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[o]" accesskey="o"><span class="vector-icon mw-ui-icon-logIn mw-ui-icon-wikimedia-logIn"></span> <span>Logga 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"> Sidor för utloggade redigerare <a href="/wiki/Hj%C3%A4lp:Introduktion" aria-label="Läs mer om redigering"><span>läs mer</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:Mina_bidrag" title="En lista över redigeringar från denna IP-adress [y]" accesskey="y"><span>Bidrag</span></a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/Special:Min_diskussion" title="Diskussion om redigeringar från det här IP-numret [n]" accesskey="n"><span>Diskussion</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="Webbplats"> <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="Innehåll" 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">Innehåll</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">flytta till sidofältet</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">dölj</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">Inledning</div> </a> </li> <li id="toc-Formulering_av_de_naturliga_talen" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Formulering_av_de_naturliga_talen"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Formulering av de naturliga talen</span> </div> </a> <ul id="toc-Formulering_av_de_naturliga_talen-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Definition_av_operatorer" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Definition_av_operatorer"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Definition av operatorer</span> </div> </a> <button aria-controls="toc-Definition_av_operatorer-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>Växla underavsnittet Definition av operatorer</span> </button> <ul id="toc-Definition_av_operatorer-sublist" class="vector-toc-list"> <li id="toc-Addition" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Addition"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Addition</span> </div> </a> <ul id="toc-Addition-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Multiplikation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Multiplikation"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Multiplikation</span> </div> </a> <ul id="toc-Multiplikation-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Axiomatisering_med_första_ordningens_logik" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Axiomatisering_med_första_ordningens_logik"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Axiomatisering med första ordningens logik</span> </div> </a> <ul id="toc-Axiomatisering_med_första_ordningens_logik-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Källor" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Källor"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Källor</span> </div> </a> <button aria-controls="toc-Källor-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>Växla underavsnittet Källor</span> </button> <ul id="toc-Källor-sublist" class="vector-toc-list"> <li id="toc-Noter" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Noter"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Noter</span> </div> </a> <ul id="toc-Noter-sublist" class="vector-toc-list"> </ul> </li> </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="Innehåll" 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="Växla innehållsförteckningen" > <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">Växla innehållsförteckningen</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">Peanoaritmetik</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="Gå till en artikel på ett annat språk. Tillgänglig på 19 språk" > <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-19" 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">19 språk</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-be badge-Q70894304 mw-list-item" title=""><a href="https://be.wikipedia.org/wiki/%D0%90%D1%80%D1%8B%D1%84%D0%BC%D0%B5%D1%82%D1%8B%D0%BA%D0%B0_%D0%9F%D0%B5%D0%B0%D0%BD%D0%B0" title="Арыфметыка Пеана – belarusiska" lang="be" hreflang="be" data-title="Арыфметыка Пеана" data-language-autonym="Беларуская" data-language-local-name="belarusiska" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-ca badge-Q70894304 mw-list-item" title=""><a href="https://ca.wikipedia.org/wiki/Aritm%C3%A8tica_de_Peano" title="Aritmètica de Peano – katalanska" lang="ca" hreflang="ca" data-title="Aritmètica de Peano" data-language-autonym="Català" data-language-local-name="katalanska" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%A4%D0%BE%D1%80%D0%BC%D0%B0%D0%BB%D0%BB%C4%95_%D0%B0%D1%80%D0%B8%D1%84%D0%BC%D0%B5%D1%82%D0%B8%D0%BA%D0%B0" title="Формаллĕ арифметика – tjuvasjiska" lang="cv" hreflang="cv" data-title="Формаллĕ арифметика" data-language-autonym="Чӑвашла" data-language-local-name="tjuvasjiska" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Peanova_aritmetika" title="Peanova aritmetika – tjeckiska" lang="cs" hreflang="cs" data-title="Peanova aritmetika" data-language-autonym="Čeština" data-language-local-name="tjeckiska" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Peano-Arithmetik" title="Peano-Arithmetik – tyska" lang="de" hreflang="de" data-title="Peano-Arithmetik" data-language-autonym="Deutsch" data-language-local-name="tyska" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-en badge-Q70894304 mw-list-item" title=""><a href="https://en.wikipedia.org/wiki/Peano_arithmetic" title="Peano arithmetic – engelska" lang="en" hreflang="en" data-title="Peano arithmetic" data-language-autonym="English" data-language-local-name="engelska" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-es badge-Q70894304 mw-list-item" title=""><a href="https://es.wikipedia.org/wiki/Aritmetica_de_Peano" title="Aritmetica de Peano – spanska" lang="es" hreflang="es" data-title="Aritmetica de Peano" data-language-autonym="Español" data-language-local-name="spanska" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fr badge-Q70894304 mw-list-item" title=""><a href="https://fr.wikipedia.org/wiki/Arithm%C3%A9tique_de_Peano" title="Arithmétique de Peano – franska" lang="fr" hreflang="fr" data-title="Arithmétique de Peano" data-language-autonym="Français" data-language-local-name="franska" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ko badge-Q70894304 mw-list-item" title=""><a href="https://ko.wikipedia.org/wiki/%ED%8E%98%EC%95%84%EB%85%B8_%EC%82%B0%EC%88%A0" title="페아노 산술 – koreanska" lang="ko" hreflang="ko" data-title="페아노 산술" data-language-autonym="한국어" data-language-local-name="koreanska" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-id badge-Q70894304 mw-list-item" title=""><a href="https://id.wikipedia.org/wiki/Aritmetika_Peano" title="Aritmetika Peano – indonesiska" lang="id" hreflang="id" data-title="Aritmetika Peano" data-language-autonym="Bahasa Indonesia" data-language-local-name="indonesiska" 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/Aritmetica_di_Peano" title="Aritmetica di Peano – italienska" lang="it" hreflang="it" data-title="Aritmetica di Peano" data-language-autonym="Italiano" data-language-local-name="italienska" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Peano-aritmetika" title="Peano-aritmetika – ungerska" lang="hu" hreflang="hu" data-title="Peano-aritmetika" data-language-autonym="Magyar" data-language-local-name="ungerska" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-nl badge-Q70894304 mw-list-item" title=""><a href="https://nl.wikipedia.org/wiki/Peano-rekenkunde" title="Peano-rekenkunde – nederländska" lang="nl" hreflang="nl" data-title="Peano-rekenkunde" data-language-autonym="Nederlands" data-language-local-name="nederländska" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja badge-Q70894304 mw-list-item" title=""><a href="https://ja.wikipedia.org/wiki/%E3%83%9A%E3%82%A2%E3%83%8E%E7%AE%97%E8%A1%93" title="ペアノ算術 – japanska" lang="ja" hreflang="ja" data-title="ペアノ算術" data-language-autonym="日本語" data-language-local-name="japanska" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-pt badge-Q70894304 mw-list-item" title=""><a href="https://pt.wikipedia.org/wiki/Aritm%C3%A9tica_de_Peano" title="Aritmética de Peano – portugisiska" lang="pt" hreflang="pt" data-title="Aritmética de Peano" data-language-autonym="Português" data-language-local-name="portugisiska" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru badge-Q70894304 mw-list-item" title=""><a href="https://ru.wikipedia.org/wiki/%D0%90%D1%80%D0%B8%D1%84%D0%BC%D0%B5%D1%82%D0%B8%D0%BA%D0%B0_%D0%9F%D0%B5%D0%B0%D0%BD%D0%BE" title="Арифметика Пеано – ryska" lang="ru" hreflang="ru" data-title="Арифметика Пеано" data-language-autonym="Русский" data-language-local-name="ryska" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Peanova_aritmetika" title="Peanova aritmetika – slovakiska" lang="sk" hreflang="sk" data-title="Peanova aritmetika" data-language-autonym="Slovenčina" data-language-local-name="slovakiska" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%A4%D0%BE%D1%80%D0%BC%D0%B0%D0%BB%D1%8C%D0%BD%D0%B0_%D0%B0%D1%80%D0%B8%D1%84%D0%BC%D0%B5%D1%82%D0%B8%D0%BA%D0%B0" title="Формальна арифметика – ukrainska" lang="uk" hreflang="uk" data-title="Формальна арифметика" data-language-autonym="Українська" data-language-local-name="ukrainska" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-zh badge-Q70894304 mw-list-item" title=""><a href="https://zh.wikipedia.org/wiki/%E7%9A%AE%E4%BA%9A%E8%AF%BA%E7%AE%97%E6%9C%AF" title="皮亚诺算术 – kinesiska" lang="zh" hreflang="zh" data-title="皮亚诺算术" data-language-autonym="中文" data-language-local-name="kinesiska" class="interlanguage-link-target"><span>中文</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit 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data-event-name="pinnable-header.vector-appearance.unpin">dölj</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">Från Wikipedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="sv" dir="ltr"><p><b>Peanoaritmetik</b> (ibland förkortat <b>PA</b>) är inom <a href="/wiki/Talteori" title="Talteori">talteorin</a> ett sätt att konstruera de <a href="/wiki/Naturligt_tal" class="mw-redirect" title="Naturligt tal">naturliga talen</a> <i>N</i>, samt addition och multiplikation med hjälp av <a href="/wiki/Peanos_axiom" title="Peanos axiom">Peanos axiom</a>. Peanoaritmetiken kan ses som en <a href="/wiki/Slutenhet_(matematik)" title="Slutenhet (matematik)">sluten</a> <a href="/wiki/Algebraisk_struktur" title="Algebraisk struktur">algebraisk struktur</a> (<i>N</i>, +, 0, ·, 1), där 0 och 1 är neutrala element för addition respektive multiplikation. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Formulering_av_de_naturliga_talen">Formulering av de naturliga talen</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Peanoaritmetik&veaction=edit&section=1" title="Redigera avsnitt: Formulering av de naturliga talen" class="mw-editsection-visualeditor"><span>redigera</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Peanoaritmetik&action=edit&section=1" title="Redigera avsnitts källkod: Formulering av de naturliga talen"><span>redigera wikitext</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Peanoaritmetiken behandlar de aritmetiska egenskaperna hos de naturliga talen, ofta representerade av en mängd <i>N</i> (ibland dubbeltecknat, <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>). Peanos axiom definierar dessa tal genom en konstant 0 och en ”efterföljarfunktion” (eng. <a href="https://en.wikipedia.org/wiki/successor_function" class="extiw" title="en:successor function">successor function</a>) <i>S</i> sådan att <i>S</i>(<i>n</i>) ger det på <i>n</i> efterföljande talet (<i>S</i>(0) ger 1, <i>S</i>(1) ger 2, etc.). I moderna anpassningar har Peanos ursprungliga nio axiom reducerats till fem, ty fyra av dem endast beskriver underförstådda egenskaper hos likhet. Dessa fem är: </p> <dl><dd><ol><li>0 är ett naturligt tal.</li> <li>För varje naturligt tal <i>n</i> så är <i>S</i>(<i>n</i>) ett naturligt tal. Den naturliga talmängden är sluten under <i>S</i>.</li> <li>För alla naturliga tal <i>m</i> och <i>n</i>, så är <i>S</i>(<i>m</i>)=<i>S</i>(<i>n</i>) om och endast om <i>m</i> = <i>n</i>. <i>S</i> är en <a href="/wiki/Injektiv_funktion" title="Injektiv funktion">injektiv funktion</a>.</li> <li>Det finns inget naturligt tal <i>n</i> sådant att <i>S</i>(<i>n</i>) = 0.</li> <li>Om <i>K</i> är en mängd sådan att</li></ol> <dl><dd><dl><dd><ul><li><i>K</i> innehåller 0, och</li> <li>för varje naturligt tal <i>n</i>, om <i>K</i> innehåller <i>n</i> så innehåller <i>K</i> också <i>S</i>(<i>n</i>)</li></ul></dd> <dd>så innehåller <i>K</i> alla naturliga tal.</dd></dl></dd></dl></dd></dl> <p>För att visa att <i>K</i> innehåller <i>S</i>(<i>n</i>), så räcker det att veta att <i>K</i> innehåller <i>n</i>. Det femte axiomet ger alltså ett sätt att definiera den naturliga talmängden genom <a href="/wiki/Matematisk_induktion" title="Matematisk induktion">matematisk induktion</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Definition_av_operatorer">Definition av operatorer</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Peanoaritmetik&veaction=edit&section=2" title="Redigera avsnitt: Definition av operatorer" class="mw-editsection-visualeditor"><span>redigera</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Peanoaritmetik&action=edit&section=2" title="Redigera avsnitts källkod: Definition av operatorer"><span>redigera wikitext</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Med hjälp av efterföljarfunktionen <i>S</i> kan man definiera addition och multiplikation som <a href="/wiki/Bin%C3%A4r_operator" title="Binär operator">binära funktioner</a>. De tar två element i <i>N</i> och ger tillbaka ett tredje; de naturliga talen är således slutna under addition och multiplikation. </p> <div class="mw-heading mw-heading3"><h3 id="Addition">Addition</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Peanoaritmetik&veaction=edit&section=3" title="Redigera avsnitt: Addition" class="mw-editsection-visualeditor"><span>redigera</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Peanoaritmetik&action=edit&section=3" title="Redigera avsnitts källkod: Addition"><span>redigera wikitext</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Addition definieras rekursivt som </p> <dl><dd><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+0=a,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>+</mo> <mn>0</mn> <mo>=</mo> <mi>a</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a+0=a,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/813799da4092f843d6f9e96d10967e9faa6729a9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.208ex; height:2.509ex;" alt="{\displaystyle a+0=a,}"></span></dd> <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+S(b)=S(a+b).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>+</mo> <mi>S</mi> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>S</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a+S(b)=S(a+b).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04216b7fb2ae7d97e77433e68f5149f781a050ca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.498ex; height:2.843ex;" alt="{\displaystyle a+S(b)=S(a+b).}"></span></dd></dl></dd></dl> <p>Till exempel, </p> <dl><dd><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 3+2=3+S(1)=S(3+1)=S(3+S(0))=S(S(3+0))=S(S(3))=S(4)=5.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3</mn> <mo>+</mo> <mn>2</mn> <mo>=</mo> <mn>3</mn> <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> <mn>3</mn> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mi>S</mi> <mo stretchy="false">(</mo> <mn>3</mn> <mo>+</mo> <mi>S</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mi>S</mi> <mo stretchy="false">(</mo> <mi>S</mi> <mo stretchy="false">(</mo> <mn>3</mn> <mo>+</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mi>S</mi> <mo stretchy="false">(</mo> <mi>S</mi> <mo stretchy="false">(</mo> <mn>3</mn> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mi>S</mi> <mo stretchy="false">(</mo> <mn>4</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>5.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3+2=3+S(1)=S(3+1)=S(3+S(0))=S(S(3+0))=S(S(3))=S(4)=5.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b8908815e4c779f3c1e68336f7196f42e370f6f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:81.427ex; height:2.843ex;" alt="{\displaystyle 3+2=3+S(1)=S(3+1)=S(3+S(0))=S(S(3+0))=S(S(3))=S(4)=5.}"></span></dd></dl></dd></dl> <p><br /> </p> <div class="mw-heading mw-heading3"><h3 id="Multiplikation">Multiplikation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Peanoaritmetik&veaction=edit&section=4" title="Redigera avsnitt: Multiplikation" class="mw-editsection-visualeditor"><span>redigera</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Peanoaritmetik&action=edit&section=4" title="Redigera avsnitts källkod: Multiplikation"><span>redigera wikitext</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Givet addition definieras multiplikation rekursivt som </p> <dl><dd><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 0=0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>⋅</mo> <mn>0</mn> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\cdot 0=0,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/540d7a7b6f63e28d8cc6fa99854ac2f598f40570" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.979ex; height:2.509ex;" alt="{\displaystyle a\cdot 0=0,}"></span></dd> <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(b)=a+(a\cdot b).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>⋅</mo> <mi>S</mi> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\cdot S(b)=a+(a\cdot b).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd9b68253f2690a93db39b24f9fbabeb46d8c73a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.746ex; height:2.843ex;" alt="{\displaystyle a\cdot S(b)=a+(a\cdot b).}"></span></dd></dl></dd></dl> <p>Till exempel, </p> <dl><dd><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 4\cdot 1=4\cdot S(0)=4+(4\cdot 0)=4+0=4.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>4</mn> <mo>⋅</mo> <mn>1</mn> <mo>=</mo> <mn>4</mn> <mo>⋅</mo> <mi>S</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>4</mn> <mo>+</mo> <mo stretchy="false">(</mo> <mn>4</mn> <mo>⋅</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>4</mn> <mo>+</mo> <mn>0</mn> <mo>=</mo> <mn>4.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 4\cdot 1=4\cdot S(0)=4+(4\cdot 0)=4+0=4.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89068f1318950608be87fc8b6e6e9d39b9678458" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:40.501ex; height:2.843ex;" alt="{\displaystyle 4\cdot 1=4\cdot S(0)=4+(4\cdot 0)=4+0=4.}"></span></dd></dl></dd></dl> <p>Då man låter 0 vara det <a href="/wiki/Neutralt_element" title="Neutralt element">neutrala elementet</a> för addition följer att 1 är det neutrala elementet för multiplikation. Vidare uppvisar både addition och multiplikation här <a href="/wiki/Kommutativitet" title="Kommutativitet">kommutativa</a> och <a href="/wiki/Associativitet" title="Associativitet">associativa</a> egenskaper, och multiplikation är <a href="/wiki/Distributivitet" title="Distributivitet">distributiv</a> över addition. Den algebraiska strukturen (N, +, 0, ·, 1) bildar således en kommutativ halv<a href="/wiki/Ring_(matematik)" title="Ring (matematik)">ring</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Axiomatisering_med_första_ordningens_logik"><span id="Axiomatisering_med_f.C3.B6rsta_ordningens_logik"></span>Axiomatisering med första ordningens logik</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Peanoaritmetik&veaction=edit&section=5" title="Redigera avsnitt: Axiomatisering med första ordningens logik" class="mw-editsection-visualeditor"><span>redigera</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Peanoaritmetik&action=edit&section=5" title="Redigera avsnitts källkod: Axiomatisering med första ordningens logik"><span>redigera wikitext</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Inom logisk analys är det önskvärt att uttrycka Peanoaritmetik med hjälp av <a href="/wiki/F%C3%B6rsta_ordningens_logik" title="Första ordningens logik">första ordningens logik</a>, då det underlättar vid exempelvis <a href="/wiki/Bevisteori" title="Bevisteori">bevisteori</a>. Det är då nödvändigt att låta addition och multiplikation vara en del av axiomen, istället för att definiera dem med hjälp av efterföljarfunktionen. De nya axiomen består delvis av några av de ursprungliga axiomen, delvis av definitionerna för addition och multiplikation ovan. </p> <dl><dd><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\in N.\ S(x)\neq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mo>∈</mo> <mi>N</mi> <mo>.</mo> <mtext> </mtext> <mi>S</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>≠</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall x\in N.\ S(x)\neq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/291a1f32127831f245b4b6ec7532d135f1fc690e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.04ex; height:2.843ex;" alt="{\displaystyle \forall x\in N.\ S(x)\neq 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\in N.\ 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> <mo>∈</mo> <mi>N</mi> <mo>.</mo> <mtext> </mtext> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall x,y\in N.\ S(x)=S(y)\Rightarrow x=y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a433020a10e3a40862f6ceafb062039bbe80dcd5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.729ex; height:2.843ex;" alt="{\displaystyle \forall x,y\in N.\ 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\in N.\ x+0=x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mo>∈</mo> <mi>N</mi> <mo>.</mo> <mtext> </mtext> <mi>x</mi> <mo>+</mo> <mn>0</mn> <mo>=</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall x\in N.\ x+0=x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/908140ebb816e8362fa3949cc115fb56aa948725" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:18.902ex; height:2.343ex;" alt="{\displaystyle \forall x\in N.\ 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\in N.\ 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> <mo>∈</mo> <mi>N</mi> <mo>.</mo> <mtext> </mtext> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall x,y\in N.\ x+S(y)=S(x+y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd22f1e32dd4868976b74b7f006805b1ace8c14d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.697ex; height:2.843ex;" alt="{\displaystyle \forall x,y\in N.\ 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\in N.\ x\cdot 0=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mo>∈</mo> <mi>N</mi> <mo>.</mo> <mtext> </mtext> <mi>x</mi> <mo>⋅</mo> <mn>0</mn> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall x\in N.\ x\cdot 0=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a431adfbfe6b02f49c9a50afe9ec3f8be1472273" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:17.573ex; height:2.176ex;" alt="{\displaystyle \forall x\in N.\ 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\in N.\ x\cdot S(y)=x+(x\cdot y)}"> <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>N</mi> <mo>.</mo> <mtext> </mtext> <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> <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\in N.\ x\cdot S(y)=x+(x\cdot y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b477aa463f67df7a218974673bbeecf01d88d19c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.045ex; height:2.843ex;" alt="{\displaystyle \forall x,y\in N.\ x\cdot S(y)=x+(x\cdot y)}"></span></li></ol></dd></dl> <p>Att konstatera att en mängd som innehåller 0 och efterföljaren av alla tal i mängden innehåller alla naturliga tal är inte heller möjligt utan <a href="/wiki/Andra_ordningens_logik" title="Andra ordningens logik">andra ordningens logik</a>. Lösningen är att omvandla det ursprungliga femte axiomet till ett första ordningens <i>axiomschema</i> som gäller för alla naturliga tal: </p> <dl><dd><ul><li>Om φ(<i>n</i>) är en sats sådan att <ul><li>φ(<i>n</i>) = 0 är sann, och</li> <li>för varje naturligt tal <i>n</i>, om φ(<i>n</i>) är sann, så är också φ(<i>S</i>(<i>n</i>)) sann</li></ul></li></ul> <dl><dd>så är φ(<i>n</i>) sann för alla naturliga tal.</dd></dl></dd></dl> <p>Detta ger ett axiom för varje tal <i>n</i> och totalt ett oändligt men <a href="/wiki/Uppr%C3%A4knelig" class="mw-redirect" title="Uppräknelig">uppräkneligt</a> antal axiom; man säger då att systemet inte är <i>ändligt axiomatiserat</i>. </p> <div class="mw-heading mw-heading2"><h2 id="Källor"><span id="K.C3.A4llor"></span>Källor</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Peanoaritmetik&veaction=edit&section=6" title="Redigera avsnitt: Källor" class="mw-editsection-visualeditor"><span>redigera</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Peanoaritmetik&action=edit&section=6" title="Redigera avsnitts källkod: Källor"><span>redigera wikitext</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Hofstadter, Douglas R., 1985. <i>Gödel, Escher, Bach : ett evigt gyllene band</i>. Svensk uppl. Stockholm: Bromberg.</li> <li>van Oosten, Jaap, <i>Introduction to Peano Arithmetic - Gödel Incompleteness and Nonstandard Models</i> <a rel="nofollow" class="external free" href="http://www.staff.science.uu.nl/~ooste110/syllabi/peanomoeder.pdf">http://www.staff.science.uu.nl/~ooste110/syllabi/peanomoeder.pdf</a> (Hämtad 5 maj 2015)</li> <li>Weisstein, Eric W. "Peano Arithmetic." From MathWorld--A Wolfram Web Resource. <a rel="nofollow" class="external free" href="http://mathworld.wolfram.com/PeanoArithmetic.html">http://mathworld.wolfram.com/PeanoArithmetic.html</a> (Hämtad 5 maj 2015)</li> <li>Weisstein, Eric W. "Peano's Axioms." From MathWorld--A Wolfram Web Resource. <a rel="nofollow" class="external free" href="http://mathworld.wolfram.com/PeanosAxioms.html">http://mathworld.wolfram.com/PeanosAxioms.html</a> (Hämtad 5 maj 2015)</li></ul> <div class="mw-heading mw-heading3"><h3 id="Noter">Noter</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Peanoaritmetik&veaction=edit&section=7" title="Redigera avsnitt: Noter" class="mw-editsection-visualeditor"><span>redigera</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Peanoaritmetik&action=edit&section=7" title="Redigera avsnitts källkod: Noter"><span>redigera wikitext</span></a><span class="mw-editsection-bracket">]</span></span></div>    </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="">Hämtad från ”<a dir="ltr" href="https://sv.wikipedia.org/w/index.php?title=Peanoaritmetik&oldid=34240589">https://sv.wikipedia.org/w/index.php?title=Peanoaritmetik&oldid=34240589</a>”</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Wikipedia:Kategorier" title="Wikipedia:Kategorier">Kategori</a>: <ul><li><a href="/wiki/Kategori:Talteori" title="Kategori:Talteori">Talteori</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"> Sidan redigerades senast den 24 april 2016 kl. 22.20.</li> <li id="footer-info-copyright">Wikipedias text är tillgänglig under licensen <a rel="nofollow" class="external text" href="//creativecommons.org/licenses/by-sa/4.0/deed.sv">Creative Commons Erkännande-dela-lika 4.0 Unported</a>. 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