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In formal set theory, a cardinal number (also called "the cardinality") is a type of number defined in such a way that any method of counting sets using it gives the same result. (This is not true for the ordinal numbers.) In fact, the cardinal numbers are obtained by collecting all ordinal numbers which are obtainable by counting a given set. A set has aleph_0 (aleph-0)..." /> <meta name="DC.Subject" scheme="MathWorld" content="Mathematics:Foundations of Mathematics:Set Theory:Cardinal Numbers" /> <meta name="DC.Subject" scheme="MathWorld" content="Mathematics:History and Terminology:Disciplinary Terminology:Biological Terminology" /> <meta name="DC.Subject" scheme="MathWorld" content="Mathematics:History and Terminology:Disciplinary Terminology:Religious Terminology" /> <meta name="DC.Subject" scheme="MSC_2000" content="03E10" /> <meta name="DC.Rights" content="Copyright 1999-2024 Wolfram Research, Inc. 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In formal set theory, a cardinal number (also called "the cardinality") is a type of number defined in such a way that any method of counting sets using it gives the same result. (This is not true for the ordinal numbers.) In fact, the cardinal numbers are obtained by collecting all ordinal numbers which are obtainable by counting a given set. A set has aleph_0 (aleph-0)..."> <meta name="twitter:card" content="summary_large_image"> <meta name="twitter:site" content="@WolframResearch"> <meta name="twitter:title" content="Cardinal Number -- from Wolfram MathWorld"> <meta name="twitter:description" content="In common usage, a cardinal number is a number used in counting (a counting number), such as 1, 2, 3, .... In formal set theory, a cardinal number (also called "the cardinality") is a type of number defined in such a way that any method of counting sets using it gives the same result. (This is not true for the ordinal numbers.) In fact, the cardinal numbers are obtained by collecting all ordinal numbers which are obtainable by counting a given set. 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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/CardinalNumbers.html">Cardinal Numbers</a> </li> </ul><ul class="breadcrumb"> <li> <a href="/topics/HistoryandTerminology.html">History and Terminology</a> </li> <li> <a href="/topics/DisciplinaryTerminology.html">Disciplinary Terminology</a> </li> <li> <a href="/topics/BiologicalTerminology.html">Biological Terminology</a> </li> </ul><ul class="breadcrumb"> <li> <a href="/topics/HistoryandTerminology.html">History and Terminology</a> </li> <li> <a href="/topics/DisciplinaryTerminology.html">Disciplinary Terminology</a> </li> <li> <a href="/topics/ReligiousTerminology.html">Religious Terminology</a> </li> </ul></nav>   <h1>Cardinal Number</h1>  <hr class="margin-t-1-8 margin-b-3-4">   <div class="entry-content"> <p> In common usage, a cardinal number is a number used in counting (a <a href="/CountingNumber.html">counting number</a>), such as 1, 2, 3, .... </p> <p> In formal <a href="/SetTheory.html">set theory</a>, a cardinal number (also called "the cardinality") is a type of number defined in such a way that any method of counting <a href="/Set.html">sets</a> using it gives the same result. (This is not true for the <a href="/OrdinalNumber.html">ordinal numbers</a>.) In fact, the cardinal numbers are obtained by collecting all <a href="/OrdinalNumber.html">ordinal numbers</a> which are obtainable by counting a given set. A set has <img src="/images/equations/CardinalNumber/Inline1.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="18" height="22" alt="aleph_0" /> (<a href="/Aleph-0.html">aleph-0</a>) members if it can be put into a <a href="/One-to-OneCorrespondence.html">one-to-one correspondence</a> with the finite <a href="/OrdinalNumber.html">ordinal numbers</a>. The cardinality of a set is also frequently referred to as the "power" of a set (Moore 1982, Dauben 1990, Suppes 1972). </p> <p> In Georg Cantor's original notation, the symbol for a <a href="/Set.html">set</a> <img src="/images/equations/CardinalNumber/Inline2.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="12" height="21" alt="A" /> annotated with a single overbar <img src="/images/equations/CardinalNumber/Inline3.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="12" height="21" alt="A^_" /> indicated <img src="/images/equations/CardinalNumber/Inline4.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="12" height="21" alt="A" /> stripped of any structure besides order, hence it represented the <a href="/OrderType.html">order type</a> of the set. A double overbar <img src="/images/equations/CardinalNumber/Inline5.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="12" height="23" alt="A^_^_" /> then indicated stripping the order from the set and thus indicated the cardinal number of the set. However, in modern notation, the symbol <img src="/images/equations/CardinalNumber/Inline6.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="20" height="21" alt="|A|" /> is used to denote the cardinal number of set. </p> <p> Cantor, the father of modern <a href="/SetTheory.html">set theory</a>, noticed that while the <a href="/OrdinalNumber.html">ordinal numbers</a> <img src="/images/equations/CardinalNumber/Inline7.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="40" height="21" alt="omega+1" />, <img src="/images/equations/CardinalNumber/Inline8.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="40" height="21" alt="omega+2" />, ... were bigger than omega in the sense of order, they were not bigger in the sense of <a href="/Equipollent.html">equipollence</a>. This led him to study what would come to be called cardinal numbers. He called the ordinals <img src="/images/equations/CardinalNumber/Inline9.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="12" height="21" alt="omega" />, <img src="/images/equations/CardinalNumber/Inline10.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="40" height="21" alt="omega+1" />, ... that are equipollent to the integers "the second number class" (as opposed to the finite ordinals, which he called the "first number class"). Cantor showed </p> <p> 1. The second number class is bigger than the first. </p> <p> 2. There is no class bigger than the first number class and smaller than the second. </p> <p> 3. The class of real numbers is bigger than the first number class. </p> <p> One of the first serious mathematical definitions of cardinal was the one devised by Gottlob Frege and Bertrand Russell, who defined a cardinal number <img src="/images/equations/CardinalNumber/Inline11.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="20" height="21" alt="|A|" /> as the set of all sets <a href="/Equipollent.html">equipollent</a> to <img src="/images/equations/CardinalNumber/Inline12.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="12" height="21" alt="A" />. (Moore 1982, p. 153; Suppes 1972, p. 109). Unfortunately, the objects produced by this definition are not sets in the sense of <a href="/Zermelo-FraenkelSetTheory.html">Zermelo-Fraenkel set theory</a>, but rather "<a href="/ProperClass.html">proper classes</a>" in the terminology of von Neumann. </p> <p> Tarski (1924) proposed to instead define a cardinal number by stating that every set <img src="/images/equations/CardinalNumber/Inline13.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="12" height="21" alt="A" /> is associated with a cardinal number <img src="/images/equations/CardinalNumber/Inline14.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="20" height="21" alt="|A|" />, and two sets <img src="/images/equations/CardinalNumber/Inline15.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="12" height="21" alt="A" /> and <img src="/images/equations/CardinalNumber/Inline16.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="11" height="21" alt="B" /> have the same cardinal number <a href="/Iff.html">iff</a> they are <a href="/Equipollent.html">equipollent</a> (Moore 1982, pp. 52 and 214; Rubin 1967, p. 266; Suppes 1972, p. 111). The problem is that this definition requires a special axiom to guarantee that cardinals exist. </p> <p> A. P. Morse and Dana Scott defined cardinal number by letting <img src="/images/equations/CardinalNumber/Inline17.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="12" height="21" alt="A" /> be any set, then calling <img src="/images/equations/CardinalNumber/Inline18.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="20" height="21" alt="|A|" /> the set of all sets <a href="/Equipollent.html">equipollent</a> to <img src="/images/equations/CardinalNumber/Inline19.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="12" height="21" alt="A" /> and of least possible <a href="/Rank.html">rank</a> (Rubin 1967, p. 270). </p> <p> It is possible to associate cardinality with a specific set, but the process required either the <a href="/AxiomofFoundation.html">axiom of foundation</a> or the <a href="/AxiomofChoice.html">axiom of choice</a>. However, these are two of the more controversial <a href="/Zermelo-FraenkelAxioms.html">Zermelo-Fraenkel axioms</a>. With the <a href="/AxiomofChoice.html">axiom of choice</a>, the cardinals can be enumerated through the ordinals. In fact, the two can be put into <a href="/One-to-OneCorrespondence.html">one-to-one correspondence</a>. The <a href="/AxiomofChoice.html">axiom of choice</a> implies that every set can be <a href="/WellOrderedSet.html">well ordered</a> and can therefore be associated with an <a href="/OrdinalNumber.html">ordinal number</a>. </p> <p> This leads to the definition of cardinal number for a <a href="/Set.html">set</a> <img src="/images/equations/CardinalNumber/Inline20.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="12" height="21" alt="A" /> as the least <a href="/OrdinalNumber.html">ordinal number</a> <img src="/images/equations/CardinalNumber/Inline21.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="9" height="21" alt="b" /> such that <img src="/images/equations/CardinalNumber/Inline22.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="12" height="21" alt="A" /> and <img src="/images/equations/CardinalNumber/Inline23.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="9" height="21" alt="b" /> are <a href="/Equipollent.html">equipollent</a>. In this model, the cardinal numbers are just the <a href="/InitialOrdinal.html">initial ordinals</a>. This definition obviously depends on the <a href="/AxiomofChoice.html">axiom of choice</a>, because if the <a href="/AxiomofChoice.html">axiom of choice</a> is not true, then there are sets that cannot be well ordered. Cantor believed that every set could be well ordered and used this correspondence to define the <img src="/images/equations/CardinalNumber/Inline24.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="11" height="21" alt="aleph" />s ("alephs"). For any <a href="/OrdinalNumber.html">ordinal number</a> <img src="/images/equations/CardinalNumber/Inline25.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="10" height="21" alt="alpha" />, <img src="/images/equations/CardinalNumber/Inline26.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="60" height="21" alt="aleph_alpha=omega_alpha" />. </p> <p> An <a href="/InaccessibleCardinal.html">inaccessible cardinal</a> cannot be expressed in terms of a smaller number of smaller cardinals. </p> </div>  <hr class="margin-b-1-1-4"> <div class="c-777 entry-secondary-content">  <h2>See also</h2><a href="/Aleph.html">Aleph</a>, <a href="/Aleph-0.html">Aleph-0</a>, <a href="/Aleph-1.html">Aleph-1</a>, <a href="/Cantor-DedekindAxiom.html">Cantor-Dedekind Axiom</a>, <a href="/CantorDiagonalMethod.html">Cantor Diagonal Method</a>, <a href="/CardinalAddition.html">Cardinal Addition</a>, <a href="/CardinalExponentiation.html">Cardinal Exponentiation</a>, <a href="/CardinalMultiplication.html">Cardinal Multiplication</a>, <a href="/Continuum.html">Continuum</a>, <a href="/ContinuumHypothesis.html">Continuum Hypothesis</a>, <a href="/Equipollent.html">Equipollent</a>, <a href="/InaccessibleCardinal.html">Inaccessible Cardinal</a>, <a href="/Infinity.html">Infinity</a>, <a href="/OrdinalNumber.html">Ordinal Number</a>, <a href="/PowerSet.html">Power Set</a>, <a href="/SurrealNumber.html">Surreal Number</a>, <a href="/UncountablyInfinite.html">Uncountably Infinite</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=aleph-0"> aleph-0 </a> </li> <li> <a target="_blank" href="http://www.wolframalpha.com/input/?i=cardinality+of+the+integers"> cardinality of the integers </a> </li> <li><a target="_blank" href="https://www.wolframalpha.com/input/?i=165+million">165 million</a></li> </ul> </div> </div>   <h2>References</h2><cite>Cantor, G. <i>Über unendliche, lineare Punktmannigfaltigkeiten, Arbeiten zur Mengenlehre aus dem Jahren 1872-1884.</i> Leipzig, Germany: Teubner, 1884.</cite><cite>Conway, J. H. and Guy, R. K. "Cardinal Numbers." In <i><a href="http://www.amazon.com/exec/obidos/ASIN/038797993X/ref=nosim/ericstreasuretro">The Book of Numbers.</a></i> New York: Springer-Verlag, pp. 277-282, 1996.</cite><cite>Courant, R. and Robbins, H. "Cantor's 'Cardinal Numbers.' " §2.4.3 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. 83-86, 1996.</cite><cite>Dauben, J. W. <i><a href="http://www.amazon.com/exec/obidos/ASIN/0691024472/ref=nosim/ericstreasuretro">Georg Cantor: His Mathematics and Philosophy of the Infinite.</a></i> Princeton, NJ: Princeton University Press, 1990.</cite><cite>Ferreirós, J. "The Notion of Cardinality and the Continuum Hypothesis." Ch. 6 in <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, pp. 171-214, 1999.</cite><cite>Moore, G. H. <i><a href="http://www.amazon.com/exec/obidos/ASIN/0387906703/ref=nosim/ericstreasuretro">Zermelo's Axiom of Choice: Its Origin, Development, and Influence.</a></i> New York: Springer-Verlag, 1982.</cite><cite>Rubin, J. E. <i><a href="http://www.amazon.com/exec/obidos/ASIN/B0006BQH7S/ref=nosim/ericstreasuretro">Set Theory for the Mathematician.</a></i> New York: Holden-Day, 1967.</cite><cite>Suppes, P. <i><a href="http://www.amazon.com/exec/obidos/ASIN/0486616304/ref=nosim/ericstreasuretro">Axiomatic Set Theory.</a></i> New York: Dover, 1972.</cite><cite>Tarski, A. "Sur quelques théorèmes qui équivalent à l'axiome du choix." <i>Fund. Math.</i> <b>5</b>, 147-154, 1924.</cite><h2>Referenced on Wolfram|Alpha</h2><a href="http://www.wolframalpha.com/entities/mathworld/cardinal_number/va/9f/37/" title="Cardinal Number" target="_blank">Cardinal Number</a>   <h2>Cite this as:</h2> <p> <a href="/about/author.html">Weisstein, Eric W.</a> "Cardinal Number." From <a href="/"><i>MathWorld</i></a>--A Wolfram Web Resource. <a href="https://mathworld.wolfram.com/CardinalNumber.html">https://mathworld.wolfram.com/CardinalNumber.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/CardinalNumbers.html">Cardinal Numbers</a> </li> </ul><ul class="breadcrumb"> <li> <a href="/topics/HistoryandTerminology.html">History and Terminology</a> </li> <li> <a href="/topics/DisciplinaryTerminology.html">Disciplinary Terminology</a> </li> <li> <a href="/topics/BiologicalTerminology.html">Biological Terminology</a> </li> </ul><ul class="breadcrumb"> <li> <a href="/topics/HistoryandTerminology.html">History and Terminology</a> </li> <li> <a href="/topics/DisciplinaryTerminology.html">Disciplinary Terminology</a> </li> <li> <a href="/topics/ReligiousTerminology.html">Religious Terminology</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>