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class="vector-toc-link" href="#Foundational_theories"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2.1</span> <span>Foundational theories</span> </div> </a> <ul id="toc-Foundational_theories-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-20th_century" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#20th_century"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>20th century</span> </div> </a> <ul id="toc-20th_century-sublist" class="vector-toc-list"> <li id="toc-Set_theory_and_paradoxes" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Set_theory_and_paradoxes"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3.1</span> <span>Set theory and paradoxes</span> </div> </a> <ul id="toc-Set_theory_and_paradoxes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Symbolic_logic" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Symbolic_logic"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3.2</span> <span>Symbolic logic</span> </div> </a> <ul id="toc-Symbolic_logic-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Beginnings_of_the_other_branches" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Beginnings_of_the_other_branches"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3.3</span> <span>Beginnings of the other branches</span> </div> </a> <ul id="toc-Beginnings_of_the_other_branches-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-Formal_logical_systems" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Formal_logical_systems"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Formal logical systems</span> </div> </a> <button aria-controls="toc-Formal_logical_systems-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 Formal logical systems subsection</span> </button> <ul id="toc-Formal_logical_systems-sublist" class="vector-toc-list"> <li id="toc-First-order_logic" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#First-order_logic"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>First-order logic</span> </div> </a> <ul id="toc-First-order_logic-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Other_classical_logics" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Other_classical_logics"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Other classical logics</span> </div> </a> <ul id="toc-Other_classical_logics-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Nonclassical_and_modal_logic" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Nonclassical_and_modal_logic"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>Nonclassical and modal logic</span> </div> </a> <ul id="toc-Nonclassical_and_modal_logic-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Algebraic_logic" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Algebraic_logic"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.4</span> <span>Algebraic logic</span> </div> </a> <ul id="toc-Algebraic_logic-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Set_theory" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Set_theory"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Set theory</span> </div> </a> <ul id="toc-Set_theory-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Model_theory" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Model_theory"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Model theory</span> </div> </a> <ul id="toc-Model_theory-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Recursion_theory" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Recursion_theory"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Recursion theory</span> </div> </a> <button aria-controls="toc-Recursion_theory-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 Recursion theory subsection</span> </button> <ul id="toc-Recursion_theory-sublist" class="vector-toc-list"> <li id="toc-Algorithmically_unsolvable_problems" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Algorithmically_unsolvable_problems"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.1</span> <span>Algorithmically unsolvable problems</span> </div> </a> <ul id="toc-Algorithmically_unsolvable_problems-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Proof_theory_and_constructive_mathematics" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Proof_theory_and_constructive_mathematics"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Proof theory and constructive mathematics</span> </div> </a> <ul id="toc-Proof_theory_and_constructive_mathematics-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Applications" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Applications"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Applications</span> </div> </a> <ul id="toc-Applications-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Connections_with_computer_science" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Connections_with_computer_science"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Connections with computer science</span> </div> </a> <ul id="toc-Connections_with_computer_science-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Foundations_of_mathematics" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Foundations_of_mathematics"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>Foundations of mathematics</span> </div> </a> <ul id="toc-Foundations_of_mathematics-sublist" class="vector-toc-list"> </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">11</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> <span class="vector-toc-numb">12</span> <span>Notes</span> </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">13</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-Undergraduate_texts" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Undergraduate_texts"> <div class="vector-toc-text"> <span class="vector-toc-numb">13.1</span> <span>Undergraduate texts</span> </div> </a> <ul id="toc-Undergraduate_texts-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Graduate_texts" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Graduate_texts"> <div class="vector-toc-text"> <span class="vector-toc-numb">13.2</span> <span>Graduate texts</span> </div> </a> <ul id="toc-Graduate_texts-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Research_papers,_monographs,_texts,_and_surveys" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Research_papers,_monographs,_texts,_and_surveys"> <div class="vector-toc-text"> <span class="vector-toc-numb">13.3</span> <span>Research papers, monographs, texts, and surveys</span> </div> </a> <ul id="toc-Research_papers,_monographs,_texts,_and_surveys-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Classical_papers,_texts,_and_collections" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Classical_papers,_texts,_and_collections"> <div class="vector-toc-text"> <span class="vector-toc-numb">13.4</span> <span>Classical papers, texts, and collections</span> </div> </a> <ul id="toc-Classical_papers,_texts,_and_collections-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">14</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" 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type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 85 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-85" 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">85 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/Wiskundige_logika" title="Wiskundige logika – Afrikaans" lang="af" hreflang="af" data-title="Wiskundige logika" data-language-autonym="Afrikaans" data-language-local-name="Afrikaans" class="interlanguage-link-target"><span>Afrikaans</span></a></li><li class="interlanguage-link interwiki-als mw-list-item"><a href="https://als.wikipedia.org/wiki/Mathematische_Logik" title="Mathematische Logik – Alemannic" lang="gsw" hreflang="gsw" data-title="Mathematische Logik" data-language-autonym="Alemannisch" data-language-local-name="Alemannic" class="interlanguage-link-target"><span>Alemannisch</span></a></li><li class="interlanguage-link interwiki-am mw-list-item"><a href="https://am.wikipedia.org/wiki/%E1%88%92%E1%88%B3%E1%89%A3%E1%8B%8A_%E1%88%A5%E1%8A%90_%E1%8A%A0%E1%88%9D%E1%8A%AD%E1%8A%95%E1%8B%AE" title="ሒሳባዊ ሥነ አምክንዮ – Amharic" lang="am" hreflang="am" data-title="ሒሳባዊ ሥነ አምክንዮ" data-language-autonym="አማርኛ" data-language-local-name="Amharic" class="interlanguage-link-target"><span>አማርኛ</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%85%D9%86%D8%B7%D9%82_%D8%B1%D9%8A%D8%A7%D8%B6%D9%8A" title="منطق رياضي – Arabic" lang="ar" hreflang="ar" data-title="منطق رياضي" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-an mw-list-item"><a href="https://an.wikipedia.org/wiki/Lochica_matematica" title="Lochica matematica – Aragonese" lang="an" hreflang="an" data-title="Lochica matematica" data-language-autonym="Aragonés" data-language-local-name="Aragonese" class="interlanguage-link-target"><span>Aragonés</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/L%C3%B3xica_matem%C3%A1tica" title="Lóxica matemática – Asturian" lang="ast" hreflang="ast" data-title="Lóxica matemática" data-language-autonym="Asturianu" data-language-local-name="Asturian" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Riyazi_m%C9%99ntiq" title="Riyazi məntiq – Azerbaijani" lang="az" hreflang="az" data-title="Riyazi məntiq" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%97%E0%A6%BE%E0%A6%A3%E0%A6%BF%E0%A6%A4%E0%A6%BF%E0%A6%95_%E0%A6%AF%E0%A7%81%E0%A6%95%E0%A7%8D%E0%A6%A4%E0%A6%BF%E0%A6%AC%E0%A6%BF%E0%A6%9C%E0%A7%8D%E0%A6%9E%E0%A6%BE%E0%A6%A8" title="গাণিতিক যুক্তিবিজ্ঞান – Bangla" lang="bn" hreflang="bn" data-title="গাণিতিক যুক্তিবিজ্ঞান" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-zh-min-nan mw-list-item"><a href="https://zh-min-nan.wikipedia.org/wiki/S%C3%B2%CD%98-l%C3%AD_su-l%C3%AD" title="Sò͘-lí su-lí – Minnan" lang="nan" hreflang="nan" data-title="Sò͘-lí su-lí" data-language-autonym="閩南語 / Bân-lâm-gú" data-language-local-name="Minnan" class="interlanguage-link-target"><span>閩南語 / Bân-lâm-gú</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%9C%D0%B0%D1%82%D1%8D%D0%BC%D0%B0%D1%82%D1%8B%D1%87%D0%BD%D0%B0%D1%8F_%D0%BB%D0%BE%D0%B3%D1%96%D0%BA%D0%B0" title="Матэматычная логіка – Belarusian" lang="be" hreflang="be" data-title="Матэматычная логіка" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-be-x-old mw-list-item"><a href="https://be-tarask.wikipedia.org/wiki/%D0%9C%D0%B0%D1%82%D1%8D%D0%BC%D0%B0%D1%82%D1%8B%D1%87%D0%BD%D0%B0%D1%8F_%D0%BB%D1%91%D0%B3%D1%96%D0%BA%D0%B0" title="Матэматычная лёгіка – Belarusian (Taraškievica orthography)" lang="be-tarask" hreflang="be-tarask" data-title="Матэматычная лёгіка" data-language-autonym="Беларуская (тарашкевіца)" data-language-local-name="Belarusian (Taraškievica orthography)" class="interlanguage-link-target"><span>Беларуская (тарашкевіца)</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%9C%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%B0_%D0%BB%D0%BE%D0%B3%D0%B8%D0%BA%D0%B0" title="Математическа логика – Bulgarian" lang="bg" hreflang="bg" data-title="Математическа логика" 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/Matemati%C4%8Dka_logika" title="Matematička logika – Bosnian" lang="bs" hreflang="bs" data-title="Matematička logika" 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/L%C3%B2gica_matem%C3%A0tica" title="Lògica matemàtica – Catalan" lang="ca" hreflang="ca" data-title="Lògica matemàtica" 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%9C%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%C4%83%D0%BB%D0%BB%D0%B0_%D0%BB%D0%BE%D0%B3%D0%B8%D0%BA%D0%B0" title="Математикăлла логика – Chuvash" lang="cv" hreflang="cv" data-title="Математикăлла логика" 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/Matematick%C3%A1_logika" title="Matematická logika – Czech" lang="cs" hreflang="cs" data-title="Matematická logika" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Rhesymeg_mathemateg" title="Rhesymeg mathemateg – Welsh" lang="cy" hreflang="cy" data-title="Rhesymeg mathemateg" data-language-autonym="Cymraeg" data-language-local-name="Welsh" class="interlanguage-link-target"><span>Cymraeg</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Matematisk_logik" title="Matematisk logik – Danish" lang="da" hreflang="da" data-title="Matematisk logik" 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/Mathematische_Logik" title="Mathematische Logik – German" lang="de" hreflang="de" data-title="Mathematische Logik" 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/Matemaatiline_loogika" title="Matemaatiline loogika – Estonian" lang="et" hreflang="et" data-title="Matemaatiline loogika" 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%9C%CE%B1%CE%B8%CE%B7%CE%BC%CE%B1%CF%84%CE%B9%CE%BA%CE%AE_%CE%BB%CE%BF%CE%B3%CE%B9%CE%BA%CE%AE" title="Μαθηματική λογική – Greek" lang="el" hreflang="el" data-title="Μαθηματική λογική" 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/L%C3%B3gica_matem%C3%A1tica" title="Lógica matemática – Spanish" lang="es" hreflang="es" data-title="Lógica matemática" 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/Matematika_logiko" title="Matematika logiko – Esperanto" lang="eo" hreflang="eo" data-title="Matematika logiko" 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/Logika_matematiko" title="Logika matematiko – Basque" lang="eu" hreflang="eu" data-title="Logika matematiko" 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%85%D9%86%D8%B7%D9%82_%D8%B1%DB%8C%D8%A7%D8%B6%DB%8C" title="منطق ریاضی – Persian" lang="fa" hreflang="fa" data-title="منطق ریاضی" 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/Logique_math%C3%A9matique" title="Logique mathématique – French" lang="fr" hreflang="fr" data-title="Logique mathématique" 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/Loighic_mhatamaitici%C3%BAil" title="Loighic mhatamaiticiúil – Irish" lang="ga" hreflang="ga" data-title="Loighic mhatamaiticiúil" data-language-autonym="Gaeilge" data-language-local-name="Irish" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-gd mw-list-item"><a href="https://gd.wikipedia.org/wiki/Loidig_mhatamataigeach" title="Loidig mhatamataigeach – Scottish Gaelic" lang="gd" hreflang="gd" data-title="Loidig mhatamataigeach" data-language-autonym="Gàidhlig" data-language-local-name="Scottish Gaelic" class="interlanguage-link-target"><span>Gàidhlig</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/L%C3%B3xica_matem%C3%A1tica" title="Lóxica matemática – Galician" lang="gl" hreflang="gl" data-title="Lóxica matemática" 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/%EC%88%98%EB%A6%AC_%EB%85%BC%EB%A6%AC%ED%95%99" title="수리 논리학 – Korean" lang="ko" hreflang="ko" data-title="수리 논리학" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%84%D5%A1%D5%A9%D5%A5%D5%B4%D5%A1%D5%BF%D5%AB%D5%AF%D5%A1%D5%AF%D5%A1%D5%B6_%D5%BF%D6%80%D5%A1%D5%B4%D5%A1%D5%A2%D5%A1%D5%B6%D5%B8%D6%82%D5%A9%D5%B5%D5%B8%D6%82%D5%B6" title="Մաթեմատիկական տրամաբանություն – Armenian" lang="hy" hreflang="hy" data-title="Մաթեմատիկական տրամաբանություն" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%97%E0%A4%A3%E0%A4%BF%E0%A4%A4%E0%A5%80%E0%A4%AF_%E0%A4%A4%E0%A4%B0%E0%A5%8D%E0%A4%95%E0%A4%B6%E0%A4%BE%E0%A4%B8%E0%A5%8D%E0%A4%A4%E0%A5%8D%E0%A4%B0" title="गणितीय तर्कशास्त्र – Hindi" lang="hi" hreflang="hi" data-title="गणितीय तर्कशास्त्र" 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/Matemati%C4%8Dka_logika" title="Matematička logika – Croatian" lang="hr" hreflang="hr" data-title="Matematička logika" 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/Matematikala_logiko" title="Matematikala logiko – Ido" lang="io" hreflang="io" data-title="Matematikala logiko" 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/Logika_matematika" title="Logika matematika – Indonesian" lang="id" hreflang="id" data-title="Logika matematika" 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-is mw-list-item"><a href="https://is.wikipedia.org/wiki/St%C3%A6r%C3%B0fr%C3%A6%C3%B0ileg_r%C3%B6kfr%C3%A6%C3%B0i" title="Stærðfræðileg rökfræði – Icelandic" lang="is" hreflang="is" data-title="Stærðfræðileg rökfræði" data-language-autonym="Íslenska" data-language-local-name="Icelandic" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Logica_matematica" title="Logica matematica – Italian" lang="it" hreflang="it" data-title="Logica matematica" 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%9C%D7%95%D7%92%D7%99%D7%A7%D7%94_%D7%9E%D7%AA%D7%9E%D7%98%D7%99%D7%AA" title="לוגיקה מתמטית – Hebrew" lang="he" hreflang="he" data-title="לוגיקה מתמטית" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-jv mw-list-item"><a href="https://jv.wikipedia.org/wiki/Logika_mat%C3%A9matika" title="Logika matématika – Javanese" lang="jv" hreflang="jv" data-title="Logika matématika" data-language-autonym="Jawa" data-language-local-name="Javanese" class="interlanguage-link-target"><span>Jawa</span></a></li><li class="interlanguage-link interwiki-kn mw-list-item"><a href="https://kn.wikipedia.org/wiki/%E0%B2%97%E0%B2%A3%E0%B2%BF%E0%B2%A4_%E0%B2%A4%E0%B2%B0%E0%B3%8D%E0%B2%95%E0%B2%B6%E0%B2%BE%E0%B2%B8%E0%B3%8D%E0%B2%A4%E0%B3%8D%E0%B2%B0" title="ಗಣಿತ ತರ್ಕಶಾಸ್ತ್ರ – Kannada" lang="kn" hreflang="kn" data-title="ಗಣಿತ ತರ್ಕಶಾಸ್ತ್ರ" 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%9B%E1%83%90%E1%83%97%E1%83%94%E1%83%9B%E1%83%90%E1%83%A2%E1%83%98%E1%83%99%E1%83%A3%E1%83%A0%E1%83%98_%E1%83%9A%E1%83%9D%E1%83%92%E1%83%98%E1%83%99%E1%83%90" title="მათემატიკური ლოგიკა – Georgian" lang="ka" hreflang="ka" data-title="მათემატიკური ლოგიკა" data-language-autonym="ქართული" data-language-local-name="Georgian" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%9C%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0%D0%BB%D1%8B%D2%9B_%D0%BB%D0%BE%D0%B3%D0%B8%D0%BA%D0%B0" title="Математикалық логика – Kazakh" lang="kk" hreflang="kk" data-title="Математикалық логика" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-ky mw-list-item"><a href="https://ky.wikipedia.org/wiki/%D0%9C%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0%D0%BB%D1%8B%D0%BA_%D0%BB%D0%BE%D0%B3%D0%B8%D0%BA%D0%B0" title="Математикалык логика – Kyrgyz" lang="ky" hreflang="ky" data-title="Математикалык логика" data-language-autonym="Кыргызча" data-language-local-name="Kyrgyz" class="interlanguage-link-target"><span>Кыргызча</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Logica_mathematica" title="Logica mathematica – Latin" lang="la" hreflang="la" data-title="Logica mathematica" data-language-autonym="Latina" data-language-local-name="Latin" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Matem%C4%81tisk%C4%81_lo%C4%A3ika" title="Matemātiskā loģika – Latvian" lang="lv" hreflang="lv" data-title="Matemātiskā loģika" data-language-autonym="Latviešu" data-language-local-name="Latvian" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-lb mw-list-item"><a href="https://lb.wikipedia.org/wiki/Mathematesch_Logik" title="Mathematesch Logik – Luxembourgish" lang="lb" hreflang="lb" data-title="Mathematesch Logik" data-language-autonym="Lëtzebuergesch" data-language-local-name="Luxembourgish" class="interlanguage-link-target"><span>Lëtzebuergesch</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Simbolin%C4%97_logika" title="Simbolinė logika – Lithuanian" lang="lt" hreflang="lt" data-title="Simbolinė logika" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-lij mw-list-item"><a href="https://lij.wikipedia.org/wiki/Logica_matematica" title="Logica matematica – Ligurian" lang="lij" hreflang="lij" data-title="Logica matematica" data-language-autonym="Ligure" data-language-local-name="Ligurian" class="interlanguage-link-target"><span>Ligure</span></a></li><li class="interlanguage-link interwiki-lg mw-list-item"><a href="https://lg.wikipedia.org/wiki/Okusengekensonga_okw%27ekibalo(Mathematical_logic)" title="Okusengekensonga okw'ekibalo(Mathematical logic) – Ganda" lang="lg" hreflang="lg" data-title="Okusengekensonga okw'ekibalo(Mathematical logic)" data-language-autonym="Luganda" data-language-local-name="Ganda" class="interlanguage-link-target"><span>Luganda</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Matematikai_logika" title="Matematikai logika – Hungarian" lang="hu" hreflang="hu" data-title="Matematikai logika" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%9C%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D1%87%D0%BA%D0%B0_%D0%BB%D0%BE%D0%B3%D0%B8%D0%BA%D0%B0" title="Математичка логика – Macedonian" lang="mk" hreflang="mk" data-title="Математичка логика" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Logik_matematik" title="Logik matematik – Malay" lang="ms" hreflang="ms" data-title="Logik matematik" data-language-autonym="Bahasa Melayu" data-language-local-name="Malay" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-mwl mw-list-item"><a href="https://mwl.wikipedia.org/wiki/L%C3%B3gica_matem%C3%A1tica" title="Lógica matemática – Mirandese" lang="mwl" hreflang="mwl" data-title="Lógica matemática" data-language-autonym="Mirandés" data-language-local-name="Mirandese" class="interlanguage-link-target"><span>Mirandés</span></a></li><li class="interlanguage-link interwiki-my mw-list-item"><a href="https://my.wikipedia.org/wiki/%E1%80%9E%E1%80%84%E1%80%BA%E1%80%B9%E1%80%81%E1%80%BB%E1%80%AC%E1%80%9A%E1%80%AF%E1%80%90%E1%80%B9%E1%80%90%E1%80%AD%E1%80%97%E1%80%B1%E1%80%92" title="သင်္ချာယုတ္တိဗေဒ – Burmese" lang="my" hreflang="my" data-title="သင်္ချာယုတ္တိဗေဒ" 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/Wiskundige_logica" title="Wiskundige logica – Dutch" lang="nl" hreflang="nl" data-title="Wiskundige logica" 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/%E6%95%B0%E7%90%86%E8%AB%96%E7%90%86%E5%AD%A6" title="数理論理学 – Japanese" lang="ja" hreflang="ja" data-title="数理論理学" 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/Matematisk_logikk" title="Matematisk logikk – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Matematisk logikk" 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/Matematisk_logikk" title="Matematisk logikk – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Matematisk logikk" 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-om mw-list-item"><a href="https://om.wikipedia.org/wiki/Yaayaa_Herregaa" title="Yaayaa Herregaa – Oromo" lang="om" hreflang="om" data-title="Yaayaa Herregaa" data-language-autonym="Oromoo" data-language-local-name="Oromo" class="interlanguage-link-target"><span>Oromoo</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Matematik_mantiq" title="Matematik mantiq – Uzbek" lang="uz" hreflang="uz" data-title="Matematik mantiq" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="Uzbek" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-pnb mw-list-item"><a href="https://pnb.wikipedia.org/wiki/%D8%B1%DB%8C%D8%A7%D8%B6%DB%8C%D8%A7%D8%AA%DB%8C_%D9%85%D9%86%D8%B7%D9%82" title="ریاضیاتی منطق – Western Punjabi" lang="pnb" hreflang="pnb" data-title="ریاضیاتی منطق" data-language-autonym="پنجابی" data-language-local-name="Western Punjabi" class="interlanguage-link-target"><span>پنجابی</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Logika_matematyczna" title="Logika matematyczna – Polish" lang="pl" hreflang="pl" data-title="Logika matematyczna" 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/L%C3%B3gica_matem%C3%A1tica" title="Lógica matemática – Portuguese" lang="pt" hreflang="pt" data-title="Lógica matemática" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Logic%C4%83_matematic%C4%83" title="Logică matematică – Romanian" lang="ro" hreflang="ro" data-title="Logică matematică" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9C%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%B0%D1%8F_%D0%BB%D0%BE%D0%B3%D0%B8%D0%BA%D0%B0" title="Математическая логика – Russian" lang="ru" hreflang="ru" data-title="Математическая логика" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sco mw-list-item"><a href="https://sco.wikipedia.org/wiki/Mathematical_logic" title="Mathematical logic – Scots" lang="sco" hreflang="sco" data-title="Mathematical logic" data-language-autonym="Scots" data-language-local-name="Scots" class="interlanguage-link-target"><span>Scots</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Logjika_matematikore" title="Logjika matematikore – Albanian" lang="sq" hreflang="sq" data-title="Logjika matematikore" data-language-autonym="Shqip" data-language-local-name="Albanian" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-si mw-list-item"><a href="https://si.wikipedia.org/wiki/%E0%B6%9C%E0%B6%AB%E0%B7%92%E0%B6%AD%E0%B6%B8%E0%B6%BA_%E0%B6%AD%E0%B6%BB%E0%B7%8A%E0%B6%9A%E0%B6%AB%E0%B6%BA" title="ගණිතමය තර්කණය – Sinhala" lang="si" hreflang="si" data-title="ගණිතමය තර්කණය" data-language-autonym="සිංහල" data-language-local-name="Sinhala" class="interlanguage-link-target"><span>සිංහල</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Mathematical_logic" title="Mathematical logic – Simple English" lang="en-simple" hreflang="en-simple" data-title="Mathematical logic" data-language-autonym="Simple English" 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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"><span>For Quine's theory sometimes called "Mathematical Logic", see <a href="/wiki/New_Foundations" title="New Foundations">New Foundations</a>.</span> <span>For other uses, see <a href="/wiki/Logic_(disambiguation)" class="mw-disambig" title="Logic (disambiguation)">Logic (disambiguation)</a>.</span></div> <p class="mw-empty-elt"> </p> <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 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style="padding-bottom:0.35em;"> <ul><li><a href="/wiki/History_of_mathematics" title="History of mathematics">History</a></li> <li><a href="/wiki/Lists_of_mathematics_topics" title="Lists of mathematics topics">Index</a></li></ul></td></tr><tr><td class="sidebar-content-with-subgroup"> <table class="sidebar-subgroup"><tbody><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible"><div class="sidebar-list-title" style="border-top:1px solid #aaa;background:#ddddff;text-align:center;;color: var(--color-base)"><a href="/wiki/Areas_of_mathematics" class="mw-redirect" title="Areas of mathematics">Areas</a></div><div class="sidebar-list-content mw-collapsible-content hlist"> <ul><li><a href="/wiki/Number_theory" title="Number theory">Number theory</a></li> <li><a href="/wiki/Geometry" title="Geometry">Geometry</a></li> <li><a href="/wiki/Algebra" title="Algebra">Algebra</a></li> <li><a href="/wiki/Calculus" title="Calculus">Calculus</a> and <a 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src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/20px-Nuvola_apps_edu_mathematics_blue-p.svg.png" decoding="async" width="20" height="20" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/30px-Nuvola_apps_edu_mathematics_blue-p.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/40px-Nuvola_apps_edu_mathematics_blue-p.svg.png 2x" data-file-width="128" data-file-height="128" /></a></span> <a href="/wiki/Portal:Mathematics" title="Portal:Mathematics">Mathematics Portal</a></th></tr><tr><td class="sidebar-navbar"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Math_topics_sidebar" title="Template:Math topics sidebar"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Math_topics_sidebar" title="Template talk:Math topics sidebar"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Math_topics_sidebar" title="Special:EditPage/Template:Math topics sidebar"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <p><b>Mathematical logic</b> is the study of <a href="/wiki/Logic#Formal_logic" title="Logic">formal logic</a> within <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>. Major subareas include <a href="/wiki/Model_theory" title="Model theory">model theory</a>, <a href="/wiki/Proof_theory" title="Proof theory">proof theory</a>, <a href="/wiki/Set_theory" title="Set theory">set theory</a>, and <a href="/wiki/Recursion_theory" class="mw-redirect" title="Recursion theory">recursion theory</a> (also known as computability theory). Research in mathematical logic commonly addresses the mathematical properties of formal systems of logic such as their expressive or deductive power. However, it can also include uses of logic to characterize correct mathematical reasoning or to establish <a href="/wiki/Foundations_of_mathematics" title="Foundations of mathematics">foundations of mathematics</a>. </p><p>Since its inception, mathematical logic has both contributed to and been motivated by the study of foundations of mathematics. This study began in the late 19th century with the development of <a href="/wiki/Axiom" title="Axiom">axiomatic</a> frameworks for <a href="/wiki/Geometry" title="Geometry">geometry</a>, <a href="/wiki/Arithmetic" title="Arithmetic">arithmetic</a>, and <a href="/wiki/Mathematical_analysis" title="Mathematical analysis">analysis</a>. In the early 20th century it was shaped by <a href="/wiki/David_Hilbert" title="David Hilbert">David Hilbert</a>'s <a href="/wiki/Hilbert%27s_program" title="Hilbert's program">program</a> to prove the consistency of foundational theories. Results of <a href="/wiki/Kurt_G%C3%B6del" title="Kurt Gödel">Kurt Gödel</a>, <a href="/wiki/Gerhard_Gentzen" title="Gerhard Gentzen">Gerhard Gentzen</a>, and others provided partial resolution to the program, and clarified the issues involved in proving consistency. Work in set theory showed that almost all ordinary mathematics can be formalized in terms of sets, although there are some theorems that cannot be proven in common axiom systems for set theory. Contemporary work in the foundations of mathematics often focuses on establishing which parts of mathematics can be formalized in particular formal systems (as in <a href="/wiki/Reverse_mathematics" title="Reverse mathematics">reverse mathematics</a>) rather than trying to find theories in which all of mathematics can be developed. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Subfields_and_scope">Subfields and scope</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_logic&action=edit&section=1" title="Edit section: Subfields and scope"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <i>Handbook of Mathematical Logic</i><sup id="cite_ref-FOOTNOTEBarwise1989_1-0" class="reference"><a href="#cite_note-FOOTNOTEBarwise1989-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> in 1977 makes a rough division of contemporary mathematical logic into four areas: </p> <ol><li><a href="/wiki/Set_theory" title="Set theory">set theory</a></li> <li><a href="/wiki/Model_theory" title="Model theory">model theory</a></li> <li><a href="/wiki/Recursion_theory" class="mw-redirect" title="Recursion theory">recursion theory</a>, and</li> <li><a href="/wiki/Proof_theory" title="Proof theory">proof theory</a> and <a href="/wiki/Constructive_mathematics" class="mw-redirect" title="Constructive mathematics">constructive mathematics</a> (considered as parts of a single area).</li></ol> <p>Additionally, sometimes the field of <a href="/wiki/Computational_complexity_theory" title="Computational complexity theory">computational complexity theory</a> is also included as part of mathematical logic.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> Each area has a distinct focus, although many techniques and results are shared among multiple areas. The borderlines amongst these fields, and the lines separating mathematical logic and other fields of mathematics, are not always sharp. <a href="/wiki/G%C3%B6del%27s_incompleteness_theorem" class="mw-redirect" title="Gödel's incompleteness theorem">Gödel's incompleteness theorem</a> marks not only a milestone in recursion theory and proof theory, but has also led to <a href="/wiki/L%C3%B6b%27s_theorem" title="Löb's theorem">Löb's theorem</a> in modal logic. The method of <a href="/wiki/Forcing_(mathematics)" title="Forcing (mathematics)">forcing</a> is employed in set theory, model theory, and recursion theory, as well as in the study of intuitionistic mathematics. </p><p>The mathematical field of <a href="/wiki/Category_theory" title="Category theory">category theory</a> uses many formal axiomatic methods, and includes the study of <a href="/wiki/Categorical_logic" title="Categorical logic">categorical logic</a>, but category theory is not ordinarily considered a subfield of mathematical logic. Because of its applicability in diverse fields of mathematics, mathematicians including <a href="/wiki/Saunders_Mac_Lane" title="Saunders Mac Lane">Saunders Mac Lane</a> have proposed category theory as a foundational system for mathematics, independent of set theory. These foundations use <a href="/wiki/Topos" title="Topos">toposes</a>, which resemble generalized models of set theory that may employ classical or nonclassical logic. </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=Mathematical_logic&action=edit&section=2" title="Edit section: History"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Mathematical logic emerged in the mid-19th century as a subfield of mathematics, reflecting the confluence of two traditions: formal philosophical logic and mathematics.<sup id="cite_ref-FOOTNOTEFerreirós2001443_3-0" class="reference"><a href="#cite_note-FOOTNOTEFerreirós2001443-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> Mathematical logic, also called 'logistic', 'symbolic logic', the '<a href="/wiki/Boolean_algebra" title="Boolean algebra">algebra of logic</a>', and, more recently, simply 'formal logic', is the set of logical theories elaborated in the course of the nineteenth century with the aid of an artificial notation and a rigorously deductive method.<sup id="cite_ref-FOOTNOTEBochenski1959Sec._0.1,_p._1_4-0" class="reference"><a href="#cite_note-FOOTNOTEBochenski1959Sec._0.1,_p._1-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> Before this emergence, logic was studied with <a href="/wiki/Rhetoric" title="Rhetoric">rhetoric</a>, with <i>calculationes</i>,<sup id="cite_ref-FOOTNOTESwineshead1498_5-0" class="reference"><a href="#cite_note-FOOTNOTESwineshead1498-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> through the <a href="/wiki/Syllogism" title="Syllogism">syllogism</a>, and with <a href="/wiki/Philosophy" title="Philosophy">philosophy</a>. The first half of the 20th century saw an explosion of fundamental results, accompanied by vigorous debate over the foundations of mathematics. </p> <div class="mw-heading mw-heading3"><h3 id="Early_history">Early history</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_logic&action=edit&section=3" title="Edit section: Early history"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Further information: <a href="/wiki/History_of_logic" title="History of logic">History of logic</a></div> <p>Theories of logic were developed in many cultures in history, including <a href="/wiki/Logic_in_China" title="Logic in China">China</a>, <a href="/wiki/Logic_in_India" class="mw-redirect" title="Logic in India">India</a>, <a href="/wiki/Logic_in_Greece" class="mw-redirect" title="Logic in Greece">Greece</a> and the <a href="/wiki/Logic_in_Islamic_philosophy" title="Logic in Islamic philosophy">Islamic world</a>. Greek methods, particularly <a href="/wiki/Aristotelian_logic" class="mw-redirect" title="Aristotelian logic">Aristotelian logic</a> (or term logic) as found in the <i><a href="/wiki/Organon" title="Organon">Organon</a></i>, found wide application and acceptance in Western science and mathematics for millennia.<sup id="cite_ref-FOOTNOTEBoehner1950xiv_6-0" class="reference"><a href="#cite_note-FOOTNOTEBoehner1950xiv-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> The <a href="/wiki/Stoicism" title="Stoicism">Stoics</a>, especially <a href="/wiki/Chrysippus" title="Chrysippus">Chrysippus</a>, began the development of <a href="/wiki/Predicate_logic" class="mw-redirect" title="Predicate logic">predicate logic</a>. In 18th-century Europe, attempts to treat the operations of formal logic in a symbolic or algebraic way had been made by philosophical mathematicians including <a href="/wiki/Gottfried_Wilhelm_Leibniz" title="Gottfried Wilhelm Leibniz">Leibniz</a> and <a href="/wiki/Johann_Heinrich_Lambert" title="Johann Heinrich Lambert">Lambert</a>, but their labors remained isolated and little known. </p> <div class="mw-heading mw-heading3"><h3 id="19th_century">19th century</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_logic&action=edit&section=4" title="Edit section: 19th century"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In the middle of the nineteenth century, <a href="/wiki/George_Boole" title="George Boole">George Boole</a> and then <a href="/wiki/Augustus_De_Morgan" title="Augustus De Morgan">Augustus De Morgan</a> presented systematic mathematical treatments of logic. Their work, building on work by algebraists such as <a href="/wiki/George_Peacock_(mathematician)" class="mw-redirect" title="George Peacock (mathematician)">George Peacock</a>, extended the traditional Aristotelian doctrine of logic into a sufficient framework for the study of <a href="/wiki/Foundations_of_mathematics" title="Foundations of mathematics">foundations of mathematics</a>.<sup id="cite_ref-FOOTNOTEKatz1998686_7-0" class="reference"><a href="#cite_note-FOOTNOTEKatz1998686-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> In 1847, <a href="/wiki/Vatroslav_Berti%C4%87" title="Vatroslav Bertić">Vatroslav Bertić</a> made substantial work on algebraization of logic, independently from Boole.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Charles_Sanders_Peirce" title="Charles Sanders Peirce">Charles Sanders Peirce</a> later built upon the work of Boole to develop a logical system for relations and quantifiers, which he published in several papers from 1870 to 1885. </p><p><a href="/wiki/Gottlob_Frege" title="Gottlob Frege">Gottlob Frege</a> presented an independent development of logic with quantifiers in his <i><a href="/wiki/Begriffsschrift" title="Begriffsschrift">Begriffsschrift</a></i>, published in 1879, a work generally considered as marking a turning point in the history of logic. Frege's work remained obscure, however, until <a href="/wiki/Bertrand_Russell" title="Bertrand Russell">Bertrand Russell</a> began to promote it near the turn of the century. The two-dimensional notation Frege developed was never widely adopted and is unused in contemporary texts. </p><p>From 1890 to 1905, <a href="/wiki/Ernst_Schr%C3%B6der_(mathematician)" title="Ernst Schröder (mathematician)">Ernst Schröder</a> published <i>Vorlesungen über die Algebra der Logik</i> in three volumes. This work summarized and extended the work of Boole, De Morgan, and Peirce, and was a comprehensive reference to symbolic logic as it was understood at the end of the 19th century. </p> <div class="mw-heading mw-heading4"><h4 id="Foundational_theories">Foundational theories</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_logic&action=edit&section=5" title="Edit section: Foundational theories"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Concerns that mathematics had not been built on a proper foundation led to the development of axiomatic systems for fundamental areas of mathematics such as arithmetic, analysis, and geometry. </p><p>In logic, the term <i>arithmetic</i> refers to the theory of the <a href="/wiki/Natural_number" title="Natural number">natural numbers</a>. <a href="/wiki/Giuseppe_Peano" title="Giuseppe Peano">Giuseppe Peano</a><sup id="cite_ref-FOOTNOTEPeano1889_9-0" class="reference"><a href="#cite_note-FOOTNOTEPeano1889-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> published a set of axioms for arithmetic that came to bear his name (<a href="/wiki/Peano_axioms" title="Peano axioms">Peano axioms</a>), using a variation of the logical system of Boole and Schröder but adding quantifiers. Peano was unaware of Frege's work at the time. Around the same time <a href="/wiki/Richard_Dedekind" title="Richard Dedekind">Richard Dedekind</a> showed that the natural numbers are uniquely characterized by their <a href="/wiki/Mathematical_induction" title="Mathematical induction">induction</a> properties. Dedekind proposed a different characterization, which lacked the formal logical character of Peano's axioms.<sup id="cite_ref-FOOTNOTEDedekind1888_10-0" class="reference"><a href="#cite_note-FOOTNOTEDedekind1888-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> Dedekind's work, however, proved theorems inaccessible in Peano's system, including the uniqueness of the set of natural numbers (up to isomorphism) and the recursive definitions of addition and multiplication from the <a href="/wiki/Successor_function" title="Successor function">successor function</a> and mathematical induction. </p><p>In the mid-19th century, flaws in Euclid's axioms for geometry became known.<sup id="cite_ref-FOOTNOTEKatz1998774_11-0" class="reference"><a href="#cite_note-FOOTNOTEKatz1998774-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> In addition to the independence of the <a href="/wiki/Parallel_postulate" title="Parallel postulate">parallel postulate</a>, established by <a href="/wiki/Nikolai_Lobachevsky" title="Nikolai Lobachevsky">Nikolai Lobachevsky</a> in 1826,<sup id="cite_ref-FOOTNOTELobachevsky1840_12-0" class="reference"><a href="#cite_note-FOOTNOTELobachevsky1840-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> mathematicians discovered that certain theorems taken for granted by Euclid were not in fact provable from his axioms. Among these is the theorem that a line contains at least two points, or that circles of the same radius whose centers are separated by that radius must intersect. Hilbert<sup id="cite_ref-FOOTNOTEHilbert1899_13-0" class="reference"><a href="#cite_note-FOOTNOTEHilbert1899-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> developed a complete set of <a href="/wiki/Hilbert%27s_axioms" title="Hilbert's axioms">axioms for geometry</a>, building on <a href="/wiki/Pasch%27s_axiom" title="Pasch's axiom">previous work</a> by Pasch.<sup id="cite_ref-FOOTNOTEPasch1882_14-0" class="reference"><a href="#cite_note-FOOTNOTEPasch1882-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> The success in axiomatizing geometry motivated Hilbert to seek complete axiomatizations of other areas of mathematics, such as the natural numbers and the <a href="/wiki/Real_line" class="mw-redirect" title="Real line">real line</a>. This would prove to be a major area of research in the first half of the 20th century. </p><p>The 19th century saw great advances in the theory of <a href="/wiki/Real_analysis" title="Real analysis">real analysis</a>, including theories of convergence of functions and <a href="/wiki/Fourier_series" title="Fourier series">Fourier series</a>. Mathematicians such as <a href="/wiki/Karl_Weierstrass" title="Karl Weierstrass">Karl Weierstrass</a> began to construct functions that stretched intuition, such as <a href="/wiki/Continuous,_nowhere_differentiable_function" class="mw-redirect" title="Continuous, nowhere differentiable function">nowhere-differentiable continuous functions</a>. Previous conceptions of a function as a rule for computation, or a smooth graph, were no longer adequate. Weierstrass began to advocate the <a href="/wiki/Arithmetization_of_analysis" title="Arithmetization of analysis">arithmetization of analysis</a>, which sought to axiomatize analysis using properties of the natural numbers. The modern <a href="/wiki/(%CE%B5,_%CE%B4)-definition_of_limit" class="mw-redirect" title="(ε, δ)-definition of limit">(ε, δ)-definition of limit</a> and <a href="/wiki/Continuous_function" title="Continuous function">continuous functions</a> was already developed by <a href="/wiki/Bernard_Bolzano" title="Bernard Bolzano">Bolzano</a> in 1817,<sup id="cite_ref-FOOTNOTEFelscher2000_15-0" class="reference"><a href="#cite_note-FOOTNOTEFelscher2000-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> but remained relatively unknown. <a href="/wiki/Cauchy" class="mw-redirect" title="Cauchy">Cauchy</a> in 1821 defined continuity in terms of <a href="/wiki/Infinitesimal" title="Infinitesimal">infinitesimals</a> (see Cours d'Analyse, page 34). In 1858, Dedekind proposed a definition of the real numbers in terms of <a href="/wiki/Dedekind_cuts" class="mw-redirect" title="Dedekind cuts">Dedekind cuts</a> of rational numbers, a definition still employed in contemporary texts.<sup id="cite_ref-FOOTNOTEDedekind1872_16-0" class="reference"><a href="#cite_note-FOOTNOTEDedekind1872-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> </p><p><a href="/wiki/Georg_Cantor" title="Georg Cantor">Georg Cantor</a> developed the fundamental concepts of infinite set theory. His early results developed the theory of <a href="/wiki/Cardinality" title="Cardinality">cardinality</a> and <a href="/wiki/Cantor%27s_first_uncountability_proof" class="mw-redirect" title="Cantor's first uncountability proof">proved</a> that the reals and the natural numbers have different cardinalities.<sup id="cite_ref-FOOTNOTECantor1874_17-0" class="reference"><a href="#cite_note-FOOTNOTECantor1874-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> Over the next twenty years, Cantor developed a theory of <a href="/wiki/Transfinite_number" title="Transfinite number">transfinite numbers</a> in a series of publications. In 1891, he published a new proof of the uncountability of the real numbers that introduced the <a href="/wiki/Cantor%27s_diagonal_argument" title="Cantor's diagonal argument">diagonal argument</a>, and used this method to prove <a href="/wiki/Cantor%27s_theorem" title="Cantor's theorem">Cantor's theorem</a> that no set can have the same cardinality as its <a href="/wiki/Powerset" class="mw-redirect" title="Powerset">powerset</a>. Cantor believed that every set could be <a href="/wiki/Well-ordered" class="mw-redirect" title="Well-ordered">well-ordered</a>, but was unable to produce a proof for this result, leaving it as an open problem in 1895.<sup id="cite_ref-FOOTNOTEKatz1998807_18-0" class="reference"><a href="#cite_note-FOOTNOTEKatz1998807-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="20th_century">20th century</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_logic&action=edit&section=6" title="Edit section: 20th century"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In the early decades of the 20th century, the main areas of study were set theory and formal logic. The discovery of paradoxes in informal set theory caused some to wonder whether mathematics itself is inconsistent, and to look for proofs of consistency. </p><p>In 1900, <a href="/wiki/David_Hilbert" title="David Hilbert">Hilbert</a> posed a famous list of <a href="/wiki/Hilbert%27s_problems" title="Hilbert's problems">23 problems</a> for the next century. The first two of these were to resolve the <a href="/wiki/Continuum_hypothesis" title="Continuum hypothesis">continuum hypothesis</a> and prove the consistency of elementary arithmetic, respectively; the tenth was to produce a method that could decide whether a multivariate polynomial equation over the <a href="/wiki/Integer" title="Integer">integers</a> has a solution. Subsequent work to resolve these problems shaped the direction of mathematical logic, as did the effort to resolve Hilbert's <i><a href="/wiki/Entscheidungsproblem" title="Entscheidungsproblem">Entscheidungsproblem</a></i>, posed in 1928. This problem asked for a procedure that would decide, given a formalized mathematical statement, whether the statement is true or false. </p> <div class="mw-heading mw-heading4"><h4 id="Set_theory_and_paradoxes">Set theory and paradoxes</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_logic&action=edit&section=7" title="Edit section: Set theory and paradoxes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Ernst_Zermelo" title="Ernst Zermelo">Ernst Zermelo</a> gave a proof that <a href="/wiki/Well-ordering_theorem" title="Well-ordering theorem">every set could be well-ordered</a>, a result <a href="/wiki/Georg_Cantor" title="Georg Cantor">Georg Cantor</a> had been unable to obtain.<sup id="cite_ref-FOOTNOTEZermelo1904_19-0" class="reference"><a href="#cite_note-FOOTNOTEZermelo1904-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> To achieve the proof, Zermelo introduced the <a href="/wiki/Axiom_of_choice" title="Axiom of choice">axiom of choice</a>, which drew heated debate and research among mathematicians and the pioneers of set theory. The immediate criticism of the method led Zermelo to publish a second exposition of his result, directly addressing criticisms of his proof.<sup id="cite_ref-FOOTNOTEZermelo1908a_20-0" class="reference"><a href="#cite_note-FOOTNOTEZermelo1908a-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> This paper led to the general acceptance of the axiom of choice in the mathematics community. </p><p>Skepticism about the axiom of choice was reinforced by recently discovered paradoxes in <a href="/wiki/Naive_set_theory" title="Naive set theory">naive set theory</a>. <a href="/wiki/Cesare_Burali-Forti" title="Cesare Burali-Forti">Cesare Burali-Forti</a><sup id="cite_ref-FOOTNOTEBurali-Forti1897_21-0" class="reference"><a href="#cite_note-FOOTNOTEBurali-Forti1897-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> was the first to state a paradox: the <a href="/wiki/Burali-Forti_paradox" title="Burali-Forti paradox">Burali-Forti paradox</a> shows that the collection of all <a href="/wiki/Ordinal_number" title="Ordinal number">ordinal numbers</a> cannot form a set. Very soon thereafter, <a href="/wiki/Bertrand_Russell" title="Bertrand Russell">Bertrand Russell</a> discovered <a href="/wiki/Russell%27s_paradox" title="Russell's paradox">Russell's paradox</a> in 1901, and <a href="/wiki/Jules_Richard_(mathematician)" title="Jules Richard (mathematician)">Jules Richard</a> discovered <a href="/wiki/Richard%27s_paradox" title="Richard's paradox">Richard's paradox</a>.<sup id="cite_ref-FOOTNOTERichard1905_22-0" class="reference"><a href="#cite_note-FOOTNOTERichard1905-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> </p><p>Zermelo provided the first set of axioms for set theory.<sup id="cite_ref-FOOTNOTEZermelo1908b_23-0" class="reference"><a href="#cite_note-FOOTNOTEZermelo1908b-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup> These axioms, together with the additional <a href="/wiki/Axiom_of_replacement" class="mw-redirect" title="Axiom of replacement">axiom of replacement</a> proposed by <a href="/wiki/Abraham_Fraenkel" title="Abraham Fraenkel">Abraham Fraenkel</a>, are now called <a href="/wiki/Zermelo%E2%80%93Fraenkel_set_theory" title="Zermelo–Fraenkel set theory">Zermelo–Fraenkel set theory</a> (ZF). Zermelo's axioms incorporated the principle of <a href="/wiki/Limitation_of_size" title="Limitation of size">limitation of size</a> to avoid Russell's paradox. </p><p>In 1910, the first volume of <i><a href="/wiki/Principia_Mathematica" title="Principia Mathematica">Principia Mathematica</a></i> by Russell and <a href="/wiki/Alfred_North_Whitehead" title="Alfred North Whitehead">Alfred North Whitehead</a> was published. This seminal work developed the theory of functions and cardinality in a completely formal framework of <a href="/wiki/Type_theory" title="Type theory">type theory</a>, which Russell and Whitehead developed in an effort to avoid the paradoxes. <i>Principia Mathematica</i> is considered one of the most influential works of the 20th century, although the framework of type theory did not prove popular as a foundational theory for mathematics.<sup id="cite_ref-FOOTNOTEFerreirós2001445_24-0" class="reference"><a href="#cite_note-FOOTNOTEFerreirós2001445-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup> </p><p>Fraenkel<sup id="cite_ref-FOOTNOTEFraenkel1922_25-0" class="reference"><a href="#cite_note-FOOTNOTEFraenkel1922-25"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup> proved that the axiom of choice cannot be proved from the axioms of Zermelo's set theory with <a href="/wiki/Urelements" class="mw-redirect" title="Urelements">urelements</a>. Later work by <a href="/wiki/Paul_Cohen" title="Paul Cohen">Paul Cohen</a><sup id="cite_ref-FOOTNOTECohen1966_26-0" class="reference"><a href="#cite_note-FOOTNOTECohen1966-26"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup> showed that the addition of urelements is not needed, and the axiom of choice is unprovable in ZF. Cohen's proof developed the method of <a href="/wiki/Forcing_(mathematics)" title="Forcing (mathematics)">forcing</a>, which is now an important tool for establishing <a href="/wiki/Independence_result" class="mw-redirect" title="Independence result">independence results</a> in set theory.<sup id="cite_ref-27" class="reference"><a href="#cite_note-27"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading4"><h4 id="Symbolic_logic">Symbolic logic</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_logic&action=edit&section=8" title="Edit section: Symbolic logic"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Leopold_L%C3%B6wenheim" title="Leopold Löwenheim">Leopold Löwenheim</a><sup id="cite_ref-FOOTNOTELöwenheim1915_28-0" class="reference"><a href="#cite_note-FOOTNOTELöwenheim1915-28"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup> and <a href="/wiki/Thoralf_Skolem" title="Thoralf Skolem">Thoralf Skolem</a><sup id="cite_ref-FOOTNOTESkolem1920_29-0" class="reference"><a href="#cite_note-FOOTNOTESkolem1920-29"><span class="cite-bracket">[</span>29<span class="cite-bracket">]</span></a></sup> obtained the <a href="/wiki/L%C3%B6wenheim%E2%80%93Skolem_theorem" title="Löwenheim–Skolem theorem">Löwenheim–Skolem theorem</a>, which says that <a href="/wiki/First-order_logic" title="First-order logic">first-order logic</a> cannot control the <a href="/wiki/Cardinal_number" title="Cardinal number">cardinalities</a> of infinite structures. Skolem realized that this theorem would apply to first-order formalizations of set theory, and that it implies any such formalization has a <a href="/wiki/Countable" class="mw-redirect" title="Countable">countable</a> <a href="/wiki/Structure_(mathematical_logic)" title="Structure (mathematical logic)">model</a>. This counterintuitive fact became known as <a href="/wiki/Skolem%27s_paradox" title="Skolem's paradox">Skolem's paradox</a>. </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Young_Kurt_G%C3%B6del_as_a_student_in_1925.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/9e/Young_Kurt_G%C3%B6del_as_a_student_in_1925.jpg/220px-Young_Kurt_G%C3%B6del_as_a_student_in_1925.jpg" decoding="async" width="220" height="291" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/9e/Young_Kurt_G%C3%B6del_as_a_student_in_1925.jpg/330px-Young_Kurt_G%C3%B6del_as_a_student_in_1925.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/9e/Young_Kurt_G%C3%B6del_as_a_student_in_1925.jpg/440px-Young_Kurt_G%C3%B6del_as_a_student_in_1925.jpg 2x" data-file-width="674" data-file-height="893" /></a><figcaption>Portrait of young <a href="/wiki/Kurt_G%C3%B6del" title="Kurt Gödel">Kurt Gödel</a> as a student in <a href="/wiki/Vienna" title="Vienna">Vienna</a>,1925.</figcaption></figure> <p>In his doctoral thesis, <a href="/wiki/Kurt_G%C3%B6del" title="Kurt Gödel">Kurt Gödel</a> proved the <a href="/wiki/Completeness_theorem" class="mw-redirect" title="Completeness theorem">completeness theorem</a>, which establishes a correspondence between syntax and semantics in first-order logic.<sup id="cite_ref-FOOTNOTEGödel1929_30-0" class="reference"><a href="#cite_note-FOOTNOTEGödel1929-30"><span class="cite-bracket">[</span>30<span class="cite-bracket">]</span></a></sup> Gödel used the completeness theorem to prove the <a href="/wiki/Compactness_theorem" title="Compactness theorem">compactness theorem</a>, demonstrating the finitary nature of first-order <a href="/wiki/Logical_consequence" title="Logical consequence">logical consequence</a>. These results helped establish first-order logic as the dominant logic used by mathematicians. </p><p>In 1931, Gödel published <i><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>, which proved the incompleteness (in a different meaning of the word) of all sufficiently strong, effective first-order theories. This result, known as <a href="/wiki/G%C3%B6del%27s_incompleteness_theorem" class="mw-redirect" title="Gödel's incompleteness theorem">Gödel's incompleteness theorem</a>, establishes severe limitations on axiomatic foundations for mathematics, striking a strong blow to Hilbert's program. It showed the impossibility of providing a consistency proof of arithmetic within any formal theory of arithmetic. Hilbert, however, did not acknowledge the importance of the incompleteness theorem for some time.<sup id="cite_ref-HilbertBernays1934_PlusNote_31-0" class="reference"><a href="#cite_note-HilbertBernays1934_PlusNote-31"><span class="cite-bracket">[</span>a<span class="cite-bracket">]</span></a></sup> </p><p>Gödel's theorem shows that a <a href="/wiki/Consistency" title="Consistency">consistency</a> proof of any sufficiently strong, effective axiom system cannot be obtained in the system itself, if the system is consistent, nor in any weaker system. This leaves open the possibility of consistency proofs that cannot be formalized within the system they consider. Gentzen proved the consistency of arithmetic using a finitistic system together with a principle of <a href="/wiki/Transfinite_induction" title="Transfinite induction">transfinite induction</a>.<sup id="cite_ref-FOOTNOTEGentzen1936_32-0" class="reference"><a href="#cite_note-FOOTNOTEGentzen1936-32"><span class="cite-bracket">[</span>31<span class="cite-bracket">]</span></a></sup> Gentzen's result introduced the ideas of <a href="/wiki/Cut_elimination" class="mw-redirect" title="Cut elimination">cut elimination</a> and <a href="/wiki/Proof-theoretic_ordinal" class="mw-redirect" title="Proof-theoretic ordinal">proof-theoretic ordinals</a>, which became key tools in proof theory. Gödel gave a different consistency proof, which reduces the consistency of classical arithmetic to that of intuitionistic arithmetic in higher types.<sup id="cite_ref-FOOTNOTEGödel1958_33-0" class="reference"><a href="#cite_note-FOOTNOTEGödel1958-33"><span class="cite-bracket">[</span>32<span class="cite-bracket">]</span></a></sup> </p><p>The first textbook on symbolic logic for the layman was written by <a href="/wiki/Lewis_Carroll" title="Lewis Carroll">Lewis Carroll</a>,<sup id="cite_ref-34" class="reference"><a href="#cite_note-34"><span class="cite-bracket">[</span>33<span class="cite-bracket">]</span></a></sup> author of <i><a href="/wiki/Alice%27s_Adventures_in_Wonderland" title="Alice's Adventures in Wonderland">Alice's Adventures in Wonderland</a></i>, in 1896.<sup id="cite_ref-FOOTNOTECarroll1896_35-0" class="reference"><a href="#cite_note-FOOTNOTECarroll1896-35"><span class="cite-bracket">[</span>34<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading4"><h4 id="Beginnings_of_the_other_branches">Beginnings of the other branches</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_logic&action=edit&section=9" title="Edit section: Beginnings of the other branches"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Alfred_Tarski" title="Alfred Tarski">Alfred Tarski</a> developed the basics of <a href="/wiki/Model_theory" title="Model theory">model theory</a>. </p><p>Beginning in 1935, a group of prominent mathematicians collaborated under the pseudonym <a href="/wiki/Nicolas_Bourbaki" title="Nicolas Bourbaki">Nicolas Bourbaki</a> to publish <i><a href="/wiki/%C3%89l%C3%A9ments_de_math%C3%A9matique" title="Éléments de mathématique">Éléments de mathématique</a></i>, a series of encyclopedic mathematics texts. These texts, written in an austere and axiomatic style, emphasized rigorous presentation and set-theoretic foundations. Terminology coined by these texts, such as the words <a href="/wiki/Bijection,_injection,_and_surjection" class="mw-redirect" title="Bijection, injection, and surjection"><i>bijection</i>, <i>injection</i>, and <i>surjection</i></a>, and the set-theoretic foundations the texts employed, were widely adopted throughout mathematics. </p><p>The study of computability came to be known as recursion theory or <a href="/wiki/Computability_theory" title="Computability theory">computability theory</a>, because early formalizations by Gödel and Kleene relied on recursive definitions of functions.<sup id="cite_ref-36" class="reference"><a href="#cite_note-36"><span class="cite-bracket">[</span>b<span class="cite-bracket">]</span></a></sup> When these definitions were shown equivalent to Turing's formalization involving <a href="/wiki/Turing_machine" title="Turing machine">Turing machines</a>, it became clear that a new concept – the <a href="/wiki/Computable_function" title="Computable function">computable function</a> – had been discovered, and that this definition was robust enough to admit numerous independent characterizations. In his work on the incompleteness theorems in 1931, Gödel lacked a rigorous concept of an effective formal system; he immediately realized that the new definitions of computability could be used for this purpose, allowing him to state the incompleteness theorems in generality that could only be implied in the original paper. </p><p>Numerous results in recursion theory were obtained in the 1940s by <a href="/wiki/Stephen_Cole_Kleene" title="Stephen Cole Kleene">Stephen Cole Kleene</a> and <a href="/wiki/Emil_Leon_Post" title="Emil Leon Post">Emil Leon Post</a>. Kleene<sup id="cite_ref-FOOTNOTEKleene1943_37-0" class="reference"><a href="#cite_note-FOOTNOTEKleene1943-37"><span class="cite-bracket">[</span>35<span class="cite-bracket">]</span></a></sup> introduced the concepts of relative computability, foreshadowed by Turing,<sup id="cite_ref-FOOTNOTETuring1939_38-0" class="reference"><a href="#cite_note-FOOTNOTETuring1939-38"><span class="cite-bracket">[</span>36<span class="cite-bracket">]</span></a></sup> and the <a href="/wiki/Arithmetical_hierarchy" title="Arithmetical hierarchy">arithmetical hierarchy</a>. Kleene later generalized recursion theory to higher-order functionals. Kleene and <a href="/wiki/Georg_Kreisel" title="Georg Kreisel">Georg Kreisel</a> studied formal versions of intuitionistic mathematics, particularly in the context of proof theory. </p> <div class="mw-heading mw-heading2"><h2 id="Formal_logical_systems">Formal logical systems <span class="anchor" id="Formal_logic"></span></h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_logic&action=edit&section=10" title="Edit section: Formal logical systems"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1246091330"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><table class="sidebar nomobile nowraplinks"><tbody><tr><th class="sidebar-title" style="font-size: 130%; margin: 6px 0px 6px 0px; background: #ddf;"><a href="/wiki/Logical_connective" title="Logical connective">Logical connectives</a></th></tr><tr><td class="sidebar-content"> <table style="width:100%;border-collapse:collapse;border-spacing:0px 0px;border:none;line-height:1.3em;"><tbody><tr style="vertical-align:top"><td style="text-align:left;"> <a href="/wiki/Negation" title="Negation">NOT</a></td><td style="text-align:right;font-size:125%;line-height:0.8em;vertical-align:middle;white-space:nowrap;font-family:serif;"> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \neg A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">¬</mi> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \neg A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/195aae731102b36b14a902a091d04ac5c6a5af49" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.293ex; height:2.176ex;" alt="{\displaystyle \neg A}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6daf6db742ace65252b589963f7e7a07603ccb56" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:3.551ex; height:2.343ex;" alt="{\displaystyle -A}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {A}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>A</mi> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {A}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92efef0e89bdc77f6a848764195ef5b9d9bfcc6a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.858ex; height:3.009ex;" alt="{\displaystyle {\overline {A}}}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sim A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∼</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sim A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c79bf96c247833282e773fae43602343150c1665" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.196ex; height:2.176ex;" alt="{\displaystyle \sim A}"></span></td></tr><tr style="vertical-align:top"><td style="text-align:left;"> <a href="/wiki/Logical_conjunction" title="Logical conjunction">AND</a></td><td style="text-align:right;font-size:125%;line-height:0.8em;vertical-align:middle;white-space:nowrap;font-family:serif;"> <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 A\land B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>∧</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\land B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/74954195333a8593163b93a9688695b8dc74da55" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.09ex; height:2.176ex;" alt="{\displaystyle A\land B}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\cdot B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>⋅</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\cdot B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a90e903f21f11a0f4ab3caca1e6943ba7a9849" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.186ex; height:2.176ex;" alt="{\displaystyle A\cdot B}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle AB}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle AB}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b04153f9681e5b06066357774475c04aaef3a8bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.507ex; height:2.176ex;" alt="{\displaystyle AB}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\&B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mi mathvariant="normal">&</mi> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\&B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f65ee34a8896390e0d1f193c137d9eb64815c1a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.315ex; height:2.176ex;" alt="{\displaystyle A\&B}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\&\&B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mi mathvariant="normal">&</mi> <mi mathvariant="normal">&</mi> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\&\&B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1ab3517dd859d7eaa8f3f1656c8125f99ada1470" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.123ex; height:2.176ex;" alt="{\displaystyle A\&\&B}"></span></td></tr><tr style="vertical-align:top"><td style="text-align:left;"> <a href="/wiki/Sheffer_stroke" title="Sheffer stroke">NAND</a></td><td style="text-align:right;font-size:125%;line-height:0.8em;vertical-align:middle;white-space:nowrap;font-family:serif;"> <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 A{\overline {\land }}B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mover> <mo>∧</mo> <mo accent="false">¯</mo> </mover> </mrow> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A{\overline {\land }}B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0167d56141342887a74d56a036e6fbbad7172b0f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.172ex; height:2.843ex;" alt="{\displaystyle A{\overline {\land }}B}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\uparrow B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo stretchy="false">↑</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\uparrow B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5723e9ef44d446f4410c273b056d7c7c8e6f2564" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.96ex; height:2.509ex;" alt="{\displaystyle A\uparrow B}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\mid B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>∣</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\mid B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1400973074b4691cc0638a68118716a2b218fce2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.444ex; height:2.843ex;" alt="{\displaystyle A\mid B}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {A\cdot B}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <mo>⋅</mo> <mi>B</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {A\cdot B}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/225f35bb78e90b9126458f1bc6bf1ed3f0724bbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.301ex; height:3.009ex;" alt="{\displaystyle {\overline {A\cdot B}}}"></span></td></tr><tr style="vertical-align:top"><td style="text-align:left;"> <a href="/wiki/Logical_disjunction" title="Logical disjunction">OR</a></td><td style="text-align:right;font-size:125%;line-height:0.8em;vertical-align:middle;white-space:nowrap;font-family:serif;"> <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 A\lor B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>∨</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\lor B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9b9c9c90857c12727201dd9e47a4e7c8658fdbc5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.09ex; height:2.176ex;" alt="{\displaystyle A\lor B}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A+B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>+</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A+B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4279cdbd3cb8ec4c3423065d9a7d83a82cfc89e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.348ex; height:2.343ex;" alt="{\displaystyle A+B}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\mid B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>∣</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\mid B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1400973074b4691cc0638a68118716a2b218fce2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.444ex; height:2.843ex;" alt="{\displaystyle A\mid B}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\parallel B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>∥</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\parallel B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04bd0239f9b74f7ef1520ba4e30454b06e695289" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.96ex; height:2.843ex;" alt="{\displaystyle A\parallel B}"></span></td></tr><tr style="vertical-align:top"><td style="text-align:left;"> <a href="/wiki/Logical_NOR" title="Logical NOR">NOR</a></td><td style="text-align:right;font-size:125%;line-height:0.8em;vertical-align:middle;white-space:nowrap;font-family:serif;"> <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 A{\overline {\lor }}B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mover> <mo>∨</mo> <mo accent="false">¯</mo> </mover> </mrow> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A{\overline {\lor }}B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/591afbca3e984765b18abb189f4bb1b88116c400" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.172ex; height:2.843ex;" alt="{\displaystyle A{\overline {\lor }}B}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\downarrow B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo stretchy="false">↓</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\downarrow B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c5e77260e67880093dafe958880ea02f5026164" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.96ex; height:2.509ex;" alt="{\displaystyle A\downarrow B}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {A+B}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <mo>+</mo> <mi>B</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {A+B}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/08840f8e2022f127fc459d801a8f8ce93f65f55a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.462ex; height:3.176ex;" alt="{\displaystyle {\overline {A+B}}}"></span></td></tr><tr style="vertical-align:top"><td style="text-align:left;"> <a href="/wiki/XNOR_gate" title="XNOR gate">XNOR</a></td><td style="text-align:right;font-size:125%;line-height:0.8em;vertical-align:middle;white-space:nowrap;font-family:serif;"> <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 A\ {\text{XNOR}}\ B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mtext>XNOR</mtext> </mrow> <mtext> </mtext> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\ {\text{XNOR}}\ B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54fbdc2cf9fad58dda134c0c9baa55a4712b3955" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.673ex; height:2.176ex;" alt="{\displaystyle A\ {\text{XNOR}}\ B}"></span></td></tr><tr style="vertical-align:top"><td style="text-align:left;"> └ <a href="/wiki/Logical_biconditional" title="Logical biconditional">equivalent</a></td><td style="text-align:right;font-size:125%;line-height:0.8em;vertical-align:middle;white-space:nowrap;font-family:serif;"> <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 A\equiv B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>≡</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\equiv B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b933daba3ef47ec3b4f3097ea6e741b85149707" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.606ex; height:2.176ex;" alt="{\displaystyle A\equiv B}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\Leftrightarrow B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo stretchy="false">⇔</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\Leftrightarrow B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce08fffb4d36ba12921b8b3e06228887015b2b8e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.121ex; height:2.176ex;" alt="{\displaystyle A\Leftrightarrow B}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\leftrightharpoons B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo stretchy="false">⇋</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\leftrightharpoons B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a8639312e9e120cd65c98fc48a6d5256d57288c6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.121ex; height:2.176ex;" alt="{\displaystyle A\leftrightharpoons B}"></span></td></tr><tr style="vertical-align:top"><td style="text-align:left;"> <a href="/wiki/Exclusive_or" title="Exclusive or">XOR</a></td><td style="text-align:right;font-size:125%;line-height:0.8em;vertical-align:middle;white-space:nowrap;font-family:serif;"> <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 A{\underline {\lor }}B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <munder> <mo>∨</mo> <mo>_</mo> </munder> </mrow> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A{\underline {\lor }}B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/803e1413692912954b90e99694e10c728e27a153" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.562ex; margin-bottom: -0.776ex; width:5.06ex; height:3.176ex;" alt="{\displaystyle A{\underline {\lor }}B}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\oplus B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>⊕</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\oplus B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0512d6bdd29ff000dea0bf68b853618dcaabc3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.348ex; height:2.343ex;" alt="{\displaystyle A\oplus B}"></span></td></tr><tr style="vertical-align:top"><td style="text-align:left;"> └nonequivalent</td><td style="text-align:right;font-size:125%;line-height:0.8em;vertical-align:middle;white-space:nowrap;font-family:serif;"> <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 A\not \equiv B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>≢</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\not \equiv B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0339ed15cd58f7263c5eec8e5628168aa6006200" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.607ex; height:2.676ex;" alt="{\displaystyle A\not \equiv B}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\not \Leftrightarrow B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>⇎</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\not \Leftrightarrow B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47bfd97641b05a7f7fc0bcd02e83fa6532c62bb4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.121ex; height:2.176ex;" alt="{\displaystyle A\not \Leftrightarrow B}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\nleftrightarrow B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>↮</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\nleftrightarrow B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/926467de6fd4709bdbc59c3168a21298cdf0d26c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.121ex; height:2.176ex;" alt="{\displaystyle A\nleftrightarrow B}"></span></td></tr><tr style="vertical-align:top"><td style="text-align:left;"> <a href="/wiki/Material_conditional" title="Material conditional">implies</a></td><td style="text-align:right;font-size:125%;line-height:0.8em;vertical-align:middle;white-space:nowrap;font-family:serif;"> <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 A\Rightarrow B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo stretchy="false">⇒</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\Rightarrow B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e560143d45c97e6387c7c3aa90e9d7745002228" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.121ex; height:2.176ex;" alt="{\displaystyle A\Rightarrow B}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\supset B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>⊃</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\supset B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee952838d8b3e67045072a8f2b71e7fc0467dea6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.606ex; height:2.176ex;" alt="{\displaystyle A\supset B}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\rightarrow B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo stretchy="false">→</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\rightarrow B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/23efef033def56a67de7ded823f14626de26d174" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.121ex; height:2.176ex;" alt="{\displaystyle A\rightarrow B}"></span></td></tr><tr style="vertical-align:top"><td style="text-align:left;"> <a href="/wiki/Converse_(logic)" title="Converse (logic)">converse</a></td><td style="text-align:right;font-size:125%;line-height:0.8em;vertical-align:middle;white-space:nowrap;font-family:serif;"> <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 A\Leftarrow B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo stretchy="false">⇐</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\Leftarrow B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c8569d34530d97f701080546ca0f20c0defadf8f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.121ex; height:2.176ex;" alt="{\displaystyle A\Leftarrow B}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\subset B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>⊂</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\subset B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/010e98bb4c817357e3ef7e8fa7fbe2385b2aec6e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.606ex; height:2.176ex;" alt="{\displaystyle A\subset B}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\leftarrow B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo stretchy="false">←</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\leftarrow B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/456da65c891c438fea04d7e40283b67d600fe92d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.121ex; height:2.176ex;" alt="{\displaystyle A\leftarrow B}"></span></td></tr></tbody></table></td> </tr><tr><th class="sidebar-heading" style="background: #eef; text-align: center;"> Related concepts</th></tr><tr><td class="sidebar-content"> <div class="hlist" style="line-height:1.3em;"><ul><li><a href="/wiki/Propositional_calculus" title="Propositional calculus">Propositional calculus</a></li><li><a href="/wiki/First-order_logic" title="First-order logic">Predicate logic</a></li><li><a href="/wiki/Boolean_algebra" title="Boolean algebra">Boolean algebra</a></li><li><a href="/wiki/Truth_table" title="Truth table">Truth table</a></li><li><a href="/wiki/Truth_function" title="Truth function">Truth function</a></li><li><a href="/wiki/Boolean_function" title="Boolean function">Boolean function</a></li><li><a href="/wiki/Functional_completeness" title="Functional completeness">Functional completeness</a></li><li><a href="/wiki/Scope_(logic)" title="Scope (logic)">Scope (logic)</a></li></ul></div></td> </tr><tr><th class="sidebar-heading" style="background: #eef; text-align: center;"> Applications</th></tr><tr><td class="sidebar-content"> <div class="hlist"><ul><li><a href="/wiki/Logic_gate" title="Logic gate">Digital logic</a></li><li><a href="/wiki/Programming_language" title="Programming language">Programming languages</a></li><li><a class="mw-selflink selflink">Mathematical logic</a></li><li><a href="/wiki/Philosophy_of_logic" title="Philosophy of logic">Philosophy of logic</a></li></ul></div></td> </tr><tr><td class="sidebar-below hlist" style="background: #eef; text-align: center;"> <span class="noviewer" typeof="mw:File"><span title="Category"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/16px-Symbol_category_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/23px-Symbol_category_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/31px-Symbol_category_class.svg.png 2x" data-file-width="180" data-file-height="185" /></span></span> <a href="/wiki/Category:Logical_connectives" title="Category:Logical connectives">Category</a></td></tr><tr><td class="sidebar-navbar"><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:Logical_connectives_sidebar" title="Template:Logical connectives sidebar"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/w/index.php?title=Template_talk:Logical_connectives_sidebar&action=edit&redlink=1" class="new" title="Template talk:Logical connectives sidebar (page does not exist)"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Logical_connectives_sidebar" title="Special:EditPage/Template:Logical connectives sidebar"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <p>At its core, mathematical logic deals with mathematical concepts expressed using <a href="/wiki/Logical_system" class="mw-redirect" title="Logical system">formal logical systems</a>. These systems, though they differ in many details, share the common property of considering only expressions in a fixed <a href="/wiki/Formal_language" title="Formal language">formal language</a>. The systems of <a href="/wiki/Propositional_logic" class="mw-redirect" title="Propositional logic">propositional logic</a> and <a href="/wiki/First-order_logic" title="First-order logic">first-order logic</a> are the most widely studied today, because of their applicability to <a href="/wiki/Foundations_of_mathematics" title="Foundations of mathematics">foundations of mathematics</a> and because of their desirable proof-theoretic properties.<sup id="cite_ref-FerreirósSurveys_39-0" class="reference"><a href="#cite_note-FerreirósSurveys-39"><span class="cite-bracket">[</span>c<span class="cite-bracket">]</span></a></sup> Stronger classical logics such as <a href="/wiki/Second-order_logic" title="Second-order logic">second-order logic</a> or <a href="/wiki/Infinitary_logic" title="Infinitary logic">infinitary logic</a> are also studied, along with <a href="/wiki/Non-classical_logic" title="Non-classical logic">Non-classical logics</a> such as <a href="/wiki/Intuitionistic_logic" title="Intuitionistic logic">intuitionistic logic</a>. </p> <div class="mw-heading mw-heading3"><h3 id="First-order_logic">First-order logic</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_logic&action=edit&section=11" title="Edit section: First-order logic"><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/First-order_logic" title="First-order logic">First-order logic</a></div> <p><b>First-order logic</b> is a particular <a href="/wiki/Logical_system" class="mw-redirect" title="Logical system">formal system of logic</a>. Its <a href="/wiki/Syntax" title="Syntax">syntax</a> involves only finite expressions as <a href="/wiki/Well-formed_formula" title="Well-formed formula">well-formed formulas</a>, while its <a href="/wiki/First-order_logic#Semantics" title="First-order logic">semantics</a> are characterized by the limitation of all <a href="/wiki/Quantifiers_(logic)" class="mw-redirect" title="Quantifiers (logic)">quantifiers</a> to a fixed <a href="/wiki/Domain_of_discourse" title="Domain of discourse">domain of discourse</a>. </p><p>Early results from formal logic established limitations of first-order logic. The <a href="/wiki/L%C3%B6wenheim%E2%80%93Skolem_theorem" title="Löwenheim–Skolem theorem">Löwenheim–Skolem theorem</a> (1919) showed that if a set of sentences in a countable first-order language has an infinite model then it has at least one model of each infinite cardinality. This shows that it is impossible for a set of first-order axioms to characterize the natural numbers, the real numbers, or any other infinite structure up to <a href="/wiki/Isomorphism" title="Isomorphism">isomorphism</a>. As the goal of early foundational studies was to produce axiomatic theories for all parts of mathematics, this limitation was particularly stark. </p><p><a href="/wiki/G%C3%B6del%27s_completeness_theorem" title="Gödel's completeness theorem">Gödel's completeness theorem</a> established the equivalence between semantic and syntactic definitions of <a href="/wiki/Logical_consequence" title="Logical consequence">logical consequence</a> in first-order logic.<sup id="cite_ref-FOOTNOTEGödel1929_30-1" class="reference"><a href="#cite_note-FOOTNOTEGödel1929-30"><span class="cite-bracket">[</span>30<span class="cite-bracket">]</span></a></sup> It shows that if a particular sentence is true in every model that satisfies a particular set of axioms, then there must be a finite deduction of the sentence from the axioms. The <a href="/wiki/Compactness_theorem" title="Compactness theorem">compactness theorem</a> first appeared as a lemma in Gödel's proof of the completeness theorem, and it took many years before logicians grasped its significance and began to apply it routinely. It says that a set of sentences has a model if and only if every finite subset has a model, or in other words that an inconsistent set of formulas must have a finite inconsistent subset. The completeness and compactness theorems allow for sophisticated analysis of logical consequence in first-order logic and the development of <a href="/wiki/Model_theory" title="Model theory">model theory</a>, and they are a key reason for the prominence of first-order logic in mathematics. </p><p><a href="/wiki/G%C3%B6del%27s_incompleteness_theorems" title="Gödel's incompleteness theorems">Gödel's incompleteness theorems</a> establish additional limits on first-order axiomatizations.<sup id="cite_ref-FOOTNOTEGödel1931_40-0" class="reference"><a href="#cite_note-FOOTNOTEGödel1931-40"><span class="cite-bracket">[</span>37<span class="cite-bracket">]</span></a></sup> The <b>first incompleteness theorem</b> states that for any consistent, effectively given (defined below) logical system that is capable of interpreting arithmetic, there exists a statement that is true (in the sense that it holds for the natural numbers) but not provable within that logical system (and which indeed may fail in some <a href="/wiki/Non-standard_model_of_arithmetic" title="Non-standard model of arithmetic">non-standard models of arithmetic</a> which may be consistent with the logical system). For example, in every logical system capable of expressing the <a href="/wiki/Peano_axioms" title="Peano axioms">Peano axioms</a>, the Gödel sentence holds for the natural numbers but cannot be proved. </p><p>Here a logical system is said to be effectively given if it is possible to decide, given any formula in the language of the system, whether the formula is an axiom, and one which can express the Peano axioms is called "sufficiently strong." When applied to first-order logic, the first incompleteness theorem implies that any sufficiently strong, consistent, effective first-order theory has models that are not <a href="/wiki/Elementary_substructure" class="mw-redirect" title="Elementary substructure">elementarily equivalent</a>, a stronger limitation than the one established by the Löwenheim–Skolem theorem. The <b>second incompleteness theorem</b> states that no sufficiently strong, consistent, effective axiom system for arithmetic can prove its own consistency, which has been interpreted to show that <a href="/wiki/Hilbert%27s_program" title="Hilbert's program">Hilbert's program</a> cannot be reached. </p> <div class="mw-heading mw-heading3"><h3 id="Other_classical_logics">Other classical logics</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_logic&action=edit&section=12" title="Edit section: Other classical logics"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Many logics besides first-order logic are studied. These include <a href="/wiki/Infinitary_logics" class="mw-redirect" title="Infinitary logics">infinitary logics</a>, which allow for formulas to provide an infinite amount of information, and <a href="/wiki/Higher-order_logic" title="Higher-order logic">higher-order logics</a>, which include a portion of set theory directly in their semantics. </p><p>The most well studied infinitary logic is <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 L_{\omega _{1},\omega }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mi>ω</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L_{\omega _{1},\omega }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/037b84dcf7eb85313f964632c939bce6464a657c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.149ex; height:2.843ex;" alt="{\displaystyle L_{\omega _{1},\omega }}"></span>. In this logic, quantifiers may only be nested to finite depths, as in first-order logic, but formulas may have finite or countably infinite conjunctions and disjunctions within them. Thus, for example, it is possible to say that an object is a whole number using a formula of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L_{\omega _{1},\omega }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mi>ω</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L_{\omega _{1},\omega }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/037b84dcf7eb85313f964632c939bce6464a657c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.149ex; height:2.843ex;" alt="{\displaystyle L_{\omega _{1},\omega }}"></span> such as </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x=0)\lor (x=1)\lor (x=2)\lor \cdots .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>=</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>∨</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>=</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>∨</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>=</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>∨</mo> <mo>⋯</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x=0)\lor (x=1)\lor (x=2)\lor \cdots .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f1ab1d9af54fdcb661a5ef228fcef266605adf56" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.705ex; height:2.843ex;" alt="{\displaystyle (x=0)\lor (x=1)\lor (x=2)\lor \cdots .}"></span></dd></dl> <p>Higher-order logics allow for quantification not only of elements of the <a href="/wiki/Domain_of_discourse" title="Domain of discourse">domain of discourse</a>, but subsets of the domain of discourse, sets of such subsets, and other objects of higher type. The semantics are defined so that, rather than having a separate domain for each higher-type quantifier to range over, the quantifiers instead range over all objects of the appropriate type. The logics studied before the development of first-order logic, for example Frege's logic, had similar set-theoretic aspects. Although higher-order logics are more expressive, allowing complete axiomatizations of structures such as the natural numbers, they do not satisfy analogues of the completeness and compactness theorems from first-order logic, and are thus less amenable to proof-theoretic analysis. </p><p>Another type of logics are <b><style data-mw-deduplicate="TemplateStyles:r1238216509">.mw-parser-output .vanchor>:target~.vanchor-text{background-color:#b1d2ff}@media screen{html.skin-theme-clientpref-night .mw-parser-output .vanchor>:target~.vanchor-text{background-color:#0f4dc9}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .vanchor>:target~.vanchor-text{background-color:#0f4dc9}}</style><span class="vanchor"><span id="fixed-point_logic"></span><span class="vanchor-text">fixed-point logic</span></span>s</b> that allow <a href="/wiki/Inductive_definition" class="mw-redirect" title="Inductive definition">inductive definitions</a>, like one writes for <a href="/wiki/Primitive_recursive_function" title="Primitive recursive function">primitive recursive functions</a>. </p><p>One can formally define an extension of first-order logic — a notion which encompasses all logics in this section because they behave like first-order logic in certain fundamental ways, but does not encompass all logics in general, e.g. it does not encompass intuitionistic, modal or <a href="/wiki/Fuzzy_logic" title="Fuzzy logic">fuzzy logic</a>. </p><p><a href="/wiki/Lindstr%C3%B6m%27s_theorem" title="Lindström's theorem">Lindström's theorem</a> implies that the only extension of first-order logic satisfying both the <a href="/wiki/Compactness_theorem" title="Compactness theorem">compactness theorem</a> and the <a href="/wiki/L%C3%B6wenheim%E2%80%93Skolem_theorem#Downward_part" title="Löwenheim–Skolem theorem">downward Löwenheim–Skolem theorem</a> is first-order logic. </p> <div class="mw-heading mw-heading3"><h3 id="Nonclassical_and_modal_logic">Nonclassical and modal logic</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_logic&action=edit&section=13" title="Edit section: Nonclassical and modal logic"><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/Non-classical_logic" title="Non-classical logic">Non-classical logic</a></div> <p><a href="/wiki/Modal_logic" title="Modal logic">Modal logics</a> include additional modal operators, such as an operator which states that a particular formula is not only true, but necessarily true. Although modal logic is not often used to axiomatize mathematics, it has been used to study the properties of first-order provability<sup id="cite_ref-FOOTNOTESolovay1976_41-0" class="reference"><a href="#cite_note-FOOTNOTESolovay1976-41"><span class="cite-bracket">[</span>38<span class="cite-bracket">]</span></a></sup> and set-theoretic forcing.<sup id="cite_ref-FOOTNOTEHamkinsLöwe2007_42-0" class="reference"><a href="#cite_note-FOOTNOTEHamkinsLöwe2007-42"><span class="cite-bracket">[</span>39<span class="cite-bracket">]</span></a></sup> </p><p><a href="/wiki/Intuitionistic_logic" title="Intuitionistic logic">Intuitionistic logic</a> was developed by Heyting to study Brouwer's program of intuitionism, in which Brouwer himself avoided formalization. Intuitionistic logic specifically does not include the <a href="/wiki/Law_of_the_excluded_middle" class="mw-redirect" title="Law of the excluded middle">law of the excluded middle</a>, which states that each sentence is either true or its negation is true. Kleene's work with the proof theory of intuitionistic logic showed that constructive information can be recovered from intuitionistic proofs. For example, any provably total function in intuitionistic arithmetic is <a href="/wiki/Computable" class="mw-redirect" title="Computable">computable</a>; this is not true in classical theories of arithmetic such as <a href="/wiki/Peano_arithmetic" class="mw-redirect" title="Peano arithmetic">Peano arithmetic</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Algebraic_logic">Algebraic logic</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_logic&action=edit&section=14" title="Edit section: Algebraic logic"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Algebraic_logic" title="Algebraic logic">Algebraic logic</a> uses the methods of <a href="/wiki/Abstract_algebra" title="Abstract algebra">abstract algebra</a> to study the semantics of formal logics. A fundamental example is the use of <a href="/wiki/Boolean_algebra_(structure)" title="Boolean algebra (structure)">Boolean algebras</a> to represent <a href="/wiki/Truth_value" title="Truth value">truth values</a> in classical propositional logic, and the use of <a href="/wiki/Heyting_algebra" title="Heyting algebra">Heyting algebras</a> to represent truth values in intuitionistic propositional logic. Stronger logics, such as first-order logic and higher-order logic, are studied using more complicated algebraic structures such as <a href="/wiki/Cylindric_algebra" title="Cylindric algebra">cylindric algebras</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Set_theory">Set theory</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_logic&action=edit&section=15" title="Edit section: Set theory"><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/Set_theory" title="Set theory">Set theory</a></div> <p><b><a href="/wiki/Set_theory" title="Set theory">Set theory</a></b> is the study of <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">sets</a>, which are abstract collections of objects. Many of the basic notions, such as ordinal and cardinal numbers, were developed informally by Cantor before formal axiomatizations of set theory were developed. The <a href="/wiki/Zermelo_set_theory" title="Zermelo set theory">first such axiomatization</a>, due to Zermelo,<sup id="cite_ref-FOOTNOTEZermelo1908b_23-1" class="reference"><a href="#cite_note-FOOTNOTEZermelo1908b-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup> was extended slightly to become <a href="/wiki/Zermelo%E2%80%93Fraenkel_set_theory" title="Zermelo–Fraenkel set theory">Zermelo–Fraenkel set theory</a> (ZF), which is now the most widely used foundational theory for mathematics. </p><p>Other formalizations of set theory have been proposed, including <a href="/wiki/Von_Neumann%E2%80%93Bernays%E2%80%93G%C3%B6del_set_theory" title="Von Neumann–Bernays–Gödel set theory">von Neumann–Bernays–Gödel set theory</a> (NBG), <a href="/wiki/Morse%E2%80%93Kelley_set_theory" title="Morse–Kelley set theory">Morse–Kelley set theory</a> (MK), and <a href="/wiki/New_Foundations" title="New Foundations">New Foundations</a> (NF). Of these, ZF, NBG, and MK are similar in describing a <a href="/wiki/Cumulative_hierarchy" title="Cumulative hierarchy">cumulative hierarchy</a> of sets. New Foundations takes a different approach; it allows objects such as the set of all sets at the cost of restrictions on its set-existence axioms. The system of <a href="/wiki/Kripke%E2%80%93Platek_set_theory" title="Kripke–Platek set theory">Kripke–Platek set theory</a> is closely related to generalized recursion theory. </p><p>Two famous statements in set theory are the <a href="/wiki/Axiom_of_choice" title="Axiom of choice">axiom of choice</a> and the <a href="/wiki/Continuum_hypothesis" title="Continuum hypothesis">continuum hypothesis</a>. The axiom of choice, first stated by Zermelo,<sup id="cite_ref-FOOTNOTEZermelo1904_19-1" class="reference"><a href="#cite_note-FOOTNOTEZermelo1904-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> was proved independent of ZF by Fraenkel,<sup id="cite_ref-FOOTNOTEFraenkel1922_25-1" class="reference"><a href="#cite_note-FOOTNOTEFraenkel1922-25"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup> but has come to be widely accepted by mathematicians. It states that given a collection of nonempty sets there is a single set <i>C</i> that contains exactly one element from each set in the collection. The set <i>C</i> is said to "choose" one element from each set in the collection. While the ability to make such a choice is considered obvious by some, since each set in the collection is nonempty, the lack of a general, concrete rule by which the choice can be made renders the axiom nonconstructive. <a href="/wiki/Stefan_Banach" title="Stefan Banach">Stefan Banach</a> and <a href="/wiki/Alfred_Tarski" title="Alfred Tarski">Alfred Tarski</a> showed that the axiom of choice can be used to decompose a solid ball into a finite number of pieces which can then be rearranged, with no scaling, to make two solid balls of the original size.<sup id="cite_ref-FOOTNOTEBanachTarski1924_43-0" class="reference"><a href="#cite_note-FOOTNOTEBanachTarski1924-43"><span class="cite-bracket">[</span>40<span class="cite-bracket">]</span></a></sup> This theorem, known as the <a href="/wiki/Banach%E2%80%93Tarski_paradox" title="Banach–Tarski paradox">Banach–Tarski paradox</a>, is one of many counterintuitive results of the axiom of choice. </p><p>The continuum hypothesis, first proposed as a conjecture by Cantor, was listed by <a href="/wiki/David_Hilbert" title="David Hilbert">David Hilbert</a> as one of his 23 problems in 1900. Gödel showed that the continuum hypothesis cannot be disproven from the axioms of Zermelo–Fraenkel set theory (with or without the axiom of choice), by developing the <a href="/wiki/Constructible_universe" title="Constructible universe">constructible universe</a> of set theory in which the continuum hypothesis must hold. In 1963, <a href="/wiki/Paul_Cohen" title="Paul Cohen">Paul Cohen</a> showed that the continuum hypothesis cannot be proven from the axioms of Zermelo–Fraenkel set theory.<sup id="cite_ref-FOOTNOTECohen1966_26-1" class="reference"><a href="#cite_note-FOOTNOTECohen1966-26"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup> This independence result did not completely settle Hilbert's question, however, as it is possible that new axioms for set theory could resolve the hypothesis. Recent work along these lines has been conducted by <a href="/wiki/W._Hugh_Woodin" title="W. Hugh Woodin">W. Hugh Woodin</a>, although its importance is not yet clear.<sup id="cite_ref-FOOTNOTEWoodin2001_44-0" class="reference"><a href="#cite_note-FOOTNOTEWoodin2001-44"><span class="cite-bracket">[</span>41<span class="cite-bracket">]</span></a></sup> </p><p>Contemporary research in set theory includes the study of <a href="/wiki/Large_cardinal" title="Large cardinal">large cardinals</a> and <a href="/wiki/Determinacy" title="Determinacy">determinacy</a>. Large cardinals are <a href="/wiki/Cardinal_numbers" class="mw-redirect" title="Cardinal numbers">cardinal numbers</a> with particular properties so strong that the existence of such cardinals cannot be proved in ZFC. The existence of the smallest large cardinal typically studied, an <a href="/wiki/Inaccessible_cardinal" title="Inaccessible cardinal">inaccessible cardinal</a>, already implies the consistency of ZFC. Despite the fact that large cardinals have extremely high <a href="/wiki/Cardinality" title="Cardinality">cardinality</a>, their existence has many ramifications for the structure of the real line. <i>Determinacy</i> refers to the possible existence of winning strategies for certain two-player games (the games are said to be <i>determined</i>). The existence of these strategies implies structural properties of the real line and other <a href="/wiki/Polish_space" title="Polish space">Polish spaces</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Model_theory">Model theory</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_logic&action=edit&section=16" title="Edit section: Model theory"><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/Model_theory" title="Model theory">Model theory</a></div> <p><b><a href="/wiki/Model_theory" title="Model theory">Model theory</a></b> studies the models of various formal theories. Here a <a href="/wiki/Theory_(mathematical_logic)" title="Theory (mathematical logic)">theory</a> is a set of formulas in a particular formal logic and <a href="/wiki/Signature_(logic)" title="Signature (logic)">signature</a>, while a <a href="/wiki/Structure_(mathematical_logic)" title="Structure (mathematical logic)">model</a> is a structure that gives a concrete interpretation of the theory. Model theory is closely related to <a href="/wiki/Universal_algebra" title="Universal algebra">universal algebra</a> and <a href="/wiki/Algebraic_geometry" title="Algebraic geometry">algebraic geometry</a>, although the methods of model theory focus more on logical considerations than those fields. </p><p>The set of all models of a particular theory is called an <a href="/wiki/Elementary_class" title="Elementary class">elementary class</a>; classical model theory seeks to determine the properties of models in a particular elementary class, or determine whether certain classes of structures form elementary classes. </p><p>The method of <a href="/wiki/Quantifier_elimination" title="Quantifier elimination">quantifier elimination</a> can be used to show that definable sets in particular theories cannot be too complicated. Tarski established quantifier elimination for <a href="/wiki/Real-closed_field" class="mw-redirect" title="Real-closed field">real-closed fields</a>, a result which also shows the theory of the field of real numbers is <a href="/wiki/Decidable_set" class="mw-redirect" title="Decidable set">decidable</a>.<sup id="cite_ref-FOOTNOTETarski1948_45-0" class="reference"><a href="#cite_note-FOOTNOTETarski1948-45"><span class="cite-bracket">[</span>42<span class="cite-bracket">]</span></a></sup> He also noted that his methods were equally applicable to algebraically closed fields of arbitrary characteristic. A modern subfield developing from this is concerned with <a href="/wiki/O-minimal_theory" title="O-minimal theory">o-minimal structures</a>. </p><p><a href="/wiki/Morley%27s_categoricity_theorem" class="mw-redirect" title="Morley's categoricity theorem">Morley's categoricity theorem</a>, proved by <a href="/wiki/Michael_D._Morley" title="Michael D. Morley">Michael D. Morley</a>,<sup id="cite_ref-FOOTNOTEMorley1965_46-0" class="reference"><a href="#cite_note-FOOTNOTEMorley1965-46"><span class="cite-bracket">[</span>43<span class="cite-bracket">]</span></a></sup> states that if a first-order theory in a countable language is categorical in some uncountable cardinality, i.e. all models of this cardinality are isomorphic, then it is categorical in all uncountable cardinalities. </p><p>A trivial consequence of the <a href="/wiki/Continuum_hypothesis" title="Continuum hypothesis">continuum hypothesis</a> is that a complete theory with less than continuum many nonisomorphic countable models can have only countably many. <a href="/wiki/Vaught_conjecture" title="Vaught conjecture">Vaught's conjecture</a>, named after <a href="/wiki/Robert_Lawson_Vaught" title="Robert Lawson Vaught">Robert Lawson Vaught</a>, says that this is true even independently of the continuum hypothesis. Many special cases of this conjecture have been established. </p> <div class="mw-heading mw-heading2"><h2 id="Recursion_theory">Recursion theory</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_logic&action=edit&section=17" title="Edit section: Recursion theory"><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/Recursion_theory" class="mw-redirect" title="Recursion theory">Recursion theory</a></div> <p><b><a href="/wiki/Recursion_theory" class="mw-redirect" title="Recursion theory">Recursion theory</a></b>, also called <b>computability theory</b>, studies the properties of <a href="/wiki/Computable_function" title="Computable function">computable functions</a> and the <a href="/wiki/Turing_degree" title="Turing degree">Turing degrees</a>, which divide the uncomputable functions into sets that have the same level of uncomputability. Recursion theory also includes the study of generalized computability and definability. Recursion theory grew from the work of <a href="/wiki/R%C3%B3zsa_P%C3%A9ter" title="Rózsa Péter">Rózsa Péter</a>, <a href="/wiki/Alonzo_Church" title="Alonzo Church">Alonzo Church</a> and <a href="/wiki/Alan_Turing" title="Alan Turing">Alan Turing</a> in the 1930s, which was greatly extended by <a href="/wiki/Stephen_Cole_Kleene" title="Stephen Cole Kleene">Kleene</a> and <a href="/wiki/Emil_Leon_Post" title="Emil Leon Post">Post</a> in the 1940s.<sup id="cite_ref-FOOTNOTESoare2011_47-0" class="reference"><a href="#cite_note-FOOTNOTESoare2011-47"><span class="cite-bracket">[</span>44<span class="cite-bracket">]</span></a></sup> </p><p>Classical recursion theory focuses on the computability of functions from the natural numbers to the natural numbers. The fundamental results establish a robust, canonical class of computable functions with numerous independent, equivalent characterizations using <a href="/wiki/Turing_machine" title="Turing machine">Turing machines</a>, <a href="/wiki/Lambda_calculus" title="Lambda calculus">λ calculus</a>, and other systems. More advanced results concern the structure of the Turing degrees and the <a href="/wiki/Lattice_(order)" title="Lattice (order)">lattice</a> of <a href="/wiki/Recursively_enumerable_set" class="mw-redirect" title="Recursively enumerable set">recursively enumerable sets</a>. </p><p>Generalized recursion theory extends the ideas of recursion theory to computations that are no longer necessarily finite. It includes the study of computability in higher types as well as areas such as <a href="/wiki/Hyperarithmetical_theory" title="Hyperarithmetical theory">hyperarithmetical theory</a> and <a href="/wiki/Alpha_recursion_theory" title="Alpha recursion theory">α-recursion theory</a>. </p><p>Contemporary research in recursion theory includes the study of applications such as <a href="/wiki/Algorithmic_randomness" class="mw-redirect" title="Algorithmic randomness">algorithmic randomness</a>, <a href="/wiki/Computable_model_theory" title="Computable model theory">computable model theory</a>, and <a href="/wiki/Reverse_mathematics" title="Reverse mathematics">reverse mathematics</a>, as well as new results in pure recursion theory. </p> <div class="mw-heading mw-heading3"><h3 id="Algorithmically_unsolvable_problems">Algorithmically unsolvable problems</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_logic&action=edit&section=18" title="Edit section: Algorithmically unsolvable problems"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>An important subfield of recursion theory studies algorithmic unsolvability; a <a href="/wiki/Decision_problem" title="Decision problem">decision problem</a> or <a href="/wiki/Function_problem" title="Function problem">function problem</a> is <b>algorithmically unsolvable</b> if there is no possible computable algorithm that returns the correct answer for all legal inputs to the problem. The first results about unsolvability, obtained independently by Church and Turing in 1936, showed that the <a href="/wiki/Entscheidungsproblem" title="Entscheidungsproblem">Entscheidungsproblem</a> is algorithmically unsolvable. Turing proved this by establishing the unsolvability of the <a href="/wiki/Halting_problem" title="Halting problem">halting problem</a>, a result with far-ranging implications in both recursion theory and computer science. </p><p>There are many known examples of undecidable problems from ordinary mathematics. The <a href="/wiki/Word_problem_for_groups" title="Word problem for groups">word problem for groups</a> was proved algorithmically unsolvable by <a href="/wiki/Pyotr_Novikov" title="Pyotr Novikov">Pyotr Novikov</a> in 1955 and independently by W. Boone in 1959. The <a href="/wiki/Busy_beaver" title="Busy beaver">busy beaver</a> problem, developed by <a href="/wiki/Tibor_Rad%C3%B3" title="Tibor Radó">Tibor Radó</a> in 1962, is another well-known example. </p><p><a href="/wiki/Hilbert%27s_tenth_problem" title="Hilbert's tenth problem">Hilbert's tenth problem</a> asked for an algorithm to determine whether a multivariate polynomial equation with integer coefficients has a solution in the integers. Partial progress was made by <a href="/wiki/Julia_Robinson" title="Julia Robinson">Julia Robinson</a>, <a href="/wiki/Martin_Davis_(mathematician)" title="Martin Davis (mathematician)">Martin Davis</a> and <a href="/wiki/Hilary_Putnam" title="Hilary Putnam">Hilary Putnam</a>. The algorithmic unsolvability of the problem was proved by <a href="/wiki/Yuri_Matiyasevich" title="Yuri Matiyasevich">Yuri Matiyasevich</a> in 1970.<sup id="cite_ref-FOOTNOTEDavis1973_48-0" class="reference"><a href="#cite_note-FOOTNOTEDavis1973-48"><span class="cite-bracket">[</span>45<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Proof_theory_and_constructive_mathematics">Proof theory and constructive mathematics</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_logic&action=edit&section=19" title="Edit section: Proof theory and constructive mathematics"><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_theory" title="Proof theory">Proof theory</a></div> <p><b><a href="/wiki/Proof_theory" title="Proof theory">Proof theory</a></b> is the study of formal proofs in various logical deduction systems. These proofs are represented as formal mathematical objects, facilitating their analysis by mathematical techniques. Several deduction systems are commonly considered, including <a href="/wiki/Hilbert-style_deduction_system" class="mw-redirect" title="Hilbert-style deduction system">Hilbert-style deduction systems</a>, systems of <a href="/wiki/Natural_deduction" title="Natural deduction">natural deduction</a>, and the <a href="/wiki/Sequent_calculus" title="Sequent calculus">sequent calculus</a> developed by Gentzen. </p><p>The study of <b>constructive mathematics</b>, in the context of mathematical logic, includes the study of systems in non-classical logic such as intuitionistic logic, as well as the study of <a href="/wiki/Impredicativity" title="Impredicativity">predicative</a> systems. An early proponent of predicativism was <a href="/wiki/Hermann_Weyl" title="Hermann Weyl">Hermann Weyl</a>, who showed it is possible to develop a large part of real analysis using only predicative methods.<sup id="cite_ref-FOOTNOTEWeyl1918_49-0" class="reference"><a href="#cite_note-FOOTNOTEWeyl1918-49"><span class="cite-bracket">[</span>46<span class="cite-bracket">]</span></a></sup> </p><p>Because proofs are entirely finitary, whereas truth in a structure is not, it is common for work in constructive mathematics to emphasize provability. The relationship between provability in classical (or nonconstructive) systems and provability in intuitionistic (or constructive, respectively) systems is of particular interest. Results such as the <a href="/wiki/G%C3%B6del%E2%80%93Gentzen_negative_translation" class="mw-redirect" title="Gödel–Gentzen negative translation">Gödel–Gentzen negative translation</a> show that it is possible to embed (or <i><a href="/wiki/Logic_translation" title="Logic translation">translate</a></i>) classical logic into intuitionistic logic, allowing some properties about intuitionistic proofs to be transferred back to classical proofs. </p><p>Recent developments in proof theory include the study of <a href="/wiki/Proof_mining" title="Proof mining">proof mining</a> by <a href="/wiki/Ulrich_Kohlenbach" title="Ulrich Kohlenbach">Ulrich Kohlenbach</a> and the study of <a href="/wiki/Proof-theoretic_ordinal" class="mw-redirect" title="Proof-theoretic ordinal">proof-theoretic ordinals</a> by <a href="/w/index.php?title=Michael_Rathjen&action=edit&redlink=1" class="new" title="Michael Rathjen (page does not exist)">Michael Rathjen</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_logic&action=edit&section=20" title="Edit section: Applications"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>"Mathematical logic has been successfully applied not only to mathematics and its foundations (<a href="/wiki/Gottlob_Frege" title="Gottlob Frege">G. Frege</a>, <a href="/wiki/Bertrand_Russell" title="Bertrand Russell">B. Russell</a>, <a href="/wiki/David_Hilbert" title="David Hilbert">D. Hilbert</a>, <a href="/wiki/Paul_Bernays" title="Paul Bernays">P. Bernays</a>, <a href="/wiki/Heinrich_Scholz" title="Heinrich Scholz">H. Scholz</a>, <a href="/wiki/Rudolf_Carnap" title="Rudolf Carnap">R. Carnap</a>, <a href="/wiki/Stanislaw_Lesniewski" class="mw-redirect" title="Stanislaw Lesniewski">S. Lesniewski</a>, <a href="/wiki/Thoralf_Skolem" title="Thoralf Skolem">T. Skolem</a>), but also to physics (R. Carnap, A. Dittrich, B. Russell, <a href="/wiki/Claude_Shannon" title="Claude Shannon">C. E. Shannon</a>, <a href="/wiki/Alfred_North_Whitehead" title="Alfred North Whitehead">A. N. Whitehead</a>, <a href="/wiki/Hans_Reichenbach" title="Hans Reichenbach">H. Reichenbach</a>, P. Fevrier), to biology (<a href="/wiki/Joseph_Henry_Woodger" title="Joseph Henry Woodger">J. H. Woodger</a>, <a href="/wiki/Alfred_Tarski" title="Alfred Tarski">A. Tarski</a>), to psychology (<a href="/wiki/Frederic_Fitch" title="Frederic Fitch">F. B. Fitch</a>, <a href="/wiki/Carl_Gustav_Hempel" title="Carl Gustav Hempel">C. G. Hempel</a>), to law and morals (<a href="/wiki/Karl_Menger" title="Karl Menger">K. Menger</a>, U. Klug, P. Oppenheim), to economics (<a href="/wiki/John_von_Neumann" title="John von Neumann">J. Neumann</a>, <a href="/wiki/Oskar_Morgenstern" title="Oskar Morgenstern">O. Morgenstern</a>), to practical questions (<a href="/wiki/Edmund_Berkeley" title="Edmund Berkeley">E. C. Berkeley</a>, E. Stamm), and even to metaphysics (J. [Jan] Salamucha, H. Scholz, <a href="/wiki/Jozef_Maria_Bochenski" class="mw-redirect" title="Jozef Maria Bochenski">J. M. Bochenski</a>). Its applications to the history of logic have proven extremely fruitful (<a href="/wiki/Jan_Lukasiewicz" class="mw-redirect" title="Jan Lukasiewicz">J. Lukasiewicz</a>, H. Scholz, <a href="/wiki/Benson_Mates" title="Benson Mates">B. Mates</a>, A. Becker, <a href="/wiki/Ernest_Addison_Moody" title="Ernest Addison Moody">E. Moody</a>, J. Salamucha, K. Duerr, Z. Jordan, <a href="/wiki/Philotheus_Boehner" title="Philotheus Boehner">P. Boehner</a>, J. M. Bochenski, S. [Stanislaw] T. Schayer, <a href="/wiki/Daniel_H._H._Ingalls_Sr." title="Daniel H. H. Ingalls Sr.">D. Ingalls</a>)."<sup id="cite_ref-FOOTNOTEBochenski1959Sec._0.3,_p._2_50-0" class="reference"><a href="#cite_note-FOOTNOTEBochenski1959Sec._0.3,_p._2-50"><span class="cite-bracket">[</span>47<span class="cite-bracket">]</span></a></sup> "Applications have also been made to theology (F. Drewnowski, J. Salamucha, I. Thomas)."<sup id="cite_ref-FOOTNOTEBochenski1959Sec._0.3,_p._2_50-1" class="reference"><a href="#cite_note-FOOTNOTEBochenski1959Sec._0.3,_p._2-50"><span class="cite-bracket">[</span>47<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Connections_with_computer_science">Connections with computer science</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_logic&action=edit&section=21" title="Edit section: Connections with computer science"><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/Logic_in_computer_science" title="Logic in computer science">Logic in computer science</a></div> <p>The study of <a href="/wiki/Computability_theory_(computer_science)" class="mw-redirect" title="Computability theory (computer science)">computability theory in computer science</a> is closely related to the study of computability in mathematical logic. There is a difference of emphasis, however. <a href="/wiki/Computer_science" title="Computer science">Computer scientists</a> often focus on concrete programming languages and <a href="/wiki/Feasible_computability" class="mw-redirect" title="Feasible computability">feasible computability</a>, while researchers in mathematical logic often focus on computability as a theoretical concept and on noncomputability. </p><p>The theory of <a href="/wiki/Program_semantics" class="mw-redirect" title="Program semantics">semantics of programming languages</a> is related to <a href="/wiki/Model_theory" title="Model theory">model theory</a>, as is <a href="/wiki/Program_verification" class="mw-redirect" title="Program verification">program verification</a> (in particular, <a href="/wiki/Model_checking" title="Model checking">model checking</a>). The <a href="/wiki/Curry%E2%80%93Howard_correspondence" title="Curry–Howard correspondence">Curry–Howard correspondence</a> between proofs and programs relates to <a href="/wiki/Proof_theory" title="Proof theory">proof theory</a>, especially <a href="/wiki/Intuitionistic_logic" title="Intuitionistic logic">intuitionistic logic</a>. Formal calculi such as the <a href="/wiki/Lambda_calculus" title="Lambda calculus">lambda calculus</a> and <a href="/wiki/Combinatory_logic" title="Combinatory logic">combinatory logic</a> are now studied as idealized <a href="/wiki/Programming_languages" class="mw-redirect" title="Programming languages">programming languages</a>. </p><p>Computer science also contributes to mathematics by developing techniques for the automatic checking or even finding of proofs, such as <a href="/wiki/Automated_theorem_proving" title="Automated theorem proving">automated theorem proving</a> and <a href="/wiki/Logic_programming" title="Logic programming">logic programming</a>. </p><p><a href="/wiki/Descriptive_complexity_theory" title="Descriptive complexity theory">Descriptive complexity theory</a> relates logics to <a href="/wiki/Computational_complexity_theory" title="Computational complexity theory">computational complexity</a>. The first significant result in this area, <a href="/wiki/Fagin%27s_theorem" title="Fagin's theorem">Fagin's theorem</a> (1974) established that <a href="/wiki/NP_(complexity)" title="NP (complexity)">NP</a> is precisely the set of languages expressible by sentences of existential <a href="/wiki/Second-order_logic" title="Second-order logic">second-order logic</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Foundations_of_mathematics">Foundations of mathematics</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_logic&action=edit&section=22" title="Edit section: Foundations of mathematics"><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/Foundations_of_mathematics" title="Foundations of mathematics">Foundations of mathematics</a></div> <p>In the 19th century, mathematicians became aware of logical gaps and inconsistencies in their field. It was shown that <a href="/wiki/Euclid" title="Euclid">Euclid</a>'s axioms for geometry, which had been taught for centuries as an example of the axiomatic method, were incomplete. The use of <a href="/wiki/Infinitesimal" title="Infinitesimal">infinitesimals</a>, and the very definition of <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">function</a>, came into question in analysis, as pathological examples such as Weierstrass' nowhere-<a href="/wiki/Differentiable_function" title="Differentiable function">differentiable</a> continuous function were discovered. </p><p>Cantor's study of arbitrary infinite sets also drew criticism. <a href="/wiki/Leopold_Kronecker" title="Leopold Kronecker">Leopold Kronecker</a> famously stated "God made the integers; all else is the work of man," endorsing a return to the study of finite, concrete objects in mathematics. Although Kronecker's argument was carried forward by constructivists in the 20th century, the mathematical community as a whole rejected them. <a href="/wiki/David_Hilbert" title="David Hilbert">David Hilbert</a> argued in favor of the study of the infinite, saying "No one shall expel us from the Paradise that Cantor has created." </p><p>Mathematicians began to search for axiom systems that could be used to formalize large parts of mathematics. In addition to removing ambiguity from previously naive terms such as function, it was hoped that this axiomatization would allow for consistency proofs. In the 19th century, the main method of proving the consistency of a set of axioms was to provide a model for it. Thus, for example, <a href="/wiki/Non-Euclidean_geometry" title="Non-Euclidean geometry">non-Euclidean geometry</a> can be proved consistent by defining <i>point</i> to mean a point on a fixed sphere and <i>line</i> to mean a <a href="/wiki/Great_circle" title="Great circle">great circle</a> on the sphere. The resulting structure, a model of <a href="/wiki/Elliptic_geometry" title="Elliptic geometry">elliptic geometry</a>, satisfies the axioms of plane geometry except the parallel postulate. </p><p>With the development of formal logic, Hilbert asked whether it would be possible to prove that an axiom system is consistent by analyzing the structure of possible proofs in the system, and showing through this analysis that it is impossible to prove a contradiction. This idea led to the study of <a href="/wiki/Proof_theory" title="Proof theory">proof theory</a>. Moreover, Hilbert proposed that the analysis should be entirely concrete, using the term <i>finitary</i> to refer to the methods he would allow but not precisely defining them. This project, known as <a href="/wiki/Hilbert%27s_program" title="Hilbert's program">Hilbert's program</a>, was seriously affected by Gödel's incompleteness theorems, which show that the consistency of formal theories of arithmetic cannot be established using methods formalizable in those theories. Gentzen showed that it is possible to produce a proof of the consistency of arithmetic in a finitary system augmented with axioms of <a href="/wiki/Transfinite_induction" title="Transfinite induction">transfinite induction</a>, and the techniques he developed to do so were seminal in proof theory. </p><p>A second thread in the history of foundations of mathematics involves <a href="/wiki/Nonclassical_logic" class="mw-redirect" title="Nonclassical logic">nonclassical logics</a> and <a href="/wiki/Constructive_mathematics" class="mw-redirect" title="Constructive mathematics">constructive mathematics</a>. The study of constructive mathematics includes many different programs with various definitions of <i>constructive</i>. At the most accommodating end, proofs in ZF set theory that do not use the axiom of choice are called constructive by many mathematicians. More limited versions of constructivism limit themselves to <a href="/wiki/Natural_numbers" class="mw-redirect" title="Natural numbers">natural numbers</a>, <a href="/wiki/Number-theoretic_function" class="mw-redirect" title="Number-theoretic function">number-theoretic functions</a>, and sets of natural numbers (which can be used to represent real numbers, facilitating the study of <a href="/wiki/Mathematical_analysis" title="Mathematical analysis">mathematical analysis</a>). A common idea is that a concrete means of computing the values of the function must be known before the function itself can be said to exist. </p><p>In the early 20th century, <a href="/wiki/Luitzen_Egbertus_Jan_Brouwer" class="mw-redirect" title="Luitzen Egbertus Jan Brouwer">Luitzen Egbertus Jan Brouwer</a> founded <a href="/wiki/Intuitionism" title="Intuitionism">intuitionism</a> as a part of <a href="/wiki/Philosophy_of_mathematics" title="Philosophy of mathematics">philosophy of mathematics</a>. This philosophy, poorly understood at first, stated that in order for a mathematical statement to be true to a mathematician, that person must be able to <i>intuit</i> the statement, to not only believe its truth but understand the reason for its truth. A consequence of this definition of truth was the rejection of the <a href="/wiki/Law_of_the_excluded_middle" class="mw-redirect" title="Law of the excluded middle">law of the excluded middle</a>, for there are statements that, according to Brouwer, could not be claimed to be true while their negations also could not be claimed true. Brouwer's philosophy was influential, and the cause of bitter disputes among prominent mathematicians. Kleene and Kreisel would later study formalized versions of intuitionistic logic (Brouwer rejected formalization, and presented his work in unformalized natural language). With the advent of the <a href="/wiki/BHK_interpretation" class="mw-redirect" title="BHK interpretation">BHK interpretation</a> and <a href="/wiki/Kripke_model" class="mw-redirect" title="Kripke model">Kripke models</a>, intuitionism became easier to reconcile with classical mathematics. </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=Mathematical_logic&action=edit&section=23" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239009302">.mw-parser-output .portalbox{padding:0;margin:0.5em 0;display:table;box-sizing:border-box;max-width:175px;list-style:none}.mw-parser-output .portalborder{border:1px solid var(--border-color-base,#a2a9b1);padding:0.1em;background:var(--background-color-neutral-subtle,#f8f9fa)}.mw-parser-output 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symbols">List of logic symbols</a></li> <li><a href="/wiki/List_of_mathematical_logic_topics" title="List of mathematical logic topics">List of mathematical logic topics</a></li> <li><a href="/wiki/List_of_set_theory_topics" title="List of set theory topics">List of set theory topics</a></li> <li><a href="/wiki/Mereology" title="Mereology">Mereology</a></li> <li><a href="/wiki/Propositional_calculus" title="Propositional calculus">Propositional calculus</a></li> <li><a href="/wiki/Well-formed_formula" title="Well-formed formula">Well-formed formula</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_logic&action=edit&section=24" title="Edit section: Notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist reflist-lower-alpha"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-HilbertBernays1934_PlusNote-31"><span class="mw-cite-backlink"><b><a href="#cite_ref-HilbertBernays1934_PlusNote_31-0">^</a></b></span> <span class="reference-text">In the foreword to the 1934 first edition of "<a href="/wiki/Grundlagen_der_Mathematik" title="Grundlagen der Mathematik">Grundlagen der Mathematik</a>" (<a href="#CITEREFHilbertBernays1934">Hilbert & Bernays 1934</a>), Bernays wrote the following, which is reminiscent of the famous note by <a href="/wiki/Gottlob_Frege#Work_as_a_logician" title="Gottlob Frege">Frege</a> when informed of Russell's paradox.<blockquote style="margin:0; padding:0;"><p>"Die Ausführung dieses Vorhabens hat eine wesentliche Verzögerung dadurch erfahren, daß in einem Stadium, in dem die Darstellung schon ihrem Abschuß nahe war, durch das Erscheinen der Arbeiten von Herbrand und von Gödel eine veränderte Situation im Gebiet der Beweistheorie entstand, welche die Berücksichtigung neuer Einsichten zur Aufgabe machte. Dabei ist der Umfang des Buches angewachsen, so daß eine Teilung in zwei Bände angezeigt erschien."</p></blockquote> Translation: <blockquote style="margin:0; padding:0;"><p>"Carrying out this plan [by Hilbert for an exposition on proof theory for mathematical logic] has experienced an essential delay because, at the stage at which the exposition was already near to its conclusion, there occurred an altered situation in the area of proof theory due to the appearance of works by Herbrand and Gödel, which necessitated the consideration of new insights. Thus the scope of this book has grown, so that a division into two volumes seemed advisable."</p></blockquote> So certainly Hilbert was aware of the importance of Gödel's work by 1934. The second volume in 1939 included a form of Gentzen's consistency proof for arithmetic.</span> </li> <li id="cite_note-36"><span class="mw-cite-backlink"><b><a href="#cite_ref-36">^</a></b></span> <span class="reference-text">A detailed study of this terminology is given by <a href="#CITEREFSoare1996">Soare 1996</a>.</span> </li> <li id="cite_note-FerreirósSurveys-39"><span class="mw-cite-backlink"><b><a href="#cite_ref-FerreirósSurveys_39-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFFerreirós2001">Ferreirós 2001</a> surveys the rise of first-order logic over other formal logics in the early 20th century.</span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_logic&action=edit&section=25" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626"><div class="reflist reflist-columns references-column-width" style="column-width: 15em;"> <ol class="references"> <li id="cite_note-FOOTNOTEBarwise1989-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEBarwise1989_1-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFBarwise1989">Barwise (1989)</a>.</span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.jaist.ac.jp/CTFM/CTFM2014/submissions/CTFM2014_booklet.pdf">"Computability Theory and Foundations of Mathematics / February, 17th – 20th, 2014 / Tokyo Institute of Technology, Tokyo, Japan"</a> <span class="cs1-format">(PDF)</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Computability+Theory+and+Foundations+of+Mathematics+%2F+February%2C+17th+%E2%80%93+20th%2C+2014+%2F+Tokyo+Institute+of+Technology%2C+Tokyo%2C+Japan&rft_id=http%3A%2F%2Fwww.jaist.ac.jp%2FCTFM%2FCTFM2014%2Fsubmissions%2FCTFM2014_booklet.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+logic" class="Z3988"></span></span> </li> <li id="cite_note-FOOTNOTEFerreirós2001443-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEFerreirós2001443_3-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFFerreirós2001">Ferreirós (2001)</a>, p. 443.</span> </li> <li id="cite_note-FOOTNOTEBochenski1959Sec._0.1,_p._1-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEBochenski1959Sec._0.1,_p._1_4-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFBochenski1959">Bochenski (1959)</a>, Sec. 0.1, p. 1.</span> </li> <li id="cite_note-FOOTNOTESwineshead1498-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTESwineshead1498_5-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFSwineshead1498">Swineshead (1498)</a>.</span> </li> <li id="cite_note-FOOTNOTEBoehner1950xiv-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEBoehner1950xiv_6-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFBoehner1950">Boehner (1950)</a>, p. xiv.</span> </li> <li id="cite_note-FOOTNOTEKatz1998686-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEKatz1998686_7-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFKatz1998">Katz (1998)</a>, p. 686.</span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.enciklopedija.hr/natuknica.aspx?id=7222">"Bertić, Vatroslav | Hrvatska enciklopedija"</a>. <i>www.enciklopedija.hr</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2023-05-01</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=www.enciklopedija.hr&rft.atitle=Berti%C4%87%2C+Vatroslav+%7C+Hrvatska+enciklopedija&rft_id=https%3A%2F%2Fwww.enciklopedija.hr%2Fnatuknica.aspx%3Fid%3D7222&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+logic" class="Z3988"></span></span> </li> <li id="cite_note-FOOTNOTEPeano1889-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEPeano1889_9-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFPeano1889">Peano (1889)</a>.</span> </li> <li id="cite_note-FOOTNOTEDedekind1888-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEDedekind1888_10-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFDedekind1888">Dedekind (1888)</a>.</span> </li> <li id="cite_note-FOOTNOTEKatz1998774-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEKatz1998774_11-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFKatz1998">Katz (1998)</a>, p. 774.</span> </li> <li id="cite_note-FOOTNOTELobachevsky1840-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTELobachevsky1840_12-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFLobachevsky1840">Lobachevsky (1840)</a>.</span> </li> <li id="cite_note-FOOTNOTEHilbert1899-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEHilbert1899_13-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFHilbert1899">Hilbert (1899)</a>.</span> </li> <li id="cite_note-FOOTNOTEPasch1882-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEPasch1882_14-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFPasch1882">Pasch (1882)</a>.</span> </li> <li id="cite_note-FOOTNOTEFelscher2000-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEFelscher2000_15-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFFelscher2000">Felscher (2000)</a>.</span> </li> <li id="cite_note-FOOTNOTEDedekind1872-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEDedekind1872_16-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFDedekind1872">Dedekind (1872)</a>.</span> </li> <li id="cite_note-FOOTNOTECantor1874-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTECantor1874_17-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFCantor1874">Cantor (1874)</a>.</span> </li> <li id="cite_note-FOOTNOTEKatz1998807-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEKatz1998807_18-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFKatz1998">Katz (1998)</a>, p. 807.</span> </li> <li id="cite_note-FOOTNOTEZermelo1904-19"><span class="mw-cite-backlink">^ <a href="#cite_ref-FOOTNOTEZermelo1904_19-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-FOOTNOTEZermelo1904_19-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><a href="#CITEREFZermelo1904">Zermelo (1904)</a>.</span> </li> <li id="cite_note-FOOTNOTEZermelo1908a-20"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEZermelo1908a_20-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFZermelo1908a">Zermelo (1908a)</a>.</span> </li> <li id="cite_note-FOOTNOTEBurali-Forti1897-21"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEBurali-Forti1897_21-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFBurali-Forti1897">Burali-Forti (1897)</a>.</span> </li> <li id="cite_note-FOOTNOTERichard1905-22"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTERichard1905_22-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFRichard1905">Richard (1905)</a>.</span> </li> <li id="cite_note-FOOTNOTEZermelo1908b-23"><span class="mw-cite-backlink">^ <a href="#cite_ref-FOOTNOTEZermelo1908b_23-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-FOOTNOTEZermelo1908b_23-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><a href="#CITEREFZermelo1908b">Zermelo (1908b)</a>.</span> </li> <li id="cite_note-FOOTNOTEFerreirós2001445-24"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEFerreirós2001445_24-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFFerreirós2001">Ferreirós (2001)</a>, p. 445.</span> </li> <li id="cite_note-FOOTNOTEFraenkel1922-25"><span class="mw-cite-backlink">^ <a href="#cite_ref-FOOTNOTEFraenkel1922_25-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-FOOTNOTEFraenkel1922_25-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><a href="#CITEREFFraenkel1922">Fraenkel (1922)</a>.</span> </li> <li id="cite_note-FOOTNOTECohen1966-26"><span class="mw-cite-backlink">^ <a href="#cite_ref-FOOTNOTECohen1966_26-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-FOOTNOTECohen1966_26-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><a href="#CITEREFCohen1966">Cohen (1966)</a>.</span> </li> <li id="cite_note-27"><span class="mw-cite-backlink"><b><a href="#cite_ref-27">^</a></b></span> <span class="reference-text">See also <a href="#CITEREFCohen2008">Cohen 2008</a>.</span> </li> <li id="cite_note-FOOTNOTELöwenheim1915-28"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTELöwenheim1915_28-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFLöwenheim1915">Löwenheim (1915)</a>.</span> </li> <li id="cite_note-FOOTNOTESkolem1920-29"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTESkolem1920_29-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFSkolem1920">Skolem (1920)</a>.</span> </li> <li id="cite_note-FOOTNOTEGödel1929-30"><span class="mw-cite-backlink">^ <a href="#cite_ref-FOOTNOTEGödel1929_30-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-FOOTNOTEGödel1929_30-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><a href="#CITEREFGödel1929">Gödel (1929)</a>.</span> </li> <li id="cite_note-FOOTNOTEGentzen1936-32"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEGentzen1936_32-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFGentzen1936">Gentzen (1936)</a>.</span> </li> <li id="cite_note-FOOTNOTEGödel1958-33"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEGödel1958_33-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFGödel1958">Gödel (1958)</a>.</span> </li> <li id="cite_note-34"><span class="mw-cite-backlink"><b><a href="#cite_ref-34">^</a></b></span> <span class="reference-text">Lewis Carroll: SYMBOLIC LOGIC Part I Elementary. pub. Macmillan 1896. Available online at: <a rel="nofollow" class="external free" href="https://archive.org/details/symboliclogic00carr">https://archive.org/details/symboliclogic00carr</a></span> </li> <li id="cite_note-FOOTNOTECarroll1896-35"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTECarroll1896_35-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFCarroll1896">Carroll (1896)</a>.</span> </li> <li id="cite_note-FOOTNOTEKleene1943-37"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEKleene1943_37-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFKleene1943">Kleene (1943)</a>.</span> </li> <li id="cite_note-FOOTNOTETuring1939-38"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTETuring1939_38-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFTuring1939">Turing (1939)</a>.</span> </li> <li id="cite_note-FOOTNOTEGödel1931-40"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEGödel1931_40-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFGödel1931">Gödel (1931)</a>.</span> </li> <li id="cite_note-FOOTNOTESolovay1976-41"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTESolovay1976_41-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFSolovay1976">Solovay (1976)</a>.</span> </li> <li id="cite_note-FOOTNOTEHamkinsLöwe2007-42"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEHamkinsLöwe2007_42-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFHamkinsLöwe2007">Hamkins & Löwe (2007)</a>.</span> </li> <li id="cite_note-FOOTNOTEBanachTarski1924-43"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEBanachTarski1924_43-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFBanachTarski1924">Banach & Tarski (1924)</a>.</span> </li> <li id="cite_note-FOOTNOTEWoodin2001-44"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEWoodin2001_44-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFWoodin2001">Woodin (2001)</a>.</span> </li> <li id="cite_note-FOOTNOTETarski1948-45"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTETarski1948_45-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFTarski1948">Tarski (1948)</a>.</span> </li> <li id="cite_note-FOOTNOTEMorley1965-46"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEMorley1965_46-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFMorley1965">Morley (1965)</a>.</span> </li> <li id="cite_note-FOOTNOTESoare2011-47"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTESoare2011_47-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFSoare2011">Soare (2011)</a>.</span> </li> <li id="cite_note-FOOTNOTEDavis1973-48"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEDavis1973_48-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFDavis1973">Davis (1973)</a>.</span> </li> <li id="cite_note-FOOTNOTEWeyl1918-49"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEWeyl1918_49-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFWeyl1918">Weyl 1918</a>.</span> </li> <li id="cite_note-FOOTNOTEBochenski1959Sec._0.3,_p._2-50"><span class="mw-cite-backlink">^ <a href="#cite_ref-FOOTNOTEBochenski1959Sec._0.3,_p._2_50-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-FOOTNOTEBochenski1959Sec._0.3,_p._2_50-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><a href="#CITEREFBochenski1959">Bochenski (1959)</a>, Sec. 0.3, p. 2.</span> </li> </ol></div> <div class="mw-heading mw-heading3"><h3 id="Undergraduate_texts">Undergraduate texts</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_logic&action=edit&section=26" title="Edit section: Undergraduate texts"><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="CITEREFWalicki2011" class="citation book cs1">Walicki, Michał (2011). <i>Introduction to Mathematical Logic</i>. <a href="/wiki/Singapore" title="Singapore">Singapore</a>: <a href="/wiki/World_Scientific_Publishing" class="mw-redirect" title="World Scientific Publishing">World Scientific Publishing</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9789814343879" title="Special:BookSources/9789814343879"><bdi>9789814343879</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+Mathematical+Logic&rft.place=Singapore&rft.pub=World+Scientific+Publishing&rft.date=2011&rft.isbn=9789814343879&rft.aulast=Walicki&rft.aufirst=Micha%C5%82&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+logic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBoolosBurgessJeffrey2002" class="citation book cs1"><a href="/wiki/George_Boolos" title="George Boolos">Boolos, George</a>; Burgess, John; <a href="/wiki/Richard_Jeffrey" title="Richard Jeffrey">Jeffrey, Richard</a> (2002). <i>Computability and Logic</i> (4th ed.). <a href="/wiki/Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780521007580" title="Special:BookSources/9780521007580"><bdi>9780521007580</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Computability+and+Logic&rft.edition=4th&rft.pub=Cambridge+University+Press&rft.date=2002&rft.isbn=9780521007580&rft.aulast=Boolos&rft.aufirst=George&rft.au=Burgess%2C+John&rft.au=Jeffrey%2C+Richard&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+logic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCrossleyAshBrickhillStillwell1972" class="citation book cs1">Crossley, J.N.; Ash, C.J.; Brickhill, C.J.; Stillwell, J.C.; Williams, N.H. (1972). <i>What is mathematical logic?</i>. London, Oxford, New York City: <a href="/wiki/Oxford_University_Press" title="Oxford University Press">Oxford University Press</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780198880875" title="Special:BookSources/9780198880875"><bdi>9780198880875</bdi></a>. <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:0251.02001">0251.02001</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=What+is+mathematical+logic%3F&rft.place=London%2C+Oxford%2C+New+York+City&rft.pub=Oxford+University+Press&rft.date=1972&rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A0251.02001%23id-name%3DZbl&rft.isbn=9780198880875&rft.aulast=Crossley&rft.aufirst=J.N.&rft.au=Ash%2C+C.J.&rft.au=Brickhill%2C+C.J.&rft.au=Stillwell%2C+J.C.&rft.au=Williams%2C+N.H.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+logic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEnderton2001" class="citation book cs1">Enderton, Herbert (2001). <i>A mathematical introduction to logic</i> (2nd ed.). <a href="/wiki/Boston" title="Boston">Boston</a> MA: <a href="/wiki/Academic_Press" title="Academic Press">Academic Press</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-12-238452-3" title="Special:BookSources/978-0-12-238452-3"><bdi>978-0-12-238452-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+mathematical+introduction+to+logic&rft.place=Boston+MA&rft.edition=2nd&rft.pub=Academic+Press&rft.date=2001&rft.isbn=978-0-12-238452-3&rft.aulast=Enderton&rft.aufirst=Herbert&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+logic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFisher1982" class="citation book cs1">Fisher, Alec (1982). <i>Formal Number Theory and Computability: A Workbook</i>. (suitable as a first course for independent study) (1st ed.). Oxford University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-853188-3" title="Special:BookSources/978-0-19-853188-3"><bdi>978-0-19-853188-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Formal+Number+Theory+and+Computability%3A+A+Workbook&rft.edition=1st&rft.pub=Oxford+University+Press&rft.date=1982&rft.isbn=978-0-19-853188-3&rft.aulast=Fisher&rft.aufirst=Alec&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+logic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHamilton1988" class="citation book cs1">Hamilton, A.G. (1988). <i>Logic for Mathematicians</i> (2nd ed.). Cambridge University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-36865-0" title="Special:BookSources/978-0-521-36865-0"><bdi>978-0-521-36865-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Logic+for+Mathematicians&rft.edition=2nd&rft.pub=Cambridge+University+Press&rft.date=1988&rft.isbn=978-0-521-36865-0&rft.aulast=Hamilton&rft.aufirst=A.G.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+logic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEbbinghausFlumThomas1994" class="citation book cs1">Ebbinghaus, H.-D.; Flum, J.; Thomas, W. (1994). <a rel="nofollow" class="external text" href="https://www.springer.com/mathematics/book/978-0-387-94258-2"><i>Mathematical Logic</i></a> (2nd ed.). <a href="/wiki/New_York_City" title="New York City">New York City</a>: <a href="/wiki/Springer_Science%2BBusiness_Media" title="Springer Science+Business Media">Springer</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780387942582" title="Special:BookSources/9780387942582"><bdi>9780387942582</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=New+York+City&rft.edition=2nd&rft.pub=Springer&rft.date=1994&rft.isbn=9780387942582&rft.aulast=Ebbinghaus&rft.aufirst=H.-D.&rft.au=Flum%2C+J.&rft.au=Thomas%2C+W.&rft_id=https%3A%2F%2Fwww.springer.com%2Fmathematics%2Fbook%2F978-0-387-94258-2&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+logic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKatz1964" class="citation book cs1">Katz, Robert (1964). <i>Axiomatic Analysis</i>. <a href="/wiki/Boston" title="Boston">Boston</a> MA: <a href="/wiki/D._C._Heath_and_Company" title="D. C. Heath and Company">D. C. Heath and Company</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Axiomatic+Analysis&rft.place=Boston+MA&rft.pub=D.+C.+Heath+and+Company&rft.date=1964&rft.aulast=Katz&rft.aufirst=Robert&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+logic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMendelson1997" class="citation book cs1"><a href="/wiki/Elliott_Mendelson" title="Elliott Mendelson">Mendelson, Elliott</a> (1997). <i>Introduction to Mathematical Logic</i> (4th ed.). London: <a href="/wiki/Chapman_%26_Hall" title="Chapman & Hall">Chapman & Hall</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-412-80830-2" title="Special:BookSources/978-0-412-80830-2"><bdi>978-0-412-80830-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Mathematical+Logic&rft.place=London&rft.edition=4th&rft.pub=Chapman+%26+Hall&rft.date=1997&rft.isbn=978-0-412-80830-2&rft.aulast=Mendelson&rft.aufirst=Elliott&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+logic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRautenberg2010" class="citation book cs1"><a href="/wiki/Wolfgang_Rautenberg" title="Wolfgang Rautenberg">Rautenberg, Wolfgang</a> (2010). <i>A Concise Introduction to Mathematical Logic</i> (3rd ed.). <a href="/wiki/New_York_City" title="New York City">New York City</a>: <a href="/wiki/Springer_Science%2BBusiness_Media" title="Springer Science+Business Media">Springer</a>. <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%2F978-1-4419-1221-3">10.1007/978-1-4419-1221-3</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9781441912206" title="Special:BookSources/9781441912206"><bdi>9781441912206</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+Concise+Introduction+to+Mathematical+Logic&rft.place=New+York+City&rft.edition=3rd&rft.pub=Springer&rft.date=2010&rft_id=info%3Adoi%2F10.1007%2F978-1-4419-1221-3&rft.isbn=9781441912206&rft.aulast=Rautenberg&rft.aufirst=Wolfgang&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+logic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSchwichtenberg2003–2004" class="citation book cs1"><a href="/wiki/Helmut_Schwichtenberg" title="Helmut Schwichtenberg">Schwichtenberg, Helmut</a> (2003–2004). <a rel="nofollow" class="external text" href="http://www.mathematik.uni-muenchen.de/~schwicht/lectures/logic/ws03/ml.pdf"><i>Mathematical Logic</i></a> <span class="cs1-format">(PDF)</span>. <a href="/wiki/Munich" title="Munich">Munich</a>: Mathematisches Institut der Universität München<span class="reference-accessdate">. Retrieved <span class="nowrap">2016-02-24</span></span>.</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=Munich&rft.pub=Mathematisches+Institut+der+Universit%C3%A4t+M%C3%BCnchen&rft.date=2003%2F2004&rft.aulast=Schwichtenberg&rft.aufirst=Helmut&rft_id=http%3A%2F%2Fwww.mathematik.uni-muenchen.de%2F~schwicht%2Flectures%2Flogic%2Fws03%2Fml.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+logic" class="Z3988"></span></li> <li>Shawn Hedman, <i>A first course in logic: an introduction to model theory, proof theory, computability, and complexity</i>, <a href="/wiki/Oxford_University_Press" title="Oxford University Press">Oxford University Press</a>, 2004, <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/0-19-852981-3" title="Special:BookSources/0-19-852981-3">0-19-852981-3</a>. Covers logics in close relation with <a href="/wiki/Computability_theory" title="Computability theory">computability theory</a> and <a href="/wiki/Computational_complexity_theory" title="Computational complexity theory">complexity theory</a></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFvan_Dalen2013" class="citation book cs1">van Dalen, Dirk (2013). <i>Logic and Structure</i>. Universitext. Berlin: <a href="/wiki/Springer_Science%2BBusiness_Media" title="Springer Science+Business Media">Springer</a>. <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%2F978-1-4471-4558-5">10.1007/978-1-4471-4558-5</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4471-4557-8" title="Special:BookSources/978-1-4471-4557-8"><bdi>978-1-4471-4557-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Logic+and+Structure&rft.place=Berlin&rft.series=Universitext&rft.pub=Springer&rft.date=2013&rft_id=info%3Adoi%2F10.1007%2F978-1-4471-4558-5&rft.isbn=978-1-4471-4557-8&rft.aulast=van+Dalen&rft.aufirst=Dirk&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+logic" class="Z3988"></span></li></ul> <div class="mw-heading mw-heading3"><h3 id="Graduate_texts">Graduate texts</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_logic&action=edit&section=27" title="Edit section: Graduate texts"><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="CITEREFHinman2005" class="citation book cs1">Hinman, Peter G. (2005). <i>Fundamentals of mathematical logic</i>. A K Peters, Ltd. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/1-56881-262-0" title="Special:BookSources/1-56881-262-0"><bdi>1-56881-262-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Fundamentals+of+mathematical+logic&rft.pub=A+K+Peters%2C+Ltd.&rft.date=2005&rft.isbn=1-56881-262-0&rft.aulast=Hinman&rft.aufirst=Peter+G.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+logic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAndrews2002" class="citation book cs1">Andrews, Peter B. (2002). <i>An Introduction to Mathematical Logic and Type Theory: To Truth Through Proof</i> (2nd ed.). <a href="/wiki/Boston" title="Boston">Boston</a>: Kluwer Academic Publishers. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4020-0763-7" title="Special:BookSources/978-1-4020-0763-7"><bdi>978-1-4020-0763-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=An+Introduction+to+Mathematical+Logic+and+Type+Theory%3A+To+Truth+Through+Proof&rft.place=Boston&rft.edition=2nd&rft.pub=Kluwer+Academic+Publishers&rft.date=2002&rft.isbn=978-1-4020-0763-7&rft.aulast=Andrews&rft.aufirst=Peter+B.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+logic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBarwise1989" class="citation book cs1"><a href="/wiki/Jon_Barwise" title="Jon Barwise">Barwise, Jon</a>, ed. (1989). <i>Handbook of Mathematical Logic</i>. Studies in Logic and the Foundations of Mathematics. <a href="/wiki/Amsterdam" title="Amsterdam">Amsterdam</a>: <a href="/wiki/Elsevier" title="Elsevier">Elsevier</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780444863881" title="Special:BookSources/9780444863881"><bdi>9780444863881</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+Mathematical+Logic&rft.place=Amsterdam&rft.series=Studies+in+Logic+and+the+Foundations+of+Mathematics&rft.pub=Elsevier&rft.date=1989&rft.isbn=9780444863881&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+logic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHodges1997" class="citation book cs1"><a href="/wiki/Wilfrid_Hodges" title="Wilfrid Hodges">Hodges, Wilfrid</a> (1997). <i>A shorter model theory</i>. <a href="/wiki/Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780521587136" title="Special:BookSources/9780521587136"><bdi>9780521587136</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+shorter+model+theory&rft.pub=Cambridge+University+Press&rft.date=1997&rft.isbn=9780521587136&rft.aulast=Hodges&rft.aufirst=Wilfrid&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+logic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJech2003" class="citation book cs1"><a href="/wiki/Thomas_Jech" title="Thomas Jech">Jech, Thomas</a> (2003). <i>Set Theory: Millennium Edition</i>. Springer Monographs in Mathematics. Berlin, New York: <a href="/wiki/Springer_Science%2BBusiness_Media" title="Springer Science+Business Media">Springer</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9783540440857" title="Special:BookSources/9783540440857"><bdi>9783540440857</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Set+Theory%3A+Millennium+Edition&rft.place=Berlin%2C+New+York&rft.series=Springer+Monographs+in+Mathematics&rft.pub=Springer&rft.date=2003&rft.isbn=9783540440857&rft.aulast=Jech&rft.aufirst=Thomas&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+logic" class="Z3988"></span></li> <li><a href="/wiki/Stephen_Cole_Kleene" title="Stephen Cole Kleene">Kleene, Stephen Cole</a>.(1952), <i><a rel="nofollow" class="external text" href="https://books.google.com/books?id=HZAjPwAACAAJ&source=gbs_ViewAPI">Introduction to Metamathematics.</a></i> New York: Van Nostrand. (Ishi Press: 2009 reprint).</li> <li><a href="/wiki/Stephen_Cole_Kleene" title="Stephen Cole Kleene">Kleene, Stephen Cole</a>. (1967), <i><a rel="nofollow" class="external text" href="http://worldcat.org/oclc/523472">Mathematical Logic.</a></i> John Wiley. Dover reprint, 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="CITEREFShoenfield2001" class="citation book cs1"><a href="/wiki/Joseph_R._Shoenfield" title="Joseph R. Shoenfield">Shoenfield, Joseph R.</a> (2001) [1967]. <i>Mathematical Logic</i> (2nd ed.). <a href="/wiki/A_K_Peters" 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/9781568811352" title="Special:BookSources/9781568811352"><bdi>9781568811352</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.edition=2nd&rft.pub=A+K+Peters&rft.date=2001&rft.isbn=9781568811352&rft.aulast=Shoenfield&rft.aufirst=Joseph+R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+logic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTroelstraSchwichtenberg2000" class="citation book cs1"><a href="/wiki/A._S._Troelstra" class="mw-redirect" title="A. S. Troelstra">Troelstra, Anne Sjerp</a>; Schwichtenberg, Helmut (2000). <i>Basic Proof Theory</i>. Cambridge Tracts in Theoretical Computer Science (2nd ed.). Cambridge University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-77911-1" title="Special:BookSources/978-0-521-77911-1"><bdi>978-0-521-77911-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Basic+Proof+Theory&rft.series=Cambridge+Tracts+in+Theoretical+Computer+Science&rft.edition=2nd&rft.pub=Cambridge+University+Press&rft.date=2000&rft.isbn=978-0-521-77911-1&rft.aulast=Troelstra&rft.aufirst=Anne+Sjerp&rft.au=Schwichtenberg%2C+Helmut&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+logic" class="Z3988"></span></li></ul> <div class="mw-heading mw-heading3"><h3 id="Research_papers,_monographs,_texts,_and_surveys"><span id="Research_papers.2C_monographs.2C_texts.2C_and_surveys"></span>Research papers, monographs, texts, and surveys</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_logic&action=edit&section=28" title="Edit section: Research papers, monographs, texts, and surveys"><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="CITEREFAugusto2017" class="citation book cs1">Augusto, Luis M. (2017). <a rel="nofollow" class="external text" href="http://www.collegepublications.co.uk/logic/mlf/?00029"><i>Logical consequences. Theory and applications: An introduction</i></a>. London: College Publications. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-84890-236-7" title="Special:BookSources/978-1-84890-236-7"><bdi>978-1-84890-236-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Logical+consequences.+Theory+and+applications%3A+An+introduction&rft.place=London&rft.pub=College+Publications&rft.date=2017&rft.isbn=978-1-84890-236-7&rft.aulast=Augusto&rft.aufirst=Luis+M.&rft_id=http%3A%2F%2Fwww.collegepublications.co.uk%2Flogic%2Fmlf%2F%3F00029&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+logic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBoehner1950" class="citation book cs1"><a href="/wiki/Philotheus_Boehner" title="Philotheus Boehner">Boehner, Philotheus</a> (1950). <i>Medieval Logic</i>. Manchester.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Medieval+Logic&rft.place=Manchester&rft.date=1950&rft.aulast=Boehner&rft.aufirst=Philotheus&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+logic" class="Z3988"></span><span class="cs1-maint citation-comment"><code class="cs1-code">{{<a href="/wiki/Template:Cite_book" title="Template:Cite book">cite book</a>}}</code>: CS1 maint: location missing publisher (<a href="/wiki/Category:CS1_maint:_location_missing_publisher" title="Category:CS1 maint: location missing publisher">link</a>)</span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCohen1966" class="citation book cs1"><a href="/wiki/Paul_Cohen" title="Paul Cohen">Cohen, Paul J.</a> (1966). <i>Set Theory and the Continuum Hypothesis</i>. <a href="/wiki/Menlo_Park,_California" title="Menlo Park, California">Menlo Park CA</a>: W. A. Benjamin.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Set+Theory+and+the+Continuum+Hypothesis&rft.place=Menlo+Park+CA&rft.pub=W.+A.+Benjamin&rft.date=1966&rft.aulast=Cohen&rft.aufirst=Paul+J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+logic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCohen2008" class="citation book cs1"><a href="/wiki/Paul_Cohen" title="Paul Cohen">Cohen, Paul J.</a> (2008) [1966]. <i>Set theory and the continuum hypothesis</i>. <a href="/wiki/Mineola,_New_York" title="Mineola, New York">Mineola NY</a>: Dover Publications. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780486469218" title="Special:BookSources/9780486469218"><bdi>9780486469218</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Set+theory+and+the+continuum+hypothesis&rft.place=Mineola+NY&rft.pub=Dover+Publications&rft.date=2008&rft.isbn=9780486469218&rft.aulast=Cohen&rft.aufirst=Paul+J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+logic" class="Z3988"></span></li> <li>J.D. Sneed, <i>The Logical Structure of Mathematical Physics</i>. Reidel, Dordrecht, 1971 (revised edition 1979).</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDavis1973" class="citation journal cs1"><a href="/wiki/Martin_Davis_(mathematician)" title="Martin Davis (mathematician)">Davis, Martin</a> (1973). "<a href="/wiki/Hilbert%27s_tenth_problem" title="Hilbert's tenth problem">Hilbert's tenth problem</a> is unsolvable". <i><a href="/wiki/American_Mathematical_Monthly" class="mw-redirect" title="American Mathematical Monthly">The American Mathematical Monthly</a></i>. <b>80</b> (3): 233–269. <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%2F2318447">10.2307/2318447</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/2318447">2318447</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+American+Mathematical+Monthly&rft.atitle=Hilbert%27s+tenth+problem+is+unsolvable&rft.volume=80&rft.issue=3&rft.pages=233-269&rft.date=1973&rft_id=info%3Adoi%2F10.2307%2F2318447&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2318447%23id-name%3DJSTOR&rft.aulast=Davis&rft.aufirst=Martin&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+logic" class="Z3988"></span><br /> Reprinted as an appendix in <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMartin_Davis1985" class="citation book cs1">Martin Davis (1985). <i>Computability and Unsolvability</i>. Dover. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780486614717" title="Special:BookSources/9780486614717"><bdi>9780486614717</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Computability+and+Unsolvability&rft.pub=Dover&rft.date=1985&rft.isbn=9780486614717&rft.au=Martin+Davis&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+logic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFelscher2000" class="citation journal cs1">Felscher, Walter (2000). "Bolzano, Cauchy, Epsilon, Delta". <i>The American Mathematical Monthly</i>. <b>107</b> (9): 844–862. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2695743">10.2307/2695743</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/2695743">2695743</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+American+Mathematical+Monthly&rft.atitle=Bolzano%2C+Cauchy%2C+Epsilon%2C+Delta&rft.volume=107&rft.issue=9&rft.pages=844-862&rft.date=2000&rft_id=info%3Adoi%2F10.2307%2F2695743&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2695743%23id-name%3DJSTOR&rft.aulast=Felscher&rft.aufirst=Walter&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+logic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFerreirós2001" class="citation journal cs1">Ferreirós, José (2001). <a rel="nofollow" class="external text" href="https://idus.us.es/xmlui/bitstream/11441/38373/1/The%20road%20to%20modern%20logic.pdf">"The Road to Modern Logic-An Interpretation"</a> <span class="cs1-format">(PDF)</span>. <i>Bulletin of Symbolic Logic</i>. <b>7</b> (4): 441–484. <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%2F2687794">10.2307/2687794</a>. <a href="/wiki/Hdl_(identifier)" class="mw-redirect" title="Hdl (identifier)">hdl</a>:<a rel="nofollow" class="external text" href="https://hdl.handle.net/11441%2F38373">11441/38373</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/2687794">2687794</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:43258676">43258676</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Bulletin+of+Symbolic+Logic&rft.atitle=The+Road+to+Modern+Logic-An+Interpretation&rft.volume=7&rft.issue=4&rft.pages=441-484&rft.date=2001&rft_id=info%3Ahdl%2F11441%2F38373&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A43258676%23id-name%3DS2CID&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2687794%23id-name%3DJSTOR&rft_id=info%3Adoi%2F10.2307%2F2687794&rft.aulast=Ferreir%C3%B3s&rft.aufirst=Jos%C3%A9&rft_id=https%3A%2F%2Fidus.us.es%2Fxmlui%2Fbitstream%2F11441%2F38373%2F1%2FThe%2520road%2520to%2520modern%2520logic.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+logic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHamkinsLöwe2007" class="citation journal cs1">Hamkins, Joel David; Löwe, Benedikt (2007). "The modal logic of forcing". <i>Transactions of the American Mathematical Society</i>. <b>360</b> (4): 1793–1818. <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/0509616">math/0509616</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/10.1090%2Fs0002-9947-07-04297-3">10.1090/s0002-9947-07-04297-3</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:14724471">14724471</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Transactions+of+the+American+Mathematical+Society&rft.atitle=The+modal+logic+of+forcing&rft.volume=360&rft.issue=4&rft.pages=1793-1818&rft.date=2007&rft_id=info%3Aarxiv%2Fmath%2F0509616&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A14724471%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1090%2Fs0002-9947-07-04297-3&rft.aulast=Hamkins&rft.aufirst=Joel+David&rft.au=L%C3%B6we%2C+Benedikt&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+logic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKatz1998" class="citation book cs1">Katz, Victor J. (1998). <i>A History of Mathematics</i>. Addison–Wesley. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780321016188" title="Special:BookSources/9780321016188"><bdi>9780321016188</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+History+of+Mathematics&rft.pub=Addison%E2%80%93Wesley&rft.date=1998&rft.isbn=9780321016188&rft.aulast=Katz&rft.aufirst=Victor+J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+logic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMorley1965" class="citation journal cs1"><a href="/wiki/Michael_D._Morley" title="Michael D. Morley">Morley, Michael</a> (1965). <a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F1994188">"Categoricity in Power"</a>. <i><a href="/wiki/Transactions_of_the_American_Mathematical_Society" title="Transactions of the American Mathematical Society">Transactions of the American Mathematical Society</a></i>. <b>114</b> (2): 514–538. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F1994188">10.2307/1994188</a></span>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/1994188">1994188</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Transactions+of+the+American+Mathematical+Society&rft.atitle=Categoricity+in+Power&rft.volume=114&rft.issue=2&rft.pages=514-538&rft.date=1965&rft_id=info%3Adoi%2F10.2307%2F1994188&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F1994188%23id-name%3DJSTOR&rft.aulast=Morley&rft.aufirst=Michael&rft_id=https%3A%2F%2Fdoi.org%2F10.2307%252F1994188&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+logic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSoare1996" class="citation journal cs1">Soare, Robert I. (1996). "Computability and recursion". <i>Bulletin of Symbolic Logic</i>. <b>2</b> (3): 284–321. <a href="/wiki/CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.35.5803">10.1.1.35.5803</a></span>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F420992">10.2307/420992</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/420992">420992</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:5894394">5894394</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Bulletin+of+Symbolic+Logic&rft.atitle=Computability+and+recursion&rft.volume=2&rft.issue=3&rft.pages=284-321&rft.date=1996&rft_id=https%3A%2F%2Fciteseerx.ist.psu.edu%2Fviewdoc%2Fsummary%3Fdoi%3D10.1.1.35.5803%23id-name%3DCiteSeerX&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A5894394%23id-name%3DS2CID&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F420992%23id-name%3DJSTOR&rft_id=info%3Adoi%2F10.2307%2F420992&rft.aulast=Soare&rft.aufirst=Robert+I.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+logic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSolovay1976" class="citation journal cs1"><a href="/wiki/Robert_M._Solovay" title="Robert M. Solovay">Solovay, Robert M.</a> (1976). "Provability Interpretations of Modal Logic". <i><a href="/wiki/Israel_Journal_of_Mathematics" title="Israel Journal of Mathematics">Israel Journal of Mathematics</a></i>. <b>25</b> (3–4): 287–304. <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%2FBF02757006">10.1007/BF02757006</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:121226261">121226261</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=Provability+Interpretations+of+Modal+Logic&rft.volume=25&rft.issue=3%E2%80%934&rft.pages=287-304&rft.date=1976&rft_id=info%3Adoi%2F10.1007%2FBF02757006&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A121226261%23id-name%3DS2CID&rft.aulast=Solovay&rft.aufirst=Robert+M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+logic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWoodin2001" class="citation journal cs1"><a href="/wiki/W._Hugh_Woodin" title="W. Hugh Woodin">Woodin, W. Hugh</a> (2001). <a rel="nofollow" class="external text" href="https://www.ams.org/notices/200106/fea-woodin.pdf">"The Continuum Hypothesis, Part I"</a> <span class="cs1-format">(PDF)</span>. <i>Notices of the American Mathematical Society</i>. <b>48</b> (6).</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=The+Continuum+Hypothesis%2C+Part+I&rft.volume=48&rft.issue=6&rft.date=2001&rft.aulast=Woodin&rft.aufirst=W.+Hugh&rft_id=https%3A%2F%2Fwww.ams.org%2Fnotices%2F200106%2Ffea-woodin.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+logic" class="Z3988"></span></li></ul> <div class="mw-heading mw-heading3"><h3 id="Classical_papers,_texts,_and_collections"><span id="Classical_papers.2C_texts.2C_and_collections"></span>Classical papers, texts, and collections</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_logic&action=edit&section=29" title="Edit section: Classical papers, texts, and collections"><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="CITEREFBanachTarski1924" class="citation journal cs1 cs1-prop-foreign-lang-source"><a href="/wiki/Stefan_Banach" title="Stefan Banach">Banach, Stefan</a>; <a href="/wiki/Alfred_Tarski" title="Alfred Tarski">Tarski, Alfred</a> (1924). <a rel="nofollow" class="external text" href="http://matwbn.icm.edu.pl/ksiazki/fm/fm6/fm6127.pdf">"Sur la décomposition des ensembles de points en parties respectivement congruentes"</a> <span class="cs1-format">(PDF)</span>. <i><a href="/wiki/Fundamenta_Mathematicae" title="Fundamenta Mathematicae">Fundamenta Mathematicae</a></i> (in French). <b>6</b>: 244–277. <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/10.4064%2Ffm-6-1-244-277">10.4064/fm-6-1-244-277</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Fundamenta+Mathematicae&rft.atitle=Sur+la+d%C3%A9composition+des+ensembles+de+points+en+parties+respectivement+congruentes&rft.volume=6&rft.pages=244-277&rft.date=1924&rft_id=info%3Adoi%2F10.4064%2Ffm-6-1-244-277&rft.aulast=Banach&rft.aufirst=Stefan&rft.au=Tarski%2C+Alfred&rft_id=http%3A%2F%2Fmatwbn.icm.edu.pl%2Fksiazki%2Ffm%2Ffm6%2Ffm6127.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+logic" class="Z3988"></span></li></ul> <p><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBochenski1959" class="citation book cs1">Bochenski, Jozef Maria, ed. (1959). <i>A Precis of Mathematical Logic</i>. Synthese Library, Vol. 1. Translated by Otto Bird. <a href="/wiki/Dordrecht" title="Dordrecht">Dordrecht</a>: <a href="/wiki/Springer_Science%2BBusiness_Media" title="Springer Science+Business Media">Springer</a>. <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%2F978-94-017-0592-9">10.1007/978-94-017-0592-9</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9789048183296" title="Special:BookSources/9789048183296"><bdi>9789048183296</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+Precis+of+Mathematical+Logic&rft.place=Dordrecht&rft.series=Synthese+Library%2C+Vol.+1&rft.pub=Springer&rft.date=1959&rft_id=info%3Adoi%2F10.1007%2F978-94-017-0592-9&rft.isbn=9789048183296&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+logic" class="Z3988"></span> </p> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBurali-Forti1897" class="citation book cs1">Burali-Forti, Cesare (1897). <i>A question on transfinite numbers</i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+question+on+transfinite+numbers&rft.date=1897&rft.aulast=Burali-Forti&rft.aufirst=Cesare&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+logic" class="Z3988"></span> Reprinted in <a href="#CITEREFvan_Heijenoort1976">van Heijenoort 1976</a>, pp. 104–111</li></ul> <p><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCantor1874" class="citation journal cs1">Cantor, Georg (1874). <a rel="nofollow" class="external text" href="http://gdz.sub.uni-goettingen.de/download/PPN243919689_0077/PPN243919689_0077___LOG_0014.pdf">"Ueber eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen"</a> <span class="cs1-format">(PDF)</span>. <i><a href="/wiki/Journal_f%C3%BCr_die_Reine_und_Angewandte_Mathematik" class="mw-redirect" title="Journal für die Reine und Angewandte Mathematik">Journal für die Reine und Angewandte Mathematik</a></i>. <b>1874</b> (77): 258–262. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1515%2Fcrll.1874.77.258">10.1515/crll.1874.77.258</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:199545885">199545885</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+f%C3%BCr+die+Reine+und+Angewandte+Mathematik&rft.atitle=Ueber+eine+Eigenschaft+des+Inbegriffes+aller+reellen+algebraischen+Zahlen&rft.volume=1874&rft.issue=77&rft.pages=258-262&rft.date=1874&rft_id=info%3Adoi%2F10.1515%2Fcrll.1874.77.258&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A199545885%23id-name%3DS2CID&rft.aulast=Cantor&rft.aufirst=Georg&rft_id=http%3A%2F%2Fgdz.sub.uni-goettingen.de%2Fdownload%2FPPN243919689_0077%2FPPN243919689_0077___LOG_0014.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+logic" class="Z3988"></span> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCarroll1896" class="citation book cs1"><a href="/wiki/Lewis_Carroll" title="Lewis Carroll">Carroll, Lewis</a> (1896). <a rel="nofollow" class="external text" href="https://www.gutenberg.org/ebooks/28696"><i>Symbolic Logic</i></a>. Kessinger Legacy Reprints. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9781163444955" title="Special:BookSources/9781163444955"><bdi>9781163444955</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Symbolic+Logic&rft.pub=Kessinger+Legacy+Reprints&rft.date=1896&rft.isbn=9781163444955&rft.aulast=Carroll&rft.aufirst=Lewis&rft_id=https%3A%2F%2Fwww.gutenberg.org%2Febooks%2F28696&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+logic" class="Z3988"></span> </p> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDedekind1872" class="citation book cs1 cs1-prop-foreign-lang-source"><a href="/wiki/Richard_Dedekind" title="Richard Dedekind">Dedekind, Richard</a> (1872). <i>Stetigkeit und irrationale Zahlen</i> (in German).</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Stetigkeit+und+irrationale+Zahlen&rft.date=1872&rft.aulast=Dedekind&rft.aufirst=Richard&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+logic" class="Z3988"></span> English translation as: "Consistency and irrational numbers".</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDedekind1888" class="citation book cs1"><a href="/wiki/Richard_Dedekind" title="Richard Dedekind">Dedekind, Richard</a> (1888). <i>Was sind und was sollen die Zahlen?</i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Was+sind+und+was+sollen+die+Zahlen%3F&rft.date=1888&rft.aulast=Dedekind&rft.aufirst=Richard&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+logic" class="Z3988"></span> Two English translations: <ul><li>1963 (1901). <i>Essays on the Theory of Numbers</i>. Beman, W. W., ed. and trans. Dover.</li> <li>1996. In <i>From Kant to Hilbert: A Source Book in the Foundations of Mathematics</i>, 2 vols, Ewald, William B., ed., <a href="/wiki/Oxford_University_Press" title="Oxford University Press">Oxford University Press</a>: 787–832.</li></ul></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFraenkel1922" class="citation book cs1 cs1-prop-foreign-lang-source"><a href="/wiki/Abraham_Fraenkel" title="Abraham Fraenkel">Fraenkel, Abraham A.</a> (1922). "Der Begriff 'definit' und die Unabhängigkeit des Auswahlsaxioms". <i>Sitzungsberichte der Preussischen Akademie der Wissenschaften, Physikalisch-mathematische Klasse</i> (in German). pp. 253–257.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Der+Begriff+%27definit%27+und+die+Unabh%C3%A4ngigkeit+des+Auswahlsaxioms&rft.btitle=Sitzungsberichte+der+Preussischen+Akademie+der+Wissenschaften%2C+Physikalisch-mathematische+Klasse&rft.pages=253-257&rft.date=1922&rft.aulast=Fraenkel&rft.aufirst=Abraham+A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+logic" class="Z3988"></span> Reprinted in English translation as "The notion of 'definite' and the independence of the axiom of choice" in <a href="#CITEREFvan_Heijenoort1976">van Heijenoort 1976</a>, pp. 284–289.</li> <li><a href="/wiki/Gottlob_Frege" title="Gottlob Frege">Frege, Gottlob</a> (1879), <i><a href="/wiki/Begriffsschrift" title="Begriffsschrift">Begriffsschrift</a>, eine der arithmetischen nachgebildete Formelsprache des reinen Denkens</i>. Halle a. S.: Louis Nebert. Translation: <i>Concept Script, a formal language of pure thought modelled upon that of arithmetic</i>, by S. Bauer-Mengelberg in <a href="#CITEREFvan_Heijenoort1976">van Heijenoort 1976</a>.</li> <li><a href="/wiki/Gottlob_Frege" title="Gottlob Frege">Frege, Gottlob</a> (1884), <i>Die Grundlagen der Arithmetik: eine logisch-mathematische Untersuchung über den Begriff der Zahl</i>. Breslau: W. Koebner. Translation: <a href="/wiki/J._L._Austin" title="J. L. Austin">J. L. Austin</a>, 1974. <i>The Foundations of Arithmetic: A logico-mathematical enquiry into the concept of number</i>, 2nd ed. Blackwell.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGentzen1936" class="citation journal cs1"><a href="/wiki/Gerhard_Gentzen" title="Gerhard Gentzen">Gentzen, Gerhard</a> (1936). "Die Widerspruchsfreiheit der reinen Zahlentheorie". <i>Mathematische Annalen</i>. <b>112</b>: 132–213. <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%2FBF01565428">10.1007/BF01565428</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:122719892">122719892</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Mathematische+Annalen&rft.atitle=Die+Widerspruchsfreiheit+der+reinen+Zahlentheorie&rft.volume=112&rft.pages=132-213&rft.date=1936&rft_id=info%3Adoi%2F10.1007%2FBF01565428&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A122719892%23id-name%3DS2CID&rft.aulast=Gentzen&rft.aufirst=Gerhard&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+logic" class="Z3988"></span> Reprinted in English translation in Gentzen's <i>Collected works</i>, M. E. Szabo, ed., North-Holland, Amsterdam, 1969.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGödel1929" class="citation book cs1"><a href="/wiki/Kurt_G%C3%B6del" title="Kurt Gödel">Gödel, Kurt</a> (1929). <i>Über die Vollständigkeit des Logikkalküls</i> [<i>Completeness of the logical calculus</i>]. doctoral dissertation. University Of Vienna.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=%C3%9Cber+die+Vollst%C3%A4ndigkeit+des+Logikkalk%C3%BCls&rft.series=doctoral+dissertation&rft.pub=University+Of+Vienna&rft.date=1929&rft.aulast=G%C3%B6del&rft.aufirst=Kurt&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+logic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGödel1930" class="citation journal cs1 cs1-prop-foreign-lang-source"><a href="/wiki/Kurt_G%C3%B6del" title="Kurt Gödel">Gödel, Kurt</a> (1930). "Die Vollständigkeit der Axiome des logischen Funktionen-kalküls" [The completeness of the axioms of the calculus of logical functions]. <i>Monatshefte für Mathematik und Physik</i> (in German). <b>37</b>: 349–360. <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%2FBF01696781">10.1007/BF01696781</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:123343522">123343522</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Monatshefte+f%C3%BCr+Mathematik+und+Physik&rft.atitle=Die+Vollst%C3%A4ndigkeit+der+Axiome+des+logischen+Funktionen-kalk%C3%BCls&rft.volume=37&rft.pages=349-360&rft.date=1930&rft_id=info%3Adoi%2F10.1007%2FBF01696781&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A123343522%23id-name%3DS2CID&rft.aulast=G%C3%B6del&rft.aufirst=Kurt&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+logic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGödel1931" class="citation journal cs1 cs1-prop-foreign-lang-source"><a href="/wiki/Kurt_G%C3%B6del" title="Kurt Gödel">Gödel, Kurt</a> (1931). "Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I" [<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>Monatshefte für Mathematik und Physik</i> (in German). <b>38</b> (1): 173–198. <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>. <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:197663120">197663120</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Monatshefte+f%C3%BCr+Mathematik+und+Physik&rft.atitle=%C3%9Cber+formal+unentscheidbare+S%C3%A4tze+der+Principia+Mathematica+und+verwandter+Systeme+I&rft.volume=38&rft.issue=1&rft.pages=173-198&rft.date=1931&rft_id=info%3Adoi%2F10.1007%2FBF01700692&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A197663120%23id-name%3DS2CID&rft.aulast=G%C3%B6del&rft.aufirst=Kurt&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+logic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGödel1958" class="citation journal cs1 cs1-prop-foreign-lang-source"><a href="/wiki/Kurt_G%C3%B6del" title="Kurt Gödel">Gödel, Kurt</a> (1958). <a rel="nofollow" class="external text" href="https://doi.org/10.1111%2Fj.1746-8361.1958.tb01464.x">"Über eine bisher noch nicht benützte Erweiterung des finiten Standpunktes"</a>. <i>Dialectica</i> (in German). <b>12</b> (3–4): 280–287. <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/10.1111%2Fj.1746-8361.1958.tb01464.x">10.1111/j.1746-8361.1958.tb01464.x</a></span>.</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=%C3%9Cber+eine+bisher+noch+nicht+ben%C3%BCtzte+Erweiterung+des+finiten+Standpunktes&rft.volume=12&rft.issue=3%E2%80%934&rft.pages=280-287&rft.date=1958&rft_id=info%3Adoi%2F10.1111%2Fj.1746-8361.1958.tb01464.x&rft.aulast=G%C3%B6del&rft.aufirst=Kurt&rft_id=https%3A%2F%2Fdoi.org%2F10.1111%252Fj.1746-8361.1958.tb01464.x&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+logic" class="Z3988"></span> Reprinted in English translation in Gödel's <i>Collected Works</i>, vol II, <a href="/wiki/Solomon_Feferman" title="Solomon Feferman">Solomon Feferman</a> et al., eds. Oxford University Press, 1993.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFvan_Heijenoort1976" class="citation book cs1"><a href="/wiki/Jean_van_Heijenoort" title="Jean van Heijenoort">van Heijenoort, Jean</a>, ed. (1976) [1967]. <i>From Frege to Gödel: A Source Book in Mathematical Logic, 1879–1931</i> (3rd ed.). <a href="/wiki/Cambridge,_Massachusetts" title="Cambridge, Massachusetts">Cambridge MA</a>: <a href="/wiki/Harvard_University_Press" title="Harvard University Press">Harvard University Press</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780674324497" title="Special:BookSources/9780674324497"><bdi>9780674324497</bdi></a>. (pbk.).</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=From+Frege+to+G%C3%B6del%3A+A+Source+Book+in+Mathematical+Logic%2C+1879%E2%80%931931&rft.place=Cambridge+MA&rft.edition=3rd&rft.pub=Harvard+University+Press&rft.date=1976&rft.isbn=9780674324497&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+logic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHilbert1899" class="citation book cs1 cs1-prop-foreign-lang-source"><a href="/wiki/David_Hilbert" title="David Hilbert">Hilbert, David</a> (1899). <i>Grundlagen der Geometrie</i> (in German). <a href="/wiki/Leipzig" title="Leipzig">Leipzig</a>: Teubner.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Grundlagen+der+Geometrie&rft.place=Leipzig&rft.pub=Teubner&rft.date=1899&rft.aulast=Hilbert&rft.aufirst=David&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+logic" class="Z3988"></span> English 1902 edition (<i>The Foundations of Geometry</i>) republished 1980, Open Court, Chicago.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHilbert1929" class="citation journal cs1"><a href="/wiki/David_Hilbert" title="David Hilbert">Hilbert, David</a> (1929). <a rel="nofollow" class="external text" href="http://gdz.sub.uni-goettingen.de/index.php?id=11&PPN=GDZPPN002273500&L=1">"Probleme der Grundlegung der Mathematik"</a>. <i>Mathematische Annalen</i>. <b>102</b>: 1–9. <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%2FBF01782335">10.1007/BF01782335</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:122870563">122870563</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Mathematische+Annalen&rft.atitle=Probleme+der+Grundlegung+der+Mathematik&rft.volume=102&rft.pages=1-9&rft.date=1929&rft_id=info%3Adoi%2F10.1007%2FBF01782335&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A122870563%23id-name%3DS2CID&rft.aulast=Hilbert&rft.aufirst=David&rft_id=http%3A%2F%2Fgdz.sub.uni-goettingen.de%2Findex.php%3Fid%3D11%26PPN%3DGDZPPN002273500%26L%3D1&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+logic" class="Z3988"></span> Lecture given at the International Congress of Mathematicians, 3 September 1928. Published in English translation as "The Grounding of Elementary Number Theory", in Mancosu 1998, pp. 266–273.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHilbertBernays1934" class="citation book cs1"><a href="/wiki/David_Hilbert" title="David Hilbert">Hilbert, David</a>; <a href="/wiki/Paul_Bernays" title="Paul Bernays">Bernays, Paul</a> (1934). <i>Grundlagen der Mathematik. I</i>. Die Grundlehren der mathematischen Wissenschaften. Vol. 40. Berlin, New York City: <a href="/wiki/Springer_Science%2BBusiness_Media" title="Springer Science+Business Media">Springer</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9783540041344" title="Special:BookSources/9783540041344"><bdi>9783540041344</bdi></a>. <a href="/wiki/JFM_(identifier)" class="mw-redirect" title="JFM (identifier)">JFM</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:60.0017.02">60.0017.02</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=0237246">0237246</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Grundlagen+der+Mathematik.+I&rft.place=Berlin%2C+New+York+City&rft.series=Die+Grundlehren+der+mathematischen+Wissenschaften&rft.pub=Springer&rft.date=1934&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D0237246%23id-name%3DMR&rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A60.0017.02%23id-name%3DJFM&rft.isbn=9783540041344&rft.aulast=Hilbert&rft.aufirst=David&rft.au=Bernays%2C+Paul&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+logic" 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_Kleene" class="mw-redirect" title="Stephen Kleene">Kleene, Stephen Cole</a> (1943). <a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F1990131">"Recursive Predicates and Quantifiers"</a>. <i>Transactions of the American Mathematical Society</i>. <b>53</b> (1): 41–73. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F1990131">10.2307/1990131</a></span>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/1990131">1990131</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Transactions+of+the+American+Mathematical+Society&rft.atitle=Recursive+Predicates+and+Quantifiers&rft.volume=53&rft.issue=1&rft.pages=41-73&rft.date=1943&rft_id=info%3Adoi%2F10.2307%2F1990131&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F1990131%23id-name%3DJSTOR&rft.aulast=Kleene&rft.aufirst=Stephen+Cole&rft_id=https%3A%2F%2Fdoi.org%2F10.2307%252F1990131&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+logic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLobachevsky1840" class="citation book cs1 cs1-prop-foreign-lang-source"><a href="/wiki/Nikolai_Lobachevsky" title="Nikolai Lobachevsky">Lobachevsky, Nikolai</a> (1840). <i>Geometrishe Untersuchungen zur Theorie der Parellellinien</i> (in German).</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Geometrishe+Untersuchungen+zur+Theorie+der+Parellellinien&rft.date=1840&rft.aulast=Lobachevsky&rft.aufirst=Nikolai&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+logic" class="Z3988"></span> Reprinted in English translation as <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRobert_Bonola1955" class="citation book cs1">Robert Bonola, ed. (1955). "Geometric Investigations on the Theory of Parallel Lines". <i>Non-Euclidean Geometry</i>. Dover. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-486-60027-0" title="Special:BookSources/0-486-60027-0"><bdi>0-486-60027-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Geometric+Investigations+on+the+Theory+of+Parallel+Lines&rft.btitle=Non-Euclidean+Geometry&rft.pub=Dover&rft.date=1955&rft.isbn=0-486-60027-0&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+logic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLöwenheim1915" class="citation journal cs1 cs1-prop-foreign-lang-source"><a href="/wiki/Leopold_L%C3%B6wenheim" title="Leopold Löwenheim">Löwenheim, Leopold</a> (1915). <a rel="nofollow" class="external text" href="http://gdz.sub.uni-goettingen.de/index.php?id=11&PPN=GDZPPN002266121&L=1">"Über Möglichkeiten im Relativkalkül"</a>. <i>Mathematische Annalen</i> (in German). <b>76</b> (4): 447–470. <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%2FBF01458217">10.1007/BF01458217</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/0025-5831">0025-5831</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:116581304">116581304</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Mathematische+Annalen&rft.atitle=%C3%9Cber+M%C3%B6glichkeiten+im+Relativkalk%C3%BCl&rft.volume=76&rft.issue=4&rft.pages=447-470&rft.date=1915&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A116581304%23id-name%3DS2CID&rft.issn=0025-5831&rft_id=info%3Adoi%2F10.1007%2FBF01458217&rft.aulast=L%C3%B6wenheim&rft.aufirst=Leopold&rft_id=http%3A%2F%2Fgdz.sub.uni-goettingen.de%2Findex.php%3Fid%3D11%26PPN%3DGDZPPN002266121%26L%3D1&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+logic" class="Z3988"></span> Translated as "On possibilities in the calculus of relatives" in <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJean_van_Heijenoort1967" class="citation book cs1"><a href="/wiki/Jean_van_Heijenoort" title="Jean van Heijenoort">Jean van Heijenoort</a> (1967). <i>A Source Book in Mathematical Logic, 1879–1931</i>. Harvard Univ. Press. pp. 228–251.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Source+Book+in+Mathematical+Logic%2C+1879%E2%80%931931&rft.pages=228-251&rft.pub=Harvard+Univ.+Press&rft.date=1967&rft.au=Jean+van+Heijenoort&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+logic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMancosu1998" class="citation book cs1"><a href="/w/index.php?title=Paolo_Mancosu&action=edit&redlink=1" class="new" title="Paolo Mancosu (page does not exist)">Mancosu, Paolo</a>, ed. (1998). <i>From Brouwer to Hilbert. The Debate on the Foundations of Mathematics in the 1920s</i>. Oxford University Press.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=From+Brouwer+to+Hilbert.+The+Debate+on+the+Foundations+of+Mathematics+in+the+1920s&rft.pub=Oxford+University+Press&rft.date=1998&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+logic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPasch1882" class="citation book cs1"><a href="/wiki/Moritz_Pasch" title="Moritz Pasch">Pasch, Moritz</a> (1882). <i>Vorlesungen über neuere Geometrie</i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Vorlesungen+%C3%BCber+neuere+Geometrie&rft.date=1882&rft.aulast=Pasch&rft.aufirst=Moritz&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+logic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPeano1889" class="citation book cs1 cs1-prop-foreign-lang-source"><a href="/wiki/Giuseppe_Peano" title="Giuseppe Peano">Peano, Giuseppe</a> (1889). <a href="/wiki/Arithmetices_principia,_nova_methodo_exposita" title="Arithmetices principia, nova methodo exposita"><i>Arithmetices principia, nova methodo exposita</i></a> (in Lithuanian).</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Arithmetices+principia%2C+nova+methodo+exposita&rft.date=1889&rft.aulast=Peano&rft.aufirst=Giuseppe&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+logic" class="Z3988"></span> Excerpt reprinted in English translation as "The principles of arithmetic, presented by a new method"in <a href="#CITEREFvan_Heijenoort1976">van Heijenoort 1976</a>, pp. 83–97.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRichard1905" class="citation journal cs1 cs1-prop-foreign-lang-source">Richard, Jules (1905). "Les principes des mathématiques et le problème des ensembles". <i>Revue Générale des Sciences Pures et Appliquées</i> (in French). <b>16</b>: 541.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Revue+G%C3%A9n%C3%A9rale+des+Sciences+Pures+et+Appliqu%C3%A9es&rft.atitle=Les+principes+des+math%C3%A9matiques+et+le+probl%C3%A8me+des+ensembles&rft.volume=16&rft.pages=541&rft.date=1905&rft.aulast=Richard&rft.aufirst=Jules&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+logic" class="Z3988"></span> Reprinted in English translation as "The principles of mathematics and the problems of sets" in <a href="#CITEREFvan_Heijenoort1976">van Heijenoort 1976</a>, pp. 142–144.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSkolem1920" class="citation journal cs1 cs1-prop-foreign-lang-source"><a href="/wiki/Thoralf_Skolem" title="Thoralf Skolem">Skolem, Thoralf</a> (1920). "Logisch-kombinatorische Untersuchungen über die Erfüllbarkeit oder Beweisbarkeit mathematischer Sätze nebst einem Theoreme über dichte Mengen". <i>Videnskapsselskapet Skrifter, I. Matematisk-naturvidenskabelig Klasse</i> (in German). <b>6</b>: 1–36.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Videnskapsselskapet+Skrifter%2C+I.+Matematisk-naturvidenskabelig+Klasse&rft.atitle=Logisch-kombinatorische+Untersuchungen+%C3%BCber+die+Erf%C3%BCllbarkeit+oder+Beweisbarkeit+mathematischer+S%C3%A4tze+nebst+einem+Theoreme+%C3%BCber+dichte+Mengen&rft.volume=6&rft.pages=1-36&rft.date=1920&rft.aulast=Skolem&rft.aufirst=Thoralf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+logic" class="Z3988"></span></li></ul> <p><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSoare2011" class="citation web cs1">Soare, Robert Irving (22 December 2011). <a rel="nofollow" class="external text" href="http://www.people.cs.uchicago.edu/~soare/Turing/frontice.pdf">"Computability Theory and Applications: The Art of Classical Computability"</a> <span class="cs1-format">(PDF)</span>. <i>Department of Mathematics</i>. University of Chicago<span class="reference-accessdate">. Retrieved <span class="nowrap">23 August</span> 2017</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Department+of+Mathematics&rft.atitle=Computability+Theory+and+Applications%3A+The+Art+of+Classical+Computability&rft.date=2011-12-22&rft.aulast=Soare&rft.aufirst=Robert+Irving&rft_id=http%3A%2F%2Fwww.people.cs.uchicago.edu%2F~soare%2FTuring%2Ffrontice.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+logic" class="Z3988"></span> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSwineshead1498" class="citation book cs1 cs1-prop-foreign-lang-source"><a href="/wiki/Richard_Swineshead" title="Richard Swineshead">Swineshead, Richard</a> (1498). <i>Calculationes Suiseth Anglici</i> (in Lithuanian). Papie: Per Franciscum Gyrardengum.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Calculationes+Suiseth+Anglici&rft.pub=Papie%3A+Per+Franciscum+Gyrardengum&rft.date=1498&rft.aulast=Swineshead&rft.aufirst=Richard&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+logic" class="Z3988"></span> </p> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTarski1948" class="citation book cs1"><a href="/wiki/Alfred_Tarski" title="Alfred Tarski">Tarski, Alfred</a> (1948). <i>A decision method for elementary algebra and geometry</i>. <a href="/wiki/Santa_Monica,_California" title="Santa Monica, California">Santa Monica CA</a>: <a href="/wiki/RAND_Corporation" title="RAND Corporation">RAND Corporation</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+decision+method+for+elementary+algebra+and+geometry&rft.place=Santa+Monica+CA&rft.pub=RAND+Corporation&rft.date=1948&rft.aulast=Tarski&rft.aufirst=Alfred&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+logic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTuring1939" class="citation journal cs1"><a href="/wiki/A._M._Turing" class="mw-redirect" title="A. M. Turing">Turing, Alan M.</a> (1939). "Systems of Logic Based on Ordinals". <i><a href="/wiki/Proceedings_of_the_London_Mathematical_Society" class="mw-redirect" title="Proceedings of the London Mathematical Society">Proceedings of the London Mathematical Society</a></i>. <b>45</b> (2): 161–228. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1112%2Fplms%2Fs2-45.1.161">10.1112/plms/s2-45.1.161</a>. <a href="/wiki/Hdl_(identifier)" class="mw-redirect" title="Hdl (identifier)">hdl</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://hdl.handle.net/21.11116%2F0000-0001-91CE-3">21.11116/0000-0001-91CE-3</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Proceedings+of+the+London+Mathematical+Society&rft.atitle=Systems+of+Logic+Based+on+Ordinals&rft.volume=45&rft.issue=2&rft.pages=161-228&rft.date=1939&rft_id=info%3Ahdl%2F21.11116%2F0000-0001-91CE-3&rft_id=info%3Adoi%2F10.1112%2Fplms%2Fs2-45.1.161&rft.aulast=Turing&rft.aufirst=Alan+M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+logic" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeyl1918" class="citation book cs1 cs1-prop-foreign-lang-source">Weyl, Hermann (1918). <i>Das Kontinuum. Kritische Untersuchungen über die Grundlagen der Analysis</i> (in German). Leipzig.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Das+Kontinuum.+Kritische+Untersuchungen+%C3%BCber+die+Grundlagen+der+Analysis&rft.place=Leipzig&rft.date=1918&rft.aulast=Weyl&rft.aufirst=Hermann&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+logic" class="Z3988"></span><span class="cs1-maint citation-comment"><code class="cs1-code">{{<a href="/wiki/Template:Cite_book" title="Template:Cite book">cite book</a>}}</code>: CS1 maint: location missing publisher (<a href="/wiki/Category:CS1_maint:_location_missing_publisher" title="Category:CS1 maint: location missing publisher">link</a>)</span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFZermelo1904" class="citation journal cs1 cs1-prop-foreign-lang-source"><a href="/wiki/Ernst_Zermelo" title="Ernst Zermelo">Zermelo, Ernst</a> (1904). <a rel="nofollow" class="external text" href="http://gdz.sub.uni-goettingen.de/index.php?id=11&PPN=GDZPPN002260018&L=1">"Beweis, daß jede Menge wohlgeordnet werden kann"</a>. <i>Mathematische Annalen</i> (in German). <b>59</b> (4): 514–516. <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%2FBF01445300">10.1007/BF01445300</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:124189935">124189935</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Mathematische+Annalen&rft.atitle=Beweis%2C+da%C3%9F+jede+Menge+wohlgeordnet+werden+kann&rft.volume=59&rft.issue=4&rft.pages=514-516&rft.date=1904&rft_id=info%3Adoi%2F10.1007%2FBF01445300&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A124189935%23id-name%3DS2CID&rft.aulast=Zermelo&rft.aufirst=Ernst&rft_id=http%3A%2F%2Fgdz.sub.uni-goettingen.de%2Findex.php%3Fid%3D11%26PPN%3DGDZPPN002260018%26L%3D1&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+logic" class="Z3988"></span> Reprinted in English translation as "Proof that every set can be well-ordered" in <a href="#CITEREFvan_Heijenoort1976">van Heijenoort 1976</a>, pp. 139–141.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFZermelo1908a" class="citation journal cs1 cs1-prop-foreign-lang-source"><a href="/wiki/Ernst_Zermelo" title="Ernst Zermelo">Zermelo, Ernst</a> (1908a). <a rel="nofollow" class="external text" href="http://gdz.sub.uni-goettingen.de/index.php?id=11&PPN=GDZPPN002261952&L=1">"Neuer Beweis für die Möglichkeit einer Wohlordnung"</a>. <i><a href="/wiki/Mathematische_Annalen" title="Mathematische Annalen">Mathematische Annalen</a></i> (in German). <b>65</b>: 107–128. <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%2FBF01450054">10.1007/BF01450054</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/0025-5831">0025-5831</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:119924143">119924143</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Mathematische+Annalen&rft.atitle=Neuer+Beweis+f%C3%BCr+die+M%C3%B6glichkeit+einer+Wohlordnung&rft.volume=65&rft.pages=107-128&rft.date=1908&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A119924143%23id-name%3DS2CID&rft.issn=0025-5831&rft_id=info%3Adoi%2F10.1007%2FBF01450054&rft.aulast=Zermelo&rft.aufirst=Ernst&rft_id=http%3A%2F%2Fgdz.sub.uni-goettingen.de%2Findex.php%3Fid%3D11%26PPN%3DGDZPPN002261952%26L%3D1&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+logic" class="Z3988"></span> Reprinted in English translation as "A new proof of the possibility of a well-ordering" in <a href="#CITEREFvan_Heijenoort1976">van Heijenoort 1976</a>, pp. 183–198.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFZermelo1908b" class="citation journal cs1"><a href="/wiki/Ernst_Zermelo" title="Ernst Zermelo">Zermelo, Ernst</a> (1908b). <a rel="nofollow" class="external text" href="http://gdz.sub.uni-goettingen.de/index.php?id=11&PPN=PPN235181684_0065&DMDID=DMDLOG_0018&L=1">"Untersuchungen über die Grundlagen der Mengenlehre"</a>. <i>Mathematische Annalen</i>. <b>65</b> (2): 261–281. <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%2FBF01449999">10.1007/BF01449999</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:120085563">120085563</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Mathematische+Annalen&rft.atitle=Untersuchungen+%C3%BCber+die+Grundlagen+der+Mengenlehre&rft.volume=65&rft.issue=2&rft.pages=261-281&rft.date=1908&rft_id=info%3Adoi%2F10.1007%2FBF01449999&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A120085563%23id-name%3DS2CID&rft.aulast=Zermelo&rft.aufirst=Ernst&rft_id=http%3A%2F%2Fgdz.sub.uni-goettingen.de%2Findex.php%3Fid%3D11%26PPN%3DPPN235181684_0065%26DMDID%3DDMDLOG_0018%26L%3D1&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+logic" 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=Mathematical_logic&action=edit&section=30" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation cs2"><a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Mathematical_logic">"Mathematical logic"</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=Mathematical+logic&rft.btitle=Encyclopedia+of+Mathematics&rft.pub=EMS+Press&rft.date=2001&rft_id=https%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%3Ftitle%3DMathematical_logic&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+logic" class="Z3988"></span></li> <li><a rel="nofollow" class="external text" href="http://www.quantrelog.se/pvlmatrix/index_main.htm">Polyvalued logic and Quantity Relation Logic</a></li> <li><i><a rel="nofollow" class="external text" href="http://www.fecundity.com/logic/">forall x: an introduction to formal logic</a></i>, a free textbook by <span class="nowrap">P. D. Magnus</span>.</li> <li><i><a rel="nofollow" class="external text" href="http://euclid.trentu.ca/math/sb/pcml/">A Problem Course in Mathematical Logic</a></i>, a free textbook by Stefan Bilaniuk.</li> <li>Detlovs, Vilnis, and Podnieks, Karlis (University of Latvia), <i><a rel="nofollow" class="external text" href="http://www.ltn.lv/~podnieks/mlog/ml.htm">Introduction to Mathematical Logic.</a></i> (hyper-textbook).</li> <li>In the <a href="/wiki/Stanford_Encyclopedia_of_Philosophy" title="Stanford Encyclopedia of Philosophy">Stanford Encyclopedia of Philosophy</a>: <dl><dd><a rel="nofollow" class="external text" href="http://plato.stanford.edu/entries/logic-classical/">Classical Logic</a> by <a href="/wiki/Stewart_Shapiro" title="Stewart Shapiro">Stewart Shapiro</a>.</dd> <dd><a rel="nofollow" class="external text" href="http://plato.stanford.edu/entries/modeltheory-fo/">First-order Model Theory</a> by <a href="/wiki/Wilfrid_Hodges" title="Wilfrid Hodges">Wilfrid Hodges</a>.</dd></dl></li> <li>In the <a rel="nofollow" class="external text" href="http://www.ucl.ac.uk/philosophy/LPSG/">London Philosophy Study Guide</a>: <dl><dd><a rel="nofollow" class="external text" href="http://www.ucl.ac.uk/philosophy/LPSG/MathLogic.htm">Mathematical Logic</a></dd> <dd><a rel="nofollow" class="external text" href="http://www.ucl.ac.uk/philosophy/LPSG/SetTheory.htm">Set Theory & Further Logic</a></dd> <dd><a rel="nofollow" class="external text" href="http://www.ucl.ac.uk/philosophy/LPSG/PhilMath.htm">Philosophy of Mathematics</a></dd></dl></li> <li><a rel="nofollow" class="external text" href="http://oldwww.ma.man.ac.uk/~jeff/">School of Mathematics, University of Manchester, Prof. Jeff Paris’s Mathematical Logic (course material and unpublished papers)</a></li></ul> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1236075235">.mw-parser-output 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title="Graph theory">Graph theory</a></li> <li><a href="/wiki/Order_theory" title="Order theory">Order theory</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Geometry" title="Geometry">Geometry</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/Algebraic_geometry" title="Algebraic geometry">Algebraic</a></li> <li><a href="/wiki/Analytic_geometry" title="Analytic geometry">Analytic</a></li> <li><a href="/wiki/Arithmetic_geometry" title="Arithmetic geometry">Arithmetic</a></li> <li><a href="/wiki/Differential_geometry" title="Differential geometry">Differential</a></li> <li><a href="/wiki/Discrete_geometry" title="Discrete geometry">Discrete</a></li> <li><a href="/wiki/Euclidean_geometry" title="Euclidean geometry">Euclidean</a></li> <li><a href="/wiki/Finite_geometry" title="Finite geometry">Finite</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Number_theory" title="Number theory">Number theory</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/Arithmetic" title="Arithmetic">Arithmetic</a></li> <li><a href="/wiki/Algebraic_number_theory" title="Algebraic number theory">Algebraic number theory</a></li> <li><a href="/wiki/Analytic_number_theory" title="Analytic number theory">Analytic number theory</a></li> <li><a href="/wiki/Diophantine_geometry" title="Diophantine geometry">Diophantine geometry</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Topology" title="Topology">Topology</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/General_topology" title="General topology">General</a></li> <li><a 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art">Mathematics and art</a></li> <li><a href="/wiki/Mathematics_education" title="Mathematics education">Mathematics education</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div> <ul><li><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" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/24px-Nuvola_apps_edu_mathematics_blue-p.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/32px-Nuvola_apps_edu_mathematics_blue-p.svg.png 2x" data-file-width="128" data-file-height="128" /></a></span> </span><a href="/wiki/Portal:Mathematics" 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href="/wiki/Printed_circuit_board" title="Printed circuit board">Printed circuit board</a></li> <li><a href="/wiki/Peripheral" title="Peripheral">Peripheral</a></li> <li><a href="/wiki/Integrated_circuit" title="Integrated circuit">Integrated circuit</a></li> <li><a href="/wiki/Very_Large_Scale_Integration" class="mw-redirect" title="Very Large Scale Integration">Very Large Scale Integration</a></li> <li><a href="/wiki/System_on_a_chip" title="System on a chip">Systems on Chip (SoCs)</a></li> <li><a href="/wiki/Green_computing" title="Green computing">Energy consumption (Green computing)</a></li> <li><a href="/wiki/Electronic_design_automation" title="Electronic design automation">Electronic design automation</a></li> <li><a href="/wiki/Hardware_acceleration" title="Hardware acceleration">Hardware acceleration</a></li> <li><a href="/wiki/Processor_(computing)" title="Processor (computing)">Processor</a></li> <li><a href="/wiki/List_of_computer_size_categories" title="List of computer size categories">Size</a> / <a href="/wiki/Form_factor_(design)" title="Form factor (design)">Form</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Computer systems organization</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/Computer_architecture" title="Computer architecture">Computer architecture</a></li> <li><a href="/wiki/Computational_complexity" title="Computational complexity">Computational complexity</a></li> <li><a href="/wiki/Dependability" title="Dependability">Dependability</a></li> <li><a href="/wiki/Embedded_system" title="Embedded system">Embedded system</a></li> <li><a href="/wiki/Real-time_computing" title="Real-time computing">Real-time computing</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Computer_network" title="Computer network">Networks</a></th><td 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(computing)">Interpreter</a></li> <li><a href="/wiki/Middleware" title="Middleware">Middleware</a></li> <li><a href="/wiki/Virtual_machine" title="Virtual machine">Virtual machine</a></li> <li><a href="/wiki/Operating_system" title="Operating system">Operating system</a></li> <li><a href="/wiki/Software_quality" title="Software quality">Software quality</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Programming_language_theory" title="Programming language theory">Software notations</a> and <a href="/wiki/Programming_tool" title="Programming tool">tools</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/Programming_paradigm" title="Programming paradigm">Programming paradigm</a></li> <li><a href="/wiki/Programming_language" title="Programming language">Programming language</a></li> <li><a href="/wiki/Compiler_construction" class="mw-redirect" title="Compiler construction">Compiler</a></li> <li><a href="/wiki/Domain-specific_language" title="Domain-specific language">Domain-specific language</a></li> <li><a href="/wiki/Modeling_language" title="Modeling language">Modeling language</a></li> <li><a href="/wiki/Software_framework" title="Software framework">Software framework</a></li> <li><a href="/wiki/Integrated_development_environment" title="Integrated development environment">Integrated development environment</a></li> <li><a href="/wiki/Software_configuration_management" title="Software configuration management">Software configuration management</a></li> <li><a href="/wiki/Library_(computing)" title="Library (computing)">Software library</a></li> <li><a href="/wiki/Software_repository" title="Software repository">Software repository</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Software_development" title="Software development">Software development</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/Control_variable_(programming)" class="mw-redirect" title="Control variable (programming)">Control variable</a></li> <li><a href="/wiki/Software_development_process" title="Software development process">Software development process</a></li> <li><a href="/wiki/Requirements_analysis" title="Requirements analysis">Requirements analysis</a></li> <li><a href="/wiki/Software_design" title="Software design">Software design</a></li> <li><a href="/wiki/Software_construction" title="Software construction">Software construction</a></li> <li><a href="/wiki/Software_deployment" title="Software deployment">Software deployment</a></li> <li><a href="/wiki/Software_engineering" title="Software engineering">Software engineering</a></li> <li><a href="/wiki/Software_maintenance" title="Software maintenance">Software maintenance</a></li> <li><a href="/wiki/Programming_team" title="Programming team">Programming team</a></li> <li><a href="/wiki/Open-source_software" title="Open-source software">Open-source model</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Theory_of_computation" title="Theory of computation">Theory of computation</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/Model_of_computation" title="Model of computation">Model of computation</a> <ul><li><a href="/wiki/Stochastic_computing" title="Stochastic computing">Stochastic</a></li></ul></li> <li><a href="/wiki/Formal_language" title="Formal language">Formal language</a></li> <li><a href="/wiki/Automata_theory" title="Automata theory">Automata theory</a></li> <li><a href="/wiki/Computability_theory" title="Computability theory">Computability theory</a></li> <li><a href="/wiki/Computational_complexity_theory" title="Computational complexity theory">Computational complexity theory</a></li> <li><a href="/wiki/Logic_in_computer_science" title="Logic in computer science">Logic</a></li> <li><a href="/wiki/Semantics_(computer_science)" title="Semantics (computer science)">Semantics</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Algorithm" title="Algorithm">Algorithms</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/Algorithm_design" class="mw-redirect" title="Algorithm design">Algorithm design</a></li> <li><a href="/wiki/Analysis_of_algorithms" title="Analysis of algorithms">Analysis of algorithms</a></li> <li><a href="/wiki/Algorithmic_efficiency" title="Algorithmic efficiency">Algorithmic efficiency</a></li> <li><a href="/wiki/Randomized_algorithm" title="Randomized algorithm">Randomized algorithm</a></li> <li><a href="/wiki/Computational_geometry" title="Computational geometry">Computational geometry</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Mathematics of <a href="/wiki/Computing" title="Computing">computing</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/Discrete_mathematics" title="Discrete mathematics">Discrete mathematics</a></li> <li><a href="/wiki/Probability" title="Probability">Probability</a></li> <li><a href="/wiki/Statistics" title="Statistics">Statistics</a></li> <li><a href="/wiki/Mathematical_software" title="Mathematical software">Mathematical software</a></li> <li><a href="/wiki/Information_theory" title="Information theory">Information theory</a></li> <li><a href="/wiki/Mathematical_analysis" title="Mathematical analysis">Mathematical analysis</a></li> <li><a href="/wiki/Numerical_analysis" title="Numerical analysis">Numerical analysis</a></li> <li><a href="/wiki/Theoretical_computer_science" title="Theoretical computer science">Theoretical computer science</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Information_system" title="Information system">Information systems</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/Database" title="Database">Database management system</a></li> <li><a href="/wiki/Computer_data_storage" title="Computer data storage">Information storage systems</a></li> <li><a href="/wiki/Enterprise_information_system" title="Enterprise information system">Enterprise information system</a></li> <li><a href="/wiki/Social_software" title="Social software">Social information systems</a></li> <li><a href="/wiki/Geographic_information_system" title="Geographic information system">Geographic information system</a></li> <li><a href="/wiki/Decision_support_system" title="Decision support system">Decision support system</a></li> <li><a href="/wiki/Process_control" class="mw-redirect" title="Process control">Process control system</a></li> <li><a href="/wiki/Multimedia_database" title="Multimedia database">Multimedia information system</a></li> <li><a href="/wiki/Data_mining" title="Data mining">Data mining</a></li> <li><a href="/wiki/Digital_library" title="Digital library">Digital library</a></li> <li><a href="/wiki/Computing_platform" title="Computing platform">Computing platform</a></li> <li><a href="/wiki/Digital_marketing" title="Digital marketing">Digital marketing</a></li> <li><a href="/wiki/World_Wide_Web" title="World Wide Web">World Wide Web</a></li> <li><a href="/wiki/Information_retrieval" title="Information retrieval">Information retrieval</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Computer_security" title="Computer security">Security</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/Cryptography" title="Cryptography">Cryptography</a></li> <li><a href="/wiki/Formal_methods" title="Formal methods">Formal methods</a></li> <li><a href="/wiki/Security_hacker" title="Security hacker">Security hacker</a></li> <li><a href="/wiki/Security_service_(telecommunication)" title="Security service (telecommunication)">Security services</a></li> <li><a href="/wiki/Intrusion_detection_system" title="Intrusion detection system">Intrusion detection system</a></li> <li><a href="/wiki/Hardware_security" title="Hardware security">Hardware security</a></li> <li><a href="/wiki/Network_security" title="Network security">Network security</a></li> <li><a href="/wiki/Information_security" title="Information security">Information security</a></li> <li><a href="/wiki/Application_security" title="Application security">Application security</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Human%E2%80%93computer_interaction" title="Human–computer interaction">Human–computer interaction</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/Interaction_design" title="Interaction design">Interaction design</a></li> <li><a href="/wiki/Social_computing" title="Social computing">Social computing</a></li> <li><a href="/wiki/Ubiquitous_computing" title="Ubiquitous computing">Ubiquitous computing</a></li> <li><a href="/wiki/Visualization_(graphics)" title="Visualization (graphics)">Visualization</a></li> <li><a href="/wiki/Computer_accessibility" title="Computer accessibility">Accessibility</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Concurrency_(computer_science)" title="Concurrency (computer science)">Concurrency</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/Concurrent_computing" title="Concurrent computing">Concurrent computing</a></li> <li><a href="/wiki/Parallel_computing" title="Parallel computing">Parallel computing</a></li> <li><a href="/wiki/Distributed_computing" title="Distributed computing">Distributed computing</a></li> <li><a href="/wiki/Multithreading_(computer_architecture)" title="Multithreading (computer architecture)">Multithreading</a></li> <li><a href="/wiki/Multiprocessing" title="Multiprocessing">Multiprocessing</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Artificial_intelligence" title="Artificial intelligence">Artificial intelligence</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/Natural_language_processing" title="Natural language processing">Natural language processing</a></li> <li><a href="/wiki/Knowledge_representation_and_reasoning" title="Knowledge representation and reasoning">Knowledge representation and reasoning</a></li> <li><a href="/wiki/Computer_vision" title="Computer vision">Computer vision</a></li> <li><a href="/wiki/Automated_planning_and_scheduling" title="Automated planning and scheduling">Automated planning and scheduling</a></li> <li><a href="/wiki/Mathematical_optimization" title="Mathematical optimization">Search methodology</a></li> <li><a href="/wiki/Control_theory" title="Control theory">Control method</a></li> <li><a href="/wiki/Philosophy_of_artificial_intelligence" title="Philosophy of artificial intelligence">Philosophy of artificial intelligence</a></li> <li><a href="/wiki/Distributed_artificial_intelligence" title="Distributed artificial intelligence">Distributed artificial intelligence</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Machine_learning" title="Machine learning">Machine learning</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/Supervised_learning" title="Supervised learning">Supervised learning</a></li> <li><a href="/wiki/Unsupervised_learning" title="Unsupervised learning">Unsupervised learning</a></li> <li><a href="/wiki/Reinforcement_learning" title="Reinforcement learning">Reinforcement learning</a></li> <li><a href="/wiki/Multi-task_learning" title="Multi-task learning">Multi-task learning</a></li> <li><a href="/wiki/Cross-validation_(statistics)" title="Cross-validation (statistics)">Cross-validation</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Computer_graphics" title="Computer graphics">Graphics</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/Computer_animation" title="Computer animation">Animation</a></li> <li><a href="/wiki/Rendering_(computer_graphics)" title="Rendering (computer graphics)">Rendering</a></li> <li><a href="/wiki/Photograph_manipulation" title="Photograph manipulation">Photograph manipulation</a></li> <li><a href="/wiki/Graphics_processing_unit" title="Graphics processing unit">Graphics processing unit</a></li> <li><a href="/wiki/Mixed_reality" title="Mixed reality">Mixed reality</a></li> <li><a href="/wiki/Virtual_reality" title="Virtual reality">Virtual reality</a></li> <li><a href="/wiki/Image_compression" title="Image compression">Image compression</a></li> <li><a href="/wiki/Solid_modeling" title="Solid modeling">Solid modeling</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Applied computing</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/Quantum_Computing" class="mw-redirect" title="Quantum Computing">Quantum Computing</a></li> <li><a href="/wiki/E-commerce" title="E-commerce">E-commerce</a></li> <li><a href="/wiki/Enterprise_software" title="Enterprise software">Enterprise software</a></li> <li><a href="/wiki/Computational_mathematics" title="Computational mathematics">Computational mathematics</a></li> <li><a href="/wiki/Computational_physics" title="Computational physics">Computational physics</a></li> <li><a href="/wiki/Computational_chemistry" title="Computational chemistry">Computational chemistry</a></li> <li><a href="/wiki/Computational_biology" title="Computational biology">Computational biology</a></li> <li><a href="/wiki/Computational_social_science" title="Computational social science">Computational social science</a></li> <li><a href="/wiki/Computational_engineering" title="Computational engineering">Computational engineering</a></li> <li><a href="/wiki/Template:Differentiable_computing" title="Template:Differentiable computing">Differentiable computing</a></li> <li><a href="/wiki/Health_informatics" title="Health informatics">Computational healthcare</a></li> <li><a href="/wiki/Digital_art" title="Digital art">Digital art</a></li> <li><a href="/wiki/Electronic_publishing" title="Electronic publishing">Electronic publishing</a></li> <li><a href="/wiki/Cyberwarfare" title="Cyberwarfare">Cyberwarfare</a></li> <li><a href="/wiki/Electronic_voting" title="Electronic voting">Electronic voting</a></li> <li><a href="/wiki/Video_game" title="Video game">Video games</a></li> <li><a href="/wiki/Word_processor" title="Word processor">Word processing</a></li> <li><a href="/wiki/Operations_research" title="Operations research">Operations research</a></li> <li><a href="/wiki/Educational_technology" title="Educational technology">Educational technology</a></li> <li><a href="/wiki/Document_management_system" title="Document management system">Document management</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div> <ul><li><span class="noviewer" typeof="mw:File"><span title="Category"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/16px-Symbol_category_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/23px-Symbol_category_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/31px-Symbol_category_class.svg.png 2x" data-file-width="180" data-file-height="185" /></span></span> <a href="/wiki/Category:Computer_science" title="Category:Computer science">Category</a></li> <li><span class="noviewer" typeof="mw:File"><span title="Outline"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/41/Global_thinking.svg/10px-Global_thinking.svg.png" decoding="async" width="10" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/41/Global_thinking.svg/15px-Global_thinking.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/41/Global_thinking.svg/21px-Global_thinking.svg.png 2x" data-file-width="130" data-file-height="200" /></span></span> <a href="/wiki/Outline_of_computer_science" title="Outline of computer science">Outline</a></li> <li><span class="noviewer" typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/e/e0/Symbol_question.svg/16px-Symbol_question.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/e/e0/Symbol_question.svg/23px-Symbol_question.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/e/e0/Symbol_question.svg/31px-Symbol_question.svg.png 2x" data-file-width="180" data-file-height="185" /></span></span> <a href="/wiki/Template:Glossaries_of_computers" title="Template:Glossaries of computers">Glossaries</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="Logic" style="padding:3px"><table class="nowraplinks hlist 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:Logic" title="Template:Logic"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Logic" title="Template talk:Logic"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Logic" title="Special:EditPage/Template:Logic"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Logic" style="font-size:114%;margin:0 4em"><a href="/wiki/Logic" title="Logic">Logic</a></div></th></tr><tr><td class="navbox-abovebelow" colspan="2"><div> <ul><li><a href="/wiki/Outline_of_logic" title="Outline of logic">Outline</a></li> <li><a href="/wiki/History_of_logic" title="History of logic">History</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Major fields</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/Logic_in_computer_science" title="Logic in computer science">Computer science</a></li> <li><a href="/wiki/Formal_semantics_(natural_language)" title="Formal semantics (natural language)">Formal semantics (natural language)</a></li> <li><a href="/wiki/Inference" title="Inference">Inference</a></li> <li><a href="/wiki/Philosophy_of_logic" title="Philosophy of logic">Philosophy of logic</a></li> <li><a href="/wiki/Formal_proof" title="Formal proof">Proof</a></li> <li><a href="/wiki/Semantics_of_logic" title="Semantics of logic">Semantics of logic</a></li> <li><a href="/wiki/Syntax_(logic)" title="Syntax (logic)">Syntax</a></li></ul> </div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">Logics</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/Classical_logic" title="Classical logic">Classical</a></li> <li><a href="/wiki/Informal_logic" title="Informal logic">Informal</a> <ul><li><a href="/wiki/Critical_thinking" title="Critical thinking">Critical thinking</a></li> <li><a href="/wiki/Reason" title="Reason">Reason</a></li></ul></li> <li><a class="mw-selflink selflink">Mathematical</a></li> <li><a href="/wiki/Non-classical_logic" title="Non-classical logic">Non-classical</a></li> <li><a href="/wiki/Philosophical_logic" title="Philosophical logic">Philosophical</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Theories</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Argumentation_theory" title="Argumentation theory">Argumentation</a></li> <li><a href="/wiki/Metalogic" title="Metalogic">Metalogic</a></li> <li><a href="/wiki/Metamathematics" title="Metamathematics">Metamathematics</a></li> <li><a href="/wiki/Set_theory" title="Set theory">Set</a></li></ul> </div></td></tr></tbody></table><div> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Foundations</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/Abductive_reasoning" title="Abductive reasoning">Abduction</a></li> <li><a href="/wiki/Analytic%E2%80%93synthetic_distinction" title="Analytic–synthetic distinction">Analytic and synthetic propositions</a></li> <li><a href="/wiki/Antecedent_(logic)" title="Antecedent (logic)">Antecedent</a></li> <li><a href="/wiki/Consequent" title="Consequent">Consequent</a></li> <li><a href="/wiki/Contradiction" title="Contradiction">Contradiction</a> <ul><li><a href="/wiki/Paradox" title="Paradox">Paradox</a></li> <li><a href="/wiki/Antinomy" title="Antinomy">Antinomy</a></li></ul></li> <li><a href="/wiki/Deductive_reasoning" title="Deductive reasoning">Deduction</a></li> <li><a href="/wiki/Deductive_closure" title="Deductive closure">Deductive closure</a></li> <li><a href="/wiki/Definition" title="Definition">Definition</a></li> <li><a href="/wiki/Description" title="Description">Description</a></li> <li><a href="/wiki/Logical_consequence" title="Logical consequence">Entailment</a> <ul><li><a href="/wiki/Entailment_(linguistics)" title="Entailment (linguistics)">Linguistic</a></li></ul></li> <li><a href="/wiki/Logical_form" title="Logical form">Form</a></li> <li><a href="/wiki/Inductive_reasoning" title="Inductive reasoning">Induction</a></li> <li><a href="/wiki/Logical_truth" title="Logical truth">Logical truth</a></li> <li><a href="/wiki/Name" title="Name">Name</a></li> <li><a href="/wiki/Necessity_and_sufficiency" title="Necessity and sufficiency">Necessity and sufficiency</a></li> <li><a href="/wiki/Premise" title="Premise">Premise</a></li> <li><a href="/wiki/Probability" title="Probability">Probability</a></li> <li><a href="/wiki/Proposition" title="Proposition">Proposition</a></li> <li><a href="/wiki/Reference" title="Reference">Reference</a></li> <li><a href="/wiki/Statement_(logic)" title="Statement (logic)">Statement</a></li> <li><a href="/wiki/Substitution_(logic)" title="Substitution (logic)">Substitution</a></li> <li><a href="/wiki/Truth" title="Truth">Truth</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%">Lists</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Index_of_logic_articles" title="Index of logic articles">topics</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/List_of_mathematical_logic_topics" title="List of mathematical logic topics">Mathematical logic</a></li> <li><a href="/wiki/List_of_Boolean_algebra_topics" title="List of Boolean algebra topics">Boolean algebra</a></li> <li><a href="/wiki/List_of_set_theory_topics" title="List of set theory topics">Set theory</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;">other</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/List_of_logicians" title="List of logicians">Logicians</a></li> <li><a href="/wiki/List_of_rules_of_inference" title="List of rules of inference">Rules of inference</a></li> <li><a href="/wiki/List_of_paradoxes" title="List of paradoxes">Paradoxes</a></li> <li><a href="/wiki/List_of_fallacies" title="List of fallacies">Fallacies</a></li> <li><a href="/wiki/List_of_logic_symbols" title="List of logic symbols">Logic symbols</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div> <ul><li><span class="nowrap"><span class="noviewer" typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/cd/Socrates.png/18px-Socrates.png" decoding="async" width="18" height="28" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/cd/Socrates.png/27px-Socrates.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/cd/Socrates.png/36px-Socrates.png 2x" data-file-width="326" data-file-height="500" /></span></span> </span><a href="/wiki/Portal:Philosophy" title="Portal:Philosophy">Philosophy portal</a></li> <li><a href="/wiki/Category:Logic" title="Category:Logic">Category</a></li> <li><a href="/wiki/Wikipedia:WikiProject_Logic" title="Wikipedia:WikiProject Logic">WikiProject</a> (<a href="/wiki/Wikipedia_talk:WikiProject_Logic" title="Wikipedia talk:WikiProject Logic">talk</a>)</li> <li><a class="external text" href="https://en.wikipedia.org/w/index.php?title=Special:Recentchangeslinked&target=Template:Logic&hidebots=0">changes</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"></div><div role="navigation" class="navbox" aria-labelledby="Mathematical_logic" style="padding:3px"><table class="nowraplinks mw-collapsible mw-collapsed navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Mathematical_logic" title="Template:Mathematical logic"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Mathematical_logic" title="Template talk:Mathematical logic"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Mathematical_logic" title="Special:EditPage/Template:Mathematical logic"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Mathematical_logic" style="font-size:114%;margin:0 4em"><a class="mw-selflink selflink">Mathematical logic</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">General</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Axiom" title="Axiom">Axiom</a> <ul><li><a href="/wiki/List_of_axioms" title="List of axioms">list</a></li></ul></li> <li><a href="/wiki/Cardinality" title="Cardinality">Cardinality</a></li> <li><a href="/wiki/First-order_logic" title="First-order logic">First-order logic</a></li> <li><a href="/wiki/Formal_proof" title="Formal proof">Formal proof</a></li> <li><a href="/wiki/Formal_semantics_(logic)" class="mw-redirect" title="Formal semantics (logic)">Formal semantics</a></li> <li><a href="/wiki/Foundations_of_mathematics" title="Foundations of mathematics">Foundations of mathematics</a></li> <li><a href="/wiki/Information_theory" title="Information theory">Information theory</a></li> <li><a href="/wiki/Lemma_(mathematics)" title="Lemma (mathematics)">Lemma</a></li> <li><a href="/wiki/Logical_consequence" title="Logical consequence">Logical consequence</a></li> <li><a href="/wiki/Structure_(mathematical_logic)" title="Structure (mathematical logic)">Model</a></li> <li><a href="/wiki/Theorem" title="Theorem">Theorem</a></li> <li><a href="/wiki/Theory_(mathematical_logic)" title="Theory (mathematical logic)">Theory</a></li> <li><a href="/wiki/Type_theory" title="Type theory">Type theory</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Theorems (<a href="/wiki/Category:Theorems_in_the_foundations_of_mathematics" title="Category:Theorems in the foundations of mathematics">list</a>)<br /> and <a href="/wiki/Paradoxes_of_set_theory" title="Paradoxes of set theory">paradoxes</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/G%C3%B6del%27s_completeness_theorem" title="Gödel's completeness theorem">Gödel's completeness</a> and <a href="/wiki/G%C3%B6del%27s_incompleteness_theorems" title="Gödel's incompleteness theorems">incompleteness theorems</a></li> <li><a href="/wiki/Tarski%27s_undefinability_theorem" title="Tarski's undefinability theorem">Tarski's undefinability</a></li> <li><a href="/wiki/Banach%E2%80%93Tarski_paradox" title="Banach–Tarski paradox">Banach–Tarski paradox</a></li> <li>Cantor's <a href="/wiki/Cantor%27s_theorem" title="Cantor's theorem">theorem,</a> <a href="/wiki/Cantor%27s_paradox" title="Cantor's paradox">paradox</a> and <a href="/wiki/Cantor%27s_diagonal_argument" title="Cantor's diagonal argument">diagonal argument</a></li> <li><a href="/wiki/Compactness_theorem" title="Compactness theorem">Compactness</a></li> <li><a href="/wiki/Halting_problem" title="Halting problem">Halting problem</a></li> <li><a href="/wiki/Lindstr%C3%B6m%27s_theorem" title="Lindström's theorem">Lindström's</a></li> <li><a href="/wiki/L%C3%B6wenheim%E2%80%93Skolem_theorem" title="Löwenheim–Skolem theorem">Löwenheim–Skolem</a></li> <li><a href="/wiki/Russell%27s_paradox" title="Russell's paradox">Russell's paradox</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Logic" title="Logic">Logics</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th id="Traditional" scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Term_logic" title="Term logic">Traditional</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Classical_logic" title="Classical logic">Classical logic</a></li> <li><a href="/wiki/Logical_truth" title="Logical truth">Logical truth</a></li> <li><a href="/wiki/Tautology_(logic)" title="Tautology (logic)">Tautology</a></li> <li><a href="/wiki/Proposition" title="Proposition">Proposition</a></li> <li><a href="/wiki/Inference" title="Inference">Inference</a></li> <li><a href="/wiki/Logical_equivalence" title="Logical equivalence">Logical equivalence</a></li> <li><a href="/wiki/Consistency" title="Consistency">Consistency</a> <ul><li><a href="/wiki/Equiconsistency" title="Equiconsistency">Equiconsistency</a></li></ul></li> <li><a href="/wiki/Argument" title="Argument">Argument</a></li> <li><a href="/wiki/Soundness" title="Soundness">Soundness</a></li> <li><a href="/wiki/Validity_(logic)" title="Validity (logic)">Validity</a></li> <li><a href="/wiki/Syllogism" title="Syllogism">Syllogism</a></li> <li><a href="/wiki/Square_of_opposition" title="Square of opposition">Square of opposition</a></li> <li><a href="/wiki/Venn_diagram" title="Venn diagram">Venn diagram</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Propositional_calculus" title="Propositional calculus">Propositional</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Boolean_algebra" title="Boolean algebra">Boolean algebra</a></li> <li><a href="/wiki/Boolean_function" title="Boolean function">Boolean functions</a></li> <li><a href="/wiki/Logical_connective" title="Logical connective">Logical connectives</a></li> <li><a href="/wiki/Propositional_calculus" title="Propositional calculus">Propositional calculus</a></li> <li><a href="/wiki/Propositional_formula" title="Propositional formula">Propositional formula</a></li> <li><a href="/wiki/Truth_table" title="Truth table">Truth tables</a></li> <li><a href="/wiki/Many-valued_logic" title="Many-valued logic">Many-valued logic</a> <ul><li><a href="/wiki/Three-valued_logic" title="Three-valued logic">3</a></li> <li><a href="/wiki/Finite-valued_logic" title="Finite-valued logic">finite</a></li> <li><a href="/wiki/Infinite-valued_logic" title="Infinite-valued logic">∞</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Predicate_logic" class="mw-redirect" title="Predicate logic">Predicate</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/First-order_logic" title="First-order logic">First-order</a> <ul><li><a href="/wiki/List_of_first-order_theories" title="List of first-order theories"><span style="font-size:85%;">list</span></a></li></ul></li> <li><a href="/wiki/Second-order_logic" title="Second-order logic">Second-order</a> <ul><li><a href="/wiki/Monadic_second-order_logic" title="Monadic second-order logic">Monadic</a></li></ul></li> <li><a href="/wiki/Higher-order_logic" title="Higher-order logic">Higher-order</a></li> <li><a href="/wiki/Fixed-point_logic" title="Fixed-point logic">Fixed-point</a></li> <li><a href="/wiki/Free_logic" title="Free logic">Free</a></li> <li><a href="/wiki/Quantifier_(logic)" title="Quantifier (logic)">Quantifiers</a></li> <li><a href="/wiki/Predicate_(mathematical_logic)" title="Predicate (mathematical logic)">Predicate</a></li> <li><a href="/wiki/Monadic_predicate_calculus" title="Monadic predicate calculus">Monadic predicate calculus</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Set_theory" title="Set theory">Set theory</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><td colspan="2" class="navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Zermelo%E2%80%93Fraenkel_set_theory" title="Zermelo–Fraenkel set theory">Set</a> <ul><li><a href="/wiki/Hereditary_set" title="Hereditary set">hereditary</a></li></ul></li> <li><a href="/wiki/Class_(set_theory)" title="Class (set theory)">Class</a></li> <li>(<a href="/wiki/Urelement" title="Urelement">Ur-</a>)<a href="/wiki/Element_(mathematics)" title="Element (mathematics)">Element</a></li> <li><a href="/wiki/Ordinal_number" title="Ordinal number">Ordinal number</a></li> <li><a href="/wiki/Extensionality" title="Extensionality">Extensionality</a></li> <li><a href="/wiki/Forcing_(mathematics)" title="Forcing (mathematics)">Forcing</a></li> <li><a href="/wiki/Relation_(mathematics)" title="Relation (mathematics)">Relation</a> <ul><li><a href="/wiki/Equivalence_relation" title="Equivalence relation">equivalence</a></li> <li><a href="/wiki/Partition_of_a_set" title="Partition of a set">partition</a></li></ul></li> <li>Set operations: <ul><li><a href="/wiki/Intersection_(set_theory)" title="Intersection (set theory)">intersection</a></li> <li><a href="/wiki/Union_(set_theory)" title="Union (set theory)">union</a></li> <li><a href="/wiki/Complement_(set_theory)" title="Complement (set theory)">complement</a></li> <li><a href="/wiki/Cartesian_product" title="Cartesian product">Cartesian product</a></li> <li><a href="/wiki/Power_set" title="Power set">power set</a></li> <li><a href="/wiki/List_of_set_identities_and_relations" title="List of set identities and relations">identities</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Types of <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">sets</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Countable_set" title="Countable set">Countable</a></li> <li><a href="/wiki/Uncountable_set" title="Uncountable set">Uncountable</a></li> <li><a href="/wiki/Empty_set" title="Empty set">Empty</a></li> <li><a href="/wiki/Inhabited_set" title="Inhabited set">Inhabited</a></li> <li><a href="/wiki/Singleton_(mathematics)" title="Singleton (mathematics)">Singleton</a></li> <li><a href="/wiki/Finite_set" title="Finite set">Finite</a></li> <li><a href="/wiki/Infinite_set" title="Infinite set">Infinite</a></li> <li><a href="/wiki/Transitive_set" title="Transitive set">Transitive</a></li> <li><a href="/wiki/Ultrafilter_(set_theory)" class="mw-redirect" title="Ultrafilter (set theory)">Ultrafilter</a></li> <li><a href="/wiki/Recursive_set" class="mw-redirect" title="Recursive set">Recursive</a></li> <li><a href="/wiki/Fuzzy_set" title="Fuzzy set">Fuzzy</a></li> <li><a href="/wiki/Universal_set" title="Universal set">Universal</a></li> <li><a href="/wiki/Universe_(mathematics)" title="Universe (mathematics)">Universe</a> <ul><li><a href="/wiki/Constructible_universe" title="Constructible universe">constructible</a></li> <li><a href="/wiki/Grothendieck_universe" title="Grothendieck universe">Grothendieck</a></li> <li><a href="/wiki/Von_Neumann_universe" title="Von Neumann universe">Von Neumann</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Map_(mathematics)" title="Map (mathematics)">Maps</a> and <a href="/wiki/Cardinality" title="Cardinality">cardinality</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Function_(mathematics)" title="Function (mathematics)">Function</a>/<a href="/wiki/Map_(mathematics)" title="Map (mathematics)">Map</a> <ul><li><a href="/wiki/Domain_of_a_function" title="Domain of a function">domain</a></li> <li><a href="/wiki/Codomain" title="Codomain">codomain</a></li> <li><a href="/wiki/Image_(mathematics)" title="Image (mathematics)">image</a></li></ul></li> <li><a href="/wiki/Injective_function" title="Injective function">In</a>/<a href="/wiki/Surjective_function" title="Surjective function">Sur</a>/<a href="/wiki/Bijection" title="Bijection">Bi</a>-jection</li> <li><a href="/wiki/Schr%C3%B6der%E2%80%93Bernstein_theorem" title="Schröder–Bernstein theorem">Schröder–Bernstein theorem</a></li> <li><a href="/wiki/Isomorphism" title="Isomorphism">Isomorphism</a></li> <li><a href="/wiki/G%C3%B6del_numbering" title="Gödel numbering">Gödel numbering</a></li> <li><a href="/wiki/Enumeration" title="Enumeration">Enumeration</a></li> <li><a href="/wiki/Large_cardinal" title="Large cardinal">Large cardinal</a> <ul><li><a href="/wiki/Inaccessible_cardinal" title="Inaccessible cardinal">inaccessible</a></li></ul></li> <li><a href="/wiki/Aleph_number" title="Aleph number">Aleph number</a></li> <li><a href="/wiki/Operation_(mathematics)" title="Operation (mathematics)">Operation</a> <ul><li><a href="/wiki/Binary_operation" title="Binary operation">binary</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Set theories</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Zermelo%E2%80%93Fraenkel_set_theory" title="Zermelo–Fraenkel set theory">Zermelo–Fraenkel</a> <ul><li><a href="/wiki/Axiom_of_choice" title="Axiom of choice">axiom of choice</a></li> <li><a href="/wiki/Continuum_hypothesis" title="Continuum hypothesis">continuum hypothesis</a></li></ul></li> <li><a href="/wiki/General_set_theory" title="General set theory">General</a></li> <li><a href="/wiki/Kripke%E2%80%93Platek_set_theory" title="Kripke–Platek set theory">Kripke–Platek</a></li> <li><a href="/wiki/Morse%E2%80%93Kelley_set_theory" title="Morse–Kelley set theory">Morse–Kelley</a></li> <li><a href="/wiki/Naive_set_theory" title="Naive set theory">Naive</a></li> <li><a href="/wiki/New_Foundations" title="New Foundations">New Foundations</a></li> <li><a href="/wiki/Tarski%E2%80%93Grothendieck_set_theory" title="Tarski–Grothendieck set theory">Tarski–Grothendieck</a></li> <li><a href="/wiki/Von_Neumann%E2%80%93Bernays%E2%80%93G%C3%B6del_set_theory" title="Von Neumann–Bernays–Gödel set theory">Von Neumann–Bernays–Gödel</a></li> <li><a href="/wiki/Ackermann_set_theory" title="Ackermann set theory">Ackermann</a></li> <li><a href="/wiki/Constructive_set_theory" title="Constructive set theory">Constructive</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Formal_system" title="Formal system">Formal systems</a> (<a href="/wiki/List_of_formal_systems" title="List of formal systems"><span style="font-size:85%;">list</span></a>),<br /><a href="/wiki/Formal_language" title="Formal language">language</a> and <a href="/wiki/Syntax_(logic)" title="Syntax (logic)">syntax</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><td colspan="2" class="navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Alphabet_(formal_languages)" title="Alphabet (formal languages)">Alphabet</a></li> <li><a href="/wiki/Arity" title="Arity">Arity</a></li> <li><a href="/wiki/Automata_theory" title="Automata theory">Automata</a></li> <li><a href="/wiki/Axiom_schema" title="Axiom schema">Axiom schema</a></li> <li><a href="/wiki/Expression_(mathematics)" title="Expression (mathematics)">Expression</a> <ul><li><a href="/wiki/Ground_expression" title="Ground expression">ground</a></li></ul></li> <li><a href="/wiki/Extension_by_new_constant_and_function_names" title="Extension by new constant and function names">Extension</a> <ul><li><a href="/wiki/Extension_by_definitions" title="Extension by definitions">by definition</a></li> <li><a href="/wiki/Conservative_extension" title="Conservative extension">conservative</a></li></ul></li> <li><a href="/wiki/Finitary_relation" title="Finitary relation">Relation</a></li> <li><a href="/wiki/Formation_rule" title="Formation rule">Formation rule</a></li> <li><a href="/wiki/Formal_grammar" title="Formal grammar">Grammar</a></li> <li><a href="/wiki/Well-formed_formula" title="Well-formed formula">Formula</a> <ul><li><a href="/wiki/Atomic_formula" title="Atomic formula">atomic</a></li> <li><a href="/wiki/Sentence_(mathematical_logic)" title="Sentence (mathematical logic)">closed</a></li> <li><a href="/wiki/Ground_formula" class="mw-redirect" title="Ground formula">ground</a></li> <li><a href="/wiki/Open_formula" title="Open formula">open</a></li></ul></li> <li><a href="/wiki/Free_variables_and_bound_variables" title="Free variables and bound variables">Free/bound variable</a></li> <li><a href="/wiki/Formal_language" title="Formal language">Language</a></li> <li><a href="/wiki/Metalanguage" title="Metalanguage">Metalanguage</a></li> <li><a href="/wiki/Logical_connective" title="Logical connective">Logical connective</a> <ul><li><a href="/wiki/Negation" title="Negation">¬</a></li> <li><a href="/wiki/Logical_disjunction" title="Logical disjunction">∨</a></li> <li><a href="/wiki/Logical_conjunction" title="Logical conjunction">∧</a></li> <li><a href="/wiki/Material_conditional" title="Material conditional">→</a></li> <li><a href="/wiki/Logical_biconditional" title="Logical biconditional">↔</a></li> <li><a href="/wiki/Logical_equality" title="Logical equality">=</a></li></ul></li> <li><a href="/wiki/Predicate_(mathematical_logic)" title="Predicate (mathematical logic)">Predicate</a> <ul><li><a href="/wiki/Functional_predicate" title="Functional predicate">functional</a></li> <li><a href="/wiki/Predicate_variable" title="Predicate variable">variable</a></li> <li><a href="/wiki/Propositional_variable" title="Propositional variable">propositional variable</a></li></ul></li> <li><a href="/wiki/Formal_proof" title="Formal proof">Proof</a></li> <li><a href="/wiki/Quantifier_(logic)" title="Quantifier (logic)">Quantifier</a> <ul><li><a href="/wiki/Existential_quantification" title="Existential quantification">∃</a></li> <li><a href="/wiki/Uniqueness_quantification" title="Uniqueness quantification">!</a></li> <li><a href="/wiki/Universal_quantification" title="Universal quantification">∀</a></li> <li><a href="/wiki/Quantifier_rank" title="Quantifier rank">rank</a></li></ul></li> <li><a href="/wiki/Sentence_(mathematical_logic)" title="Sentence (mathematical logic)">Sentence</a> <ul><li><a href="/wiki/Atomic_sentence" title="Atomic sentence">atomic</a></li> <li><a href="/wiki/Spectrum_of_a_sentence" title="Spectrum of a sentence">spectrum</a></li></ul></li> <li><a href="/wiki/Signature_(logic)" title="Signature (logic)">Signature</a></li> <li><a href="/wiki/String_(formal_languages)" class="mw-redirect" title="String (formal languages)">String</a></li> <li><a href="/wiki/Substitution_(logic)" title="Substitution (logic)">Substitution</a></li> <li><a href="/wiki/Symbol_(formal)" title="Symbol (formal)">Symbol</a> <ul><li><a href="/wiki/Uninterpreted_function" title="Uninterpreted function">function</a></li> <li><a href="/wiki/Logical_constant" title="Logical constant">logical/constant</a></li> <li><a href="/wiki/Non-logical_symbol" title="Non-logical symbol">non-logical</a></li> <li><a href="/wiki/Variable_(mathematics)" title="Variable (mathematics)">variable</a></li></ul></li> <li><a href="/wiki/Term_(logic)" title="Term (logic)">Term</a></li> <li><a href="/wiki/Theory_(mathematical_logic)" title="Theory (mathematical logic)">Theory</a> <ul><li><a href="/wiki/List_of_mathematical_theories" title="List of mathematical theories"><span style="font-size:85%;">list</span></a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><span class="nowrap">Example <a href="/wiki/Axiomatic_system" title="Axiomatic system">axiomatic<br />systems</a> <span style="font-size:85%;">(<a href="/wiki/List_of_first-order_theories" title="List of first-order theories">list</a>)</span></span></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li>of <a href="/wiki/True_arithmetic" title="True arithmetic">arithmetic</a>: <ul><li><a href="/wiki/Peano_axioms" title="Peano axioms">Peano</a></li> <li><a href="/wiki/Second-order_arithmetic" title="Second-order arithmetic">second-order</a></li> <li><a href="/wiki/Elementary_function_arithmetic" title="Elementary function arithmetic">elementary function</a></li> <li><a href="/wiki/Primitive_recursive_arithmetic" title="Primitive recursive arithmetic">primitive recursive</a></li> <li><a href="/wiki/Robinson_arithmetic" title="Robinson arithmetic">Robinson</a></li> <li><a href="/wiki/Skolem_arithmetic" title="Skolem arithmetic">Skolem</a></li></ul></li> <li>of the <a href="/wiki/Construction_of_the_real_numbers" title="Construction of the real numbers">real numbers</a> <ul><li><a href="/wiki/Tarski%27s_axiomatization_of_the_reals" title="Tarski's axiomatization of the reals">Tarski's axiomatization</a></li></ul></li> <li>of <a href="/wiki/Axiomatization_of_Boolean_algebras" class="mw-redirect" title="Axiomatization of Boolean algebras">Boolean algebras</a> <ul><li><a href="/wiki/Boolean_algebras_canonically_defined" title="Boolean algebras canonically defined">canonical</a></li> <li><a href="/wiki/Minimal_axioms_for_Boolean_algebra" title="Minimal axioms for Boolean algebra">minimal axioms</a></li></ul></li> <li>of <a href="/wiki/Foundations_of_geometry" title="Foundations of geometry">geometry</a>: <ul><li><a href="/wiki/Euclidean_geometry" title="Euclidean geometry">Euclidean</a>: <ul><li><a href="/wiki/Euclid%27s_Elements" title="Euclid's Elements"><i>Elements</i></a></li> <li><a href="/wiki/Hilbert%27s_axioms" title="Hilbert's axioms">Hilbert's</a></li> <li><a href="/wiki/Tarski%27s_axioms" title="Tarski's axioms">Tarski's</a></li></ul></li> <li><a href="/wiki/Non-Euclidean_geometry" title="Non-Euclidean geometry">non-Euclidean</a></li></ul></li></ul> <ul><li><i><a href="/wiki/Principia_Mathematica" title="Principia Mathematica">Principia Mathematica</a></i></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Proof_theory" title="Proof theory">Proof theory</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Formal_proof" title="Formal proof">Formal proof</a></li> <li><a href="/wiki/Natural_deduction" title="Natural deduction">Natural deduction</a></li> <li><a href="/wiki/Logical_consequence" title="Logical consequence">Logical consequence</a></li> <li><a href="/wiki/Rule_of_inference" title="Rule of inference">Rule of inference</a></li> <li><a href="/wiki/Sequent_calculus" title="Sequent calculus">Sequent calculus</a></li> <li><a href="/wiki/Theorem" title="Theorem">Theorem</a></li> <li><a href="/wiki/Formal_system" title="Formal system">Systems</a> <ul><li><a href="/wiki/Axiomatic_system" title="Axiomatic system">axiomatic</a></li> <li><a href="/wiki/Deductive_system" class="mw-redirect" title="Deductive system">deductive</a></li> <li><a href="/wiki/Hilbert_system" title="Hilbert system">Hilbert</a> <ul><li><a href="/wiki/List_of_Hilbert_systems" class="mw-redirect" title="List of Hilbert systems">list</a></li></ul></li></ul></li> <li><a href="/wiki/Complete_theory" title="Complete theory">Complete theory</a></li> <li><a href="/wiki/Independence_(mathematical_logic)" title="Independence (mathematical logic)">Independence</a> (<a href="/wiki/List_of_statements_independent_of_ZFC" title="List of statements independent of ZFC">from ZFC</a>)</li> <li><a href="/wiki/Proof_of_impossibility" title="Proof of impossibility">Proof of impossibility</a></li> <li><a href="/wiki/Ordinal_analysis" title="Ordinal analysis">Ordinal analysis</a></li> <li><a href="/wiki/Reverse_mathematics" title="Reverse mathematics">Reverse mathematics</a></li> <li><a href="/wiki/Self-verifying_theories" title="Self-verifying theories">Self-verifying theories</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Model_theory" title="Model theory">Model theory</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Interpretation_(logic)" title="Interpretation (logic)">Interpretation</a> <ul><li><a href="/wiki/Interpretation_function" class="mw-redirect" title="Interpretation function">function</a></li> <li><a href="/wiki/Interpretation_(model_theory)" title="Interpretation (model theory)">of models</a></li></ul></li> <li><a href="/wiki/Structure_(mathematical_logic)" title="Structure (mathematical logic)">Model</a> <ul><li><a href="/wiki/Elementary_equivalence" title="Elementary equivalence">equivalence</a></li> <li><a href="/wiki/Finite_model_theory" title="Finite model theory">finite</a></li> <li><a href="/wiki/Saturated_model" title="Saturated model">saturated</a></li> <li><a href="/wiki/Spectrum_of_a_theory" title="Spectrum of a theory">spectrum</a></li> <li><a href="/wiki/Substructure_(mathematics)" title="Substructure (mathematics)">submodel</a></li></ul></li> <li><a href="/wiki/Non-standard_model" title="Non-standard model">Non-standard model</a> <ul><li><a href="/wiki/Non-standard_model_of_arithmetic" title="Non-standard model of arithmetic">of arithmetic</a></li></ul></li> <li><a href="/wiki/Diagram_(mathematical_logic)" title="Diagram (mathematical logic)">Diagram</a> <ul><li><a href="/wiki/Elementary_diagram" title="Elementary diagram">elementary</a></li></ul></li> <li><a href="/wiki/Categorical_theory" title="Categorical theory">Categorical theory</a></li> <li><a href="/wiki/Model_complete_theory" title="Model complete theory">Model complete theory</a></li> <li><a href="/wiki/Satisfiability" title="Satisfiability">Satisfiability</a></li> <li><a href="/wiki/Semantics_of_logic" title="Semantics of logic">Semantics of logic</a></li> <li><a href="/wiki/Strength_(mathematical_logic)" title="Strength (mathematical logic)">Strength</a></li> <li><a href="/wiki/Theories_of_truth" class="mw-redirect" title="Theories of truth">Theories of truth</a> <ul><li><a href="/wiki/Semantic_theory_of_truth" title="Semantic theory of truth">semantic</a></li> <li><a href="/wiki/Tarski%27s_theory_of_truth" class="mw-redirect" title="Tarski's theory of truth">Tarski's</a></li> <li><a href="/wiki/Kripke%27s_theory_of_truth" class="mw-redirect" title="Kripke's theory of truth">Kripke's</a></li></ul></li> <li><a href="/wiki/T-schema" title="T-schema">T-schema</a></li> <li><a href="/wiki/Transfer_principle" title="Transfer principle">Transfer principle</a></li> <li><a href="/wiki/Truth_predicate" title="Truth predicate">Truth predicate</a></li> <li><a href="/wiki/Truth_value" title="Truth value">Truth value</a></li> <li><a href="/wiki/Type_(model_theory)" title="Type (model theory)">Type</a></li> <li><a href="/wiki/Ultraproduct" title="Ultraproduct">Ultraproduct</a></li> <li><a href="/wiki/Validity_(logic)" title="Validity (logic)">Validity</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Computability_theory" title="Computability theory">Computability theory</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Church_encoding" title="Church encoding">Church encoding</a></li> <li><a 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" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/24px-Nuvola_apps_edu_mathematics_blue-p.svg.png 1.5x, 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href="https://d-nb.info/gnd/4037951-6">Germany</a></span></li><li><span class="uid"><span class="rt-commentedText tooltip tooltip-dotted" title="Algebraic logic"><a rel="nofollow" class="external text" href="https://id.loc.gov/authorities/sh85003435">United States</a></span></span></li><li><span class="uid"><span class="rt-commentedText tooltip tooltip-dotted" title="Logique mathématique"><a rel="nofollow" class="external text" href="https://catalogue.bnf.fr/ark:/12148/cb11965690r">France</a></span></span></li><li><span class="uid"><span class="rt-commentedText tooltip tooltip-dotted" title="Logique mathématique"><a rel="nofollow" class="external text" href="https://data.bnf.fr/ark:/12148/cb11965690r">BnF data</a></span></span></li><li><span class="uid"><a rel="nofollow" class="external text" href="https://id.ndl.go.jp/auth/ndlna/00565709">Japan</a></span></li><li><span class="uid"><span class="rt-commentedText tooltip tooltip-dotted" title="symbolická logika"><a rel="nofollow" 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