing_consistency"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Expressing consistency</span> </div> </a> <ul id="toc-Expressing_consistency-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-The_Hilbert–Bernays_conditions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#The_Hilbert–Bernays_conditions"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>The Hilbert–Bernays conditions</span> </div> </a> <ul id="toc-The_Hilbert–Bernays_conditions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Implications_for_consistency_proofs" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Implications_for_consistency_proofs"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>Implications for consistency proofs</span> </div> </a> <ul id="toc-Implications_for_consistency_proofs-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Examples_of_undecidable_statements" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Examples_of_undecidable_statements"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Examples of undecidable statements</span> </div> </a> <button aria-controls="toc-Examples_of_undecidable_statements-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 Examples of undecidable statements subsection</span> </button> <ul id="toc-Examples_of_undecidable_statements-sublist" class="vector-toc-list"> <li id="toc-Undecidable_statements_provable_in_larger_systems" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Undecidable_statements_provable_in_larger_systems"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Undecidable statements provable in larger systems</span> </div> </a> <ul id="toc-Undecidable_statements_provable_in_larger_systems-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Relationship_with_computability" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Relationship_with_computability"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Relationship with computability</span> </div> </a> <ul id="toc-Relationship_with_computability-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Proof_sketch_for_the_first_theorem" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Proof_sketch_for_the_first_theorem"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Proof sketch for the first theorem</span> </div> </a> <button aria-controls="toc-Proof_sketch_for_the_first_theorem-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 Proof sketch for the first theorem subsection</span> </button> <ul id="toc-Proof_sketch_for_the_first_theorem-sublist" class="vector-toc-list"> <li id="toc-Arithmetization_of_syntax" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Arithmetization_of_syntax"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.1</span> <span>Arithmetization of syntax</span> </div> </a> <ul id="toc-Arithmetization_of_syntax-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Construction_of_a_statement_about_"provability"" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Construction_of_a_statement_about_"provability""> <div class="vector-toc-text"> <span class="vector-toc-numb">6.2</span> <span>Construction of a statement about "provability"</span> </div> </a> <ul id="toc-Construction_of_a_statement_about_"provability"-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Diagonalization" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Diagonalization"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.3</span> <span>Diagonalization</span> </div> </a> <ul id="toc-Diagonalization-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Proof_via_Berry's_paradox" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Proof_via_Berry's_paradox"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.4</span> <span>Proof via Berry's paradox</span> </div> </a> <ul id="toc-Proof_via_Berry's_paradox-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Computer_verified_proofs" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Computer_verified_proofs"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.5</span> <span>Computer verified proofs</span> </div> </a> <ul id="toc-Computer_verified_proofs-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Proof_sketch_for_the_second_theorem" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Proof_sketch_for_the_second_theorem"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Proof sketch for the second theorem</span> </div> </a> <ul id="toc-Proof_sketch_for_the_second_theorem-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Discussion_and_implications" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Discussion_and_implications"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Discussion and implications</span> </div> </a> <button aria-controls="toc-Discussion_and_implications-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 Discussion and implications subsection</span> </button> <ul id="toc-Discussion_and_implications-sublist" class="vector-toc-list"> <li id="toc-Consequences_for_logicism_and_Hilbert's_second_problem" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Consequences_for_logicism_and_Hilbert's_second_problem"> <div class="vector-toc-text"> <span class="vector-toc-numb">8.1</span> <span>Consequences for logicism and Hilbert's second problem</span> </div> </a> <ul id="toc-Consequences_for_logicism_and_Hilbert's_second_problem-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Minds_and_machines" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Minds_and_machines"> <div class="vector-toc-text"> <span class="vector-toc-numb">8.2</span> <span>Minds and machines</span> </div> </a> <ul id="toc-Minds_and_machines-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Paraconsistent_logic" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Paraconsistent_logic"> <div class="vector-toc-text"> <span class="vector-toc-numb">8.3</span> <span>Paraconsistent logic</span> </div> </a> <ul id="toc-Paraconsistent_logic-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Appeals_to_the_incompleteness_theorems_in_other_fields" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Appeals_to_the_incompleteness_theorems_in_other_fields"> <div class="vector-toc-text"> <span class="vector-toc-numb">8.4</span> <span>Appeals to the incompleteness theorems in other fields</span> </div> </a> <ul id="toc-Appeals_to_the_incompleteness_theorems_in_other_fields-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-History" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#History"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</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-Announcement" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Announcement"> <div class="vector-toc-text"> <span class="vector-toc-numb">9.1</span> <span>Announcement</span> </div> </a> <ul id="toc-Announcement-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Generalization_and_acceptance" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Generalization_and_acceptance"> <div class="vector-toc-text"> <span class="vector-toc-numb">9.2</span> <span>Generalization and acceptance</span> </div> </a> <ul id="toc-Generalization_and_acceptance-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Criticisms" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Criticisms"> <div class="vector-toc-text"> <span class="vector-toc-numb">9.3</span> <span>Criticisms</span> </div> </a> <ul id="toc-Criticisms-sublist" class="vector-toc-list"> <li id="toc-Finsler" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Finsler"> <div class="vector-toc-text"> <span class="vector-toc-numb">9.3.1</span> <span>Finsler</span> </div> </a> <ul id="toc-Finsler-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Zermelo" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Zermelo"> <div class="vector-toc-text"> <span class="vector-toc-numb">9.3.2</span> <span>Zermelo</span> </div> </a> <ul id="toc-Zermelo-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Wittgenstein" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Wittgenstein"> <div class="vector-toc-text"> <span class="vector-toc-numb">9.3.3</span> <span>Wittgenstein</span> </div> </a> <ul id="toc-Wittgenstein-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">11</span> <span>References</span> </div> </a> <button aria-controls="toc-References-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 References subsection</span> </button> <ul id="toc-References-sublist" class="vector-toc-list"> <li id="toc-Citations" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Citations"> <div class="vector-toc-text"> <span class="vector-toc-numb">11.1</span> <span>Citations</span> </div> </a> <ul id="toc-Citations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Articles_by_Gödel" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Articles_by_Gödel"> <div class="vector-toc-text"> <span class="vector-toc-numb">11.2</span> <span>Articles by Gödel</span> </div> </a> <ul id="toc-Articles_by_Gödel-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Translations,_during_his_lifetime,_of_Gödel's_paper_into_English" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Translations,_during_his_lifetime,_of_Gödel's_paper_into_English"> <div class="vector-toc-text"> <span class="vector-toc-numb">11.3</span> <span>Translations, during his lifetime, of Gödel's paper into English</span> </div> </a> <ul id="toc-Translations,_during_his_lifetime,_of_Gödel's_paper_into_English-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Articles_by_others" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Articles_by_others"> <div class="vector-toc-text"> <span class="vector-toc-numb">11.4</span> <span>Articles by others</span> </div> </a> <ul id="toc-Articles_by_others-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Books_about_the_theorems" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Books_about_the_theorems"> <div class="vector-toc-text"> <span class="vector-toc-numb">11.5</span> <span>Books about the theorems</span> </div> </a> <ul id="toc-Books_about_the_theorems-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Miscellaneous_references" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Miscellaneous_references"> <div class="vector-toc-text"> <span class="vector-toc-numb">11.6</span> <span>Miscellaneous references</span> </div> </a> <ul id="toc-Miscellaneous_references-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">12</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Gödel's incompleteness theorems</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 54 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-54" 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">54 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-als mw-list-item"><a href="https://als.wikipedia.org/wiki/G%C3%B6delscher_Unvollst%C3%A4ndigkeitssatz" title="Gödelscher Unvollständigkeitssatz – Alemannic" lang="gsw" hreflang="gsw" data-title="Gödelscher Unvollständigkeitssatz" data-language-autonym="Alemannisch" data-language-local-name="Alemannic" class="interlanguage-link-target"><span>Alemannisch</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%85%D8%A8%D8%B1%D9%87%D9%86%D8%A7%D8%AA_%D8%B9%D8%AF%D9%85_%D8%A7%D9%84%D8%A7%D9%83%D8%AA%D9%85%D8%A7%D9%84_%D9%84%D8%BA%D9%88%D8%AF%D9%84" title="مبرهنات عدم الاكتمال لغودل – Arabic" lang="ar" hreflang="ar" data-title="مبرهنات عدم الاكتمال لغودل" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Teoremas_de_incompletitud_de_G%C3%B6del" title="Teoremas de incompletitud de Gödel – Asturian" lang="ast" hreflang="ast" data-title="Teoremas de incompletitud de Gödel" data-language-autonym="Asturianu" data-language-local-name="Asturian" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%A2%D0%B5%D0%BE%D1%80%D0%B5%D0%BC%D0%B0_%D0%BD%D0%B0_%D0%93%D1%8C%D0%BE%D0%B4%D0%B5%D0%BB_%D0%B7%D0%B0_%D0%BD%D0%B5%D0%BF%D1%8A%D0%BB%D0%BD%D0%BE%D1%82%D0%B0" title="Теорема на Гьодел за непълнота – Bulgarian" lang="bg" hreflang="bg" data-title="Теорема на Гьодел за непълнота" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/G%C3%B6delove_teoreme_nepotpunosti" title="Gödelove teoreme nepotpunosti – Bosnian" lang="bs" hreflang="bs" data-title="Gödelove teoreme nepotpunosti" data-language-autonym="Bosanski" data-language-local-name="Bosnian" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Teorema_d%27incompletesa_de_G%C3%B6del" title="Teorema d'incompletesa de Gödel – Catalan" lang="ca" hreflang="ca" data-title="Teorema d'incompletesa de Gödel" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%93%D1%91%D0%B4%D0%B5%D0%BB%C4%95%D0%BD_%D1%82%D1%83%D0%BB%D0%BB%D0%B8%D0%BC%D0%B0%D1%80%D0%BB%C4%83%D1%85_%D1%82%D0%B5%D0%BE%D1%80%D0%B5%D0%BC%D0%B8" title="Гёделĕн туллимарлăх теореми – Chuvash" lang="cv" hreflang="cv" data-title="Гёделĕн туллимарлăх теореми" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/G%C3%B6delovy_v%C4%9Bty_o_ne%C3%BAplnosti" title="Gödelovy věty o neúplnosti – Czech" lang="cs" hreflang="cs" data-title="Gödelovy věty o neúplnosti" 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/G%C3%B6dels_ufuldst%C3%A6ndighedss%C3%A6tning" title="Gödels ufuldstændighedssætning – Danish" lang="da" hreflang="da" data-title="Gödels ufuldstændighedssætning" 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/G%C3%B6delscher_Unvollst%C3%A4ndigkeitssatz" title="Gödelscher Unvollständigkeitssatz – German" lang="de" hreflang="de" data-title="Gödelscher Unvollständigkeitssatz" 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/G%C3%B6deli_mittet%C3%A4ielikkuse_teoreemid" title="Gödeli mittetäielikkuse teoreemid – Estonian" lang="et" hreflang="et" data-title="Gödeli mittetäielikkuse teoreemid" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%98%CE%B5%CF%89%CF%81%CE%AE%CE%BC%CE%B1%CF%84%CE%B1_%CE%BC%CE%B7_%CF%80%CE%BB%CE%B7%CF%81%CF%8C%CF%84%CE%B7%CF%84%CE%B1%CF%82_%CF%84%CE%BF%CF%85_%CE%93%CE%BA%CE%AD%CE%BD%CF%84%CE%B5%CE%BB" title="Θεωρήματα μη πληρότητας του Γκέντελ – Greek" lang="el" hreflang="el" data-title="Θεωρήματα μη πληρότητας του Γκέντελ" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Teoremas_de_incompletitud_de_G%C3%B6del" title="Teoremas de incompletitud de Gödel – Spanish" lang="es" hreflang="es" data-title="Teoremas de incompletitud de Gödel" 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/Teoremoj_de_nekompleteco" title="Teoremoj de nekompleteco – Esperanto" lang="eo" hreflang="eo" data-title="Teoremoj de nekompleteco" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/G%C3%B6delen_ez-osotasunaren_teoremak" title="Gödelen ez-osotasunaren teoremak – Basque" lang="eu" hreflang="eu" data-title="Gödelen ez-osotasunaren teoremak" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%82%D8%B6%D8%A7%DB%8C%D8%A7%DB%8C_%D9%86%D8%A7%D8%AA%D9%85%D8%A7%D9%85%DB%8C%D8%AA_%DA%AF%D9%88%D8%AF%D9%84" title="قضایای ناتمامیت گودل – Persian" lang="fa" hreflang="fa" data-title="قضایای ناتمامیت گودل" 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%A9or%C3%A8mes_d%27incompl%C3%A9tude_de_G%C3%B6del" title="Théorèmes d'incomplétude de Gödel – French" lang="fr" hreflang="fr" data-title="Théorèmes d'incomplétude de Gödel" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Cruth%C3%BA_G%C3%B6del" title="Cruthú Gödel – Irish" lang="ga" hreflang="ga" data-title="Cruthú Gödel" data-language-autonym="Gaeilge" data-language-local-name="Irish" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Teorema_da_incompletude_de_G%C3%B6del" title="Teorema da incompletude de Gödel – Galician" lang="gl" hreflang="gl" data-title="Teorema da incompletude de Gödel" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EA%B4%B4%EB%8D%B8%EC%9D%98_%EB%B6%88%EC%99%84%EC%A0%84%EC%84%B1_%EC%A0%95%EB%A6%AC" title="괴델의 불완전성 정리 – Korean" lang="ko" hreflang="ko" data-title="괴델의 불완전성 정리" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%97%E0%A4%BE%E0%A4%A1%E0%A5%87%E0%A4%B2_%E0%A4%95%E0%A4%BE_%E0%A4%85%E0%A4%AA%E0%A5%82%E0%A4%B0%E0%A5%8D%E0%A4%A3%E0%A4%A4%E0%A4%BE_%E0%A4%AA%E0%A5%8D%E0%A4%B0%E0%A4%AE%E0%A5%87%E0%A4%AF" title="गाडेल का अपूर्णता प्रमेय – Hindi" lang="hi" hreflang="hi" data-title="गाडेल का अपूर्णता प्रमेय" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/G%C3%B6delovi_teoremi_nepotpunosti" title="Gödelovi teoremi nepotpunosti – Croatian" lang="hr" hreflang="hr" data-title="Gödelovi teoremi nepotpunosti" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Godel-teorio" title="Godel-teorio – Ido" lang="io" hreflang="io" data-title="Godel-teorio" data-language-autonym="Ido" data-language-local-name="Ido" class="interlanguage-link-target"><span>Ido</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Teorema_ketaklengkapan_G%C3%B6del" title="Teorema ketaklengkapan Gödel – Indonesian" lang="id" hreflang="id" data-title="Teorema ketaklengkapan Gödel" 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/Teoremi_di_incompletezza_di_G%C3%B6del" title="Teoremi di incompletezza di Gödel – Italian" lang="it" hreflang="it" data-title="Teoremi di incompletezza di Gödel" 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%9E%D7%A9%D7%A4%D7%98%D7%99_%D7%94%D7%90%D7%99-%D7%A9%D7%9C%D7%9E%D7%95%D7%AA_%D7%A9%D7%9C_%D7%92%D7%93%D7%9C" title="משפטי האי-שלמות של גדל – Hebrew" lang="he" hreflang="he" data-title="משפטי האי-שלמות של גדל" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-kn mw-list-item"><a href="https://kn.wikipedia.org/wiki/%E0%B2%97%E0%B3%8B%E0%B2%A1%E0%B3%86%E0%B2%B2%E0%B3%8D%E2%80%8C%E0%B2%B0_%E0%B2%85%E0%B2%AA%E0%B3%82%E0%B2%B0%E0%B3%8D%E0%B2%A3%E0%B2%A4%E0%B3%86%E0%B2%AF_%E0%B2%AA%E0%B3%8D%E0%B2%B0%E0%B2%AE%E0%B3%87%E0%B2%AF" title="ಗೋಡೆಲ್‌ರ ಅಪೂರ್ಣತೆಯ ಪ್ರಮೇಯ – Kannada" lang="kn" hreflang="kn" data-title="ಗೋಡೆಲ್‌ರ ಅಪೂರ್ಣತೆಯ ಪ್ರಮೇಯ" data-language-autonym="ಕನ್ನಡ" data-language-local-name="Kannada" class="interlanguage-link-target"><span>ಕನ್ನಡ</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%92%E1%83%9D%E1%83%93%E1%83%94%E1%83%9A%E1%83%98%E1%83%A1_%E1%83%90%E1%83%A0%E1%83%90%E1%83%A1%E1%83%A0%E1%83%A3%E1%83%9A%E1%83%9D%E1%83%91%E1%83%98%E1%83%A1_%E1%83%97%E1%83%94%E1%83%9D%E1%83%A0%E1%83%94%E1%83%9B%E1%83%94%E1%83%91%E1%83%98" title="გოდელის არასრულობის თეორემები – Georgian" lang="ka" hreflang="ka" data-title="გოდელის არასრულობის თეორემები" data-language-autonym="ქართული" data-language-local-name="Georgian" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Theoremata_G%C3%B6del_de_imperfectione" title="Theoremata Gödel de imperfectione – Latin" lang="la" hreflang="la" data-title="Theoremata Gödel de imperfectione" data-language-autonym="Latina" data-language-local-name="Latin" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/G%C3%B6del_els%C5%91_nemteljess%C3%A9gi_t%C3%A9tele" title="Gödel első nemteljességi tétele – Hungarian" lang="hu" hreflang="hu" data-title="Gödel első nemteljességi tétele" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-mn mw-list-item"><a href="https://mn.wikipedia.org/wiki/%D0%91%D2%AF%D1%80%D1%8D%D0%BD_%D0%B1%D1%83%D1%81_%D0%B1%D0%B0%D0%B9%D0%B4%D0%BB%D1%8B%D0%BD_%D1%82%D1%83%D1%85%D0%B0%D0%B9_%D0%93%D1%91%D0%B4%D1%8D%D0%BB%D1%8B%D0%BD_%D1%82%D0%B5%D0%BE%D1%80%D0%B5%D0%BC%D1%83%D1%83%D0%B4" title="Бүрэн бус байдлын тухай Гёдэлын теоремууд – Mongolian" lang="mn" hreflang="mn" data-title="Бүрэн бус байдлын тухай Гёдэлын теоремууд" data-language-autonym="Монгол" data-language-local-name="Mongolian" class="interlanguage-link-target"><span>Монгол</span></a></li><li class="interlanguage-link interwiki-my mw-list-item"><a href="https://my.wikipedia.org/wiki/%E1%80%82%E1%80%B0%E1%80%92%E1%80%9A%E1%80%BA%E1%81%8F_%E1%80%99%E1%80%95%E1%80%BC%E1%80%8A%E1%80%B7%E1%80%BA%E1%80%85%E1%80%AF%E1%80%B6%E1%80%81%E1%80%BC%E1%80%84%E1%80%BA%E1%80%B8%E1%80%9E%E1%80%AE%E1%80%A1%E1%80%AD%E1%80%AF%E1%80%9B%E1%80%99%E1%80%BA%E1%80%99%E1%80%BB%E1%80%AC%E1%80%B8" title="ဂူဒယ်၏ မပြည့်စုံခြင်းသီအိုရမ်များ – Burmese" lang="my" hreflang="my" data-title="ဂူဒယ်၏ မပြည့်စုံခြင်းသီအိုရမ်များ" data-language-autonym="မြန်မာဘာသာ" data-language-local-name="Burmese" class="interlanguage-link-target"><span>မြန်မာဘာသာ</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Onvolledigheidsstellingen_van_G%C3%B6del" title="Onvolledigheidsstellingen van Gödel – Dutch" lang="nl" hreflang="nl" data-title="Onvolledigheidsstellingen van Gödel" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%82%B2%E3%83%BC%E3%83%87%E3%83%AB%E3%81%AE%E4%B8%8D%E5%AE%8C%E5%85%A8%E6%80%A7%E5%AE%9A%E7%90%86" title="ゲーデルの不完全性定理 – Japanese" lang="ja" hreflang="ja" data-title="ゲーデルの不完全性定理" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/G%C3%B6dels_ufullstendighetsteoremer" title="Gödels ufullstendighetsteoremer – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Gödels ufullstendighetsteoremer" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Ufullstendigheitsteorema" title="Ufullstendigheitsteorema – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Ufullstendigheitsteorema" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-nov mw-list-item"><a href="https://nov.wikipedia.org/wiki/Teoreme_de_G%C3%B6del" title="Teoreme de Gödel – Novial" lang="nov" hreflang="nov" data-title="Teoreme de Gödel" data-language-autonym="Novial" data-language-local-name="Novial" class="interlanguage-link-target"><span>Novial</span></a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%97%E0%A9%8B%E0%A8%87%E0%A8%A1%E0%A8%B2_%E0%A8%A6%E0%A9%80%E0%A8%86%E0%A8%82_%E0%A8%85%E0%A8%AA%E0%A9%82%E0%A8%B0%E0%A8%A8%E0%A8%A4%E0%A8%BE_%E0%A8%A6%E0%A9%80%E0%A8%86%E0%A8%82_%E0%A8%A5%E0%A8%BF%E0%A8%8A%E0%A8%B0%E0%A8%AE%E0%A8%BE%E0%A8%82" title="ਗੋਇਡਲ ਦੀਆਂ ਅਪੂਰਨਤਾ ਦੀਆਂ ਥਿਊਰਮਾਂ – Punjabi" lang="pa" hreflang="pa" data-title="ਗੋਇਡਲ ਦੀਆਂ ਅਪੂਰਨਤਾ ਦੀਆਂ ਥਿਊਰਮਾਂ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="Punjabi" class="interlanguage-link-target"><span>ਪੰਜਾਬੀ</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Twierdzenia_G%C3%B6dla" title="Twierdzenia Gödla – Polish" lang="pl" hreflang="pl" data-title="Twierdzenia Gödla" 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/Teoremas_da_incompletude_de_G%C3%B6del" title="Teoremas da incompletude de Gödel – Portuguese" lang="pt" hreflang="pt" data-title="Teoremas da incompletude de Gödel" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A2%D0%B5%D0%BE%D1%80%D0%B5%D0%BC%D1%8B_%D0%93%D1%91%D0%B4%D0%B5%D0%BB%D1%8F_%D0%BE_%D0%BD%D0%B5%D0%BF%D0%BE%D0%BB%D0%BD%D0%BE%D1%82%D0%B5" title="Теоремы Гёделя о неполноте – Russian" lang="ru" hreflang="ru" data-title="Теоремы Гёделя о неполноте" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-scn mw-list-item"><a href="https://scn.wikipedia.org/wiki/Tiurema_d%27incumplitizza_di_G%C3%B6del" title="Tiurema d'incumplitizza di Gödel – 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							cdx-button--size-medium cdx-button--icon-only cdx-dialog__header__close-button" aria-label="Close" onclick="document.getElementById("mw-fr-revision-details").style.display = "none";" type="submit"><span class="cdx-icon cdx-icon--medium
							cdx-fr-css-icon--close"></span></button></header><div class="cdx-dialog__body">This is the <a href="/wiki/Wikipedia:Pending_changes" title="Wikipedia:Pending changes">latest accepted revision</a>, <a class="external text" href="https://en.wikipedia.org/w/index.php?title=Special:Log&type=review&page=G%C3%B6del%27s_incompleteness_theorems">reviewed</a> on <i>29 November 2024</i>.</div></div><div tabindex="0"></div></div></div></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Limitative results in mathematical logic</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">For the earlier theory about the correspondence between truth and provability, see <a href="/wiki/G%C3%B6del%27s_completeness_theorem" title="Gödel's completeness theorem">Gödel's completeness theorem</a>.</div> <p><b>Gödel's incompleteness theorems</b> are two <a href="/wiki/Theorem" title="Theorem">theorems</a> of <a href="/wiki/Mathematical_logic" title="Mathematical logic">mathematical logic</a> that are concerned with the limits of provability in formal axiomatic theories. These results, published by <a href="/wiki/Kurt_G%C3%B6del" title="Kurt Gödel">Kurt Gödel</a> in 1931, are important both in mathematical logic and in the <a href="/wiki/Philosophy_of_mathematics" title="Philosophy of mathematics">philosophy of mathematics</a>. The theorems are widely, but not universally, interpreted as showing that <a href="/wiki/Hilbert%27s_program" title="Hilbert's program">Hilbert's program</a> to find a complete and consistent set of <a href="/wiki/Axiom" title="Axiom">axioms</a> for all <a href="/wiki/Mathematics" title="Mathematics">mathematics</a> is impossible. </p><p>The first incompleteness theorem states that no <a href="/wiki/Consistency" title="Consistency">consistent system</a> of <a href="/wiki/Axiom" title="Axiom">axioms</a> whose theorems can be listed by an <a href="/wiki/Effective_procedure" class="mw-redirect" title="Effective procedure">effective procedure</a> (i.e. an <a href="/wiki/Algorithm" title="Algorithm">algorithm</a>) is capable of <a href="/wiki/Mathematical_proof" title="Mathematical proof">proving</a> all truths about the arithmetic of <a href="/wiki/Natural_number" title="Natural number">natural numbers</a>. For any such consistent formal system, there will always be statements about natural numbers that are true, but that are unprovable within the system. </p><p>The second incompleteness theorem, an extension of the first, shows that the system cannot demonstrate its own consistency. </p><p>Employing a <a href="/wiki/Cantor%27s_diagonal_argument" title="Cantor's diagonal argument">diagonal argument</a>, Gödel's incompleteness theorems were the first of several closely related theorems on the limitations of formal systems. They were followed by <a href="/wiki/Tarski%27s_undefinability_theorem" title="Tarski's undefinability theorem">Tarski's undefinability theorem</a> on the formal undefinability of truth, <a href="/wiki/Alonzo_Church" title="Alonzo Church">Church</a>'s proof that Hilbert's <i><a href="/wiki/Entscheidungsproblem" title="Entscheidungsproblem">Entscheidungsproblem</a></i> is unsolvable, and <a href="/wiki/Alan_Turing" title="Alan Turing">Turing</a>'s theorem that there is no algorithm to solve the <a href="/wiki/Halting_problem" title="Halting problem">halting problem</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Formal_systems:_completeness,_consistency,_and_effective_axiomatization"><span id="Formal_systems:_completeness.2C_consistency.2C_and_effective_axiomatization"></span>Formal systems: completeness, consistency, and effective axiomatization</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=G%C3%B6del%27s_incompleteness_theorems&action=edit&section=1" title="Edit section: Formal systems: completeness, consistency, and effective axiomatization"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The incompleteness theorems apply to <a href="/wiki/Formal_system" title="Formal system">formal systems</a> that are of sufficient complexity to express the basic arithmetic of the natural numbers and which are consistent and effectively axiomatized. Particularly in the context of <a href="/wiki/First-order_logic" title="First-order logic">first-order logic</a>, formal systems are also called <i>formal theories</i>. In general, a formal system is a deductive apparatus that consists of a particular set of axioms along with rules of symbolic manipulation (or rules of inference) that allow for the derivation of new theorems from the axioms. One example of such a system is first-order <a href="/wiki/Peano_arithmetic" class="mw-redirect" title="Peano arithmetic">Peano arithmetic</a>, a system in which all variables are intended to denote natural numbers. In other systems, such as <a href="/wiki/Set_theory" title="Set theory">set theory</a>, only some sentences of the formal system express statements about the natural numbers. The incompleteness theorems are about formal provability <i>within</i> these systems, rather than about "provability" in an informal sense. </p><p>There are several properties that a formal system may have, including completeness, consistency, and the existence of an effective axiomatization. The incompleteness theorems show that systems which contain a sufficient amount of arithmetic cannot possess all three of these properties. </p> <div class="mw-heading mw-heading3"><h3 id="Effective_axiomatization">Effective axiomatization</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=G%C3%B6del%27s_incompleteness_theorems&action=edit&section=2" title="Edit section: Effective axiomatization"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A formal system is said to be <i>effectively axiomatized</i> (also called <i>effectively generated</i>) if its set of theorems is <a href="/wiki/Recursively_enumerable" class="mw-redirect" title="Recursively enumerable">recursively enumerable</a>. This means that there is a computer program that, in principle, could enumerate all the theorems of the system without listing any statements that are not theorems. Examples of effectively generated theories include Peano arithmetic and <a href="/wiki/Zermelo%E2%80%93Fraenkel_set_theory" title="Zermelo–Fraenkel set theory">Zermelo–Fraenkel set theory</a> (ZFC).<sup id="cite_ref-FOOTNOTEFranzén2005112_1-0" class="reference"><a href="#cite_note-FOOTNOTEFranzén2005112-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><p>The theory known as <a href="/wiki/True_arithmetic" title="True arithmetic">true arithmetic</a> consists of all true statements about the standard integers in the language of Peano arithmetic. This theory is consistent and complete, and contains a sufficient amount of arithmetic. However, it does not have a recursively enumerable set of axioms, and thus does not satisfy the hypotheses of the incompleteness theorems. </p> <div class="mw-heading mw-heading3"><h3 id="Completeness">Completeness</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=G%C3%B6del%27s_incompleteness_theorems&action=edit&section=3" title="Edit section: Completeness"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A set of axioms is (<i>syntactically</i>, or <i>negation</i>-) <a href="/wiki/Complete_theory" title="Complete theory">complete</a> if, for any statement in the axioms' language, that statement or its negation is provable from the axioms.<sup id="cite_ref-FOOTNOTESmith200724_2-0" class="reference"><a href="#cite_note-FOOTNOTESmith200724-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> This is the notion relevant for Gödel's first Incompleteness theorem. It is not to be confused with <i>semantic</i> completeness, which means that the set of axioms proves all the semantic tautologies of the given language. In his <a href="/wiki/G%C3%B6del%27s_completeness_theorem" title="Gödel's completeness theorem">completeness theorem</a> (not to be confused with the incompleteness theorems described here), Gödel proved that first-order logic is <i>semantically</i> complete. But it is not syntactically complete, since there are sentences expressible in the language of first-order logic that can be neither proved nor disproved from the axioms of logic alone. </p><p>In a system of mathematics, thinkers such as Hilbert believed that it was just a matter of time to find such an axiomatization that would allow one to either prove or disprove (by proving its negation) every mathematical formula. </p><p>A formal system might be syntactically incomplete by design, as logics generally are. Or it may be incomplete simply because not all the necessary axioms have been discovered or included. For example, <a href="/wiki/Euclidean_geometry" title="Euclidean geometry">Euclidean geometry</a> without the <a href="/wiki/Parallel_postulate" title="Parallel postulate">parallel postulate</a> is incomplete, because some statements in the language (such as the parallel postulate itself) can not be proved from the remaining axioms. Similarly, the theory of <a href="/wiki/Dense_linear_order" class="mw-redirect" title="Dense linear order">dense linear orders</a> is not complete, but becomes complete with an extra axiom stating that there are no endpoints in the order. The <a href="/wiki/Continuum_hypothesis" title="Continuum hypothesis">continuum hypothesis</a> is a statement in the language of <a href="/wiki/Zermelo%E2%80%93Fraenkel_set_theory" title="Zermelo–Fraenkel set theory">ZFC</a> that is not provable within ZFC, so ZFC is not complete. In this case, there is no obvious candidate for a new axiom that resolves the issue. </p><p>The theory of first-order <a href="/wiki/Peano_arithmetic" class="mw-redirect" title="Peano arithmetic">Peano arithmetic</a> seems consistent. Assuming this is indeed the case, note that it has an infinite but recursively enumerable set of axioms, and can encode enough arithmetic for the hypotheses of the incompleteness theorem. Thus by the first incompleteness theorem, Peano Arithmetic is not complete. The theorem gives an explicit example of a statement of arithmetic that is neither provable nor disprovable in Peano's arithmetic. Moreover, this statement is true in the usual <a href="/wiki/Model_theory" title="Model theory">model</a>. In addition, no effectively axiomatized, consistent extension of Peano arithmetic can be complete. </p> <div class="mw-heading mw-heading3"><h3 id="Consistency">Consistency</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=G%C3%B6del%27s_incompleteness_theorems&action=edit&section=4" title="Edit section: Consistency"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A set of axioms is (simply) <a href="/wiki/Consistency" title="Consistency">consistent</a> if there is no statement such that both the statement and its negation are provable from the axioms, and <i>inconsistent</i> otherwise. That is to say, a consistent axiomatic system is one that is free from contradiction. </p><p>Peano arithmetic is provably consistent from ZFC, but not from within itself. Similarly, ZFC is not provably consistent from within itself, but ZFC + "there exists an <a href="/wiki/Inaccessible_cardinal" title="Inaccessible cardinal">inaccessible cardinal</a>" proves ZFC is consistent because if <span class="texhtml mvar" style="font-style:italic;">κ</span> is the least such cardinal, then <span class="texhtml"><i>V</i><sub><span class="texhtml mvar" style="font-style:italic;">κ</span></sub></span> sitting inside the <a href="/wiki/Von_Neumann_universe" title="Von Neumann universe">von Neumann universe</a> is a <a href="/wiki/Inner_model" title="Inner model">model</a> of ZFC, and a theory is consistent if and only if it has a model. </p><p>If one takes all statements in the language of <a href="/wiki/Peano_arithmetic" class="mw-redirect" title="Peano arithmetic">Peano arithmetic</a> as axioms, then this theory is complete, has a recursively enumerable set of axioms, and can describe addition and multiplication. However, it is not consistent. </p><p>Additional examples of inconsistent theories arise from the <a href="/wiki/Na%C3%AFve_set_theory#Paradoxes_in_early_set_theory" class="mw-redirect" title="Naïve set theory">paradoxes</a> that result when the <a href="/wiki/Axiom_schema_of_specification#Unrestricted_comprehension" title="Axiom schema of specification">axiom schema of unrestricted comprehension</a> is assumed in set theory. </p> <div class="mw-heading mw-heading3"><h3 id="Systems_which_contain_arithmetic">Systems which contain arithmetic</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=G%C3%B6del%27s_incompleteness_theorems&action=edit&section=5" title="Edit section: Systems which contain arithmetic"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The incompleteness theorems apply only to formal systems which are able to prove a sufficient collection of facts about the natural numbers. One sufficient collection is the set of theorems of <a href="/wiki/Robinson_arithmetic" title="Robinson arithmetic">Robinson arithmetic</a> <span class="texhtml mvar" style="font-style:italic;">Q</span>. Some systems, such as Peano arithmetic, can directly express statements about natural numbers. Others, such as ZFC set theory, are able to interpret statements about natural numbers into their language. Either of these options is appropriate for the incompleteness theorems. </p><p>The theory of <a href="/wiki/Algebraically_closed_field" title="Algebraically closed field">algebraically closed fields</a> of a given <a href="/wiki/Characteristic_(algebra)" title="Characteristic (algebra)">characteristic</a> is complete, consistent, and has an infinite but recursively enumerable set of axioms. However it is not possible to encode the integers into this theory, and the theory cannot describe arithmetic of integers. A similar example is the theory of <a href="/wiki/Real_closed_field" title="Real closed field">real closed fields</a>, which is essentially equivalent to <a href="/wiki/Tarski%27s_axioms" title="Tarski's axioms">Tarski's axioms</a> for <a href="/wiki/Euclidean_geometry" title="Euclidean geometry">Euclidean geometry</a>. So Euclidean geometry itself (in Tarski's formulation) is an example of a complete, consistent, effectively axiomatized theory. </p><p>The system of <a href="/wiki/Presburger_arithmetic" title="Presburger arithmetic">Presburger arithmetic</a> consists of a set of axioms for the natural numbers with just the addition operation (multiplication is omitted). Presburger arithmetic is complete, consistent, and recursively enumerable and can encode addition but not multiplication of natural numbers, showing that for Gödel's theorems one needs the theory to encode not just addition but also multiplication. </p><p><a href="/wiki/Dan_Willard" title="Dan Willard">Dan Willard</a> (<a href="#CITEREFWillard2001">2001</a>) has studied some weak families of arithmetic systems which allow enough arithmetic as relations to formalise Gödel numbering, but which are not strong enough to have multiplication as a function, and so fail to prove the second incompleteness theorem; that is to say, these systems are consistent and capable of proving their own consistency (see <a href="/wiki/Self-verifying_theories" title="Self-verifying theories">self-verifying theories</a>). </p> <div class="mw-heading mw-heading3"><h3 id="Conflicting_goals">Conflicting goals</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=G%C3%B6del%27s_incompleteness_theorems&action=edit&section=6" title="Edit section: Conflicting goals"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In choosing a set of axioms, one goal is to be able to prove as many correct results as possible, without proving any incorrect results. For example, we could imagine a set of true axioms which allow us to prove every true arithmetical claim about the natural numbers (<a href="#CITEREFSmith2007">Smith 2007</a>, p. 2). In the standard system of first-order logic, an inconsistent set of axioms will prove every statement in its language (this is sometimes called the <a href="/wiki/Principle_of_explosion" title="Principle of explosion">principle of explosion</a>), and is thus automatically complete. A set of axioms that is both complete and consistent, however, proves a <a href="/wiki/Maximal_set" title="Maximal set">maximal set</a> of non-<a href="/wiki/Contradiction" title="Contradiction">contradictory</a> theorems.<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 2023)">citation needed</span></a></i>]</sup> </p><p>The pattern illustrated in the previous sections with Peano arithmetic, ZFC, and ZFC + "there exists an inaccessible cardinal" cannot generally be broken. Here ZFC + "there exists an inaccessible cardinal" cannot from itself, be proved consistent. It is also not complete, as illustrated by the continuum hypothesis, which is unresolvable<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> in ZFC + "there exists an inaccessible cardinal". </p><p>The first incompleteness theorem shows that, in formal systems that can express basic arithmetic, a complete and consistent finite list of axioms can never be created: each time an additional, consistent statement is added as an axiom, there are other true statements that still cannot be proved, even with the new axiom. If an axiom is ever added that makes the system complete, it does so at the cost of making the system inconsistent. It is not even possible for an infinite list of axioms to be complete, consistent, and effectively axiomatized. </p> <div class="mw-heading mw-heading2"><h2 id="First_incompleteness_theorem">First incompleteness theorem</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=G%C3%B6del%27s_incompleteness_theorems&action=edit&section=7" title="Edit section: First incompleteness theorem"><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/Proof_sketch_for_G%C3%B6del%27s_first_incompleteness_theorem" title="Proof sketch for Gödel's first incompleteness theorem">Proof sketch for Gödel's first incompleteness theorem</a></div> <p><b>Gödel's first incompleteness theorem</b> first appeared as "Theorem VI" in Gödel's 1931 paper "<a href="/wiki/On_Formally_Undecidable_Propositions_of_Principia_Mathematica_and_Related_Systems" title="On Formally Undecidable Propositions of Principia Mathematica and Related Systems">On Formally Undecidable Propositions of Principia Mathematica and Related Systems</a> I". The hypotheses of the theorem were improved shortly thereafter by J. Barkley Rosser (<a href="#CITEREFRosser1936">1936</a>) using <a href="/wiki/Rosser%27s_trick" title="Rosser's trick">Rosser's trick</a>. The resulting theorem (incorporating Rosser's improvement) may be paraphrased in English as follows, where "formal system" includes the assumption that the system is effectively generated. </p> <blockquote><p><b>First Incompleteness Theorem</b>: "Any consistent formal system <span class="texhtml mvar" style="font-style:italic;">F</span> within which a certain amount of elementary arithmetic can be carried out is incomplete; i.e. there are statements of the language of <span class="texhtml mvar" style="font-style:italic;">F</span> which can neither be proved nor disproved in <span class="texhtml mvar" style="font-style:italic;">F</span>." (Raatikainen 2020)</p></blockquote> <p>The unprovable statement <span class="texhtml"><i>G</i><sub><i>F</i></sub></span> referred to by the theorem is often referred to as "the Gödel sentence" for the system <span class="texhtml mvar" style="font-style:italic;">F</span>. The proof constructs a particular Gödel sentence for the system <span class="texhtml mvar" style="font-style:italic;">F</span>, but there are infinitely many statements in the language of the system that share the same properties, such as the conjunction of the Gödel sentence and any <a href="/wiki/Logically_valid" class="mw-redirect" title="Logically valid">logically valid</a> sentence. </p><p>Each effectively generated system has its own Gödel sentence. It is possible to define a larger system <span class="texhtml mvar" style="font-style:italic;">F'</span> that contains the whole of <span class="texhtml mvar" style="font-style:italic;">F</span> plus <span class="texhtml"><i>G</i><sub><i>F</i></sub></span> as an additional axiom. This will not result in a complete system, because Gödel's theorem will also apply to <span class="texhtml mvar" style="font-style:italic;">F'</span>, and thus <span class="texhtml mvar" style="font-style:italic;">F'</span> also cannot be complete. In this case, <span class="texhtml"><i>G</i><sub><i>F</i></sub></span> is indeed a theorem in <span class="texhtml mvar" style="font-style:italic;">F'</span>, because it is an axiom. Because <span class="texhtml"><i>G</i><sub><i>F</i></sub></span> states only that it is not provable in <span class="texhtml mvar" style="font-style:italic;">F</span>, no contradiction is presented by its provability within <span class="texhtml mvar" style="font-style:italic;">F'</span>. However, because the incompleteness theorem applies to <span class="texhtml mvar" style="font-style:italic;">F'</span>, there will be a new Gödel statement <span class="texhtml"><i>G</i><sub><i>F</i>'</sub></span> for <span class="texhtml mvar" style="font-style:italic;">F'</span>, showing that <span class="texhtml mvar" style="font-style:italic;">F'</span> is also incomplete. <span class="texhtml"><i>G</i><sub><i>F</i>'</sub></span> will differ from <span class="texhtml"><i>G</i><sub><i>F</i></sub></span> in that <span class="texhtml"><i>G</i><sub><i>F</i>'</sub></span> will refer to <span class="texhtml mvar" style="font-style:italic;">F'</span>, rather than <span class="texhtml mvar" style="font-style:italic;">F</span>. </p> <div class="mw-heading mw-heading3"><h3 id="Syntactic_form_of_the_Gödel_sentence"><span id="Syntactic_form_of_the_G.C3.B6del_sentence"></span>Syntactic form of the Gödel sentence</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=G%C3%B6del%27s_incompleteness_theorems&action=edit&section=8" title="Edit section: Syntactic form of the Gödel sentence"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The Gödel sentence is designed to refer, indirectly, to itself. The sentence states that, when a particular sequence of steps is used to construct another sentence, that constructed sentence will not be provable in <span class="texhtml mvar" style="font-style:italic;">F</span>. However, the sequence of steps is such that the constructed sentence turns out to be <span class="texhtml"><i>G</i><sub><i>F</i></sub></span> itself. In this way, the Gödel sentence <span class="texhtml"><i>G</i><sub><i>F</i></sub></span> indirectly states its own unprovability within <span class="texhtml mvar" style="font-style:italic;">F</span> (<a href="#CITEREFSmith2007">Smith 2007</a>, p. 135). </p><p>To prove the first incompleteness theorem, Gödel demonstrated that the notion of provability within a system could be expressed purely in terms of arithmetical functions that operate on <a href="/wiki/G%C3%B6del_number" class="mw-redirect" title="Gödel number">Gödel numbers</a> of sentences of the system. Therefore, the system, which can prove certain facts about numbers, can also indirectly prove facts about its own statements, provided that it is effectively generated. Questions about the provability of statements within the system are represented as questions about the arithmetical properties of numbers themselves, which would be decidable by the system if it were complete. </p><p>Thus, although the Gödel sentence refers indirectly to sentences of the system <span class="texhtml mvar" style="font-style:italic;">F</span>, when read as an arithmetical statement the Gödel sentence directly refers only to natural numbers. It asserts that no natural number has a particular property, where that property is given by a <a href="/wiki/Primitive_recursive_function" title="Primitive recursive function">primitive recursive</a> relation (<a href="#CITEREFSmith2007">Smith 2007</a>, p. 141). As such, the Gödel sentence can be written in the language of arithmetic with a simple syntactic form. In particular, it can be expressed as a formula in the language of arithmetic consisting of a number of leading universal quantifiers followed by a quantifier-free body (these formulas are at level <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 \Pi _{1}^{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi mathvariant="normal">Π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Pi _{1}^{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b0b3c0e79d8a5db977a9838be477eb3e30348937" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.797ex; height:3.176ex;" alt="{\displaystyle \Pi _{1}^{0}}"></span> of the <a href="/wiki/Arithmetical_hierarchy" title="Arithmetical hierarchy">arithmetical hierarchy</a>). Via the <a href="/wiki/MRDP_theorem" class="mw-redirect" title="MRDP theorem">MRDP theorem</a>, the Gödel sentence can be re-written as a statement that a particular polynomial in many variables with integer coefficients never takes the value zero when integers are substituted for its variables (<a href="#CITEREFFranzén2005">Franzén 2005</a>, p. 71). </p> <div class="mw-heading mw-heading3"><h3 id="Truth_of_the_Gödel_sentence"><span id="Truth_of_the_G.C3.B6del_sentence"></span>Truth of the Gödel sentence</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=G%C3%B6del%27s_incompleteness_theorems&action=edit&section=9" title="Edit section: Truth of the Gödel sentence"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The first incompleteness theorem shows that the Gödel sentence <span class="texhtml"><i>G</i><sub><i>F</i></sub></span> of an appropriate formal theory <span class="texhtml mvar" style="font-style:italic;">F</span> is unprovable in <span class="texhtml mvar" style="font-style:italic;">F</span>. Because, when interpreted as a statement about arithmetic, this unprovability is exactly what the sentence (indirectly) asserts, the Gödel sentence is, in fact, true (<a href="#CITEREFSmoryński1977">Smoryński 1977</a>, p. 825; also see <a href="#CITEREFFranzén2005">Franzén 2005</a>, pp. 28–33). For this reason, the sentence <span class="texhtml"><i>G</i><sub><i>F</i></sub></span> is often said to be "true but unprovable." (<a href="#CITEREFRaatikainen2020">Raatikainen 2020</a>). However, since the Gödel sentence cannot itself formally specify its intended interpretation, the truth of the sentence <span class="texhtml"><i>G</i><sub><i>F</i></sub></span> may only be arrived at via a meta-analysis from outside the system. In general, this meta-analysis can be carried out within the weak formal system known as <a href="/wiki/Primitive_recursive_arithmetic" title="Primitive recursive arithmetic">primitive recursive arithmetic</a>, which proves the implication <span class="texhtml"><i>Con</i>(<i>F</i>)→<i>G</i><sub>F</sub></span>, where <span class="texhtml"><i>Con</i>(<i>F</i>)</span> is a canonical sentence asserting the consistency of <span class="texhtml mvar" style="font-style:italic;">F</span> (<a href="#CITEREFSmoryński1977">Smoryński 1977</a>, p. 840, <a href="#CITEREFKikuchiTanaka1994">Kikuchi & Tanaka 1994</a>, p. 403). </p><p>Although the Gödel sentence of a consistent theory is true as a statement about the <a href="/wiki/Intended_interpretation" class="mw-redirect" title="Intended interpretation">intended interpretation</a> of arithmetic, the Gödel sentence will be false in some <a href="/wiki/Peano_axioms#Nonstandard_models" title="Peano axioms">nonstandard models of arithmetic</a>, as a consequence of Gödel's <a href="/wiki/Completeness_theorem" class="mw-redirect" title="Completeness theorem">completeness theorem</a> (<a href="#CITEREFFranzén2005">Franzén 2005</a>, p. 135). That theorem shows that, when a sentence is independent of a theory, the theory will have models in which the sentence is true and models in which the sentence is false. As described earlier, the Gödel sentence of a system <span class="texhtml mvar" style="font-style:italic;">F</span> is an arithmetical statement which claims that no number exists with a particular property. The incompleteness theorem shows that this claim will be independent of the system <span class="texhtml mvar" style="font-style:italic;">F</span>, and the truth of the Gödel sentence follows from the fact that no standard natural number has the property in question. Any model in which the Gödel sentence is false must contain some element which satisfies the property within that model. Such a model must be "nonstandard" – it must contain elements that do not correspond to any standard natural number (<a href="#CITEREFRaatikainen2020">Raatikainen 2020</a>, <a href="#CITEREFFranzén2005">Franzén 2005</a>, p. 135). </p> <div class="mw-heading mw-heading3"><h3 id="Relationship_with_the_liar_paradox">Relationship with the liar paradox</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=G%C3%B6del%27s_incompleteness_theorems&action=edit&section=10" title="Edit section: Relationship with the liar paradox"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Gödel specifically cites <a href="/wiki/Richard%27s_paradox" title="Richard's paradox">Richard's paradox</a> and the <a href="/wiki/Liar_paradox" title="Liar paradox">liar paradox</a> as semantical analogues to his syntactical incompleteness result in the introductory section of "<a href="/wiki/On_Formally_Undecidable_Propositions_in_Principia_Mathematica_and_Related_Systems_I" class="mw-redirect" title="On Formally Undecidable Propositions in Principia Mathematica and Related Systems I">On Formally Undecidable Propositions in Principia Mathematica and Related Systems I</a>". The <a href="/wiki/Liar_paradox" title="Liar paradox">liar paradox</a> is the sentence "This sentence is false." An analysis of the liar sentence shows that it cannot be true (for then, as it asserts, it is false), nor can it be false (for then, it is true). A Gödel sentence <span class="texhtml mvar" style="font-style:italic;">G</span> for a system <span class="texhtml mvar" style="font-style:italic;">F</span> makes a similar assertion to the liar sentence, but with truth replaced by provability: <span class="texhtml mvar" style="font-style:italic;">G</span> says "<span class="texhtml mvar" style="font-style:italic;">G</span> is not provable in the system <span class="texhtml mvar" style="font-style:italic;">F</span>." The analysis of the truth and provability of <span class="texhtml mvar" style="font-style:italic;">G</span> is a formalized version of the analysis of the truth of the liar sentence. </p><p>It is not possible to replace "not provable" with "false" in a Gödel sentence because the predicate "<span class="texhtml mvar" style="font-style:italic;">Q</span> is the <a href="/wiki/G%C3%B6del_number" class="mw-redirect" title="Gödel number">Gödel number</a> of a false formula" cannot be represented as a formula of arithmetic. This result, known as <a href="/wiki/Tarski%27s_undefinability_theorem" title="Tarski's undefinability theorem">Tarski's undefinability theorem</a>, was discovered independently both by Gödel, when he was working on the proof of the incompleteness theorem, and by the theorem's namesake, <a href="/wiki/Alfred_Tarski" title="Alfred Tarski">Alfred Tarski</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Extensions_of_Gödel's_original_result"><span id="Extensions_of_G.C3.B6del.27s_original_result"></span>Extensions of Gödel's original result</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=G%C3%B6del%27s_incompleteness_theorems&action=edit&section=11" title="Edit section: Extensions of Gödel's original result"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Compared to the theorems stated in Gödel's 1931 paper, many contemporary statements of the incompleteness theorems are more general in two ways. These generalized statements are phrased to apply to a broader class of systems, and they are phrased to incorporate weaker consistency assumptions. </p><p>Gödel demonstrated the incompleteness of the system of <i><a href="/wiki/Principia_Mathematica" title="Principia Mathematica">Principia Mathematica</a></i>, a particular system of arithmetic, but a parallel demonstration could be given for any effective system of a certain expressiveness. Gödel commented on this fact in the introduction to his paper, but restricted the proof to one system for concreteness. In modern statements of the theorem, it is common to state the effectiveness and expressiveness conditions as hypotheses for the incompleteness theorem, so that it is not limited to any particular formal system. The terminology used to state these conditions was not yet developed in 1931 when Gödel published his results. </p><p>Gödel's original statement and proof of the incompleteness theorem requires the assumption that the system is not just consistent but <i><a href="/wiki/Omega-consistent" class="mw-redirect" title="Omega-consistent">ω-consistent</a></i>. A system is <b>ω-consistent</b> if it is not ω-inconsistent, and is ω-inconsistent if there is a predicate <span class="texhtml mvar" style="font-style:italic;">P</span> such that for every specific natural number <span class="texhtml mvar" style="font-style:italic;">m</span> the system proves <span class="texhtml">~<i>P</i>(<i>m</i>)</span>, and yet the system also proves that there exists a natural number <span class="texhtml mvar" style="font-style:italic;">n</span> such that <span class="texhtml mvar" style="font-style:italic;">P</span>(<span class="texhtml mvar" style="font-style:italic;">n</span>). That is, the system says that a number with property <span class="texhtml mvar" style="font-style:italic;">P</span> exists while denying that it has any specific value. The ω-consistency of a system implies its consistency, but consistency does not imply ω-consistency. <a href="/wiki/J._Barkley_Rosser" title="J. Barkley Rosser">J. Barkley Rosser</a> (<a href="#CITEREFRosser1936">1936</a>) strengthened the incompleteness theorem by finding a variation of the proof (<a href="/wiki/Rosser%27s_trick" title="Rosser's trick">Rosser's trick</a>) that only requires the system to be consistent, rather than ω-consistent. This is mostly of technical interest, because all true formal theories of arithmetic (theories whose axioms are all true statements about natural numbers) are ω-consistent, and thus Gödel's theorem as originally stated applies to them. The stronger version of the incompleteness theorem that only assumes consistency, rather than ω-consistency, is now commonly known as Gödel's incompleteness theorem and as the Gödel–Rosser theorem. </p> <div class="mw-heading mw-heading2"><h2 id="Second_incompleteness_theorem">Second incompleteness theorem</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=G%C3%B6del%27s_incompleteness_theorems&action=edit&section=12" title="Edit section: Second incompleteness theorem"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>For each formal system <span class="texhtml mvar" style="font-style:italic;">F</span> containing basic arithmetic, it is possible to canonically define a formula Cons(<span class="texhtml mvar" style="font-style:italic;">F</span>) expressing the consistency of <span class="texhtml mvar" style="font-style:italic;">F</span>. This formula expresses the property that "there does not exist a natural number coding a formal derivation within the system <span class="texhtml mvar" style="font-style:italic;">F</span> whose conclusion is a syntactic contradiction." The syntactic contradiction is often taken to be "0=1", in which case Cons(<span class="texhtml mvar" style="font-style:italic;">F</span>) states "there is no natural number that codes a derivation of '0=1' from the axioms of <span class="texhtml mvar" style="font-style:italic;">F</span>." </p><p><b>Gödel's second incompleteness theorem</b> shows that, under general assumptions, this canonical consistency statement Cons(<span class="texhtml mvar" style="font-style:italic;">F</span>) will not be provable in <span class="texhtml mvar" style="font-style:italic;">F</span>. The theorem first appeared as "Theorem XI" in Gödel's 1931 paper "<a href="/wiki/On_Formally_Undecidable_Propositions_in_Principia_Mathematica_and_Related_Systems_I" class="mw-redirect" title="On Formally Undecidable Propositions in Principia Mathematica and Related Systems I">On Formally Undecidable Propositions in Principia Mathematica and Related Systems I</a>". In the following statement, the term "formalized system" also includes an assumption that <span class="texhtml mvar" style="font-style:italic;">F</span> is effectively axiomatized. This theorem states that for any consistent system <i>F</i> within which a certain amount of elementary arithmetic can be carried out, the consistency of <i>F</i> cannot be proved in <i>F</i> itself.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> This theorem is stronger than the first incompleteness theorem because the statement constructed in the first incompleteness theorem does not directly express the consistency of the system. The proof of the second incompleteness theorem is obtained by formalizing the proof of the first incompleteness theorem within the system <span class="texhtml mvar" style="font-style:italic;">F</span> itself. </p> <div class="mw-heading mw-heading3"><h3 id="Expressing_consistency">Expressing consistency</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=G%C3%B6del%27s_incompleteness_theorems&action=edit&section=13" title="Edit section: Expressing consistency"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>There is a technical subtlety in the second incompleteness theorem regarding the method of expressing the consistency of <span class="texhtml mvar" style="font-style:italic;">F</span> as a formula in the language of <span class="texhtml mvar" style="font-style:italic;">F</span>. There are many ways to express the consistency of a system, and not all of them lead to the same result. The formula Cons(<span class="texhtml mvar" style="font-style:italic;">F</span>) from the second incompleteness theorem is a particular expression of consistency. </p><p>Other formalizations of the claim that <span class="texhtml mvar" style="font-style:italic;">F</span> is consistent may be inequivalent in <span class="texhtml mvar" style="font-style:italic;">F</span>, and some may even be provable. For example, first-order Peano arithmetic (PA) can prove that "the largest consistent <a href="/wiki/Subset" title="Subset">subset</a> of PA" is consistent. But, because PA is consistent, the largest consistent subset of PA is just PA, so in this sense PA "proves that it is consistent". What PA does not prove is that the largest consistent subset of PA is, in fact, the whole of PA. (The term "largest consistent subset of PA" is meant here to be the largest consistent initial segment of the axioms of PA under some particular effective enumeration.) </p> <div class="mw-heading mw-heading3"><h3 id="The_Hilbert–Bernays_conditions"><span id="The_Hilbert.E2.80.93Bernays_conditions"></span>The Hilbert–Bernays conditions</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=G%C3%B6del%27s_incompleteness_theorems&action=edit&section=14" title="Edit section: The Hilbert–Bernays conditions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The standard proof of the second incompleteness theorem assumes that the provability predicate <span class="texhtml"><i>Prov</i><sub>A</sub>(<i>P</i>)</span> satisfies the <a href="/wiki/Hilbert%E2%80%93Bernays_provability_conditions" title="Hilbert–Bernays provability conditions">Hilbert–Bernays provability conditions</a>. Letting <span class="texhtml">#(<i>P</i>)</span> represent the Gödel number of a formula <span class="texhtml mvar" style="font-style:italic;">P</span>, the provability conditions say: </p> <ol><li>If <span class="texhtml mvar" style="font-style:italic;">F</span> proves <span class="texhtml mvar" style="font-style:italic;">P</span>, then <span class="texhtml mvar" style="font-style:italic;">F</span> proves <span class="texhtml"><i>Prov</i><sub>A</sub>(#(<i>P</i>))</span>.</li> <li><span class="texhtml mvar" style="font-style:italic;">F</span> proves 1.; that is, <span class="texhtml mvar" style="font-style:italic;">F</span> proves <span class="texhtml"><i>Prov</i><sub>A</sub>(#(<i>P</i>)) → <i>Prov</i><sub>A</sub>(#(<i>Prov</i><sub>A</sub>(#(<i>P</i>))))</span>.</li> <li><span class="texhtml mvar" style="font-style:italic;">F</span> proves <span class="texhtml"><i>Prov</i><sub>A</sub>(#(<i>P</i> → <i>Q</i>)) ∧ <i>Prov</i><sub>A</sub>(#(<i>P</i>)) → <i>Prov</i><sub>A</sub>(#(<i>Q</i>))</span>   (analogue of <a href="/wiki/Modus_ponens" title="Modus ponens">modus ponens</a>).</li></ol> <p>There are systems, such as Robinson arithmetic, which are strong enough to meet the assumptions of the first incompleteness theorem, but which do not prove the Hilbert–Bernays conditions. Peano arithmetic, however, is strong enough to verify these conditions, as are all theories stronger than Peano arithmetic. </p> <div class="mw-heading mw-heading3"><h3 id="Implications_for_consistency_proofs">Implications for consistency proofs</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=G%C3%B6del%27s_incompleteness_theorems&action=edit&section=15" title="Edit section: Implications for consistency proofs"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Gödel's second incompleteness theorem also implies that a system <span class="texhtml"><i>F</i><sub>1</sub></span> satisfying the technical conditions outlined above cannot prove the consistency of any system <span class="texhtml"><i>F</i><sub>2</sub></span> that proves the consistency of <span class="texhtml"><i>F</i><sub>1</sub></span>. This is because such a system <span class="texhtml"><i>F</i><sub>1</sub></span> can prove that if <span class="texhtml"><i>F</i><sub>2</sub></span> proves the consistency of <span class="texhtml"><i>F</i><sub>1</sub></span>, then <span class="texhtml"><i>F</i><sub>1</sub></span> is in fact consistent. For the claim that <span class="texhtml"><i>F</i><sub>1</sub></span> is consistent has form "for all numbers <span class="texhtml mvar" style="font-style:italic;">n</span>, <span class="texhtml mvar" style="font-style:italic;">n</span> has the decidable property of not being a code for a proof of contradiction in <span class="texhtml"><i>F</i><sub>1</sub></span>". If <span class="texhtml"><i>F</i><sub>1</sub></span> were in fact inconsistent, then <span class="texhtml"><i>F</i><sub>2</sub></span> would prove for some <span class="texhtml mvar" style="font-style:italic;">n</span> that <span class="texhtml mvar" style="font-style:italic;">n</span> is the code of a contradiction in <span class="texhtml"><i>F</i><sub>1</sub></span>. But if <span class="texhtml"><i>F</i><sub>2</sub></span> also proved that <span class="texhtml"><i>F</i><sub>1</sub></span> is consistent (that is, that there is no such <span class="texhtml mvar" style="font-style:italic;">n</span>), then it would itself be inconsistent. This reasoning can be formalized in <span class="texhtml"><i>F</i><sub>1</sub></span> to show that if <span class="texhtml"><i>F</i><sub>2</sub></span> is consistent, then <span class="texhtml"><i>F</i><sub>1</sub></span> is consistent. Since, by second incompleteness theorem, <span class="texhtml"><i>F</i><sub>1</sub></span> does not prove its consistency, it cannot prove the consistency of <span class="texhtml"><i>F</i><sub>2</sub></span> either. </p><p>This corollary of the second incompleteness theorem shows that there is no hope of proving, for example, the consistency of Peano arithmetic using any finitistic means that can be formalized in a system the consistency of which is provable in Peano arithmetic (PA). For example, the system of <a href="/wiki/Primitive_recursive_arithmetic" title="Primitive recursive arithmetic">primitive recursive arithmetic</a> (PRA), which is widely accepted as an accurate formalization of finitistic mathematics, is provably consistent in PA. Thus PRA cannot prove the consistency of PA. This fact is generally seen to imply that <a href="/wiki/Hilbert%27s_program" title="Hilbert's program">Hilbert's program</a>, which aimed to justify the use of "ideal" (infinitistic) mathematical principles in the proofs of "real" (finitistic) mathematical statements by giving a finitistic proof that the ideal principles are consistent, cannot be carried out.<sup id="cite_ref-FOOTNOTEFranzén2005106_5-0" class="reference"><a href="#cite_note-FOOTNOTEFranzén2005106-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p><p>The corollary also indicates the epistemological relevance of the second incompleteness theorem. It would provide no interesting information if a system <span class="texhtml mvar" style="font-style:italic;">F</span> proved its consistency. This is because inconsistent theories prove everything, including their consistency. Thus a consistency proof of <span class="texhtml mvar" style="font-style:italic;">F</span> in <span class="texhtml mvar" style="font-style:italic;">F</span> would give us no clue as to whether <span class="texhtml mvar" style="font-style:italic;">F</span> is consistent; no doubts about the consistency of <span class="texhtml mvar" style="font-style:italic;">F</span> would be resolved by such a consistency proof. The interest in consistency proofs lies in the possibility of proving the consistency of a system <span class="texhtml mvar" style="font-style:italic;">F</span> in some system <span class="texhtml mvar" style="font-style:italic;">F'</span> that is in some sense less doubtful than <span class="texhtml mvar" style="font-style:italic;">F</span> itself, for example, weaker than <span class="texhtml mvar" style="font-style:italic;">F</span>. For many naturally occurring theories <span class="texhtml mvar" style="font-style:italic;">F</span> and <span class="texhtml mvar" style="font-style:italic;">F'</span>, such as <span class="texhtml mvar" style="font-style:italic;">F</span> = Zermelo–Fraenkel set theory and <span class="texhtml mvar" style="font-style:italic;">F'</span> = primitive recursive arithmetic, the consistency of <span class="texhtml mvar" style="font-style:italic;">F'</span> is provable in <span class="texhtml mvar" style="font-style:italic;">F</span>, and thus <span class="texhtml mvar" style="font-style:italic;">F'</span> cannot prove the consistency of <span class="texhtml mvar" style="font-style:italic;">F</span> by the above corollary of the second incompleteness theorem. </p><p>The second incompleteness theorem does not rule out altogether the possibility of proving the consistency of a different system with different axioms. For example, <a href="/wiki/Gerhard_Gentzen" title="Gerhard Gentzen">Gerhard Gentzen</a> proved the consistency of Peano arithmetic in a different system that includes an axiom asserting that the <a href="/wiki/Ordinal_number" title="Ordinal number">ordinal</a> called <span class="texhtml"><i>ε</i><sub>0</sub></span> is <a href="/wiki/Wellfounded" class="mw-redirect" title="Wellfounded">wellfounded</a>; see <a href="/wiki/Gentzen%27s_consistency_proof" title="Gentzen's consistency proof">Gentzen's consistency proof</a>. Gentzen's theorem spurred the development of <a href="/wiki/Ordinal_analysis" title="Ordinal analysis">ordinal analysis</a> in proof theory. </p> <div class="mw-heading mw-heading2"><h2 id="Examples_of_undecidable_statements">Examples of undecidable statements</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=G%C3%B6del%27s_incompleteness_theorems&action=edit&section=16" title="Edit section: Examples of undecidable statements"><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/List_of_statements_independent_of_ZFC" title="List of statements independent of ZFC">List of statements independent of ZFC</a></div> <p>There are two distinct senses of the word "undecidable" in mathematics and computer science. The first of these is the <a href="/wiki/Proof_theory" title="Proof theory">proof-theoretic</a> sense used in relation to Gödel's theorems, that of a statement being neither provable nor refutable in a specified <a href="/wiki/Deductive_system" class="mw-redirect" title="Deductive system">deductive system</a>. The second sense, which will not be discussed here, is used in relation to <a href="/wiki/Computability_theory" title="Computability theory">computability theory</a> and applies not to statements but to <a href="/wiki/Decision_problem" title="Decision problem">decision problems</a>, which are countably infinite sets of questions each requiring a yes or no answer. Such a problem is said to be undecidable if there is no <a href="/wiki/Computable_function" title="Computable function">computable function</a> that correctly answers every question in the problem set (see <a href="/wiki/Undecidable_problem" title="Undecidable problem">undecidable problem</a>). </p><p>Because of the two meanings of the word undecidable, the term <a href="/wiki/Independence_(mathematical_logic)" title="Independence (mathematical logic)">independent</a> is sometimes used instead of undecidable for the "neither provable nor refutable" sense. </p><p>Undecidability of a statement in a particular deductive system does not, in and of itself, address the question of whether the <a href="/wiki/Truth_value" title="Truth value">truth value</a> of the statement is well-defined, or whether it can be determined by other means. Undecidability only implies that the particular deductive system being considered does not prove the truth or falsity of the statement. Whether there exist so-called "absolutely undecidable" statements, whose truth value can never be known or is ill-specified, is a controversial point in the <a href="/wiki/Philosophy_of_mathematics" title="Philosophy of mathematics">philosophy of mathematics</a>. </p><p>The combined work of Gödel and <a href="/wiki/Paul_Cohen_(mathematician)" class="mw-redirect" title="Paul Cohen (mathematician)">Paul Cohen</a> has given two concrete examples of undecidable statements (in the first sense of the term): The <a href="/wiki/Continuum_hypothesis" title="Continuum hypothesis">continuum hypothesis</a> can neither be proved nor refuted in <a href="/wiki/ZFC" class="mw-redirect" title="ZFC">ZFC</a> (the standard axiomatization of <a href="/wiki/Set_theory" title="Set theory">set theory</a>), and the <a href="/wiki/Axiom_of_choice" title="Axiom of choice">axiom of choice</a> can neither be proved nor refuted in ZF (which is all the ZFC axioms <i>except</i> the axiom of choice). These results do not require the incompleteness theorem. Gödel proved in 1940 that neither of these statements could be disproved in ZF or ZFC set theory. In the 1960s, Cohen proved that neither is provable from ZF, and the continuum hypothesis cannot be proved from ZFC. </p><p><a href="#CITEREFShelah1974">Shelah (1974)</a> showed that the <a href="/wiki/Whitehead_problem" title="Whitehead problem">Whitehead problem</a> in <a href="/wiki/Group_theory" title="Group theory">group theory</a> is undecidable, in the first sense of the term, in standard set theory.<sup id="cite_ref-FOOTNOTEShelah1974_6-0" class="reference"><a href="#cite_note-FOOTNOTEShelah1974-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p><p><a href="/wiki/Gregory_Chaitin" title="Gregory Chaitin">Gregory Chaitin</a> produced undecidable statements in <a href="/wiki/Algorithmic_information_theory" title="Algorithmic information theory">algorithmic information theory</a> and proved another incompleteness theorem in that setting. <a href="/wiki/Chaitin%27s_incompleteness_theorem" class="mw-redirect" title="Chaitin's incompleteness theorem">Chaitin's incompleteness theorem</a> states that for any system that can represent enough arithmetic, there is an upper bound <span class="texhtml mvar" style="font-style:italic;">c</span> such that no specific number can be proved in that system to have <a href="/wiki/Kolmogorov_complexity" title="Kolmogorov complexity">Kolmogorov complexity</a> greater than <span class="texhtml mvar" style="font-style:italic;">c</span>. While Gödel's theorem is related to the <a href="/wiki/Liar_paradox" title="Liar paradox">liar paradox</a>, Chaitin's result is related to <a href="/wiki/Berry%27s_paradox" class="mw-redirect" title="Berry's paradox">Berry's paradox</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Undecidable_statements_provable_in_larger_systems">Undecidable statements provable in larger systems</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=G%C3%B6del%27s_incompleteness_theorems&action=edit&section=17" title="Edit section: Undecidable statements provable in larger systems"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>These are natural mathematical equivalents of the Gödel "true but undecidable" sentence. They can be proved in a larger system which is generally accepted as a valid form of reasoning, but are undecidable in a more limited system such as Peano Arithmetic. </p><p>In 1977, <a href="/wiki/Jeff_Paris_(mathematician)" title="Jeff Paris (mathematician)">Paris</a> and <a href="/wiki/Leo_Harrington" title="Leo Harrington">Harrington</a> proved that the <a href="/wiki/Paris%E2%80%93Harrington_theorem" title="Paris–Harrington theorem">Paris–Harrington principle</a>, a version of the infinite <a href="/wiki/Ramsey_theorem" class="mw-redirect" title="Ramsey theorem">Ramsey theorem</a>, is undecidable in (first-order) <a href="/wiki/Peano_arithmetic" class="mw-redirect" title="Peano arithmetic">Peano arithmetic</a>, but can be proved in the stronger system of <a href="/wiki/Second-order_arithmetic" title="Second-order arithmetic">second-order arithmetic</a>. Kirby and Paris later showed that <a href="/wiki/Goodstein%27s_theorem" title="Goodstein's theorem">Goodstein's theorem</a>, a statement about sequences of natural numbers somewhat simpler than the Paris–Harrington principle, is also undecidable in Peano arithmetic. </p><p><a href="/wiki/Kruskal%27s_tree_theorem" title="Kruskal's tree theorem">Kruskal's tree theorem</a>, which has applications in computer science, is also undecidable from Peano arithmetic but provable in set theory. In fact Kruskal's tree theorem (or its finite form) is undecidable in a much stronger system ATR<sub>0</sub> codifying the principles acceptable based on a philosophy of mathematics called <a href="/wiki/Impredicativity" title="Impredicativity">predicativism</a>.<sup id="cite_ref-Simpson2009_7-0" class="reference"><a href="#cite_note-Simpson2009-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> The related but more general <a href="/wiki/Graph_minor_theorem" class="mw-redirect" title="Graph minor theorem">graph minor theorem</a> (2003) has consequences for <a href="/wiki/Computational_complexity_theory" title="Computational complexity theory">computational complexity theory</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Relationship_with_computability">Relationship with computability</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=G%C3%B6del%27s_incompleteness_theorems&action=edit&section=18" title="Edit section: Relationship with computability"><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/Halting_problem#Gödel's_incompleteness_theorems" title="Halting problem">Halting problem § Gödel's incompleteness theorems</a></div> <p>The incompleteness theorem is closely related to several results about <a href="/wiki/Undecidable_set" class="mw-redirect" title="Undecidable set">undecidable sets</a> in <a href="/wiki/Recursion_theory" class="mw-redirect" title="Recursion theory">recursion theory</a>. </p><p><a href="#CITEREFKleene1943">Kleene (1943)</a> presented a proof of Gödel's incompleteness theorem using basic results of computability theory. One such result shows that the <a href="/wiki/Halting_problem" title="Halting problem">halting problem</a> is undecidable: no computer program can correctly determine, given any program <span class="texhtml mvar" style="font-style:italic;">P</span> as input, whether <span class="texhtml mvar" style="font-style:italic;">P</span> eventually halts when run with a particular given input. Kleene showed that the existence of a complete effective system of arithmetic with certain consistency properties would force the halting problem to be decidable, a contradiction.<sup id="cite_ref-FOOTNOTEKleene1943_8-0" class="reference"><a href="#cite_note-FOOTNOTEKleene1943-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> This method of proof has also been presented by <a href="#CITEREFShoenfield1967">Shoenfield (1967)</a>; <a href="#CITEREFCharlesworth1981">Charlesworth (1981)</a>; and <a href="#CITEREFHopcroftUllman1979">Hopcroft & Ullman (1979)</a>.<sup id="cite_ref-FOOTNOTEShoenfield1967132Charlesworth1981HopcroftUllman1979_9-0" class="reference"><a href="#cite_note-FOOTNOTEShoenfield1967132Charlesworth1981HopcroftUllman1979-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> </p><p><a href="#CITEREFFranzén2005">Franzén (2005)</a> explains how <a href="/wiki/Matiyasevich%27s_theorem" class="mw-redirect" title="Matiyasevich's theorem">Matiyasevich's solution</a> to <a href="/wiki/Hilbert%27s_10th_problem" class="mw-redirect" title="Hilbert's 10th problem">Hilbert's 10th problem</a> can be used to obtain a proof to Gödel's first incompleteness theorem.<sup id="cite_ref-FOOTNOTEFranzén200573_10-0" class="reference"><a href="#cite_note-FOOTNOTEFranzén200573-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Yuri_Matiyasevich" title="Yuri Matiyasevich">Matiyasevich</a> proved that there is no algorithm that, given a multivariate polynomial <span class="texhtml"><i>p</i>(<i>x</i><sub>1</sub>, <i>x</i><sub>2</sub>,...,<i>x</i><sub>k</sub>)</span> with integer coefficients, determines whether there is an integer solution to the equation <span class="texhtml mvar" style="font-style:italic;">p</span> = 0. Because polynomials with integer coefficients, and integers themselves, are directly expressible in the language of arithmetic, if a multivariate integer polynomial equation <span class="texhtml mvar" style="font-style:italic;">p</span> = 0 does have a solution in the integers then any sufficiently strong system of arithmetic <span class="texhtml mvar" style="font-style:italic;">T</span> will prove this. Moreover, suppose the system <span class="texhtml mvar" style="font-style:italic;">T</span> is ω-consistent. In that case, it will never prove that a particular polynomial equation has a solution when there is no solution in the integers. Thus, if <span class="texhtml mvar" style="font-style:italic;">T</span> were complete and ω-consistent, it would be possible to determine algorithmically whether a polynomial equation has a solution by merely enumerating proofs of <span class="texhtml mvar" style="font-style:italic;">T</span> until either "<span class="texhtml mvar" style="font-style:italic;">p</span> has a solution" or "<span class="texhtml mvar" style="font-style:italic;">p</span> has no solution" is found, in contradiction to Matiyasevich's theorem. Hence it follows that <span class="texhtml mvar" style="font-style:italic;">T</span> cannot be ω-consistent and complete. Moreover, for each consistent effectively generated system <span class="texhtml mvar" style="font-style:italic;">T</span>, it is possible to effectively generate a multivariate polynomial <span class="texhtml mvar" style="font-style:italic;">p</span> over the integers such that the equation <span class="texhtml mvar" style="font-style:italic;">p</span> = 0 has no solutions over the integers, but the lack of solutions cannot be proved in <span class="texhtml mvar" style="font-style:italic;">T</span>.<sup id="cite_ref-FOOTNOTEDavis2006416Jones1980_11-0" class="reference"><a href="#cite_note-FOOTNOTEDavis2006416Jones1980-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> </p><p><a href="#CITEREFSmoryński1977">Smoryński (1977)</a> shows how the existence of <a href="/wiki/Recursively_inseparable_sets" class="mw-redirect" title="Recursively inseparable sets">recursively inseparable sets</a> can be used to prove the first incompleteness theorem. This proof is often extended to show that systems such as Peano arithmetic are <a href="/wiki/Essentially_undecidable" class="mw-redirect" title="Essentially undecidable">essentially undecidable</a>.<sup id="cite_ref-FOOTNOTESmoryński1977842Kleene1967274_12-0" class="reference"><a href="#cite_note-FOOTNOTESmoryński1977842Kleene1967274-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> </p><p><a href="/wiki/Chaitin%27s_incompleteness_theorem" class="mw-redirect" title="Chaitin's incompleteness theorem">Chaitin's incompleteness theorem</a> gives a different method of producing independent sentences, based on <a href="/wiki/Kolmogorov_complexity" title="Kolmogorov complexity">Kolmogorov complexity</a>. Like the proof presented by Kleene that was mentioned above, Chaitin's theorem only applies to theories with the additional property that all their axioms are true in the standard model of the natural numbers. Gödel's incompleteness theorem is distinguished by its applicability to consistent theories that nonetheless include false statements in the standard model; these theories are known as <a href="/wiki/%CE%A9-consistent_theory" title="Ω-consistent theory">ω-inconsistent</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Proof_sketch_for_the_first_theorem">Proof sketch for the first theorem</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=G%C3%B6del%27s_incompleteness_theorems&action=edit&section=19" title="Edit section: Proof sketch for the first theorem"><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/Proof_sketch_for_G%C3%B6del%27s_first_incompleteness_theorem" title="Proof sketch for Gödel's first incompleteness theorem">Proof sketch for Gödel's first incompleteness theorem</a></div> <p>The <a href="/wiki/Proof_by_contradiction" title="Proof by contradiction">proof by contradiction</a> has three essential parts. To begin, choose a formal system that meets the proposed criteria: </p> <ol><li>Statements in the system can be represented by natural numbers (known as Gödel numbers). The significance of this is that properties of statements—such as their truth and falsehood—will be equivalent to determining whether their Gödel numbers have certain properties, and that properties of the statements can therefore be demonstrated by examining their Gödel numbers. This part culminates in the construction of a formula expressing the idea that <i>"statement <span class="texhtml mvar" style="font-style:italic;">S</span> is provable in the system"</i> (which can be applied to any statement "<span class="texhtml mvar" style="font-style:italic;">S</span>" in the system).</li> <li>In the formal system it is possible to construct a number whose matching statement, when interpreted, is <a href="/wiki/Self_reference" class="mw-redirect" title="Self reference">self-referential</a> and essentially says that it (i.e. the statement itself) is unprovable. This is done using a technique called "<a href="/wiki/Diagonal_lemma" title="Diagonal lemma">diagonalization</a>" (so-called because of its origins as <a href="/wiki/Cantor%27s_diagonal_argument" title="Cantor's diagonal argument">Cantor's diagonal argument</a>).</li> <li>Within the formal system this statement permits a demonstration that it is neither provable nor disprovable in the system, and therefore the system cannot in fact be ω-consistent. Hence the original assumption that the proposed system met the criteria is false.</li></ol> <div class="mw-heading mw-heading3"><h3 id="Arithmetization_of_syntax">Arithmetization of syntax</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=G%C3%B6del%27s_incompleteness_theorems&action=edit&section=20" title="Edit section: Arithmetization of syntax"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The main problem in fleshing out the proof described above is that it seems at first that to construct a statement <span class="texhtml mvar" style="font-style:italic;">p</span> that is equivalent to "<span class="texhtml mvar" style="font-style:italic;">p</span> cannot be proved", <span class="texhtml mvar" style="font-style:italic;">p</span> would somehow have to contain a reference to <span class="texhtml mvar" style="font-style:italic;">p</span>, which could easily give rise to an infinite regress. Gödel's technique is to show that statements can be matched with numbers (often called the arithmetization of <a href="/wiki/Syntax" title="Syntax">syntax</a>) in such a way that <i>"proving a statement"</i> can be replaced with <i>"testing whether a number has a given property"</i>. This allows a self-referential formula to be constructed in a way that avoids any infinite regress of definitions. The same technique was later used by <a href="/wiki/Alan_Turing" title="Alan Turing">Alan Turing</a> in his work on the <i><a href="/wiki/Entscheidungsproblem" title="Entscheidungsproblem">Entscheidungsproblem</a></i>. </p><p>In simple terms, a method can be devised so that every formula or statement that can be formulated in the system gets a unique number, called its <a href="/wiki/G%C3%B6del_number" class="mw-redirect" title="Gödel number">Gödel number</a>, in such a way that it is possible to mechanically convert back and forth between formulas and Gödel numbers. The numbers involved might be very long indeed (in terms of number of digits), but this is not a barrier; all that matters is that such numbers can be constructed. A simple example is how English can be stored as a <a href="/wiki/Character_encoding" title="Character encoding">sequence of numbers for each letter</a> and then combined into a single larger number: </p> <dl><dd><ul><li>The word <b><code>hello</code></b> is encoded as 104-101-108-108-111 in <a href="/wiki/ASCII" title="ASCII">ASCII</a>, which can be converted into the number 104101108108111.</li> <li>The logical statement <b><code>x=y => y=x</code></b> is encoded as 120-061-121-032-061-062-032-121-061-120 in <a href="/wiki/ASCII" title="ASCII">ASCII</a>, which can be converted into the number 120061121032061062032121061120.</li></ul></dd></dl> <p>In principle, proving a statement true or false can be shown to be equivalent to proving that the number matching the statement does or does not have a given property. Because the formal system is strong enough to support reasoning about <i>numbers in general</i>, it can support reasoning about <i>numbers that represent formulae and statements</i> as well. Crucially, because the system can support reasoning about <i>properties of numbers</i>, the results are equivalent to reasoning about <i>provability of their equivalent statements</i>. </p> <div class="mw-heading mw-heading3"><h3 id="Construction_of_a_statement_about_"provability""><span id="Construction_of_a_statement_about_.22provability.22"></span>Construction of a statement about "provability"</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=G%C3%B6del%27s_incompleteness_theorems&action=edit&section=21" title="Edit section: Construction of a statement about "provability""><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Having shown that in principle the system can indirectly make statements about provability, by analyzing properties of those numbers representing statements it is now possible to show how to create a statement that actually does this. </p><p>A formula <span class="texhtml"><i>F</i>(<i>x</i>)</span> that contains exactly one free variable <span class="texhtml mvar" style="font-style:italic;">x</span> is called a <i>statement form</i> or <i>class-sign</i>. As soon as <span class="texhtml mvar" style="font-style:italic;">x</span> is replaced by a specific number, the statement form turns into a <i><a href="/wiki/Bona_fide" class="mw-redirect" title="Bona fide">bona fide</a></i> statement, and it is then either provable in the system, or not. For certain formulas one can show that for every natural number <span class="texhtml mvar" style="font-style:italic;">n</span>, <span class="nowrap">⁠<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 F(n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F(n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b15eed45079a46ea02630c05f701f5a14148efca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.945ex; height:2.843ex;" alt="{\displaystyle F(n)}"></span>⁠</span> is true if and only if it can be proved (the precise requirement in the original proof is weaker, but for the proof sketch this will suffice). In particular, this is true for every specific arithmetic operation between a finite number of natural numbers, such as "2<span class="nowrap"> × </span>3 = 6". </p><p>Statement forms themselves are not statements and therefore cannot be proved or disproved. But every statement form <span class="texhtml"><i>F</i>(<i>x</i>)</span> can be assigned a Gödel number denoted by <span class="texhtml"><b>G</b>(<i>F</i>)</span>. The choice of the free variable used in the form <span class="texhtml mvar" style="font-style:italic;">F</span>(<span class="texhtml mvar" style="font-style:italic;">x</span>) is not relevant to the assignment of the Gödel number <span class="texhtml"><b>G</b>(<i>F</i>)</span>. </p><p><span class="anchor" id="Bew"></span>The notion of provability itself can also be encoded by Gödel numbers, in the following way: since a proof is a list of statements which obey certain rules, the Gödel number of a proof can be defined. Now, for every statement <span class="texhtml mvar" style="font-style:italic;">p</span>, one may ask whether a number <span class="texhtml mvar" style="font-style:italic;">x</span> is the Gödel number of its proof. The relation between the Gödel number of <span class="texhtml mvar" style="font-style:italic;">p</span> and <span class="texhtml mvar" style="font-style:italic;">x</span>, the potential Gödel number of its proof, is an arithmetical relation between two numbers. Therefore, there is a statement form <span class="texhtml"><i>Bew</i>(<i>y</i>)</span> that uses this arithmetical relation to state that a Gödel number of a proof of <span class="texhtml mvar" style="font-style:italic;">y</span> exists: </p> <dl><dd><span class="texhtml"><i>Bew</i>(<i>y</i>) = ∃ <i>x</i></span> (<span class="texhtml mvar" style="font-style:italic;">y</span> is the Gödel number of a formula and <span class="texhtml mvar" style="font-style:italic;">x</span> is the Gödel number of a proof of the formula encoded by <span class="texhtml mvar" style="font-style:italic;">y</span>).</dd></dl> <p>The name <b>Bew</b> is short for <i>beweisbar</i>, the German word for "provable"; this name was originally used by Gödel to denote the provability formula just described. Note that "<span class="texhtml"><i>Bew</i>(<i>y</i>)</span>" is merely an abbreviation that represents a particular, very long, formula in the original language of <span class="texhtml mvar" style="font-style:italic;">T</span>; the string "<span class="texhtml">Bew</span>" itself is not claimed to be part of this language. </p><p>An important feature of the formula <span class="texhtml"><i>Bew</i>(<i>y</i>)</span> is that if a statement <span class="texhtml mvar" style="font-style:italic;">p</span> is provable in the system then <span class="texhtml"><i>Bew</i>(<b>G</b>(<i>p</i>))</span> is also provable. This is because any proof of <span class="texhtml mvar" style="font-style:italic;">p</span> would have a corresponding Gödel number, the existence of which causes <span class="texhtml">Bew(<b>G</b>(<i>p</i>))</span> to be satisfied. </p> <div class="mw-heading mw-heading3"><h3 id="Diagonalization">Diagonalization</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=G%C3%B6del%27s_incompleteness_theorems&action=edit&section=22" title="Edit section: Diagonalization"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The next step in the proof is to obtain a statement which, indirectly, asserts its own unprovability. Although Gödel constructed this statement directly, the existence of at least one such statement follows from the <a href="/wiki/Diagonal_lemma" title="Diagonal lemma">diagonal lemma</a>, which says that for any sufficiently strong formal system and any statement form <span class="texhtml mvar" style="font-style:italic;">F</span> there is a statement <span class="texhtml mvar" style="font-style:italic;">p</span> such that the system proves </p> <dl><dd><span class="texhtml"><i>p</i> ↔ <i>F</i>(<b>G</b>(<i>p</i>))</span>.</dd></dl> <p>By letting <span class="texhtml mvar" style="font-style:italic;">F</span> be the negation of <span class="texhtml"><i>Bew</i>(<i>x</i>)</span>, we obtain the theorem </p> <dl><dd><span class="texhtml"><i>p</i> ↔ ~<i>Bew</i>(<b>G</b>(<i>p</i>))</span></dd></dl> <p>and the <span class="texhtml mvar" style="font-style:italic;">p</span> defined by this roughly states that its own Gödel number is the Gödel number of an unprovable formula. </p><p>The statement <span class="texhtml mvar" style="font-style:italic;">p</span> is not literally equal to <span class="texhtml">~<i>Bew</i>(<b>G</b>(<i>p</i>))</span>; rather, <span class="texhtml mvar" style="font-style:italic;">p</span> states that if a certain calculation is performed, the resulting Gödel number will be that of an unprovable statement. But when this calculation is performed, the resulting Gödel number turns out to be the Gödel number of <span class="texhtml mvar" style="font-style:italic;">p</span> itself. This is similar to the following sentence in English: </p> <dl><dd>", when preceded by itself in quotes, is unprovable.", when preceded by itself in quotes, is unprovable.</dd></dl> <p>This sentence does not directly refer to itself, but when the stated transformation is made the original sentence is obtained as a result, and thus this sentence indirectly asserts its own unprovability. The proof of the diagonal lemma employs a similar method. </p><p>Now, assume that the axiomatic system is <a href="/wiki/Omega-consistent" class="mw-redirect" title="Omega-consistent">ω-consistent</a>, and let <span class="texhtml mvar" style="font-style:italic;">p</span> be the statement obtained in the previous section. </p><p>If <span class="texhtml mvar" style="font-style:italic;">p</span> were provable, then <span class="texhtml"><i>Bew</i>(<b>G</b>(<i>p</i>))</span> would be provable, as argued above. But <span class="texhtml mvar" style="font-style:italic;">p</span> asserts the negation of <span class="texhtml"><i>Bew</i>(<b>G</b>(<i>p</i>))</span>. Thus the system would be inconsistent, proving both a statement and its negation. This contradiction shows that <span class="texhtml mvar" style="font-style:italic;">p</span> cannot be provable. </p><p>If the negation of <span class="texhtml mvar" style="font-style:italic;">p</span> were provable, then <span class="texhtml"><i>Bew</i>(<b>G</b>(<i>p</i>))</span> would be provable (because <span class="texhtml mvar" style="font-style:italic;">p</span> was constructed to be equivalent to the negation of <span class="texhtml"><i>Bew</i>(<b>G</b>(<i>p</i>))</span>). However, for each specific number <span class="texhtml mvar" style="font-style:italic;">x</span>, <span class="texhtml mvar" style="font-style:italic;">x</span> cannot be the Gödel number of the proof of <span class="texhtml mvar" style="font-style:italic;">p</span>, because <span class="texhtml mvar" style="font-style:italic;">p</span> is not provable (from the previous paragraph). Thus on one hand the system proves there is a number with a certain property (that it is the Gödel number of the proof of <span class="texhtml mvar" style="font-style:italic;">p</span>), but on the other hand, for every specific number <span class="texhtml mvar" style="font-style:italic;">x</span>, we can prove that it does not have this property. This is impossible in an ω-consistent system. Thus the negation of <span class="texhtml mvar" style="font-style:italic;">p</span> is not provable. </p><p>Thus the statement <span class="texhtml mvar" style="font-style:italic;">p</span> is undecidable in our axiomatic system: it can neither be proved nor disproved within the system. </p><p>In fact, to show that <span class="texhtml mvar" style="font-style:italic;">p</span> is not provable only requires the assumption that the system is consistent. The stronger assumption of ω-consistency is required to show that the negation of <span class="texhtml mvar" style="font-style:italic;">p</span> is not provable. Thus, if <span class="texhtml mvar" style="font-style:italic;">p</span> is constructed for a particular system: </p> <ul><li>If the system is ω-consistent, it can prove neither <span class="texhtml mvar" style="font-style:italic;">p</span> nor its negation, and so <span class="texhtml mvar" style="font-style:italic;">p</span> is undecidable.</li> <li>If the system is consistent, it may have the same situation, or it may prove the negation of <span class="texhtml mvar" style="font-style:italic;">p</span>. In the later case, we have a statement ("not <span class="texhtml mvar" style="font-style:italic;">p</span>") which is false but provable, and the system is not ω-consistent.</li></ul> <p>If one tries to "add the missing axioms" to avoid the incompleteness of the system, then one has to add either <span class="texhtml mvar" style="font-style:italic;">p</span> or "not <span class="texhtml mvar" style="font-style:italic;">p</span>" as axioms. But then the definition of "being a Gödel number of a proof" of a statement changes. which means that the formula <span class="texhtml"><i>Bew</i>(<i>x</i>)</span> is now different. Thus when we apply the diagonal lemma to this new Bew, we obtain a new statement <span class="texhtml mvar" style="font-style:italic;">p</span>, different from the previous one, which will be undecidable in the new system if it is ω-consistent. </p> <div class="mw-heading mw-heading3"><h3 id="Proof_via_Berry's_paradox"><span id="Proof_via_Berry.27s_paradox"></span>Proof via Berry's paradox</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=G%C3%B6del%27s_incompleteness_theorems&action=edit&section=23" title="Edit section: Proof via Berry's paradox"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="#CITEREFBoolos1989">Boolos (1989)</a> sketches an alternative proof of the first incompleteness theorem that uses <a href="/wiki/Berry%27s_paradox" class="mw-redirect" title="Berry's paradox">Berry's paradox</a> rather than the <a href="/wiki/Liar_paradox" title="Liar paradox">liar paradox</a> to construct a true but unprovable formula. A similar proof method was independently discovered by <a href="/wiki/Saul_Kripke" title="Saul Kripke">Saul Kripke</a>.<sup id="cite_ref-FOOTNOTEBoolos1998383_13-0" class="reference"><a href="#cite_note-FOOTNOTEBoolos1998383-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> Boolos's proof proceeds by constructing, for any <a href="/wiki/Computably_enumerable" class="mw-redirect" title="Computably enumerable">computably enumerable</a> set <span class="texhtml mvar" style="font-style:italic;">S</span> of true sentences of arithmetic, another sentence which is true but not contained in <span class="texhtml mvar" style="font-style:italic;">S</span>. This gives the first incompleteness theorem as a corollary. According to Boolos, this proof is interesting because it provides a "different sort of reason" for the incompleteness of effective, consistent theories of arithmetic.<sup id="cite_ref-FOOTNOTEBoolos1998388_14-0" class="reference"><a href="#cite_note-FOOTNOTEBoolos1998388-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Computer_verified_proofs">Computer verified proofs</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=G%C3%B6del%27s_incompleteness_theorems&action=edit&section=24" title="Edit section: Computer verified proofs"><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/Automated_theorem_proving" title="Automated theorem proving">Automated theorem proving</a></div> <p>The incompleteness theorems are among a relatively small number of nontrivial theorems that have been transformed into formalized theorems that can be completely verified by <a href="/wiki/Proof_assistant" title="Proof assistant">proof assistant</a> software. Gödel's original proofs of the incompleteness theorems, like most mathematical proofs, were written in <a href="/wiki/Natural_language" title="Natural language">natural language</a> intended for human readers. </p><p>Computer-verified proofs of versions of the first incompleteness theorem were announced by <a href="/wiki/Natarajan_Shankar" title="Natarajan Shankar">Natarajan Shankar</a> in 1986 using <a href="/wiki/Nqthm" title="Nqthm">Nqthm</a> (<a href="#CITEREFShankar1994">Shankar 1994</a>), by Russell O'Connor in 2003 using <a href="/wiki/Coq_(software)" title="Coq (software)">Coq</a> (<a href="#CITEREFO'Connor2005">O'Connor 2005</a>) and by John Harrison in 2009 using <a href="/wiki/HOL_Light" title="HOL Light">HOL Light</a> (<a href="#CITEREFHarrison2009">Harrison 2009</a>). A computer-verified proof of both incompleteness theorems was announced by <a href="/wiki/Lawrence_Paulson" title="Lawrence Paulson">Lawrence Paulson</a> in 2013 using <a href="/wiki/Isabelle_theorem_prover" class="mw-redirect" title="Isabelle theorem prover">Isabelle</a> (<a href="#CITEREFPaulson2014">Paulson 2014</a>). </p> <div class="mw-heading mw-heading2"><h2 id="Proof_sketch_for_the_second_theorem">Proof sketch for the second theorem</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=G%C3%B6del%27s_incompleteness_theorems&action=edit&section=25" title="Edit section: Proof sketch for the second theorem"><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/Hilbert%E2%80%93Bernays_provability_conditions" title="Hilbert–Bernays provability conditions">Hilbert–Bernays provability conditions</a></div> <p>The main difficulty in proving the second incompleteness theorem is to show that various facts about provability used in the proof of the first incompleteness theorem can be formalized within a system <span class="texhtml mvar" style="font-style:italic;">S</span> using a formal predicate <span class="texhtml mvar" style="font-style:italic;"><i>P</i></span> for provability. Once this is done, the second incompleteness theorem follows by formalizing the entire proof of the first incompleteness theorem within the system <span class="texhtml mvar" style="font-style:italic;">S</span> itself. </p><p>Let <span class="texhtml mvar" style="font-style:italic;">p</span> stand for the undecidable sentence constructed above, and assume for purposes of obtaining a contradiction that the consistency of the system <span class="texhtml mvar" style="font-style:italic;">S</span> can be proved from within the system <span class="texhtml mvar" style="font-style:italic;">S</span> itself. This is equivalent to proving the statement "System <span class="texhtml mvar" style="font-style:italic;">S</span> is consistent". Now consider the statement <span class="texhtml mvar" style="font-style:italic;">c</span>, where <span class="texhtml mvar" style="font-style:italic;">c</span> = "If the system <span class="texhtml mvar" style="font-style:italic;">S</span> is consistent, then <span class="texhtml mvar" style="font-style:italic;">p</span> is not provable". The proof of sentence <span class="texhtml mvar" style="font-style:italic;">c</span> can be formalized within the system <span class="texhtml mvar" style="font-style:italic;">S</span>, and therefore the statement <span class="texhtml mvar" style="font-style:italic;">c</span>, "<span class="texhtml mvar" style="font-style:italic;">p</span> is not provable", (or identically, "not <span class="texhtml"><i>P</i>(<i>p</i>)</span>") can be proved in the system <span class="texhtml mvar" style="font-style:italic;">S</span>. </p><p>Observe then, that if we can prove that the system <span class="texhtml mvar" style="font-style:italic;">S</span> is consistent (ie. the statement in the hypothesis of <span class="texhtml mvar" style="font-style:italic;">c</span>), then we have proved that <span class="texhtml mvar" style="font-style:italic;">p</span> is not provable. But this is a contradiction since by the 1st Incompleteness Theorem, this sentence (ie. what is implied in the sentence <span class="texhtml mvar" style="font-style:italic;">c</span>, ""<span class="texhtml mvar" style="font-style:italic;">p</span>" is not provable") is what we construct to be unprovable. Notice that this is why we require formalizing the first Incompleteness Theorem in <span class="texhtml mvar" style="font-style:italic;">S</span>: to prove the 2nd Incompleteness Theorem, we obtain a contradiction with the 1st Incompleteness Theorem which can do only by showing that the theorem holds in <span class="texhtml mvar" style="font-style:italic;">S</span>. So we cannot prove that the system <span class="texhtml mvar" style="font-style:italic;">S</span> is consistent. And the 2nd Incompleteness Theorem statement follows. </p> <div class="mw-heading mw-heading2"><h2 id="Discussion_and_implications">Discussion and implications</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=G%C3%B6del%27s_incompleteness_theorems&action=edit&section=26" title="Edit section: Discussion and implications"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The incompleteness results affect the <a href="/wiki/Philosophy_of_mathematics" title="Philosophy of mathematics">philosophy of mathematics</a>, particularly versions of <a href="/wiki/Symbolic_Logic" class="mw-redirect" title="Symbolic Logic">formalism</a>, which use a single system of formal logic to define their principles. </p> <div class="mw-heading mw-heading3"><h3 id="Consequences_for_logicism_and_Hilbert's_second_problem"><span id="Consequences_for_logicism_and_Hilbert.27s_second_problem"></span>Consequences for logicism and Hilbert's second problem</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=G%C3%B6del%27s_incompleteness_theorems&action=edit&section=27" title="Edit section: Consequences for logicism and Hilbert's second problem"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The incompleteness theorem is sometimes thought to have severe consequences for the program of <a href="/wiki/Logicism" title="Logicism">logicism</a> proposed by <a href="/wiki/Gottlob_Frege" title="Gottlob Frege">Gottlob Frege</a> and <a href="/wiki/Bertrand_Russell" title="Bertrand Russell">Bertrand Russell</a>, which aimed to define the natural numbers in terms of logic.<sup id="cite_ref-FOOTNOTEHellman1981451–468_15-0" class="reference"><a href="#cite_note-FOOTNOTEHellman1981451–468-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Bob_Hale_(philosopher)" title="Bob Hale (philosopher)">Bob Hale</a> and <a href="/wiki/Crispin_Wright" title="Crispin Wright">Crispin Wright</a> argue that it is not a problem for logicism because the incompleteness theorems apply equally to first-order logic as they do to arithmetic. They argue that only those who believe that the natural numbers are to be defined in terms of first order logic have this problem. </p><p>Many logicians believe that Gödel's incompleteness theorems struck a fatal blow to <a href="/wiki/David_Hilbert" title="David Hilbert">David Hilbert</a>'s <a href="/wiki/Hilbert%27s_second_problem" title="Hilbert's second problem">second problem</a>, which asked for a finitary consistency proof for mathematics. The second incompleteness theorem, in particular, is often viewed as making the problem impossible. Not all mathematicians agree with this analysis, however, and the status of Hilbert's second problem is not yet decided (see "<a href="/wiki/Hilbert%27s_second_problem#Modern_viewpoints_on_the_status_of_the_problem" title="Hilbert's second problem">Modern viewpoints on the status of the problem</a>"). </p> <div class="mw-heading mw-heading3"><h3 id="Minds_and_machines">Minds and machines</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=G%C3%B6del%27s_incompleteness_theorems&action=edit&section=28" title="Edit section: Minds and machines"><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/Mechanism_(philosophy)#Gödelian_arguments" title="Mechanism (philosophy)">Mechanism (philosophy) § Gödelian arguments</a></div> <p>Authors including the philosopher <a href="/wiki/John_Lucas_(philosopher)" title="John Lucas (philosopher)">J. R. Lucas</a> and physicist <a href="/wiki/Roger_Penrose" title="Roger Penrose">Roger Penrose</a> have debated what, if anything, Gödel's incompleteness theorems imply about human intelligence. Much of the debate centers on whether the human mind is equivalent to a <a href="/wiki/Turing_machine" title="Turing machine">Turing machine</a>, or by the <a href="/wiki/Church%E2%80%93Turing_thesis" title="Church–Turing thesis">Church–Turing thesis</a>, any finite machine at all. If it is, and if the machine is consistent, then Gödel's incompleteness theorems would apply to it. </p><p><a href="#CITEREFPutnam1960">Putnam (1960)</a> suggested that while Gödel's theorems cannot be applied to humans, since they make mistakes and are therefore inconsistent, it may be applied to the human faculty of science or mathematics in general. Assuming that it is consistent, either its consistency cannot be proved or it cannot be represented by a Turing machine.<sup id="cite_ref-FOOTNOTEPutnam1960_16-0" class="reference"><a href="#cite_note-FOOTNOTEPutnam1960-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> </p><p><a href="#CITEREFWigderson2010">Wigderson (2010)</a> has proposed that the concept of mathematical "knowability" should be based on <a href="/wiki/Computational_complexity_theory" title="Computational complexity theory">computational complexity</a> rather than logical decidability. He writes that "when <i>knowability</i> is interpreted by modern standards, namely via computational complexity, the Gödel phenomena are very much with us."<sup id="cite_ref-FOOTNOTEWigderson2010_17-0" class="reference"><a href="#cite_note-FOOTNOTEWigderson2010-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> </p><p><a href="/wiki/Douglas_Hofstadter" title="Douglas Hofstadter">Douglas Hofstadter</a>, in his books <i><a href="/wiki/G%C3%B6del,_Escher,_Bach" title="Gödel, Escher, Bach">Gödel, Escher, Bach</a></i> and <i><a href="/wiki/I_Am_a_Strange_Loop" title="I Am a Strange Loop">I Am a Strange Loop</a></i>, cites Gödel's theorems as an example of what he calls a <i>strange loop</i>, a hierarchical, self-referential structure existing within an axiomatic formal system. He argues that this is the same kind of structure that gives rise to consciousness, the sense of "I", in the human mind. While the self-reference in Gödel's theorem comes from the Gödel sentence asserting its unprovability within the formal system of Principia Mathematica, the self-reference in the human mind comes from how the brain abstracts and categorises stimuli into "symbols", or groups of neurons which respond to concepts, in what is effectively also a formal system, eventually giving rise to symbols modeling the concept of the very entity doing the perception. Hofstadter argues that a strange loop in a sufficiently complex formal system can give rise to a "downward" or "upside-down" causality, a situation in which the normal hierarchy of cause-and-effect is flipped upside-down. In the case of Gödel's theorem, this manifests, in short, as the following: </p> <blockquote><p> Merely from knowing the formula's meaning, one can infer its truth or falsity without any effort to derive it in the old-fashioned way, which requires one to trudge methodically "upwards" from the axioms. This is not just peculiar; it is astonishing. Normally, one cannot merely look at what a mathematical conjecture says and simply appeal to the content of that statement on its own to deduce whether the statement is true or false.<sup id="cite_ref-FOOTNOTEHofstadter2007_18-0" class="reference"><a href="#cite_note-FOOTNOTEHofstadter2007-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> </p></blockquote> <p>In the case of the mind, a far more complex formal system, this "downward causality" manifests, in Hofstadter's view, as the ineffable human instinct that the causality of our minds lies on the high level of desires, concepts, personalities, thoughts, and ideas, rather than on the low level of interactions between neurons or even fundamental particles, even though according to physics the latter seems to possess the causal power. </p> <blockquote><p> There is thus a curious upside-downness to our normal human way of perceiving the world: we are built to perceive “big stuff” rather than “small stuff”, even though the domain of the tiny seems to be where the actual motors driving reality reside.<sup id="cite_ref-FOOTNOTEHofstadter2007_18-1" class="reference"><a href="#cite_note-FOOTNOTEHofstadter2007-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> </p></blockquote> <div class="mw-heading mw-heading3"><h3 id="Paraconsistent_logic">Paraconsistent logic</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=G%C3%B6del%27s_incompleteness_theorems&action=edit&section=29" title="Edit section: Paraconsistent logic"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Although Gödel's theorems are usually studied in the context of classical logic, they also have a role in the study of <a href="/wiki/Paraconsistent_logic" title="Paraconsistent logic">paraconsistent logic</a> and of inherently contradictory statements (<i><a href="/wiki/Dialetheia" class="mw-redirect" title="Dialetheia">dialetheia</a></i>). Priest (<a href="#CITEREFPriest1984">1984</a>, <a href="#CITEREFPriest2006">2006</a>) argues that replacing the notion of formal proof in Gödel's theorem with the usual notion of informal proof can be used to show that naive mathematics is inconsistent, and uses this as evidence for <a href="/wiki/Dialetheism" title="Dialetheism">dialetheism</a>.<sup id="cite_ref-FOOTNOTEPriest1984Priest2006_19-0" class="reference"><a href="#cite_note-FOOTNOTEPriest1984Priest2006-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> The cause of this inconsistency is the inclusion of a truth predicate for a system within the language of the system.<sup id="cite_ref-FOOTNOTEPriest200647_20-0" class="reference"><a href="#cite_note-FOOTNOTEPriest200647-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> <a href="#CITEREFShapiro2002">Shapiro (2002)</a> gives a more mixed appraisal of the applications of Gödel's theorems to dialetheism.<sup id="cite_ref-FOOTNOTEShapiro2002_21-0" class="reference"><a href="#cite_note-FOOTNOTEShapiro2002-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Appeals_to_the_incompleteness_theorems_in_other_fields">Appeals to the incompleteness theorems in other fields</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=G%C3%B6del%27s_incompleteness_theorems&action=edit&section=30" title="Edit section: Appeals to the incompleteness theorems in other fields"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Appeals and analogies are sometimes made to the incompleteness of theorems in support of arguments that go beyond mathematics and logic. Several authors have commented negatively on such extensions and interpretations, including <a href="#CITEREFFranzén2005">Franzén (2005)</a>, <a href="#CITEREFRaatikainen2005">Raatikainen (2005)</a>, <a href="#CITEREFSokalBricmont1999">Sokal & Bricmont (1999)</a>; and <a href="#CITEREFStangroomBenson2006">Stangroom & Benson (2006)</a>.<sup id="cite_ref-FOOTNOTEFranzén2005Raatikainen2005SokalBricmont1999StangroomBenson2006_22-0" class="reference"><a href="#cite_note-FOOTNOTEFranzén2005Raatikainen2005SokalBricmont1999StangroomBenson2006-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> <a href="#CITEREFSokalBricmont1999">Sokal & Bricmont (1999)</a> and <a href="#CITEREFStangroomBenson2006">Stangroom & Benson (2006)</a>, for example, quote from <a href="/wiki/Rebecca_Goldstein" title="Rebecca Goldstein">Rebecca Goldstein</a>'s comments on the disparity between Gödel's avowed <a href="/wiki/Mathematical_Platonism" class="mw-redirect" title="Mathematical Platonism">Platonism</a> and the <a href="/wiki/Anti-realist" class="mw-redirect" title="Anti-realist">anti-realist</a> uses to which his ideas are sometimes put. <a href="#CITEREFSokalBricmont1999">Sokal & Bricmont (1999)</a> criticize <a href="/wiki/R%C3%A9gis_Debray" title="Régis Debray">Régis Debray</a>'s invocation of the theorem in the context of sociology; Debray has defended this use as metaphorical (ibid.).<sup id="cite_ref-FOOTNOTESokalBricmont1999StangroomBenson200610SokalBricmont1999187_23-0" class="reference"><a href="#cite_note-FOOTNOTESokalBricmont1999StangroomBenson200610SokalBricmont1999187-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="History">History</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=G%C3%B6del%27s_incompleteness_theorems&action=edit&section=31" title="Edit section: History"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>After Gödel published his proof of the <a href="/wiki/Completeness_theorem" class="mw-redirect" title="Completeness theorem">completeness theorem</a> as his doctoral thesis in 1929, he turned to a second problem for his <a href="/wiki/Habilitation" title="Habilitation">habilitation</a>. His original goal was to obtain a positive solution to <a href="/wiki/Hilbert%27s_second_problem" title="Hilbert's second problem">Hilbert's second problem</a>.<sup id="cite_ref-FOOTNOTEDawson199763_24-0" class="reference"><a href="#cite_note-FOOTNOTEDawson199763-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup> At the time, theories of natural numbers and real numbers similar to <a href="/wiki/Second-order_arithmetic" title="Second-order arithmetic">second-order arithmetic</a> were known as "analysis", while theories of natural numbers alone were known as "arithmetic". </p><p>Gödel was not the only person working on the consistency problem. <a href="/wiki/Wilhelm_Ackermann" title="Wilhelm Ackermann">Ackermann</a> had published a flawed consistency proof for analysis in 1925, in which he attempted to use the method of <a href="/wiki/Epsilon_calculus" title="Epsilon calculus">ε-substitution</a> originally developed by Hilbert. Later that year, <a href="/wiki/John_von_Neumann" title="John von Neumann">von Neumann</a> was able to correct the proof for a system of arithmetic without any axioms of induction. By 1928, Ackermann had communicated a modified proof to Bernays; this modified proof led Hilbert to announce his belief in 1929 that the consistency of arithmetic had been demonstrated and that a consistent proof of analysis would likely soon follow. After the publication of the incompleteness theorems showed that Ackermann's modified proof must be erroneous, von Neumann produced a concrete example showing that its main technique was unsound.<sup id="cite_ref-FOOTNOTEZach2007418Zach200333_25-0" class="reference"><a href="#cite_note-FOOTNOTEZach2007418Zach200333-25"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup> </p><p>In the course of his research, Gödel discovered that although a sentence, asserting its falsehood leads to paradox, a sentence that asserts its non-provability does not. In particular, Gödel was aware of the result now called <a href="/wiki/Tarski%27s_indefinability_theorem" class="mw-redirect" title="Tarski's indefinability theorem">Tarski's indefinability theorem</a>, although he never published it. Gödel announced his first incompleteness theorem to Carnap, Feigel, and Waismann on August 26, 1930; all four would attend the <a href="/wiki/Second_Conference_on_the_Epistemology_of_the_Exact_Sciences" title="Second Conference on the Epistemology of the Exact Sciences">Second Conference on the Epistemology of the Exact Sciences</a>, a key conference in <a href="/wiki/K%C3%B6nigsberg" title="Königsberg">Königsberg</a> the following week. </p> <div class="mw-heading mw-heading3"><h3 id="Announcement">Announcement</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=G%C3%B6del%27s_incompleteness_theorems&action=edit&section=32" title="Edit section: Announcement"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The 1930 <a href="/wiki/Second_Conference_on_the_Epistemology_of_the_Exact_Sciences" title="Second Conference on the Epistemology of the Exact Sciences">Königsberg conference</a> was a joint meeting of three academic societies, with many of the key logicians of the time in attendance. Carnap, Heyting, and von Neumann delivered one-hour addresses on the mathematical philosophies of logicism, intuitionism, and formalism, respectively.<sup id="cite_ref-FOOTNOTEDawson199669_26-0" class="reference"><a href="#cite_note-FOOTNOTEDawson199669-26"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup> The conference also included Hilbert's retirement address, as he was leaving his position at the University of Göttingen. Hilbert used the speech to argue his belief that all mathematical problems can be solved. He ended his address by saying, </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>For the mathematician there is no <i><a href="/wiki/Ignoramus_et_ignorabimus" title="Ignoramus et ignorabimus">Ignorabimus</a></i>, and, in my opinion, not at all for natural science either. ... The true reason why [no one] has succeeded in finding an unsolvable problem is, in my opinion, that there is no unsolvable problem. In contrast to the foolish <i>Ignorabimus</i>, our credo avers: We must know. We shall know!</p></blockquote> <p>This speech quickly became known as a summary of Hilbert's beliefs on mathematics (its final six words, "<i>Wir müssen wissen. Wir werden wissen!</i>", were used as Hilbert's epitaph in 1943). Although Gödel was likely in attendance for Hilbert's address, the two never met face to face.<sup id="cite_ref-FOOTNOTEDawson199672_27-0" class="reference"><a href="#cite_note-FOOTNOTEDawson199672-27"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup> </p><p>Gödel announced his first incompleteness theorem at a roundtable discussion session on the third day of the conference. The announcement drew little attention apart from that of von Neumann, who pulled Gödel aside for a conversation. Later that year, working independently with knowledge of the first incompleteness theorem, von Neumann obtained a proof of the second incompleteness theorem, which he announced to Gödel in a letter dated November 20, 1930.<sup id="cite_ref-FOOTNOTEDawson199670_28-0" class="reference"><a href="#cite_note-FOOTNOTEDawson199670-28"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup> Gödel had independently obtained the second incompleteness theorem and included it in his submitted manuscript, which was received by <i>Monatshefte für Mathematik</i> on November 17, 1930. </p><p>Gödel's paper was published in the <i>Monatshefte</i> in 1931 under the title "Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I" ("<a href="/wiki/On_Formally_Undecidable_Propositions_in_Principia_Mathematica_and_Related_Systems_I" class="mw-redirect" title="On Formally Undecidable Propositions in Principia Mathematica and Related Systems I">On Formally Undecidable Propositions in Principia Mathematica and Related Systems I</a>"). As the title implies, Gödel originally planned to publish a second part of the paper in the next volume of the <i>Monatshefte</i>; the prompt acceptance of the first paper was one reason he changed his plans.<sup id="cite_ref-FOOTNOTEvan_Heijenoort1967page_328,_footnote_68a_29-0" class="reference"><a href="#cite_note-FOOTNOTEvan_Heijenoort1967page_328,_footnote_68a-29"><span class="cite-bracket">[</span>29<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Generalization_and_acceptance">Generalization and acceptance</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=G%C3%B6del%27s_incompleteness_theorems&action=edit&section=33" title="Edit section: Generalization and acceptance"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Gödel gave a series of lectures on his theorems at Princeton in 1933–1934 to an audience that included Church, Kleene, and Rosser. By this time, Gödel had grasped that the key property his theorems required is that the system must be effective (at the time, the term "general recursive" was used). Rosser proved in 1936 that the hypothesis of ω-consistency, which was an integral part of Gödel's original proof, could be replaced by simple consistency if the Gödel sentence was changed appropriately. These developments left the incompleteness theorems in essentially their modern form. </p><p>Gentzen published his <a href="/wiki/Gentzen%27s_consistency_proof" title="Gentzen's consistency proof">consistency proof</a> for first-order arithmetic in 1936. Hilbert accepted this proof as "finitary" although (as Gödel's theorem had already shown) it cannot be formalized within the system of arithmetic that is being proved consistent. </p><p>The impact of the incompleteness theorems on Hilbert's program was quickly realized. Bernays included a full proof of the incompleteness theorems in the second volume of <i>Grundlagen der Mathematik</i> (<a href="#CITEREFBernays1939">1939</a>), along with additional results of Ackermann on the ε-substitution method and Gentzen's consistency proof of arithmetic. This was the first full published proof of the second incompleteness theorem. </p> <div class="mw-heading mw-heading3"><h3 id="Criticisms">Criticisms</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=G%C3%B6del%27s_incompleteness_theorems&action=edit&section=34" title="Edit section: Criticisms"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading4"><h4 id="Finsler">Finsler</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=G%C3%B6del%27s_incompleteness_theorems&action=edit&section=35" title="Edit section: Finsler"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="#CITEREFFinsler1926">Finsler (1926)</a> used a version of <a href="/wiki/Richard%27s_paradox" title="Richard's paradox">Richard's paradox</a> to construct an expression that was false but unprovable in a particular, informal framework he had developed.<sup id="cite_ref-FOOTNOTEFinsler1926_30-0" class="reference"><a href="#cite_note-FOOTNOTEFinsler1926-30"><span class="cite-bracket">[</span>30<span class="cite-bracket">]</span></a></sup> Gödel was unaware of this paper when he proved the incompleteness theorems (Collected Works Vol. IV., p. 9). Finsler wrote to Gödel in 1931 to inform him about this paper, which Finsler felt had priority for an incompleteness theorem. Finsler's methods did not rely on formalized provability and had only a superficial resemblance to Gödel's work.<sup id="cite_ref-FOOTNOTEvan_Heijenoort1967328_31-0" class="reference"><a href="#cite_note-FOOTNOTEvan_Heijenoort1967328-31"><span class="cite-bracket">[</span>31<span class="cite-bracket">]</span></a></sup> Gödel read the paper but found it deeply flawed, and his response to Finsler laid out concerns about the lack of formalization.<sup id="cite_ref-FOOTNOTEDawson199689_32-0" class="reference"><a href="#cite_note-FOOTNOTEDawson199689-32"><span class="cite-bracket">[</span>32<span class="cite-bracket">]</span></a></sup> Finsler continued to argue for his philosophy of mathematics, which eschewed formalization, for the remainder of his career. </p> <div class="mw-heading mw-heading4"><h4 id="Zermelo">Zermelo</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=G%C3%B6del%27s_incompleteness_theorems&action=edit&section=36" title="Edit section: Zermelo"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In September 1931, <a href="/wiki/Ernst_Zermelo" title="Ernst Zermelo">Ernst Zermelo</a> wrote to Gödel to announce what he described as an "essential gap" in Gödel's argument.<sup id="cite_ref-FOOTNOTEDawson199676_33-0" class="reference"><a href="#cite_note-FOOTNOTEDawson199676-33"><span class="cite-bracket">[</span>33<span class="cite-bracket">]</span></a></sup> In October, Gödel replied with a 10-page letter, where he pointed out that Zermelo mistakenly assumed that the notion of truth in a system is definable in that system; it is not true in general by <a href="/wiki/Tarski%27s_undefinability_theorem" title="Tarski's undefinability theorem">Tarski's undefinability theorem</a>.<sup id="cite_ref-FOOTNOTEDawson199676Grattan-Guinness2005512–513_34-0" class="reference"><a href="#cite_note-FOOTNOTEDawson199676Grattan-Guinness2005512–513-34"><span class="cite-bracket">[</span>34<span class="cite-bracket">]</span></a></sup> However, Zermelo did not relent and published his criticisms in print with "a rather scathing paragraph on his young competitor".<sup id="cite_ref-FOOTNOTEGrattan-Guinness2005513_35-0" class="reference"><a href="#cite_note-FOOTNOTEGrattan-Guinness2005513-35"><span class="cite-bracket">[</span>35<span class="cite-bracket">]</span></a></sup> Gödel decided that pursuing the matter further was pointless, and Carnap agreed.<sup id="cite_ref-FOOTNOTEDawson199677_36-0" class="reference"><a href="#cite_note-FOOTNOTEDawson199677-36"><span class="cite-bracket">[</span>36<span class="cite-bracket">]</span></a></sup> Much of Zermelo's subsequent work was related to logic stronger than first-order logic, with which he hoped to show both the consistency and categoricity of mathematical theories. </p> <div class="mw-heading mw-heading4"><h4 id="Wittgenstein">Wittgenstein</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=G%C3%B6del%27s_incompleteness_theorems&action=edit&section=37" title="Edit section: Wittgenstein"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Ludwig_Wittgenstein" title="Ludwig Wittgenstein">Ludwig Wittgenstein</a> wrote several passages about the incompleteness theorems that were published posthumously in his 1953 <i><a href="/wiki/Remarks_on_the_Foundations_of_Mathematics" title="Remarks on the Foundations of Mathematics">Remarks on the Foundations of Mathematics</a></i>, particularly, one section sometimes called the "notorious paragraph" where he seems to confuse the notions of "true" and "provable" in Russell's system. Gödel was a member of the <a href="/wiki/Vienna_Circle" title="Vienna Circle">Vienna Circle</a> during the period in which Wittgenstein's early <a href="/wiki/Ideal_language_philosophy" title="Ideal language philosophy">ideal language philosophy</a> and <a href="/wiki/Tractatus_Logico-Philosophicus" title="Tractatus Logico-Philosophicus">Tractatus Logico-Philosophicus</a> dominated the circle's thinking. There has been some controversy about whether Wittgenstein misunderstood the incompleteness theorem or just expressed himself unclearly. Writings in Gödel's <a href="/wiki/Nachlass" title="Nachlass">Nachlass</a> express the belief that Wittgenstein misread his ideas. </p><p>Multiple commentators have read Wittgenstein as misunderstanding <a href="/wiki/G%C3%B6del" class="mw-redirect" title="Gödel">Gödel</a>, although <a href="#CITEREFFloydPutnam2000">Floyd & Putnam (2000)</a> as well as <a href="#CITEREFPriest2004">Priest (2004)</a> have provided textual readings arguing that most commentary misunderstands Wittgenstein.<sup id="cite_ref-FOOTNOTERodych2003FloydPutnam2000Priest2004_37-0" class="reference"><a href="#cite_note-FOOTNOTERodych2003FloydPutnam2000Priest2004-37"><span class="cite-bracket">[</span>37<span class="cite-bracket">]</span></a></sup> On their release, Bernays, Dummett, and Kreisel wrote separate reviews on Wittgenstein's remarks, all of which were extremely negative.<sup id="cite_ref-FOOTNOTEBerto2009208_38-0" class="reference"><a href="#cite_note-FOOTNOTEBerto2009208-38"><span class="cite-bracket">[</span>38<span class="cite-bracket">]</span></a></sup> The unanimity of this criticism caused Wittgenstein's remarks on the incompleteness theorems to have little impact on the logic community. In 1972, Gödel stated: "Has Wittgenstein lost his mind? Does he mean it seriously? He intentionally utters trivially nonsensical statements", and wrote to <a href="/wiki/Karl_Menger" title="Karl Menger">Karl Menger</a> that Wittgenstein's comments demonstrate a misunderstanding of the incompleteness theorems writing: </p> <blockquote><p> It is clear from the passages you cite that Wittgenstein did <i>not</i> understand [the first incompleteness theorem] (or pretended not to understand it). He interpreted it as a kind of logical paradox, while in fact is just the opposite, namely a mathematical theorem within an absolutely uncontroversial part of mathematics (finitary number theory or combinatorics).<sup id="cite_ref-FOOTNOTEWang1996179_39-0" class="reference"><a href="#cite_note-FOOTNOTEWang1996179-39"><span class="cite-bracket">[</span>39<span class="cite-bracket">]</span></a></sup> </p></blockquote> <p>Since the publication of Wittgenstein's <i>Nachlass</i> in 2000, a series of papers in philosophy have sought to evaluate whether the original criticism of Wittgenstein's remarks was justified. <a href="#CITEREFFloydPutnam2000">Floyd & Putnam (2000)</a> argue that Wittgenstein had a more complete understanding of the incompleteness theorem than was previously assumed. They are particularly concerned with the interpretation of a Gödel sentence for an ω-inconsistent system as saying "I am not provable", since the system has no models in which the provability predicate corresponds to actual provability. <a href="#CITEREFRodych2003">Rodych (2003)</a> argues that their interpretation of Wittgenstein is not historically justified. <a href="#CITEREFBerto2009">Berto (2009)</a> explores the relationship between Wittgenstein's writing and theories of paraconsistent logic.<sup id="cite_ref-FOOTNOTEFloydPutnam2000Rodych2003Berto2009_40-0" class="reference"><a href="#cite_note-FOOTNOTEFloydPutnam2000Rodych2003Berto2009-40"><span class="cite-bracket">[</span>40<span class="cite-bracket">]</span></a></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=G%C3%B6del%27s_incompleteness_theorems&action=edit&section=38" title="Edit section: See 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Machines and Gödel">Minds, Machines and Gödel</a></i></li> <li><a href="/wiki/Non-standard_model_of_arithmetic" title="Non-standard model of arithmetic">Non-standard model of arithmetic</a></li> <li><a href="/wiki/Proof_theory" title="Proof theory">Proof theory</a></li> <li><a href="/wiki/Provability_logic" title="Provability logic">Provability logic</a></li> <li><a href="/wiki/Quining" class="mw-redirect" title="Quining">Quining</a></li> <li><a href="/wiki/Tarski%27s_undefinability_theorem" title="Tarski's undefinability theorem">Tarski's undefinability theorem</a></li> <li><a href="/wiki/Theory_of_everything#Gödel's_incompleteness_theorem" title="Theory of everything">Theory of everything#Gödel's incompleteness theorem</a></li> <li><a href="/wiki/Typographical_Number_Theory" title="Typographical Number Theory">Typographical Number Theory</a></li></ul> </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=G%C3%B6del%27s_incompleteness_theorems&action=edit&section=39" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Citations">Citations</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=G%C3%B6del%27s_incompleteness_theorems&action=edit&section=40" title="Edit section: Citations"><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-FOOTNOTEFranzén2005112-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEFranzén2005112_1-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFFranzén2005">Franzén 2005</a>, p. 112.</span> </li> <li id="cite_note-FOOTNOTESmith200724-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTESmith200724_2-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFSmith2007">Smith 2007</a>, p. 24.</span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text">in technical terms: <a href="/wiki/Independence_(mathematical_logic)" title="Independence (mathematical logic)">independent</a>; see <a href="/wiki/Continuum_hypothesis#Independence_from_ZFC" title="Continuum hypothesis">Continuum hypothesis#Independence from ZFC</a></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><a href="#CITEREFRaatikainen2020">Raatikainen 2020</a> : "Assume <span class="texhtml mvar" style="font-style:italic;">F</span> is a consistent formalized system which contains elementary arithmetic. Then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F\not \vdash {\text{Cons}}(F)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo>⊬</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>Cons</mtext> </mrow> <mo stretchy="false">(</mo> <mi>F</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F\not \vdash {\text{Cons}}(F)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add12801bef0001bba9f27303881f322448f02cc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.051ex; height:2.843ex;" alt="{\displaystyle F\not \vdash {\text{Cons}}(F)}"></span>."</span> </li> <li id="cite_note-FOOTNOTEFranzén2005106-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEFranzén2005106_5-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFFranzén2005">Franzén 2005</a>, p. 106.</span> </li> <li id="cite_note-FOOTNOTEShelah1974-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEShelah1974_6-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFShelah1974">Shelah 1974</a>.</span> </li> <li id="cite_note-Simpson2009-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-Simpson2009_7-0">^</a></b></span> <span class="reference-text">S. G. Simpson, <i>Subsystems of Second-Order Arithmetic</i> (2009). Perspectives in Logic, ISBN 9780521884396.</span> </li> <li id="cite_note-FOOTNOTEKleene1943-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEKleene1943_8-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFKleene1943">Kleene 1943</a>.</span> </li> <li id="cite_note-FOOTNOTEShoenfield1967132Charlesworth1981HopcroftUllman1979-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEShoenfield1967132Charlesworth1981HopcroftUllman1979_9-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFShoenfield1967">Shoenfield 1967</a>, p. 132; <a href="#CITEREFCharlesworth1981">Charlesworth 1981</a>; <a href="#CITEREFHopcroftUllman1979">Hopcroft & Ullman 1979</a>.</span> </li> <li id="cite_note-FOOTNOTEFranzén200573-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEFranzén200573_10-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFFranzén2005">Franzén 2005</a>, p. 73.</span> </li> <li id="cite_note-FOOTNOTEDavis2006416Jones1980-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEDavis2006416Jones1980_11-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFDavis2006">Davis 2006</a>, p. 416; <a href="#CITEREFJones1980">Jones 1980</a>.</span> </li> <li id="cite_note-FOOTNOTESmoryński1977842Kleene1967274-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTESmoryński1977842Kleene1967274_12-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFSmoryński1977">Smoryński 1977</a>, p. 842; <a href="#CITEREFKleene1967">Kleene 1967</a>, p. 274.</span> </li> <li id="cite_note-FOOTNOTEBoolos1998383-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEBoolos1998383_13-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFBoolos1998">Boolos 1998</a>, p. 383.</span> </li> <li id="cite_note-FOOTNOTEBoolos1998388-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEBoolos1998388_14-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFBoolos1998">Boolos 1998</a>, p. 388.</span> </li> <li id="cite_note-FOOTNOTEHellman1981451–468-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEHellman1981451–468_15-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFHellman1981">Hellman 1981</a>, pp. 451–468.</span> </li> <li id="cite_note-FOOTNOTEPutnam1960-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEPutnam1960_16-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFPutnam1960">Putnam 1960</a>.</span> </li> <li id="cite_note-FOOTNOTEWigderson2010-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEWigderson2010_17-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFWigderson2010">Wigderson 2010</a>.</span> </li> <li id="cite_note-FOOTNOTEHofstadter2007-18"><span class="mw-cite-backlink">^ <a href="#cite_ref-FOOTNOTEHofstadter2007_18-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-FOOTNOTEHofstadter2007_18-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><a href="#CITEREFHofstadter2007">Hofstadter 2007</a>.</span> </li> <li id="cite_note-FOOTNOTEPriest1984Priest2006-19"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEPriest1984Priest2006_19-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFPriest1984">Priest 1984</a>; <a href="#CITEREFPriest2006">Priest 2006</a>.</span> </li> <li id="cite_note-FOOTNOTEPriest200647-20"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEPriest200647_20-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFPriest2006">Priest 2006</a>, p. 47.</span> </li> <li id="cite_note-FOOTNOTEShapiro2002-21"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEShapiro2002_21-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFShapiro2002">Shapiro 2002</a>.</span> </li> <li id="cite_note-FOOTNOTEFranzén2005Raatikainen2005SokalBricmont1999StangroomBenson2006-22"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEFranzén2005Raatikainen2005SokalBricmont1999StangroomBenson2006_22-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFFranzén2005">Franzén 2005</a>; <a href="#CITEREFRaatikainen2005">Raatikainen 2005</a>; <a href="#CITEREFSokalBricmont1999">Sokal & Bricmont 1999</a>; <a href="#CITEREFStangroomBenson2006">Stangroom & Benson 2006</a>.</span> </li> <li id="cite_note-FOOTNOTESokalBricmont1999StangroomBenson200610SokalBricmont1999187-23"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTESokalBricmont1999StangroomBenson200610SokalBricmont1999187_23-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFSokalBricmont1999">Sokal & Bricmont 1999</a>; <a href="#CITEREFStangroomBenson2006">Stangroom & Benson 2006</a>, p. 10; <a href="#CITEREFSokalBricmont1999">Sokal & Bricmont 1999</a>, p. 187.</span> </li> <li id="cite_note-FOOTNOTEDawson199763-24"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEDawson199763_24-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFDawson1997">Dawson 1997</a>, p. 63.</span> </li> <li id="cite_note-FOOTNOTEZach2007418Zach200333-25"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEZach2007418Zach200333_25-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFZach2007">Zach 2007</a>, p. 418; <a href="#CITEREFZach2003">Zach 2003</a>, p. 33.</span> </li> <li id="cite_note-FOOTNOTEDawson199669-26"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEDawson199669_26-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFDawson1996">Dawson 1996</a>, p. 69.</span> </li> <li id="cite_note-FOOTNOTEDawson199672-27"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEDawson199672_27-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFDawson1996">Dawson 1996</a>, p. 72.</span> </li> <li id="cite_note-FOOTNOTEDawson199670-28"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEDawson199670_28-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFDawson1996">Dawson 1996</a>, p. 70.</span> </li> <li id="cite_note-FOOTNOTEvan_Heijenoort1967page_328,_footnote_68a-29"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEvan_Heijenoort1967page_328,_footnote_68a_29-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFvan_Heijenoort1967">van Heijenoort 1967</a>, page 328, footnote 68a.</span> </li> <li id="cite_note-FOOTNOTEFinsler1926-30"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEFinsler1926_30-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFFinsler1926">Finsler 1926</a>.</span> </li> <li id="cite_note-FOOTNOTEvan_Heijenoort1967328-31"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEvan_Heijenoort1967328_31-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFvan_Heijenoort1967">van Heijenoort 1967</a>, p. 328.</span> </li> <li id="cite_note-FOOTNOTEDawson199689-32"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEDawson199689_32-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFDawson1996">Dawson 1996</a>, p. 89.</span> </li> <li id="cite_note-FOOTNOTEDawson199676-33"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEDawson199676_33-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFDawson1996">Dawson 1996</a>, p. 76.</span> </li> <li id="cite_note-FOOTNOTEDawson199676Grattan-Guinness2005512–513-34"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEDawson199676Grattan-Guinness2005512–513_34-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFDawson1996">Dawson 1996</a>, p. 76; <a href="#CITEREFGrattan-Guinness2005">Grattan-Guinness 2005</a>, pp. 512–513.</span> </li> <li id="cite_note-FOOTNOTEGrattan-Guinness2005513-35"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEGrattan-Guinness2005513_35-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFGrattan-Guinness2005">Grattan-Guinness 2005</a>, pp. 513.</span> </li> <li id="cite_note-FOOTNOTEDawson199677-36"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEDawson199677_36-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFDawson1996">Dawson 1996</a>, p. 77.</span> </li> <li id="cite_note-FOOTNOTERodych2003FloydPutnam2000Priest2004-37"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTERodych2003FloydPutnam2000Priest2004_37-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFRodych2003">Rodych 2003</a>; <a href="#CITEREFFloydPutnam2000">Floyd & Putnam 2000</a>; <a href="#CITEREFPriest2004">Priest 2004</a>.</span> </li> <li id="cite_note-FOOTNOTEBerto2009208-38"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEBerto2009208_38-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFBerto2009">Berto 2009</a>, p. 208.</span> </li> <li id="cite_note-FOOTNOTEWang1996179-39"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEWang1996179_39-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFWang1996">Wang 1996</a>, p. 179.</span> </li> <li id="cite_note-FOOTNOTEFloydPutnam2000Rodych2003Berto2009-40"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEFloydPutnam2000Rodych2003Berto2009_40-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFFloydPutnam2000">Floyd & Putnam 2000</a>; <a href="#CITEREFRodych2003">Rodych 2003</a>; <a href="#CITEREFBerto2009">Berto 2009</a>.</span> </li> </ol></div></div> <div class="mw-heading mw-heading3"><h3 id="Articles_by_Gödel"><span id="Articles_by_G.C3.B6del"></span>Articles by Gödel</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=G%C3%B6del%27s_incompleteness_theorems&action=edit&section=41" title="Edit section: Articles by Gödel"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Kurt Gödel, 1931, "Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme, I", <a href="/wiki/Monatshefte_f%C3%BCr_Mathematik_und_Physik" class="mw-redirect" title="Monatshefte für Mathematik und Physik">Monatshefte für Mathematik und Physik</a>, v. 38 n. 1, pp. 173–198. <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><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%2FBF01700692">10.1007/BF01700692</a></li> <li>—, 1931, "Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme, I", in <a href="/wiki/Solomon_Feferman" title="Solomon Feferman">Solomon Feferman</a>, ed., 1986. <i>Kurt Gödel Collected works, Vol. I</i>. Oxford University Press, pp. 144–195. <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/978-0195147209" title="Special:BookSources/978-0195147209">978-0195147209</a>. The original German with a facing English translation, preceded by an introductory note by <a href="/wiki/Stephen_Cole_Kleene" title="Stephen Cole Kleene">Stephen Cole Kleene</a>.</li> <li>—, 1951, "Some basic theorems on the foundations of mathematics and their implications", in <a href="/wiki/Solomon_Feferman" title="Solomon Feferman">Solomon Feferman</a>, ed., 1995. <i>Kurt Gödel Collected works, Vol. III</i>, Oxford University Press, pp. 304–323. <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/978-0195147223" title="Special:BookSources/978-0195147223">978-0195147223</a>.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Translations,_during_his_lifetime,_of_Gödel's_paper_into_English"><span id="Translations.2C_during_his_lifetime.2C_of_G.C3.B6del.27s_paper_into_English"></span>Translations, during his lifetime, of Gödel's paper into English</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=G%C3%B6del%27s_incompleteness_theorems&action=edit&section=42" title="Edit section: Translations, during his lifetime, of Gödel's paper into English"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>None of the following agree in all translated words and in typography. The typography is a serious matter, because Gödel expressly wished to emphasize "those metamathematical notions that had been defined in their usual sense before . . ." (<a href="#CITEREFvan_Heijenoort1967">van Heijenoort 1967</a>, p. 595). Three translations exist. Of the first John Dawson states that: "The Meltzer translation was seriously deficient and received a devastating review in the <i>Journal of Symbolic Logic</i>; "Gödel also complained about Braithwaite's commentary (<a href="#CITEREFDawson1997">Dawson 1997</a>, p. 216). "Fortunately, the Meltzer translation was soon supplanted by a better one prepared by Elliott Mendelson for Martin Davis's anthology <i>The Undecidable</i> . . . he found the translation "not quite so good" as he had expected . . . [but because of time constraints he] agreed to its publication" (ibid). (In a footnote Dawson states that "he would regret his compliance, for the published volume was marred throughout by sloppy typography and numerous misprints" (ibid)). Dawson states that "The translation that Gödel favored was that by Jean van Heijenoort" (ibid). For the serious student another version exists as a set of lecture notes recorded by Stephen Kleene and J. B. Rosser "during lectures given by Gödel at to the Institute for Advanced Study during the spring of 1934" (cf commentary by <a href="#CITEREFDavis1965">Davis 1965</a>, p. 39 and beginning on p. 41); this version is titled "On Undecidable Propositions of Formal Mathematical Systems". In their order of publication: </p> <ul><li>B. Meltzer (translation) and <a href="/wiki/R._B._Braithwaite" title="R. B. Braithwaite">R. B. Braithwaite</a> (Introduction), 1962. <i>On Formally Undecidable Propositions of Principia Mathematica and Related Systems</i>, Dover Publications, New York (Dover edition 1992), <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/0-486-66980-7" title="Special:BookSources/0-486-66980-7">0-486-66980-7</a> (pbk.) This contains a useful translation of Gödel's German abbreviations on pp. 33–34. As noted above, typography, translation and commentary is suspect. Unfortunately, this translation was reprinted with all its suspect content by</li></ul> <dl><dd><ul><li><a href="/wiki/Stephen_Hawking" title="Stephen Hawking">Stephen Hawking</a> editor, 2005. <i>God Created the Integers: The Mathematical Breakthroughs That Changed History</i>, Running Press, Philadelphia, <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/0-7624-1922-9" title="Special:BookSources/0-7624-1922-9">0-7624-1922-9</a>. Gödel's paper appears starting on p. 1097, with Hawking's commentary starting on p. 1089.</li></ul></dd></dl> <ul><li><a href="/wiki/Martin_Davis_(mathematician)" title="Martin Davis (mathematician)">Martin Davis</a> editor, 1965. <i>The Undecidable: Basic Papers on Undecidable Propositions, Unsolvable problems and Computable Functions</i>, Raven Press, New York, no ISBN. Gödel's paper begins on page 5, preceded by one page of commentary.</li> <li><a href="/wiki/Jean_van_Heijenoort" title="Jean van Heijenoort">Jean van Heijenoort</a> editor, 1967, 3rd edition 1967. <i>From Frege to Gödel: A Source Book in Mathematical Logic, 1879-1931</i>, Harvard University Press, Cambridge Mass., <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/0-674-32449-8" title="Special:BookSources/0-674-32449-8">0-674-32449-8</a> (pbk). van Heijenoort did the translation. He states that "Professor Gödel approved the translation, which in many places was accommodated to his wishes." (p. 595). Gödel's paper begins on p. 595; van Heijenoort's commentary begins on p. 592.</li> <li>Martin Davis editor, 1965, ibid. "On Undecidable Propositions of Formal Mathematical Systems." A copy with Gödel's corrections of errata and Gödel's added notes begins on page 41, preceded by two pages of Davis's commentary. Until Davis included this in his volume this lecture existed only as mimeographed notes.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Articles_by_others">Articles by others</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=G%C3%B6del%27s_incompleteness_theorems&action=edit&section=43" title="Edit section: Articles by others"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBoolos1989" class="citation journal cs1"><a href="/wiki/George_Boolos" title="George Boolos">Boolos, George</a> (1989). "A New Proof of the Gödel Incompleteness Theorem". <i><a href="/wiki/Notices_of_the_American_Mathematical_Society" title="Notices of the American Mathematical Society">Notices of the American Mathematical Society</a></i>. <b>36</b>: 388–390, 676. <q>reprinted in <a href="#CITEREFBoolos1998">Boolos (1998</a>, pp. 383–388)</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Notices+of+the+American+Mathematical+Society&rft.atitle=A+New+Proof+of+the+G%C3%B6del+Incompleteness+Theorem&rft.volume=36&rft.pages=388-390%2C+676&rft.date=1989&rft.aulast=Boolos&rft.aufirst=George&rfr_id=info%3Asid%2Fen.wikipedia.org%3AG%C3%B6del%27s+incompleteness+theorems" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBoolos1998" class="citation book cs1"><a href="/wiki/George_Boolos" title="George Boolos">Boolos, George</a> (1998). <i>Logic, logic, and logic</i>. <a href="/wiki/Harvard_University_Press" title="Harvard University Press">Harvard University Press</a>. p. 443. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-674-53766-1" title="Special:BookSources/0-674-53766-1"><bdi>0-674-53766-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Logic%2C+logic%2C+and+logic&rft.pages=443&rft.pub=Harvard+University+Press&rft.date=1998&rft.isbn=0-674-53766-1&rft.aulast=Boolos&rft.aufirst=George&rfr_id=info%3Asid%2Fen.wikipedia.org%3AG%C3%B6del%27s+incompleteness+theorems" class="Z3988"></span></li> <li>Bernd Buldt, 2014, "<a rel="nofollow" class="external text" href="http://opus.ipfw.edu/cgi/viewcontent.cgi?article=1297&context=philos_facpubs">The Scope of Gödel's First Incompleteness Theorem</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20160306110140/http://opus.ipfw.edu/cgi/viewcontent.cgi?article=1297&context=philos_facpubs">Archived</a> 2016-03-06 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>", <i><a href="/wiki/Logica_Universalis" title="Logica Universalis">Logica Universalis</a></i>, v. 8, pp. 499–552. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs11787-014-0107-3">10.1007/s11787-014-0107-3</a></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCharlesworth1981" class="citation journal cs1">Charlesworth, Arthur (1981). "A Proof of Godel's Theorem in Terms of Computer Programs". <i>Mathematics Magazine</i>. <b>54</b> (3): 109–121. <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%2F2689794">10.2307/2689794</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/2689794">2689794</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Mathematics+Magazine&rft.atitle=A+Proof+of+Godel%27s+Theorem+in+Terms+of+Computer+Programs&rft.volume=54&rft.issue=3&rft.pages=109-121&rft.date=1981&rft_id=info%3Adoi%2F10.2307%2F2689794&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2689794%23id-name%3DJSTOR&rft.aulast=Charlesworth&rft.aufirst=Arthur&rfr_id=info%3Asid%2Fen.wikipedia.org%3AG%C3%B6del%27s+incompleteness+theorems" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDavis1965" class="citation book cs1">Davis, Martin (1965). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=euBQAAAAMAAJ"><i>The Undecidable: Basic Papers on Undecidable Propositions, Unsolvable Problems and Computable Functions</i></a>. Raven Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-911216-01-1" title="Special:BookSources/978-0-911216-01-1"><bdi>978-0-911216-01-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Undecidable%3A+Basic+Papers+on+Undecidable+Propositions%2C+Unsolvable+Problems+and+Computable+Functions&rft.pub=Raven+Press&rft.date=1965&rft.isbn=978-0-911216-01-1&rft.aulast=Davis&rft.aufirst=Martin&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DeuBQAAAAMAAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3AG%C3%B6del%27s+incompleteness+theorems" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDavis2006" class="citation journal cs1"><a href="/wiki/Martin_Davis_(mathematician)" title="Martin Davis (mathematician)">Davis, Martin</a> (2006). <a rel="nofollow" class="external text" href="https://www.ams.org/notices/200604/fea-davis.pdf">"The Incompleteness Theorem"</a> <span class="cs1-format">(PDF)</span>. <i>Notices of the AMS</i>. <b>53</b> (4): 414.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Notices+of+the+AMS&rft.atitle=The+Incompleteness+Theorem&rft.volume=53&rft.issue=4&rft.pages=414&rft.date=2006&rft.aulast=Davis&rft.aufirst=Martin&rft_id=https%3A%2F%2Fwww.ams.org%2Fnotices%2F200604%2Ffea-davis.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AG%C3%B6del%27s+incompleteness+theorems" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFinsler1926" class="citation journal cs1">Finsler, Paul (1926). <a rel="nofollow" class="external text" href="http://www.digizeitschriften.de/dms/resolveppn/?PPN=GDZPPN002369001">"Formale Beweise und die Entscheidbarkeit"</a>. <i>Mathematische Zeitschrift</i>. <b>25</b>: 676–682. <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%2Fbf01283861">10.1007/bf01283861</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:121054124">121054124</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Mathematische+Zeitschrift&rft.atitle=Formale+Beweise+und+die+Entscheidbarkeit&rft.volume=25&rft.pages=676-682&rft.date=1926&rft_id=info%3Adoi%2F10.1007%2Fbf01283861&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A121054124%23id-name%3DS2CID&rft.aulast=Finsler&rft.aufirst=Paul&rft_id=http%3A%2F%2Fwww.digizeitschriften.de%2Fdms%2Fresolveppn%2F%3FPPN%3DGDZPPN002369001&rfr_id=info%3Asid%2Fen.wikipedia.org%3AG%C3%B6del%27s+incompleteness+theorems" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGrattan-Guinness2005" class="citation book cs1"><a href="/wiki/Ivor_Grattan-Guinness" title="Ivor Grattan-Guinness">Grattan-Guinness, Ivor</a>, ed. (2005). <i>Landmark Writings in Western Mathematics 1640-1940</i>. Elsevier. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780444508713" title="Special:BookSources/9780444508713"><bdi>9780444508713</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Landmark+Writings+in+Western+Mathematics+1640-1940&rft.pub=Elsevier&rft.date=2005&rft.isbn=9780444508713&rfr_id=info%3Asid%2Fen.wikipedia.org%3AG%C3%B6del%27s+incompleteness+theorems" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFvan_Heijenoort1967" class="citation book cs1"><a href="/wiki/Jean_van_Heijenoort" title="Jean van Heijenoort">van Heijenoort, Jean</a> (1967). "Gödel's Theorem". In Edwards, Paul (ed.). <i><a href="/wiki/The_Encyclopedia_of_Philosophy" title="The Encyclopedia of Philosophy">Encyclopedia of Philosophy</a></i>. Vol. 3. Macmillan. pp. 348–357.</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%27s+Theorem&rft.btitle=Encyclopedia+of+Philosophy&rft.pages=348-357&rft.pub=Macmillan&rft.date=1967&rft.aulast=van+Heijenoort&rft.aufirst=Jean&rfr_id=info%3Asid%2Fen.wikipedia.org%3AG%C3%B6del%27s+incompleteness+theorems" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHellman1981" class="citation journal cs1"><a href="/wiki/Geoffrey_Hellman" title="Geoffrey Hellman">Hellman, Geoffrey</a> (1981). "How to Gödel a Frege-Russell: Gödel's Incompleteness Theorems and Logicism". <i>Noûs</i>. <b>15</b> (4 - Special Issue on Philosophy of Mathematics): 451–468. <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%2F2214847">10.2307/2214847</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0029-4624">0029-4624</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/2214847">2214847</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=No%C3%BBs&rft.atitle=How+to+G%C3%B6del+a+Frege-Russell%3A+G%C3%B6del%27s+Incompleteness+Theorems+and+Logicism&rft.volume=15&rft.issue=4+-+Special+Issue+on+Philosophy+of+Mathematics&rft.pages=451-468&rft.date=1981&rft.issn=0029-4624&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2214847%23id-name%3DJSTOR&rft_id=info%3Adoi%2F10.2307%2F2214847&rft.aulast=Hellman&rft.aufirst=Geoffrey&rfr_id=info%3Asid%2Fen.wikipedia.org%3AG%C3%B6del%27s+incompleteness+theorems" class="Z3988"></span></li> <li><a href="/wiki/David_Hilbert" title="David Hilbert">David Hilbert</a>, 1900, "<a rel="nofollow" class="external text" href="http://aleph0.clarku.edu/~djoyce/hilbert/problems.html#prob2">Mathematical Problems.</a>" English translation of a lecture delivered before the International Congress of Mathematicians at Paris, containing Hilbert's statement of his Second Problem.</li> <li>Martin Hirzel, 2000, "<a rel="nofollow" class="external text" href="https://web.archive.org/web/20040916041216/http://www.research.ibm.com/people/h/hirzel/papers/canon00-goedel.pdf">On formally undecidable propositions of Principia Mathematica and related systems I.</a>." An English translation of Gödel's paper. Archived from <a rel="nofollow" class="external text" href="http://www.research.ibm.com/people/h/hirzel/papers/canon00-goedel.pdf">the original</a>. Sept. 16, 2004.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKikuchiTanaka1994" class="citation journal cs1">Kikuchi, Makoto; Tanaka, Kazuyuki (July 1994). <a rel="nofollow" class="external text" href="https://doi.org/10.1305%2Fndjfl%2F1040511346">"On Formalization of Model-Theoretic Proofs of Gödel's Theorems"</a>. <i>Notre Dame Journal of Formal Logic</i>. <b>35</b> (3): 403–412. <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/10.1305%2Fndjfl%2F1040511346">10.1305/ndjfl/1040511346</a></span>. <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1326122">1326122</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Notre+Dame+Journal+of+Formal+Logic&rft.atitle=On+Formalization+of+Model-Theoretic+Proofs+of+G%C3%B6del%27s+Theorems&rft.volume=35&rft.issue=3&rft.pages=403-412&rft.date=1994-07&rft_id=info%3Adoi%2F10.1305%2Fndjfl%2F1040511346&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1326122%23id-name%3DMR&rft.aulast=Kikuchi&rft.aufirst=Makoto&rft.au=Tanaka%2C+Kazuyuki&rft_id=https%3A%2F%2Fdoi.org%2F10.1305%252Fndjfl%252F1040511346&rfr_id=info%3Asid%2Fen.wikipedia.org%3AG%C3%B6del%27s+incompleteness+theorems" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKleene1943" class="citation journal cs1"><a href="/wiki/Stephen_Cole_Kleene" title="Stephen Cole Kleene">Kleene, S. 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Retrieved <span class="nowrap">November 7,</span> 2022</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=G%C3%B6del%27s+Incompleteness+Theorems&rft.pub=Stanford+Encyclopedia+of+Philosophy&rft.date=2020&rft.aulast=Raatikainen&rft.aufirst=Panu&rft_id=https%3A%2F%2Fplato.stanford.edu%2Fentries%2Fgoedel-incompleteness%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AG%C3%B6del%27s+incompleteness+theorems" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRaatikainen2005" class="citation journal cs1">Raatikainen, Panu (2005). <a rel="nofollow" class="external text" href="https://www.cairn.info/revue-internationale-de-philosophie-2005-4-page-513.htm">"On the philosophical relevance of Gödel's incompleteness theorems"</a>. <i>Revue Internationale de Philosophie</i>. <b>59</b> (4): 513–534. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.3917%2Frip.234.0513">10.3917/rip.234.0513</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:52083793">52083793</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Revue+Internationale+de+Philosophie&rft.atitle=On+the+philosophical+relevance+of+G%C3%B6del%27s+incompleteness+theorems&rft.volume=59&rft.issue=4&rft.pages=513-534&rft.date=2005&rft_id=info%3Adoi%2F10.3917%2Frip.234.0513&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A52083793%23id-name%3DS2CID&rft.aulast=Raatikainen&rft.aufirst=Panu&rft_id=https%3A%2F%2Fwww.cairn.info%2Frevue-internationale-de-philosophie-2005-4-page-513.htm&rfr_id=info%3Asid%2Fen.wikipedia.org%3AG%C3%B6del%27s+incompleteness+theorems" class="Z3988"></span></li> <li><a href="/wiki/John_Barkley_Rosser" class="mw-redirect" title="John Barkley Rosser">John Barkley Rosser</a>, 1936, "Extensions of some theorems of Gödel and Church", reprinted from the <i>Journal of Symbolic Logic</i>, v. 1 (1936) pp. 87–91, in Martin Davis 1965, <i>The Undecidable</i> (loc. cit.) pp. 230–235.</li> <li>—, 1939, "An Informal Exposition of proofs of Gödel's Theorem and Church's Theorem", Reprinted from the <i>Journal of Symbolic Logic</i>, v. 4 (1939) pp. 53–60, in Martin Davis 1965, <i>The Undecidable</i> (loc. cit.) pp. 223–230</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSmoryński1977" class="citation book cs1">Smoryński, C. (1977). "The incompleteness theorems". In <a href="/wiki/Jon_Barwise" title="Jon Barwise">Jon Barwise</a> (ed.). <i>Handbook of mathematical logic</i>. Amsterdam: North-Holland Pub. Co. pp. 821–866. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-444-86388-1" title="Special:BookSources/978-0-444-86388-1"><bdi>978-0-444-86388-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=The+incompleteness+theorems&rft.btitle=Handbook+of+mathematical+logic&rft.place=Amsterdam&rft.pages=821-866&rft.pub=North-Holland+Pub.+Co&rft.date=1977&rft.isbn=978-0-444-86388-1&rft.aulast=Smory%C5%84ski&rft.aufirst=C.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AG%C3%B6del%27s+incompleteness+theorems" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWillard2001" class="citation journal cs1">Willard, Dan E. (2001). "Self-Verifying Axiom Systems, the Incompleteness Theorem and Related Reflection Principles". <i>Journal of Symbolic Logic</i>. <b>66</b> (2): 536–596. <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%2F2695030">10.2307/2695030</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/2695030">2695030</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=Self-Verifying+Axiom+Systems%2C+the+Incompleteness+Theorem+and+Related+Reflection+Principles&rft.volume=66&rft.issue=2&rft.pages=536-596&rft.date=2001&rft_id=info%3Adoi%2F10.2307%2F2695030&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2695030%23id-name%3DJSTOR&rft.aulast=Willard&rft.aufirst=Dan+E.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AG%C3%B6del%27s+incompleteness+theorems" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFZach2003" class="citation journal cs1"><a href="/wiki/Richard_Zach" title="Richard Zach">Zach, Richard</a> (2003). <a rel="nofollow" class="external text" href="http://people.ucalgary.ca/~rzach/static/conprf.pdf">"The Practice of Finitism: Epsilon Calculus and Consistency Proofs in Hilbert's Program"</a> <span class="cs1-format">(PDF)</span>. <i><a href="/wiki/Synthese" title="Synthese">Synthese</a></i>. <b>137</b> (1). Springer Science and Business Media LLC: 211–259. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/math/0102189">math/0102189</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/10.1023%2Fa%3A1026247421383">10.1023/a:1026247421383</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0039-7857">0039-7857</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:16657040">16657040</a>.</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=The+Practice+of+Finitism%3A+Epsilon+Calculus+and+Consistency+Proofs+in+Hilbert%27s+Program&rft.volume=137&rft.issue=1&rft.pages=211-259&rft.date=2003&rft_id=info%3Aarxiv%2Fmath%2F0102189&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A16657040%23id-name%3DS2CID&rft.issn=0039-7857&rft_id=info%3Adoi%2F10.1023%2Fa%3A1026247421383&rft.aulast=Zach&rft.aufirst=Richard&rft_id=http%3A%2F%2Fpeople.ucalgary.ca%2F~rzach%2Fstatic%2Fconprf.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AG%C3%B6del%27s+incompleteness+theorems" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFZach2005" class="citation book cs1"><a href="/wiki/Richard_Zach" title="Richard Zach">Zach, Richard</a> (2005). "Kurt Gödel, paper on the incompleteness theorems (1931)". In <a href="/wiki/Ivor_Grattan-Guinness" title="Ivor Grattan-Guinness">Grattan-Guinness, Ivor</a> (ed.). <a rel="nofollow" class="external text" href="https://philarchive.org/rec/ZACKGP"><i>Landmark Writings in Western Mathematics 1640-1940</i></a>. Elsevier. pp. 917–925. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fb978-044450871-3%2F50152-2">10.1016/b978-044450871-3/50152-2</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780444508713" title="Special:BookSources/9780444508713"><bdi>9780444508713</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Kurt+G%C3%B6del%2C+paper+on+the+incompleteness+theorems+%281931%29&rft.btitle=Landmark+Writings+in+Western+Mathematics+1640-1940&rft.pages=917-925&rft.pub=Elsevier&rft.date=2005&rft_id=info%3Adoi%2F10.1016%2Fb978-044450871-3%2F50152-2&rft.isbn=9780444508713&rft.aulast=Zach&rft.aufirst=Richard&rft_id=https%3A%2F%2Fphilarchive.org%2Frec%2FZACKGP&rfr_id=info%3Asid%2Fen.wikipedia.org%3AG%C3%B6del%27s+incompleteness+theorems" class="Z3988"></span></li></ul> <div class="mw-heading mw-heading3"><h3 id="Books_about_the_theorems">Books about the theorems</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=G%C3%B6del%27s_incompleteness_theorems&action=edit&section=44" title="Edit section: Books about the theorems"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Francesco Berto. <i>There's Something about Gödel: The Complete Guide to the Incompleteness Theorem</i> John Wiley and Sons. 2010.</li> <li>Norbert Domeisen, 1990. <a rel="nofollow" class="external text" href="https://archive.today/20240527193320/https://www.webcitation.org/6gQ72rSwF?url=http://www.textarchiv.homepage.bluewin.ch/Antinomien/Logik_der_Antinomien.html"><i>Logik der Antinomien</i></a>. Bern: Peter Lang. 142 S. 1990. <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/3-261-04214-1" title="Special:BookSources/3-261-04214-1">3-261-04214-1</a>. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:0724.03003">0724.03003</a>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFranzén2005" class="citation book cs1"><a href="/wiki/Torkel_Franz%C3%A9n" title="Torkel Franzén">Franzén, Torkel</a> (2005). <i>Gödel's theorem : an incomplete guide to its use and abuse</i>. Wellesley, MA: A K Peters. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/1-56881-238-8" title="Special:BookSources/1-56881-238-8"><bdi>1-56881-238-8</bdi></a>. <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=2146326">2146326</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=G%C3%B6del%27s+theorem+%3A+an+incomplete+guide+to+its+use+and+abuse&rft.place=Wellesley%2C+MA&rft.pub=A+K+Peters&rft.date=2005&rft.isbn=1-56881-238-8&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D2146326%23id-name%3DMR&rft.aulast=Franz%C3%A9n&rft.aufirst=Torkel&rfr_id=info%3Asid%2Fen.wikipedia.org%3AG%C3%B6del%27s+incompleteness+theorems" class="Z3988"></span></li> <li><a href="/wiki/Douglas_Hofstadter" title="Douglas Hofstadter">Douglas Hofstadter</a>, 1979. <i><a href="/wiki/G%C3%B6del,_Escher,_Bach" title="Gödel, Escher, Bach">Gödel, Escher, Bach: An Eternal Golden Braid</a></i>. Vintage Books. <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/0-465-02685-0" title="Special:BookSources/0-465-02685-0">0-465-02685-0</a>. 1999 reprint: <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/0-465-02656-7" title="Special:BookSources/0-465-02656-7">0-465-02656-7</a>. <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=530196">530196</a></li> <li> —, 2007. <i><a href="/wiki/I_Am_a_Strange_Loop" title="I Am a Strange Loop">I Am a Strange Loop</a></i>. Basic Books. <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/978-0-465-03078-1" title="Special:BookSources/978-0-465-03078-1">978-0-465-03078-1</a>. <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/0-465-03078-5" title="Special:BookSources/0-465-03078-5">0-465-03078-5</a>. <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=2360307">2360307</a></li> <li><a href="/wiki/Stanley_Jaki" title="Stanley Jaki">Stanley Jaki</a>, OSB, 2005. <i>The drama of the quantities</i>. <a rel="nofollow" class="external text" href="http://www.realviewbooks.com/">Real View Books.</a></li> <li><a href="/wiki/Per_Lindstr%C3%B6m" title="Per Lindström">Per Lindström</a>, 1997. <i><a rel="nofollow" class="external text" href="https://projecteuclid.org/euclid.lnl/1235416274">Aspects of Incompleteness</a></i>, Lecture Notes in Logic v. 10.</li> <li><a href="/wiki/J.R._Lucas" class="mw-redirect" title="J.R. Lucas">J.R. Lucas</a>, FBA, 1970. <i>The Freedom of the Will</i>. Clarendon Press, Oxford, 1970.</li> <li><a href="/wiki/Adrian_William_Moore" title="Adrian William Moore">Adrian William Moore</a>, 2022. <i>Gödel´s Theorem: A Very Short Introduction</i>. Oxford University Press, Oxford, 2022.</li> <li><a href="/wiki/Ernest_Nagel" title="Ernest Nagel">Ernest Nagel</a>, <a href="/wiki/James_R._Newman" title="James R. Newman">James Roy Newman</a>, Douglas Hofstadter, 2002 (1958). <i>Gödel's Proof</i>, revised ed. <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/0-8147-5816-9" title="Special:BookSources/0-8147-5816-9">0-8147-5816-9</a>. <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1871678">1871678</a></li> <li><a href="/wiki/Rudy_Rucker" title="Rudy Rucker">Rudy Rucker</a>, 1995 (1982). <i>Infinity and the Mind: The Science and Philosophy of the Infinite</i>. Princeton Univ. Press. <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=658492">658492</a></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSmith2007" class="citation book cs1">Smith, Peter (2007). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20051023200804/http://www.godelbook.net/"><i>An introduction to Gödel's Theorems</i></a>. Cambridge, U.K.: Cambridge University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-67453-9" title="Special:BookSources/978-0-521-67453-9"><bdi>978-0-521-67453-9</bdi></a>. <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=2384958">2384958</a>. Archived from <a rel="nofollow" class="external text" href="http://www.godelbook.net/">the original</a> on 2005-10-23<span class="reference-accessdate">. Retrieved <span class="nowrap">2005-10-29</span></span>.</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+G%C3%B6del%27s+Theorems&rft.place=Cambridge%2C+U.K.&rft.pub=Cambridge+University+Press&rft.date=2007&rft.isbn=978-0-521-67453-9&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D2384958%23id-name%3DMR&rft.aulast=Smith&rft.aufirst=Peter&rft_id=http%3A%2F%2Fwww.godelbook.net%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AG%C3%B6del%27s+incompleteness+theorems" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFShankar1994" class="citation book cs1">Shankar, N. (1994). <i>Metamathematics, machines, and Gödel's proof</i>. Cambridge tracts in theoretical computer science. Vol. 38. Cambridge: Cambridge University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-521-58533-3" title="Special:BookSources/0-521-58533-3"><bdi>0-521-58533-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Metamathematics%2C+machines%2C+and+G%C3%B6del%27s+proof&rft.place=Cambridge&rft.series=Cambridge+tracts+in+theoretical+computer+science&rft.pub=Cambridge+University+Press&rft.date=1994&rft.isbn=0-521-58533-3&rft.aulast=Shankar&rft.aufirst=N.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AG%C3%B6del%27s+incompleteness+theorems" class="Z3988"></span></li> <li><a href="/wiki/Raymond_Smullyan" title="Raymond Smullyan">Raymond Smullyan</a>, 1987. <i>Forever Undecided</i> <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/0192801414" title="Special:BookSources/0192801414">0192801414</a> - puzzles based on undecidability in formal systems</li> <li>—, 1992. <i>Godel's Incompleteness Theorems</i>. Oxford Univ. Press. <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/0195046722" title="Special:BookSources/0195046722">0195046722</a></li> <li>—, 1994. <i>Diagonalization and Self-Reference</i>. Oxford Univ. Press. <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1318913">1318913</a>. <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/0198534507" title="Special:BookSources/0198534507">0198534507</a></li> <li>—, 2013. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=xUapAAAAQBAJ"><i>The Godelian Puzzle Book: Puzzles, Paradoxes and Proofs</i></a>. Courier Corporation. <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/978-0-486-49705-1" title="Special:BookSources/978-0-486-49705-1">978-0-486-49705-1</a>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWang1996" class="citation book cs1"><a href="/wiki/Hao_Wang_(academic)" title="Hao Wang (academic)">Wang, Hao</a> (1996). <i>A Logical Journey: From Gödel to Philosophy</i>. MIT Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-262-23189-1" title="Special:BookSources/0-262-23189-1"><bdi>0-262-23189-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Logical+Journey%3A+From+G%C3%B6del+to+Philosophy&rft.pub=MIT+Press&rft.date=1996&rft.isbn=0-262-23189-1&rft.aulast=Wang&rft.aufirst=Hao&rfr_id=info%3Asid%2Fen.wikipedia.org%3AG%C3%B6del%27s+incompleteness+theorems" class="Z3988"></span> <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1433803">1433803</a></li></ul> <div class="mw-heading mw-heading3"><h3 id="Miscellaneous_references">Miscellaneous references</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=G%C3%B6del%27s_incompleteness_theorems&action=edit&section=45" title="Edit section: Miscellaneous references"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBerto2009" class="citation journal cs1">Berto, Francesco (2009). "The Gödel Paradox and Wittgenstein's Reasons". <i><a href="/wiki/Philosophia_Mathematica" title="Philosophia Mathematica">Philosophia Mathematica</a></i>. <b>III</b> (17).</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Philosophia+Mathematica&rft.atitle=The+G%C3%B6del+Paradox+and+Wittgenstein%27s+Reasons&rft.volume=III&rft.issue=17&rft.date=2009&rft.aulast=Berto&rft.aufirst=Francesco&rfr_id=info%3Asid%2Fen.wikipedia.org%3AG%C3%B6del%27s+incompleteness+theorems" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDawson1996" class="citation book cs1">Dawson, John W. Jr. (1996). <i>Logical dilemmas: The life and work of Kurt Gödel</i>. Taylor & Francis. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-56881-025-6" title="Special:BookSources/978-1-56881-025-6"><bdi>978-1-56881-025-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Logical+dilemmas%3A+The+life+and+work+of+Kurt+G%C3%B6del&rft.pub=Taylor+%26+Francis&rft.date=1996&rft.isbn=978-1-56881-025-6&rft.aulast=Dawson&rft.aufirst=John+W.+Jr.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AG%C3%B6del%27s+incompleteness+theorems" class="Z3988"></span></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: <a href="/wiki/A._K._Peters" class="mw-redirect" title="A. K. Peters">A. K. Peters</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-56881-256-4" title="Special:BookSources/978-1-56881-256-4"><bdi>978-1-56881-256-4</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/36104240">36104240</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Logical+dilemmas%3A+The+life+and+work+of+Kurt+G%C3%B6del&rft.place=Wellesley%2C+Massachusetts&rft.pub=A.+K.+Peters&rft.date=1997&rft_id=info%3Aoclcnum%2F36104240&rft.isbn=978-1-56881-256-4&rft.aulast=Dawson&rft.aufirst=John+W.+Jr.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AG%C3%B6del%27s+incompleteness+theorems" class="Z3988"></span></li> <li><a href="/wiki/Rebecca_Goldstein" title="Rebecca Goldstein">Rebecca Goldstein</a>, 2005, <i>Incompleteness: the Proof and Paradox of Kurt Gödel</i>, W. W. Norton & Company. <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/0-393-05169-2" title="Special:BookSources/0-393-05169-2">0-393-05169-2</a></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFloydPutnam2000" class="citation journal cs1">Floyd, Juliet; Putnam, Hilary (2000). "A Note on Wittgenstein's "Notorious Paragraph" about the Godel Theorem". <i><a href="/wiki/The_Journal_of_Philosophy" title="The Journal of Philosophy">The Journal of Philosophy</a></i>. <b>97</b> (11). JSTOR: 624–632. <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%2F2678455">10.2307/2678455</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0022-362X">0022-362X</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/2678455">2678455</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+Philosophy&rft.atitle=A+Note+on+Wittgenstein%27s+%22Notorious+Paragraph%22+about+the+Godel+Theorem&rft.volume=97&rft.issue=11&rft.pages=624-632&rft.date=2000&rft.issn=0022-362X&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2678455%23id-name%3DJSTOR&rft_id=info%3Adoi%2F10.2307%2F2678455&rft.aulast=Floyd&rft.aufirst=Juliet&rft.au=Putnam%2C+Hilary&rfr_id=info%3Asid%2Fen.wikipedia.org%3AG%C3%B6del%27s+incompleteness+theorems" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHarrison2009" class="citation book cs1">Harrison, J. (2009). <i>Handbook of practical logic and automated reasoning</i>. Cambridge: Cambridge University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0521899574" title="Special:BookSources/978-0521899574"><bdi>978-0521899574</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Handbook+of+practical+logic+and+automated+reasoning&rft.place=Cambridge&rft.pub=Cambridge+University+Press&rft.date=2009&rft.isbn=978-0521899574&rft.aulast=Harrison&rft.aufirst=J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AG%C3%B6del%27s+incompleteness+theorems" class="Z3988"></span></li> <li><a href="/wiki/David_Hilbert" title="David Hilbert">David Hilbert</a> and <a href="/wiki/Paul_Bernays" title="Paul Bernays">Paul Bernays</a>, <i><a href="/wiki/Grundlagen_der_Mathematik" title="Grundlagen der Mathematik">Grundlagen der Mathematik</a></i>, Springer-Verlag.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHopcroftUllman1979" class="citation book cs1"><a href="/wiki/John_Hopcroft" title="John Hopcroft">Hopcroft, John E.</a>; <a href="/wiki/Jeffrey_Ullman" title="Jeffrey Ullman">Ullman, Jeffrey</a> (1979). <i><a href="/wiki/Introduction_to_Automata_Theory,_Languages,_and_Computation" title="Introduction to Automata Theory, Languages, and Computation">Introduction to Automata Theory, Languages, and Computation</a></i>. Reading, Mass.: Addison-Wesley. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-201-02988-X" title="Special:BookSources/0-201-02988-X"><bdi>0-201-02988-X</bdi></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+Automata+Theory%2C+Languages%2C+and+Computation&rft.place=Reading%2C+Mass.&rft.pub=Addison-Wesley&rft.date=1979&rft.isbn=0-201-02988-X&rft.aulast=Hopcroft&rft.aufirst=John+E.&rft.au=Ullman%2C+Jeffrey&rfr_id=info%3Asid%2Fen.wikipedia.org%3AG%C3%B6del%27s+incompleteness+theorems" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHofstadter2007" class="citation book cs1"><a href="/wiki/Douglas_Hofstadter" title="Douglas Hofstadter">Hofstadter, Douglas R.</a> (2007) [2003]. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20190508182834/https://publicism.info/philosophy/strange/14.html">"Chapter 12. On Downward Causality"</a>. <a href="/wiki/I_Am_a_Strange_Loop" title="I Am a Strange Loop"><i>I Am a Strange Loop</i></a>. Basic Books. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-465-03078-1" title="Special:BookSources/978-0-465-03078-1"><bdi>978-0-465-03078-1</bdi></a>. Archived from <a rel="nofollow" class="external text" href="https://publicism.info/philosophy/strange/14.html">the original</a> on 2019-05-08<span class="reference-accessdate">. Retrieved <span class="nowrap">2018-10-24</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Chapter+12.+On+Downward+Causality&rft.btitle=I+Am+a+Strange+Loop&rft.pub=Basic+Books&rft.date=2007&rft.isbn=978-0-465-03078-1&rft.aulast=Hofstadter&rft.aufirst=Douglas+R.&rft_id=https%3A%2F%2Fpublicism.info%2Fphilosophy%2Fstrange%2F14.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AG%C3%B6del%27s+incompleteness+theorems" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJones1980" class="citation journal cs1">Jones, James P. (1980). <a rel="nofollow" class="external text" href="https://www.ams.org/bull/1980-03-02/S0273-0979-1980-14832-6/S0273-0979-1980-14832-6.pdf">"Undecidable Diophantine Equations"</a> <span class="cs1-format">(PDF)</span>. <i>Bulletin of the American Mathematical Society</i>. <b>3</b> (2): 859–862. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1090%2FS0273-0979-1980-14832-6">10.1090/S0273-0979-1980-14832-6</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+the+American+Mathematical+Society&rft.atitle=Undecidable+Diophantine+Equations&rft.volume=3&rft.issue=2&rft.pages=859-862&rft.date=1980&rft_id=info%3Adoi%2F10.1090%2FS0273-0979-1980-14832-6&rft.aulast=Jones&rft.aufirst=James+P.&rft_id=https%3A%2F%2Fwww.ams.org%2Fbull%2F1980-03-02%2FS0273-0979-1980-14832-6%2FS0273-0979-1980-14832-6.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AG%C3%B6del%27s+incompleteness+theorems" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKleene1967" class="citation book cs1"><a href="/wiki/Stephen_Cole_Kleene" title="Stephen Cole Kleene">Kleene, Stephen Cole</a> (1967). <i>Mathematical Logic</i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mathematical+Logic&rft.date=1967&rft.aulast=Kleene&rft.aufirst=Stephen+Cole&rfr_id=info%3Asid%2Fen.wikipedia.org%3AG%C3%B6del%27s+incompleteness+theorems" class="Z3988"></span> Reprinted by Dover, 2002. <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/0-486-42533-9" title="Special:BookSources/0-486-42533-9">0-486-42533-9</a></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFO'Connor2005" class="citation book cs1">O'Connor, Russell (2005). "Essential Incompleteness of Arithmetic Verified by Coq". <i>Theorem Proving in Higher Order Logics</i>. Lecture Notes in Computer Science. Vol. 3603. pp. 245–260. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/cs/0505034">cs/0505034</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%2F11541868_16">10.1007/11541868_16</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-28372-0" title="Special:BookSources/978-3-540-28372-0"><bdi>978-3-540-28372-0</bdi></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:15610367">15610367</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Essential+Incompleteness+of+Arithmetic+Verified+by+Coq&rft.btitle=Theorem+Proving+in+Higher+Order+Logics&rft.series=Lecture+Notes+in+Computer+Science&rft.pages=245-260&rft.date=2005&rft_id=info%3Aarxiv%2Fcs%2F0505034&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A15610367%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1007%2F11541868_16&rft.isbn=978-3-540-28372-0&rft.aulast=O%27Connor&rft.aufirst=Russell&rfr_id=info%3Asid%2Fen.wikipedia.org%3AG%C3%B6del%27s+incompleteness+theorems" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPaulson2014" class="citation journal cs1"><a href="/wiki/Lawrence_Paulson" title="Lawrence Paulson">Paulson, Lawrence</a> (2014). <a rel="nofollow" class="external text" href="https://www.repository.cam.ac.uk/handle/1810/245422">"A machine-assisted proof of Gödel's incompleteness theorems for the theory of hereditarily finite sets"</a>. <i>Review of Symbolic Logic</i>. <b>7</b> (3): 484–498. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/2104.14260">2104.14260</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/10.1017%2FS1755020314000112">10.1017/S1755020314000112</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:13913592">13913592</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Review+of+Symbolic+Logic&rft.atitle=A+machine-assisted+proof+of+G%C3%B6del%27s+incompleteness+theorems+for+the+theory+of+hereditarily+finite+sets&rft.volume=7&rft.issue=3&rft.pages=484-498&rft.date=2014&rft_id=info%3Aarxiv%2F2104.14260&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A13913592%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1017%2FS1755020314000112&rft.aulast=Paulson&rft.aufirst=Lawrence&rft_id=https%3A%2F%2Fwww.repository.cam.ac.uk%2Fhandle%2F1810%2F245422&rfr_id=info%3Asid%2Fen.wikipedia.org%3AG%C3%B6del%27s+incompleteness+theorems" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPriest1984" class="citation journal cs1"><a href="/wiki/Graham_Priest" title="Graham Priest">Priest, Graham</a> (1984). "Logic of Paradox Revisited". <i>Journal of Philosophical Logic</i>. <b>13</b> (2): 153–179. <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%2FBF00453020">10.1007/BF00453020</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+Philosophical+Logic&rft.atitle=Logic+of+Paradox+Revisited&rft.volume=13&rft.issue=2&rft.pages=153-179&rft.date=1984&rft_id=info%3Adoi%2F10.1007%2FBF00453020&rft.aulast=Priest&rft.aufirst=Graham&rfr_id=info%3Asid%2Fen.wikipedia.org%3AG%C3%B6del%27s+incompleteness+theorems" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPriest2004" class="citation book cs1">Priest, Graham (2004). "Wittgenstein's Remarks on Gödel's Theorem". In Max Kölbel (ed.). <i>Wittgenstein's lasting significance</i>. Psychology Press. pp. 207–227. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-134-40617-3" title="Special:BookSources/978-1-134-40617-3"><bdi>978-1-134-40617-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Wittgenstein%27s+Remarks+on+G%C3%B6del%27s+Theorem&rft.btitle=Wittgenstein%27s+lasting+significance&rft.pages=207-227&rft.pub=Psychology+Press&rft.date=2004&rft.isbn=978-1-134-40617-3&rft.aulast=Priest&rft.aufirst=Graham&rfr_id=info%3Asid%2Fen.wikipedia.org%3AG%C3%B6del%27s+incompleteness+theorems" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPriest2006" class="citation book cs1"><a href="/wiki/Graham_Priest" title="Graham Priest">Priest, Graham</a> (2006). <i>In Contradiction: A Study of the Transconsistent</i>. Oxford University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-19-926329-9" title="Special:BookSources/0-19-926329-9"><bdi>0-19-926329-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=In+Contradiction%3A+A+Study+of+the+Transconsistent&rft.pub=Oxford+University+Press&rft.date=2006&rft.isbn=0-19-926329-9&rft.aulast=Priest&rft.aufirst=Graham&rfr_id=info%3Asid%2Fen.wikipedia.org%3AG%C3%B6del%27s+incompleteness+theorems" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPutnam1960" class="citation book cs1"><a href="/wiki/Hilary_Putnam" title="Hilary Putnam">Putnam, Hilary</a> (1960). "Minds and Machines". In <a href="/wiki/Sidney_Hook" title="Sidney Hook">Sidney Hook</a> (ed.). <i>Dimensions of Mind: A Symposium</i>. New York University Press.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Minds+and+Machines&rft.btitle=Dimensions+of+Mind%3A+A+Symposium&rft.pub=New+York+University+Press&rft.date=1960&rft.aulast=Putnam&rft.aufirst=Hilary&rfr_id=info%3Asid%2Fen.wikipedia.org%3AG%C3%B6del%27s+incompleteness+theorems" class="Z3988"></span> Reprinted in Anderson, A. R., ed., 1964. <i>Minds and Machines</i>. Prentice-Hall: 77.</li> <li><a href="/wiki/Wolfgang_Rautenberg" title="Wolfgang Rautenberg">Wolfgang Rautenberg</a>, 2010, <i><a rel="nofollow" class="external text" href="https://link.springer.com/book/10.1007%2F978-1-4419-1221-3">A Concise Introduction to Mathematical Logic</a></i>, 3rd. ed., Springer, <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/978-1-4419-1220-6" title="Special:BookSources/978-1-4419-1220-6">978-1-4419-1220-6</a></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRodych2003" class="citation journal cs1">Rodych, Victor (2003). "Misunderstanding Gödel: New Arguments about Wittgenstein and New Remarks by Wittgenstein". <i>Dialectica</i>. <b>57</b> (3): 279–313. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1111%2Fj.1746-8361.2003.tb00272.x">10.1111/j.1746-8361.2003.tb00272.x</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Dialectica&rft.atitle=Misunderstanding+G%C3%B6del%3A+New+Arguments+about+Wittgenstein+and+New+Remarks+by+Wittgenstein&rft.volume=57&rft.issue=3&rft.pages=279-313&rft.date=2003&rft_id=info%3Adoi%2F10.1111%2Fj.1746-8361.2003.tb00272.x&rft.aulast=Rodych&rft.aufirst=Victor&rfr_id=info%3Asid%2Fen.wikipedia.org%3AG%C3%B6del%27s+incompleteness+theorems" class="Z3988"></span> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1111%2Fj.1746-8361.2003.tb00272.x">10.1111/j.1746-8361.2003.tb00272.x</a></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFShelah1974" class="citation journal cs1"><a href="/wiki/Saharon_Shelah" title="Saharon Shelah">Shelah, Saharon</a> (1974). <a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF02757281">"Infinite Abelian groups, Whitehead problem and some constructions"</a>. <i><a href="/wiki/Israel_Journal_of_Mathematics" title="Israel Journal of Mathematics">Israel Journal of Mathematics</a></i>. <b>18</b> (3): 243–256. <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/10.1007%2FBF02757281">10.1007/BF02757281</a></span>. <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0357114">0357114</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Israel+Journal+of+Mathematics&rft.atitle=Infinite+Abelian+groups%2C+Whitehead+problem+and+some+constructions&rft.volume=18&rft.issue=3&rft.pages=243-256&rft.date=1974&rft_id=info%3Adoi%2F10.1007%2FBF02757281&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D0357114%23id-name%3DMR&rft.aulast=Shelah&rft.aufirst=Saharon&rft_id=https%3A%2F%2Fdoi.org%2F10.1007%252FBF02757281&rfr_id=info%3Asid%2Fen.wikipedia.org%3AG%C3%B6del%27s+incompleteness+theorems" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFShapiro2002" class="citation journal cs1"><a href="/wiki/Stewart_Shapiro" title="Stewart Shapiro">Shapiro, Stewart</a> (2002). "Incompleteness and Inconsistency". <i>Mind</i>. <b>111</b> (444): 817–32. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1093%2Fmind%2F111.444.817">10.1093/mind/111.444.817</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Mind&rft.atitle=Incompleteness+and+Inconsistency&rft.volume=111&rft.issue=444&rft.pages=817-32&rft.date=2002&rft_id=info%3Adoi%2F10.1093%2Fmind%2F111.444.817&rft.aulast=Shapiro&rft.aufirst=Stewart&rfr_id=info%3Asid%2Fen.wikipedia.org%3AG%C3%B6del%27s+incompleteness+theorems" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSokalBricmont1999" class="citation book cs1"><a href="/wiki/Alan_Sokal" title="Alan Sokal">Sokal, Alan</a>; <a href="/wiki/Jean_Bricmont" title="Jean Bricmont">Bricmont, Jean</a> (1999). <i><a href="/wiki/Fashionable_Nonsense" title="Fashionable Nonsense">Fashionable Nonsense</a>: Postmodern Intellectuals' Abuse of Science</i>. Picador. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-312-20407-8" title="Special:BookSources/0-312-20407-8"><bdi>0-312-20407-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Fashionable+Nonsense%3A+Postmodern+Intellectuals%27+Abuse+of+Science&rft.pub=Picador&rft.date=1999&rft.isbn=0-312-20407-8&rft.aulast=Sokal&rft.aufirst=Alan&rft.au=Bricmont%2C+Jean&rfr_id=info%3Asid%2Fen.wikipedia.org%3AG%C3%B6del%27s+incompleteness+theorems" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFShoenfield1967" class="citation book cs1"><a href="/wiki/Joseph_R._Shoenfield" title="Joseph R. Shoenfield">Shoenfield, Joseph R.</a> (1967). <i>Mathematical logic</i>. Natick, Mass.: Association for Symbolic Logic (published 2001). <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-56881-135-2" title="Special:BookSources/978-1-56881-135-2"><bdi>978-1-56881-135-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mathematical+logic&rft.place=Natick%2C+Mass.&rft.pub=Association+for+Symbolic+Logic&rft.date=1967&rft.isbn=978-1-56881-135-2&rft.aulast=Shoenfield&rft.aufirst=Joseph+R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AG%C3%B6del%27s+incompleteness+theorems" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStangroomBenson2006" class="citation book cs1"><a href="/wiki/Jeremy_Stangroom" title="Jeremy Stangroom">Stangroom, Jeremy</a>; <a href="/wiki/Ophelia_Benson" title="Ophelia Benson">Benson, Ophelia</a> (2006). <i>Why Truth Matters</i>. Continuum. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-8264-9528-1" title="Special:BookSources/0-8264-9528-1"><bdi>0-8264-9528-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Why+Truth+Matters&rft.pub=Continuum&rft.date=2006&rft.isbn=0-8264-9528-1&rft.aulast=Stangroom&rft.aufirst=Jeremy&rft.au=Benson%2C+Ophelia&rfr_id=info%3Asid%2Fen.wikipedia.org%3AG%C3%B6del%27s+incompleteness+theorems" class="Z3988"></span></li> <li>George Tourlakis, <i>Lectures in Logic and Set Theory, Volume 1, Mathematical Logic</i>, Cambridge University Press, 2003. <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/978-0-521-75373-9" title="Special:BookSources/978-0-521-75373-9">978-0-521-75373-9</a></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWigderson2010" class="citation book cs1"><a href="/wiki/Avi_Wigderson" title="Avi Wigderson">Wigderson, Avi</a> (2010). <a rel="nofollow" class="external text" href="http://www.math.ias.edu/~avi/BOOKS/Godel_Widgerson_Text.pdf">"The Gödel Phenomena in Mathematics: A Modern View"</a> <span class="cs1-format">(PDF)</span>. <i>Kurt Gödel and the Foundations of Mathematics: Horizons of Truth</i>. Cambridge University Press.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=The+G%C3%B6del+Phenomena+in+Mathematics%3A+A+Modern+View&rft.btitle=Kurt+G%C3%B6del+and+the+Foundations+of+Mathematics%3A+Horizons+of+Truth&rft.pub=Cambridge+University+Press&rft.date=2010&rft.aulast=Wigderson&rft.aufirst=Avi&rft_id=http%3A%2F%2Fwww.math.ias.edu%2F~avi%2FBOOKS%2FGodel_Widgerson_Text.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AG%C3%B6del%27s+incompleteness+theorems" class="Z3988"></span></li> <li><a href="/wiki/Hao_Wang_(academic)" title="Hao Wang (academic)">Hao Wang</a>, 1996, <i>A Logical Journey: From Gödel to Philosophy</i>, The MIT Press, Cambridge MA, <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/0-262-23189-1" title="Special:BookSources/0-262-23189-1">0-262-23189-1</a>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFZach2007" class="citation book cs1">Zach, Richard (2007). "Hilbert's Program Then and Now". In Jacquette, Dale (ed.). <i>Philosophy of logic</i>. Handbook of the Philosophy of Science. Vol. 5. Amsterdam: Elsevier. pp. 411–447. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/math/0508572">math/0508572</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/10.1016%2Fb978-044451541-4%2F50014-2">10.1016/b978-044451541-4/50014-2</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-444-51541-4" title="Special:BookSources/978-0-444-51541-4"><bdi>978-0-444-51541-4</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/162131413">162131413</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:291599">291599</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Hilbert%27s+Program+Then+and+Now&rft.btitle=Philosophy+of+logic&rft.place=Amsterdam&rft.series=Handbook+of+the+Philosophy+of+Science&rft.pages=411-447&rft.pub=Elsevier&rft.date=2007&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A291599%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1016%2Fb978-044451541-4%2F50014-2&rft_id=info%3Aoclcnum%2F162131413&rft_id=info%3Aarxiv%2Fmath%2F0508572&rft.isbn=978-0-444-51541-4&rft.aulast=Zach&rft.aufirst=Richard&rfr_id=info%3Asid%2Fen.wikipedia.org%3AG%C3%B6del%27s+incompleteness+theorems" class="Z3988"></span></li></ul> <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=G%C3%B6del%27s_incompleteness_theorems&action=edit&section=46" 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://www.bbc.co.uk/programmes/b00dshx3">Godel's Incompleteness Theorems</a> on <a href="/wiki/In_Our_Time_(radio_series)" title="In Our Time (radio series)"><i>In Our Time</i></a> at the <a href="/wiki/BBC" title="BBC">BBC</a></li> <li><a rel="nofollow" class="external text" href="https://plato.stanford.edu/entries/goedel/">"Kurt Gödel"</a> entry by <a href="/wiki/Juliette_Kennedy" title="Juliette Kennedy">Juliette Kennedy</a> in the <i><a href="/wiki/Stanford_Encyclopedia_of_Philosophy" title="Stanford Encyclopedia of Philosophy">Stanford Encyclopedia of Philosophy</a></i>, July 5, 2011.</li> <li><a rel="nofollow" class="external text" href="https://plato.stanford.edu/entries/goedel-incompleteness/">"Gödel's Incompleteness Theorems"</a> entry by Panu Raatikainen in the <i><a href="/wiki/Stanford_Encyclopedia_of_Philosophy" title="Stanford Encyclopedia of Philosophy">Stanford Encyclopedia of Philosophy</a></i>, November 11, 2013.</li> <li><a rel="nofollow" class="external text" href="http://plato.stanford.edu/entries/logic-paraconsistent/#AriGodThe"><i>Paraconsistent Logic § Arithmetic and Gödel's Theorem</i></a> entry in the <a href="/wiki/Stanford_Encyclopedia_of_Philosophy" title="Stanford Encyclopedia of Philosophy">Stanford Encyclopedia of Philosophy</a>.</li> <li>MacTutor biographies: <ul><li><a rel="nofollow" class="external text" href="http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Godel.html">Kurt Gödel.</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20051013055626/http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Godel.html">Archived</a> 2005-10-13 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a></li> <li><a rel="nofollow" class="external text" href="http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Gentzen.html">Gerhard Gentzen.</a></li></ul></li> <li><a rel="nofollow" class="external text" href="http://podnieks.id.lv/gt.html">What is Mathematics:Gödel's Theorem and Around</a> by <i>Karlis Podnieks</i>. An online free book.</li> <li><a rel="nofollow" class="external text" href="http://blog.plover.com/math/Gdl-Smullyan.html">World's shortest explanation of Gödel's theorem</a> using a printing machine as an example.</li> <li><a rel="nofollow" class="external text" href="http://www.radiolab.org/story/161758-break-cycle/">October 2011 RadioLab episode</a> about/including Gödel's Incompleteness theorem</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation cs2"><a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Gödel_incompleteness_theorem">"Gödel incompleteness theorem"</a>, <i><a href="/wiki/Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a></i>, <a href="/wiki/European_Mathematical_Society" title="European Mathematical Society">EMS Press</a>, 2001 [1994]</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+incompleteness+theorem&rft.btitle=Encyclopedia+of+Mathematics&rft.pub=EMS+Press&rft.date=2001&rft_id=https%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%3Ftitle%3DG%C3%B6del_incompleteness_theorem&rfr_id=info%3Asid%2Fen.wikipedia.org%3AG%C3%B6del%27s+incompleteness+theorems" class="Z3988"></span></li> <li><a rel="nofollow" class="external text" href="https://www.quantamagazine.org/how-godels-incompleteness-theorems-work-20200714/">How Gödel's Proof Works</a> by <a href="/wiki/Natalie_Wolchover" title="Natalie Wolchover">Natalie Wolchover</a>, <a href="/wiki/Quanta_Magazine" title="Quanta Magazine">Quanta Magazine</a>, July 14, 2020.</li> <li><a rel="nofollow" class="external autonumber" href="https://www.isa-afp.org/entries/Incompleteness.html">[1]</a> and <a rel="nofollow" class="external autonumber" href="https://www.isa-afp.org/entries/Goedel_Incompleteness.html">[2]</a> Gödel's incompleteness theorems formalised in Isabelle/HOL</li></ul> <p>` </p> <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 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class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 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title="Metamathematics">metamathematics</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0;padding-left:2.0em;padding-right:2.0em;"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Cantor%27s_theorem" title="Cantor's theorem">Cantor's theorem</a></li> <li><i><a href="/wiki/Entscheidungsproblem" title="Entscheidungsproblem">Entscheidungsproblem</a></i></li> <li><a href="/wiki/Church%E2%80%93Turing_thesis" title="Church–Turing thesis">Church–Turing thesis</a></li> <li><a href="/wiki/Consistency" title="Consistency">Consistency</a></li> <li><a href="/wiki/Effective_method" title="Effective method">Effective method</a></li> <li><a href="/wiki/Foundations_of_mathematics" title="Foundations of mathematics">Foundations of mathematics</a> <ul><li><a href="/wiki/Foundations_of_geometry" title="Foundations of geometry">of geometry</a></li></ul></li> <li><a href="/wiki/G%C3%B6del%27s_completeness_theorem" title="Gödel's completeness theorem">Gödel's completeness theorem</a></li> <li><a class="mw-selflink selflink">Gödel's incompleteness theorems</a></li> <li><a href="/wiki/Soundness" title="Soundness">Soundness</a></li> <li><a href="/wiki/Completeness_(logic)" title="Completeness (logic)">Completeness</a></li> <li><a href="/wiki/Decidability_(logic)" title="Decidability (logic)">Decidability</a></li> <li><a href="/wiki/Interpretation_(logic)" title="Interpretation (logic)">Interpretation</a></li> <li><a href="/wiki/L%C3%B6wenheim%E2%80%93Skolem_theorem" title="Löwenheim–Skolem theorem">Löwenheim–Skolem theorem</a></li> <li><a href="/wiki/Metatheorem" title="Metatheorem">Metatheorem</a></li> <li><a href="/wiki/Satisfiability" title="Satisfiability">Satisfiability</a></li> <li><a href="/wiki/Independence_(mathematical_logic)" title="Independence (mathematical logic)">Independence</a></li> <li><a href="/wiki/Type%E2%80%93token_distinction" title="Type–token distinction">Type–token distinction</a></li> <li><a href="/wiki/Use%E2%80%93mention_distinction" title="Use–mention distinction">Use–mention distinction</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 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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 class="mw-selflink selflink">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 href="/wiki/Church%E2%80%93Turing_thesis" title="Church–Turing thesis">Church–Turing thesis</a></li> <li><a href="/wiki/Computably_enumerable_set" title="Computably enumerable set">Computably enumerable</a></li> <li><a href="/wiki/Computable_function" title="Computable function">Computable function</a></li> <li><a href="/wiki/Computable_set" title="Computable set">Computable set</a></li> <li><a href="/wiki/Decision_problem" title="Decision problem">Decision problem</a> <ul><li><a href="/wiki/Decidability_(logic)" title="Decidability (logic)">decidable</a></li> <li><a href="/wiki/Undecidable_problem" title="Undecidable problem">undecidable</a></li> <li><a href="/wiki/P_(complexity)" title="P (complexity)">P</a></li> <li><a href="/wiki/NP_(complexity)" title="NP (complexity)">NP</a></li> <li><a href="/wiki/P_versus_NP_problem" title="P versus NP problem">P versus NP problem</a></li></ul></li> <li><a href="/wiki/Kolmogorov_complexity" title="Kolmogorov complexity">Kolmogorov complexity</a></li> <li><a href="/wiki/Lambda_calculus" title="Lambda calculus">Lambda calculus</a></li> <li><a href="/wiki/Primitive_recursive_function" title="Primitive recursive function">Primitive recursive function</a></li> <li><a href="/wiki/Recursion" title="Recursion">Recursion</a></li> <li><a href="/wiki/Recursive_set" class="mw-redirect" title="Recursive set">Recursive set</a></li> <li><a href="/wiki/Turing_machine" title="Turing machine">Turing machine</a></li> <li><a href="/wiki/Type_theory" title="Type theory">Type theory</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Abstract_logic" title="Abstract logic">Abstract logic</a></li> <li><a href="/wiki/Algebraic_logic" title="Algebraic logic">Algebraic logic</a></li> <li><a href="/wiki/Automated_theorem_proving" title="Automated theorem proving">Automated theorem proving</a></li> <li><a href="/wiki/Category_theory" title="Category theory">Category theory</a></li> <li><a href="/wiki/Concrete_category" title="Concrete category">Concrete</a>/<a href="/wiki/Category_(mathematics)" title="Category (mathematics)">Abstract category</a></li> <li><a href="/wiki/Category_of_sets" title="Category of sets">Category of sets</a></li> <li><a href="/wiki/History_of_logic" title="History of logic">History of logic</a></li> <li><a href="/wiki/History_of_mathematical_logic" class="mw-redirect" title="History of mathematical logic">History of mathematical logic</a> <ul><li><a href="/wiki/Timeline_of_mathematical_logic" title="Timeline of mathematical logic">timeline</a></li></ul></li> <li><a href="/wiki/Logicism" title="Logicism">Logicism</a></li> <li><a href="/wiki/Mathematical_object" title="Mathematical object">Mathematical object</a></li> <li><a href="/wiki/Philosophy_of_mathematics" title="Philosophy of mathematics">Philosophy of mathematics</a></li> <li><a href="/wiki/Supertask" title="Supertask">Supertask</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div><b><span class="nowrap"><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Nuvola_apps_edu_mathematics_blue-p.svg" class="mw-file-description"><img alt="icon" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/16px-Nuvola_apps_edu_mathematics_blue-p.svg.png" decoding="async" width="16" height="16" class="mw-file-element" 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