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<!doctype html> <html lang="en" class="foundationsofmathematics"> <head> <title>Set Theory -- from Wolfram MathWorld</title> <meta name="DC.Title" content="Set Theory" /> <meta name="DC.Creator" content="Weisstein, Eric W." /> <meta name="DC.Description" content="Set theory is the mathematical theory of sets. Set theory is closely associated with the branch of mathematics known as logic. There are a number of different versions of set theory, each with its own rules and axioms. In order of increasing consistency strength, several versions of set theory include Peano arithmetic (ordinary algebra), second-order arithmetic (analysis), Zermelo-Fraenkel set theory, Mahlo, weakly compact, hyper-Mahlo, ineffable, measurable, Ramsey, supercompact, huge, and..." /> <meta name="description" content="Set theory is the mathematical theory of sets. Set theory is closely associated with the branch of mathematics known as logic. 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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/SetTheory.html">Set Theory</a> </li> <li> <a href="/topics/GeneralSetTheory.html">General Set Theory</a> </li> </ul></nav>   <h1>Set Theory</h1>  <hr class="margin-t-1-8 margin-b-3-4">   <div class="entry-content"> <p> Set theory is the mathematical theory of <a href="/Set.html">sets</a>. Set theory is closely associated with the branch of mathematics known as <a href="/Logic.html">logic</a>. </p> <p> There are a number of different versions of set theory, each with its own rules and <a href="/Axiom.html">axioms</a>. In order of increasing <a href="/ConsistencyStrength.html">consistency strength</a>, several versions of set theory include <a href="/PeanoArithmetic.html">Peano arithmetic</a> (ordinary <a href="/Algebra.html">algebra</a>), second-order arithmetic (<a href="/Analysis.html">analysis</a>), <a href="/Zermelo-FraenkelSetTheory.html">Zermelo-Fraenkel set theory</a>, Mahlo, weakly compact, hyper-Mahlo, ineffable, measurable, Ramsey, supercompact, huge, and <img src="/images/equations/SetTheory/Inline1.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="9" height="21" alt="n" />-huge set theory. </p> </div>  <hr class="margin-b-1-1-4"> <div class="c-777 entry-secondary-content">  <h2>See also</h2><a href="/AbstractAlgebra.html">Abstract Algebra</a>, <a href="/Analysis.html">Analysis</a>, <a href="/AxiomaticSetTheory.html">Axiomatic Set Theory</a>, <a href="/ConsistencyStrength.html">Consistency Strength</a>, <a href="/ContinuumHypothesis.html">Continuum Hypothesis</a>, <a href="/DescriptiveSetTheory.html">Descriptive Set Theory</a>, <a href="/Impredicative.html">Impredicative</a>, <a href="/KuratowskisClosure-ComplementProblem.html">Kuratowski's Closure-Complement Problem</a>, <a href="/NaiveSetTheory.html">Naive Set Theory</a>, <a href="/PeanoArithmetic.html">Peano Arithmetic</a>, <a href="/Sentence.html">Sentence</a>, <a href="/Set.html">Set</a>, <a href="/Theory.html">Theory</a>, <a href="/Zermelo-FraenkelAxioms.html">Zermelo-Fraenkel Axioms</a>, <a href="/Zermelo-FraenkelSetTheory.html">Zermelo-Fraenkel Set Theory</a>, <a href="/ZermeloSetTheory.html">Zermelo Set Theory</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=set+theory"> set theory </a> </li> <li><a target="_blank" href="https://www.wolframalpha.com/input/?i=2009%21">2009!</a></li> <li><a target="_blank" href="https://www.wolframalpha.com/input/?i=foci+of+hyperbola+with+semiaxes+3%2C4">foci of hyperbola with semiaxes 3,4</a></li> </ul> </div> </div>   <h2>References</h2><cite>Courant, R. and Robbins, H. "The Algebra of Sets." Supplement to Ch. 2 in <i><a href="http://www.amazon.com/exec/obidos/ASIN/0195105192/ref=nosim/ericstreasuretro">What Is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed.</a></i> Oxford, England: Oxford University Press, pp. 108-116, 1996.</cite><cite>Devlin, K. <i><a href="http://www.amazon.com/exec/obidos/ASIN/0387940944/ref=nosim/ericstreasuretro">The Joy of Sets: Fundamentals of Contemporary Set Theory, 2nd ed.</a></i> New York: Springer-Verlag, 1993.</cite><cite>Ferreirós, J. <i><a href="http://www.amazon.com/exec/obidos/ASIN/0817657495/ref=nosim/ericstreasuretro">Labyrinth of Thought: A History of Set Theory and Its Role in Modern Mathematics.</a></i> Basel, Switzerland: Birkhäuser, 1999.</cite><cite>Halmos, P. R. <i><a href="http://www.amazon.com/exec/obidos/ASIN/0387900926/ref=nosim/ericstreasuretro">Naive Set Theory.</a></i> New York: Springer-Verlag, 1974.</cite><cite>MacTutor History of Mathematics Archive. "The Beginnings of Set Theory." <a href="http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Beginnings_of_set_theory.html">http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Beginnings_of_set_theory.html</a>.</cite><cite>MathPages. "Set Theory and Foundations." <a href="http://www.mathpages.com/home/ifoundat.htm">http://www.mathpages.com/home/ifoundat.htm</a>.</cite><cite>Stewart, I. <i><a href="http://www.amazon.com/exec/obidos/ASIN/0192832026/ref=nosim/ericstreasuretro">The Problems of Mathematics, 2nd ed.</a></i> Oxford: Oxford University Press, p. 96, 1987.</cite><cite>Weisstein, E. W. "Books about Set Theory." <a href="http://www.ericweisstein.com/encyclopedias/books/SetTheory.html">http://www.ericweisstein.com/encyclopedias/books/SetTheory.html</a>.</cite><h2>Referenced on Wolfram|Alpha</h2><a href="http://www.wolframalpha.com/entities/mathworld/set_theory/38/ni/lm/" title="Set Theory" target="_blank">Set Theory</a>   <h2>Cite this as:</h2> <p> <a href="/about/author.html">Weisstein, Eric W.</a> "Set Theory." From <a href="/"><i>MathWorld</i></a>--A Wolfram Web Resource. <a href="https://mathworld.wolfram.com/SetTheory.html">https://mathworld.wolfram.com/SetTheory.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/SetTheory.html">Set Theory</a> </li> <li> <a href="/topics/GeneralSetTheory.html">General Set Theory</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/">13,209 Entries</a></li> <li class="display-n__600"><a href="/whatsnew/">Last Updated: Tue Nov 26 2024</a></li>  <li><a href="https://www.wolfram.com" target="_blank">©1999–2024 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>