CINXE.COM

<!DOCTYPE html> <html class="client-nojs vector-feature-language-in-header-enabled vector-feature-language-in-main-page-header-disabled vector-feature-sticky-header-disabled vector-feature-page-tools-pinned-disabled vector-feature-toc-pinned-clientpref-1 vector-feature-main-menu-pinned-disabled vector-feature-limited-width-clientpref-1 vector-feature-limited-width-content-enabled vector-feature-custom-font-size-clientpref-1 vector-feature-appearance-pinned-clientpref-1 vector-feature-night-mode-enabled skin-theme-clientpref-day vector-toc-available" lang="en" dir="ltr"> <head> <meta charset="UTF-8"> <title>Peano axioms - Wikipedia</title> <script>(function(){var className="client-js vector-feature-language-in-header-enabled vector-feature-language-in-main-page-header-disabled vector-feature-sticky-header-disabled vector-feature-page-tools-pinned-disabled vector-feature-toc-pinned-clientpref-1 vector-feature-main-menu-pinned-disabled vector-feature-limited-width-clientpref-1 vector-feature-limited-width-content-enabled vector-feature-custom-font-size-clientpref-1 vector-feature-appearance-pinned-clientpref-1 vector-feature-night-mode-enabled skin-theme-clientpref-day vector-toc-available";var cookie=document.cookie.match(/(?:^|; )enwikimwclientpreferences=([^;]+)/);if(cookie){cookie[1].split('%2C').forEach(function(pref){className=className.replace(new RegExp('(^| )'+pref.replace(/-clientpref-\w+$|[^\w-]+/g,'')+'-clientpref-\\w+( |$)'),'$1'+pref+'$2');});}document.documentElement.className=className;}());RLCONF={"wgBreakFrames":false,"wgSeparatorTransformTable":["",""],"wgDigitTransformTable":["",""],"wgDefaultDateFormat":"dmy", "wgMonthNames":["","January","February","March","April","May","June","July","August","September","October","November","December"],"wgRequestId":"6ae2d4de-7576-4065-b31d-1c8429733cfb","wgCanonicalNamespace":"","wgCanonicalSpecialPageName":false,"wgNamespaceNumber":0,"wgPageName":"Peano_axioms","wgTitle":"Peano axioms","wgCurRevisionId":1255254109,"wgRevisionId":1255254109,"wgArticleId":25005,"wgIsArticle":true,"wgIsRedirect":false,"wgAction":"view","wgUserName":null,"wgUserGroups":["*"],"wgCategories":["Webarchive template wayback links","Articles with short description","Short description is different from Wikidata","Pages with Italian IPA","Articles containing Latin-language text","Articles needing additional references from May 2024","All articles needing additional references","All articles with unsourced statements","Articles with unsourced statements from January 2022","Articles with unsourced statements from May 2024","Articles containing German-language text", "CS1 maint: multiple names: authors list","CS1 maint: location missing publisher","Articles with Internet Encyclopedia of Philosophy links","Wikipedia articles incorporating text from PlanetMath","Pages that use a deprecated format of the math tags","1889 introductions","Mathematical axioms","Formal theories of arithmetic","Logic in computer science","Mathematical logic"],"wgPageViewLanguage":"en","wgPageContentLanguage":"en","wgPageContentModel":"wikitext","wgRelevantPageName":"Peano_axioms","wgRelevantArticleId":25005,"wgIsProbablyEditable":true,"wgRelevantPageIsProbablyEditable":true,"wgRestrictionEdit":[],"wgRestrictionMove":[],"wgRedirectedFrom":"Peano_arithmetic","wgNoticeProject":"wikipedia","wgCiteReferencePreviewsActive":false,"wgFlaggedRevsParams":{"tags":{"status":{"levels":1}}},"wgMediaViewerOnClick":true,"wgMediaViewerEnabledByDefault":true,"wgPopupsFlags":0,"wgVisualEditor":{"pageLanguageCode":"en","pageLanguageDir":"ltr","pageVariantFallbacks":"en"}, "wgMFDisplayWikibaseDescriptions":{"search":true,"watchlist":true,"tagline":false,"nearby":true},"wgWMESchemaEditAttemptStepOversample":false,"wgWMEPageLength":50000,"wgInternalRedirectTargetUrl":"/wiki/Peano_axioms","wgRelatedArticlesCompat":[],"wgCentralAuthMobileDomain":false,"wgEditSubmitButtonLabelPublish":true,"wgULSPosition":"interlanguage","wgULSisCompactLinksEnabled":false,"wgVector2022LanguageInHeader":true,"wgULSisLanguageSelectorEmpty":false,"wgWikibaseItemId":"Q842755","wgCheckUserClientHintsHeadersJsApi":["brands","architecture","bitness","fullVersionList","mobile","model","platform","platformVersion"],"GEHomepageSuggestedEditsEnableTopics":true,"wgGETopicsMatchModeEnabled":false,"wgGEStructuredTaskRejectionReasonTextInputEnabled":false,"wgGELevelingUpEnabledForUser":false};RLSTATE={"ext.globalCssJs.user.styles":"ready","site.styles":"ready","user.styles":"ready","ext.globalCssJs.user":"ready","user":"ready","user.options":"loading","ext.cite.styles":"ready", "ext.math.styles":"ready","skins.vector.search.codex.styles":"ready","skins.vector.styles":"ready","skins.vector.icons":"ready","ext.wikimediamessages.styles":"ready","ext.visualEditor.desktopArticleTarget.noscript":"ready","ext.uls.interlanguage":"ready","wikibase.client.init":"ready","ext.wikimediaBadges":"ready"};RLPAGEMODULES=["mediawiki.action.view.redirect","ext.cite.ux-enhancements","mediawiki.page.media","ext.scribunto.logs","site","mediawiki.page.ready","mediawiki.toc","skins.vector.js","ext.centralNotice.geoIP","ext.centralNotice.startUp","ext.gadget.ReferenceTooltips","ext.gadget.switcher","ext.urlShortener.toolbar","ext.centralauth.centralautologin","mmv.bootstrap","ext.popups","ext.visualEditor.desktopArticleTarget.init","ext.visualEditor.targetLoader","ext.echo.centralauth","ext.eventLogging","ext.wikimediaEvents","ext.navigationTiming","ext.uls.interface","ext.cx.eventlogging.campaigns","ext.cx.uls.quick.actions","wikibase.client.vector-2022","ext.checkUser.clientHints", "ext.quicksurveys.init","ext.growthExperiments.SuggestedEditSession","wikibase.sidebar.tracking"];</script> <script>(RLQ=window.RLQ||[]).push(function(){mw.loader.impl(function(){return["user.options@12s5i",function($,jQuery,require,module){mw.user.tokens.set({"patrolToken":"+\\","watchToken":"+\\","csrfToken":"+\\"}); }];});});</script> <link rel="stylesheet" href="/w/load.php?lang=en&modules=ext.cite.styles%7Cext.math.styles%7Cext.uls.interlanguage%7Cext.visualEditor.desktopArticleTarget.noscript%7Cext.wikimediaBadges%7Cext.wikimediamessages.styles%7Cskins.vector.icons%2Cstyles%7Cskins.vector.search.codex.styles%7Cwikibase.client.init&only=styles&skin=vector-2022"> <script async="" src="/w/load.php?lang=en&modules=startup&only=scripts&raw=1&skin=vector-2022"></script> <meta name="ResourceLoaderDynamicStyles" content=""> <link rel="stylesheet" href="/w/load.php?lang=en&modules=site.styles&only=styles&skin=vector-2022"> <meta name="generator" content="MediaWiki 1.44.0-wmf.4"> <meta name="referrer" content="origin"> <meta name="referrer" content="origin-when-cross-origin"> <meta name="robots" content="max-image-preview:standard"> <meta name="format-detection" content="telephone=no"> <meta name="viewport" content="width=1120"> <meta property="og:title" content="Peano axioms - Wikipedia"> <meta property="og:type" content="website"> <link rel="preconnect" href="//upload.wikimedia.org"> <link rel="alternate" media="only screen and (max-width: 640px)" href="//en.m.wikipedia.org/wiki/Peano_axioms"> <link rel="alternate" type="application/x-wiki" title="Edit this page" href="/w/index.php?title=Peano_axioms&action=edit"> <link rel="apple-touch-icon" href="/static/apple-touch/wikipedia.png"> <link rel="icon" href="/static/favicon/wikipedia.ico"> <link rel="search" type="application/opensearchdescription+xml" href="/w/rest.php/v1/search" title="Wikipedia (en)"> <link rel="EditURI" type="application/rsd+xml" href="//en.wikipedia.org/w/api.php?action=rsd"> <link rel="canonical" href="https://en.wikipedia.org/wiki/Peano_axioms"> <link rel="license" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en"> <link rel="alternate" type="application/atom+xml" title="Wikipedia Atom feed" href="/w/index.php?title=Special:RecentChanges&feed=atom"> <link rel="dns-prefetch" href="//meta.wikimedia.org" /> <link rel="dns-prefetch" href="//login.wikimedia.org"> </head> <body class="skin--responsive skin-vector skin-vector-search-vue mediawiki ltr sitedir-ltr mw-hide-empty-elt ns-0 ns-subject mw-editable page-Peano_axioms rootpage-Peano_axioms skin-vector-2022 action-view"><a class="mw-jump-link" href="#bodyContent">Jump to content</a> <div class="vector-header-container"> <header class="vector-header mw-header"> <div class="vector-header-start"> <nav class="vector-main-menu-landmark" aria-label="Site"> <div id="vector-main-menu-dropdown" class="vector-dropdown vector-main-menu-dropdown vector-button-flush-left vector-button-flush-right" > <input type="checkbox" id="vector-main-menu-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-main-menu-dropdown" class="vector-dropdown-checkbox " aria-label="Main menu" > <label id="vector-main-menu-dropdown-label" for="vector-main-menu-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-menu mw-ui-icon-wikimedia-menu"></span> <span class="vector-dropdown-label-text">Main menu</span> </label> <div class="vector-dropdown-content"> <div id="vector-main-menu-unpinned-container" class="vector-unpinned-container"> <div id="vector-main-menu" class="vector-main-menu vector-pinnable-element"> <div class="vector-pinnable-header vector-main-menu-pinnable-header vector-pinnable-header-unpinned" data-feature-name="main-menu-pinned" data-pinnable-element-id="vector-main-menu" data-pinned-container-id="vector-main-menu-pinned-container" data-unpinned-container-id="vector-main-menu-unpinned-container" > <div class="vector-pinnable-header-label">Main menu</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-main-menu.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-main-menu.unpin">hide</button> </div> <div id="p-navigation" class="vector-menu mw-portlet mw-portlet-navigation" > <div class="vector-menu-heading"> Navigation </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-mainpage-description" class="mw-list-item"><a href="/wiki/Main_Page" title="Visit the main page [z]" accesskey="z"><span>Main page</span></a></li><li id="n-contents" class="mw-list-item"><a href="/wiki/Wikipedia:Contents" title="Guides to browsing Wikipedia"><span>Contents</span></a></li><li id="n-currentevents" class="mw-list-item"><a href="/wiki/Portal:Current_events" title="Articles related to current events"><span>Current events</span></a></li><li id="n-randompage" class="mw-list-item"><a href="/wiki/Special:Random" title="Visit a randomly selected article [x]" accesskey="x"><span>Random article</span></a></li><li id="n-aboutsite" class="mw-list-item"><a href="/wiki/Wikipedia:About" title="Learn about Wikipedia and how it works"><span>About Wikipedia</span></a></li><li id="n-contactpage" class="mw-list-item"><a href="//en.wikipedia.org/wiki/Wikipedia:Contact_us" title="How to contact Wikipedia"><span>Contact us</span></a></li> </ul> </div> </div> <div id="p-interaction" class="vector-menu mw-portlet mw-portlet-interaction" > <div class="vector-menu-heading"> Contribute </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-help" class="mw-list-item"><a href="/wiki/Help:Contents" title="Guidance on how to use and edit Wikipedia"><span>Help</span></a></li><li id="n-introduction" class="mw-list-item"><a href="/wiki/Help:Introduction" title="Learn how to edit Wikipedia"><span>Learn to edit</span></a></li><li id="n-portal" class="mw-list-item"><a href="/wiki/Wikipedia:Community_portal" title="The hub for editors"><span>Community portal</span></a></li><li id="n-recentchanges" class="mw-list-item"><a href="/wiki/Special:RecentChanges" title="A list of recent changes to Wikipedia [r]" accesskey="r"><span>Recent changes</span></a></li><li id="n-upload" class="mw-list-item"><a href="/wiki/Wikipedia:File_upload_wizard" title="Add images or other media for use on Wikipedia"><span>Upload file</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> <a href="/wiki/Main_Page" class="mw-logo"> <img class="mw-logo-icon" src="/static/images/icons/wikipedia.png" alt="" aria-hidden="true" height="50" width="50"> <span class="mw-logo-container skin-invert"> <img class="mw-logo-wordmark" alt="Wikipedia" src="/static/images/mobile/copyright/wikipedia-wordmark-en.svg" style="width: 7.5em; height: 1.125em;"> <img class="mw-logo-tagline" alt="The Free Encyclopedia" src="/static/images/mobile/copyright/wikipedia-tagline-en.svg" width="117" height="13" style="width: 7.3125em; height: 0.8125em;"> </span> </a> </div> <div class="vector-header-end"> <div id="p-search" role="search" class="vector-search-box-vue vector-search-box-collapses vector-search-box-show-thumbnail vector-search-box-auto-expand-width vector-search-box"> <a href="/wiki/Special:Search" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only search-toggle" title="Search Wikipedia [f]" accesskey="f"><span class="vector-icon mw-ui-icon-search mw-ui-icon-wikimedia-search"></span> <span>Search</span> </a> <div class="vector-typeahead-search-container"> <div class="cdx-typeahead-search cdx-typeahead-search--show-thumbnail cdx-typeahead-search--auto-expand-width"> <form action="/w/index.php" id="searchform" class="cdx-search-input cdx-search-input--has-end-button"> <div id="simpleSearch" class="cdx-search-input__input-wrapper" data-search-loc="header-moved"> <div class="cdx-text-input cdx-text-input--has-start-icon"> <input class="cdx-text-input__input" type="search" name="search" placeholder="Search Wikipedia" aria-label="Search Wikipedia" autocapitalize="sentences" title="Search Wikipedia [f]" accesskey="f" id="searchInput" > <span class="cdx-text-input__icon cdx-text-input__start-icon"></span> </div> <input type="hidden" name="title" value="Special:Search"> </div> <button class="cdx-button cdx-search-input__end-button">Search</button> </form> </div> </div> </div> <nav class="vector-user-links vector-user-links-wide" aria-label="Personal tools"> <div class="vector-user-links-main"> <div id="p-vector-user-menu-preferences" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <div id="p-vector-user-menu-userpage" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <nav class="vector-appearance-landmark" aria-label="Appearance"> <div id="vector-appearance-dropdown" class="vector-dropdown " title="Change the appearance of the page's font size, width, and color" > <input type="checkbox" id="vector-appearance-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-appearance-dropdown" class="vector-dropdown-checkbox " aria-label="Appearance" > <label id="vector-appearance-dropdown-label" for="vector-appearance-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-appearance mw-ui-icon-wikimedia-appearance"></span> <span class="vector-dropdown-label-text">Appearance</span> </label> <div class="vector-dropdown-content"> <div id="vector-appearance-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <div id="p-vector-user-menu-notifications" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <div id="p-vector-user-menu-overflow" class="vector-menu mw-portlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="https://donate.wikimedia.org/wiki/Special:FundraiserRedirector?utm_source=donate&utm_medium=sidebar&utm_campaign=C13_en.wikipedia.org&uselang=en" class=""><span>Donate</span></a> </li> <li id="pt-createaccount-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=Special:CreateAccount&returnto=Peano+axioms" title="You are encouraged to create an account and log in; however, it is not mandatory" class=""><span>Create account</span></a> </li> <li id="pt-login-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=Special:UserLogin&returnto=Peano+axioms" title="You're encouraged to log in; however, it's not mandatory. [o]" accesskey="o" class=""><span>Log in</span></a> </li> </ul> </div> </div> </div> <div id="vector-user-links-dropdown" class="vector-dropdown vector-user-menu vector-button-flush-right vector-user-menu-logged-out" title="Log in and more options" > <input type="checkbox" id="vector-user-links-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-user-links-dropdown" class="vector-dropdown-checkbox " aria-label="Personal tools" > <label id="vector-user-links-dropdown-label" for="vector-user-links-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-ellipsis mw-ui-icon-wikimedia-ellipsis"></span> <span class="vector-dropdown-label-text">Personal tools</span> </label> <div class="vector-dropdown-content"> <div id="p-personal" class="vector-menu mw-portlet mw-portlet-personal user-links-collapsible-item" title="User menu" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport" class="user-links-collapsible-item mw-list-item"><a href="https://donate.wikimedia.org/wiki/Special:FundraiserRedirector?utm_source=donate&utm_medium=sidebar&utm_campaign=C13_en.wikipedia.org&uselang=en"><span>Donate</span></a></li><li id="pt-createaccount" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Special:CreateAccount&returnto=Peano+axioms" title="You are encouraged to create an account and log in; however, it is not mandatory"><span class="vector-icon mw-ui-icon-userAdd mw-ui-icon-wikimedia-userAdd"></span> <span>Create account</span></a></li><li id="pt-login" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Special:UserLogin&returnto=Peano+axioms" title="You're encouraged to log in; however, it's not mandatory. [o]" accesskey="o"><span class="vector-icon mw-ui-icon-logIn mw-ui-icon-wikimedia-logIn"></span> <span>Log in</span></a></li> </ul> </div> </div> <div id="p-user-menu-anon-editor" class="vector-menu mw-portlet mw-portlet-user-menu-anon-editor" > <div class="vector-menu-heading"> Pages for logged out editors <a href="/wiki/Help:Introduction" aria-label="Learn more about editing"><span>learn more</span></a> </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-anoncontribs" class="mw-list-item"><a href="/wiki/Special:MyContributions" title="A list of edits made from this IP address [y]" accesskey="y"><span>Contributions</span></a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/Special:MyTalk" title="Discussion about edits from this IP address [n]" accesskey="n"><span>Talk</span></a></li> </ul> </div> </div> </div> </div> </nav> </div> </header> </div> <div class="mw-page-container"> <div class="mw-page-container-inner"> <div class="vector-sitenotice-container"> <div id="siteNotice"></div> </div> <div class="vector-column-start"> <div class="vector-main-menu-container"> <div id="mw-navigation"> <nav id="mw-panel" class="vector-main-menu-landmark" aria-label="Site"> <div id="vector-main-menu-pinned-container" class="vector-pinned-container"> </div> </nav> </div> </div> <div class="vector-sticky-pinned-container"> <nav id="mw-panel-toc" aria-label="Contents" data-event-name="ui.sidebar-toc" class="mw-table-of-contents-container vector-toc-landmark"> <div id="vector-toc-pinned-container" class="vector-pinned-container"> <div id="vector-toc" class="vector-toc vector-pinnable-element"> <div class="vector-pinnable-header vector-toc-pinnable-header vector-pinnable-header-pinned" data-feature-name="toc-pinned" data-pinnable-element-id="vector-toc" > <h2 class="vector-pinnable-header-label">Contents</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">hide</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">(Top)</div> </a> </li> <li id="toc-Historical_second-order_formulation" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Historical_second-order_formulation"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Historical second-order formulation</span> </div> </a> <button aria-controls="toc-Historical_second-order_formulation-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 Historical second-order formulation subsection</span> </button> <ul id="toc-Historical_second-order_formulation-sublist" class="vector-toc-list"> <li id="toc-Defining_arithmetic_operations_and_relations" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Defining_arithmetic_operations_and_relations"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>Defining arithmetic operations and relations</span> </div> </a> <ul id="toc-Defining_arithmetic_operations_and_relations-sublist" class="vector-toc-list"> <li id="toc-Addition" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Addition"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1.1</span> <span>Addition</span> </div> </a> <ul id="toc-Addition-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Multiplication" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Multiplication"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1.2</span> <span>Multiplication</span> </div> </a> <ul id="toc-Multiplication-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Inequalities" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Inequalities"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1.3</span> <span>Inequalities</span> </div> </a> <ul id="toc-Inequalities-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Models" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Models"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2</span> <span>Models</span> </div> </a> <ul id="toc-Models-sublist" class="vector-toc-list"> <li id="toc-Set-theoretic_models" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Set-theoretic_models"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2.1</span> <span>Set-theoretic models</span> </div> </a> <ul id="toc-Set-theoretic_models-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Interpretation_in_category_theory" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Interpretation_in_category_theory"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2.2</span> <span>Interpretation in category theory</span> </div> </a> <ul id="toc-Interpretation_in_category_theory-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Consistency" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Consistency"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.3</span> <span>Consistency</span> </div> </a> <ul id="toc-Consistency-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Peano_arithmetic_as_first-order_theory" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Peano_arithmetic_as_first-order_theory"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Peano arithmetic as first-order theory</span> </div> </a> <button aria-controls="toc-Peano_arithmetic_as_first-order_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 Peano arithmetic as first-order theory subsection</span> </button> <ul id="toc-Peano_arithmetic_as_first-order_theory-sublist" class="vector-toc-list"> <li id="toc-Equivalent_axiomatizations" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Equivalent_axiomatizations"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Equivalent axiomatizations</span> </div> </a> <ul id="toc-Equivalent_axiomatizations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Undecidability_and_incompleteness" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Undecidability_and_incompleteness"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Undecidability and incompleteness</span> </div> </a> <ul id="toc-Undecidability_and_incompleteness-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Nonstandard_models" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Nonstandard_models"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Nonstandard models</span> </div> </a> <ul id="toc-Nonstandard_models-sublist" class="vector-toc-list"> <li id="toc-Overspill" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Overspill"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3.1</span> <span>Overspill</span> </div> </a> <ul id="toc-Overspill-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Notes</span> </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>References</span> </div> </a> <button aria-controls="toc-References-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle References subsection</span> </button> <ul id="toc-References-sublist" class="vector-toc-list"> <li id="toc-Citations" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Citations"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>Citations</span> </div> </a> <ul id="toc-Citations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Sources" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Sources"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.2</span> <span>Sources</span> </div> </a> <ul id="toc-Sources-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Further_reading" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Further_reading"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Further reading</span> </div> </a> <ul id="toc-Further_reading-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Peano axioms</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 36 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-36" 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">36 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%85%D8%B3%D9%84%D9%85%D8%A7%D8%AA_%D8%A8%D9%8A%D8%A7%D9%86%D9%88" title="مسلمات بيانو – Arabic" lang="ar" hreflang="ar" data-title="مسلمات بيانو" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Peano_aksiomlar%C4%B1" title="Peano aksiomları – Azerbaijani" lang="az" hreflang="az" data-title="Peano aksiomları" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%90%D0%BA%D1%81%D1%96%D1%91%D0%BC%D1%8B_%D0%9F%D0%B5%D0%B0%D0%BD%D0%B0" title="Аксіёмы Пеана – Belarusian" lang="be" hreflang="be" data-title="Аксіёмы Пеана" 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%90%D0%BA%D1%81%D1%96%D1%91%D0%BC%D1%8B_%D0%9F%D1%8D%D0%B0%D0%BD%D0%B0" title="Аксіёмы Пэана – Belarusian (Taraškievica orthography)" lang="be-tarask" hreflang="be-tarask" data-title="Аксіёмы Пэана" 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%90%D0%BA%D1%81%D0%B8%D0%BE%D0%BC%D0%B8_%D0%BD%D0%B0_%D0%9F%D0%B5%D0%B0%D0%BD%D0%BE" title="Аксиоми на Пеано – Bulgarian" lang="bg" hreflang="bg" data-title="Аксиоми на Пеано" 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/Peanovi_aksiomi" title="Peanovi aksiomi – Bosnian" lang="bs" hreflang="bs" data-title="Peanovi aksiomi" 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/Axiomes_de_Peano" title="Axiomes de Peano – Catalan" lang="ca" hreflang="ca" data-title="Axiomes de Peano" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Peanovy_axiomy" title="Peanovy axiomy – Czech" lang="cs" hreflang="cs" data-title="Peanovy axiomy" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Peanos_aksiomer" title="Peanos aksiomer – Danish" lang="da" hreflang="da" data-title="Peanos aksiomer" 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/Peano-Axiome" title="Peano-Axiome – German" lang="de" hreflang="de" data-title="Peano-Axiome" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%91%CE%BE%CE%B9%CF%8E%CE%BC%CE%B1%CF%84%CE%B1_%CE%A0%CE%B5%CE%AC%CE%BD%CE%BF" title="Αξιώματα Πεάνο – Greek" lang="el" hreflang="el" data-title="Αξιώματα Πεάνο" 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/Axiomas_de_Peano" title="Axiomas de Peano – Spanish" lang="es" hreflang="es" data-title="Axiomas de Peano" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Peanoren_axiomak" title="Peanoren axiomak – Basque" lang="eu" hreflang="eu" data-title="Peanoren axiomak" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%A7%D8%B5%D9%88%D9%84_%D9%85%D9%88%D8%B6%D9%88%D8%B9%D9%87_%D9%BE%D8%A6%D8%A7%D9%86%D9%88" title="اصول موضوعه پئانو – Persian" lang="fa" hreflang="fa" data-title="اصول موضوعه پئانو" 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/Axiomes_de_Peano" title="Axiomes de Peano – French" lang="fr" hreflang="fr" data-title="Axiomes de Peano" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%ED%8E%98%EC%95%84%EB%85%B8_%EA%B3%B5%EB%A6%AC%EA%B3%84" title="페아노 공리계 – Korean" lang="ko" hreflang="ko" data-title="페아노 공리계" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%AA%E0%A4%BF%E0%A4%AF%E0%A4%BE%E0%A4%A8%E0%A5%8B_%E0%A4%95%E0%A5%87_%E0%A4%85%E0%A4%AD%E0%A4%BF%E0%A4%97%E0%A5%83%E0%A4%B9%E0%A5%80%E0%A4%A4" title="पियानो के अभिगृहीत – Hindi" lang="hi" hreflang="hi" data-title="पियानो के अभिगृहीत" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Aksioma_Peano" title="Aksioma Peano – Indonesian" lang="id" hreflang="id" data-title="Aksioma Peano" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Assiomi_di_Peano" title="Assiomi di Peano – Italian" lang="it" hreflang="it" data-title="Assiomi di Peano" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%9E%D7%A2%D7%A8%D7%9B%D7%AA_%D7%A4%D7%90%D7%A0%D7%95" title="מערכת פאנו – Hebrew" lang="he" hreflang="he" data-title="מערכת פאנו" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%9F%D0%B5%D0%B0%D0%BD%D0%BE_%D0%B0%D0%BA%D1%81%D0%B8%D0%BE%D0%BC%D0%B0%D0%BB%D0%B0%D1%80%D1%8B" title="Пеано аксиомалары – Kazakh" lang="kk" hreflang="kk" data-title="Пеано аксиомалары" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Peano_aksiomas" title="Peano aksiomas – Latvian" lang="lv" hreflang="lv" data-title="Peano aksiomas" data-language-autonym="Latviešu" data-language-local-name="Latvian" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Giuseppe_Peano#A_természetes_számok_Peano-axiómái" title="Giuseppe Peano – Hungarian" lang="hu" hreflang="hu" data-title="Giuseppe Peano" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Axioma%27s_van_Peano" title="Axioma's van Peano – Dutch" lang="nl" hreflang="nl" data-title="Axioma's van Peano" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%83%9A%E3%82%A2%E3%83%8E%E3%81%AE%E5%85%AC%E7%90%86" title="ペアノの公理 – Japanese" lang="ja" hreflang="ja" data-title="ペアノの公理" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Peanos_aksiomer" title="Peanos aksiomer – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Peanos aksiomer" 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-pms mw-list-item"><a href="https://pms.wikipedia.org/wiki/Assi%C3%B2ma_%C3%ABd_Peano" title="Assiòma ëd Peano – Piedmontese" lang="pms" hreflang="pms" data-title="Assiòma ëd Peano" data-language-autonym="Piemontèis" data-language-local-name="Piedmontese" class="interlanguage-link-target"><span>Piemontèis</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Axiomas_de_Peano" title="Axiomas de Peano – Portuguese" lang="pt" hreflang="pt" data-title="Axiomas de Peano" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%90%D0%BA%D1%81%D0%B8%D0%BE%D0%BC%D1%8B_%D0%9F%D0%B5%D0%B0%D0%BD%D0%BE" title="Аксиомы Пеано – Russian" lang="ru" hreflang="ru" data-title="Аксиомы Пеано" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%9F%D0%B5%D0%B0%D0%BD%D0%BE%D0%B2%D0%B5_%D0%B0%D0%BA%D1%81%D0%B8%D0%BE%D0%BC%D0%B5" title="Пеанове аксиоме – Serbian" lang="sr" hreflang="sr" data-title="Пеанове аксиоме" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Peanon_aksioomat" title="Peanon aksioomat – Finnish" lang="fi" hreflang="fi" data-title="Peanon aksioomat" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Peanos_axiom" title="Peanos axiom – Swedish" lang="sv" hreflang="sv" data-title="Peanos axiom" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Peano_aksiyomlar%C4%B1" title="Peano aksiyomları – Turkish" lang="tr" hreflang="tr" data-title="Peano aksiyomları" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%90%D0%BA%D1%81%D1%96%D0%BE%D0%BC%D0%B8_%D0%9F%D0%B5%D0%B0%D0%BD%D0%BE" title="Аксіоми Пеано – Ukrainian" lang="uk" hreflang="uk" data-title="Аксіоми Пеано" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/H%E1%BB%87_ti%C3%AAn_%C4%91%E1%BB%81_Peano" title="Hệ tiên đề Peano – Vietnamese" lang="vi" hreflang="vi" data-title="Hệ tiên đề Peano" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E7%9A%AE%E4%BA%9A%E8%AF%BA%E5%85%AC%E7%90%86" title="皮亚诺公理 – Chinese" lang="zh" hreflang="zh" data-title="皮亚诺公理" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target"><span>中文</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q842755#sitelinks-wikipedia" title="Edit interlanguage links" class="wbc-editpage">Edit links</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="Namespaces"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-nstab-main" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Peano_axioms" title="View the content page [c]" accesskey="c"><span>Article</span></a></li><li id="ca-talk" class="vector-tab-noicon mw-list-item"><a href="/wiki/Talk:Peano_axioms" rel="discussion" title="Discuss improvements to the content page [t]" accesskey="t"><span>Talk</span></a></li> </ul> </div> </div> <div id="vector-variants-dropdown" class="vector-dropdown emptyPortlet" > <input type="checkbox" id="vector-variants-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-variants-dropdown" class="vector-dropdown-checkbox " aria-label="Change language variant" > <label id="vector-variants-dropdown-label" for="vector-variants-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">English</span> </label> <div class="vector-dropdown-content"> <div id="p-variants" class="vector-menu mw-portlet mw-portlet-variants emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> </div> </div> </nav> </div> <div id="right-navigation" class="vector-collapsible"> <nav aria-label="Views"> <div id="p-views" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-views" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-view" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Peano_axioms"><span>Read</span></a></li><li id="ca-edit" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Peano_axioms&action=edit" title="Edit this page [e]" accesskey="e"><span>Edit</span></a></li><li id="ca-history" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Peano_axioms&action=history" title="Past revisions of this page [h]" accesskey="h"><span>View history</span></a></li> </ul> </div> </div> </nav> <nav class="vector-page-tools-landmark" aria-label="Page tools"> <div id="vector-page-tools-dropdown" class="vector-dropdown vector-page-tools-dropdown" > <input type="checkbox" id="vector-page-tools-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-tools-dropdown" class="vector-dropdown-checkbox " aria-label="Tools" > <label id="vector-page-tools-dropdown-label" for="vector-page-tools-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">Tools</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-tools-unpinned-container" class="vector-unpinned-container"> <div id="vector-page-tools" class="vector-page-tools vector-pinnable-element"> <div class="vector-pinnable-header vector-page-tools-pinnable-header vector-pinnable-header-unpinned" data-feature-name="page-tools-pinned" data-pinnable-element-id="vector-page-tools" data-pinned-container-id="vector-page-tools-pinned-container" data-unpinned-container-id="vector-page-tools-unpinned-container" > <div class="vector-pinnable-header-label">Tools</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-page-tools.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-page-tools.unpin">hide</button> </div> <div id="p-cactions" class="vector-menu mw-portlet mw-portlet-cactions emptyPortlet vector-has-collapsible-items" title="More options" > <div class="vector-menu-heading"> Actions </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-more-view" class="selected vector-more-collapsible-item mw-list-item"><a href="/wiki/Peano_axioms"><span>Read</span></a></li><li id="ca-more-edit" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Peano_axioms&action=edit" title="Edit this page [e]" accesskey="e"><span>Edit</span></a></li><li id="ca-more-history" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Peano_axioms&action=history"><span>View history</span></a></li> </ul> </div> </div> <div id="p-tb" class="vector-menu mw-portlet mw-portlet-tb" > <div class="vector-menu-heading"> General </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-whatlinkshere" class="mw-list-item"><a href="/wiki/Special:WhatLinksHere/Peano_axioms" title="List of all English Wikipedia pages containing links to this page [j]" accesskey="j"><span>What links here</span></a></li><li id="t-recentchangeslinked" class="mw-list-item"><a href="/wiki/Special:RecentChangesLinked/Peano_axioms" rel="nofollow" title="Recent changes in pages linked from this page [k]" accesskey="k"><span>Related changes</span></a></li><li id="t-upload" class="mw-list-item"><a href="/wiki/Wikipedia:File_Upload_Wizard" title="Upload files [u]" accesskey="u"><span>Upload file</span></a></li><li id="t-specialpages" class="mw-list-item"><a href="/wiki/Special:SpecialPages" title="A list of all special pages [q]" accesskey="q"><span>Special pages</span></a></li><li id="t-permalink" class="mw-list-item"><a href="/w/index.php?title=Peano_axioms&oldid=1255254109" title="Permanent link to this revision of this page"><span>Permanent link</span></a></li><li id="t-info" class="mw-list-item"><a href="/w/index.php?title=Peano_axioms&action=info" title="More information about this page"><span>Page information</span></a></li><li id="t-cite" class="mw-list-item"><a href="/w/index.php?title=Special:CiteThisPage&page=Peano_axioms&id=1255254109&wpFormIdentifier=titleform" title="Information on how to cite this page"><span>Cite this page</span></a></li><li id="t-urlshortener" class="mw-list-item"><a href="/w/index.php?title=Special:UrlShortener&url=https%3A%2F%2Fen.wikipedia.org%2Fwiki%2FPeano_axioms"><span>Get shortened URL</span></a></li><li id="t-urlshortener-qrcode" class="mw-list-item"><a href="/w/index.php?title=Special:QrCode&url=https%3A%2F%2Fen.wikipedia.org%2Fwiki%2FPeano_axioms"><span>Download QR code</span></a></li> </ul> </div> </div> <div id="p-coll-print_export" class="vector-menu mw-portlet mw-portlet-coll-print_export" > <div class="vector-menu-heading"> Print/export </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="coll-download-as-rl" class="mw-list-item"><a href="/w/index.php?title=Special:DownloadAsPdf&page=Peano_axioms&action=show-download-screen" title="Download this page as a PDF file"><span>Download as PDF</span></a></li><li id="t-print" class="mw-list-item"><a href="/w/index.php?title=Peano_axioms&printable=yes" title="Printable version of this page [p]" accesskey="p"><span>Printable version</span></a></li> </ul> </div> </div> <div id="p-wikibase-otherprojects" class="vector-menu mw-portlet mw-portlet-wikibase-otherprojects" > <div class="vector-menu-heading"> In other projects </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-wikibase" class="wb-otherproject-link wb-otherproject-wikibase-dataitem mw-list-item"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q842755" title="Structured data on this page hosted by Wikidata [g]" accesskey="g"><span>Wikidata item</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> </div> </div> </div> <div class="vector-column-end"> <div class="vector-sticky-pinned-container"> <nav class="vector-page-tools-landmark" aria-label="Page tools"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="Appearance"> <div id="vector-appearance-pinned-container" class="vector-pinned-container"> <div id="vector-appearance" class="vector-appearance vector-pinnable-element"> <div class="vector-pinnable-header vector-appearance-pinnable-header vector-pinnable-header-pinned" data-feature-name="appearance-pinned" data-pinnable-element-id="vector-appearance" data-pinned-container-id="vector-appearance-pinned-container" data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">Appearance</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">hide</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"><span class="mw-redirectedfrom">(Redirected from <a href="/w/index.php?title=Peano_arithmetic&redirect=no" class="mw-redirect" title="Peano arithmetic">Peano arithmetic</a>)</span></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Axioms for the natural numbers</div> <p>In <a href="/wiki/Mathematical_logic" title="Mathematical logic">mathematical logic</a>, the <b>Peano axioms</b> (<span class="rt-commentedText nowrap"><span class="IPA nopopups noexcerpt" lang="en-fonipa"><a href="/wiki/Help:IPA/English" title="Help:IPA/English">/<span style="border-bottom:1px dotted"><span title="'p' in 'pie'">p</span><span title="/i/: 'y' in 'happy'">i</span><span title="/ˈ/: primary stress follows">ˈ</span><span title="/ɑː/: 'a' in 'father'">ɑː</span><span title="'n' in 'nigh'">n</span><span title="/oʊ/: 'o' in 'code'">oʊ</span></span>/</a></span></span>,<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> <span class="IPA nowrap" lang="it-Latn-fonipa"><a href="/wiki/Help:IPA/Italian" title="Help:IPA/Italian">[peˈaːno]</a></span>), also known as the <b>Dedekind–Peano axioms</b> or the <b>Peano postulates</b>, are <a href="/wiki/Axiom" title="Axiom">axioms</a> for the <a href="/wiki/Natural_number" title="Natural number">natural numbers</a> presented by the 19th-century Italian mathematician <a href="/wiki/Giuseppe_Peano" title="Giuseppe Peano">Giuseppe Peano</a>. These axioms have been used nearly unchanged in a number of <a href="/wiki/Metamathematics" title="Metamathematics">metamathematical</a> investigations, including research into fundamental questions of whether <a href="/wiki/Number_theory" title="Number theory">number theory</a> is <a href="/wiki/Consistency_proof" class="mw-redirect" title="Consistency proof">consistent</a> and <a href="/wiki/Completeness_(logic)" title="Completeness (logic)">complete</a>. </p><p>The <a href="/wiki/Axiomatization" class="mw-redirect" title="Axiomatization">axiomatization</a> of <a href="/wiki/Arithmetic" title="Arithmetic">arithmetic</a> provided by Peano axioms is commonly called <b>Peano arithmetic</b>. </p><p>The importance of formalizing <a href="/wiki/Arithmetic" title="Arithmetic">arithmetic</a> was not well appreciated until the work of <a href="/wiki/Hermann_Grassmann" title="Hermann Grassmann">Hermann Grassmann</a>, who showed in the 1860s that many facts in arithmetic could be derived from more basic facts about the <a href="/wiki/Successor_function" title="Successor function">successor operation</a> and <a href="/wiki/Mathematical_induction" title="Mathematical induction">induction</a>.<sup id="cite_ref-FOOTNOTEGrassmann1861_2-0" class="reference"><a href="#cite_note-FOOTNOTEGrassmann1861-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-FOOTNOTEWang1957145,_147"It_is_rather_well-known,_through_Peano's_own_acknowledgement,_that_Peano_[…]_made_extensive_use_of_Grassmann's_work_in_his_development_of_the_axioms._It_is_not_so_well-known_that_Grassmann_had_essentially_the_characterization_of_the_set_of_all_integers,_now_customary_in_texts_of_modern_algebra,_that_it_forms_an_ordered_[[integral_domain]]_in_wihich_each_set_of_positive_elements_has_a_least_member._[…]_[Grassmann's_book]_was_probably_the_first_serious_and_rather_successful_attempt_to_put_numbers_on_a_more_or_less_axiomatic_basis."_3-0" class="reference"><a href="#cite_note-FOOTNOTEWang1957145,_147"It_is_rather_well-known,_through_Peano's_own_acknowledgement,_that_Peano_[…]_made_extensive_use_of_Grassmann's_work_in_his_development_of_the_axioms._It_is_not_so_well-known_that_Grassmann_had_essentially_the_characterization_of_the_set_of_all_integers,_now_customary_in_texts_of_modern_algebra,_that_it_forms_an_ordered_[[integral_domain]]_in_wihich_each_set_of_positive_elements_has_a_least_member._[…]_[Grassmann's_book]_was_probably_the_first_serious_and_rather_successful_attempt_to_put_numbers_on_a_more_or_less_axiomatic_basis."-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> In 1881, <a href="/wiki/Charles_Sanders_Peirce" title="Charles Sanders Peirce">Charles Sanders Peirce</a> provided an <a href="/wiki/Axiomatic_system#Axiomatization" title="Axiomatic system">axiomatization</a> of natural-number arithmetic.<sup id="cite_ref-FOOTNOTEPeirce1881_4-0" class="reference"><a href="#cite_note-FOOTNOTEPeirce1881-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-FOOTNOTEShields1997_5-0" class="reference"><a href="#cite_note-FOOTNOTEShields1997-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> In 1888, <a href="/wiki/Richard_Dedekind" title="Richard Dedekind">Richard Dedekind</a> proposed another axiomatization of natural-number arithmetic, and in 1889, Peano published a simplified version of them as a collection of axioms in his book <i>The principles of arithmetic presented by a new method</i> (<a href="/wiki/Latin_language" class="mw-redirect" title="Latin language">Latin</a>: <i lang="la"><a href="/wiki/Arithmetices_principia,_nova_methodo_exposita" title="Arithmetices principia, nova methodo exposita">Arithmetices principia, nova methodo exposita</a></i>). </p><p>The nine Peano axioms contain three types of statements. The first axiom asserts the existence of at least one member of the set of natural numbers. The next four are general statements about <a href="/wiki/Equality_(mathematics)" title="Equality (mathematics)">equality</a>; in modern treatments these are often not taken as part of the Peano axioms, but rather as axioms of the "underlying logic".<sup id="cite_ref-FOOTNOTEVan_Heijenoort196794_6-0" class="reference"><a href="#cite_note-FOOTNOTEVan_Heijenoort196794-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> The next three axioms are <a href="/wiki/First-order_logic" title="First-order logic">first-order</a> statements about natural numbers expressing the fundamental properties of the successor operation. The ninth, final, axiom is a <a href="/wiki/Second-order_logic" title="Second-order logic">second-order</a> statement of the principle of mathematical induction over the natural numbers, which makes this formulation close to <a href="/wiki/Second-order_arithmetic" title="Second-order arithmetic">second-order arithmetic</a>. A weaker first-order system is obtained by explicitly adding the addition and multiplication operation symbols and replacing the <a href="/wiki/Second-order_arithmetic#Induction_and_comprehension_schema" title="Second-order arithmetic">second-order induction</a> axiom with a first-order <a href="/wiki/Axiom_schema" title="Axiom schema">axiom schema</a>. The term <i>Peano arithmetic</i> is sometimes used for specifically naming this restricted system. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Historical_second-order_formulation">Historical second-order formulation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Peano_axioms&action=edit&section=1" title="Edit section: Historical second-order formulation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1251242444">.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output .ambox.mbox-small-left{margin:4px 1em 4px 0;overflow:hidden;width:238px;border-collapse:collapse;font-size:88%;line-height:1.25em}.mw-parser-output .ambox-speedy{border-left:10px solid #b32424;background-color:#fee7e6}.mw-parser-output .ambox-delete{border-left:10px solid #b32424}.mw-parser-output .ambox-content{border-left:10px solid #f28500}.mw-parser-output .ambox-style{border-left:10px solid #fc3}.mw-parser-output .ambox-move{border-left:10px solid #9932cc}.mw-parser-output .ambox-protection{border-left:10px solid #a2a9b1}.mw-parser-output .ambox .mbox-text{border:none;padding:0.25em 0.5em;width:100%}.mw-parser-output .ambox .mbox-image{border:none;padding:2px 0 2px 0.5em;text-align:center}.mw-parser-output .ambox .mbox-imageright{border:none;padding:2px 0.5em 2px 0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .mbox-image-div{width:52px}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}@media print{body.ns-0 .mw-parser-output .ambox{display:none!important}}</style><table class="box-More_citations_needed plainlinks metadata ambox ambox-content ambox-Refimprove" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><a href="/wiki/File:Question_book-new.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/50px-Question_book-new.svg.png" decoding="async" width="50" height="39" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/75px-Question_book-new.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/100px-Question_book-new.svg.png 2x" data-file-width="512" data-file-height="399" /></a></span></div></td><td class="mbox-text"><div class="mbox-text-span">This article <b>needs additional citations for <a href="/wiki/Wikipedia:Verifiability" title="Wikipedia:Verifiability">verification</a></b>.<span class="hide-when-compact"> Please help <a href="/wiki/Special:EditPage/Peano_axioms" title="Special:EditPage/Peano axioms">improve this article</a> by <a href="/wiki/Help:Referencing_for_beginners" title="Help:Referencing for beginners">adding citations to reliable sources</a>. Unsourced material may be challenged and removed.<br /><small><span class="plainlinks"><i>Find sources:</i> <a rel="nofollow" class="external text" href="https://www.google.com/search?as_eq=wikipedia&q=%22Peano+axioms%22">"Peano axioms"</a> – <a rel="nofollow" class="external text" href="https://www.google.com/search?tbm=nws&q=%22Peano+axioms%22+-wikipedia&tbs=ar:1">news</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.google.com/search?&q=%22Peano+axioms%22&tbs=bkt:s&tbm=bks">newspapers</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.google.com/search?tbs=bks:1&q=%22Peano+axioms%22+-wikipedia">books</a> <b>·</b> <a rel="nofollow" class="external text" href="https://scholar.google.com/scholar?q=%22Peano+axioms%22">scholar</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.jstor.org/action/doBasicSearch?Query=%22Peano+axioms%22&acc=on&wc=on">JSTOR</a></span></small></span> <span class="date-container"><i>(<span class="date">May 2024</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <p>When Peano formulated his axioms, the language of <a href="/wiki/Mathematical_logic" title="Mathematical logic">mathematical logic</a> was in its infancy. The system of logical notation he created to present the axioms did not prove to be popular, although it was the genesis of the modern notation for <a href="/wiki/Element_(mathematics)" title="Element (mathematics)">set membership</a> (∈, which comes from Peano's ε). Peano maintained a clear distinction between mathematical and logical symbols, which was not yet common in mathematics; such a separation had first been introduced in the <i><a href="/wiki/Begriffsschrift" title="Begriffsschrift">Begriffsschrift</a></i> by <a href="/wiki/Gottlob_Frege" title="Gottlob Frege">Gottlob Frege</a>, published in 1879.<sup id="cite_ref-FOOTNOTEVan_Heijenoort19672_7-0" class="reference"><a href="#cite_note-FOOTNOTEVan_Heijenoort19672-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> Peano was unaware of Frege's work and independently recreated his logical apparatus based on the work of <a href="/wiki/George_Boole" title="George Boole">Boole</a> and <a href="/wiki/Ernst_Schr%C3%B6der_(mathematician)" title="Ernst Schröder (mathematician)">Schröder</a>.<sup id="cite_ref-FOOTNOTEVan_Heijenoort196783_8-0" class="reference"><a href="#cite_note-FOOTNOTEVan_Heijenoort196783-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </p><p>The Peano axioms define the arithmetical properties of <i><a href="/wiki/Natural_numbers" class="mw-redirect" title="Natural numbers">natural numbers</a></i>, usually represented as a <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">set</a> <b>N</b> or <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {N} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {N} .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/682f44bd6a1ea39ecf1e21a8290b9d5b2f504505" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.325ex; height:2.176ex;" alt="{\displaystyle \mathbb {N} .}"></span> The <a href="/wiki/Non-logical_symbol" title="Non-logical symbol">non-logical symbols</a> for the axioms consist of a constant symbol 0 and a unary function symbol <i>S</i>. </p><p>The first axiom states that the constant 0 is a natural number: </p> <div><ol start="1"><li>0 is a natural number.</li></ol></div> <p>Peano's original formulation of the axioms used 1 instead of 0 as the "first" natural number,<sup id="cite_ref-FOOTNOTEPeano18891_9-0" class="reference"><a href="#cite_note-FOOTNOTEPeano18891-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> while the axioms in <i><a href="/wiki/Formulario_mathematico" title="Formulario mathematico">Formulario mathematico</a></i> include zero.<sup id="cite_ref-FOOTNOTEPeano190827_10-0" class="reference"><a href="#cite_note-FOOTNOTEPeano190827-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> </p><p>The next four axioms describe the <a href="/wiki/Equality_(mathematics)" title="Equality (mathematics)">equality</a> <a href="/wiki/Relation_(mathematics)" title="Relation (mathematics)">relation</a>. Since they are logically valid in first-order logic with equality, they are not considered to be part of "the Peano axioms" in modern treatments.<sup id="cite_ref-FOOTNOTEVan_Heijenoort196783_8-1" class="reference"><a href="#cite_note-FOOTNOTEVan_Heijenoort196783-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </p> <div><ol start="2"><li>For every natural number <i>x</i>, <span class="nowrap"><i>x</i> = <i>x</i></span>. That is, equality is <a href="/wiki/Reflexive_relation" title="Reflexive relation">reflexive</a>.</li><li>For all natural numbers <i>x</i> and <i>y</i>, if <span class="nowrap"><i>x</i> = <i>y</i></span>, then <span class="nowrap"><i>y</i> = <i>x</i></span>. That is, equality is <a href="/wiki/Symmetric_relation" title="Symmetric relation">symmetric</a>.</li><li>For all natural numbers <i>x</i>, <i>y</i> and <i>z</i>, if <i>x</i> = <i>y</i> and <i>y</i> = <i>z</i>, then <span class="nowrap"><i>x</i> = <i>z</i></span>. That is, equality is <a href="/wiki/Transitive_relation" title="Transitive relation">transitive</a>.</li><li>For all <i>a</i> and <i>b</i>, if <i>b</i> is a natural number and <span class="nowrap"><i>a</i> = <i>b</i></span>, then <i>a</i> is also a natural number. That is, the natural numbers are <a href="/wiki/Closure_(mathematics)" title="Closure (mathematics)">closed</a> under equality.</li></ol></div> <p>The remaining axioms define the arithmetical properties of the natural numbers. The naturals are assumed to be closed under a single-valued "<a href="/wiki/Successor_function" title="Successor function">successor</a>" <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">function</a> <i>S</i>. </p> <div><ol start="6"><li>For every natural number <i>n</i>, <i>S</i>(<i>n</i>) is a natural number. That is, the natural numbers are <a href="/wiki/Closure_(mathematics)" title="Closure (mathematics)">closed</a> under <i>S</i>.</li><li>For all natural numbers <i>m</i> and <i>n</i>, if <span class="nowrap"><i>S</i>(<i>m</i>) = <i>S</i>(<i>n</i>)</span>, then <span class="nowrap"><i>m</i> = <i>n</i></span>. That is, <i>S</i> is an <a href="/wiki/Injective_function" title="Injective function">injection</a>.</li><li>For every natural number <i>n</i>, <span class="nowrap"><i>S</i>(<i>n</i>) = 0</span> is false. That is, there is no natural number whose successor is 0.</li></ol></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Domino_effect_visualizing_exclusion_of_junk_term_by_induction_axiom.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/79/Domino_effect_visualizing_exclusion_of_junk_term_by_induction_axiom.jpg/400px-Domino_effect_visualizing_exclusion_of_junk_term_by_induction_axiom.jpg" decoding="async" width="400" height="300" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/79/Domino_effect_visualizing_exclusion_of_junk_term_by_induction_axiom.jpg/600px-Domino_effect_visualizing_exclusion_of_junk_term_by_induction_axiom.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/79/Domino_effect_visualizing_exclusion_of_junk_term_by_induction_axiom.jpg/800px-Domino_effect_visualizing_exclusion_of_junk_term_by_induction_axiom.jpg 2x" data-file-width="3200" data-file-height="2400" /></a><figcaption>The chain of light dominoes on the right, starting with the nearest, can represent the set <b>N</b> of natural numbers.<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>note 1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> However, axioms 1–8 are <i>also</i> satisfied by the set of all dominoes — whether light or dark — taken together.<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>note 2<span class="cite-bracket">]</span></a></sup> The 9th axiom (<a href="/wiki/Mathematical_induction" title="Mathematical induction">induction</a>) limits <b>N</b> to the chain of light pieces ("no junk") as only light dominoes will fall when the nearest is toppled. <sup id="cite_ref-FOOTNOTEMeseguerGoguen1986sections_2.3_(p._464)_and_4.1_(p._471)_15-0" class="reference"><a href="#cite_note-FOOTNOTEMeseguerGoguen1986sections_2.3_(p._464)_and_4.1_(p._471)-15"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup></figcaption></figure> <p>Axioms 1, 6, 7, 8 define a <a href="/wiki/Unary_numeral_system" title="Unary numeral system">unary representation</a> of the intuitive notion of natural numbers: the number 1 can be defined as <i>S</i>(0), 2 as <i>S</i>(<i>S</i>(0)), etc. However, considering the notion of natural numbers as being defined by these axioms, axioms 1, 6, 7, 8 do not imply that the successor function generates all the natural numbers different from 0. </p><p>The intuitive notion that each natural number can be obtained by applying <i>successor</i> sufficiently many times to zero requires an additional axiom, which is sometimes called the <i><a href="/wiki/Axiom_of_induction" class="mw-redirect" title="Axiom of induction">axiom of induction</a></i>. </p> <div><ol start="9"><li>If <i>K</i> is a set such that: <ul><li>0 is in <i>K</i>, and</li> <li>for every natural number <i>n</i>, <i>n</i> being in <i>K</i> implies that <i>S</i>(<i>n</i>) is in <i>K</i>,</li></ul> then <i>K</i> contains every natural number.</li></ol></div> <p>The induction axiom is sometimes stated in the following form: </p> <div><ol start="9"><li>If <i>φ</i> is a unary <a href="/wiki/Predicate_(mathematics)" class="mw-redirect" title="Predicate (mathematics)">predicate</a> such that: <ul><li><i>φ</i>(0) is true, and</li> <li>for every natural number <i>n</i>, <i>φ</i>(<i>n</i>) being true implies that <i>φ</i>(<i>S</i>(<i>n</i>)) is true,</li></ul> then <i>φ</i>(<i>n</i>) is true for every natural number <i>n</i>.</li></ol></div> <p>In Peano's original formulation, the induction axiom is a <a href="/wiki/Second-order_logic" title="Second-order logic">second-order axiom</a>. It is now common to replace this second-order principle with a weaker <a href="/wiki/First-order_logic" title="First-order logic">first-order</a> induction scheme. There are important differences between the second-order and first-order formulations, as discussed in the section <a href="#Peano_arithmetic_as_first-order_theory">§ Peano arithmetic as first-order theory</a> below. </p> <div class="mw-heading mw-heading3"><h3 id="Defining_arithmetic_operations_and_relations">Defining arithmetic operations and relations</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Peano_axioms&action=edit&section=2" title="Edit section: Defining arithmetic operations and relations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>If we use the second-order induction axiom, it is possible to define <a href="/wiki/Addition" title="Addition">addition</a>, <a href="/wiki/Multiplication" title="Multiplication">multiplication</a>, and <a href="/wiki/Total_order" title="Total order">total (linear) ordering</a> on <a href="/wiki/Natural_numbers#Notation" class="mw-redirect" title="Natural numbers"><b>N</b></a> directly using the axioms. However, <span class="citation-needed-content" style="padding-left:0.1em; padding-right:0.1em; color:var(--color-subtle, #54595d); border:1px solid var(--border-color-subtle, #c8ccd1);">with first-order induction, this is not possible</span><sup class="noprint Inline-Template Template-Fact" style="margin-left:0.1em; white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (January 2022)">citation needed</span></a></i>]</sup> and addition and multiplication are often added as axioms. The respective functions and relations are constructed in <a href="/wiki/Set_theory" title="Set theory">set theory</a> or <a href="/wiki/Second-order_logic" title="Second-order logic">second-order logic</a>, and can be shown to be unique using the Peano axioms. </p> <div class="mw-heading mw-heading4"><h4 id="Addition">Addition</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Peano_axioms&action=edit&section=3" title="Edit section: Addition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Addition_in_N" class="mw-redirect" title="Addition in N">Addition</a> is a function that <a href="/wiki/Map_(mathematics)" title="Map (mathematics)">maps</a> two natural numbers (two elements of <b>N</b>) to another one. It is defined <a href="/wiki/Recursion" title="Recursion">recursively</a> 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 {\begin{aligned}a+0&=a,&{\textrm {(1)}}\\a+S(b)&=S(a+b).&{\textrm {(2)}}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>a</mi> <mo>+</mo> <mn>0</mn> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>a</mi> <mo>,</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext>(1)</mtext> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>a</mi> <mo>+</mo> <mi>S</mi> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>S</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext>(2)</mtext> </mrow> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}a+0&=a,&{\textrm {(1)}}\\a+S(b)&=S(a+b).&{\textrm {(2)}}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/80857d5980826ae352be5a7cd8eb9cb70bdf5843" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:28.866ex; height:6.176ex;" alt="{\displaystyle {\begin{aligned}a+0&=a,&{\textrm {(1)}}\\a+S(b)&=S(a+b).&{\textrm {(2)}}\end{aligned}}}"></span></dd></dl> <p>For example: </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 {\begin{aligned}a+1&=a+S(0)&{\mbox{by definition}}\\&=S(a+0)&{\mbox{using (2)}}\\&=S(a),&{\mbox{using (1)}}\\\\a+2&=a+S(1)&{\mbox{by definition}}\\&=S(a+1)&{\mbox{using (2)}}\\&=S(S(a))&{\mbox{using }}a+1=S(a)\\\\a+3&=a+S(2)&{\mbox{by definition}}\\&=S(a+2)&{\mbox{using (2)}}\\&=S(S(S(a)))&{\mbox{using }}a+2=S(S(a))\\{\text{etc.}}&\\\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>a</mi> <mo>+</mo> <mn>1</mn> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>a</mi> <mo>+</mo> <mi>S</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>by definition</mtext> </mstyle> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mi>S</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>using (2)</mtext> </mstyle> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mi>S</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>using (1)</mtext> </mstyle> </mrow> </mtd> </mtr> <mtr> <mtd /> </mtr> <mtr> <mtd> <mi>a</mi> <mo>+</mo> <mn>2</mn> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>a</mi> <mo>+</mo> <mi>S</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>by definition</mtext> </mstyle> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mi>S</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>using (2)</mtext> </mstyle> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mi>S</mi> <mo stretchy="false">(</mo> <mi>S</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>using </mtext> </mstyle> </mrow> <mi>a</mi> <mo>+</mo> <mn>1</mn> <mo>=</mo> <mi>S</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd /> </mtr> <mtr> <mtd> <mi>a</mi> <mo>+</mo> <mn>3</mn> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>a</mi> <mo>+</mo> <mi>S</mi> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>by definition</mtext> </mstyle> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mi>S</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>using (2)</mtext> </mstyle> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mi>S</mi> <mo stretchy="false">(</mo> <mi>S</mi> <mo stretchy="false">(</mo> <mi>S</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>using </mtext> </mstyle> </mrow> <mi>a</mi> <mo>+</mo> <mn>2</mn> <mo>=</mo> <mi>S</mi> <mo stretchy="false">(</mo> <mi>S</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>etc.</mtext> </mrow> </mtd> <mtd /> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}a+1&=a+S(0)&{\mbox{by definition}}\\&=S(a+0)&{\mbox{using (2)}}\\&=S(a),&{\mbox{using (1)}}\\\\a+2&=a+S(1)&{\mbox{by definition}}\\&=S(a+1)&{\mbox{using (2)}}\\&=S(S(a))&{\mbox{using }}a+1=S(a)\\\\a+3&=a+S(2)&{\mbox{by definition}}\\&=S(a+2)&{\mbox{using (2)}}\\&=S(S(S(a)))&{\mbox{using }}a+2=S(S(a))\\{\text{etc.}}&\\\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f8b0d0ca513dfd9807949cb1391d025f3d48a7f8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -18.005ex; width:46.952ex; height:37.176ex;" alt="{\displaystyle {\begin{aligned}a+1&=a+S(0)&{\mbox{by definition}}\\&=S(a+0)&{\mbox{using (2)}}\\&=S(a),&{\mbox{using (1)}}\\\\a+2&=a+S(1)&{\mbox{by definition}}\\&=S(a+1)&{\mbox{using (2)}}\\&=S(S(a))&{\mbox{using }}a+1=S(a)\\\\a+3&=a+S(2)&{\mbox{by definition}}\\&=S(a+2)&{\mbox{using (2)}}\\&=S(S(S(a)))&{\mbox{using }}a+2=S(S(a))\\{\text{etc.}}&\\\end{aligned}}}"></span></dd></dl> <p>To prove commutativity of addition, first prove <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 0+b=b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>+</mo> <mi>b</mi> <mo>=</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0+b=b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b94da230b13c8c5ca70da8d0e129e8d422995d0f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.096ex; height:2.343ex;" alt="{\displaystyle 0+b=b}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S(a)+b=S(a+b)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>b</mi> <mo>=</mo> <mi>S</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S(a)+b=S(a+b)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b0760daf95c9641adec1b43071372c780486b2da" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.851ex; height:2.843ex;" alt="{\displaystyle S(a)+b=S(a+b)}"></span>, each by induction on <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 b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"></span>. Using both results, then prove <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=b+a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mo>=</mo> <mi>b</mi> <mo>+</mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a+b=b+a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/684f43b5094501674e8314be5e24a80ee64682e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:13.234ex; height:2.343ex;" alt="{\displaystyle a+b=b+a}"></span> by induction on <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 b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"></span>. The <a href="/wiki/Mathematical_structure" title="Mathematical structure">structure</a> <span class="nowrap">(<b>N</b>, +)</span> is a <a href="/wiki/Commutative" class="mw-redirect" title="Commutative">commutative</a> <a href="/wiki/Monoid" title="Monoid">monoid</a> with identity element 0. <span class="nowrap">(<b>N</b>, +)</span> is also a <a href="/wiki/Cancellation_property" title="Cancellation property">cancellative</a> <a href="/wiki/Magma_(algebra)" title="Magma (algebra)">magma</a>, and thus <a href="/wiki/Embedding" title="Embedding">embeddable</a> in a <a href="/wiki/Group_(mathematics)" title="Group (mathematics)">group</a>. The smallest group embedding <b>N</b> is the <a href="/wiki/Integer" title="Integer">integers</a>.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (May 2024)">citation needed</span></a></i>]</sup> </p> <div class="mw-heading mw-heading4"><h4 id="Multiplication">Multiplication</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Peano_axioms&action=edit&section=4" title="Edit section: Multiplication"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Similarly, <a href="/wiki/Multiplication" title="Multiplication">multiplication</a> is a function mapping two natural numbers to another one. Given addition, it is defined recursively 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 {\begin{aligned}a\cdot 0&=0,\\a\cdot S(b)&=a+(a\cdot b).\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>a</mi> <mo>⋅</mo> <mn>0</mn> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mi>a</mi> <mo>⋅</mo> <mi>S</mi> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>a</mi> <mo>+</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>⋅</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}a\cdot 0&=0,\\a\cdot S(b)&=a+(a\cdot b).\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3f34d9dba1c07730a645b39c0840bf4b00ae6727" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:21.498ex; height:5.843ex;" alt="{\displaystyle {\begin{aligned}a\cdot 0&=0,\\a\cdot S(b)&=a+(a\cdot b).\end{aligned}}}"></span></dd></dl> <p>It is easy to see that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S(0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S(0)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f113163faf55307ea92fcdbbbf83372c4be3cd3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.471ex; height:2.843ex;" alt="{\displaystyle S(0)}"></span> is the multiplicative <a href="/wiki/Identity_element" title="Identity element">right identity</a>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\cdot S(0)=a+(a\cdot 0)=a+0=a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>⋅</mo> <mi>S</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mi>a</mi> <mo>+</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>⋅</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mi>a</mi> <mo>+</mo> <mn>0</mn> <mo>=</mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\cdot S(0)=a+(a\cdot 0)=a+0=a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d71e13dc67f46b095fe1dfa9223c270de0e9ed5a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.089ex; height:2.843ex;" alt="{\displaystyle a\cdot S(0)=a+(a\cdot 0)=a+0=a}"></span></dd></dl> <p>To show that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S(0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S(0)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f113163faf55307ea92fcdbbbf83372c4be3cd3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.471ex; height:2.843ex;" alt="{\displaystyle S(0)}"></span> is also the multiplicative left identity requires the induction axiom due to the way multiplication is defined: </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S(0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S(0)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f113163faf55307ea92fcdbbbf83372c4be3cd3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.471ex; height:2.843ex;" alt="{\displaystyle S(0)}"></span> is the left identity of 0: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S(0)\cdot 0=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>⋅</mo> <mn>0</mn> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S(0)\cdot 0=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cac925900a1a15e9e96375bd448091ea6434222e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.573ex; height:2.843ex;" alt="{\displaystyle S(0)\cdot 0=0}"></span>.</li> <li>If <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S(0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S(0)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f113163faf55307ea92fcdbbbf83372c4be3cd3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.471ex; height:2.843ex;" alt="{\displaystyle S(0)}"></span> is the left identity 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 a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <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/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span> (that 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 S(0)\cdot a=a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>⋅</mo> <mi>a</mi> <mo>=</mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S(0)\cdot a=a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dca83da5c480941706a3a0756bec47e64bf78250" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.708ex; height:2.843ex;" alt="{\displaystyle S(0)\cdot a=a}"></span>), then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S(0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S(0)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f113163faf55307ea92fcdbbbf83372c4be3cd3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.471ex; height:2.843ex;" alt="{\displaystyle S(0)}"></span> is also the left identity of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S(a)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S(a)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e073905684e0b1c2f60a76511348f4b610ad3ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.538ex; height:2.843ex;" alt="{\displaystyle S(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 S(0)\cdot S(a)=S(0)+S(0)\cdot a=S(0)+a=a+S(0)=S(a+0)=S(a)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>⋅</mo> <mi>S</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>S</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mi>S</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>⋅</mo> <mi>a</mi> <mo>=</mo> <mi>S</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mi>a</mi> <mo>=</mo> <mi>a</mi> <mo>+</mo> <mi>S</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mi>S</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mi>S</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S(0)\cdot S(a)=S(0)+S(0)\cdot a=S(0)+a=a+S(0)=S(a+0)=S(a)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/18c2803e47444808d97d527652a05b555648e054" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:71.034ex; height:2.843ex;" alt="{\displaystyle S(0)\cdot S(a)=S(0)+S(0)\cdot a=S(0)+a=a+S(0)=S(a+0)=S(a)}"></span>, using commutativity of addition.</li></ul> <p>Therefore, by the induction axiom <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S(0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S(0)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f113163faf55307ea92fcdbbbf83372c4be3cd3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.471ex; height:2.843ex;" alt="{\displaystyle S(0)}"></span> is the multiplicative left identity of all natural numbers. Moreover, it can be shown<sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> that multiplication is commutative and <a href="/wiki/Distributive_law" class="mw-redirect" title="Distributive law">distributes over</a> addition: </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 a\cdot (b+c)=(a\cdot b)+(a\cdot c)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>⋅</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mo>+</mo> <mi>c</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>⋅</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>⋅</mo> <mi>c</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\cdot (b+c)=(a\cdot b)+(a\cdot c)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0969d65db9f1f1097aa4f72bcddac8c46f1ca6ef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.943ex; height:2.843ex;" alt="{\displaystyle a\cdot (b+c)=(a\cdot b)+(a\cdot c)}"></span>.</dd></dl> <p>Thus, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\mathbb {N} ,+,0,\cdot ,S(0))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mo>,</mo> <mo>+</mo> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mo>⋅</mo> <mo>,</mo> <mi>S</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\mathbb {N} ,+,0,\cdot ,S(0))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba608f5bc28cfc7c1044eecf4a6608b67a9fdf49" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.712ex; height:2.843ex;" alt="{\displaystyle (\mathbb {N} ,+,0,\cdot ,S(0))}"></span> is a commutative <a href="/wiki/Semiring" title="Semiring">semiring</a>. </p> <div class="mw-heading mw-heading4"><h4 id="Inequalities">Inequalities</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Peano_axioms&action=edit&section=5" title="Edit section: Inequalities"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The usual <a href="/wiki/Total_order" title="Total order">total order</a> relation ≤ on natural numbers can be defined as follows, assuming 0 is a natural number: </p> <dl><dd>For all <span class="nowrap"><i>a</i>, <i>b</i> ∈ <b>N</b></span>, <span class="nowrap"><i>a</i> ≤ <i>b</i></span> if and only if there exists some <span class="nowrap"><i>c</i> ∈ <b>N</b></span> such that <span class="nowrap"><i>a</i> + <i>c</i> = <i>b</i></span>.</dd></dl> <p>This relation is stable under addition and multiplication: for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a,b,c\in \mathbb {N} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>,</mo> <mi>c</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a,b,c\in \mathbb {N} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/adecc9a338fe8edb4b1b7b58197c5dfebd73f10f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.821ex; height:2.509ex;" alt="{\displaystyle a,b,c\in \mathbb {N} }"></span>, if <span class="nowrap"><i>a</i> ≤ <i>b</i></span>, then: </p> <ul><li><i>a</i> + <i>c</i> ≤ <i>b</i> + <i>c</i>, and</li> <li><i>a</i> · <i>c</i> ≤ <i>b</i> · <i>c</i>.</li></ul> <p>Thus, the structure <span class="nowrap">(<b>N</b>, +, ·, 1, 0, ≤)</span> is an <a href="/wiki/Ordered_ring" title="Ordered ring">ordered semiring</a>; because there is no natural number between 0 and 1, it is a discrete ordered semiring. </p><p>The axiom of induction is sometimes stated in the following form that uses a stronger hypothesis, making use of the order relation "≤": </p> <dl><dd>For any <a href="/wiki/Predicate_(mathematics)" class="mw-redirect" title="Predicate (mathematics)">predicate</a> <i>φ</i>, if <ul><li><i>φ</i>(0) is true, and</li> <li>for every <span class="nowrap"><i>n</i> ∈ <b>N</b></span>, if <i>φ</i>(<i>k</i>) is true for every <span class="nowrap"><i>k</i> ∈ <b>N</b></span> such that <span class="nowrap"><i>k</i> ≤ <i>n</i></span>, then <i>φ</i>(<i>S</i>(<i>n</i>)) is true,</li> <li>then for every <span class="nowrap"><i>n</i> ∈ <b>N</b></span>, <i>φ</i>(<i>n</i>) is true.</li></ul></dd></dl> <p>This form of the induction axiom, called <i>strong induction</i>, is a consequence of the standard formulation, but is often better suited for reasoning about the ≤ order. For example, to show that the naturals are <a href="/wiki/Well-order" title="Well-order">well-ordered</a>—every <a href="/wiki/Empty_set" title="Empty set">nonempty</a> <a href="/wiki/Subset" title="Subset">subset</a> of <b>N</b> has a <a href="/wiki/Least_element" class="mw-redirect" title="Least element">least element</a>—one can reason as follows. Let a nonempty <span class="nowrap"><i>X</i> ⊆ <b>N</b></span> be given and assume <i>X</i> has no least element. </p> <ul><li>Because 0 is the least element of <b>N</b>, it must be that <span class="nowrap">0 ∉ <i>X</i></span>.</li> <li>For any <span class="nowrap"><i>n</i> ∈ <b>N</b></span>, suppose for every <span class="nowrap"><i>k</i> ≤ <i>n</i></span>, <span class="nowrap"><i>k</i> ∉ <i>X</i></span>. Then <span class="nowrap"><i>S</i>(<i>n</i>) ∉ <i>X</i></span>, for otherwise it would be the least element of <i>X</i>.</li></ul> <p>Thus, by the strong induction principle, for every <span class="nowrap"><i>n</i> ∈ <b>N</b></span>, <span class="nowrap"><i>n</i> ∉ <i>X</i></span>. Thus, <span class="nowrap"><i>X</i> ∩ <b>N</b> = ∅</span>, which <a href="/wiki/Contradiction" title="Contradiction">contradicts</a> <i>X</i> being a nonempty subset of <b>N</b>. Thus <i>X</i> has a least element. </p> <div class="mw-heading mw-heading3"><h3 id="Models">Models</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Peano_axioms&action=edit&section=6" title="Edit section: Models"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A <a href="/wiki/Model_theory" title="Model theory">model</a> of the Peano axioms is a triple <span class="nowrap">(<b>N</b>, 0, <i>S</i>)</span>, where <b>N</b> is a (necessarily infinite) set, <span class="nowrap">0 ∈ <b>N</b></span> and <span class="nowrap"><i>S</i>: <b>N</b> → <b>N</b></span> satisfies the axioms above. <a href="/wiki/Richard_Dedekind" title="Richard Dedekind">Dedekind</a> proved in his 1888 book, <i>The Nature and Meaning of Numbers</i> (<a href="/wiki/German_language" title="German language">German</a>: <i lang="de">Was sind und was sollen die Zahlen?</i>, i.e., "What are the numbers and what are they good for?") that any two models of the Peano axioms (including the second-order induction axiom) are <a href="/wiki/Isomorphism" title="Isomorphism">isomorphic</a>. In particular, given two models <span class="nowrap">(<b>N</b><sub><i>A</i></sub>, 0<sub><i>A</i></sub>, <i>S</i><sub><i>A</i></sub>)</span> and <span class="nowrap">(<b>N</b><sub><i>B</i></sub>, 0<sub><i>B</i></sub>, <i>S</i><sub><i>B</i></sub>)</span> of the Peano axioms, there is a unique <a href="/wiki/Homomorphism" title="Homomorphism">homomorphism</a> <span class="nowrap"><i>f</i> : <b>N</b><sub><i>A</i></sub> → <b>N</b><sub><i>B</i></sub></span> satisfying </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 {\begin{aligned}f(0_{A})&=0_{B}\\f(S_{A}(n))&=S_{B}(f(n))\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <msub> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}f(0_{A})&=0_{B}\\f(S_{A}(n))&=S_{B}(f(n))\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e70e0f0f1cc309315e4cffb0f3ac1d02fe190d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:22.228ex; height:6.176ex;" alt="{\displaystyle {\begin{aligned}f(0_{A})&=0_{B}\\f(S_{A}(n))&=S_{B}(f(n))\end{aligned}}}"></span></dd></dl> <p>and it is a <a href="/wiki/Bijective_function" class="mw-redirect" title="Bijective function">bijection</a>. This means that the second-order Peano axioms are <a href="/wiki/Morley%27s_categoricity_theorem" class="mw-redirect" title="Morley's categoricity theorem">categorical</a>. (This is not the case with any first-order reformulation of the Peano axioms, below.) </p> <div class="mw-heading mw-heading4"><h4 id="Set-theoretic_models">Set-theoretic models</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Peano_axioms&action=edit&section=7" title="Edit section: Set-theoretic models"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Set-theoretic_definition_of_natural_numbers" title="Set-theoretic definition of natural numbers">Set-theoretic definition of natural numbers</a></div> <p>The Peano axioms can be derived from <a href="/wiki/Set_theory" title="Set theory">set theoretic</a> constructions of the <a href="/wiki/Natural_number" title="Natural number">natural numbers</a> and axioms of set theory such as <a href="/wiki/Zermelo%E2%80%93Fraenkel_set_theory" title="Zermelo–Fraenkel set theory">ZF</a>.<sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> The standard construction of the naturals, due to <a href="/wiki/John_von_Neumann" title="John von Neumann">John von Neumann</a>, starts from a definition of 0 as the empty set, ∅, and an operator <i>s</i> on sets defined 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 s(a)=a\cup \{a\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>a</mi> <mo>∪</mo> <mo fence="false" stretchy="false">{</mo> <mi>a</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s(a)=a\cup \{a\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a14a11619d33d05c3a39a213dcd03b7b6ca96dae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.595ex; height:2.843ex;" alt="{\displaystyle s(a)=a\cup \{a\}}"></span></dd></dl> <p>The set of natural numbers <b>N</b> is defined as the intersection of all sets <a href="/wiki/Closure_(mathematics)" title="Closure (mathematics)">closed</a> under <i>s</i> that contain the empty set. Each natural number is equal (as a set) to the set of natural numbers less than it: </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 {\begin{aligned}0&=\emptyset \\1&=s(0)=s(\emptyset )=\emptyset \cup \{\emptyset \}=\{\emptyset \}=\{0\}\\2&=s(1)=s(\{0\})=\{0\}\cup \{\{0\}\}=\{0,\{0\}\}=\{0,1\}\\3&=s(2)=s(\{0,1\})=\{0,1\}\cup \{\{0,1\}\}=\{0,1,\{0,1\}\}=\{0,1,2\}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi mathvariant="normal">∅</mi> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>s</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mi>s</mi> <mo stretchy="false">(</mo> <mi mathvariant="normal">∅</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi mathvariant="normal">∅</mi> <mo>∪</mo> <mo fence="false" stretchy="false">{</mo> <mi mathvariant="normal">∅</mi> <mo fence="false" stretchy="false">}</mo> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi mathvariant="normal">∅</mi> <mo fence="false" stretchy="false">}</mo> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>s</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mi>s</mi> <mo stretchy="false">(</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> <mo>∪</mo> <mo fence="false" stretchy="false">{</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> <mo fence="false" stretchy="false">}</mo> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo>,</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> <mo fence="false" stretchy="false">}</mo> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo fence="false" stretchy="false">}</mo> </mtd> </mtr> <mtr> <mtd> <mn>3</mn> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>s</mi> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mi>s</mi> <mo stretchy="false">(</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo fence="false" stretchy="false">}</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo fence="false" stretchy="false">}</mo> <mo>∪</mo> <mo fence="false" stretchy="false">{</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo fence="false" stretchy="false">}</mo> <mo fence="false" stretchy="false">}</mo> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo fence="false" stretchy="false">}</mo> <mo fence="false" stretchy="false">}</mo> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo fence="false" stretchy="false">}</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}0&=\emptyset \\1&=s(0)=s(\emptyset )=\emptyset \cup \{\emptyset \}=\{\emptyset \}=\{0\}\\2&=s(1)=s(\{0\})=\{0\}\cup \{\{0\}\}=\{0,\{0\}\}=\{0,1\}\\3&=s(2)=s(\{0,1\})=\{0,1\}\cup \{\{0,1\}\}=\{0,1,\{0,1\}\}=\{0,1,2\}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec4e0820b48f8d7f1803aa86f3b3b50b93182deb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.505ex; width:66.609ex; height:12.176ex;" alt="{\displaystyle {\begin{aligned}0&=\emptyset \\1&=s(0)=s(\emptyset )=\emptyset \cup \{\emptyset \}=\{\emptyset \}=\{0\}\\2&=s(1)=s(\{0\})=\{0\}\cup \{\{0\}\}=\{0,\{0\}\}=\{0,1\}\\3&=s(2)=s(\{0,1\})=\{0,1\}\cup \{\{0,1\}\}=\{0,1,\{0,1\}\}=\{0,1,2\}\end{aligned}}}"></span></dd></dl> <p>and so on. The set <b>N</b> together with 0 and the <a href="/wiki/Successor_function" title="Successor function">successor function</a> <span class="nowrap"><i>s</i> : <b>N</b> → <b>N</b></span> satisfies the Peano axioms. </p><p>Peano arithmetic is <a href="/wiki/Equiconsistent" class="mw-redirect" title="Equiconsistent">equiconsistent</a> with several weak systems of set theory.<sup id="cite_ref-FOOTNOTETarskiGivant1987Section_7.6_18-0" class="reference"><a href="#cite_note-FOOTNOTETarskiGivant1987Section_7.6-18"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> One such system is ZFC with the <a href="/wiki/Axiom_of_infinity" title="Axiom of infinity">axiom of infinity</a> replaced by its negation. Another such system consists of <a href="/wiki/General_set_theory" title="General set theory">general set theory</a> (<a href="/wiki/Axiom_of_extensionality" title="Axiom of extensionality">extensionality</a>, existence of the <a href="/wiki/Empty_set" title="Empty set">empty set</a>, and the <a href="/wiki/General_set_theory" title="General set theory">axiom of adjunction</a>), augmented by an axiom schema stating that a property that holds for the empty set and holds of an adjunction whenever it holds of the adjunct must hold for all sets. </p> <div class="mw-heading mw-heading4"><h4 id="Interpretation_in_category_theory">Interpretation in category theory</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Peano_axioms&action=edit&section=8" title="Edit section: Interpretation in category theory"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The Peano axioms can also be understood using <a href="/wiki/Category_theory" title="Category theory">category theory</a>. Let <i>C</i> be a <a href="/wiki/Category_(mathematics)" title="Category (mathematics)">category</a> with <a href="/wiki/Terminal_object" class="mw-redirect" title="Terminal object">terminal object</a> 1<sub><i>C</i></sub>, and define the category of <a href="/wiki/Pointed_unary_system" class="mw-redirect" title="Pointed unary system">pointed unary systems</a>, US<sub>1</sub>(<i>C</i>) as follows: </p> <ul><li>The objects of US<sub>1</sub>(<i>C</i>) are triples <span class="nowrap">(<i>X</i>, 0<sub><i>X</i></sub>, <i>S</i><sub><i>X</i></sub>)</span> where <i>X</i> is an object of <i>C</i>, and <span class="nowrap">0<sub><i>X</i></sub> : 1<sub><i>C</i></sub> → <i>X</i></span> and <span class="nowrap"><i>S</i><sub><i>X</i></sub> : <i>X</i> → <i>X</i></span> are <i>C</i>-morphisms.</li> <li>A morphism <i>φ</i> : (<i>X</i>, 0<sub><i>X</i></sub>, <i>S</i><sub><i>X</i></sub>) → (<i>Y</i>, 0<sub><i>Y</i></sub>, <i>S</i><sub><i>Y</i></sub>) is a <i>C</i>-morphism <span class="nowrap"><i>φ</i> : <i>X</i> → <i>Y</i></span> with <span class="nowrap"><i>φ</i> 0<sub><i>X</i></sub> = 0<sub><i>Y</i></sub></span> and <span class="nowrap"><i>φ</i> <i>S</i><sub><i>X</i></sub> = <i>S</i><sub><i>Y</i></sub> <i>φ</i></span>.</li></ul> <p>Then <i>C</i> is said to satisfy the Dedekind–Peano axioms if US<sub>1</sub>(<i>C</i>) has an initial object; this initial object is known as a <a href="/wiki/Natural_number_object" class="mw-redirect" title="Natural number object">natural number object</a> in <i>C</i>. If <span class="nowrap">(<b>N</b>, 0, <i>S</i>)</span> is this initial object, and <span class="nowrap">(<i>X</i>, 0<sub><i>X</i></sub>, <i>S</i><sub><i>X</i></sub>)</span> is any other object, then the unique map <span class="nowrap"><i>u</i> : (<b>N</b>, 0, <i>S</i>) → (<i>X</i>, 0<sub><i>X</i></sub>, <i>S</i><sub><i>X</i></sub>)</span> is such that </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 {\begin{aligned}u(0)&=0_{X},\\u(Sx)&=S_{X}(ux).\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>u</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <msub> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mi>u</mi> <mo stretchy="false">(</mo> <mi>S</mi> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>u</mi> <mi>x</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}u(0)&=0_{X},\\u(Sx)&=S_{X}(ux).\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab921c5cdd164343a7c45cf1919129b52eda38dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:17.991ex; height:6.176ex;" alt="{\displaystyle {\begin{aligned}u(0)&=0_{X},\\u(Sx)&=S_{X}(ux).\end{aligned}}}"></span></dd></dl> <p>This is precisely the recursive definition of 0<sub><i>X</i></sub> and <i>S</i><sub><i>X</i></sub>. </p> <div class="mw-heading mw-heading3"><h3 id="Consistency">Consistency</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Peano_axioms&action=edit&section=9" title="Edit section: Consistency"><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/Hilbert%27s_second_problem" title="Hilbert's second problem">Hilbert's second problem</a> and <a href="/wiki/Consistency" title="Consistency">Consistency</a></div> <p>When the Peano axioms were first proposed, <a href="/wiki/Bertrand_Russell" title="Bertrand Russell">Bertrand Russell</a> and others agreed that these axioms implicitly defined what we mean by a "natural number".<sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Henri_Poincar%C3%A9" title="Henri Poincaré">Henri Poincaré</a> was more cautious, saying they only defined natural numbers if they were <i>consistent</i>; if there is a proof that starts from just these axioms and derives a contradiction such as 0 = 1, then the axioms are inconsistent, and don't define anything.<sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> In 1900, <a href="/wiki/David_Hilbert" title="David Hilbert">David Hilbert</a> posed the problem of proving their consistency using only <a href="/wiki/Finitism" title="Finitism">finitistic</a> methods as the <a href="/wiki/Hilbert%27s_second_problem" title="Hilbert's second problem">second</a> of his <a href="/wiki/Hilbert%27s_problems" title="Hilbert's problems">twenty-three problems</a>.<sup id="cite_ref-FOOTNOTEHilbert1902_21-0" class="reference"><a href="#cite_note-FOOTNOTEHilbert1902-21"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> In 1931, <a href="/wiki/Kurt_G%C3%B6del" title="Kurt Gödel">Kurt Gödel</a> proved his <a href="/wiki/Second_incompleteness_theorem" class="mw-redirect" title="Second incompleteness theorem">second incompleteness theorem</a>, which shows that such a consistency proof cannot be formalized within Peano arithmetic itself, if Peano arithmetic is consistent.<sup id="cite_ref-FOOTNOTEGödel1931_22-0" class="reference"><a href="#cite_note-FOOTNOTEGödel1931-22"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> </p><p>Although it is widely claimed that Gödel's theorem rules out the possibility of a finitistic consistency proof for Peano arithmetic, this depends on exactly what one means by a finitistic proof. Gödel himself pointed out the possibility of giving a finitistic consistency proof of Peano arithmetic or stronger systems by using finitistic methods that are not formalizable in Peano arithmetic, and in 1958, Gödel published a method for proving the consistency of arithmetic using <a href="/wiki/Type_theory" title="Type theory">type theory</a>.<sup id="cite_ref-23" class="reference"><a href="#cite_note-23"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> In 1936, <a href="/wiki/Gerhard_Gentzen" title="Gerhard Gentzen">Gerhard Gentzen</a> gave <a href="/wiki/Gentzen%27s_consistency_proof" title="Gentzen's consistency proof">a proof of the consistency</a> of Peano's axioms, using <a href="/wiki/Transfinite_induction" title="Transfinite induction">transfinite induction</a> up to an <a href="/wiki/Ordinal_number" title="Ordinal number">ordinal</a> called <a href="/wiki/Epsilon_numbers_(mathematics)" class="mw-redirect" title="Epsilon numbers (mathematics)">ε<sub>0</sub></a>.<sup id="cite_ref-24" class="reference"><a href="#cite_note-24"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> Gentzen explained: "The aim of the present paper is to prove the consistency of elementary number theory or, rather, to reduce the question of consistency to certain fundamental principles". Gentzen's proof is arguably finitistic, since the transfinite ordinal ε<sub>0</sub> can be encoded in terms of finite objects (for example, as a <a href="/wiki/Turing_machine" title="Turing machine">Turing machine</a> describing a suitable order on the integers, or more abstractly as consisting of the finite <a href="/wiki/Tree_(set_theory)" title="Tree (set theory)">trees</a>, suitably linearly ordered). Whether or not Gentzen's proof meets the requirements Hilbert envisioned is unclear: there is no generally accepted definition of exactly what is meant by a finitistic proof, and Hilbert himself never gave a precise definition. </p><p>The vast majority of contemporary mathematicians believe that Peano's axioms are consistent, relying either on intuition or the acceptance of a consistency proof such as <a href="/wiki/Gentzen%27s_consistency_proof" title="Gentzen's consistency proof">Gentzen's proof</a>. A small number of philosophers and mathematicians, some of whom also advocate <a href="/wiki/Ultrafinitism" title="Ultrafinitism">ultrafinitism</a>, reject Peano's axioms because accepting the axioms amounts to accepting the infinite collection of natural numbers. In particular, addition (including the successor function) and multiplication are assumed to be <a href="/wiki/Partial_function#Total_function" title="Partial function">total</a>. Curiously, there are <a href="/wiki/Self-verifying_theories" title="Self-verifying theories">self-verifying theories</a> that are similar to PA but have subtraction and division instead of addition and multiplication, which are axiomatized in such a way to avoid proving sentences that correspond to the totality of addition and multiplication, but which are still able to prove all true <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Pi _{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Pi _{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4aab5a28da997de9084eef3e569bd1e072efc1aa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.797ex; height:2.509ex;" alt="{\displaystyle \Pi _{1}}"></span> theorems of PA, and yet can be extended to a consistent theory that proves its own consistency (stated as the non-existence of a Hilbert-style proof of "0=1").<sup id="cite_ref-FOOTNOTEWillard2001_25-0" class="reference"><a href="#cite_note-FOOTNOTEWillard2001-25"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Peano_arithmetic_as_first-order_theory">Peano arithmetic as first-order theory</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Peano_axioms&action=edit&section=10" title="Edit section: Peano arithmetic as first-order theory"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>All of the Peano axioms except the ninth axiom (the induction axiom) are statements in <a href="/wiki/First-order_logic" title="First-order logic">first-order logic</a>.<sup id="cite_ref-FOOTNOTEParteeTer_MeulenWall2012215_26-0" class="reference"><a href="#cite_note-FOOTNOTEParteeTer_MeulenWall2012215-26"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup> The arithmetical operations of addition and multiplication and the order relation can also be defined using first-order axioms. The axiom of induction above is <a href="/wiki/Second-order_logic" title="Second-order logic">second-order</a>, since it <a href="/wiki/Quantifier_(logic)" title="Quantifier (logic)">quantifies</a> over predicates (equivalently, sets of natural numbers rather than natural numbers). As an alternative one can consider a first-order <i><a href="/wiki/Axiom_schema" title="Axiom schema">axiom schema</a></i> of induction. Such a schema includes one axiom per predicate definable in the first-order language of Peano arithmetic, making it weaker than the second-order axiom.<sup id="cite_ref-FOOTNOTEHarsanyi1983_27-0" class="reference"><a href="#cite_note-FOOTNOTEHarsanyi1983-27"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup> The reason that it is weaker is that the number of predicates in first-order language is countable, whereas the number of sets of natural numbers is uncountable. Thus, there exist sets that cannot be described in first-order language (in fact, most sets have this property). </p><p>First-order axiomatizations of Peano arithmetic have another technical limitation. In second-order logic, it is possible to define the addition and multiplication operations from the <a href="/wiki/Successor_function" title="Successor function">successor operation</a>, but this cannot be done in the more restrictive setting of first-order logic. Therefore, the addition and multiplication operations are directly included in the <a href="/wiki/Signature_(logic)" title="Signature (logic)">signature</a> of Peano arithmetic, and axioms are included that relate the three operations to each other. </p><p>The following list of axioms (along with the usual axioms of equality), which contains six of the seven axioms of <a href="/wiki/Robinson_arithmetic" title="Robinson arithmetic">Robinson arithmetic</a>, is sufficient for this purpose:<sup id="cite_ref-FOOTNOTEMendelson1997155_28-0" class="reference"><a href="#cite_note-FOOTNOTEMendelson1997155-28"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup> </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall x\ (0\neq S(x))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mtext> </mtext> <mo stretchy="false">(</mo> <mn>0</mn> <mo>≠</mo> <mi>S</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall x\ (0\neq S(x))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca1c1be6a8b2ae8d33391523534ca0e095a1866a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.911ex; height:2.843ex;" alt="{\displaystyle \forall x\ (0\neq S(x))}"></span></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall x,y\ (S(x)=S(y)\Rightarrow x=y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mtext> </mtext> <mo stretchy="false">(</mo> <mi>S</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>S</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo stretchy="false">⇒</mo> <mi>x</mi> <mo>=</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall x,y\ (S(x)=S(y)\Rightarrow x=y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1fb4028b518a0bc62ed7c64f0ca2543fc1babf78" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.6ex; height:2.843ex;" alt="{\displaystyle \forall x,y\ (S(x)=S(y)\Rightarrow x=y)}"></span></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall x\ (x+0=x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mtext> </mtext> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mn>0</mn> <mo>=</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall x\ (x+0=x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/770b22ef234b5ba4abef0a8f6f056761dbf0f5f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.773ex; height:2.843ex;" alt="{\displaystyle \forall x\ (x+0=x)}"></span></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall x,y\ (x+S(y)=S(x+y))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mtext> </mtext> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>S</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>S</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall x,y\ (x+S(y)=S(x+y))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b6df0525828adaf4e0794f7ad4aed650dc61ce3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.568ex; height:2.843ex;" alt="{\displaystyle \forall x,y\ (x+S(y)=S(x+y))}"></span></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall x\ (x\cdot 0=0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mtext> </mtext> <mo stretchy="false">(</mo> <mi>x</mi> <mo>⋅</mo> <mn>0</mn> <mo>=</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall x\ (x\cdot 0=0)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1af6515eb80999b5dd22ddfa5ae5058e0e3f1989" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.444ex; height:2.843ex;" alt="{\displaystyle \forall x\ (x\cdot 0=0)}"></span></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall x,y\ (x\cdot S(y)=x\cdot y+x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mtext> </mtext> <mo stretchy="false">(</mo> <mi>x</mi> <mo>⋅</mo> <mi>S</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>x</mi> <mo>⋅</mo> <mi>y</mi> <mo>+</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall x,y\ (x\cdot S(y)=x\cdot y+x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ffa6b2809c93710fc948fb78b96cc3e77c815cb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.107ex; height:2.843ex;" alt="{\displaystyle \forall x,y\ (x\cdot S(y)=x\cdot y+x)}"></span></li></ul> <p>In addition to this list of numerical axioms, Peano arithmetic contains the induction schema, which consists of a <a href="/wiki/Recursively_enumerable_set" class="mw-redirect" title="Recursively enumerable set">recursively enumerable</a> and even decidable set of <a href="/wiki/Axioms" class="mw-redirect" title="Axioms">axioms</a>. For each formula <span class="nowrap"><i>φ</i>(<i>x</i>, <i>y</i><sub>1</sub>, ..., <i>y</i><sub><i>k</i></sub>)</span> in the language of Peano arithmetic, the <b>first-order induction axiom</b> for <i>φ</i> is the sentence </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall {\bar {y}}{\Bigg (}{\bigg (}\varphi (0,{\bar {y}})\land \forall x{\Big (}\varphi (x,{\bar {y}})\Rightarrow \varphi (S(x),{\bar {y}}){\Big )}{\bigg )}\Rightarrow \forall x\varphi (x,{\bar {y}}){\Bigg )}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>y</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.470em" minsize="2.470em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> </mrow> <mi>φ</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>y</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>∧</mo> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">(</mo> </mrow> </mrow> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>y</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mo stretchy="false">⇒</mo> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>S</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>y</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> <mo stretchy="false">⇒</mo> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>y</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.470em" minsize="2.470em">)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall {\bar {y}}{\Bigg (}{\bigg (}\varphi (0,{\bar {y}})\land \forall x{\Big (}\varphi (x,{\bar {y}})\Rightarrow \varphi (S(x),{\bar {y}}){\Big )}{\bigg )}\Rightarrow \forall x\varphi (x,{\bar {y}}){\Bigg )}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eb2e6715a396abde5f542bf3c74a8b9b31326544" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:58.65ex; height:7.509ex;" alt="{\displaystyle \forall {\bar {y}}{\Bigg (}{\bigg (}\varphi (0,{\bar {y}})\land \forall x{\Big (}\varphi (x,{\bar {y}})\Rightarrow \varphi (S(x),{\bar {y}}){\Big )}{\bigg )}\Rightarrow \forall x\varphi (x,{\bar {y}}){\Bigg )}}"></span></dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {y}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>y</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bar {y}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b298744237368f34e61ff7dc90b34016a7037af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.302ex; height:2.343ex;" alt="{\displaystyle {\bar {y}}}"></span> is an abbreviation for <i>y</i><sub>1</sub>,...,<i>y</i><sub><i>k</i></sub>. The first-order induction schema includes every instance of the first-order induction axiom; that is, it includes the induction axiom for every formula <i>φ</i>. </p> <div class="mw-heading mw-heading3"><h3 id="Equivalent_axiomatizations">Equivalent axiomatizations</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Peano_axioms&action=edit&section=11" title="Edit section: Equivalent axiomatizations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The above axiomatization of Peano arithmetic uses a signature that only has symbols for zero as well as the successor, addition, and multiplications operations. There are many other different, but equivalent, axiomatizations. One such alternative<sup id="cite_ref-FOOTNOTEKaye199116–18_29-0" class="reference"><a href="#cite_note-FOOTNOTEKaye199116–18-29"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup> uses an order relation symbol instead of the successor operation and the language of <a href="/wiki/Semiring#Discretely_ordered_semirings" title="Semiring">discretely ordered semirings</a> (axioms 1-7 for semirings, 8-10 on order, 11-13 regarding compatibility, and 14-15 for discreteness): </p> <ol><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall x,y,z\ ((x+y)+z=x+(y+z))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mtext> </mtext> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>z</mi> <mo>=</mo> <mi>x</mi> <mo>+</mo> <mo stretchy="false">(</mo> <mi>y</mi> <mo>+</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall x,y,z\ ((x+y)+z=x+(y+z))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd4620c0d31885a93e276a76477062664e63773" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.549ex; height:2.843ex;" alt="{\displaystyle \forall x,y,z\ ((x+y)+z=x+(y+z))}"></span>, i.e., addition is <a href="/wiki/Associative_property" title="Associative property">associative</a>.</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall x,y\ (x+y=y+x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mtext> </mtext> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mo>=</mo> <mi>y</mi> <mo>+</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall x,y\ (x+y=y+x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3180ef8bd8264118f81914c3d24e3bd28eef6590" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.951ex; height:2.843ex;" alt="{\displaystyle \forall x,y\ (x+y=y+x)}"></span>, i.e., addition is <a href="/wiki/Commutative_property" title="Commutative property">commutative</a>.</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall x,y,z\ ((x\cdot y)\cdot z=x\cdot (y\cdot z))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mtext> </mtext> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>⋅</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <mi>z</mi> <mo>=</mo> <mi>x</mi> <mo>⋅</mo> <mo stretchy="false">(</mo> <mi>y</mi> <mo>⋅</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall x,y,z\ ((x\cdot y)\cdot z=x\cdot (y\cdot z))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b765692ebf2ef1d751b11aeb0d58979cdedb0a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.904ex; height:2.843ex;" alt="{\displaystyle \forall x,y,z\ ((x\cdot y)\cdot z=x\cdot (y\cdot z))}"></span>, i.e., multiplication is associative.</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall x,y\ (x\cdot y=y\cdot x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mtext> </mtext> <mo stretchy="false">(</mo> <mi>x</mi> <mo>⋅</mo> <mi>y</mi> <mo>=</mo> <mi>y</mi> <mo>⋅</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall x,y\ (x\cdot y=y\cdot x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dce4987321f92156c07cd75a00f44b49f7bca46e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.629ex; height:2.843ex;" alt="{\displaystyle \forall x,y\ (x\cdot y=y\cdot x)}"></span>, i.e., multiplication is commutative.</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall x,y,z\ (x\cdot (y+z)=(x\cdot y)+(x\cdot z))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mtext> </mtext> <mo stretchy="false">(</mo> <mi>x</mi> <mo>⋅</mo> <mo stretchy="false">(</mo> <mi>y</mi> <mo>+</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>⋅</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>⋅</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall x,y,z\ (x\cdot (y+z)=(x\cdot y)+(x\cdot z))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ae41f15dc573a2b6d24e437fd72a3fc94a907381" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:37.044ex; height:2.843ex;" alt="{\displaystyle \forall x,y,z\ (x\cdot (y+z)=(x\cdot y)+(x\cdot z))}"></span>, i.e., multiplication <a href="/wiki/Distributive_property" title="Distributive property">distributes</a> over addition.</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall x\ (x+0=x\land x\cdot 0=0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mtext> </mtext> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mn>0</mn> <mo>=</mo> <mi>x</mi> <mo>∧</mo> <mi>x</mi> <mo>⋅</mo> <mn>0</mn> <mo>=</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall x\ (x+0=x\land x\cdot 0=0)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/936c8f4dba3130fdd4510c2fc86bebcc85489393" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.788ex; height:2.843ex;" alt="{\displaystyle \forall x\ (x+0=x\land x\cdot 0=0)}"></span>, i.e., zero is an <a href="/wiki/Identity_element" title="Identity element">identity</a> for addition, and an <a href="/wiki/Absorbing_element" title="Absorbing element">absorbing element</a> for multiplication (actually superfluous<sup id="cite_ref-30" class="reference"><a href="#cite_note-30"><span class="cite-bracket">[</span>note 3<span class="cite-bracket">]</span></a></sup>).</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall x\ (x\cdot 1=x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mtext> </mtext> <mo stretchy="false">(</mo> <mi>x</mi> <mo>⋅</mo> <mn>1</mn> <mo>=</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall x\ (x\cdot 1=x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/477e75d5670183c332827c6a2ce7a300d1ae0ce6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.612ex; height:2.843ex;" alt="{\displaystyle \forall x\ (x\cdot 1=x)}"></span>, i.e., one is an <a href="/wiki/Identity_element" title="Identity element">identity</a> for multiplication.</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall x,y,z\ (x<y\land y<z\Rightarrow x<z)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mtext> </mtext> <mo stretchy="false">(</mo> <mi>x</mi> <mo><</mo> <mi>y</mi> <mo>∧</mo> <mi>y</mi> <mo><</mo> <mi>z</mi> <mo stretchy="false">⇒</mo> <mi>x</mi> <mo><</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall x,y,z\ (x<y\land y<z\Rightarrow x<z)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a34ee7d42a966de9efc39290c4325ad22f3e9756" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.962ex; height:2.843ex;" alt="{\displaystyle \forall x,y,z\ (x<y\land y<z\Rightarrow x<z)}"></span>, i.e., the '<' operator is <a href="/wiki/Transitive_relation" title="Transitive relation">transitive</a>.</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall x\ (\neg (x<x))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mtext> </mtext> <mo stretchy="false">(</mo> <mi mathvariant="normal">¬</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo><</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall x\ (\neg (x<x))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d2276f62e2db0587f8e830244d178ab710157e8f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.13ex; height:2.843ex;" alt="{\displaystyle \forall x\ (\neg (x<x))}"></span>, i.e., the '<' operator is <a href="/wiki/Reflexive_relation" title="Reflexive relation">irreflexive</a>.</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall x,y\ (x<y\lor x=y\lor y<x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mtext> </mtext> <mo stretchy="false">(</mo> <mi>x</mi> <mo><</mo> <mi>y</mi> <mo>∨</mo> <mi>x</mi> <mo>=</mo> <mi>y</mi> <mo>∨</mo> <mi>y</mi> <mo><</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall x,y\ (x<y\lor x=y\lor y<x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5eee03e246af51163bffe16506783546fad62113" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.118ex; height:2.843ex;" alt="{\displaystyle \forall x,y\ (x<y\lor x=y\lor y<x)}"></span>, i.e., the ordering satisfies <a href="/wiki/Trichotomy_(mathematics)" class="mw-redirect" title="Trichotomy (mathematics)">trichotomy</a>.</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall x,y,z\ (x<y\Rightarrow x+z<y+z)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mtext> </mtext> <mo stretchy="false">(</mo> <mi>x</mi> <mo><</mo> <mi>y</mi> <mo stretchy="false">⇒</mo> <mi>x</mi> <mo>+</mo> <mi>z</mi> <mo><</mo> <mi>y</mi> <mo>+</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall x,y,z\ (x<y\Rightarrow x+z<y+z)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/537b862aef0a0cec8f18e4841a98c5a443afa3db" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.962ex; height:2.843ex;" alt="{\displaystyle \forall x,y,z\ (x<y\Rightarrow x+z<y+z)}"></span>, i.e. the ordering is preserved under addition of the same element.</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall x,y,z\ (0<z\land x<y\Rightarrow x\cdot z<y\cdot z)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mtext> </mtext> <mo stretchy="false">(</mo> <mn>0</mn> <mo><</mo> <mi>z</mi> <mo>∧</mo> <mi>x</mi> <mo><</mo> <mi>y</mi> <mo stretchy="false">⇒</mo> <mi>x</mi> <mo>⋅</mo> <mi>z</mi> <mo><</mo> <mi>y</mi> <mo>⋅</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall x,y,z\ (0<z\land x<y\Rightarrow x\cdot z<y\cdot z)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/be46f25a6b24520be2972a92a6bb88a5fb356187" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:37.571ex; height:2.843ex;" alt="{\displaystyle \forall x,y,z\ (0<z\land x<y\Rightarrow x\cdot z<y\cdot z)}"></span>, i.e. the ordering is preserved under multiplication by the same positive element.</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall x,y\ (x<y\Rightarrow \exists z\ (x+z=y))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mtext> </mtext> <mo stretchy="false">(</mo> <mi>x</mi> <mo><</mo> <mi>y</mi> <mo stretchy="false">⇒</mo> <mi mathvariant="normal">∃</mi> <mi>z</mi> <mtext> </mtext> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>z</mi> <mo>=</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall x,y\ (x<y\Rightarrow \exists z\ (x+z=y))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b4ad2edd0db588a678c1bb4a42b8584f84c2ba3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.682ex; height:2.843ex;" alt="{\displaystyle \forall x,y\ (x<y\Rightarrow \exists z\ (x+z=y))}"></span>, i.e. given any two distinct elements, the larger is the smaller plus another element.</li> <li><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 0<1\land \forall x\ (x>0\Rightarrow x\geq 1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo><</mo> <mn>1</mn> <mo>∧</mo> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mtext> </mtext> <mo stretchy="false">(</mo> <mi>x</mi> <mo>></mo> <mn>0</mn> <mo stretchy="false">⇒</mo> <mi>x</mi> <mo>≥</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0<1\land \forall x\ (x>0\Rightarrow x\geq 1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d5786f7e95a2bcc0ca8693313535657999b2131a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.813ex; height:2.843ex;" alt="{\displaystyle 0<1\land \forall x\ (x>0\Rightarrow x\geq 1)}"></span>, i.e. zero and one are distinct and there is no element between them. In other words, 0 is <a href="/wiki/Covering_relation" title="Covering relation">covered</a> by 1, which suggests that these numbers are discrete.</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall x\ (x\geq 0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mtext> </mtext> <mo stretchy="false">(</mo> <mi>x</mi> <mo>≥</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall x\ (x\geq 0)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e6a9168ad922af9613e41b6e675ef592d80a7995" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.603ex; height:2.843ex;" alt="{\displaystyle \forall x\ (x\geq 0)}"></span>, i.e. zero is the minimum element.</li></ol> <p>The theory defined by these axioms is known as <b>PA<sup>−</sup></b>. It is also incomplete and one of its important properties is that any structure <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 M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"></span> satisfying this theory has an initial segment (ordered by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \leq }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>≤</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \leq }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/440568a09c3bfdf0e1278bfa79eb137c04e94035" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \leq }"></span>) isomorphic to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {N} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {N} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fdf9a96b565ea202d0f4322e9195613fb26a9bed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {N} }"></span>. Elements in that segment are called <b>standard</b> elements, while other elements are called <b>nonstandard</b> elements. </p><p>Finally, Peano arithmetic <b>PA</b> is obtained by adding the first-order induction schema. </p> <div class="mw-heading mw-heading3"><h3 id="Undecidability_and_incompleteness">Undecidability and incompleteness</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Peano_axioms&action=edit&section=12" title="Edit section: Undecidability and incompleteness"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>According to <a href="/wiki/G%C3%B6del%27s_incompleteness_theorems" title="Gödel's incompleteness theorems">Gödel's incompleteness theorems</a>, the theory of <b>PA</b> (if consistent) is incomplete. Consequently, there are sentences of <a href="/wiki/First-order_logic" title="First-order logic">first-order logic</a> (FOL) that are true in the standard model of <b>PA</b> but are not a consequence of the FOL axiomatization. Essential incompleteness already arises for theories with weaker axioms, such as <a href="/wiki/Robinson_arithmetic" title="Robinson arithmetic">Robinson arithmetic</a>. </p><p>Closely related to the above incompleteness result (via <a href="/wiki/G%C3%B6del%27s_completeness_theorem" title="Gödel's completeness theorem">Gödel's completeness theorem</a> for FOL) it follows that there is no <a href="/wiki/Algorithm" title="Algorithm">algorithm</a> for deciding whether a given FOL sentence is a consequence of a first-order axiomatization of Peano arithmetic or not. Hence, <b>PA</b> is an example of an <a href="/wiki/Decidability_(logic)#Decidability_of_a_theory" title="Decidability (logic)">undecidable theory</a>. Undecidability arises already for the existential sentences of <b>PA</b>, due to the negative answer to <a href="/wiki/Hilbert%27s_tenth_problem" title="Hilbert's tenth problem">Hilbert's tenth problem</a>, whose proof implies that all <a href="/wiki/Computably_enumerable" class="mw-redirect" title="Computably enumerable">computably enumerable</a> sets are <a href="/wiki/Diophantine_set" title="Diophantine set">diophantine sets</a>, and thus definable by existentially quantified formulas (with free variables) of <b>PA</b>. Formulas of <b>PA</b> with higher <a href="/wiki/Quantifier_rank" title="Quantifier rank">quantifier rank</a> (more quantifier alternations) than existential formulas are more expressive, and define sets in the higher levels of the <a href="/wiki/Arithmetical_hierarchy" title="Arithmetical hierarchy">arithmetical hierarchy</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Nonstandard_models">Nonstandard models</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Peano_axioms&action=edit&section=13" title="Edit section: Nonstandard models"><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-standard_model_of_arithmetic" title="Non-standard model of arithmetic">Non-standard model of arithmetic</a></div> <p>Although the usual <a href="/wiki/Natural_number" title="Natural number">natural numbers</a> satisfy the axioms of <a href="#Equivalent_axiomatizations">PA</a>, there are other models as well (called "<a href="/wiki/Non-standard_model" title="Non-standard model">non-standard models</a>"); the <a href="/wiki/Compactness_theorem" title="Compactness theorem">compactness theorem</a> implies that the existence of nonstandard elements cannot be excluded in first-order logic.<sup id="cite_ref-FOOTNOTEHermes1973VI.4.3_31-0" class="reference"><a href="#cite_note-FOOTNOTEHermes1973VI.4.3-31"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup> The upward <a href="/wiki/L%C3%B6wenheim%E2%80%93Skolem_theorem" title="Löwenheim–Skolem theorem">Löwenheim–Skolem theorem</a> shows that there are nonstandard models of PA of all infinite cardinalities. This is not the case for the original (second-order) Peano axioms, which have only one model, up to isomorphism.<sup id="cite_ref-FOOTNOTEHermes1973VI.3.1_32-0" class="reference"><a href="#cite_note-FOOTNOTEHermes1973VI.3.1-32"><span class="cite-bracket">[</span>29<span class="cite-bracket">]</span></a></sup> This illustrates one way the first-order system PA is weaker than the second-order Peano axioms. </p><p>When interpreted as a proof within a first-order <a href="/wiki/Set_theory" title="Set theory">set theory</a>, such as <a href="/wiki/Zermelo%E2%80%93Fraenkel_set_theory" title="Zermelo–Fraenkel set theory">ZFC</a>, Dedekind's categoricity proof for PA shows that each model of set theory has a unique model of the Peano axioms, up to isomorphism, that embeds as an initial segment of all other models of PA contained within that model of set theory. In the standard model of set theory, this smallest model of PA is the standard model of PA; however, in a nonstandard model of set theory, it may be a nonstandard model of PA. This situation cannot be avoided with any first-order formalization of set theory. </p><p>It is natural to ask whether a countable nonstandard model can be explicitly constructed. The answer is affirmative as <a href="/wiki/Skolem" class="mw-redirect" title="Skolem">Skolem</a> in 1933 provided an explicit construction of such a <a href="/wiki/Non-standard_model_of_arithmetic" title="Non-standard model of arithmetic">nonstandard model</a>. On the other hand, <a href="/wiki/Tennenbaum%27s_theorem" title="Tennenbaum's theorem">Tennenbaum's theorem</a>, proved in 1959, shows that there is no countable nonstandard model of PA in which either the addition or multiplication operation is <a href="/wiki/Computable_function" title="Computable function">computable</a>.<sup id="cite_ref-FOOTNOTEKaye1991Section_11.3_33-0" class="reference"><a href="#cite_note-FOOTNOTEKaye1991Section_11.3-33"><span class="cite-bracket">[</span>30<span class="cite-bracket">]</span></a></sup> This result shows it is difficult to be completely explicit in describing the addition and multiplication operations of a countable nonstandard model of PA. There is only one possible <a href="/wiki/Order_type" title="Order type">order type</a> of a countable nonstandard model. Letting <i>ω</i> be the order type of the natural numbers, <i>ζ</i> be the order type of the integers, and <i>η</i> be the order type of the rationals, the order type of any countable nonstandard model of PA is <span class="nowrap"><i>ω</i> + <i>ζ</i>·<i>η</i></span>, which can be visualized as a copy of the natural numbers followed by a dense linear ordering of copies of the integers. </p> <div class="mw-heading mw-heading4"><h4 id="Overspill">Overspill</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Peano_axioms&action=edit&section=14" title="Edit section: Overspill"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A <b>cut</b> in a nonstandard model <i>M</i> is a nonempty subset <i>C</i> of <i>M</i> so that <i>C</i> is downward closed (<i>x</i> < <i>y</i> and <i>y</i> ∈ <i>C</i> ⇒ <i>x</i> ∈ <i>C</i>) and <i>C</i> is closed under successor. A <b>proper cut</b> is a cut that is a proper subset of <i>M</i>. Each nonstandard model has many proper cuts, including one that corresponds to the standard natural numbers. However, the induction scheme in Peano arithmetic prevents any proper cut from being definable. The overspill lemma, first proved by Abraham Robinson, formalizes this fact. </p> <style data-mw-deduplicate="TemplateStyles:r1110004140">.mw-parser-output .math_theorem{margin:1em 2em;padding:0.5em 1em 0.4em;border:1px solid #aaa;overflow:hidden}@media(max-width:500px){.mw-parser-output .math_theorem{margin:1em 0em;padding:0.5em 0.5em 0.4em}}</style><div class="math_theorem" style=""> <p><strong class="theorem-name">Overspill lemma<sup id="cite_ref-FOOTNOTEKaye199170ff._34-0" class="reference"><a href="#cite_note-FOOTNOTEKaye199170ff.-34"><span class="cite-bracket">[</span>31<span class="cite-bracket">]</span></a></sup></strong><span class="theoreme-tiret"> — </span>Let <i>M</i> be a nonstandard model of PA and let <i>C</i> be a proper cut of <i>M</i>. Suppose that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {a}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bar {a}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6513ea66e4b94de05c92bc66f00e6e021d1c2a9c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:2.009ex;" alt="{\displaystyle {\bar {a}}}"></span> is a tuple of elements of <i>M</i> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi (x,{\bar {a}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi (x,{\bar {a}})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e644eaf58026570f39d67be1b63bb2198412983c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.788ex; height:2.843ex;" alt="{\displaystyle \phi (x,{\bar {a}})}"></span> is a formula in the language of arithmetic so that </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 M\vDash \phi (b,{\bar {a}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mo>⊨</mo> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mi>b</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M\vDash \phi (b,{\bar {a}})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d25b60f6708c9fdc4c944dcc5ba1923a4ba16b2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.203ex; height:2.843ex;" alt="{\displaystyle M\vDash \phi (b,{\bar {a}})}"></span> for all <i>b</i> ∈ <i>C</i>.</dd></dl> <p>Then there is a <i>c</i> in <i>M</i> that is greater than every element of <i>C</i> such that </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 M\vDash \phi (c,{\bar {a}}).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mo>⊨</mo> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mi>c</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M\vDash \phi (c,{\bar {a}}).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9b7251bed7cd4be247843e10417a6f386bf4ff5f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.86ex; height:2.843ex;" alt="{\displaystyle M\vDash \phi (c,{\bar {a}}).}"></span></dd></dl> </div> <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=Peano_axioms&action=edit&section=15" 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 .portalbox-entry{display:table-row;font-size:85%;line-height:110%;height:1.9em;font-style:italic;font-weight:bold}.mw-parser-output .portalbox-image{display:table-cell;padding:0.2em;vertical-align:middle;text-align:center}.mw-parser-output .portalbox-link{display:table-cell;padding:0.2em 0.2em 0.2em 0.3em;vertical-align:middle}@media(min-width:720px){.mw-parser-output .portalleft{clear:left;float:left;margin:0.5em 1em 0.5em 0}.mw-parser-output .portalright{clear:right;float:right;margin:0.5em 0 0.5em 1em}}</style><ul role="navigation" aria-label="Portals" class="noprint portalbox portalborder portalright"> <li class="portalbox-entry"><span class="portalbox-image"><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><span class="portalbox-link"><a href="/wiki/Portal:Philosophy" title="Portal:Philosophy">Philosophy portal</a></span></li><li class="portalbox-entry"><span class="portalbox-image"><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/28px-Nuvola_apps_edu_mathematics_blue-p.svg.png" decoding="async" width="28" height="28" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/42px-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/56px-Nuvola_apps_edu_mathematics_blue-p.svg.png 2x" data-file-width="128" data-file-height="128" /></a></span></span><span class="portalbox-link"><a href="/wiki/Portal:Mathematics" title="Portal:Mathematics">Mathematics portal</a></span></li></ul> <ul><li><a href="/wiki/Foundations_of_mathematics" title="Foundations of mathematics">Foundations of mathematics</a></li> <li><a href="/wiki/Frege%27s_theorem" title="Frege's theorem">Frege's theorem</a></li> <li><a href="/wiki/Goodstein%27s_theorem" title="Goodstein's theorem">Goodstein's theorem</a></li> <li><a href="/wiki/Neo-logicism" class="mw-redirect" title="Neo-logicism">Neo-logicism</a></li> <li><a href="/wiki/Non-standard_model_of_arithmetic" title="Non-standard model of arithmetic">Non-standard model of arithmetic</a></li> <li><a href="/wiki/Paris%E2%80%93Harrington_theorem" title="Paris–Harrington theorem">Paris–Harrington theorem</a></li> <li><a href="/wiki/Presburger_arithmetic" title="Presburger arithmetic">Presburger arithmetic</a></li> <li><a href="/wiki/Skolem_arithmetic" title="Skolem arithmetic">Skolem arithmetic</a></li> <li><a href="/wiki/Robinson_arithmetic" title="Robinson arithmetic">Robinson arithmetic</a></li> <li><a href="/wiki/Second-order_arithmetic" title="Second-order arithmetic">Second-order arithmetic</a></li> <li><a href="/wiki/Typographical_Number_Theory" title="Typographical Number Theory">Typographical Number Theory</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=Peano_axioms&action=edit&section=16" 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"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text">the nearest light piece corresponding to 0, and a neighbor piece corresponding to successor</span> </li> <li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text">The non-contiguous set satisfies axiom 1 as it has a 0 element, 2–5 as it doesn't affect equality relations, 6 & 8 as all pieces have a successor, bar the zero element and axiom 7 as no two dominos topple, or are toppled by, the same piece.</span> </li> <li id="cite_note-30"><span class="mw-cite-backlink"><b><a href="#cite_ref-30">^</a></b></span> <span class="reference-text">"<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall x\ (x\cdot 0=0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mtext> </mtext> <mo stretchy="false">(</mo> <mi>x</mi> <mo>⋅</mo> <mn>0</mn> <mo>=</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall x\ (x\cdot 0=0)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1af6515eb80999b5dd22ddfa5ae5058e0e3f1989" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.444ex; height:2.843ex;" alt="{\displaystyle \forall x\ (x\cdot 0=0)}"></span>" can be proven from the other axioms (in first-order logic) as follows. Firstly, <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\cdot 0+x\cdot 0=x\cdot (0+0)=x\cdot 0=x\cdot 0+0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>⋅</mo> <mn>0</mn> <mo>+</mo> <mi>x</mi> <mo>⋅</mo> <mn>0</mn> <mo>=</mo> <mi>x</mi> <mo>⋅</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>+</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mi>x</mi> <mo>⋅</mo> <mn>0</mn> <mo>=</mo> <mi>x</mi> <mo>⋅</mo> <mn>0</mn> <mo>+</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\cdot 0+x\cdot 0=x\cdot (0+0)=x\cdot 0=x\cdot 0+0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/428ead53468ad9bc6c7bab82cc662f1d4b9334aa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:42.807ex; height:2.843ex;" alt="{\displaystyle x\cdot 0+x\cdot 0=x\cdot (0+0)=x\cdot 0=x\cdot 0+0}"></span> by distributivity and additive identity. Secondly, <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\cdot 0=0\lor x\cdot 0>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>⋅</mo> <mn>0</mn> <mo>=</mo> <mn>0</mn> <mo>∨</mo> <mi>x</mi> <mo>⋅</mo> <mn>0</mn> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\cdot 0=0\lor x\cdot 0>0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bceeb195e192d051edd395b84c264f2a04ef1f1e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:19.447ex; height:2.176ex;" alt="{\displaystyle x\cdot 0=0\lor x\cdot 0>0}"></span> by Axiom 15. If <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\cdot 0>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>⋅</mo> <mn>0</mn> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\cdot 0>0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/98d8681daf22fa99f492d12ca6557982c7cee0c7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.432ex; height:2.176ex;" alt="{\displaystyle x\cdot 0>0}"></span> then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\cdot 0+x\cdot 0>x\cdot 0+0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>⋅</mo> <mn>0</mn> <mo>+</mo> <mi>x</mi> <mo>⋅</mo> <mn>0</mn> <mo>></mo> <mi>x</mi> <mo>⋅</mo> <mn>0</mn> <mo>+</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\cdot 0+x\cdot 0>x\cdot 0+0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/78051fd5a888967b1915b5ab8c775bbf9fce8f3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:22.455ex; height:2.343ex;" alt="{\displaystyle x\cdot 0+x\cdot 0>x\cdot 0+0}"></span> by addition of the same element and commutativity, and hence <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\cdot 0+0>x\cdot 0+0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>⋅</mo> <mn>0</mn> <mo>+</mo> <mn>0</mn> <mo>></mo> <mi>x</mi> <mo>⋅</mo> <mn>0</mn> <mo>+</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\cdot 0+0>x\cdot 0+0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f59d1f96028909e403755d6f6a7e0fb12239cba5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:19.447ex; height:2.343ex;" alt="{\displaystyle x\cdot 0+0>x\cdot 0+0}"></span> by substitution, contradicting irreflexivity. Therefore it must be that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\cdot 0=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>⋅</mo> <mn>0</mn> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\cdot 0=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/44b1b207154bd1e280dfdeba3c7f12840a184ff2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.432ex; height:2.176ex;" alt="{\displaystyle x\cdot 0=0}"></span>.</span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Peano_axioms&action=edit&section=17" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Citations">Citations</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Peano_axioms&action=edit&section=18" title="Edit section: Citations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626"><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://www.dictionary.com/browse/peano">"Peano"</a>. <i><a href="/wiki/Random_House_Webster%27s_Unabridged_Dictionary" title="Random House Webster's Unabridged Dictionary">Random House Webster's Unabridged Dictionary</a></i>.</span> </li> <li id="cite_note-FOOTNOTEGrassmann1861-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEGrassmann1861_2-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFGrassmann1861">Grassmann 1861</a>.</span> </li> <li id="cite_note-FOOTNOTEWang1957145,_147"It_is_rather_well-known,_through_Peano's_own_acknowledgement,_that_Peano_[…]_made_extensive_use_of_Grassmann's_work_in_his_development_of_the_axioms._It_is_not_so_well-known_that_Grassmann_had_essentially_the_characterization_of_the_set_of_all_integers,_now_customary_in_texts_of_modern_algebra,_that_it_forms_an_ordered_[[integral_domain]]_in_wihich_each_set_of_positive_elements_has_a_least_member._[…]_[Grassmann's_book]_was_probably_the_first_serious_and_rather_successful_attempt_to_put_numbers_on_a_more_or_less_axiomatic_basis."-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEWang1957145,_147"It_is_rather_well-known,_through_Peano's_own_acknowledgement,_that_Peano_[…]_made_extensive_use_of_Grassmann's_work_in_his_development_of_the_axioms._It_is_not_so_well-known_that_Grassmann_had_essentially_the_characterization_of_the_set_of_all_integers,_now_customary_in_texts_of_modern_algebra,_that_it_forms_an_ordered_[[integral_domain]]_in_wihich_each_set_of_positive_elements_has_a_least_member._[…]_[Grassmann's_book]_was_probably_the_first_serious_and_rather_successful_attempt_to_put_numbers_on_a_more_or_less_axiomatic_basis."_3-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFWang1957">Wang 1957</a>, pp. 145, 147, "It is rather well-known, through Peano's own acknowledgement, that Peano […] made extensive use of Grassmann's work in his development of the axioms. It is not so well-known that Grassmann had essentially the characterization of the set of all integers, now customary in texts of modern algebra, that it forms an ordered <a href="/wiki/Integral_domain" title="Integral domain">integral domain</a> in wihich each set of positive elements has a least member. […] [Grassmann's book] was probably the first serious and rather successful attempt to put numbers on a more or less axiomatic basis.".</span> </li> <li id="cite_note-FOOTNOTEPeirce1881-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEPeirce1881_4-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFPeirce1881">Peirce 1881</a>.</span> </li> <li id="cite_note-FOOTNOTEShields1997-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEShields1997_5-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFShields1997">Shields 1997</a>.</span> </li> <li id="cite_note-FOOTNOTEVan_Heijenoort196794-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEVan_Heijenoort196794_6-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFVan_Heijenoort1967">Van Heijenoort 1967</a>, p. 94.</span> </li> <li id="cite_note-FOOTNOTEVan_Heijenoort19672-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEVan_Heijenoort19672_7-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFVan_Heijenoort1967">Van Heijenoort 1967</a>, p. 2.</span> </li> <li id="cite_note-FOOTNOTEVan_Heijenoort196783-8"><span class="mw-cite-backlink">^ <a href="#cite_ref-FOOTNOTEVan_Heijenoort196783_8-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-FOOTNOTEVan_Heijenoort196783_8-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><a href="#CITEREFVan_Heijenoort1967">Van Heijenoort 1967</a>, p. 83.</span> </li> <li id="cite_note-FOOTNOTEPeano18891-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEPeano18891_9-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFPeano1889">Peano 1889</a>, p. 1.</span> </li> <li id="cite_note-FOOTNOTEPeano190827-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEPeano190827_10-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFPeano1908">Peano 1908</a>, p. 27.</span> </li> <li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text">Matt DeVos, <a rel="nofollow" class="external text" href="https://www.sfu.ca/~mdevos/notes/graph/induction.pdf"><i>Mathematical Induction</i></a>, <a href="/wiki/Simon_Fraser_University" title="Simon Fraser University">Simon Fraser University</a></span> </li> <li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text">Gerardo con Diaz, <i><a rel="nofollow" class="external text" href="http://www.math.harvard.edu/archive/23a_fall_05/Handouts/induction.pdf">Mathematical Induction</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20130502163438/http://www.math.harvard.edu/archive/23a_fall_05/Handouts/induction.pdf">Archived</a> 2 May 2013 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a></i>, <a href="/wiki/Harvard_University" title="Harvard University">Harvard University</a></span> </li> <li id="cite_note-FOOTNOTEMeseguerGoguen1986sections_2.3_(p._464)_and_4.1_(p._471)-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEMeseguerGoguen1986sections_2.3_(p._464)_and_4.1_(p._471)_15-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFMeseguerGoguen1986">Meseguer & Goguen 1986</a>, sections 2.3 (p. 464) and 4.1 (p. 471).</span> </li> <li id="cite_note-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-16">^</a></b></span> <span class="reference-text">For formal proofs, see e.g. <a href="/wiki/File:Inductive_proofs_of_properties_of_add,_mult_from_recursive_definitions.pdf" title="File:Inductive proofs of properties of add, mult from recursive definitions.pdf">File:Inductive proofs of properties of add, mult from recursive definitions.pdf</a>.</span> </li> <li id="cite_note-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-17">^</a></b></span> <span class="reference-text"><a href="#CITEREFSuppes1960">Suppes 1960</a>, <a href="#CITEREFHatcher2014">Hatcher 2014</a></span> </li> <li id="cite_note-FOOTNOTETarskiGivant1987Section_7.6-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTETarskiGivant1987Section_7.6_18-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFTarskiGivant1987">Tarski & Givant 1987</a>, Section 7.6.</span> </li> <li id="cite_note-19"><span class="mw-cite-backlink"><b><a href="#cite_ref-19">^</a></b></span> <span class="reference-text"><a href="#CITEREFFritz1952">Fritz 1952</a>, <a rel="nofollow" class="external text" href="https://books.google.com/books?id=Yl_XAwAAQBAJ&pg=PA137,">p. 137</a><br />An illustration of 'interpretation' is Russell's own definition of 'cardinal number'. The uninterpreted system in this case is Peano's axioms for the number system, whose three primitive ideas and five axioms, Peano believed, were sufficient to enable one to derive all the properties of the system of natural numbers. Actually, Russell maintains, Peano's axioms define any progression of the form <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},x_{1},x_{2},\ldots ,x_{n},\ldots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>,</mo> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{0},x_{1},x_{2},\ldots ,x_{n},\ldots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e67e2d688614d2d85890ab707b820959fe1f13cd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:20.703ex; height:2.009ex;" alt="{\displaystyle x_{0},x_{1},x_{2},\ldots ,x_{n},\ldots }"></span> of which the series of the natural numbers is one instance.</span> </li> <li id="cite_note-20"><span class="mw-cite-backlink"><b><a href="#cite_ref-20">^</a></b></span> <span class="reference-text"><a href="#CITEREFGray2013">Gray 2013</a>, <a rel="nofollow" class="external text" href="https://archive.org/details/henripoincaresci0000gray/page/133">p. 133</a><br />So Poincaré turned to see whether logicism could generate arithmetic, more precisely, the arithmetic of ordinals. Couturat, said Poincaré, had accepted the Peano axioms as a definition of a number. But this will not do. The axioms cannot be shown to be free of contradiction by finding examples of them, and any attempt to show that they were contradiction-free by examining the totality of their implications would require the very principle of mathematical induction Couturat believed they implied. For (in a further passage dropped from S&M) either one assumed the principle in order to prove it, which would only prove that if it is true it is not self-contradictory, which says nothing; or one used the principle in another form than the one stated, in which case one must show that the number of steps in one's reasoning was an integer according to the new definition, but this could not be done (1905c, 834).</span> </li> <li id="cite_note-FOOTNOTEHilbert1902-21"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEHilbert1902_21-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFHilbert1902">Hilbert 1902</a>.</span> </li> <li id="cite_note-FOOTNOTEGödel1931-22"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEGödel1931_22-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFGödel1931">Gödel 1931</a>.</span> </li> <li id="cite_note-23"><span class="mw-cite-backlink"><b><a href="#cite_ref-23">^</a></b></span> <span class="reference-text"><a href="#CITEREFGödel1958">Gödel 1958</a></span> </li> <li id="cite_note-24"><span class="mw-cite-backlink"><b><a href="#cite_ref-24">^</a></b></span> <span class="reference-text"><a href="#CITEREFGentzen1936">Gentzen 1936</a></span> </li> <li id="cite_note-FOOTNOTEWillard2001-25"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEWillard2001_25-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFWillard2001">Willard 2001</a>.</span> </li> <li id="cite_note-FOOTNOTEParteeTer_MeulenWall2012215-26"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEParteeTer_MeulenWall2012215_26-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFParteeTer_MeulenWall2012">Partee, Ter Meulen & Wall 2012</a>, p. 215.</span> </li> <li id="cite_note-FOOTNOTEHarsanyi1983-27"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEHarsanyi1983_27-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFHarsanyi1983">Harsanyi (1983)</a>.</span> </li> <li id="cite_note-FOOTNOTEMendelson1997155-28"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEMendelson1997155_28-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFMendelson1997">Mendelson 1997</a>, p. 155.</span> </li> <li id="cite_note-FOOTNOTEKaye199116–18-29"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEKaye199116–18_29-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFKaye1991">Kaye 1991</a>, pp. 16–18.</span> </li> <li id="cite_note-FOOTNOTEHermes1973VI.4.3-31"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEHermes1973VI.4.3_31-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFHermes1973">Hermes 1973</a>, VI.4.3, presenting a theorem of <a href="/wiki/Thoralf_Skolem" title="Thoralf Skolem">Thoralf Skolem</a></span> </li> <li id="cite_note-FOOTNOTEHermes1973VI.3.1-32"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEHermes1973VI.3.1_32-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFHermes1973">Hermes 1973</a>, VI.3.1.</span> </li> <li id="cite_note-FOOTNOTEKaye1991Section_11.3-33"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEKaye1991Section_11.3_33-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFKaye1991">Kaye 1991</a>, Section 11.3.</span> </li> <li id="cite_note-FOOTNOTEKaye199170ff.-34"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEKaye199170ff._34-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFKaye1991">Kaye 1991</a>, pp. 70ff..</span> </li> </ol></div></div> <div class="mw-heading mw-heading3"><h3 id="Sources">Sources</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Peano_axioms&action=edit&section=19" title="Edit section: Sources"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239549316">.mw-parser-output .refbegin{margin-bottom:0.5em}.mw-parser-output .refbegin-hanging-indents>ul{margin-left:0}.mw-parser-output .refbegin-hanging-indents>ul>li{margin-left:0;padding-left:3.2em;text-indent:-3.2em}.mw-parser-output .refbegin-hanging-indents ul,.mw-parser-output .refbegin-hanging-indents ul li{list-style:none}@media(max-width:720px){.mw-parser-output .refbegin-hanging-indents>ul>li{padding-left:1.6em;text-indent:-1.6em}}.mw-parser-output .refbegin-columns{margin-top:0.3em}.mw-parser-output .refbegin-columns ul{margin-top:0}.mw-parser-output .refbegin-columns li{page-break-inside:avoid;break-inside:avoid-column}@media screen{.mw-parser-output .refbegin{font-size:90%}}</style><div class="refbegin" style=""> <ul><li><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFDavis1974" class="citation book cs1"><a href="/wiki/Martin_Davis_(mathematician)" title="Martin Davis (mathematician)">Davis, Martin</a> (1974). <i>Computability. Notes by Barry Jacobs</i>. <a href="/wiki/Courant_Institute_of_Mathematical_Sciences" title="Courant Institute of Mathematical Sciences">Courant Institute of Mathematical Sciences</a>, <a href="/wiki/New_York_University" title="New York University">New York University</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Computability.+Notes+by+Barry+Jacobs.&rft.pub=Courant+Institute+of+Mathematical+Sciences%2C+New+York+University&rft.date=1974&rft.aulast=Davis&rft.aufirst=Martin&rfr_id=info%3Asid%2Fen.wikipedia.org%3APeano+axioms" class="Z3988"></span></li></ul> <ul><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). <a rel="nofollow" class="external text" href="http://digisrv-1.biblio.etc.tu-bs.de:8080/docportal/servlets/MCRFileNodeServlet/DocPortal_derivate_00005731/V.C.125.pdf"><i>Was sind und was sollen die Zahlen?</i></a> [<i>What are and what should the numbers be?</i>] <span class="cs1-format">(PDF)</span>. Vieweg<span class="reference-accessdate">. Retrieved <span class="nowrap">4 July</span> 2016</span>.</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.pub=Vieweg&rft.date=1888&rft.aulast=Dedekind&rft.aufirst=Richard&rft_id=http%3A%2F%2Fdigisrv-1.biblio.etc.tu-bs.de%3A8080%2Fdocportal%2Fservlets%2FMCRFileNodeServlet%2FDocPortal_derivate_00005731%2FV.C.125.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3APeano+axioms" class="Z3988"></span> <ul><li>Two English translations: <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBeman1901" class="citation book cs1">Beman, Wooster, Woodruff (1901). <a rel="nofollow" class="external text" href="http://www.gutenberg.org/files/21016/21016-pdf.pdf"><i>Essays on the Theory of Numbers</i></a> <span class="cs1-format">(PDF)</span>. <a href="/wiki/Dover_Publications" title="Dover Publications">Dover Publications</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Essays+on+the+Theory+of+Numbers&rft.pub=Dover+Publications&rft.date=1901&rft.aulast=Beman&rft.aufirst=Wooster%2C+Woodruff&rft_id=http%3A%2F%2Fwww.gutenberg.org%2Ffiles%2F21016%2F21016-pdf.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3APeano+axioms" 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: multiple names: authors list (<a href="/wiki/Category:CS1_maint:_multiple_names:_authors_list" title="Category:CS1 maint: multiple names: authors list">link</a>)</span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEwald1996" class="citation book cs1">Ewald, William B. (1996). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=Nt_uAAAAMAAJ"><i>From Kant to Hilbert: A Source Book in the Foundations of Mathematics</i></a>. <a href="/wiki/Oxford_University_Press" title="Oxford University Press">Oxford University Press</a>. pp. 787–832. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-853271-2" title="Special:BookSources/978-0-19-853271-2"><bdi>978-0-19-853271-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=From+Kant+to+Hilbert%3A+A+Source+Book+in+the+Foundations+of+Mathematics&rft.pages=787-832&rft.pub=Oxford+University+Press&rft.date=1996&rft.isbn=978-0-19-853271-2&rft.aulast=Ewald&rft.aufirst=William+B.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DNt_uAAAAMAAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3APeano+axioms" class="Z3988"></span></li></ul></li></ul></li></ul> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFritz1952" class="citation book cs1">Fritz, Charles A. Jr. (1952). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/bertrandrussells0000frit"><i>Bertrand Russell's construction of the external world</i></a></span>. New York, Humanities Press.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Bertrand+Russell%27s+construction+of+the+external+world&rft.pub=New+York%2C+Humanities+Press&rft.date=1952&rft.aulast=Fritz&rft.aufirst=Charles+A.+Jr.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fbertrandrussells0000frit&rfr_id=info%3Asid%2Fen.wikipedia.org%3APeano+axioms" class="Z3988"></span></li></ul> <ul><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><a href="/wiki/Mathematische_Annalen" title="Mathematische Annalen">Mathematische Annalen</a></i>. <b>112</b>. Reprinted in English translation in his 1969 <i>Collected works</i>, M. E. Szabo, ed.: 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%3APeano+axioms" class="Z3988"></span></li></ul> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGödel1931" class="citation journal cs1"><a href="/wiki/Kurt_G%C3%B6del" title="Kurt Gödel">Gödel, Kurt</a> (1931). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20180411113347/http://www.w-k-essler.de/pdfs/goedel.pdf">"Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme, I"</a> <span class="cs1-format">(PDF)</span>. <i>Monatshefte für Mathematik</i>. <b>38</b>. See <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> for details on English translations.: 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>. Archived from <a rel="nofollow" class="external text" href="http://www.w-k-essler.de/pdfs/goedel.pdf">the original</a> <span class="cs1-format">(PDF)</span> on 2018-04-11<span class="reference-accessdate">. Retrieved <span class="nowrap">2013-10-31</span></span>.</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&rft.atitle=%C3%9Cber+formal+unentscheidbare+S%C3%A4tze+der+Principia+Mathematica+und+verwandter+Systeme%2C+I&rft.volume=38&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&rft_id=http%3A%2F%2Fwww.w-k-essler.de%2Fpdfs%2Fgoedel.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3APeano+axioms" class="Z3988"></span></li></ul> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGödel1958" class="citation journal cs1"><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><a href="/wiki/Dialectica" title="Dialectica">Dialectica</a></i>. <b>12</b> (3–4). Reprinted in English translation in 1990. Gödel's <i>Collected Works</i>, Vol II. <a href="/wiki/Solomon_Feferman" title="Solomon Feferman">Solomon Feferman</a> et al., eds. <a href="/wiki/Oxford_University_Press" title="Oxford University Press">Oxford University Press</a>: 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%3APeano+axioms" class="Z3988"></span></li></ul> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGrassmann1861" class="citation book cs1"><a href="/wiki/Hermann_Grassmann" title="Hermann Grassmann">Grassmann, Hermann Günther</a> (1861). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=jdQ2AAAAMAAJ"><i>Lehrbuch der Arithmetik für höhere Lehranstalten</i></a>. Verlag von Theod. Chr. Fr. Enslin.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Lehrbuch+der+Arithmetik+f%C3%BCr+h%C3%B6here+Lehranstalten&rft.pub=Verlag+von+Theod.+Chr.+Fr.+Enslin&rft.date=1861&rft.aulast=Grassmann&rft.aufirst=Hermann+G%C3%BCnther&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DjdQ2AAAAMAAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3APeano+axioms" class="Z3988"></span></li></ul> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGray2013" class="citation book cs1"><a href="/wiki/Jeremy_Gray" title="Jeremy Gray">Gray, Jeremy</a> (2013). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=w2Tya9gOKqEC&pg=PA133">"The Essayist"</a>. <i>Henri Poincaré: A scientific biography</i>. <a href="/wiki/Princeton_University_Press" title="Princeton University Press">Princeton University Press</a>. p. 133. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-691-15271-4" title="Special:BookSources/978-0-691-15271-4"><bdi>978-0-691-15271-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=The+Essayist&rft.btitle=Henri+Poincar%C3%A9%3A+A+scientific+biography&rft.pages=133&rft.pub=Princeton+University+Press&rft.date=2013&rft.isbn=978-0-691-15271-4&rft.aulast=Gray&rft.aufirst=Jeremy&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Dw2Tya9gOKqEC%26pg%3DPA133&rfr_id=info%3Asid%2Fen.wikipedia.org%3APeano+axioms" class="Z3988"></span></li></ul> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHarsanyi1983" class="citation book cs1"><a href="/wiki/John_C._Harsanyi" class="mw-redirect" title="John C. Harsanyi">Harsanyi, John C.</a> (1983). "Mathematics, the Empirical Facts, and Logical Necessity". In Hempel, Carl G.; Putnam, Hilary; Essler, Wilhelm K. (eds.). <i>Methodology, Epistemology, and Philosophy of Science</i>. pp. 167–192. <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%2F978-94-015-7676-5_8">10.1007/978-94-015-7676-5_8</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-90-481-8389-0" title="Special:BookSources/978-90-481-8389-0"><bdi>978-90-481-8389-0</bdi></a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:121297669">121297669</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Mathematics%2C+the+Empirical+Facts%2C+and+Logical+Necessity&rft.btitle=Methodology%2C+Epistemology%2C+and+Philosophy+of+Science&rft.pages=167-192&rft.date=1983&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A121297669%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1007%2F978-94-015-7676-5_8&rft.isbn=978-90-481-8389-0&rft.aulast=Harsanyi&rft.aufirst=John+C.&rfr_id=info%3Asid%2Fen.wikipedia.org%3APeano+axioms" class="Z3988"></span></li></ul> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHatcher2014" class="citation book cs1">Hatcher, William S. (2014) [1982]. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=wf3iBQAAQBAJ&pg=PP1"><i>The Logical Foundations of Mathematics</i></a>. Elsevier. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4831-8963-5" title="Special:BookSources/978-1-4831-8963-5"><bdi>978-1-4831-8963-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Logical+Foundations+of+Mathematics&rft.pub=Elsevier&rft.date=2014&rft.isbn=978-1-4831-8963-5&rft.aulast=Hatcher&rft.aufirst=William+S.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Dwf3iBQAAQBAJ%26pg%3DPP1&rfr_id=info%3Asid%2Fen.wikipedia.org%3APeano+axioms" class="Z3988"></span> Derives the Peano axioms (called <b>S</b>) from several <a href="/wiki/Axiomatic_set_theories" class="mw-redirect" title="Axiomatic set theories">axiomatic set theories</a> and from <a href="/wiki/Category_theory" title="Category theory">category theory</a>.</li></ul> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHermes1973" class="citation book cs1">Hermes, Hans (1973). <i>Introduction to Mathematical Logic</i>. Hochschultext. Springer. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/3-540-05819-2" title="Special:BookSources/3-540-05819-2"><bdi>3-540-05819-2</bdi></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/1431-4657">1431-4657</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.series=Hochschultext&rft.pub=Springer&rft.date=1973&rft.issn=1431-4657&rft.isbn=3-540-05819-2&rft.aulast=Hermes&rft.aufirst=Hans&rfr_id=info%3Asid%2Fen.wikipedia.org%3APeano+axioms" class="Z3988"></span></li></ul> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHilbert1902" class="citation journal cs1">Hilbert, David (1902). <a rel="nofollow" class="external text" href="https://www.ams.org/journals/bull/1902-08-10/S0002-9904-1902-00923-3/home.html">"Mathematische Probleme"</a> [Mathematical Problems]. <i>Bulletin of the American Mathematical Society</i>. <b>8</b> (10). Translated by Winton, Maby: 437–479. <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/10.1090%2Fs0002-9904-1902-00923-3">10.1090/s0002-9904-1902-00923-3</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Bulletin+of+the+American+Mathematical+Society&rft.atitle=Mathematische+Probleme&rft.volume=8&rft.issue=10&rft.pages=437-479&rft.date=1902&rft_id=info%3Adoi%2F10.1090%2Fs0002-9904-1902-00923-3&rft.aulast=Hilbert&rft.aufirst=David&rft_id=https%3A%2F%2Fwww.ams.org%2Fjournals%2Fbull%2F1902-08-10%2FS0002-9904-1902-00923-3%2Fhome.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3APeano+axioms" class="Z3988"></span></li></ul> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKaye1991" class="citation book cs1">Kaye, Richard (1991). <i>Models of Peano arithmetic</i>. <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/0-19-853213-X" title="Special:BookSources/0-19-853213-X"><bdi>0-19-853213-X</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Models+of+Peano+arithmetic&rft.pub=Oxford+University+Press&rft.date=1991&rft.isbn=0-19-853213-X&rft.aulast=Kaye&rft.aufirst=Richard&rfr_id=info%3Asid%2Fen.wikipedia.org%3APeano+axioms" class="Z3988"></span></li></ul> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLandau1965" class="citation book cs1"><a href="/wiki/Edmund_Landau" title="Edmund Landau">Landau, Edmund</a> (1965). <i>Grundlagen Der Analysis</i>. Derives the basic number systems from the Peano axioms. English/German vocabulary included. <a href="/wiki/American_Mathematical_Society" title="American Mathematical Society">AMS Chelsea Publishing</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8284-0141-8" title="Special:BookSources/978-0-8284-0141-8"><bdi>978-0-8284-0141-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Grundlagen+Der+Analysis&rft.pub=AMS+Chelsea+Publishing&rft.date=1965&rft.isbn=978-0-8284-0141-8&rft.aulast=Landau&rft.aufirst=Edmund&rfr_id=info%3Asid%2Fen.wikipedia.org%3APeano+axioms" class="Z3988"></span></li></ul> <ul><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> (December 1997) [December 1979]. <i>Introduction to Mathematical Logic (Discrete Mathematics and Its Applications)</i> (4th ed.). Springer. <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+%28Discrete+Mathematics+and+Its+Applications%29&rft.edition=4th&rft.pub=Springer&rft.date=1997-12&rft.isbn=978-0-412-80830-2&rft.aulast=Mendelson&rft.aufirst=Elliott&rfr_id=info%3Asid%2Fen.wikipedia.org%3APeano+axioms" class="Z3988"></span></li></ul> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMeseguerGoguen1986" class="citation book cs1">Meseguer, José; Goguen, Joseph A. (Dec 1986). "Initiality, induction, and computability". In Maurice Nivat and John C. Reynolds (ed.). <a rel="nofollow" class="external text" href="https://courses.engr.illinois.edu/cs576/sp2017/readings/background/initiality-induction-computability.pdf"><i>Algebraic Methods in Semantics</i></a> <span class="cs1-format">(PDF)</span>. Cambridge: Cambridge University Press. pp. 459–541. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-26793-9" title="Special:BookSources/978-0-521-26793-9"><bdi>978-0-521-26793-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Initiality%2C+induction%2C+and+computability&rft.btitle=Algebraic+Methods+in+Semantics&rft.place=Cambridge&rft.pages=459-541&rft.pub=Cambridge+University+Press&rft.date=1986-12&rft.isbn=978-0-521-26793-9&rft.aulast=Meseguer&rft.aufirst=Jos%C3%A9&rft.au=Goguen%2C+Joseph+A.&rft_id=https%3A%2F%2Fcourses.engr.illinois.edu%2Fcs576%2Fsp2017%2Freadings%2Fbackground%2Finitiality-induction-computability.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3APeano+axioms" class="Z3988"></span></li></ul> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFParteeTer_MeulenWall2012" class="citation book cs1">Partee, Barbara; <a href="/wiki/Alice_ter_Meulen" title="Alice ter Meulen">Ter Meulen, Alice</a>; Wall, Robert (2012). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=d5xrCQAAQBAJ"><i>Mathematical Methods in Linguistics</i></a>. Springer. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-94-009-2213-6" title="Special:BookSources/978-94-009-2213-6"><bdi>978-94-009-2213-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mathematical+Methods+in+Linguistics&rft.pub=Springer&rft.date=2012&rft.isbn=978-94-009-2213-6&rft.aulast=Partee&rft.aufirst=Barbara&rft.au=Ter+Meulen%2C+Alice&rft.au=Wall%2C+Robert&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Dd5xrCQAAQBAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3APeano+axioms" class="Z3988"></span></li></ul> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPeano1908" class="citation book cs1"><a href="/wiki/Giuseppe_Peano" title="Giuseppe Peano">Peano, Giuseppe</a> (1908). <a rel="nofollow" class="external text" href="https://archive.org/details/formulairedemat04peangoog"><i>Formulario Mathematico</i></a> (V ed.). Turin, Bocca frères, Ch. Clausen. p. 27.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Formulario+Mathematico&rft.pages=27&rft.edition=V&rft.pub=Turin%2C+Bocca+fr%C3%A8res%2C+Ch.+Clausen&rft.date=1908&rft.aulast=Peano&rft.aufirst=Giuseppe&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fformulairedemat04peangoog&rfr_id=info%3Asid%2Fen.wikipedia.org%3APeano+axioms" class="Z3988"></span></li></ul> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPeirce1881" class="citation journal cs1"><a href="/wiki/Charles_Sanders_Peirce" title="Charles Sanders Peirce">Peirce, C. S.</a> (1881). <a rel="nofollow" class="external text" href="https://archive.org/details/jstor-2369151">"On the Logic of Number"</a>. <i>American Journal of Mathematics</i>. <b>4</b> (1): 85–95. <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%2F2369151">10.2307/2369151</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/2369151">2369151</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=1507856">1507856</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=American+Journal+of+Mathematics&rft.atitle=On+the+Logic+of+Number&rft.volume=4&rft.issue=1&rft.pages=85-95&rft.date=1881&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1507856%23id-name%3DMR&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2369151%23id-name%3DJSTOR&rft_id=info%3Adoi%2F10.2307%2F2369151&rft.aulast=Peirce&rft.aufirst=C.+S.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fjstor-2369151&rfr_id=info%3Asid%2Fen.wikipedia.org%3APeano+axioms" class="Z3988"></span></li></ul> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFShields1997" class="citation book cs1">Shields, Paul (1997). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=pWjOg-zbtMAC&pg=PA43">"3. Peirce's Axiomatization of Arithmetic"</a>. In Houser, Nathan; Roberts, Don D.; Van Evra, James (eds.). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/studiesinlogicof00nath"><i>Studies in the Logic of Charles Sanders Peirce</i></a></span>. Indiana University Press. pp. 43–52. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-253-33020-3" title="Special:BookSources/0-253-33020-3"><bdi>0-253-33020-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=3.+Peirce%27s+Axiomatization+of+Arithmetic&rft.btitle=Studies+in+the+Logic+of+Charles+Sanders+Peirce&rft.pages=43-52&rft.pub=Indiana+University+Press&rft.date=1997&rft.isbn=0-253-33020-3&rft.aulast=Shields&rft.aufirst=Paul&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DpWjOg-zbtMAC%26pg%3DPA43&rfr_id=info%3Asid%2Fen.wikipedia.org%3APeano+axioms" class="Z3988"></span></li></ul> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSuppes1960" class="citation book cs1"><a href="/wiki/Patrick_Suppes" title="Patrick Suppes">Suppes, Patrick</a> (1960). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/axiomaticsettheo00supp_0"><i>Axiomatic Set Theory</i></a></span>. <a href="/wiki/Dover_Publications" title="Dover Publications">Dover Publications</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-486-61630-4" title="Special:BookSources/0-486-61630-4"><bdi>0-486-61630-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Axiomatic+Set+Theory&rft.pub=Dover+Publications&rft.date=1960&rft.isbn=0-486-61630-4&rft.aulast=Suppes&rft.aufirst=Patrick&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Faxiomaticsettheo00supp_0&rfr_id=info%3Asid%2Fen.wikipedia.org%3APeano+axioms" class="Z3988"></span> Derives the Peano axioms from <a href="/wiki/ZFC" class="mw-redirect" title="ZFC">ZFC</a></li></ul> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTarskiGivant1987" class="citation book cs1"><a href="/wiki/Alfred_Tarski" title="Alfred Tarski">Tarski, Alfred</a>; Givant, Steven (1987). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/formalizationofs0000tars"><i>A Formalization of Set Theory without Variables</i></a></span>. AMS Colloquium Publications. Vol. 41. <a href="/wiki/American_Mathematical_Society" title="American Mathematical Society">American Mathematical Society</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8218-1041-5" title="Special:BookSources/978-0-8218-1041-5"><bdi>978-0-8218-1041-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Formalization+of+Set+Theory+without+Variables&rft.series=AMS+Colloquium+Publications&rft.pub=American+Mathematical+Society&rft.date=1987&rft.isbn=978-0-8218-1041-5&rft.aulast=Tarski&rft.aufirst=Alfred&rft.au=Givant%2C+Steven&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fformalizationofs0000tars&rfr_id=info%3Asid%2Fen.wikipedia.org%3APeano+axioms" class="Z3988"></span></li></ul> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFVan_Heijenoort1967" class="citation book cs1"><a href="/wiki/Jean_van_Heijenoort" title="Jean van Heijenoort">Van Heijenoort, Jean</a> (1967). <i>From Frege to Godel: A Source Book in Mathematical Logic, 1879–1931</i>. Harvard University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-674-32449-7" title="Special:BookSources/978-0-674-32449-7"><bdi>978-0-674-32449-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=From+Frege+to+Godel%3A+A+Source+Book+in+Mathematical+Logic%2C+1879%E2%80%931931&rft.pub=Harvard+University+Press&rft.date=1967&rft.isbn=978-0-674-32449-7&rft.aulast=Van+Heijenoort&rft.aufirst=Jean&rfr_id=info%3Asid%2Fen.wikipedia.org%3APeano+axioms" class="Z3988"></span> <ul><li>Contains translations of the following two papers, with valuable commentary: <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDedekind1890" class="citation book cs1"><a href="/wiki/Richard_Dedekind" title="Richard Dedekind">Dedekind, Richard</a> (1890). <i>Letter to Keferstein</i>. On p. 100, he restates and defends his axioms of 1888. pp. 98–103.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Letter+to+Keferstein.&rft.pages=98-103&rft.date=1890&rft.aulast=Dedekind&rft.aufirst=Richard&rfr_id=info%3Asid%2Fen.wikipedia.org%3APeano+axioms" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPeano1889" class="citation book cs1"><a href="/wiki/Giuseppe_Peano" title="Giuseppe Peano">Peano, Giuseppe</a> (1889). <a rel="nofollow" class="external text" href="https://archive.org/details/arithmeticespri00peangoog"><i>Arithmetices principia, nova methodo exposita</i></a> [<i>The principles of arithmetic, presented by a new method</i>]. An excerpt of the treatise where Peano first presented his axioms, and recursively defined arithmetical operations. Fratres Bocca. pp. 83–97.</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.pages=83-97&rft.pub=Fratres+Bocca&rft.date=1889&rft.aulast=Peano&rft.aufirst=Giuseppe&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Farithmeticespri00peangoog&rfr_id=info%3Asid%2Fen.wikipedia.org%3APeano+axioms" class="Z3988"></span></li></ul></li></ul></li></ul> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFVan_Oosten1999" class="citation web cs1">Van Oosten, Jaap (June 1999). <a rel="nofollow" class="external text" href="https://webspace.science.uu.nl/~ooste110/syllabi/peanomoeder.pdf">"Introduction to Peano Arithmetic (Gödel Incompleteness and Nonstandard Models)"</a> <span class="cs1-format">(PDF)</span>. Utrecht University<span class="reference-accessdate">. Retrieved <span class="nowrap">2 September</span> 2023</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Introduction+to+Peano+Arithmetic+%28G%C3%B6del+Incompleteness+and+Nonstandard+Models%29&rft.pub=Utrecht+University&rft.date=1999-06&rft.aulast=Van+Oosten&rft.aufirst=Jaap&rft_id=https%3A%2F%2Fwebspace.science.uu.nl%2F~ooste110%2Fsyllabi%2Fpeanomoeder.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3APeano+axioms" class="Z3988"></span></li></ul> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWang1957" class="citation journal cs1"><a href="/wiki/Hao_Wang_(academic)" title="Hao Wang (academic)">Wang, Hao</a> (June 1957). "The Axiomatization of Arithmetic". <i>The Journal of Symbolic Logic</i>. <b>22</b> (2). <a href="/wiki/Association_for_Symbolic_Logic" title="Association for Symbolic Logic">Association for Symbolic Logic</a>: 145–158. <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%2F2964176">10.2307/2964176</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/2964176">2964176</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:26896458">26896458</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Journal+of+Symbolic+Logic&rft.atitle=The+Axiomatization+of+Arithmetic&rft.volume=22&rft.issue=2&rft.pages=145-158&rft.date=1957-06&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A26896458%23id-name%3DS2CID&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2964176%23id-name%3DJSTOR&rft_id=info%3Adoi%2F10.2307%2F2964176&rft.aulast=Wang&rft.aufirst=Hao&rfr_id=info%3Asid%2Fen.wikipedia.org%3APeano+axioms" class="Z3988"></span></li></ul> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWillard2001" class="citation journal cs1"><a href="/wiki/Dan_Willard" title="Dan Willard">Willard, Dan E.</a> (2001). <a rel="nofollow" class="external text" href="https://www.cs.albany.edu/~dew/m/jsl1.pdf">"Self-verifying axiom systems, the incompleteness theorem and related reflection principles"</a> <span class="cs1-format">(PDF)</span>. <i>The Journal of Symbolic Logic</i>. <b>66</b> (2): 536–596. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2695030">10.2307/2695030</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2695030">2695030</a>. <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=1833464">1833464</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:2822314">2822314</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Journal+of+Symbolic+Logic&rft.atitle=Self-verifying+axiom+systems%2C+the+incompleteness+theorem+and+related+reflection+principles&rft.volume=66&rft.issue=2&rft.pages=536-596&rft.date=2001&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A2822314%23id-name%3DS2CID&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1833464%23id-name%3DMR&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2695030%23id-name%3DJSTOR&rft_id=info%3Adoi%2F10.2307%2F2695030&rft.aulast=Willard&rft.aufirst=Dan+E.&rft_id=https%3A%2F%2Fwww.cs.albany.edu%2F~dew%2Fm%2Fjsl1.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3APeano+axioms" class="Z3988"></span></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Peano_axioms&action=edit&section=20" title="Edit section: Further reading"><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="CITEREFBuss1998" class="citation book cs1">Buss, Samuel R. (1998). "Chapter II: First-Order Proof Theory of Arithmetic". In Buss, Samuel R. (ed.). <i>Handbook of Proof Theory</i>. New York: Elsevier Science. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-444-89840-1" title="Special:BookSources/978-0-444-89840-1"><bdi>978-0-444-89840-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Chapter+II%3A+First-Order+Proof+Theory+of+Arithmetic&rft.btitle=Handbook+of+Proof+Theory&rft.place=New+York&rft.pub=Elsevier+Science&rft.date=1998&rft.isbn=978-0-444-89840-1&rft.aulast=Buss&rft.aufirst=Samuel+R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3APeano+axioms" class="Z3988"></span></li></ul> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMendelson2015" class="citation book cs1"><a href="/wiki/Elliott_Mendelson" title="Elliott Mendelson">Mendelson, Elliott</a> (June 2015) [December 1979]. <i>Introduction to Mathematical Logic (Discrete Mathematics and Its Applications)</i> (6th ed.). Chapman and Hall/CRC. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4822-3772-6" title="Special:BookSources/978-1-4822-3772-6"><bdi>978-1-4822-3772-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Mathematical+Logic+%28Discrete+Mathematics+and+Its+Applications%29&rft.edition=6th&rft.pub=Chapman+and+Hall%2FCRC&rft.date=2015-06&rft.isbn=978-1-4822-3772-6&rft.aulast=Mendelson&rft.aufirst=Elliott&rfr_id=info%3Asid%2Fen.wikipedia.org%3APeano+axioms" class="Z3988"></span></li></ul> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSmullyan2013" class="citation book cs1"><a href="/wiki/Raymond_M._Smullyan" class="mw-redirect" title="Raymond M. Smullyan">Smullyan, Raymond M.</a> (December 2013). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=xUapAAAAQBAJ"><i>The Gödelian Puzzle Book: Puzzles, Paradoxes and Proofs</i></a>. Dover Publications. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-49705-1" title="Special:BookSources/978-0-486-49705-1"><bdi>978-0-486-49705-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+G%C3%B6delian+Puzzle+Book%3A+Puzzles%2C+Paradoxes+and+Proofs&rft.pub=Dover+Publications&rft.date=2013-12&rft.isbn=978-0-486-49705-1&rft.aulast=Smullyan&rft.aufirst=Raymond+M.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DxUapAAAAQBAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3APeano+axioms" class="Z3988"></span></li></ul> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTakeuti2013" class="citation book cs1"><a href="/wiki/Gaisi_Takeuti" title="Gaisi Takeuti">Takeuti, Gaisi</a> (2013). <i>Proof theory</i> (Second ed.). Mineola, New York. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-49073-1" title="Special:BookSources/978-0-486-49073-1"><bdi>978-0-486-49073-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Proof+theory&rft.place=Mineola%2C+New+York&rft.edition=Second&rft.date=2013&rft.isbn=978-0-486-49073-1&rft.aulast=Takeuti&rft.aufirst=Gaisi&rfr_id=info%3Asid%2Fen.wikipedia.org%3APeano+axioms" 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></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=Peano_axioms&action=edit&section=21" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMurzi" class="citation encyclopaedia cs1">Murzi, Mauro. <a rel="nofollow" class="external text" href="http://www.iep.utm.edu/poincare">"Henri Poincaré"</a>. <i><a href="/wiki/Internet_Encyclopedia_of_Philosophy" title="Internet Encyclopedia of Philosophy">Internet Encyclopedia of Philosophy</a></i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Henri+Poincar%C3%A9&rft.btitle=Internet+Encyclopedia+of+Philosophy&rft.aulast=Murzi&rft.aufirst=Mauro&rft_id=http%3A%2F%2Fwww.iep.utm.edu%2Fpoincare&rfr_id=info%3Asid%2Fen.wikipedia.org%3APeano+axioms" class="Z3988"></span> Includes a discussion of Poincaré's critique of the Peano's axioms.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPodnieks2015" class="citation book cs1">Podnieks, Karlis (2015-01-25). "3. First Order Arithmetic". <a rel="nofollow" class="external text" href="https://dspace.lu.lv/dspace/handle/7/5306"><i>What is Mathematics: Gödel's Theorem and Around</i></a>. pp. 93–121.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=3.+First+Order+Arithmetic&rft.btitle=What+is+Mathematics%3A+G%C3%B6del%27s+Theorem+and+Around&rft.pages=93-121&rft.date=2015-01-25&rft.aulast=Podnieks&rft.aufirst=Karlis&rft_id=https%3A%2F%2Fdspace.lu.lv%2Fdspace%2Fhandle%2F7%2F5306&rfr_id=info%3Asid%2Fen.wikipedia.org%3APeano+axioms" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation cs2"><a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Peano_axioms">"Peano axioms"</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=Peano+axioms&rft.btitle=Encyclopedia+of+Mathematics&rft.pub=EMS+Press&rft.date=2001&rft_id=https%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%3Ftitle%3DPeano_axioms&rfr_id=info%3Asid%2Fen.wikipedia.org%3APeano+axioms" class="Z3988"></span></li> <li><span class="citation mathworld" id="Reference-Mathworld-Peano's_Axioms"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeisstein" class="citation web cs1"><a href="/wiki/Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a> <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/PeanosAxioms.html">"Peano's Axioms"</a>. <i><a href="/wiki/MathWorld" title="MathWorld">MathWorld</a></i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=MathWorld&rft.atitle=Peano%27s+Axioms&rft.au=Weisstein%2C+Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FPeanosAxioms.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3APeano+axioms" class="Z3988"></span></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBurris2001" class="citation web cs1">Burris, Stanley N. (2001). <a rel="nofollow" class="external text" href="http://www.math.uwaterloo.ca/~snburris/htdocs/scav/dedek/dedek.html">"What are numbers, and what is their meaning?: Dedekind"</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=What+are+numbers%2C+and+what+is+their+meaning%3F%3A+Dedekind&rft.date=2001&rft.aulast=Burris&rft.aufirst=Stanley+N.&rft_id=http%3A%2F%2Fwww.math.uwaterloo.ca%2F~snburris%2Fhtdocs%2Fscav%2Fdedek%2Fdedek.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3APeano+axioms" class="Z3988"></span> Commentary on Dedekind's work.</li></ul> <p><i>This article incorporates material from <a rel="nofollow" class="external text" href="https://planetmath.org/pa">PA</a> on <a href="/wiki/PlanetMath" title="PlanetMath">PlanetMath</a>, which is licensed under the <a href="/wiki/Wikipedia:CC-BY-SA" class="mw-redirect" title="Wikipedia:CC-BY-SA">Creative Commons Attribution/Share-Alike License</a>.</i> </p>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Peano_axioms&oldid=1255254109">https://en.wikipedia.org/w/index.php?title=Peano_axioms&oldid=1255254109</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a href="/wiki/Category:1889_introductions" title="Category:1889 introductions">1889 introductions</a></li><li><a href="/wiki/Category:Mathematical_axioms" title="Category:Mathematical axioms">Mathematical axioms</a></li><li><a href="/wiki/Category:Formal_theories_of_arithmetic" title="Category:Formal theories of arithmetic">Formal theories of arithmetic</a></li><li><a href="/wiki/Category:Logic_in_computer_science" title="Category:Logic in computer science">Logic in computer science</a></li><li><a href="/wiki/Category:Mathematical_logic" title="Category:Mathematical logic">Mathematical logic</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden categories: <ul><li><a href="/wiki/Category:Webarchive_template_wayback_links" title="Category:Webarchive template wayback links">Webarchive template wayback links</a></li><li><a href="/wiki/Category:Articles_with_short_description" title="Category:Articles with short description">Articles with short description</a></li><li><a href="/wiki/Category:Short_description_is_different_from_Wikidata" title="Category:Short description is different from Wikidata">Short description is different from Wikidata</a></li><li><a href="/wiki/Category:Pages_with_Italian_IPA" title="Category:Pages with Italian IPA">Pages with Italian IPA</a></li><li><a href="/wiki/Category:Articles_containing_Latin-language_text" title="Category:Articles containing Latin-language text">Articles containing Latin-language text</a></li><li><a href="/wiki/Category:Articles_needing_additional_references_from_May_2024" title="Category:Articles needing additional references from May 2024">Articles needing additional references from May 2024</a></li><li><a href="/wiki/Category:All_articles_needing_additional_references" title="Category:All articles needing additional references">All articles needing additional references</a></li><li><a href="/wiki/Category:All_articles_with_unsourced_statements" title="Category:All articles with unsourced statements">All articles with unsourced statements</a></li><li><a href="/wiki/Category:Articles_with_unsourced_statements_from_January_2022" title="Category:Articles with unsourced statements from January 2022">Articles with unsourced statements from January 2022</a></li><li><a href="/wiki/Category:Articles_with_unsourced_statements_from_May_2024" title="Category:Articles with unsourced statements from May 2024">Articles with unsourced statements from May 2024</a></li><li><a href="/wiki/Category:Articles_containing_German-language_text" title="Category:Articles containing German-language text">Articles containing German-language text</a></li><li><a href="/wiki/Category:CS1_maint:_multiple_names:_authors_list" title="Category:CS1 maint: multiple names: authors list">CS1 maint: multiple names: authors list</a></li><li><a href="/wiki/Category:CS1_maint:_location_missing_publisher" title="Category:CS1 maint: location missing publisher">CS1 maint: location missing publisher</a></li><li><a href="/wiki/Category:Articles_with_Internet_Encyclopedia_of_Philosophy_links" title="Category:Articles with Internet Encyclopedia of Philosophy links">Articles with Internet Encyclopedia of Philosophy links</a></li><li><a href="/wiki/Category:Wikipedia_articles_incorporating_text_from_PlanetMath" title="Category:Wikipedia articles incorporating text from PlanetMath">Wikipedia articles incorporating text from PlanetMath</a></li><li><a href="/wiki/Category:Pages_that_use_a_deprecated_format_of_the_math_tags" title="Category:Pages that use a deprecated format of the math tags">Pages that use a deprecated format of the math tags</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 3 November 2024, at 23:25<span class="anonymous-show"> (UTC)</span>.</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; additional terms may apply. By using this site, you agree to the <a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Terms_of_Use" class="extiw" title="foundation:Special:MyLanguage/Policy:Terms of Use">Terms of Use</a> and <a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Privacy_policy" class="extiw" title="foundation:Special:MyLanguage/Policy:Privacy policy">Privacy Policy</a>. Wikipedia® is a registered trademark of the <a rel="nofollow" class="external text" href="https://wikimediafoundation.org/">Wikimedia Foundation, Inc.</a>, a non-profit organization.</li> </ul> <ul id="footer-places"> <li id="footer-places-privacy"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Privacy_policy">Privacy policy</a></li> <li id="footer-places-about"><a href="/wiki/Wikipedia:About">About Wikipedia</a></li> <li id="footer-places-disclaimers"><a href="/wiki/Wikipedia:General_disclaimer">Disclaimers</a></li> <li id="footer-places-contact"><a href="//en.wikipedia.org/wiki/Wikipedia:Contact_us">Contact Wikipedia</a></li> <li id="footer-places-wm-codeofconduct"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Universal_Code_of_Conduct">Code of Conduct</a></li> <li id="footer-places-developers"><a href="https://developer.wikimedia.org">Developers</a></li> <li id="footer-places-statslink"><a href="https://stats.wikimedia.org/#/en.wikipedia.org">Statistics</a></li> <li id="footer-places-cookiestatement"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Cookie_statement">Cookie statement</a></li> <li id="footer-places-mobileview"><a href="//en.m.wikipedia.org/w/index.php?title=Peano_axioms&mobileaction=toggle_view_mobile" class="noprint stopMobileRedirectToggle">Mobile view</a></li> </ul> <ul id="footer-icons" class="noprint"> <li id="footer-copyrightico"><a href="https://wikimediafoundation.org/" class="cdx-button cdx-button--fake-button cdx-button--size-large cdx-button--fake-button--enabled"><img src="/static/images/footer/wikimedia-button.svg" width="84" height="29" alt="Wikimedia Foundation" loading="lazy"></a></li> <li id="footer-poweredbyico"><a href="https://www.mediawiki.org/" class="cdx-button cdx-button--fake-button cdx-button--size-large cdx-button--fake-button--enabled"><img src="/w/resources/assets/poweredby_mediawiki.svg" alt="Powered by MediaWiki" width="88" height="31" loading="lazy"></a></li> </ul> </footer> </div> </div> </div> <div class="vector-settings" id="p-dock-bottom"> <ul></ul> </div><script>(RLQ=window.RLQ||[]).push(function(){mw.config.set({"wgHostname":"mw-web.codfw.main-694cf4987f-h7rrv","wgBackendResponseTime":186,"wgPageParseReport":{"limitreport":{"cputime":"1.008","walltime":"1.341","ppvisitednodes":{"value":5059,"limit":1000000},"postexpandincludesize":{"value":104604,"limit":2097152},"templateargumentsize":{"value":6434,"limit":2097152},"expansiondepth":{"value":12,"limit":100},"expensivefunctioncount":{"value":10,"limit":500},"unstrip-depth":{"value":1,"limit":20},"unstrip-size":{"value":115805,"limit":5000000},"entityaccesscount":{"value":0,"limit":400},"timingprofile":["100.00% 1063.957 1 -total"," 23.14% 246.243 27 Template:Cite_book"," 10.93% 116.315 1 Template:IPA"," 10.69% 113.704 22 Template:Sfn"," 8.09% 86.023 1 Template:Short_description"," 6.30% 67.064 1 Template:Math_theorem"," 5.62% 59.818 2 Template:Reflist"," 5.50% 58.488 2 Template:Pagetype"," 4.88% 51.914 1 Template:Improve_references"," 4.18% 44.463 1 Template:Ambox"]},"scribunto":{"limitreport-timeusage":{"value":"0.621","limit":"10.000"},"limitreport-memusage":{"value":20251925,"limit":52428800},"limitreport-logs":"anchor_id_list = table#1 {\n [\"CITEREFBeman1901\"] = 1,\n [\"CITEREFBurris2001\"] = 1,\n [\"CITEREFBuss1998\"] = 1,\n [\"CITEREFDavis1974\"] = 1,\n [\"CITEREFDedekind1888\"] = 1,\n [\"CITEREFDedekind1890\"] = 1,\n [\"CITEREFEwald1996\"] = 1,\n [\"CITEREFFritz1952\"] = 1,\n [\"CITEREFGentzen1936\"] = 1,\n [\"CITEREFGrassmann1861\"] = 1,\n [\"CITEREFGray2013\"] = 1,\n [\"CITEREFGödel1931\"] = 1,\n [\"CITEREFGödel1958\"] = 1,\n [\"CITEREFHarsanyi1983\"] = 1,\n [\"CITEREFHatcher2014\"] = 1,\n [\"CITEREFHermes1973\"] = 1,\n [\"CITEREFHilbert1902\"] = 1,\n [\"CITEREFKaye1991\"] = 1,\n [\"CITEREFLandau1965\"] = 1,\n [\"CITEREFMendelson1997\"] = 1,\n [\"CITEREFMendelson2015\"] = 1,\n [\"CITEREFMeseguerGoguen1986\"] = 1,\n [\"CITEREFMurzi\"] = 1,\n [\"CITEREFParteeTer_MeulenWall2012\"] = 1,\n [\"CITEREFPeano1889\"] = 1,\n [\"CITEREFPeano1908\"] = 1,\n [\"CITEREFPeirce1881\"] = 1,\n [\"CITEREFPodnieks2015\"] = 1,\n [\"CITEREFShields1997\"] = 1,\n [\"CITEREFSmullyan2013\"] = 1,\n [\"CITEREFSuppes1960\"] = 1,\n [\"CITEREFTakeuti2013\"] = 1,\n [\"CITEREFTarskiGivant1987\"] = 1,\n [\"CITEREFVan_Heijenoort1967\"] = 1,\n [\"CITEREFVan_Oosten1999\"] = 1,\n [\"CITEREFWang1957\"] = 1,\n [\"CITEREFWillard2001\"] = 1,\n}\ntemplate_list = table#1 {\n [\"Citation needed span\"] = 1,\n [\"Cite IEP\"] = 1,\n [\"Cite book\"] = 27,\n [\"Cite journal\"] = 7,\n [\"Cite web\"] = 2,\n [\"Cn\"] = 1,\n [\"Further\"] = 1,\n [\"Harvnb\"] = 6,\n [\"IPA\"] = 1,\n [\"IPAc-en\"] = 1,\n [\"Improve references\"] = 1,\n [\"Langx\"] = 2,\n [\"Main\"] = 2,\n [\"Math theorem\"] = 1,\n [\"MathWorld\"] = 1,\n [\"NoteFoot\"] = 1,\n [\"NoteTag\"] = 1,\n [\"Nowrap\"] = 47,\n [\"Ordered list\"] = 5,\n [\"PlanetMath attribution\"] = 1,\n [\"Portal\"] = 1,\n [\"Refbegin\"] = 1,\n [\"Refend\"] = 1,\n [\"Reflist\"] = 1,\n [\"Section link\"] = 1,\n [\"Sfn\"] = 22,\n [\"Sfnp\"] = 1,\n [\"Short description\"] = 1,\n [\"Springer\"] = 1,\n [\"Webarchive\"] = 1,\n}\narticle_whitelist = table#1 {\n}\n"},"cachereport":{"origin":"mw-web.codfw.main-6b7f745dd4-nj7fj","timestamp":"20241125144641","ttl":2592000,"transientcontent":false}}});});</script> <script type="application/ld+json">{"@context":"https:\/\/schema.org","@type":"Article","name":"Peano axioms","url":"https:\/\/en.wikipedia.org\/wiki\/Peano_axioms","sameAs":"http:\/\/www.wikidata.org\/entity\/Q842755","mainEntity":"http:\/\/www.wikidata.org\/entity\/Q842755","author":{"@type":"Organization","name":"Contributors to Wikimedia projects"},"publisher":{"@type":"Organization","name":"Wikimedia Foundation, Inc.","logo":{"@type":"ImageObject","url":"https:\/\/www.wikimedia.org\/static\/images\/wmf-hor-googpub.png"}},"datePublished":"2001-12-31T16:25:14Z","dateModified":"2024-11-03T23:25:23Z","headline":"axiomatic system for the natural numbers"}</script> </body> </html>