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Peano-aritmetika – Wikipédia
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class="vector-toc-link"> <div class="vector-toc-text">Bevezető</div> </a> </li> <li id="toc-A_Peano-aritmetika_elsőrendű_nyelve" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#A_Peano-aritmetika_elsőrendű_nyelve"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>A Peano-aritmetika elsőrendű nyelve</span> </div> </a> <ul id="toc-A_Peano-aritmetika_elsőrendű_nyelve-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Peano-féle_axiómák_és_axiómasémák" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Peano-féle_axiómák_és_axiómasémák"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Peano-féle axiómák és axiómasémák</span> </div> </a> <ul id="toc-Peano-féle_axiómák_és_axiómasémák-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Szokásos_definíciók,_tételek" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Szokásos_definíciók,_tételek"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Szokásos definíciók, tételek</span> </div> </a> <button aria-controls="toc-Szokásos_definíciók,_tételek-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>A(z) Szokásos definíciók, tételek alszakasz kinyitása/becsukása</span> </button> <ul id="toc-Szokásos_definíciók,_tételek-sublist" class="vector-toc-list"> <li id="toc-Műveleti_tulajdonságok" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Műveleti_tulajdonságok"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Műveleti tulajdonságok</span> </div> </a> <ul id="toc-Műveleti_tulajdonságok-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Rendezés" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Rendezés"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Rendezés</span> </div> </a> <ul id="toc-Rendezés-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Erős_indukció_és_a_legkisebb_szám_elve" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Erős_indukció_és_a_legkisebb_szám_elve"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>Erős indukció és a legkisebb szám elve</span> </div> </a> <ul id="toc-Erős_indukció_és_a_legkisebb_szám_elve-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Oszthatósági_reláció" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Oszthatósági_reláció"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.4</span> <span>Oszthatósági reláció</span> </div> </a> <ul id="toc-Oszthatósági_reláció-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Maradékos_osztás" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Maradékos_osztás"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.5</span> <span>Maradékos osztás</span> </div> </a> <ul id="toc-Maradékos_osztás-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Prím-,_Felbonthatatlan-_és_RelatívPrím_predikátum" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Prím-,_Felbonthatatlan-_és_RelatívPrím_predikátum"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.6</span> <span>Prím-, Felbonthatatlan- és RelatívPrím predikátum</span> </div> </a> <ul id="toc-Prím-,_Felbonthatatlan-_és_RelatívPrím_predikátum-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Monusz" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Monusz"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.7</span> <span>Monusz</span> </div> </a> <ul id="toc-Monusz-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Korlátos_kvantorok" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Korlátos_kvantorok"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.8</span> <span>Korlátos kvantorok</span> </div> </a> <ul id="toc-Korlátos_kvantorok-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Aritmetikai_hierarchia" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Aritmetikai_hierarchia"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.9</span> <span>Aritmetikai hierarchia</span> </div> </a> <ul id="toc-Aritmetikai_hierarchia-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-További_információk" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#További_információk"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>További információk</span> </div> </a> <ul id="toc-További_információk-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Források" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Források"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Források</span> </div> </a> <ul id="toc-Források-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Jegyzetek" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Jegyzetek"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Jegyzetek</span> </div> </a> <ul id="toc-Jegyzetek-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="Tartalomjegyzék" class="vector-toc-landmark"> <div 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class="mw-page-title-main">Peano-aritmetika</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="Ugrás egy más nyelvű szócikkre. Elérhető 19 nyelven" > <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 nyelv</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-en badge-Q70894304 mw-list-item" title=""><a href="https://en.wikipedia.org/wiki/Peano_arithmetic" title="Peano arithmetic – angol" lang="en" hreflang="en" data-title="Peano arithmetic" data-language-autonym="English" data-language-local-name="angol" class="interlanguage-link-target"><span>English</span></a></li><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 – belarusz" lang="be" hreflang="be" data-title="Арыфметыка Пеана" data-language-autonym="Беларуская" data-language-local-name="belarusz" 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 – katalán" lang="ca" hreflang="ca" data-title="Aritmètica de Peano" data-language-autonym="Català" data-language-local-name="katalán" 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/Peanova_aritmetika" title="Peanova aritmetika – cseh" lang="cs" hreflang="cs" data-title="Peanova aritmetika" data-language-autonym="Čeština" data-language-local-name="cseh" class="interlanguage-link-target"><span>Čeština</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 – csuvas" lang="cv" hreflang="cv" data-title="Формаллĕ арифметика" data-language-autonym="Чӑвашла" data-language-local-name="csuvas" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Peano-Arithmetik" title="Peano-Arithmetik – német" lang="de" hreflang="de" data-title="Peano-Arithmetik" data-language-autonym="Deutsch" data-language-local-name="német" class="interlanguage-link-target"><span>Deutsch</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 – spanyol" lang="es" hreflang="es" data-title="Aritmetica de Peano" data-language-autonym="Español" data-language-local-name="spanyol" 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 – francia" lang="fr" hreflang="fr" data-title="Arithmétique de Peano" data-language-autonym="Français" data-language-local-name="francia" class="interlanguage-link-target"><span>Français</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 – indonéz" lang="id" hreflang="id" data-title="Aritmetika Peano" data-language-autonym="Bahasa Indonesia" data-language-local-name="indonéz" 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 – olasz" lang="it" hreflang="it" data-title="Aritmetica di Peano" data-language-autonym="Italiano" data-language-local-name="olasz" class="interlanguage-link-target"><span>Italiano</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 – japán" lang="ja" hreflang="ja" data-title="ペアノ算術" data-language-autonym="日本語" data-language-local-name="japán" class="interlanguage-link-target"><span>日本語</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 – koreai" lang="ko" hreflang="ko" data-title="페아노 산술" data-language-autonym="한국어" data-language-local-name="koreai" class="interlanguage-link-target"><span>한국어</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 – holland" lang="nl" hreflang="nl" data-title="Peano-rekenkunde" data-language-autonym="Nederlands" data-language-local-name="holland" class="interlanguage-link-target"><span>Nederlands</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 – portugál" lang="pt" hreflang="pt" data-title="Aritmética de Peano" data-language-autonym="Português" data-language-local-name="portugál" 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 – orosz" lang="ru" hreflang="ru" data-title="Арифметика Пеано" data-language-autonym="Русский" data-language-local-name="orosz" 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 – szlovák" lang="sk" hreflang="sk" data-title="Peanova aritmetika" data-language-autonym="Slovenčina" data-language-local-name="szlovák" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Peanoaritmetik" title="Peanoaritmetik – svéd" lang="sv" hreflang="sv" data-title="Peanoaritmetik" data-language-autonym="Svenska" data-language-local-name="svéd" class="interlanguage-link-target"><span>Svenska</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 – ukrán" lang="uk" hreflang="uk" data-title="Формальна арифметика" data-language-autonym="Українська" data-language-local-name="ukrán" 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 – kínai" lang="zh" hreflang="zh" data-title="皮亚诺算术" data-language-autonym="中文" data-language-local-name="kínai" 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 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mw-parser-output" lang="hu" dir="ltr"><p>A <b>Peano-aritmetika</b> a <a href="/wiki/Term%C3%A9szetes_sz%C3%A1mok" title="Természetes számok">természetes számok</a> egy <a href="/wiki/Els%C5%91rend%C5%B1_logika" title="Elsőrendű logika">elsőrendű</a> axiómarendszere. Szokásos jelölése: <b>PA</b>. </p><p>Első, a maitól még kissé eltérő alakját <a href="/wiki/Giuseppe_Peano" title="Giuseppe Peano">Giuseppe Peano</a> olasz matematikusnak köszönhetjük, aki 1889-ben jegyezte le axiómáit. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="A_Peano-aritmetika_elsőrendű_nyelve"><span id="A_Peano-aritmetika_els.C5.91rend.C5.B1_nyelve"></span>A Peano-aritmetika elsőrendű nyelve</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Peano-aritmetika&action=edit&section=1" title="Szakasz szerkesztése: A Peano-aritmetika elsőrendű nyelve"><span>szerkesztés</span></a><span class="mw-editsection-bracket">]</span></span></div> <table align="right" class="wikitable"> <tbody><tr> <th align="center">Rövidítések </th></tr> <tr> <td><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{array}{rcl}\scriptstyle {(x+y)}&\scriptstyle {=_{def}}&\scriptstyle {+xy}\\\scriptstyle {(x\cdot y)}&\scriptstyle {=_{def}}&\scriptstyle {\cdot xy}\\\scriptstyle {1}&\scriptstyle {=_{def}}&\scriptstyle {s0}\\\scriptstyle {2}&\scriptstyle {=_{def}}&\scriptstyle {ss0}\\\scriptstyle {3}&\scriptstyle {=_{def}}&\scriptstyle {sss0}\\\scriptstyle {\vdots }&\scriptstyle {\vdots }&\scriptstyle {\vdots }\\\scriptstyle {n}&\scriptstyle {=_{def}}&\scriptstyle {\underbrace {\scriptstyle s\dots s} _{\mathrm {n\,db} }0}\end{array}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right center left" rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> </mstyle> </mtd> <mtd> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mi>e</mi> <mi>f</mi> </mrow> </msub> </mrow> </mstyle> </mtd> <mtd> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> <mi>x</mi> <mi>y</mi> </mrow> </mstyle> </mtd> </mtr> <mtr> <mtd> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>⋅<!-- ⋅ --></mo> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> </mstyle> </mtd> <mtd> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mi>e</mi> <mi>f</mi> </mrow> </msub> </mrow> </mstyle> </mtd> <mtd> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mo>⋅<!-- ⋅ --></mo> <mi>x</mi> <mi>y</mi> </mrow> </mstyle> </mtd> </mtr> <mtr> <mtd> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </mstyle> </mtd> <mtd> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mi>e</mi> <mi>f</mi> </mrow> </msub> </mrow> </mstyle> </mtd> <mtd> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> <mn>0</mn> </mrow> </mstyle> </mtd> </mtr> <mtr> <mtd> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </mstyle> </mtd> <mtd> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mi>e</mi> <mi>f</mi> </mrow> </msub> </mrow> </mstyle> </mtd> <mtd> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> <mi>s</mi> <mn>0</mn> </mrow> </mstyle> </mtd> </mtr> <mtr> <mtd> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </mstyle> </mtd> <mtd> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mi>e</mi> <mi>f</mi> </mrow> </msub> </mrow> </mstyle> </mtd> <mtd> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> <mi>s</mi> <mi>s</mi> <mn>0</mn> </mrow> </mstyle> </mtd> </mtr> <mtr> <mtd> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mo>⋮<!-- ⋮ --></mo> </mrow> </mstyle> </mtd> <mtd> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mo>⋮<!-- ⋮ --></mo> </mrow> </mstyle> </mtd> <mtd> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mo>⋮<!-- ⋮ --></mo> </mrow> </mstyle> </mtd> </mtr> <mtr> <mtd> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </mstyle> </mtd> <mtd> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mi>e</mi> <mi>f</mi> </mrow> </msub> </mrow> </mstyle> </mtd> <mtd> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <munder> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <munder> <mstyle displaystyle="false" scriptlevel="1"> <mi>s</mi> <mo>…<!-- … --></mo> <mi>s</mi> </mstyle> <mo>⏟<!-- ⏟ --></mo> </munder> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">n</mi> <mspace width="thinmathspace" /> <mi mathvariant="normal">d</mi> <mi mathvariant="normal">b</mi> </mrow> </mrow> </munder> <mn>0</mn> </mrow> </mstyle> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{array}{rcl}\scriptstyle {(x+y)}&\scriptstyle {=_{def}}&\scriptstyle {+xy}\\\scriptstyle {(x\cdot y)}&\scriptstyle {=_{def}}&\scriptstyle {\cdot xy}\\\scriptstyle {1}&\scriptstyle {=_{def}}&\scriptstyle {s0}\\\scriptstyle {2}&\scriptstyle {=_{def}}&\scriptstyle {ss0}\\\scriptstyle {3}&\scriptstyle {=_{def}}&\scriptstyle {sss0}\\\scriptstyle {\vdots }&\scriptstyle {\vdots }&\scriptstyle {\vdots }\\\scriptstyle {n}&\scriptstyle {=_{def}}&\scriptstyle {\underbrace {\scriptstyle s\dots s} _{\mathrm {n\,db} }0}\end{array}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d9acb0a9c34f10ac0d6aee3e47145304c38e395f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -12.338ex; width:17.498ex; height:25.676ex;" alt="{\displaystyle {\begin{array}{rcl}\scriptstyle {(x+y)}&\scriptstyle {=_{def}}&\scriptstyle {+xy}\\\scriptstyle {(x\cdot y)}&\scriptstyle {=_{def}}&\scriptstyle {\cdot xy}\\\scriptstyle {1}&\scriptstyle {=_{def}}&\scriptstyle {s0}\\\scriptstyle {2}&\scriptstyle {=_{def}}&\scriptstyle {ss0}\\\scriptstyle {3}&\scriptstyle {=_{def}}&\scriptstyle {sss0}\\\scriptstyle {\vdots }&\scriptstyle {\vdots }&\scriptstyle {\vdots }\\\scriptstyle {n}&\scriptstyle {=_{def}}&\scriptstyle {\underbrace {\scriptstyle s\dots s} _{\mathrm {n\,db} }0}\end{array}}}"></span> </td></tr> </tbody></table> <p>A Peano-aritmetika nyelve a következő nem logikai jeleket tartalmazza: </p> <ul><li>A <i><a href="/wiki/0_(sz%C3%A1m)" title="0 (szám)">nullának</a></i> megfelelő konstans jel: <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 \scriptstyle {0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/757abea8b05f9281eee9cd6378f38c7a191e44d8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.822ex; height:1.676ex;" alt="{\displaystyle \scriptstyle {0}}"></span></li> <li>A <i>rákövetkezésnek</i> megfelelő egyváltozós függvényjel:<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 \scriptstyle {s}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {s}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa936bfd3f342bb212c341a7ff13c04021a74c0f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.771ex; height:1.343ex;" alt="{\displaystyle \scriptstyle {s}}"></span></li> <li>Az <i><a href="/wiki/A_term%C3%A9szetes_sz%C3%A1mok_%C3%B6sszead%C3%A1sa" title="A természetes számok összeadása">összeadásnak</a></i> megfelelő kétváltozós műveleti jel: <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 \scriptstyle {+}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {+}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d9cb3bfb2fc1868785d10c0a72ce87161c3c10b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.279ex; height:1.509ex;" alt="{\displaystyle \scriptstyle {+}}"></span></li> <li>A <i><a href="/wiki/Szorz%C3%A1s" title="Szorzás">szorzásnak</a></i> megfelelő kétváltozós műveleti jel: <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 \scriptstyle {\cdot }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mo>⋅<!-- ⋅ --></mo> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {\cdot }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d0c268adc667ee964887357252db434bc6b5d1da" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.31ex; margin-bottom: -0.482ex; width:0.457ex; height:1.009ex;" alt="{\displaystyle \scriptstyle {\cdot }}"></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="Peano-féle_axiómák_és_axiómasémák"><span id="Peano-f.C3.A9le_axi.C3.B3m.C3.A1k_.C3.A9s_axi.C3.B3mas.C3.A9m.C3.A1k"></span>Peano-féle axiómák és axiómasémák</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Peano-aritmetika&action=edit&section=2" title="Szakasz szerkesztése: Peano-féle axiómák és axiómasémák"><span>szerkesztés</span></a><span class="mw-editsection-bracket">]</span></span></div> <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 {\lnot \exists x:sx=0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">¬<!-- ¬ --></mi> <mi mathvariant="normal">∃<!-- ∃ --></mi> <mi>x</mi> <mo>:</mo> <mi>s</mi> <mi>x</mi> <mo>=</mo> <mn>0</mn> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\lnot \exists x:sx=0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/721b3d19aab0da28168d7a3b99197a5f55c3a932" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:12.791ex; height:2.176ex;" alt="{\displaystyle {\lnot \exists x:sx=0}}"></span> – A nulla semminek sem rákövetkezője.</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\forall y:(sx=sy\rightarrow x=y)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∀<!-- ∀ --></mi> <mi>x</mi> <mi mathvariant="normal">∀<!-- ∀ --></mi> <mi>y</mi> <mo>:</mo> <mo stretchy="false">(</mo> <mi>s</mi> <mi>x</mi> <mo>=</mo> <mi>s</mi> <mi>y</mi> <mo stretchy="false">→<!-- → --></mo> <mi>x</mi> <mo>=</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\forall x\forall y:(sx=sy\rightarrow x=y)}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/07d7a5a452d9598d0d89a2615a3eb8887b03bd94" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.779ex; height:2.843ex;" alt="{\displaystyle {\forall x\forall y:(sx=sy\rightarrow x=y)}}"></span> – Amiknek a rákövetkezői is azonosak, azok maguk is azonosak. (Azaz <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 \scriptstyle {s}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {s}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa936bfd3f342bb212c341a7ff13c04021a74c0f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.771ex; height:1.343ex;" alt="{\displaystyle \scriptstyle {s}}"></span> <a href="/wiki/Injekt%C3%ADv_lek%C3%A9pez%C3%A9s" title="Injektív leképezés">injektív</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:(x+0)=x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∀<!-- ∀ --></mi> <mi>x</mi> <mo>:</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mi>x</mi> </mrow> </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/423900204228d334298837f9bbb81bc0c86048dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.129ex; height:2.843ex;" alt="{\displaystyle {\forall x:(x+0)=x}}"></span> – A nullával jobbról való összegzés hatástalan. (Azaz a nulla <a href="/wiki/Z%C3%A9ruselem#Féloldali_zéruselemek" title="Zéruselem">jobb oldali</a> <a href="/wiki/Gy%C5%B1r%C5%B1_(matematika)" title="Gyűrű (matematika)">additív</a> <a href="/wiki/Z%C3%A9ruselem" title="Zéruselem">neutrális elem</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\forall y:(x+sy)=s(x+y)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∀<!-- ∀ --></mi> <mi>x</mi> <mi mathvariant="normal">∀<!-- ∀ --></mi> <mi>y</mi> <mo>:</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>s</mi> <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> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\forall x\forall y:(x+sy)=s(x+y)}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/272b6e2a370219560f2687f6bef6f9dac8b5a362" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.556ex; height:2.843ex;" alt="{\displaystyle {\forall x\forall y:(x+sy)=s(x+y)}}"></span> – a rákövetkezővel való összegzés visszavezethető az összeg rákövetkezőjére.</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"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∀<!-- ∀ --></mi> <mi>x</mi> <mo>:</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>⋅<!-- ⋅ --></mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mrow> </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/1b50601303b35428c6d2243fb792f022d9a2b2a9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.801ex; height:2.843ex;" alt="{\displaystyle {\forall x:(x\cdot 0)=0}}"></span> – A nullával jobbról való szorzás nullát ad.</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\forall y:(x\cdot sy)=(x\cdot y)+x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∀<!-- ∀ --></mi> <mi>x</mi> <mi mathvariant="normal">∀<!-- ∀ --></mi> <mi>y</mi> <mo>:</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>⋅<!-- ⋅ --></mo> <mi>s</mi> <mi>y</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> <mi>x</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\forall x\forall y:(x\cdot sy)=(x\cdot y)+x}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b2355d401eefffb1b1d2b9687156fa7648368abc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.314ex; height:2.843ex;" alt="{\displaystyle {\forall x\forall y:(x\cdot sy)=(x\cdot y)+x}}"></span> – A rákövetkezővel való szorzás visszavezethető a másik tagnak az szorzathoz való még egyszeri hozzáadására.</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 {(\varphi _{x}[0]\land \forall x(\varphi \rightarrow \varphi _{x}[sx]))\rightarrow \forall x\varphi }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <msub> <mi>φ<!-- φ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo stretchy="false">[</mo> <mn>0</mn> <mo stretchy="false">]</mo> <mo>∧<!-- ∧ --></mo> <mi mathvariant="normal">∀<!-- ∀ --></mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>φ<!-- φ --></mi> <mo stretchy="false">→<!-- → --></mo> <msub> <mi>φ<!-- φ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo stretchy="false">[</mo> <mi>s</mi> <mi>x</mi> <mo stretchy="false">]</mo> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo stretchy="false">→<!-- → --></mo> <mi mathvariant="normal">∀<!-- ∀ --></mi> <mi>x</mi> <mi>φ<!-- φ --></mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {(\varphi _{x}[0]\land \forall x(\varphi \rightarrow \varphi _{x}[sx]))\rightarrow \forall x\varphi }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2d4996863e474526d5a4e5312a36826601dc1535" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.269ex; height:2.843ex;" alt="{\displaystyle {(\varphi _{x}[0]\land \forall x(\varphi \rightarrow \varphi _{x}[sx]))\rightarrow \forall x\varphi }}"></span> – A <a href="/wiki/Teljes_indukci%C3%B3" title="Teljes indukció">teljes indukció</a> axiómasémája: Ha a <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 \scriptstyle {\varphi }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mi>φ<!-- φ --></mi> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {\varphi }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e91b58216bc9292e743f8f07287193f7ded3bdda" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.075ex; height:1.676ex;" alt="{\displaystyle \scriptstyle {\varphi }}"></span> formula igaz a nullára, továbbá a formula igazsága a rákövetkezés során öröklődik, akkor ez a formula minden számra igaz.</li></ol> <div class="mw-heading mw-heading2"><h2 id="Szokásos_definíciók,_tételek"><span id="Szok.C3.A1sos_defin.C3.ADci.C3.B3k.2C_t.C3.A9telek"></span>Szokásos definíciók, tételek</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Peano-aritmetika&action=edit&section=3" title="Szakasz szerkesztése: Szokásos definíciók, tételek"><span>szerkesztés</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Műveleti_tulajdonságok"><span id="M.C5.B1veleti_tulajdons.C3.A1gok"></span>Műveleti tulajdonságok</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Peano-aritmetika&action=edit&section=4" title="Szakasz szerkesztése: Műveleti tulajdonságok"><span>szerkesztés</span></a><span class="mw-editsection-bracket">]</span></span></div> <table align="right" class="wikitable"> <tbody><tr> <th colspan="3">Műveleti tulajdonságok tételei </th></tr> <tr> <td rowspan="2" align="center"><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 \scriptstyle {+}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {+}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d9cb3bfb2fc1868785d10c0a72ce87161c3c10b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.279ex; height:1.509ex;" alt="{\displaystyle \scriptstyle {+}}"></span> </td> <td>asszociativitás </td> <td align="left"><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 \scriptstyle {PA\vdash \forall x\forall y\forall z((x+y)+z)=(x+(y+z))}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> <mi>A</mi> <mo>⊢<!-- ⊢ --></mo> <mi mathvariant="normal">∀<!-- ∀ --></mi> <mi>x</mi> <mi mathvariant="normal">∀<!-- ∀ --></mi> <mi>y</mi> <mi mathvariant="normal">∀<!-- ∀ --></mi> <mi>z</mi> <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 stretchy="false">)</mo> <mo>=</mo> <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 stretchy="false">)</mo> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {PA\vdash \forall x\forall y\forall z((x+y)+z)=(x+(y+z))}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/08beb31701737d1c329afb4e8472fdf9a95ed2a5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:25.303ex; height:2.176ex;" alt="{\displaystyle \scriptstyle {PA\vdash \forall x\forall y\forall z((x+y)+z)=(x+(y+z))}}"></span> </td></tr> <tr> <td>kommutativitás </td> <td align="left"><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 \scriptstyle {PA\vdash \forall x\forall y(x+y)=(y+x)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> <mi>A</mi> <mo>⊢<!-- ⊢ --></mo> <mi mathvariant="normal">∀<!-- ∀ --></mi> <mi>x</mi> <mi mathvariant="normal">∀<!-- ∀ --></mi> <mi>y</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>y</mi> <mo>+</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {PA\vdash \forall x\forall y(x+y)=(y+x)}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/50203e72098dab968b65f8523a70c0a86acbfee8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.965ex; height:2.176ex;" alt="{\displaystyle \scriptstyle {PA\vdash \forall x\forall y(x+y)=(y+x)}}"></span> </td></tr> <tr> <td rowspan="3" align="center"><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 \scriptstyle {\cdot }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mo>⋅<!-- ⋅ --></mo> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {\cdot }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d0c268adc667ee964887357252db434bc6b5d1da" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.31ex; margin-bottom: -0.482ex; width:0.457ex; height:1.009ex;" alt="{\displaystyle \scriptstyle {\cdot }}"></span> </td> <td>egységelem </td> <td align="left"><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 \scriptstyle {PA\vdash \forall x(x\cdot 1)=x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> <mi>A</mi> <mo>⊢<!-- ⊢ --></mo> <mi mathvariant="normal">∀<!-- ∀ --></mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>⋅<!-- ⋅ --></mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mi>x</mi> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {PA\vdash \forall x(x\cdot 1)=x}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9e4864f78fab25b8b4b2215925a061a7e59cb4f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.043ex; height:2.176ex;" alt="{\displaystyle \scriptstyle {PA\vdash \forall x(x\cdot 1)=x}}"></span> </td></tr> <tr> <td>asszociativitás </td> <td align="left"><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 \scriptstyle {PA\vdash \forall x\forall y\forall z((x\cdot y)\cdot z)=(x\cdot (y\cdot z)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> <mi>A</mi> <mo>⊢<!-- ⊢ --></mo> <mi mathvariant="normal">∀<!-- ∀ --></mi> <mi>x</mi> <mi mathvariant="normal">∀<!-- ∀ --></mi> <mi>y</mi> <mi mathvariant="normal">∀<!-- ∀ --></mi> <mi>z</mi> <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 stretchy="false">)</mo> <mo>=</mo> <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> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {PA\vdash \forall x\forall y\forall 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/82df85743e572f7cb16b8fa8f80b5334623f04bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:21.379ex; height:2.176ex;" alt="{\displaystyle \scriptstyle {PA\vdash \forall x\forall y\forall z((x\cdot y)\cdot z)=(x\cdot (y\cdot z)}}"></span> </td></tr> <tr> <td>kommutativitás </td> <td align="left"><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 \scriptstyle {PA\vdash \forall x\forall y(x\cdot y)=(y\cdot x)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> <mi>A</mi> <mo>⊢<!-- ⊢ --></mo> <mi mathvariant="normal">∀<!-- ∀ --></mi> <mi>x</mi> <mi mathvariant="normal">∀<!-- ∀ --></mi> <mi>y</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>⋅<!-- ⋅ --></mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>y</mi> <mo>⋅<!-- ⋅ --></mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {PA\vdash \forall x\forall y(x\cdot y)=(y\cdot x)}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ae617ecb81ea174196f4e21cffdac55039ea53d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.323ex; height:2.176ex;" alt="{\displaystyle \scriptstyle {PA\vdash \forall x\forall y(x\cdot y)=(y\cdot x)}}"></span> </td></tr> <tr> <td align="center"><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 \scriptstyle {+,\cdot }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> <mo>,</mo> <mo>⋅<!-- ⋅ --></mo> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {+,\cdot }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc3142b928887e27b10d72a2d5a4b2775151f0fe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:2.193ex; height:1.676ex;" alt="{\displaystyle \scriptstyle {+,\cdot }}"></span> </td> <td>disztributivitás </td> <td align="left"><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 \scriptstyle {PA\vdash \forall x\forall y\forall z(x\cdot (y+z))=((x\cdot y)+(x\cdot z))}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> <mi>A</mi> <mo>⊢<!-- ⊢ --></mo> <mi mathvariant="normal">∀<!-- ∀ --></mi> <mi>x</mi> <mi mathvariant="normal">∀<!-- ∀ --></mi> <mi>y</mi> <mi mathvariant="normal">∀<!-- ∀ --></mi> <mi>z</mi> <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 stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</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> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {PA\vdash \forall x\forall y\forall 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/c5b07f59147b3d54e5f6d42035f4645ac4d80efb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:26.338ex; height:2.176ex;" alt="{\displaystyle \scriptstyle {PA\vdash \forall x\forall y\forall z(x\cdot (y+z))=((x\cdot y)+(x\cdot z))}}"></span> </td></tr> </tbody></table> <p>A Peano-aritmetika 3. axiómája szerint az <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 \scriptstyle {+}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {+}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d9cb3bfb2fc1868785d10c0a72ce87161c3c10b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.279ex; height:1.509ex;" alt="{\displaystyle \scriptstyle {+}}"></span> -nak van (jobb oldali) nulleleme. Levezethető továbbá a <a href="/wiki/Kommutativit%C3%A1s" title="Kommutativitás">kommutativitás</a>, <a href="/wiki/Asszociativit%C3%A1s" title="Asszociativitás">asszociativitás</a> is, azonban az összes <a href="/wiki/Csoport_(matematika)" title="Csoport (matematika)">csoportaxióma</a> még nem állítható elő, mivel szinte egy számnak sincs <a href="/wiki/Ellentett" title="Ellentett">inverze</a> – és ez rendben is van, mivel a <a href="/wiki/Negat%C3%ADv_%C3%A9s_nemnegat%C3%ADv_sz%C3%A1mok" title="Negatív és nemnegatív számok">negatív számok</a> nem természetes számok. </p><p>A szorzásnak megfelelő <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 \scriptstyle {\cdot }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mo>⋅<!-- ⋅ --></mo> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {\cdot }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d0c268adc667ee964887357252db434bc6b5d1da" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.31ex; margin-bottom: -0.482ex; width:0.457ex; height:1.009ex;" alt="{\displaystyle \scriptstyle {\cdot }}"></span> -ról is bizonyítható, hogy asszociatív, kommutatív és <a href="/wiki/Z%C3%A9ruselem" title="Zéruselem">egységelemes</a>. Inverz itt is csak kivételes esetben van. Egyfajta osztás azonban mégiscsak értelmezhető majd, ez lesz az ún. maradékos osztás, ennek azonban inkább <a href="/wiki/Sz%C3%A1melm%C3%A9let" title="Számelmélet">számelméleti</a>, mint <a href="/wiki/Algebra" title="Algebra">algebrai</a> jelentősége lesz. </p><p>Fontos tétel továbbá a két műveletet összekötő egyik irányú <a href="/wiki/Disztributivit%C3%A1s" title="Disztributivitás">disztributivitás</a>: A <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 \scriptstyle {\cdot }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mo>⋅<!-- ⋅ --></mo> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {\cdot }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d0c268adc667ee964887357252db434bc6b5d1da" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.31ex; margin-bottom: -0.482ex; width:0.457ex; height:1.009ex;" alt="{\displaystyle \scriptstyle {\cdot }}"></span> disztributív az <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 \scriptstyle {+}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {+}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d9cb3bfb2fc1868785d10c0a72ce87161c3c10b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.279ex; height:1.509ex;" alt="{\displaystyle \scriptstyle {+}}"></span> -ra nézve. </p> <div class="mw-collapsible mw-collapsed" style="clear:both;"> <div style="text-align:center; font-weight:bold;">Példa levezetésre: <i>Kommutativitás</i></div> <div class="mw-collapsible-content" style="overflow-x:auto; text-align:center;"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{array}{rcll}\scriptstyle PA\cup \{(0+x)=x\}&\scriptstyle \vdash &\scriptstyle (0+x)=x&\scriptstyle \mathrm {premissza} \\\scriptstyle PA\cup \{(0+x)=x\}&\scriptstyle \vdash &\scriptstyle s(0+x)=sx&\scriptstyle \mathrm {taut.} \\\scriptstyle PA\cup \{(0+x)=x\}&\scriptstyle \vdash &\scriptstyle s(0+x)=(0+sx)&\scriptstyle \mathrm {3.ax.} \\\scriptstyle PA\cup \{(0+x)=x\}&\scriptstyle \vdash &\scriptstyle (0+sx)=sx&\scriptstyle \mathrm {=} \\\scriptstyle PA&\scriptstyle \vdash &\scriptstyle (0+x)=x\rightarrow (0+sx)=sx&\scriptstyle \mathrm {Ded.} \\\scriptstyle PA&\scriptstyle \vdash &\scriptstyle \forall x((0+x)=x\rightarrow 0+sx=sx)&\scriptstyle \mathrm {Univ.gen.} \\\scriptstyle PA&\scriptstyle \vdash &\scriptstyle F_{x}[0]\rightarrow \forall x(F\rightarrow F_{x}[sx])\rightarrow F&\scriptstyle \mathrm {7.ax.} \\\scriptstyle PA&\scriptstyle \vdash &\scriptstyle ((0+x)=x)_{x}[0]\forall x((0+x)=x((0+x)=x)_{x}[sx])((0+x)=x)&\scriptstyle F:=(0+x)=x\\\scriptstyle PA&\scriptstyle \vdash &\scriptstyle (0+0)=0\rightarrow \forall x((0+x)=x\rightarrow (0+sx)=sx)\rightarrow (0+x)=x&\scriptstyle \mathrm {terminuscsere} \\\scriptstyle PA&\scriptstyle \vdash &\scriptstyle (0+0)=0&\scriptstyle \mathrm {3.ax.} \\\scriptstyle PA&\scriptstyle \vdash &\scriptstyle \forall x((0+x)=x\rightarrow (0+sx)=sx)\rightarrow (0+x)=x&\scriptstyle \mathrm {modusponens} \\\scriptstyle PA&\scriptstyle \vdash &\scriptstyle (0+x)=x&\scriptstyle \mathrm {modusponens} \\\scriptstyle PA&\scriptstyle \vdash &\scriptstyle (x+0)=x&\scriptstyle \mathrm {3.ax.} \\\scriptstyle PA&\scriptstyle \vdash &\scriptstyle (0+x)=(x+0)&\scriptstyle \mathrm {=} \\\scriptstyle PA&\scriptstyle \vdash &\scriptstyle \forall x(0+x)=(x+0)&\scriptstyle \mathrm {Univ.gen.} \\\scriptstyle PA\cup \{(x+y)=(y+x)\}&\scriptstyle \vdash &\scriptstyle (x+y)=(y+x)&\scriptstyle \mathrm {prem.} \\\scriptstyle PA\cup \{(x+y)=(y+x)\}&\scriptstyle \vdash &\scriptstyle s(x+y)=s(y+x)&\scriptstyle \mathrm {taut.} \\\scriptstyle PA\cup \{(x+y)=(y+x)\}&\scriptstyle \vdash &\scriptstyle s(x+y)=(x+sy)&\scriptstyle \mathrm {4.ax.} \\\scriptstyle PA\cup \{(x+y)=(y+x)\}&\scriptstyle \vdash &\scriptstyle s(y+x)=(y+sx)&\scriptstyle \mathrm {4.ax.} \\\scriptstyle PA\cup \{(x+y)=(y+x)\}&\scriptstyle \vdash &\scriptstyle (x+sy)=(sy+x)&\scriptstyle \mathrm {=} \\\scriptstyle PA&\scriptstyle \vdash &\scriptstyle (x+y)=(y+x)\rightarrow (x+sy)=(sy+x)&\scriptstyle \mathrm {Ded.} \\\scriptstyle PA&\scriptstyle \vdash &\scriptstyle \forall x(x+y)=(y+x)\rightarrow (x+sy)=(sy+x)&\scriptstyle \mathrm {Univ.gen.} \\\scriptstyle PA&\scriptstyle \vdash &\scriptstyle \forall x(x+y)=(y+x)\rightarrow \forall x((x+sy=(sy+x))&\scriptstyle \mathrm {2.ax.} \\\scriptstyle PA&\scriptstyle \vdash &\scriptstyle \forall y\forall x((x+y)=(y+x)\rightarrow \forall x(x+sy)=(sy+x)&\scriptstyle \mathrm {Univ.gen.} \\\scriptstyle PA&\scriptstyle \vdash &\scriptstyle F_{x}[0]\rightarrow \forall x(F\rightarrow F_{x}[sx]\rightarrow F&\scriptstyle \mathrm {7.ax.} \\\scriptstyle PA&\scriptstyle \vdash &\scriptstyle (\forall x(x+y)=(y+x))_{x}[0]\rightarrow \forall x(\forall x(x+y)=(y+x)\rightarrow (\forall x(x+y)=(y+x))_{x}[sx]\rightarrow \forall x(x+y)=(y+x)&\scriptstyle F:=\forall x(x+y)=(y+x)\\\scriptstyle PA&\scriptstyle \vdash &\scriptstyle \forall x(0+y)=(y+0)\rightarrow \forall x(\forall x(x+y)=(y+x)\rightarrow \forall x(sx+y)=(y+sx))\rightarrow \forall x(x+y)=(y+x)&\scriptstyle \mathrm {terminuscsere} \\\scriptstyle PA&\scriptstyle \vdash &\scriptstyle \forall y\forall x(x+y)=(y+x)\rightarrow \forall x(x+sy)=(sy+x)\rightarrow \forall x(x+y)=(y+x)&\scriptstyle \mathrm {modusponens} \\\scriptstyle PA&\scriptstyle \vdash &\scriptstyle \forall x(x+y)=(y+x)&\scriptstyle \mathrm {modusponens} \\\scriptstyle PA&\scriptstyle \vdash &\scriptstyle \forall y\forall x(x+y)=(y+x)&\scriptstyle \mathrm {Univ.gen.} \\\scriptstyle PA&\scriptstyle \vdash &\scriptstyle \forall x\forall y(x+y)=(y+x)&\scriptstyle \mathrm {kvantorcsere} \\\end{array}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right center left left" rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mstyle displaystyle="false" scriptlevel="1"> <mi>P</mi> <mi>A</mi> <mo>∪<!-- ∪ --></mo> <mo fence="false" stretchy="false">{</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>+</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>x</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mtd> <mtd> <mstyle displaystyle="false" scriptlevel="1"> <mo>⊢<!-- ⊢ --></mo> </mstyle> </mtd> <mtd> <mstyle displaystyle="false" scriptlevel="1"> <mo stretchy="false">(</mo> <mn>0</mn> <mo>+</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>x</mi> </mstyle> </mtd> <mtd> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">p</mi> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">m</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">z</mi> <mi mathvariant="normal">a</mi> </mrow> </mstyle> </mtd> </mtr> <mtr> <mtd> <mstyle displaystyle="false" scriptlevel="1"> <mi>P</mi> <mi>A</mi> <mo>∪<!-- ∪ --></mo> <mo fence="false" 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<mo>=</mo> <mo stretchy="false">(</mo> <mi>y</mi> <mo>+</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">}</mo> </mstyle> </mtd> <mtd> <mstyle displaystyle="false" scriptlevel="1"> <mo>⊢<!-- ⊢ --></mo> </mstyle> </mtd> <mtd> <mstyle displaystyle="false" scriptlevel="1"> <mi>s</mi> <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>s</mi> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mtd> <mtd> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mn>4.</mn> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">x</mi> <mo>.</mo> </mrow> </mstyle> </mtd> </mtr> <mtr> <mtd> <mstyle displaystyle="false" scriptlevel="1"> <mi>P</mi> <mi>A</mi> <mo>∪<!-- ∪ --></mo> <mo fence="false" stretchy="false">{</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>y</mi> <mo>+</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">}</mo> </mstyle> </mtd> <mtd> <mstyle displaystyle="false" scriptlevel="1"> <mo>⊢<!-- ⊢ --></mo> </mstyle> </mtd> <mtd> <mstyle displaystyle="false" scriptlevel="1"> <mi>s</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo>+</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>y</mi> <mo>+</mo> <mi>s</mi> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mtd> <mtd> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mn>4.</mn> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">x</mi> <mo>.</mo> </mrow> </mstyle> </mtd> </mtr> <mtr> <mtd> <mstyle displaystyle="false" scriptlevel="1"> <mi>P</mi> <mi>A</mi> <mo>∪<!-- ∪ --></mo> <mo fence="false" stretchy="false">{</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>y</mi> <mo>+</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">}</mo> </mstyle> </mtd> <mtd> <mstyle displaystyle="false" scriptlevel="1"> <mo>⊢<!-- ⊢ --></mo> </mstyle> </mtd> <mtd> <mstyle displaystyle="false" scriptlevel="1"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>s</mi> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>s</mi> <mi>y</mi> <mo>+</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mtd> <mtd> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mo>=</mo> </mrow> </mstyle> </mtd> </mtr> <mtr> <mtd> <mstyle displaystyle="false" scriptlevel="1"> <mi>P</mi> <mi>A</mi> </mstyle> </mtd> <mtd> <mstyle displaystyle="false" scriptlevel="1"> <mo>⊢<!-- ⊢ --></mo> </mstyle> </mtd> <mtd> <mstyle displaystyle="false" scriptlevel="1"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>y</mi> <mo>+</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→<!-- → --></mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>s</mi> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>s</mi> <mi>y</mi> <mo>+</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mtd> <mtd> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">D</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">d</mi> <mo>.</mo> </mrow> </mstyle> </mtd> </mtr> <mtr> <mtd> <mstyle displaystyle="false" scriptlevel="1"> <mi>P</mi> <mi>A</mi> </mstyle> </mtd> <mtd> <mstyle displaystyle="false" scriptlevel="1"> <mo>⊢<!-- ⊢ --></mo> </mstyle> </mtd> <mtd> <mstyle displaystyle="false" scriptlevel="1"> <mi mathvariant="normal">∀<!-- ∀ --></mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>y</mi> <mo>+</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→<!-- → --></mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>s</mi> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>s</mi> <mi>y</mi> <mo>+</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mtd> <mtd> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">U</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">v</mi> <mo>.</mo> <mi mathvariant="normal">g</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">n</mi> <mo>.</mo> </mrow> </mstyle> </mtd> </mtr> <mtr> <mtd> <mstyle displaystyle="false" scriptlevel="1"> <mi>P</mi> <mi>A</mi> </mstyle> </mtd> <mtd> <mstyle displaystyle="false" scriptlevel="1"> <mo>⊢<!-- ⊢ --></mo> </mstyle> </mtd> <mtd> <mstyle displaystyle="false" scriptlevel="1"> <mi mathvariant="normal">∀<!-- ∀ --></mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>y</mi> <mo>+</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→<!-- → --></mo> <mi mathvariant="normal">∀<!-- ∀ --></mi> <mi>x</mi> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>s</mi> <mi>y</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi>s</mi> <mi>y</mi> <mo>+</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mtd> <mtd> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mn>2.</mn> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">x</mi> <mo>.</mo> </mrow> </mstyle> </mtd> </mtr> <mtr> <mtd> <mstyle displaystyle="false" scriptlevel="1"> <mi>P</mi> <mi>A</mi> </mstyle> </mtd> <mtd> <mstyle displaystyle="false" scriptlevel="1"> <mo>⊢<!-- ⊢ --></mo> </mstyle> </mtd> <mtd> <mstyle displaystyle="false" scriptlevel="1"> <mi mathvariant="normal">∀<!-- ∀ --></mi> <mi>y</mi> <mi mathvariant="normal">∀<!-- ∀ --></mi> <mi>x</mi> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>y</mi> <mo>+</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→<!-- → --></mo> <mi mathvariant="normal">∀<!-- ∀ --></mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>s</mi> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>s</mi> <mi>y</mi> <mo>+</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mtd> <mtd> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">U</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">v</mi> <mo>.</mo> <mi mathvariant="normal">g</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">n</mi> <mo>.</mo> </mrow> </mstyle> </mtd> </mtr> <mtr> <mtd> <mstyle displaystyle="false" scriptlevel="1"> <mi>P</mi> <mi>A</mi> </mstyle> </mtd> <mtd> <mstyle displaystyle="false" scriptlevel="1"> <mo>⊢<!-- ⊢ --></mo> </mstyle> </mtd> <mtd> <mstyle displaystyle="false" scriptlevel="1"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo stretchy="false">[</mo> <mn>0</mn> <mo stretchy="false">]</mo> <mo stretchy="false">→<!-- → --></mo> <mi mathvariant="normal">∀<!-- ∀ --></mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>F</mi> <mo stretchy="false">→<!-- → --></mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo stretchy="false">[</mo> <mi>s</mi> <mi>x</mi> <mo stretchy="false">]</mo> <mo stretchy="false">→<!-- → --></mo> <mi>F</mi> </mstyle> </mtd> <mtd> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mn>7.</mn> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">x</mi> <mo>.</mo> </mrow> </mstyle> </mtd> </mtr> <mtr> <mtd> <mstyle displaystyle="false" scriptlevel="1"> <mi>P</mi> <mi>A</mi> </mstyle> </mtd> <mtd> <mstyle displaystyle="false" scriptlevel="1"> <mo>⊢<!-- ⊢ --></mo> </mstyle> </mtd> <mtd> <mstyle displaystyle="false" scriptlevel="1"> <mo stretchy="false">(</mo> <mi mathvariant="normal">∀<!-- ∀ --></mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>y</mi> <mo>+</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo stretchy="false">[</mo> <mn>0</mn> <mo stretchy="false">]</mo> <mo stretchy="false">→<!-- → --></mo> <mi mathvariant="normal">∀<!-- ∀ --></mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi mathvariant="normal">∀<!-- ∀ --></mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>y</mi> <mo>+</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→<!-- → --></mo> <mo stretchy="false">(</mo> <mi mathvariant="normal">∀<!-- ∀ --></mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>y</mi> <mo>+</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo stretchy="false">[</mo> <mi>s</mi> <mi>x</mi> <mo stretchy="false">]</mo> <mo stretchy="false">→<!-- → --></mo> <mi mathvariant="normal">∀<!-- ∀ --></mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>y</mi> <mo>+</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mtd> <mtd> <mstyle displaystyle="false" scriptlevel="1"> <mi>F</mi> <mo>:=</mo> <mi mathvariant="normal">∀<!-- ∀ --></mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>y</mi> <mo>+</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mtd> </mtr> <mtr> <mtd> <mstyle displaystyle="false" scriptlevel="1"> <mi>P</mi> <mi>A</mi> </mstyle> </mtd> <mtd> <mstyle displaystyle="false" scriptlevel="1"> <mo>⊢<!-- ⊢ --></mo> </mstyle> </mtd> <mtd> <mstyle displaystyle="false" scriptlevel="1"> <mi mathvariant="normal">∀<!-- ∀ --></mi> <mi>x</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo>+</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>y</mi> <mo>+</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo stretchy="false">→<!-- → --></mo> <mi mathvariant="normal">∀<!-- ∀ --></mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi mathvariant="normal">∀<!-- ∀ --></mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>y</mi> <mo>+</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→<!-- → --></mo> <mi mathvariant="normal">∀<!-- ∀ --></mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>y</mi> <mo>+</mo> <mi>s</mi> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo stretchy="false">→<!-- → --></mo> <mi mathvariant="normal">∀<!-- ∀ --></mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>y</mi> <mo>+</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mtd> <mtd> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">m</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">u</mi> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">e</mi> </mrow> </mstyle> </mtd> </mtr> <mtr> <mtd> <mstyle displaystyle="false" scriptlevel="1"> <mi>P</mi> <mi>A</mi> </mstyle> </mtd> <mtd> <mstyle displaystyle="false" scriptlevel="1"> <mo>⊢<!-- ⊢ --></mo> </mstyle> </mtd> <mtd> <mstyle displaystyle="false" scriptlevel="1"> <mi mathvariant="normal">∀<!-- ∀ --></mi> <mi>y</mi> <mi mathvariant="normal">∀<!-- ∀ --></mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>y</mi> <mo>+</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→<!-- → --></mo> <mi mathvariant="normal">∀<!-- ∀ --></mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>s</mi> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>s</mi> <mi>y</mi> <mo>+</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→<!-- → --></mo> <mi mathvariant="normal">∀<!-- ∀ --></mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>y</mi> <mo>+</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mtd> <mtd> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">m</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">d</mi> <mi mathvariant="normal">u</mi> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">p</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">s</mi> </mrow> </mstyle> </mtd> </mtr> <mtr> <mtd> <mstyle displaystyle="false" scriptlevel="1"> <mi>P</mi> <mi>A</mi> </mstyle> </mtd> <mtd> <mstyle displaystyle="false" scriptlevel="1"> <mo>⊢<!-- ⊢ --></mo> </mstyle> </mtd> <mtd> <mstyle displaystyle="false" scriptlevel="1"> <mi mathvariant="normal">∀<!-- ∀ --></mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>y</mi> <mo>+</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mtd> <mtd> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">m</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">d</mi> <mi mathvariant="normal">u</mi> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">p</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">s</mi> </mrow> </mstyle> </mtd> </mtr> <mtr> <mtd> <mstyle displaystyle="false" scriptlevel="1"> <mi>P</mi> <mi>A</mi> </mstyle> </mtd> <mtd> <mstyle displaystyle="false" scriptlevel="1"> <mo>⊢<!-- ⊢ --></mo> </mstyle> </mtd> <mtd> <mstyle displaystyle="false" scriptlevel="1"> <mi mathvariant="normal">∀<!-- ∀ --></mi> <mi>y</mi> <mi mathvariant="normal">∀<!-- ∀ --></mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>y</mi> <mo>+</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mtd> <mtd> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">U</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">v</mi> <mo>.</mo> <mi mathvariant="normal">g</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">n</mi> <mo>.</mo> </mrow> </mstyle> </mtd> </mtr> <mtr> <mtd> <mstyle displaystyle="false" scriptlevel="1"> <mi>P</mi> <mi>A</mi> </mstyle> </mtd> <mtd> <mstyle displaystyle="false" scriptlevel="1"> <mo>⊢<!-- ⊢ --></mo> </mstyle> </mtd> <mtd> <mstyle displaystyle="false" scriptlevel="1"> <mi mathvariant="normal">∀<!-- ∀ --></mi> <mi>x</mi> <mi mathvariant="normal">∀<!-- ∀ --></mi> <mi>y</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>y</mi> <mo>+</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mtd> <mtd> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">k</mi> <mi mathvariant="normal">v</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">e</mi> </mrow> </mstyle> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{array}{rcll}\scriptstyle PA\cup \{(0+x)=x\}&\scriptstyle \vdash &\scriptstyle (0+x)=x&\scriptstyle \mathrm {premissza} \\\scriptstyle PA\cup \{(0+x)=x\}&\scriptstyle \vdash &\scriptstyle s(0+x)=sx&\scriptstyle \mathrm {taut.} \\\scriptstyle PA\cup \{(0+x)=x\}&\scriptstyle \vdash &\scriptstyle s(0+x)=(0+sx)&\scriptstyle \mathrm {3.ax.} \\\scriptstyle PA\cup \{(0+x)=x\}&\scriptstyle \vdash &\scriptstyle (0+sx)=sx&\scriptstyle \mathrm {=} \\\scriptstyle PA&\scriptstyle \vdash &\scriptstyle (0+x)=x\rightarrow (0+sx)=sx&\scriptstyle \mathrm {Ded.} \\\scriptstyle PA&\scriptstyle \vdash &\scriptstyle \forall x((0+x)=x\rightarrow 0+sx=sx)&\scriptstyle \mathrm {Univ.gen.} \\\scriptstyle PA&\scriptstyle \vdash &\scriptstyle F_{x}[0]\rightarrow \forall x(F\rightarrow F_{x}[sx])\rightarrow F&\scriptstyle \mathrm {7.ax.} \\\scriptstyle PA&\scriptstyle \vdash &\scriptstyle ((0+x)=x)_{x}[0]\forall x((0+x)=x((0+x)=x)_{x}[sx])((0+x)=x)&\scriptstyle F:=(0+x)=x\\\scriptstyle PA&\scriptstyle \vdash &\scriptstyle (0+0)=0\rightarrow \forall x((0+x)=x\rightarrow (0+sx)=sx)\rightarrow (0+x)=x&\scriptstyle \mathrm {terminuscsere} \\\scriptstyle PA&\scriptstyle \vdash &\scriptstyle (0+0)=0&\scriptstyle \mathrm {3.ax.} \\\scriptstyle PA&\scriptstyle \vdash &\scriptstyle \forall x((0+x)=x\rightarrow (0+sx)=sx)\rightarrow (0+x)=x&\scriptstyle \mathrm {modusponens} \\\scriptstyle PA&\scriptstyle \vdash &\scriptstyle (0+x)=x&\scriptstyle \mathrm {modusponens} \\\scriptstyle PA&\scriptstyle \vdash &\scriptstyle (x+0)=x&\scriptstyle \mathrm {3.ax.} \\\scriptstyle PA&\scriptstyle \vdash &\scriptstyle (0+x)=(x+0)&\scriptstyle \mathrm {=} \\\scriptstyle PA&\scriptstyle \vdash &\scriptstyle \forall x(0+x)=(x+0)&\scriptstyle \mathrm {Univ.gen.} \\\scriptstyle PA\cup \{(x+y)=(y+x)\}&\scriptstyle \vdash &\scriptstyle (x+y)=(y+x)&\scriptstyle \mathrm {prem.} \\\scriptstyle PA\cup \{(x+y)=(y+x)\}&\scriptstyle \vdash &\scriptstyle s(x+y)=s(y+x)&\scriptstyle \mathrm {taut.} \\\scriptstyle PA\cup \{(x+y)=(y+x)\}&\scriptstyle \vdash &\scriptstyle s(x+y)=(x+sy)&\scriptstyle \mathrm {4.ax.} \\\scriptstyle PA\cup \{(x+y)=(y+x)\}&\scriptstyle \vdash &\scriptstyle s(y+x)=(y+sx)&\scriptstyle \mathrm {4.ax.} \\\scriptstyle PA\cup \{(x+y)=(y+x)\}&\scriptstyle \vdash &\scriptstyle (x+sy)=(sy+x)&\scriptstyle \mathrm {=} \\\scriptstyle PA&\scriptstyle \vdash &\scriptstyle (x+y)=(y+x)\rightarrow (x+sy)=(sy+x)&\scriptstyle \mathrm {Ded.} \\\scriptstyle PA&\scriptstyle \vdash &\scriptstyle \forall x(x+y)=(y+x)\rightarrow (x+sy)=(sy+x)&\scriptstyle \mathrm {Univ.gen.} \\\scriptstyle PA&\scriptstyle \vdash &\scriptstyle \forall x(x+y)=(y+x)\rightarrow \forall x((x+sy=(sy+x))&\scriptstyle \mathrm {2.ax.} \\\scriptstyle PA&\scriptstyle \vdash &\scriptstyle \forall y\forall x((x+y)=(y+x)\rightarrow \forall x(x+sy)=(sy+x)&\scriptstyle \mathrm {Univ.gen.} \\\scriptstyle PA&\scriptstyle \vdash &\scriptstyle F_{x}[0]\rightarrow \forall x(F\rightarrow F_{x}[sx]\rightarrow F&\scriptstyle \mathrm {7.ax.} \\\scriptstyle PA&\scriptstyle \vdash &\scriptstyle (\forall x(x+y)=(y+x))_{x}[0]\rightarrow \forall x(\forall x(x+y)=(y+x)\rightarrow (\forall x(x+y)=(y+x))_{x}[sx]\rightarrow \forall x(x+y)=(y+x)&\scriptstyle F:=\forall x(x+y)=(y+x)\\\scriptstyle PA&\scriptstyle \vdash &\scriptstyle \forall x(0+y)=(y+0)\rightarrow \forall x(\forall x(x+y)=(y+x)\rightarrow \forall x(sx+y)=(y+sx))\rightarrow \forall x(x+y)=(y+x)&\scriptstyle \mathrm {terminuscsere} \\\scriptstyle PA&\scriptstyle \vdash &\scriptstyle \forall y\forall x(x+y)=(y+x)\rightarrow \forall x(x+sy)=(sy+x)\rightarrow \forall x(x+y)=(y+x)&\scriptstyle \mathrm {modusponens} \\\scriptstyle PA&\scriptstyle \vdash &\scriptstyle \forall x(x+y)=(y+x)&\scriptstyle \mathrm {modusponens} \\\scriptstyle PA&\scriptstyle \vdash &\scriptstyle \forall y\forall x(x+y)=(y+x)&\scriptstyle \mathrm {Univ.gen.} \\\scriptstyle PA&\scriptstyle \vdash &\scriptstyle \forall x\forall y(x+y)=(y+x)&\scriptstyle \mathrm {kvantorcsere} \\\end{array}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a8f095149779629bf10f9a2a00de4bcac26ecb79" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -49.671ex; width:101.822ex; height:100.509ex;" alt="{\displaystyle {\begin{array}{rcll}\scriptstyle PA\cup \{(0+x)=x\}&\scriptstyle \vdash &\scriptstyle (0+x)=x&\scriptstyle \mathrm {premissza} \\\scriptstyle PA\cup \{(0+x)=x\}&\scriptstyle \vdash &\scriptstyle s(0+x)=sx&\scriptstyle \mathrm {taut.} \\\scriptstyle PA\cup \{(0+x)=x\}&\scriptstyle \vdash &\scriptstyle s(0+x)=(0+sx)&\scriptstyle \mathrm {3.ax.} \\\scriptstyle PA\cup \{(0+x)=x\}&\scriptstyle \vdash &\scriptstyle (0+sx)=sx&\scriptstyle \mathrm {=} \\\scriptstyle PA&\scriptstyle \vdash &\scriptstyle (0+x)=x\rightarrow (0+sx)=sx&\scriptstyle \mathrm {Ded.} \\\scriptstyle PA&\scriptstyle \vdash &\scriptstyle \forall x((0+x)=x\rightarrow 0+sx=sx)&\scriptstyle \mathrm {Univ.gen.} \\\scriptstyle PA&\scriptstyle \vdash &\scriptstyle F_{x}[0]\rightarrow \forall x(F\rightarrow F_{x}[sx])\rightarrow F&\scriptstyle \mathrm {7.ax.} \\\scriptstyle PA&\scriptstyle \vdash &\scriptstyle ((0+x)=x)_{x}[0]\forall x((0+x)=x((0+x)=x)_{x}[sx])((0+x)=x)&\scriptstyle F:=(0+x)=x\\\scriptstyle PA&\scriptstyle \vdash &\scriptstyle (0+0)=0\rightarrow \forall x((0+x)=x\rightarrow (0+sx)=sx)\rightarrow (0+x)=x&\scriptstyle \mathrm {terminuscsere} \\\scriptstyle PA&\scriptstyle \vdash &\scriptstyle (0+0)=0&\scriptstyle \mathrm {3.ax.} \\\scriptstyle PA&\scriptstyle \vdash &\scriptstyle \forall x((0+x)=x\rightarrow (0+sx)=sx)\rightarrow (0+x)=x&\scriptstyle \mathrm {modusponens} \\\scriptstyle PA&\scriptstyle \vdash &\scriptstyle (0+x)=x&\scriptstyle \mathrm {modusponens} \\\scriptstyle PA&\scriptstyle \vdash &\scriptstyle (x+0)=x&\scriptstyle \mathrm {3.ax.} \\\scriptstyle PA&\scriptstyle \vdash &\scriptstyle (0+x)=(x+0)&\scriptstyle \mathrm {=} \\\scriptstyle PA&\scriptstyle \vdash &\scriptstyle \forall x(0+x)=(x+0)&\scriptstyle \mathrm {Univ.gen.} \\\scriptstyle PA\cup \{(x+y)=(y+x)\}&\scriptstyle \vdash &\scriptstyle (x+y)=(y+x)&\scriptstyle \mathrm {prem.} \\\scriptstyle PA\cup \{(x+y)=(y+x)\}&\scriptstyle \vdash &\scriptstyle s(x+y)=s(y+x)&\scriptstyle \mathrm {taut.} \\\scriptstyle PA\cup \{(x+y)=(y+x)\}&\scriptstyle \vdash &\scriptstyle s(x+y)=(x+sy)&\scriptstyle \mathrm {4.ax.} \\\scriptstyle PA\cup \{(x+y)=(y+x)\}&\scriptstyle \vdash &\scriptstyle s(y+x)=(y+sx)&\scriptstyle \mathrm {4.ax.} \\\scriptstyle PA\cup \{(x+y)=(y+x)\}&\scriptstyle \vdash &\scriptstyle (x+sy)=(sy+x)&\scriptstyle \mathrm {=} \\\scriptstyle PA&\scriptstyle \vdash &\scriptstyle (x+y)=(y+x)\rightarrow (x+sy)=(sy+x)&\scriptstyle \mathrm {Ded.} \\\scriptstyle PA&\scriptstyle \vdash &\scriptstyle \forall x(x+y)=(y+x)\rightarrow (x+sy)=(sy+x)&\scriptstyle \mathrm {Univ.gen.} \\\scriptstyle PA&\scriptstyle \vdash &\scriptstyle \forall x(x+y)=(y+x)\rightarrow \forall x((x+sy=(sy+x))&\scriptstyle \mathrm {2.ax.} \\\scriptstyle PA&\scriptstyle \vdash &\scriptstyle \forall y\forall x((x+y)=(y+x)\rightarrow \forall x(x+sy)=(sy+x)&\scriptstyle \mathrm {Univ.gen.} \\\scriptstyle PA&\scriptstyle \vdash &\scriptstyle F_{x}[0]\rightarrow \forall x(F\rightarrow F_{x}[sx]\rightarrow F&\scriptstyle \mathrm {7.ax.} \\\scriptstyle PA&\scriptstyle \vdash &\scriptstyle (\forall x(x+y)=(y+x))_{x}[0]\rightarrow \forall x(\forall x(x+y)=(y+x)\rightarrow (\forall x(x+y)=(y+x))_{x}[sx]\rightarrow \forall x(x+y)=(y+x)&\scriptstyle F:=\forall x(x+y)=(y+x)\\\scriptstyle PA&\scriptstyle \vdash &\scriptstyle \forall x(0+y)=(y+0)\rightarrow \forall x(\forall x(x+y)=(y+x)\rightarrow \forall x(sx+y)=(y+sx))\rightarrow \forall x(x+y)=(y+x)&\scriptstyle \mathrm {terminuscsere} \\\scriptstyle PA&\scriptstyle \vdash &\scriptstyle \forall y\forall x(x+y)=(y+x)\rightarrow \forall x(x+sy)=(sy+x)\rightarrow \forall x(x+y)=(y+x)&\scriptstyle \mathrm {modusponens} \\\scriptstyle PA&\scriptstyle \vdash &\scriptstyle \forall x(x+y)=(y+x)&\scriptstyle \mathrm {modusponens} \\\scriptstyle PA&\scriptstyle \vdash &\scriptstyle \forall y\forall x(x+y)=(y+x)&\scriptstyle \mathrm {Univ.gen.} \\\scriptstyle PA&\scriptstyle \vdash &\scriptstyle \forall x\forall y(x+y)=(y+x)&\scriptstyle \mathrm {kvantorcsere} \\\end{array}}}"></span> </p> </div> </div> <div class="mw-heading mw-heading3"><h3 id="Rendezés"><span id="Rendez.C3.A9s"></span>Rendezés</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Peano-aritmetika&action=edit&section=5" title="Szakasz szerkesztése: Rendezés"><span>szerkesztés</span></a><span class="mw-editsection-bracket">]</span></span></div> <table align="right" class="wikitable"> <tbody><tr> <th colspan="3">A gyenge rendezés tételei </th></tr> <tr> <td><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 \scriptstyle {0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/757abea8b05f9281eee9cd6378f38c7a191e44d8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.822ex; height:1.676ex;" alt="{\displaystyle \scriptstyle {0}}"></span> minimális elem </td> <td align="left"><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 \scriptstyle {PA\vdash \forall x0\leq x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> <mi>A</mi> <mo>⊢<!-- ⊢ --></mo> <mi mathvariant="normal">∀<!-- ∀ --></mi> <mi>x</mi> <mn>0</mn> <mo>≤<!-- ≤ --></mo> <mi>x</mi> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {PA\vdash \forall x0\leq x}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0bb7865b21950920c4331ae20eaa178e46f02dda" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:8.366ex; height:2.009ex;" alt="{\displaystyle \scriptstyle {PA\vdash \forall x0\leq x}}"></span> </td></tr> <tr> <td><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 \scriptstyle {\leq }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mo>≤<!-- ≤ --></mo> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {\leq }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/214a1e186abadeb10df2481578bf7cf4a7c2bb96" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.279ex; height:1.843ex;" alt="{\displaystyle \scriptstyle {\leq }}"></span> reflexív </td> <td align="left"><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 \scriptstyle {PA\vdash \forall xx\leq x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> <mi>A</mi> <mo>⊢<!-- ⊢ --></mo> <mi mathvariant="normal">∀<!-- ∀ --></mi> <mi>x</mi> <mi>x</mi> <mo>≤<!-- ≤ --></mo> <mi>x</mi> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {PA\vdash \forall xx\leq x}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/538a5b05d26ee659e04cc9e020726a2c53c4b9e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:8.484ex; height:2.009ex;" alt="{\displaystyle \scriptstyle {PA\vdash \forall xx\leq x}}"></span> </td></tr> <tr> <td><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 \scriptstyle {\leq }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mo>≤<!-- ≤ --></mo> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {\leq }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/214a1e186abadeb10df2481578bf7cf4a7c2bb96" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.279ex; height:1.843ex;" alt="{\displaystyle \scriptstyle {\leq }}"></span> antiszimmetrikus </td> <td align="left"><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 \scriptstyle {PA\vdash \forall x\forall y(x\leq y\land y\leq x)\rightarrow x=y}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> <mi>A</mi> <mo>⊢<!-- ⊢ --></mo> <mi mathvariant="normal">∀<!-- ∀ --></mi> <mi>x</mi> <mi mathvariant="normal">∀<!-- ∀ --></mi> <mi>y</mi> <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> <mo stretchy="false">→<!-- → --></mo> <mi>x</mi> <mo>=</mo> <mi>y</mi> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {PA\vdash \forall x\forall y(x\leq y\land y\leq x)\rightarrow x=y}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1cc32719610889305c0fa8fe7ef0bce9665ae098" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:20.182ex; height:2.176ex;" alt="{\displaystyle \scriptstyle {PA\vdash \forall x\forall y(x\leq y\land y\leq x)\rightarrow x=y}}"></span> </td></tr> <tr> <td><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 \scriptstyle {\leq }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mo>≤<!-- ≤ --></mo> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {\leq }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/214a1e186abadeb10df2481578bf7cf4a7c2bb96" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.279ex; height:1.843ex;" alt="{\displaystyle \scriptstyle {\leq }}"></span> tranzitív </td> <td align="left"><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 \scriptstyle {PA\vdash \forall x\forall y\forall z((x\leq y\land y\leq z)\rightarrow x\leq z)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> <mi>A</mi> <mo>⊢<!-- ⊢ --></mo> <mi mathvariant="normal">∀<!-- ∀ --></mi> <mi>x</mi> <mi mathvariant="normal">∀<!-- ∀ --></mi> <mi>y</mi> <mi mathvariant="normal">∀<!-- ∀ --></mi> <mi>z</mi> <mo stretchy="false">(</mo> <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> <mo stretchy="false">→<!-- → --></mo> <mi>x</mi> <mo>≤<!-- ≤ --></mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {PA\vdash \forall x\forall y\forall z((x\leq y\land y\leq z)\rightarrow x\leq z)}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed62865b9beee51cf965a8bafba8707886cdc2f6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:22.927ex; height:2.176ex;" alt="{\displaystyle \scriptstyle {PA\vdash \forall x\forall y\forall z((x\leq y\land y\leq z)\rightarrow x\leq z)}}"></span> </td></tr> <tr> <td><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 \scriptstyle {\leq }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mo>≤<!-- ≤ --></mo> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {\leq }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/214a1e186abadeb10df2481578bf7cf4a7c2bb96" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.279ex; height:1.843ex;" alt="{\displaystyle \scriptstyle {\leq }}"></span> lineáris </td> <td align="left"><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 \scriptstyle {PA\vdash \forall x\forall y((x\leq y\vee x=y)\vee y\leq x)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> <mi>A</mi> <mo>⊢<!-- ⊢ --></mo> <mi mathvariant="normal">∀<!-- ∀ --></mi> <mi>x</mi> <mi mathvariant="normal">∀<!-- ∀ --></mi> <mi>y</mi> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>≤<!-- ≤ --></mo> <mi>y</mi> <mo>∨<!-- ∨ --></mo> <mi>x</mi> <mo>=</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>∨<!-- ∨ --></mo> <mi>y</mi> <mo>≤<!-- ≤ --></mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {PA\vdash \forall x\forall y((x\leq y\vee x=y)\vee y\leq x)}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6fa82c446f7de938d8a83b8d4ab59cbb55aed23a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:20.915ex; height:2.176ex;" alt="{\displaystyle \scriptstyle {PA\vdash \forall x\forall y((x\leq y\vee x=y)\vee y\leq x)}}"></span> </td></tr> <tr> <td><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 \scriptstyle {\leq }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mo>≤<!-- ≤ --></mo> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {\leq }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/214a1e186abadeb10df2481578bf7cf4a7c2bb96" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.279ex; height:1.843ex;" alt="{\displaystyle \scriptstyle {\leq }}"></span> és a <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 \scriptstyle {+}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {+}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d9cb3bfb2fc1868785d10c0a72ce87161c3c10b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.279ex; height:1.509ex;" alt="{\displaystyle \scriptstyle {+}}"></span> </td> <td align="left"><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 \scriptstyle {PA\vdash \forall x\forall y\forall z(x\leq y\rightarrow (x+z)\leq (y+z))}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> <mi>A</mi> <mo>⊢<!-- ⊢ --></mo> <mi mathvariant="normal">∀<!-- ∀ --></mi> <mi>x</mi> <mi mathvariant="normal">∀<!-- ∀ --></mi> <mi>y</mi> <mi mathvariant="normal">∀<!-- ∀ --></mi> <mi>z</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>≤<!-- ≤ --></mo> <mi>y</mi> <mo stretchy="false">→<!-- → --></mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>≤<!-- ≤ --></mo> <mo stretchy="false">(</mo> <mi>y</mi> <mo>+</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {PA\vdash \forall x\forall y\forall z(x\leq y\rightarrow (x+z)\leq (y+z))}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e2538323e336cff42f25f6acd8365bc8592604f0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:24.388ex; height:2.176ex;" alt="{\displaystyle \scriptstyle {PA\vdash \forall x\forall y\forall z(x\leq y\rightarrow (x+z)\leq (y+z))}}"></span> </td></tr> <tr> <td><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 \scriptstyle {\leq }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mo>≤<!-- ≤ --></mo> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {\leq }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/214a1e186abadeb10df2481578bf7cf4a7c2bb96" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.279ex; height:1.843ex;" alt="{\displaystyle \scriptstyle {\leq }}"></span> és a <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 \scriptstyle {\cdot }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mo>⋅<!-- ⋅ --></mo> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {\cdot }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d0c268adc667ee964887357252db434bc6b5d1da" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.31ex; margin-bottom: -0.482ex; width:0.457ex; height:1.009ex;" alt="{\displaystyle \scriptstyle {\cdot }}"></span> </td> <td align="left"><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 \scriptstyle {PA\vdash \forall x\forall y\forall z(x\leq y\rightarrow (x\cdot z)\leq (y\cdot z))}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> <mi>A</mi> <mo>⊢<!-- ⊢ --></mo> <mi mathvariant="normal">∀<!-- ∀ --></mi> <mi>x</mi> <mi mathvariant="normal">∀<!-- ∀ --></mi> <mi>y</mi> <mi mathvariant="normal">∀<!-- ∀ --></mi> <mi>z</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>≤<!-- ≤ --></mo> <mi>y</mi> <mo stretchy="false">→<!-- → --></mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>⋅<!-- ⋅ --></mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>≤<!-- ≤ --></mo> <mo stretchy="false">(</mo> <mi>y</mi> <mo>⋅<!-- ⋅ --></mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {PA\vdash \forall x\forall y\forall z(x\leq y\rightarrow (x\cdot z)\leq (y\cdot z))}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef05a7448d95bc5258d2e6e10894b3adbe78ce59" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:22.746ex; height:2.176ex;" alt="{\displaystyle \scriptstyle {PA\vdash \forall x\forall y\forall z(x\leq y\rightarrow (x\cdot z)\leq (y\cdot z))}}"></span> </td></tr> </tbody></table> <p>A Peano-aritmetikában definiálható a következő „nem nagyobb, mint”-nek megfelelő kétargumentumú predikátum: </p> <center><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 \scriptstyle {x\leq y\mathop {\iff } _{def}\exists z(x+z)=y}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>≤<!-- ≤ --></mo> <mi>y</mi> <msub> <mrow class="MJX-TeXAtom-OP"> <mspace width="thickmathspace" /> <mo stretchy="false">⟺<!-- ⟺ --></mo> <mspace width="thickmathspace" /> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mi>e</mi> <mi>f</mi> </mrow> </msub> <mo>⁡<!-- --></mo> <mi mathvariant="normal">∃<!-- ∃ --></mi> <mi>z</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>y</mi> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {x\leq y\mathop {\iff } _{def}\exists z(x+z)=y}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2cfc3645f2513a0b2beefec24e440590de3ab699" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.417ex; height:2.343ex;" alt="{\displaystyle \scriptstyle {x\leq y\mathop {\iff } _{def}\exists z(x+z)=y}}"></span></center> <p>Ez egy gyenge lineáris rendezés a természetes számokon, azaz </p> <ul><li><a href="/wiki/Reflex%C3%ADv_rel%C3%A1ci%C3%B3" title="Reflexív reláció">Reflexív</a>: Senki sem nagyobb magánál.</li> <li><a href="/wiki/Antiszimmetrikus_rel%C3%A1ci%C3%B3" title="Antiszimmetrikus reláció">Antiszimmetrikus</a>: Ha ketten nem nagyobbak egymásnál, akkor ugyanazok.</li> <li><a href="/wiki/Tranzit%C3%ADv_rel%C3%A1ci%C3%B3" title="Tranzitív reláció">Tranzitív</a>: Ha három közül a második nem nagyobb az elsőnél, és a harmadik nem nagyobb a másodiknál, akkor a harmadik nem nagyobb az elsőnél sem.</li> <li><a href="/wiki/Rendezett_halmaz#Definíció" title="Rendezett halmaz">Lineáris</a>: Bármely kettő közül valamelyik nem nagyobb a másiknál.</li></ul> <p>Ezen belül további jellemzők, hogy </p> <ul><li><a href="/wiki/Korl%C3%A1tos_halmaz#Számegyenes" title="Korlátos halmaz">Alulról korlátos</a>: Van olyan, aki senkinél sem nagyobb.</li> <li>A <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 \scriptstyle {0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/757abea8b05f9281eee9cd6378f38c7a191e44d8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.822ex; height:1.676ex;" alt="{\displaystyle \scriptstyle {0}}"></span> minimális eleme: 0 senkinél sem nagyobb.</li></ul> <table align="right" class="wikitable"> <tbody><tr> <th colspan="3">Szigorú rendezés tételei </th></tr> <tr> <td><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 \scriptstyle {0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/757abea8b05f9281eee9cd6378f38c7a191e44d8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.822ex; height:1.676ex;" alt="{\displaystyle \scriptstyle {0}}"></span> infimum </td> <td align="left"><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 \scriptstyle {PA\vdash \lnot \exists xx<0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> <mi>A</mi> <mo>⊢<!-- ⊢ --></mo> <mi mathvariant="normal">¬<!-- ¬ --></mi> <mi mathvariant="normal">∃<!-- ∃ --></mi> <mi>x</mi> <mi>x</mi> <mo><</mo> <mn>0</mn> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {PA\vdash \lnot \exists xx<0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/781bf77977de8401656bec3aad40a85aecd7034f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.462ex; height:1.843ex;" alt="{\displaystyle \scriptstyle {PA\vdash \lnot \exists xx<0}}"></span> </td></tr> <tr> <td><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 \scriptstyle {<}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mo><</mo> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {<}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41757f13dcd43fa627dbe59c690b638df023c342" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.279ex; height:1.509ex;" alt="{\displaystyle \scriptstyle {<}}"></span> irreflexív </td> <td align="left"><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 \scriptstyle {PA\vdash \forall x\lnot x<x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> <mi>A</mi> <mo>⊢<!-- ⊢ --></mo> <mi mathvariant="normal">∀<!-- ∀ --></mi> <mi>x</mi> <mi mathvariant="normal">¬<!-- ¬ --></mi> <mi>x</mi> <mo><</mo> <mi>x</mi> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {PA\vdash \forall x\lnot x<x}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2054da7e2f518396253f54a74d21f1d63b9f609e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.58ex; height:1.843ex;" alt="{\displaystyle \scriptstyle {PA\vdash \forall x\lnot x<x}}"></span> </td></tr> <tr> <td><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 \scriptstyle {<}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mo><</mo> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {<}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41757f13dcd43fa627dbe59c690b638df023c342" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.279ex; height:1.509ex;" alt="{\displaystyle \scriptstyle {<}}"></span> antiszimmetrikus </td> <td align="left"><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 \scriptstyle {PA\vdash \forall x\forall y(x<y\land y<x)\rightarrow x=y}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> <mi>A</mi> <mo>⊢<!-- ⊢ --></mo> <mi mathvariant="normal">∀<!-- ∀ --></mi> <mi>x</mi> <mi mathvariant="normal">∀<!-- ∀ --></mi> <mi>y</mi> <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> <mo stretchy="false">→<!-- → --></mo> <mi>x</mi> <mo>=</mo> <mi>y</mi> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {PA\vdash \forall x\forall y(x<y\land y<x)\rightarrow x=y}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd9100140f244c6e648062695f6c969f6e90a638" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:20.182ex; height:2.176ex;" alt="{\displaystyle \scriptstyle {PA\vdash \forall x\forall y(x<y\land y<x)\rightarrow x=y}}"></span> </td></tr> <tr> <td><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 \scriptstyle {<}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mo><</mo> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {<}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41757f13dcd43fa627dbe59c690b638df023c342" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.279ex; height:1.509ex;" alt="{\displaystyle \scriptstyle {<}}"></span> tranzitív </td> <td align="left"><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 \scriptstyle {PA\vdash \forall x\forall y\forall z((x<y\land y<z)\rightarrow x<z)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> <mi>A</mi> <mo>⊢<!-- ⊢ --></mo> <mi mathvariant="normal">∀<!-- ∀ --></mi> <mi>x</mi> <mi mathvariant="normal">∀<!-- ∀ --></mi> <mi>y</mi> <mi mathvariant="normal">∀<!-- ∀ --></mi> <mi>z</mi> <mo stretchy="false">(</mo> <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> <mo stretchy="false">→<!-- → --></mo> <mi>x</mi> <mo><</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {PA\vdash \forall x\forall y\forall 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/a77d6f04b3c9d2fc5470690cf6a15ce1db9135d8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:22.927ex; height:2.176ex;" alt="{\displaystyle \scriptstyle {PA\vdash \forall x\forall y\forall z((x<y\land y<z)\rightarrow x<z)}}"></span> </td></tr> <tr> <td><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 \scriptstyle {<}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mo><</mo> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {<}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41757f13dcd43fa627dbe59c690b638df023c342" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.279ex; height:1.509ex;" alt="{\displaystyle \scriptstyle {<}}"></span> lineáris </td> <td align="left"><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 \scriptstyle {PA\vdash \forall x\forall y((x<y\vee x=y)\vee x>y)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> <mi>A</mi> <mo>⊢<!-- ⊢ --></mo> <mi mathvariant="normal">∀<!-- ∀ --></mi> <mi>x</mi> <mi mathvariant="normal">∀<!-- ∀ --></mi> <mi>y</mi> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo><</mo> <mi>y</mi> <mo>∨<!-- ∨ --></mo> <mi>x</mi> <mo>=</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>∨<!-- ∨ --></mo> <mi>x</mi> <mo>></mo> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {PA\vdash \forall x\forall y((x<y\vee x=y)\vee x>y)}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/763543540a05bf46ef7e6e07cc3f0de2a232c093" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:20.915ex; height:2.176ex;" alt="{\displaystyle \scriptstyle {PA\vdash \forall x\forall y((x<y\vee x=y)\vee x>y)}}"></span> </td></tr> <tr> <td><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 \scriptstyle {<}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mo><</mo> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {<}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41757f13dcd43fa627dbe59c690b638df023c342" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.279ex; height:1.509ex;" alt="{\displaystyle \scriptstyle {<}}"></span> és a <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 \scriptstyle {+}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {+}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d9cb3bfb2fc1868785d10c0a72ce87161c3c10b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.279ex; height:1.509ex;" alt="{\displaystyle \scriptstyle {+}}"></span> </td> <td align="left"><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 \scriptstyle {PA\vdash \forall x\forall y\forall z(x<y\rightarrow (x+z)<(y+z))}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> <mi>A</mi> <mo>⊢<!-- ⊢ --></mo> <mi mathvariant="normal">∀<!-- ∀ --></mi> <mi>x</mi> <mi mathvariant="normal">∀<!-- ∀ --></mi> <mi>y</mi> <mi mathvariant="normal">∀<!-- ∀ --></mi> <mi>z</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo><</mo> <mi>y</mi> <mo stretchy="false">→<!-- → --></mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo><</mo> <mo stretchy="false">(</mo> <mi>y</mi> <mo>+</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {PA\vdash \forall x\forall y\forall z(x<y\rightarrow (x+z)<(y+z))}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0022324c4d7fa714ea0475b1d359570ab0acc464" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:24.388ex; height:2.176ex;" alt="{\displaystyle \scriptstyle {PA\vdash \forall x\forall y\forall z(x<y\rightarrow (x+z)<(y+z))}}"></span> </td></tr> <tr> <td><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 \scriptstyle {<}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mo><</mo> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {<}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41757f13dcd43fa627dbe59c690b638df023c342" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.279ex; height:1.509ex;" alt="{\displaystyle \scriptstyle {<}}"></span> és a <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 \scriptstyle {\cdot }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mo>⋅<!-- ⋅ --></mo> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {\cdot }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d0c268adc667ee964887357252db434bc6b5d1da" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.31ex; margin-bottom: -0.482ex; width:0.457ex; height:1.009ex;" alt="{\displaystyle \scriptstyle {\cdot }}"></span> </td> <td align="left"><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 \scriptstyle {PA\vdash \forall x\forall y\forall z((x<y\land 0<z)\rightarrow (x\cdot z)<(y\cdot z))}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> <mi>A</mi> <mo>⊢<!-- ⊢ --></mo> <mi mathvariant="normal">∀<!-- ∀ --></mi> <mi>x</mi> <mi mathvariant="normal">∀<!-- ∀ --></mi> <mi>y</mi> <mi mathvariant="normal">∀<!-- ∀ --></mi> <mi>z</mi> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo><</mo> <mi>y</mi> <mo>∧<!-- ∧ --></mo> <mn>0</mn> <mo><</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→<!-- → --></mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>⋅<!-- ⋅ --></mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo><</mo> <mo stretchy="false">(</mo> <mi>y</mi> <mo>⋅<!-- ⋅ --></mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {PA\vdash \forall x\forall y\forall z((x<y\land 0<z)\rightarrow (x\cdot z)<(y\cdot z))}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6ed086132c28685de155d99f9ee61212afd8499b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:27.992ex; height:2.176ex;" alt="{\displaystyle \scriptstyle {PA\vdash \forall x\forall y\forall z((x<y\land 0<z)\rightarrow (x\cdot z)<(y\cdot z))}}"></span> </td></tr> </tbody></table> <p>A reláció természetesen élesíthető „kisebb, mint”-re: </p> <center><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 \scriptstyle {x<y\mathop {\iff } _{def}x\leq y\land x\neq y}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo><</mo> <mi>y</mi> <msub> <mrow class="MJX-TeXAtom-OP"> <mspace width="thickmathspace" /> <mo stretchy="false">⟺<!-- ⟺ --></mo> <mspace width="thickmathspace" /> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mi>e</mi> <mi>f</mi> </mrow> </msub> <mo>⁡<!-- --></mo> <mi>x</mi> <mo>≤<!-- ≤ --></mo> <mi>y</mi> <mo>∧<!-- ∧ --></mo> <mi>x</mi> <mo>≠<!-- ≠ --></mo> <mi>y</mi> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {x<y\mathop {\iff } _{def}x\leq y\land x\neq y}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c3fdfae501effb3e36e3f7eaa85659f59356ea5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.539ex; height:2.343ex;" alt="{\displaystyle \scriptstyle {x<y\mathop {\iff } _{def}x\leq y\land x\neq y}}"></span></center> <p>Ez már egy totális szigorú rendezés: </p> <ul><li>Irreflexív: Senki sem kisebb magánál.</li> <li>Antiszimmetrikus: Semelyik kettő közül sem lehet egyszerre mindkettő kisebb a másiknál (ilyenkor ugyanis magánál lenne kisebb, ami tiltott).</li> <li>Tranzitív: Ha három közül a harmadik kisebb a másodiknál, és a második kisebb az elsőnél, akkor a harmadik kisebb az elsőnél is.</li> <li>lineáris: Bármely két különböző elem közül pontosan az egyik kisebb a másiknál.</li></ul> <p>Ezen belül további jellemzők, hogy </p> <ul><li>A <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 \scriptstyle {0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/757abea8b05f9281eee9cd6378f38c7a191e44d8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.822ex; height:1.676ex;" alt="{\displaystyle \scriptstyle {0}}"></span> infimum: 0-nál nincs kisebb.</li> <li>Felülről nem korlátos: Mindennél van nagyobb.</li></ul> <p>Természetesen definiálhatók a szokásos párjaik is: </p> <center><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 \scriptstyle {\begin{array}{c}\scriptstyle {x\geq y\iff _{def}\lnot x<y}\\\scriptstyle {x>y\iff _{def}\lnot x\leq y}\end{array}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>≥<!-- ≥ --></mo> <mi>y</mi> <mspace width="thickmathspace" /> <mo stretchy="false">⟺<!-- ⟺ --></mo> <msub> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mi>e</mi> <mi>f</mi> </mrow> </msub> <mi mathvariant="normal">¬<!-- ¬ --></mi> <mi>x</mi> <mo><</mo> <mi>y</mi> </mrow> </mstyle> </mtd> </mtr> <mtr> <mtd> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>></mo> <mi>y</mi> <mspace width="thickmathspace" /> <mo stretchy="false">⟺<!-- ⟺ --></mo> <msub> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mi>e</mi> <mi>f</mi> </mrow> </msub> <mi mathvariant="normal">¬<!-- ¬ --></mi> <mi>x</mi> <mo>≤<!-- ≤ --></mo> <mi>y</mi> </mrow> </mstyle> </mtd> </mtr> </mtable> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {\begin{array}{c}\scriptstyle {x\geq y\iff _{def}\lnot x<y}\\\scriptstyle {x>y\iff _{def}\lnot x\leq y}\end{array}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd1edcba2ba0a79fde743619694852dc4e8a1a45" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.018ex; margin-bottom: -0.32ex; width:14.48ex; height:5.509ex;" alt="{\displaystyle \scriptstyle {\begin{array}{c}\scriptstyle {x\geq y\iff _{def}\lnot x<y}\\\scriptstyle {x>y\iff _{def}\lnot x\leq y}\end{array}}}"></span></center> <p>További mindkettőre jellemző tulajdonság még, hogy működnek a szokásos egyenlőtlenség-rendezési szabályok: </p> <ul><li>az egyenlőtlenség mindkét oldalához hozzáadhatunk tetszőleges számot</li> <li>az egyenlőtlenség mindkét oldalát megszorozhatjuk egy (szigorú rendezés esetében nem nulla) számmal.</li></ul> <p>A rendezések kapcsán Érdemes még megjegyezni a következő formulát: </p> <center><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 \scriptstyle {\forall x(\lnot x=0\rightarrow \exists ysy=x)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∀<!-- ∀ --></mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi mathvariant="normal">¬<!-- ¬ --></mi> <mi>x</mi> <mo>=</mo> <mn>0</mn> <mo stretchy="false">→<!-- → --></mo> <mi mathvariant="normal">∃<!-- ∃ --></mi> <mi>y</mi> <mi>s</mi> <mi>y</mi> <mo>=</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {\forall x(\lnot x=0\rightarrow \exists ysy=x)}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a3eb24ddfce3f3f646ac9707c519085fdae0f721" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.452ex; height:2.176ex;" alt="{\displaystyle \scriptstyle {\forall x(\lnot x=0\rightarrow \exists ysy=x)}}"></span></center> <p>Ezt úgy lehetne mondani, hogy „minden ami nem a nulla, rákövetkezője valaminek”. A rendezés definíciói szerint ezt a következőképpen írhatjuk át: </p> <center><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 \scriptstyle {\forall x(0\leq x)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∀<!-- ∀ --></mi> <mi>x</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo>≤<!-- ≤ --></mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {\forall x(0\leq x)}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54bf7bca0c9ac16d3315289a8673308439c480c5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.174ex; height:2.176ex;" alt="{\displaystyle \scriptstyle {\forall x(0\leq x)}}"></span></center> <p>Ez a nagyon egyszerű formula azért érdekes, mert a Peano-aritmetikának vannak olyan gyengébb változatai is, melyekben ez a formula szerepel a teljes indukciós axiómaséma helyén.<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> </p> <div class="mw-heading mw-heading3"><h3 id="Erős_indukció_és_a_legkisebb_szám_elve"><span id="Er.C5.91s_indukci.C3.B3_.C3.A9s_a_legkisebb_sz.C3.A1m_elve"></span>Erős indukció és a legkisebb szám elve</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Peano-aritmetika&action=edit&section=6" title="Szakasz szerkesztése: Erős indukció és a legkisebb szám elve"><span>szerkesztés</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A Peano-aritmetikában bizonyíthatók más bizonyítási módszerek tételei is. Az egyik ilyen az ún. <i>erős indukció</i> (alakja szerint lényegében ez maga a <a href="/wiki/Transzfinit_indukci%C3%B3" title="Transzfinit indukció">transzfinit indukció</a>, csak ez nem a <a href="/wiki/Axiomatikus_halmazelm%C3%A9let" title="Axiomatikus halmazelmélet">halmazelmélet</a>). Ez azt mondja, hogy ha egy számra öröklődik egy tulajdonság az <i>összes őt megelőző</i> számról, akkor az minden számra érvényes: </p> <center><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 \scriptstyle {PA\vdash \forall x(\forall y(y<x\rightarrow F(y))\rightarrow F(x))\rightarrow \forall z(F(z))}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> <mi>A</mi> <mo>⊢<!-- ⊢ --></mo> <mi mathvariant="normal">∀<!-- ∀ --></mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi mathvariant="normal">∀<!-- ∀ --></mi> <mi>y</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo><</mo> <mi>x</mi> <mo stretchy="false">→<!-- → --></mo> <mi>F</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo stretchy="false">→<!-- → --></mo> <mi>F</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo stretchy="false">→<!-- → --></mo> <mi mathvariant="normal">∀<!-- ∀ --></mi> <mi>z</mi> <mo stretchy="false">(</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {PA\vdash \forall x(\forall y(y<x\rightarrow F(y))\rightarrow F(x))\rightarrow \forall z(F(z))}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/707d80566c6960111a6e2808e10475013cd56a70" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:30.6ex; height:2.176ex;" alt="{\displaystyle \scriptstyle {PA\vdash \forall x(\forall y(y<x\rightarrow F(y))\rightarrow F(x))\rightarrow \forall z(F(z))}}"></span></center> <p>Az egyik legfontosabb következménye ennek az ún. <i>legkisebb szám elve</i>, miszerint ha egy tulajdonság igaz minden számra, akkor van legkisebb szám is, amelyre igaz.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p> <center><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 \scriptstyle {PA\vdash \forall zF(z)\rightarrow \exists x(F(x)\land \forall y(y<x\rightarrow \lnot F(y)))}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> <mi>A</mi> <mo>⊢<!-- ⊢ --></mo> <mi mathvariant="normal">∀<!-- ∀ --></mi> <mi>z</mi> <mi>F</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→<!-- → --></mo> <mi mathvariant="normal">∃<!-- ∃ --></mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>∧<!-- ∧ --></mo> <mi mathvariant="normal">∀<!-- ∀ --></mi> <mi>y</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo><</mo> <mi>x</mi> <mo stretchy="false">→<!-- → --></mo> <mi mathvariant="normal">¬<!-- ¬ --></mi> <mi>F</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {PA\vdash \forall zF(z)\rightarrow \exists x(F(x)\land \forall y(y<x\rightarrow \lnot F(y)))}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/235459d619d1098a7a85203b5839897b64c60fbb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:29.87ex; height:2.176ex;" alt="{\displaystyle \scriptstyle {PA\vdash \forall zF(z)\rightarrow \exists x(F(x)\land \forall y(y<x\rightarrow \lnot F(y)))}}"></span></center> <p>Ez a halmazelméleti <a href="/wiki/J%C3%B3lrendez%C3%A9s" class="mw-redirect" title="Jólrendezés">jólrendezés</a> megfogalmazása: Mivel minden ilyen tulajdonságnak egy halmaz fog majd megfelelni a modellben (azok halmaza, amelyekre igaz), ez azt mondja, hogy minden ilyen halmaznak lesz egy legkisebb eleme. A jólrendezettség persze csak másodrendben definiálható tulajdonság, de ez a két formula sem egy bizonyos tétel, hanem mint a teljes indukció, tételsémák. </p><p>Ez tehát az az elv, amit akkor használunk, mikor az ún. végtelen leszállásra hivatkozva bizonyítunk. Ez a következőt jelenti: egy indirekt feltevés oda vezet, hogy ha a tulajdonság igaz egy természetes számra, akkor igaz egy nála (szigorúan) kisebb természetes számra is. Ez végtelen leszállásra vezet, azonban tudjuk, hogy a természetes számok <a href="/wiki/J%C3%B3lrendezett_halmaz" title="Jólrendezett halmaz">jólrendezett számhalmaz</a>. Innen nyerjük az ellentmondást. </p> <div class="mw-heading mw-heading3"><h3 id="Oszthatósági_reláció"><span id="Oszthat.C3.B3s.C3.A1gi_rel.C3.A1ci.C3.B3"></span>Oszthatósági reláció</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Peano-aritmetika&action=edit&section=7" title="Szakasz szerkesztése: Oszthatósági reláció"><span>szerkesztés</span></a><span class="mw-editsection-bracket">]</span></span></div> <table align="right" class="wikitable"> <tbody><tr> <th colspan="2" align="center">oszthatósági reláció tételei </th></tr> <tr> <td>reflexivitás </td> <td><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 \scriptstyle {PA\vdash d|d}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> <mi>A</mi> <mo>⊢<!-- ⊢ --></mo> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>d</mi> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {PA\vdash d|d}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/daa8905c0e109e976dd0448e8b9153fd0764811a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.648ex; height:2.176ex;" alt="{\displaystyle \scriptstyle {PA\vdash d|d}}"></span> </td></tr> <tr> <td>antiszimmetria </td> <td><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 \scriptstyle {PA\vdash \forall x\forall y((x|y\land y|x)\rightarrow x=y)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> <mi>A</mi> <mo>⊢<!-- ⊢ --></mo> <mi mathvariant="normal">∀<!-- ∀ --></mi> <mi>x</mi> <mi mathvariant="normal">∀<!-- ∀ --></mi> <mi>y</mi> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>y</mi> <mo>∧<!-- ∧ --></mo> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→<!-- → --></mo> <mi>x</mi> <mo>=</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {PA\vdash \forall x\forall y((x|y\land y|x)\rightarrow x=y)}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7f916b03b7db98409d6a3de34039a4e700afeb15" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:19.819ex; height:2.176ex;" alt="{\displaystyle \scriptstyle {PA\vdash \forall x\forall y((x|y\land y|x)\rightarrow x=y)}}"></span> </td></tr> <tr> <td>tranzitivitás </td> <td><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 \scriptstyle {PA\vdash \forall x\forall y\forall z((x|y\land y|z)\rightarrow x|z)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> <mi>A</mi> <mo>⊢<!-- ⊢ --></mo> <mi mathvariant="normal">∀<!-- ∀ --></mi> <mi>x</mi> <mi mathvariant="normal">∀<!-- ∀ --></mi> <mi>y</mi> <mi mathvariant="normal">∀<!-- ∀ --></mi> <mi>z</mi> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>y</mi> <mo>∧<!-- ∧ --></mo> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>z</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→<!-- → --></mo> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {PA\vdash \forall x\forall y\forall 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/238804cf191c84b25a9ca0d8707d9fd72e08196a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:20.463ex; height:2.176ex;" alt="{\displaystyle \scriptstyle {PA\vdash \forall x\forall y\forall z((x|y\land y|z)\rightarrow x|z)}}"></span> </td></tr> <tr> <td><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 \scriptstyle {1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab612ee1036f9fcea956f37d4dbba00bdfa98164" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.822ex; height:1.676ex;" alt="{\displaystyle \scriptstyle {1}}"></span> egység </td> <td><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 \scriptstyle {PA\vdash \forall x1|x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> <mi>A</mi> <mo>⊢<!-- ⊢ --></mo> <mi mathvariant="normal">∀<!-- ∀ --></mi> <mi>x</mi> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {PA\vdash \forall x1|x}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3b90fa031fd1fe1a4e889244ec15942b1bd2216" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.545ex; height:2.176ex;" alt="{\displaystyle \scriptstyle {PA\vdash \forall x1|x}}"></span> </td></tr> <tr> <td><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 \scriptstyle {|}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {|}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/939a919edfd505b377d8449a3c40e1139496943f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:0.457ex; height:2.176ex;" alt="{\displaystyle \scriptstyle {|}}"></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 \scriptstyle {+}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {+}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d9cb3bfb2fc1868785d10c0a72ce87161c3c10b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.279ex; height:1.509ex;" alt="{\displaystyle \scriptstyle {+}}"></span> </td> <td><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 \scriptstyle {PA\vdash (d|x\rightarrow (d|(x+y)\leftrightarrow d|y))}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> <mi>A</mi> <mo>⊢<!-- ⊢ --></mo> <mo stretchy="false">(</mo> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mo stretchy="false">→<!-- → --></mo> <mo stretchy="false">(</mo> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo stretchy="false">↔<!-- ↔ --></mo> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>y</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {PA\vdash (d|x\rightarrow (d|(x+y)\leftrightarrow d|y))}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d62e3d1eac8b5d32c8aabc46c82cee790c0044c7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:19.34ex; height:2.176ex;" alt="{\displaystyle \scriptstyle {PA\vdash (d|x\rightarrow (d|(x+y)\leftrightarrow d|y))}}"></span> </td></tr></tbody></table> <p>Bevezethető egy a <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 \scriptstyle {\leq }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mo>≤<!-- ≤ --></mo> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {\leq }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/214a1e186abadeb10df2481578bf7cf4a7c2bb96" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.279ex; height:1.843ex;" alt="{\displaystyle \scriptstyle {\leq }}"></span>-hez nagyon hasonló reláció, ami csak abban különbözik attól, hogy <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 \scriptstyle {+}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {+}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d9cb3bfb2fc1868785d10c0a72ce87161c3c10b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.279ex; height:1.509ex;" alt="{\displaystyle \scriptstyle {+}}"></span> helyett <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 \scriptstyle {\cdot }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mo>⋅<!-- ⋅ --></mo> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {\cdot }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d0c268adc667ee964887357252db434bc6b5d1da" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.31ex; margin-bottom: -0.482ex; width:0.457ex; height:1.009ex;" alt="{\displaystyle \scriptstyle {\cdot }}"></span> szerepel benne. Ez felelne meg az <a href="/wiki/Oszthat%C3%B3s%C3%A1g" title="Oszthatóság">oszthatóságnak</a>: </p> <center><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 \scriptstyle {x|y\iff _{def}\exists z(z\cdot x)=y}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>y</mi> <mspace width="thickmathspace" /> <mo stretchy="false">⟺<!-- ⟺ --></mo> <msub> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mi>e</mi> <mi>f</mi> </mrow> </msub> <mi mathvariant="normal">∃<!-- ∃ --></mi> <mi>z</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo>⋅<!-- ⋅ --></mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>y</mi> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {x|y\iff _{def}\exists z(z\cdot x)=y}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a25a87d99cd61957b5135db1a383c9159098e0e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.001ex; height:2.343ex;" alt="{\displaystyle \scriptstyle {x|y\iff _{def}\exists z(z\cdot x)=y}}"></span></center> <p>A <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 \scriptstyle {\leq }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mo>≤<!-- ≤ --></mo> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {\leq }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/214a1e186abadeb10df2481578bf7cf4a7c2bb96" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.279ex; height:1.843ex;" alt="{\displaystyle \scriptstyle {\leq }}"></span>-hez hasonlóan ez is egy parciális rendezés, azaz: </p> <ul><li>Reflexív: Minden osztója önmagának – mivel minden egyszerese önmaga.</li> <li>Antiszimmetrikus: Ha ketten osztói egymásnak, akkor ugyanarról van szó.</li> <li>Tranzitív: Ha három közül az egyik a kettő között van, Akkor az első is osztója a harmadiknak.</li></ul> <p>De természetesen – mint általában az oszthatóság – nem lineáris. </p> <div class="mw-heading mw-heading3"><h3 id="Maradékos_osztás"><span id="Marad.C3.A9kos_oszt.C3.A1s"></span>Maradékos osztás</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Peano-aritmetika&action=edit&section=8" title="Szakasz szerkesztése: Maradékos osztás"><span>szerkesztés</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Számelméleti szempontból a legfontosabb tétel a maradékos osztásért felelős tétel levezethetősége. Ugyanis ha ez létezik, akkor a (metanyelvi, modellelméleti értelemben vett) számelmélet szerint a Peano-aritmetika modellje egy euklideszi gyűrű, amiről tudható, hogy érvényes benne a <a href="/wiki/A_sz%C3%A1melm%C3%A9let_alapt%C3%A9tele" title="A számelmélet alaptétele">számelmélet alaptétele</a>, azaz minden szám lényegében egyértelműen felbontható <a href="/wiki/Pr%C3%ADmsz%C3%A1m#A_matematikai_definíció" class="mw-redirect" title="Prímszám">prímek</a> szorzatára. A számelmélet alaptétele lenne persze a legfontosabb tétel, azonban ez elsőrendben nem megfogalmazható, így a Peano-aritmetikában (az eddig látottnál is) bonyolultabb minden. </p><p>Ez a maradékos osztásért felelős tétel azt mondja ki, hogy a maradékos osztás létezik és egyértelmű: </p> <center><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 \scriptstyle {PA\vdash \forall x(d\neq 0\rightarrow \exists q\exists r(((x=q\cdot d+r\land r<d)\land \forall q'\forall r'((x=(q'\cdot d)+r'\land r'<d)\rightarrow (q=q'\land r=r')))))}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> <mi>A</mi> <mo>⊢<!-- ⊢ --></mo> <mi mathvariant="normal">∀<!-- ∀ --></mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>d</mi> <mo>≠<!-- ≠ --></mo> <mn>0</mn> <mo stretchy="false">→<!-- → --></mo> <mi mathvariant="normal">∃<!-- ∃ --></mi> <mi>q</mi> <mi mathvariant="normal">∃<!-- ∃ --></mi> <mi>r</mi> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>=</mo> <mi>q</mi> <mo>⋅<!-- ⋅ --></mo> <mi>d</mi> <mo>+</mo> <mi>r</mi> <mo>∧<!-- ∧ --></mo> <mi>r</mi> <mo><</mo> <mi>d</mi> <mo stretchy="false">)</mo> <mo>∧<!-- ∧ --></mo> <mi mathvariant="normal">∀<!-- ∀ --></mi> <msup> <mi>q</mi> <mo>′</mo> </msup> <mi mathvariant="normal">∀<!-- ∀ --></mi> <msup> <mi>r</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>=</mo> <mo stretchy="false">(</mo> <msup> <mi>q</mi> <mo>′</mo> </msup> <mo>⋅<!-- ⋅ --></mo> <mi>d</mi> <mo stretchy="false">)</mo> <mo>+</mo> <msup> <mi>r</mi> <mo>′</mo> </msup> <mo>∧<!-- ∧ --></mo> <msup> <mi>r</mi> <mo>′</mo> </msup> <mo><</mo> <mi>d</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→<!-- → --></mo> <mo stretchy="false">(</mo> <mi>q</mi> <mo>=</mo> <msup> <mi>q</mi> <mo>′</mo> </msup> <mo>∧<!-- ∧ --></mo> <mi>r</mi> <mo>=</mo> <msup> <mi>r</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {PA\vdash \forall x(d\neq 0\rightarrow \exists q\exists r(((x=q\cdot d+r\land r<d)\land \forall q'\forall r'((x=(q'\cdot d)+r'\land r'<d)\rightarrow (q=q'\land r=r')))))}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/103f3f3bf6c4c4b6f1c1581c291cabb43c708422" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:60.522ex; height:2.343ex;" alt="{\displaystyle \scriptstyle {PA\vdash \forall x(d\neq 0\rightarrow \exists q\exists r(((x=q\cdot d+r\land r<d)\land \forall q'\forall r'((x=(q'\cdot d)+r'\land r'<d)\rightarrow (q=q'\land r=r')))))}}"></span></center> <p>Ez alapján bevezethetünk egy relációt: </p> <center> <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 \scriptstyle {Rm(x,d,r)\iff _{def}((r<d\land \exists qx=((q\cdot d)+r))\vee (d=0\land r=x))}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> <mi>m</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>d</mi> <mo>,</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mspace width="thickmathspace" /> <mo stretchy="false">⟺<!-- ⟺ --></mo> <msub> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mi>e</mi> <mi>f</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mi>r</mi> <mo><</mo> <mi>d</mi> <mo>∧<!-- ∧ --></mo> <mi mathvariant="normal">∃<!-- ∃ --></mi> <mi>q</mi> <mi>x</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mi>q</mi> <mo>⋅<!-- ⋅ --></mo> <mi>d</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>∨<!-- ∨ --></mo> <mo stretchy="false">(</mo> <mi>d</mi> <mo>=</mo> <mn>0</mn> <mo>∧<!-- ∧ --></mo> <mi>r</mi> <mo>=</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {Rm(x,d,r)\iff _{def}((r<d\land \exists qx=((q\cdot d)+r))\vee (d=0\land r=x))}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d214fed16282956f105186c9d3da465aaf30645" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:40.455ex; height:2.343ex;" alt="{\displaystyle \scriptstyle {Rm(x,d,r)\iff _{def}((r<d\land \exists qx=((q\cdot d)+r))\vee (d=0\land r=x))}}"></span></center> <p>Ebből a fenti <a href="/wiki/F%C3%BCggv%C3%A9ny_(matematika)" title="Függvény (matematika)">függvényszerű</a> tulajdonság miatt származtatható egy függvény is. Ezt a függvényt a <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 \scriptstyle {rm(x,y)=z}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> <mi>m</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>z</mi> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {rm(x,y)=z}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fa1676f846f952568f7ba5a253b71adaba2f2f5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.726ex; height:2.176ex;" alt="{\displaystyle \scriptstyle {rm(x,y)=z}}"></span> alakban fogjuk használni, és ezt úgy lesz érdemes kiolvasni, hogy „az <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 \scriptstyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mi>x</mi> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a6f6eaddec053fdd1c4a73fd001e579241d248a8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.94ex; height:1.343ex;" alt="{\displaystyle \scriptstyle x}"></span>-et az <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 \scriptstyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mi>y</mi> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/696a751faf350466f822e1de24a4beb44dc43dd6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:0.817ex; height:1.676ex;" alt="{\displaystyle \scriptstyle y}"></span>-nal maradékosan osztva <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 \scriptstyle z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mi>z</mi> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea1aac04b6044a6bc2aa0ca36b4580a19019d46e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.769ex; height:1.343ex;" alt="{\displaystyle \scriptstyle z}"></span> a maradék”. </p><p><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 \scriptstyle {PA\vdash rm(x,0)=x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> <mi>A</mi> <mo>⊢<!-- ⊢ --></mo> <mi>r</mi> <mi>m</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mi>x</mi> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {PA\vdash rm(x,0)=x}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a9b0c69f53f167d7713202fb4e835d0788e0d0c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.373ex; height:2.176ex;" alt="{\displaystyle \scriptstyle {PA\vdash rm(x,0)=x}}"></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 \scriptstyle {PA\vdash (d|x\leftrightarrow rm(x,d)=0)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> <mi>A</mi> <mo>⊢<!-- ⊢ --></mo> <mo stretchy="false">(</mo> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mo stretchy="false">↔<!-- ↔ --></mo> <mi>r</mi> <mi>m</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>d</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {PA\vdash (d|x\leftrightarrow rm(x,d)=0)}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6918edebd51656d7721a557b3ff7da9cea61226c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.472ex; height:2.176ex;" alt="{\displaystyle \scriptstyle {PA\vdash (d|x\leftrightarrow rm(x,d)=0)}}"></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 \scriptstyle {PA\vdash rm(x+yd,d)=rm(x,d)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> <mi>A</mi> <mo>⊢<!-- ⊢ --></mo> <mi>r</mi> <mi>m</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mi>d</mi> <mo>,</mo> <mi>d</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>r</mi> <mi>m</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>d</mi> <mo stretchy="false">)</mo> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {PA\vdash rm(x+yd,d)=rm(x,d)}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a2482451af0f46041ae9b7a8458754b770134417" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:19.147ex; height:2.176ex;" alt="{\displaystyle \scriptstyle {PA\vdash rm(x+yd,d)=rm(x,d)}}"></span> </p> <div class="mw-heading mw-heading3"><h3 id="Prím-,_Felbonthatatlan-_és_RelatívPrím_predikátum"><span id="Pr.C3.ADm-.2C_Felbonthatatlan-_.C3.A9s_Relat.C3.ADvPr.C3.ADm_predik.C3.A1tum"></span>Prím-, Felbonthatatlan- és RelatívPrím predikátum</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Peano-aritmetika&action=edit&section=9" title="Szakasz szerkesztése: Prím-, Felbonthatatlan- és RelatívPrím predikátum"><span>szerkesztés</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A szokásos értelemben beszélhetünk irreducibilis, vagy más néven <a href="/wiki/Pr%C3%ADmsz%C3%A1m#A_matematikai_definíció" class="mw-redirect" title="Prímszám">felbonthatatlan</a> számról, </p> <center><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 \scriptstyle {Felbonthatatlan(p)\iff _{def}(p\neq 1\land \forall d(d|p\rightarrow (d=1\vee d=p)))}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mi>F</mi> <mi>e</mi> <mi>l</mi> <mi>b</mi> <mi>o</mi> <mi>n</mi> <mi>t</mi> <mi>h</mi> <mi>a</mi> <mi>t</mi> <mi>a</mi> <mi>t</mi> <mi>l</mi> <mi>a</mi> <mi>n</mi> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> <mspace width="thickmathspace" /> <mo stretchy="false">⟺<!-- ⟺ --></mo> <msub> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mi>e</mi> <mi>f</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>p</mi> <mo>≠<!-- ≠ --></mo> <mn>1</mn> <mo>∧<!-- ∧ --></mo> <mi mathvariant="normal">∀<!-- ∀ --></mi> <mi>d</mi> <mo stretchy="false">(</mo> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>p</mi> <mo stretchy="false">→<!-- → --></mo> <mo stretchy="false">(</mo> <mi>d</mi> <mo>=</mo> <mn>1</mn> <mo>∨<!-- ∨ --></mo> <mi>d</mi> <mo>=</mo> <mi>p</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {Felbonthatatlan(p)\iff _{def}(p\neq 1\land \forall d(d|p\rightarrow (d=1\vee d=p)))}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b7efdc57dbf6acdf16b952791b387d6da7e5bea6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:40.901ex; height:2.343ex;" alt="{\displaystyle \scriptstyle {Felbonthatatlan(p)\iff _{def}(p\neq 1\land \forall d(d|p\rightarrow (d=1\vee d=p)))}}"></span></center> <p>illetve prímszámról, </p> <center><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 \scriptstyle {Prime(p)\iff _{def}(p\neq 1\rightarrow \forall x\forall y(p|xy\rightarrow (p|x\vee p|y)))}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> <mi>r</mi> <mi>i</mi> <mi>m</mi> <mi>e</mi> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> <mspace width="thickmathspace" /> <mo stretchy="false">⟺<!-- ⟺ --></mo> <msub> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mi>e</mi> <mi>f</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>p</mi> <mo>≠<!-- ≠ --></mo> <mn>1</mn> <mo stretchy="false">→<!-- → --></mo> <mi mathvariant="normal">∀<!-- ∀ --></mi> <mi>x</mi> <mi mathvariant="normal">∀<!-- ∀ --></mi> <mi>y</mi> <mo stretchy="false">(</mo> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mi>y</mi> <mo stretchy="false">→<!-- → --></mo> <mo stretchy="false">(</mo> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mo>∨<!-- ∨ --></mo> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>y</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {Prime(p)\iff _{def}(p\neq 1\rightarrow \forall x\forall y(p|xy\rightarrow (p|x\vee p|y)))}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ea9e7aeca4c4f9cc31e503182a52e7bf8c32def" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.52ex; height:2.343ex;" alt="{\displaystyle \scriptstyle {Prime(p)\iff _{def}(p\neq 1\rightarrow \forall x\forall y(p|xy\rightarrow (p|x\vee p|y)))}}"></span></center> <p>továbbá <a href="/wiki/Relat%C3%ADv_pr%C3%ADmek" title="Relatív prímek">relatív prímségről</a> is: </p> <center><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 \scriptstyle {RelativPrim(a,b)\iff _{def}\forall d((d|a\land d|b)\rightarrow d=1)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> <mi>e</mi> <mi>l</mi> <mi>a</mi> <mi>t</mi> <mi>i</mi> <mi>v</mi> <mi>P</mi> <mi>r</mi> <mi>i</mi> <mi>m</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mspace width="thickmathspace" /> <mo stretchy="false">⟺<!-- ⟺ --></mo> <msub> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mi>e</mi> <mi>f</mi> </mrow> </msub> <mi mathvariant="normal">∀<!-- ∀ --></mi> <mi>d</mi> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>a</mi> <mo>∧<!-- ∧ --></mo> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>b</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→<!-- → --></mo> <mi>d</mi> <mo>=</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {RelativPrim(a,b)\iff _{def}\forall d((d|a\land d|b)\rightarrow d=1)}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b79b9481dcd5ad6a545fa1e8227b21022dfd94f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.432ex; height:2.343ex;" alt="{\displaystyle \scriptstyle {RelativPrim(a,b)\iff _{def}\forall d((d|a\land d|b)\rightarrow d=1)}}"></span> </center> <p>Ezekkel kapcsolatos tételek: </p> <ul><li>a <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 \scriptstyle {2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d1bf4c850c961bfa77cdf5be2d182e8f2037b48e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.822ex; height:1.676ex;" alt="{\displaystyle \scriptstyle {2}}"></span> a legkisebb felbonthatatlan:</li></ul> <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 \scriptstyle {PA\vdash (Felbonthatatlan(2)\land \forall x(Felbonthatatlan(x)\rightarrow 2\leq x))}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> <mi>A</mi> <mo>⊢<!-- ⊢ --></mo> <mo stretchy="false">(</mo> <mi>F</mi> <mi>e</mi> <mi>l</mi> <mi>b</mi> <mi>o</mi> <mi>n</mi> <mi>t</mi> <mi>h</mi> <mi>a</mi> <mi>t</mi> <mi>a</mi> <mi>t</mi> <mi>l</mi> <mi>a</mi> <mi>n</mi> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>∧<!-- ∧ --></mo> <mi mathvariant="normal">∀<!-- ∀ --></mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>F</mi> <mi>e</mi> <mi>l</mi> <mi>b</mi> <mi>o</mi> <mi>n</mi> <mi>t</mi> <mi>h</mi> <mi>a</mi> <mi>t</mi> <mi>a</mi> <mi>t</mi> <mi>l</mi> <mi>a</mi> <mi>n</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→<!-- → --></mo> <mn>2</mn> <mo>≤<!-- ≤ --></mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {PA\vdash (Felbonthatatlan(2)\land \forall x(Felbonthatatlan(x)\rightarrow 2\leq x))}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/532682532d09a3c3dffc91121746f79da4ad4eae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:41.564ex; height:2.176ex;" alt="{\displaystyle \scriptstyle {PA\vdash (Felbonthatatlan(2)\land \forall x(Felbonthatatlan(x)\rightarrow 2\leq x))}}"></span></dd></dl></dd></dl> <ul><li>Mindenkit oszt egy felbonthatatlan:</li></ul> <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 \scriptstyle {PA\vdash \forall x(x>1\rightarrow \exists p(Irreducible(p)\land p|x))}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> <mi>A</mi> <mo>⊢<!-- ⊢ --></mo> <mi mathvariant="normal">∀<!-- ∀ --></mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>></mo> <mn>1</mn> <mo stretchy="false">→<!-- → --></mo> <mi mathvariant="normal">∃<!-- ∃ --></mi> <mi>p</mi> <mo stretchy="false">(</mo> <mi>I</mi> <mi>r</mi> <mi>r</mi> <mi>e</mi> <mi>d</mi> <mi>u</mi> <mi>c</mi> <mi>i</mi> <mi>b</mi> <mi>l</mi> <mi>e</mi> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> <mo>∧<!-- ∧ --></mo> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {PA\vdash \forall x(x>1\rightarrow \exists p(Irreducible(p)\land p|x))}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/142ecd9f2ff933ff350c14fe889077187acc141c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:27.855ex; height:2.176ex;" alt="{\displaystyle \scriptstyle {PA\vdash \forall x(x>1\rightarrow \exists p(Irreducible(p)\land p|x))}}"></span></dd></dl></dd></dl> <ul><li>Pontosan akkor relatív prím két szám, ha nincs közös felbonthatatlan osztójuk.</li></ul> <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 \scriptstyle {PA\vdash RelativPrim(a,b)\leftrightarrow \lnot \exists x(Felbonthatatlan(x)\land (x|a\land x|b))}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> <mi>A</mi> <mo>⊢<!-- ⊢ --></mo> <mi>R</mi> <mi>e</mi> <mi>l</mi> <mi>a</mi> <mi>t</mi> <mi>i</mi> <mi>v</mi> <mi>P</mi> <mi>r</mi> <mi>i</mi> <mi>m</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo stretchy="false">↔<!-- ↔ --></mo> <mi mathvariant="normal">¬<!-- ¬ --></mi> <mi mathvariant="normal">∃<!-- ∃ --></mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>F</mi> <mi>e</mi> <mi>l</mi> <mi>b</mi> <mi>o</mi> <mi>n</mi> <mi>t</mi> <mi>h</mi> <mi>a</mi> <mi>t</mi> <mi>a</mi> <mi>t</mi> <mi>l</mi> <mi>a</mi> <mi>n</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>∧<!-- ∧ --></mo> <mo stretchy="false">(</mo> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>a</mi> <mo>∧<!-- ∧ --></mo> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>b</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {PA\vdash RelativPrim(a,b)\leftrightarrow \lnot \exists x(Felbonthatatlan(x)\land (x|a\land x|b))}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f6aa58a7b27d54d16aef81fbc3efbb3dbb53bf27" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:43.824ex; height:2.176ex;" alt="{\displaystyle \scriptstyle {PA\vdash RelativPrim(a,b)\leftrightarrow \lnot \exists x(Felbonthatatlan(x)\land (x|a\land x|b))}}"></span></dd></dl></dd></dl> <ul><li>Relatív prímeknek van szomszédos többszörösük.</li></ul> <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 \scriptstyle {PA\vdash \forall a\forall b(((a>1\land b>1)\land RelativelyPrime(a,b))\rightarrow \exists x\exists yax+1=by)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> <mi>A</mi> <mo>⊢<!-- ⊢ --></mo> <mi mathvariant="normal">∀<!-- ∀ --></mi> <mi>a</mi> <mi mathvariant="normal">∀<!-- ∀ --></mi> <mi>b</mi> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>></mo> <mn>1</mn> <mo>∧<!-- ∧ --></mo> <mi>b</mi> <mo>></mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>∧<!-- ∧ --></mo> <mi>R</mi> <mi>e</mi> <mi>l</mi> <mi>a</mi> <mi>t</mi> <mi>i</mi> <mi>v</mi> <mi>e</mi> <mi>l</mi> <mi>y</mi> <mi>P</mi> <mi>r</mi> <mi>i</mi> <mi>m</mi> <mi>e</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo stretchy="false">→<!-- → --></mo> <mi mathvariant="normal">∃<!-- ∃ --></mi> <mi>x</mi> <mi mathvariant="normal">∃<!-- ∃ --></mi> <mi>y</mi> <mi>a</mi> <mi>x</mi> <mo>+</mo> <mn>1</mn> <mo>=</mo> <mi>b</mi> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {PA\vdash \forall a\forall b(((a>1\land b>1)\land RelativelyPrime(a,b))\rightarrow \exists x\exists yax+1=by)}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce6423acb0ac66d9c9d933a09bda2b96b64edcb0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:46.089ex; height:2.176ex;" alt="{\displaystyle \scriptstyle {PA\vdash \forall a\forall b(((a>1\land b>1)\land RelativelyPrime(a,b))\rightarrow \exists x\exists yax+1=by)}}"></span></dd></dl></dd></dl> <ul><li>Minden felbonthatatlan szám prímszám.</li></ul> <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 \scriptstyle {PA\vdash \forall x(Felbonthatatlan(x)\rightarrow Prime(x))}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> <mi>A</mi> <mo>⊢<!-- ⊢ --></mo> <mi mathvariant="normal">∀<!-- ∀ --></mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>F</mi> <mi>e</mi> <mi>l</mi> <mi>b</mi> <mi>o</mi> <mi>n</mi> <mi>t</mi> <mi>h</mi> <mi>a</mi> <mi>t</mi> <mi>a</mi> <mi>t</mi> <mi>l</mi> <mi>a</mi> <mi>n</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→<!-- → --></mo> <mi>P</mi> <mi>r</mi> <mi>i</mi> <mi>m</mi> <mi>e</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {PA\vdash \forall x(Felbonthatatlan(x)\rightarrow Prime(x))}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5eaebfb7e65a9a2996dfe6c50b752a8080e5225f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:29.228ex; height:2.176ex;" alt="{\displaystyle \scriptstyle {PA\vdash \forall x(Felbonthatatlan(x)\rightarrow Prime(x))}}"></span></dd></dl></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Monusz">Monusz</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Peano-aritmetika&action=edit&section=10" title="Szakasz szerkesztése: Monusz"><span>szerkesztés</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Az összeadásnak a kivonás lenne az inverze, azonban mint tudjuk ez a természetes számok közt nem létezik. Szokás azonban mégis bevezetni egy öszvérműveletet, amit (mínusz helyett) mónusznak hívnak. Ez a kivonásféleség a következőképp működik: a szokásos kivonást adja, ha amúgyis elvégezhető lenne a kivonás, és <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 \scriptstyle 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mn>0</mn> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d9da869745acc1f1ba686e10425f3a6cc5ffaa38" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.822ex; height:1.676ex;" alt="{\displaystyle \scriptstyle 0}"></span>-t ad olyankor, mikor az egész számok köréből ismert kivonás negatív eredményt ad. </p><p>Ehhez mindenekelőtt az szükséges, hogy ez függvényszerű legyen, amit akövetkező tétel garantál: </p> <center><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 \scriptstyle {PA\vdash \forall x\forall y(y\leq x\rightarrow \exists !zx=(y+z))}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> <mi>A</mi> <mo>⊢<!-- ⊢ --></mo> <mi mathvariant="normal">∀<!-- ∀ --></mi> <mi>x</mi> <mi mathvariant="normal">∀<!-- ∀ --></mi> <mi>y</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo>≤<!-- ≤ --></mo> <mi>x</mi> <mo stretchy="false">→<!-- → --></mo> <mi mathvariant="normal">∃<!-- ∃ --></mi> <mo>!</mo> <mi>z</mi> <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> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {PA\vdash \forall x\forall y(y\leq x\rightarrow \exists !zx=(y+z))}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9799d98a61b779ddde29f707333ef771d27e3805" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:21.518ex; height:2.176ex;" alt="{\displaystyle \scriptstyle {PA\vdash \forall x\forall y(y\leq x\rightarrow \exists !zx=(y+z))}}"></span></center> <p>Ezért ha vesszük a következő relációt, </p> <center><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 \scriptstyle {Monusz(x,y,z)\iff _{def}(x=y+z\vee (x<y\land z=0))}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> <mi>o</mi> <mi>n</mi> <mi>u</mi> <mi>s</mi> <mi>z</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mspace width="thickmathspace" /> <mo stretchy="false">⟺<!-- ⟺ --></mo> <msub> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mi>e</mi> <mi>f</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>=</mo> <mi>y</mi> <mo>+</mo> <mi>z</mi> <mo>∨<!-- ∨ --></mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo><</mo> <mi>y</mi> <mo>∧<!-- ∧ --></mo> <mi>z</mi> <mo>=</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {Monusz(x,y,z)\iff _{def}(x=y+z\vee (x<y\land z=0))}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5234778beed0bad4be7accfa2d7aa66892e6431c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.014ex; height:2.343ex;" alt="{\displaystyle \scriptstyle {Monusz(x,y,z)\iff _{def}(x=y+z\vee (x<y\land z=0))}}"></span></center> <p>Akkor a tétel értelmében csinálhatunk belőle függvényt: </p> <center><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 \scriptstyle {x-y=z\iff _{def}monus(x,y)=z\iff _{def}Monusz(x,y,z)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>−<!-- − --></mo> <mi>y</mi> <mo>=</mo> <mi>z</mi> <mspace width="thickmathspace" /> <mo stretchy="false">⟺<!-- ⟺ --></mo> <msub> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mi>e</mi> <mi>f</mi> </mrow> </msub> <mi>m</mi> <mi>o</mi> <mi>n</mi> <mi>u</mi> <mi>s</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>z</mi> <mspace width="thickmathspace" /> <mo stretchy="false">⟺<!-- ⟺ --></mo> <msub> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mi>e</mi> <mi>f</mi> </mrow> </msub> <mi>M</mi> <mi>o</mi> <mi>n</mi> <mi>u</mi> <mi>s</mi> <mi>z</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {x-y=z\iff _{def}monus(x,y)=z\iff _{def}Monusz(x,y,z)}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/14d6a6442f67f0fcab8a688ce4e77f0051b1b621" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:39.397ex; height:2.343ex;" alt="{\displaystyle \scriptstyle {x-y=z\iff _{def}monus(x,y)=z\iff _{def}Monusz(x,y,z)}}"></span></center> <div class="mw-heading mw-heading3"><h3 id="Korlátos_kvantorok"><span id="Korl.C3.A1tos_kvantorok"></span>Korlátos kvantorok</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Peano-aritmetika&action=edit&section=11" title="Szakasz szerkesztése: Korlátos kvantorok"><span>szerkesztés</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A szokásos kvantifikációkat szokás gyengíteni, méghozzá a következőképpen: </p> <center> <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{array}{rcl}\scriptstyle (\forall x\leq y)F&\scriptstyle \iff _{def}&\scriptstyle \forall x(x\leq y\rightarrow F)\\\scriptstyle (\exists x\leq y)F&\scriptstyle \iff _{def}&\scriptstyle \exists x(x\leq y\land F)\\\scriptstyle (\forall x<y)F&\scriptstyle \iff _{def}&\scriptstyle \forall x(x<y\rightarrow F)\\\scriptstyle (\exists x<y)F&\scriptstyle \iff _{def}&\scriptstyle \exists x(x<y\land F)\\\end{array}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right center left" rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mstyle displaystyle="false" scriptlevel="1"> <mo stretchy="false">(</mo> <mi mathvariant="normal">∀<!-- ∀ --></mi> <mi>x</mi> <mo>≤<!-- ≤ --></mo> <mi>y</mi> <mo stretchy="false">)</mo> <mi>F</mi> </mstyle> </mtd> <mtd> <mstyle displaystyle="false" scriptlevel="1"> <mspace width="thickmathspace" /> <mo stretchy="false">⟺<!-- ⟺ --></mo> <msub> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mi>e</mi> <mi>f</mi> </mrow> </msub> </mstyle> </mtd> <mtd> <mstyle displaystyle="false" scriptlevel="1"> <mi mathvariant="normal">∀<!-- ∀ --></mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>≤<!-- ≤ --></mo> <mi>y</mi> <mo stretchy="false">→<!-- → --></mo> <mi>F</mi> <mo stretchy="false">)</mo> </mstyle> </mtd> </mtr> <mtr> <mtd> <mstyle displaystyle="false" scriptlevel="1"> <mo stretchy="false">(</mo> <mi mathvariant="normal">∃<!-- ∃ --></mi> <mi>x</mi> <mo>≤<!-- ≤ --></mo> <mi>y</mi> <mo stretchy="false">)</mo> <mi>F</mi> </mstyle> </mtd> <mtd> <mstyle displaystyle="false" scriptlevel="1"> <mspace width="thickmathspace" /> <mo stretchy="false">⟺<!-- ⟺ --></mo> <msub> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mi>e</mi> <mi>f</mi> </mrow> </msub> </mstyle> </mtd> <mtd> <mstyle displaystyle="false" scriptlevel="1"> <mi mathvariant="normal">∃<!-- ∃ --></mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>≤<!-- ≤ --></mo> <mi>y</mi> <mo>∧<!-- ∧ --></mo> <mi>F</mi> <mo stretchy="false">)</mo> </mstyle> </mtd> </mtr> <mtr> <mtd> <mstyle displaystyle="false" scriptlevel="1"> <mo stretchy="false">(</mo> <mi mathvariant="normal">∀<!-- ∀ --></mi> <mi>x</mi> <mo><</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mi>F</mi> </mstyle> </mtd> <mtd> <mstyle displaystyle="false" scriptlevel="1"> <mspace width="thickmathspace" /> <mo stretchy="false">⟺<!-- ⟺ --></mo> <msub> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mi>e</mi> <mi>f</mi> </mrow> </msub> </mstyle> </mtd> <mtd> <mstyle displaystyle="false" scriptlevel="1"> <mi mathvariant="normal">∀<!-- ∀ --></mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo><</mo> <mi>y</mi> <mo stretchy="false">→<!-- → --></mo> <mi>F</mi> <mo stretchy="false">)</mo> </mstyle> </mtd> </mtr> <mtr> <mtd> <mstyle displaystyle="false" scriptlevel="1"> <mo stretchy="false">(</mo> <mi mathvariant="normal">∃<!-- ∃ --></mi> <mi>x</mi> <mo><</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mi>F</mi> </mstyle> </mtd> <mtd> <mstyle displaystyle="false" scriptlevel="1"> <mspace width="thickmathspace" /> <mo stretchy="false">⟺<!-- ⟺ --></mo> <msub> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mi>e</mi> <mi>f</mi> </mrow> </msub> </mstyle> </mtd> <mtd> <mstyle displaystyle="false" scriptlevel="1"> <mi mathvariant="normal">∃<!-- ∃ --></mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo><</mo> <mi>y</mi> <mo>∧<!-- ∧ --></mo> <mi>F</mi> <mo stretchy="false">)</mo> </mstyle> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{array}{rcl}\scriptstyle (\forall x\leq y)F&\scriptstyle \iff _{def}&\scriptstyle \forall x(x\leq y\rightarrow F)\\\scriptstyle (\exists x\leq y)F&\scriptstyle \iff _{def}&\scriptstyle \exists x(x\leq y\land F)\\\scriptstyle (\forall x<y)F&\scriptstyle \iff _{def}&\scriptstyle \forall x(x<y\rightarrow F)\\\scriptstyle (\exists x<y)F&\scriptstyle \iff _{def}&\scriptstyle \exists x(x<y\land F)\\\end{array}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/865e4de052a8a37bff9fe7dc090c4ee811c980af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.005ex; width:27.461ex; height:13.176ex;" alt="{\displaystyle {\begin{array}{rcl}\scriptstyle (\forall x\leq y)F&\scriptstyle \iff _{def}&\scriptstyle \forall x(x\leq y\rightarrow F)\\\scriptstyle (\exists x\leq y)F&\scriptstyle \iff _{def}&\scriptstyle \exists x(x\leq y\land F)\\\scriptstyle (\forall x<y)F&\scriptstyle \iff _{def}&\scriptstyle \forall x(x<y\rightarrow F)\\\scriptstyle (\exists x<y)F&\scriptstyle \iff _{def}&\scriptstyle \exists x(x<y\land F)\\\end{array}}}"></span> </center> <p>Ahol 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 \scriptstyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mi>x</mi> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a6f6eaddec053fdd1c4a73fd001e579241d248a8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.94ex; height:1.343ex;" alt="{\displaystyle \scriptstyle x}"></span> és <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 \scriptstyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mi>y</mi> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/696a751faf350466f822e1de24a4beb44dc43dd6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:0.817ex; height:1.676ex;" alt="{\displaystyle \scriptstyle y}"></span> különböző változók kell legyenek. </p><p>Ezt a fajta kvantifikációt <i>korlátos kvantifikációnak</i> nevezzük. Egy formulát pedig <i>korlátos formulának</i> nevezünk akkor, ha benne minden kvantor korlátos. </p> <div class="mw-heading mw-heading3"><h3 id="Aritmetikai_hierarchia">Aritmetikai hierarchia</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Peano-aritmetika&action=edit&section=12" title="Szakasz szerkesztése: Aritmetikai hierarchia"><span>szerkesztés</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Az aritmetika formulahalmazában azon formuláknak, melyek <a href="/w/index.php?title=Prenex_alak&action=edit&redlink=1" class="new" title="Prenex alak (a lap nem létezik)">prenex alakjában</a> váltakoznak a kvantorok, létezik egy érdekes formulahalmaz-sorozat, egy hierarchia, amit a következőképpen definiálunk (rekurzívan): </p> <dl><dd>Bázis: A <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 \scriptstyle \Sigma _{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <msub> <mi mathvariant="normal">Σ<!-- Σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle \Sigma _{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/710f304ff7d15f864a64c32bbeb0198c2d1caad5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.018ex; height:2.009ex;" alt="{\displaystyle \scriptstyle \Sigma _{0}}"></span>-formulák és <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 _{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Π<!-- Π --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Pi _{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ff8b022bb527c2e0006fa9d6fd740ee9b9eab52" 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 _{0}}"></span>-formulák nem mások, mint a korlátos formulák.</dd> <dd>Rekurzió: <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 \scriptstyle \Sigma _{n+1}=\{(\exists x)F:F\in \Pi _{n}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <msub> <mi mathvariant="normal">Σ<!-- Σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mo stretchy="false">(</mo> <mi mathvariant="normal">∃<!-- ∃ --></mi> <mi>x</mi> <mo stretchy="false">)</mo> <mi>F</mi> <mo>:</mo> <mi>F</mi> <mo>∈<!-- ∈ --></mo> <msub> <mi mathvariant="normal">Π<!-- Π --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle \Sigma _{n+1}=\{(\exists x)F:F\in \Pi _{n}\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/126aac34d25d507da6c10b041442a2759dac5733" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.126ex; height:2.176ex;" alt="{\displaystyle \scriptstyle \Sigma _{n+1}=\{(\exists x)F:F\in \Pi _{n}\}}"></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 \scriptstyle \Pi _{n+1}=\{(\forall x)F:F\in \Sigma _{n}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <msub> <mi mathvariant="normal">Π<!-- Π --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mo stretchy="false">(</mo> <mi mathvariant="normal">∀<!-- ∀ --></mi> <mi>x</mi> <mo stretchy="false">)</mo> <mi>F</mi> <mo>:</mo> <mi>F</mi> <mo>∈<!-- ∈ --></mo> <msub> <mi mathvariant="normal">Σ<!-- Σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle \Pi _{n+1}=\{(\forall x)F:F\in \Sigma _{n}\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/acc23c0d1e325f26733d3a5c01c6254608d651aa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.126ex; height:2.176ex;" alt="{\displaystyle \scriptstyle \Pi _{n+1}=\{(\forall x)F:F\in \Sigma _{n}\}}"></span></dd></dl></dd></dl> <p>Tehát az <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 \scriptstyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mi>n</mi> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76760f2e577f85ef1b9818b3a1f7676f3378c0e7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.986ex; height:1.343ex;" alt="{\displaystyle \scriptstyle n}"></span> szám tulajdonképpen azt jelöli, hogy hány kvantor van a váltakozó kvantorú formulában. A <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 \scriptstyle \Sigma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mi mathvariant="normal">Σ<!-- Σ --></mi> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle \Sigma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce5b381f6640ccb0f35cac475885800035f066d5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.187ex; height:1.676ex;" alt="{\displaystyle \scriptstyle \Sigma }"></span> és <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 \scriptstyle \Pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mi mathvariant="normal">Π<!-- Π --></mi> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle \Pi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a98bcd15d3acd1b826ff76259e1988b46e7e1918" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.233ex; height:1.676ex;" alt="{\displaystyle \scriptstyle \Pi }"></span> pedig azt jelöli, hogy egzisztenciális vagy univerzális kvantorral kezdődik-e a váltakozó kvantorokból álló formula. </p><p>Adódik, hogy egy <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 \scriptstyle F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mi>F</mi> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle F}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2c7768d78537fd1e7e8fea3c3cbff7e64317537" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.231ex; height:1.676ex;" alt="{\displaystyle \scriptstyle F}"></span> formula pontosan akkor <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 \scriptstyle \Sigma _{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <msub> <mi mathvariant="normal">Σ<!-- Σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle \Sigma _{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5f7280ee784a3f969db2e6300eb85ad1ba409ce4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:2.151ex; height:1.843ex;" alt="{\displaystyle \scriptstyle \Sigma _{n}}"></span>-formula, ha <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 \scriptstyle \lnot F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mi mathvariant="normal">¬<!-- ¬ --></mi> <mi>F</mi> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle \lnot F}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/268dc25e7d37724b6a922da759ab180dcea6247a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.327ex; height:1.676ex;" alt="{\displaystyle \scriptstyle \lnot F}"></span> egy <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 \scriptstyle \Pi _{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <msub> <mi mathvariant="normal">Π<!-- Π --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle \Pi _{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b9b0aabba6c86d8b5294fa1e97906524bd5d2a4f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:2.197ex; height:1.843ex;" alt="{\displaystyle \scriptstyle \Pi _{n}}"></span>-beli formula. </p><p>Az egyszerre <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 \scriptstyle \Sigma _{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <msub> <mi mathvariant="normal">Σ<!-- Σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle \Sigma _{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5f7280ee784a3f969db2e6300eb85ad1ba409ce4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:2.151ex; height:1.843ex;" alt="{\displaystyle \scriptstyle \Sigma _{n}}"></span> és <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 \scriptstyle \Pi _{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <msub> <mi mathvariant="normal">Π<!-- Π --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle \Pi _{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b9b0aabba6c86d8b5294fa1e97906524bd5d2a4f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:2.197ex; height:1.843ex;" alt="{\displaystyle \scriptstyle \Pi _{n}}"></span> formulákat <i><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 \scriptstyle \Delta _{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <msub> <mi mathvariant="normal">Δ<!-- Δ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle \Delta _{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d4e3dae5c8cb14d003d32b16ab1bc9fa4d145548" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:2.334ex; height:2.009ex;" alt="{\displaystyle \scriptstyle \Delta _{n}}"></span> formuláknak</i> nevezzük. Adódik, hogy <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 \scriptstyle \Delta _{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <msub> <mi mathvariant="normal">Δ<!-- Δ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle \Delta _{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d4e3dae5c8cb14d003d32b16ab1bc9fa4d145548" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:2.334ex; height:2.009ex;" alt="{\displaystyle \scriptstyle \Delta _{n}}"></span> zárt negálásra. </p><p>A modell alaphalmazának azon részhalmazait, melyek egy szabad változós <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 \scriptstyle \Sigma _{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <msub> <mi mathvariant="normal">Σ<!-- Σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle \Sigma _{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5f7280ee784a3f969db2e6300eb85ad1ba409ce4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:2.151ex; height:1.843ex;" alt="{\displaystyle \scriptstyle \Sigma _{n}}"></span>-formulák extenziói, <i><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 \scriptstyle \Sigma _{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <msub> <mi mathvariant="normal">Σ<!-- Σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle \Sigma _{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5f7280ee784a3f969db2e6300eb85ad1ba409ce4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:2.151ex; height:1.843ex;" alt="{\displaystyle \scriptstyle \Sigma _{n}}"></span> halmaznak</i> nevezzük, amelyek pedig egy szabad változós <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 \scriptstyle \Pi _{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <msub> <mi mathvariant="normal">Π<!-- Π --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle \Pi _{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b9b0aabba6c86d8b5294fa1e97906524bd5d2a4f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:2.197ex; height:1.843ex;" alt="{\displaystyle \scriptstyle \Pi _{n}}"></span>-formulák extenziói, <i><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 \scriptstyle \Pi _{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <msub> <mi mathvariant="normal">Π<!-- Π --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle \Pi _{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b9b0aabba6c86d8b5294fa1e97906524bd5d2a4f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:2.197ex; height:1.843ex;" alt="{\displaystyle \scriptstyle \Pi _{n}}"></span> halmaznak</i>, nevezzük. </p><p>A <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 \scriptstyle \Pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mi mathvariant="normal">Π<!-- Π --></mi> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle \Pi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a98bcd15d3acd1b826ff76259e1988b46e7e1918" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.233ex; height:1.676ex;" alt="{\displaystyle \scriptstyle \Pi }"></span> és <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 \scriptstyle \Sigma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mi mathvariant="normal">Σ<!-- Σ --></mi> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle \Sigma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce5b381f6640ccb0f35cac475885800035f066d5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.187ex; height:1.676ex;" alt="{\displaystyle \scriptstyle \Sigma }"></span> jelölések egyébként régies jelölései a kvantoroknak, amik pedig arra utalnak, hogy az univerzálisan kvantifikált állítás tulajdonképpen sok állítás konjunkciója (amit a boole-algebrák nyelvén szorzással szokás jelölni), az egzisztenciálisan kvantifikált állítás pedig sok állítás diszjunkciója, amit ugyanitt összeadással szokás jelölni). Hogy miért is használunk mégis kvantorokat és nem küszöböljük ki őket ezekkel a logikai konstansokkal, az azért van, mert a 'sok' végtelent is jelenthet – végtelen hosszú formuláink pedig nincsenek. </p><p>Innen látszódik is, hogy a fent definiált korlátos kvantorok egyáltalán nem is 'igazi kvantorok', hiszen a korlátján belül végesen felsorolhatók. Ezt mondja tulajdonképpen az, hogy a következő formulák tételei a Peano-aritmetikának: </p> <center><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 \scriptstyle (\forall x\leq n)F\leftrightarrow F_{x}[0]\land F_{x}[1]\land \dots \land F_{x}[n]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mo stretchy="false">(</mo> <mi mathvariant="normal">∀<!-- ∀ --></mi> <mi>x</mi> <mo>≤<!-- ≤ --></mo> <mi>n</mi> <mo stretchy="false">)</mo> <mi>F</mi> <mo stretchy="false">↔<!-- ↔ --></mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo stretchy="false">[</mo> <mn>0</mn> <mo stretchy="false">]</mo> <mo>∧<!-- ∧ --></mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo stretchy="false">[</mo> <mn>1</mn> <mo stretchy="false">]</mo> <mo>∧<!-- ∧ --></mo> <mo>⋯<!-- ⋯ --></mo> <mo>∧<!-- ∧ --></mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo stretchy="false">[</mo> <mi>n</mi> <mo stretchy="false">]</mo> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle (\forall x\leq n)F\leftrightarrow F_{x}[0]\land F_{x}[1]\land \dots \land F_{x}[n]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/abe28bf5a25affe6bce6bc240d34236b6efaeaed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:24.814ex; height:2.176ex;" alt="{\displaystyle \scriptstyle (\forall x\leq n)F\leftrightarrow F_{x}[0]\land F_{x}[1]\land \dots \land F_{x}[n]}"></span></center> <center><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 \scriptstyle (\exists x\leq n)F\leftrightarrow F_{x}[0]\lor F_{x}[1]\lor \dots \lor F_{x}[n]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mo stretchy="false">(</mo> <mi mathvariant="normal">∃<!-- ∃ --></mi> <mi>x</mi> <mo>≤<!-- ≤ --></mo> <mi>n</mi> <mo stretchy="false">)</mo> <mi>F</mi> <mo stretchy="false">↔<!-- ↔ --></mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo stretchy="false">[</mo> <mn>0</mn> <mo stretchy="false">]</mo> <mo>∨<!-- ∨ --></mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo stretchy="false">[</mo> <mn>1</mn> <mo stretchy="false">]</mo> <mo>∨<!-- ∨ --></mo> <mo>⋯<!-- ⋯ --></mo> <mo>∨<!-- ∨ --></mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo stretchy="false">[</mo> <mi>n</mi> <mo stretchy="false">]</mo> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle (\exists x\leq n)F\leftrightarrow F_{x}[0]\lor F_{x}[1]\lor \dots \lor F_{x}[n]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8fd5df6dc94c9c076ab004c1dca62543f4d3ab7e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:24.814ex; height:2.176ex;" alt="{\displaystyle \scriptstyle (\exists x\leq n)F\leftrightarrow F_{x}[0]\lor F_{x}[1]\lor \dots \lor F_{x}[n]}"></span></center> <div class="mw-heading mw-heading2"><h2 id="További_információk"><span id="Tov.C3.A1bbi_inform.C3.A1ci.C3.B3k"></span>További információk</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Peano-aritmetika&action=edit&section=13" title="Szakasz szerkesztése: További információk"><span>szerkesztés</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="https://youproof.hu/kriptografia/11-peano-axiomarendszer-termeszetes-szam-muvelet-osszeadas-kommutativitas-asszociativitas-teljes-indukcio">Alice és Bob – 11. rész: Alice és Bob számelméletet épít</a></li> <li><a rel="nofollow" class="external text" href="https://youproof.hu/kriptografia/12-szorzas-disztributivitas-teljes-indukcio-indirekt-bizonyitas-relacio-teljes-rendezes-rendezett-halmaz">Alice és Bob – 12. rész: Alice és Bob rendet tesz</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Források"><span id="Forr.C3.A1sok"></span>Források</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Peano-aritmetika&action=edit&section=14" title="Szakasz szerkesztése: Források"><span>szerkesztés</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><cite class="book citation" style="font-style:normal">George Boolos. <i>The Logic of Provability</i>. New York: Cambridge University Press (1993). <a href="/wiki/Speci%C3%A1lis:K%C3%B6nyvforr%C3%A1sok/0-521-48325-5" title="Speciális:Könyvforrások/0-521-48325-5">ISBN 0-521-48325-5</a></cite><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Logic+of+Provability&rft.au=George+Boolos&rft.date=1993&rft.pub=Cambridge+University+Press&rft.place=New+York&rft.isbn=0-521-48325-5"><span style="display: none;"> </span></span></li> <li><cite class="book citation" style="font-style:normal">Petr Hájek, Pavel Pudlák. <i>Metamathematics of First-Order Arithmetic</i>. Berlin Heidelberg: Springer-Verlag (1993). <a href="/wiki/Speci%C3%A1lis:K%C3%B6nyvforr%C3%A1sok/0-387-50632-2" title="Speciális:Könyvforrások/0-387-50632-2">ISBN 0-387-50632-2</a></cite><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Metamathematics+of+First-Order+Arithmetic&rft.au=Petr+H%C3%A1jek%2C+Pavel+Pudl%C3%A1k&rft.date=1993&rft.pub=Springer-Verlag&rft.place=Berlin+Heidelberg&rft.isbn=0-387-50632-2"><span style="display: none;"> </span></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="Jegyzetek">Jegyzetek</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Peano-aritmetika&action=edit&section=15" title="Szakasz szerkesztése: Jegyzetek"><span>szerkesztés</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="ref-1col"><div style="-moz-column-count:2; -webkit-column-count:2; column-count:2; -webkit-column-gap: 3em; -moz-column-gap: 3em; column-gap: 3em;"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a></span> <span class="reference-text">George Boolos <i>The Logic of Provability</i>, i. m. <i>2</i> fejezet, 49. o.</span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><a href="#cite_ref-2">↑</a></span> <span class="reference-text">George Boolos <i>The Logic of Provability</i>, i. m. <i>2</i> fejezet, 23. o.</span> </li> </ol></div></div><div class="ref-1col"><div style="-moz-column-count:2; -webkit-column-count:2; column-count:2; -webkit-column-gap: 3em; -moz-column-gap: 3em; column-gap: 3em;"></div></div></div><!--esi <esi:include src="/esitest-fa8a495983347898/content" /> --><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1&useformat=desktop" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">A lap eredeti címe: „<a dir="ltr" href="https://hu.wikipedia.org/w/index.php?title=Peano-aritmetika&oldid=27066885">https://hu.wikipedia.org/w/index.php?title=Peano-aritmetika&oldid=27066885</a>”</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Wikip%C3%A9dia:Kateg%C3%B3ri%C3%A1k" title="Wikipédia:Kategóriák">Kategória</a>: <ul><li><a href="/wiki/Kateg%C3%B3ria:Matematikai_logika" title="Kategória:Matematikai logika">Matematikai logika</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"> A lap utolsó módosítása: 2024. április 17., 14:56</li> <li id="footer-info-copyright">A lap szövege <a rel="nofollow" class="external text" href="http://creativecommons.org/licenses/by-sa/4.0/deed.hu">Creative Commons Nevezd meg! – Így add tovább! 4.0</a> licenc alatt van; egyes esetekben más módon is felhasználható. 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