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Two numbers of which the successors are equal are themselves equal. 5. (induction axiom.) If a set S of numbers contains zero and also the successor of every number in S, then every number is in S. Peano's axioms are the basis for the version of number theory known as Peano arithmetic. "> <meta name="twitter:card" content="summary_large_image"> <meta name="twitter:site" content="@WolframResearch"> <meta name="twitter:title" content="Peano's Axioms -- from Wolfram MathWorld"> <meta name="twitter:description" content="1. Zero is a number. 2. If a is a number, the successor of a is a number. 3. zero is not the successor of a number. 4. Two numbers of which the successors are equal are themselves equal. 5. (induction axiom.) If a set S of numbers contains zero and also the successor of every number in S, then every number is in S. Peano's axioms are the basis for the version of number theory known as Peano arithmetic. 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method="get" action="/search/" name="search" id="search-mobile" accept-charset="UTF-8"> <input type="text" name="query" placeholder="Search" id="searchFieldMobile"> <img src="/images/header/search-icon.png" width="18" height="18" id="mobile-search" class="search-btn" title="Search" alt="Search" onclick="submitForm()"> </form> <nav class="topics-nav"> <a href="/topics/Algebra.html" id="sidebar-algebra"> Algebra </a> <a href="/topics/AppliedMathematics.html" id="sidebar-appliedmathematics"> Applied Mathematics </a> <a href="/topics/CalculusandAnalysis.html" id="sidebar-calculusandanalysis"> Calculus and Analysis </a> <a href="/topics/DiscreteMathematics.html" id="sidebar-discretemathematics"> Discrete Mathematics </a> <a href="/topics/FoundationsofMathematics.html" id="sidebar-foundationsofmathematics"> Foundations of Mathematics </a> <a href="/topics/Geometry.html" id="sidebar-geometry"> Geometry </a> <a href="/topics/HistoryandTerminology.html" id="sidebar-historyandterminology"> History and Terminology </a> <a href="/topics/NumberTheory.html" id="sidebar-numbertheory"> Number Theory </a> <a href="/topics/ProbabilityandStatistics.html" id="sidebar-probabilityandstatistics"> Probability and Statistics </a> <a href="/topics/RecreationalMathematics.html" id="sidebar-recreationalmathematics"> Recreational Mathematics </a> <a href="/topics/Topology.html" id="sidebar-topology"> Topology </a> </nav> <nav class="secondary-nav"> <a href="/letters/"> Alphabetical Index </a> <a href="/whatsnew/"> New in MathWorld </a> </nav> </section> <section id="content">  <nav class="breadcrumbs"><ul class="breadcrumb"> <li> <a href="/topics/FoundationsofMathematics.html">Foundations of Mathematics</a> </li> <li> <a href="/topics/Axioms.html">Axioms</a> </li> </ul><ul class="breadcrumb"> <li> <a href="/topics/FoundationsofMathematics.html">Foundations of Mathematics</a> </li> <li> <a href="/topics/ANewKindofScience.html">A New Kind of Science</a> </li> </ul></nav>   <h1>Peano's Axioms</h1>  <hr class="margin-t-1-8 margin-b-3-4">   <div class="entry-content"> <p> 1. Zero is a number. </p> <p> 2. If <img src="/images/equations/PeanosAxioms/Inline1.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="9" height="21" alt="a" /> is a number, the successor of <img src="/images/equations/PeanosAxioms/Inline2.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="9" height="21" alt="a" /> is a number. </p> <p> 3. <a href="/Zero.html">zero</a> is not the successor of a number. </p> <p> 4. Two numbers of which the successors are equal are themselves equal. </p> <p> 5. (<a href="/InductionAxiom.html">induction axiom</a>.) If a set <img src="/images/equations/PeanosAxioms/Inline3.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="10" height="21" alt="S" /> of numbers contains <a href="/Zero.html">zero</a> and also the successor of every number in <img src="/images/equations/PeanosAxioms/Inline4.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="10" height="21" alt="S" />, then every number is in <img src="/images/equations/PeanosAxioms/Inline5.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="10" height="21" alt="S" />. </p> <p> Peano's axioms are the basis for the version of <a href="/NumberTheory.html">number theory</a> known as <a href="/PeanoArithmetic.html">Peano arithmetic</a>. </p> </div>  <hr class="margin-b-1-1-4"> <div class="c-777 entry-secondary-content">  <h2>See also</h2><a href="/InductionAxiom.html">Induction Axiom</a>, <a href="/PeanoArithmetic.html">Peano Arithmetic</a>       <h2>Explore with Wolfram|Alpha</h2> <div id="WAwidget"> <div class="WAwidget-wrapper"> <img alt="WolframAlpha" title="WolframAlpha" src="/images/wolframalpha/WA-logo.png" width="136" height="20"> <form name="wolframalpha" action="https://www.wolframalpha.com/input/" target="_blank"> <input type="text" name="i" class="search" placeholder="Solve your math problems and get step-by-step solutions" value=""> <button type="submit" title="Evaluate on WolframAlpha"></button> </form> </div> <div class="WAwidget-wrapper try"> <p class="text-align-r"> More things to try: </p> <ul> <li> <a target="_blank" href="http://www.wolframalpha.com/input/?i=axioms"> axioms </a> </li> <li> <a target="_blank" href="http://www.wolframalpha.com/input/?i=axiom"> axiom </a> </li> <li><a target="_blank" href="https://www.wolframalpha.com/input/?i=32+coin+tosses">32 coin tosses</a></li> </ul> </div> </div>   <h2>References</h2><cite>Wolfram, S. <i><a href="http://www.amazon.com/exec/obidos/ASIN/1579550088/ref=nosim/ericstreasuretro">A New Kind of Science.</a></i> Champaign, IL: Wolfram Media, p. <a href="http://www.wolframscience.com/nksonline/page-1152a-text">1152</a>, 2002.</cite><h2>Referenced on Wolfram|Alpha</h2><a href="http://www.wolframalpha.com/entities/mathworld/peano%E2%80%99s_axioms/pp/lu/9f/" title="Peano's Axioms" target="_blank">Peano's Axioms</a>   <h2>Cite this as:</h2> <p> <a href="/about/author.html">Weisstein, Eric W.</a> "Peano's Axioms." From <a href="/"><i>MathWorld</i></a>--A Wolfram Web Resource. <a href="https://mathworld.wolfram.com/PeanosAxioms.html">https://mathworld.wolfram.com/PeanosAxioms.html</a> </p>  <h2>Subject classifications</h2><nav class="breadcrumbs"><ul class="breadcrumb"> <li> <a href="/topics/FoundationsofMathematics.html">Foundations of Mathematics</a> </li> <li> <a href="/topics/Axioms.html">Axioms</a> </li> </ul><ul class="breadcrumb"> <li> <a href="/topics/FoundationsofMathematics.html">Foundations of Mathematics</a> </li> <li> <a href="/topics/ANewKindofScience.html">A New Kind of Science</a> </li> </ul></nav>  </div> </section> </section>  </div> </main> <aside id="bottom"> <style> #bottom { padding-bottom: 65px; } #acknowledgment { display:none; } .attribution { font-size: .75rem; font-style: italic; } footer ul li:not(:last-of-type)::after { background: #a3a3a3; margin-left: .3rem; margin-right: .1rem; } @media all and (max-width: 900px) { .attribution { font-size: 12px; } } @media (max-width: 600px) { footer { max-width: 360px; } footer ul { max-width: 360px; } footer ul:nth-child(1) li:nth-child(2):after { content: ""; height: 11px; } footer ul:nth-child(1) li:nth-child(3):after { content: ""; height: 0px; } } </style> <footer> <ul> <li><a href="/about/">About MathWorld</a></li> <li><a href="/classroom/">MathWorld Classroom</a></li> <li><a href="/contact/">Contribute</a></li> <li><a href="https://www.amazon.com/exec/obidos/ASIN/1420072218/ref=nosim/weisstein-20" target="_blank">MathWorld Book</a></li> <li class="display-n display-ib__600"><a href="https://www.wolfram.com" target="_blank">wolfram.com</a></li> </ul> <ul> <li class="display-n__600"><a href="/whatsnew/">13,247 Entries</a></li> <li class="display-n__600"><a href="/whatsnew/">Last Updated: Thu Feb 27 2025</a></li>  <li><a href="https://www.wolfram.com" target="_blank">©1999–2025 Wolfram Research, Inc.</a></li> <li><a href="https://www.wolfram.com/legal/terms/mathworld.html" target="_blank">Terms of Use</a></li> </ul> <ul class="wolfram"> <li class="display-n__600 display-n__900"><a href="https://www.wolfram.com" target="_blank" aria-label="Wolfram"><img src="/images/footer/wolfram-logo.png" alt="Wolfram" title="Wolfram" width="121" height="28"></a></li> <li class="display-n__600"><a href="https://www.wolfram.com" target="_blank">wolfram.com</a></li> <li class="display-n__600"><a href="https://www.wolfram.com/education/" target="_blank">Wolfram for Education</a></li> <li class="attribution">Created, developed and nurtured by Eric Weisstein at Wolfram Research</li> </ul> </footer> <section id="acknowledgment"> <i>Created, developed and nurtured by Eric Weisstein at Wolfram Research</i> </section> </aside> <script type="text/javascript" src="/scripts/scripts.js"></script> <script src="/common/js/c2c/1.0/WolframC2C.js"></script> <script src="/common/js/c2c/1.0/WolframC2CGui.js"></script> <script src="/common/js/c2c/1.0/WolframC2CDefault.js"></script> <link rel="stylesheet" href="/common/js/c2c/1.0/WolframC2CGui.css.en"> <style> .wolfram-c2c-wrapper { padding: 0px !important; border: 0px; } .wolfram-c2c-wrapper:active { border: 0px; } .wolfram-c2c-wrapper:hover { border: 0px; } </style> <script> let c2cWrittings = new WolframC2CDefault({'triggerClass':'mathworld-c2c_above', 'uniqueIdPrefix': 'mathworld-c2c_above-'}); </script> <style> #IPstripe-outer { background: #47a2af; } #IPstripe-outer:hover { background: #0095aa; } </style> <div id="IPstripe-wrap"></div> <script src="/common/stripe/stripe.en.js"></script> </body> </html>