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''transfinite number, power in the sense of Cantor, cardinality of a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020370/c0203701.png" />''
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{{MSC|03E10}}
  
That property of the set which is intrinsic to any set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020370/c0203702.png" /> with the same cardinality as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020370/c0203703.png" />. In this connection, two sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020370/c0203704.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020370/c0203705.png" /> are said to have the same cardinality (or to be equivalent) if there is a one-to-one onto function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020370/c0203706.png" /> with domain of definition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020370/c0203707.png" /> and set of values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020370/c0203708.png" />. G. Cantor defined the cardinal number of a set as that property of it which remains after abstracting the qualitative nature of its elements and their ordering. By way of stressing the double act of abstraction, Cantor introduced the symbol <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020370/c0203709.png" /> to denote the cardinal number of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020370/c02037010.png" />. The most commonly used from among the various notations for a cardinal number are the symbols <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020370/c02037011.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020370/c02037012.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020370/c02037013.png" /> is a finite set containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020370/c02037014.png" /> elements, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020370/c02037015.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020370/c02037016.png" /> denotes the set of natural numbers, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020370/c02037017.png" /> (see [[Aleph|Aleph]]). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020370/c02037018.png" /> denotes the set of real numbers, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020370/c02037019.png" />, the power of the continuum. The set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020370/c02037020.png" /> of all subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020370/c02037021.png" /> is not equivalent to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020370/c02037022.png" /> or to any subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020370/c02037023.png" /> (Cantor's theorem). In particular, no two of the sets
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''transfinite number, power in the sense of Cantor, cardinality of a set $ A $
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020370/c02037024.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
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That property of $ A $ that is intrinsic to any set $ B $ with the same cardinality as $ A $. In this connection, two sets $ A $ and $ B $ are said to have the '''same cardinality''' (or to be '''equivalent''') if and only if there is a bijective function $ f: A \to B $ with domain $ A $ and range $ B $. G. Cantor defined the cardinal number of a set as that property of it that remains after abstracting the qualitative nature of its elements and their ordering. By way of stressing the double act of abstraction, Cantor introduced the symbol $ \overline{\overline{A}} $ to denote the cardinal number of $ A $. The most commonly used from among the various notations for a cardinal number are the symbols $ \mathsf{card}(A) $ and $ A $. If $ A $ is a finite set containing $ n $ elements, then $ \mathsf{card}(A) = n $. If $ \mathbf{Z}^{+} $ denotes the set of natural numbers, then $ \mathsf{card}(\mathbf{Z}^{+}) = \aleph_{0} $ (see [[Aleph|Aleph]]). If $ \mathbf{R} $ denotes the set of real numbers, then $ \mathsf{card}(\mathbf{R}) = \mathfrak{c} $, the power of the continuum. The set $ 2^{A} $ of all subsets of $ A $ is not equivalent to $ A $ or to any subset of $ A $ ('''Cantor’s Theorem'''). In particular, no two members of the sequence
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$$
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A, ~ 2^{A}, ~ 2^{2^{A}}, ~ 2^{2^{2^{A}}}, ~ \ldots \qquad (1)
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$$
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are equivalent. When $ A = \mathbf{Z}^{+} $, the above sequence gives rise to infinitely many distinct infinite cardinal numbers. Further cardinal numbers are obtained by taking the union $ Q $ of the sets in (1) and constructing the analogous sequence, setting $ A = Q $. This process can be continued infinitely often. The scale (class) of all infinite cardinal numbers is much richer than the scale (class) of finite cardinals. Furthermore, there are so many of them that it is not possible to form a set containing at least one of each cardinal number.
  
are equivalent. When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020370/c02037025.png" />, the above sequence gives rise to infinitely many distinct infinite cardinal numbers. Further cardinal numbers are obtained by taking the union <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020370/c02037026.png" /> of the sets in (1) and constructing the analogous sequence, setting <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020370/c02037027.png" />. This process can be continued infinitely often. The scale (class) of all infinite cardinal numbers is much richer than the scale (class) of finite cardinals. Furthermore, there are so many of them that it is not possible to form a set containing at least one of each cardinal number.
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One can define for cardinal numbers the operations of addition, multiplication, raising to a power, as well as taking the logarithm and extracting a root.
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* The cardinal number $ \delta $ is the '''sum''' of $ \alpha $ and $ \beta $, written as $ \delta = \alpha + \beta $, if and only if every set of cardinality $ \delta $ is equivalent to the [[disjoint union]] $ A \sqcup B $ of sets $ A $ and $ B $, where $ \mathsf{card}(A) = \alpha $ and $ \mathsf{card}(B) = \beta $.
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* The cardinal number $ \gamma $ is the '''product''' of $ \alpha $ and $ \beta $, written as $ \gamma = \alpha \beta $, if and only if every set of cardinality $ \gamma $ is equivalent to the Cartesian product $ A \times B $ of sets $ A $ and $ B $, where $ \mathsf{card}(A) = \alpha $ and $ \mathsf{card}(B) = \beta $.
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* Addition and multiplication of cardinal numbers is commutative and associative, and multiplication is distributive with respect to addition.
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* The cardinal number $ \kappa $ is the '''power''' with base $ \alpha $ and exponent $ \beta $, written as $ \kappa = \alpha^{\beta} $, if and only if every set of cardinality $ \kappa $ is equivalent to the set $ A^{B} $ of all functions $ f: B \to A $, where $ \mathsf{card}(A) = \alpha $ and $ \mathsf{card}(B) = \beta $.
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* The cardinal number $ \alpha $ is said to be '''smaller than or equal to''' the cardinal number $ \beta $, written as $ \alpha \leq \beta $, if and only if every set of cardinality $ \alpha $ is equivalent to some subset of a set of cardinality $ \beta $.
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* If $ \alpha \leq \beta $ and $ \beta \leq \alpha $, then $ \alpha = \beta $ (the so-called '''[[Cantor–Bernstein theorem]]'''), so that the scale of cardinal numbers is totally ordered.
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* For each cardinal number $ \beta $, the set $ \{ \alpha \mid \alpha < \beta \} $ is totally ordered, which enables one to define the '''logarithm''' $ {\log_{\alpha}}(\beta) $ of $ \beta $ to the base $ \alpha $, with $ \alpha \leq \beta $, as the smallest cardinal number $ \gamma $ such that $ \alpha^{\gamma} \geq \beta $.
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* Similarly, the $ \alpha $-th '''root''' $ \beta^{1 / \alpha} $ of the cardinal number $ \beta $ is defined as the smallest cardinal number $ \delta $ such that $ \delta^{\alpha} \geq \beta $.
  
One can define for cardinal numbers the operations of addition, multiplication, raising to a power, as well as taking the logarithm and extracting a root. Thus, the cardinal number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020370/c02037028.png" /> is the sum of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020370/c02037029.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020370/c02037030.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020370/c02037031.png" />, if each set of cardinality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020370/c02037032.png" /> can be represented as a disjoint union of sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020370/c02037033.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020370/c02037034.png" /> of cardinalities <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020370/c02037035.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020370/c02037036.png" />, respectively; the cardinal number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020370/c02037037.png" /> is the product of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020370/c02037038.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020370/c02037039.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020370/c02037040.png" />, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020370/c02037041.png" /> is the cardinal number of the Cartesian product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020370/c02037042.png" /> where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020370/c02037043.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020370/c02037044.png" />. Addition and multiplication of cardinal numbers is commutative and associative, and multiplication is distributive with respect to addition. The cardinal number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020370/c02037045.png" /> is the power with base <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020370/c02037046.png" /> and exponent <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020370/c02037047.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020370/c02037048.png" />, if every set of cardinality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020370/c02037049.png" /> is equivalent to the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020370/c02037050.png" /> of all functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020370/c02037051.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020370/c02037052.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020370/c02037053.png" />. The cardinal number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020370/c02037054.png" /> is said to be smaller or equal to the cardinal number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020370/c02037055.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020370/c02037056.png" />, if every set of cardinality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020370/c02037057.png" /> is equivalent to some subset of a set of cardinality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020370/c02037058.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020370/c02037059.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020370/c02037060.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020370/c02037061.png" /> (the Cantor–Bernstein theorem), so that the scale of cardinal numbers is totally ordered. Furthermore, for each cardinal number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020370/c02037062.png" /> the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020370/c02037063.png" /> is totally well-ordered, which enables one to define the logarithm <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020370/c02037064.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020370/c02037065.png" /> to the base <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020370/c02037066.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020370/c02037067.png" />, as the smallest cardinal number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020370/c02037068.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020370/c02037069.png" />; similarly, the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020370/c02037070.png" />-th root <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020370/c02037071.png" /> of the cardinal number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020370/c02037072.png" /> is the smallest cardinal number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020370/c02037073.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020370/c02037074.png" />.
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Any cardinal number $ \alpha $ can be identified with the smallest [[Ordinal number|ordinal number]] of cardinality $ \alpha $. In particular, $ \aleph_{0} $ corresponds to the ordinal number $ \omega_{0} $, $ \aleph_{1} $ to the ordinal $ \omega_{1} $, etc. Thus, the scale of cardinal numbers is a sub-scale of the scale of ordinal numbers. A number of properties of ordinal numbers carry over to cardinal numbers; however, these same properties can also be defined ‘intrinsically’.
  
Any cardinal number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020370/c02037075.png" /> can be identified with the smallest [[Ordinal number|ordinal number]] of cardinality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020370/c02037076.png" />. In particular, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020370/c02037077.png" /> corresponds to the ordinal number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020370/c02037078.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020370/c02037079.png" /> to the ordinal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020370/c02037080.png" />, etc. Thus, the scale of cardinal numbers is a subscale of the scale of ordinal numbers. A number of properties of ordinal numbers carry over to cardinal numbers; however, these same properties can also be defined  "intrinsically" . If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020370/c02037081.png" /> for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020370/c02037082.png" /> and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020370/c02037083.png" />, then
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If $ \alpha_{t} < \beta_{t} $ for every $ t \in T $ and if $ \mathsf{card}(T) \geq \omega_{0} $, then '''König’s Theorem''' states that
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$$
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\sum_{t \in T} \alpha_{t} < \prod_{t \in T} \beta_{t}. \qquad (2)
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$$
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If, in (2), one sets $ T = \mathbf{Z}^{+} $ and $ 1 < \alpha_{n} < \alpha_{n + 1} $, then
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$$
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\alpha_{\omega_{0}} = \alpha_{1} + \alpha_{2} + \cdots < \alpha_{1} \alpha_{2} \cdots. \qquad (3)
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$$
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In particular, for any $ \alpha \geq 2 $, it is impossible to express the power $ \alpha^{\omega_{0}} $ as the sum of an infinite increasing sequence of length $ \omega_{0} $, all terms of which are less than $ \alpha^{\omega_{0}} $.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020370/c02037084.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
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For each cardinal number $ \alpha $, the '''cofinal character''' of $ \alpha $, denoted by $ \mathsf{cf}(\alpha) $, is defined as the smallest cardinal number $ \gamma $ such that $ \alpha $ can be written as $ \displaystyle \sum_{t \in T} \beta_{t} $ for suitable $ \beta_{t} < \alpha $ and $ \mathsf{card}(T) = \gamma $.
  
(König's theorem). If in (2) one sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020370/c02037085.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020370/c02037086.png" />, then
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If $ \mathsf{cf}(\alpha) = \alpha $, then $ \alpha $ is called '''regular''', otherwise it is called '''singular'''. For each cardinal number $ \alpha $, the smallest cardinal number $ \alpha^{+} $ greater than $ \alpha $ is regular (granted the axiom of choice). An example of a singular cardinal number is the cardinal number $ \alpha_{\omega_{0}} $ on the left-hand side of (3) under the condition that $ \omega_{0} < \alpha_{1} $. In this case,
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$$
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\mathsf{cf}(\alpha_{\omega_{0}}) = \omega_{0} < \alpha_{\omega_{0}}.
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$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020370/c02037087.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>
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A cardinal number $ \alpha $ is called a '''limit cardinal number''' if and only if for any $ \beta < \alpha $, there exists a $ \gamma $ such that $ \beta < \gamma < \alpha $. Examples of limit cardinal numbers are $ \omega_{0} $ and $ \alpha_{\omega_{0}} $, while $ \omega_{1} $ is a non-limit cardinal number. A regular limit cardinal number is called '''weakly inaccessible'''. A cardinal number $ \alpha $ is said to be a '''strong limit cardinal''' if and only if for any $ \beta < \alpha $, we have $ 2^{\beta} < \alpha $. A strong regular limit cardinal number is called '''strongly inaccessible'''. It follows from the generalized [[Continuum hypothesis|continuum hypothesis]] ($ \mathsf{GCH} $) that the classes of strongly- (or weakly-) inaccessible cardinal numbers coincide. The classes of inaccessible cardinal numbers can be classified further (the so-called '''Mahlo Scheme'''), which leads to the definition of hyper-inaccessible cardinal numbers. The assertion that strongly- (or weakly-) inaccessible cardinal numbers exist happens to be independent of the usual axioms of axiomatic set theory.
  
In particular, for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020370/c02037088.png" /> it is impossible to express the power <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020370/c02037089.png" /> as the sum of an infinite increasing sequence of length <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020370/c02037090.png" /> all terms of which are less than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020370/c02037091.png" />. For each cardinal number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020370/c02037092.png" />, the cofinal character of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020370/c02037093.png" />, denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020370/c02037094.png" />, is defined as the smallest cardinal number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020370/c02037095.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020370/c02037096.png" /> can be written as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020370/c02037097.png" /> for suitable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020370/c02037098.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020370/c02037099.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020370/c020370100.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020370/c020370101.png" /> is called regular, otherwise it is called singular. For each cardinal number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020370/c020370102.png" />, the smallest cardinal number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020370/c020370103.png" /> greater than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020370/c020370104.png" /> is regular (granted the axiom of choice). An example of a singular cardinal number is the cardinal number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020370/c020370105.png" /> on the left-hand side of (3) under the condition that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020370/c020370106.png" />. In this case
+
A cardinal number $ \alpha $ is said to be '''measurable''' (more precisely, '''$ \{ 0,1 \} $-measurable''') if and only if there exists a set $ A $ of cardinality $ \alpha $ and a function $ \mu: 2^{A} \to \{ 0,1 \} $ with the following properties:
 +
* $ \mu(A) = 1 $.
 +
* $ \mu(\{ a \}) = 0 $ for any $ a \in A $.
 +
* If $ (X_{n})_{n \in \omega_{0}} $ is a sequence of pairwise disjoint subsets of $ A $, then $ \displaystyle \mu \! \left( \bigcup_{n \in \omega_{0}} X_{n} \right) = \sum_{n \in \omega_{0}} \mu(X_{n}) $.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020370/c020370107.png" /></td> </tr></table>
+
Every cardinal number less than the first uncountable strongly-inaccessible cardinal number is non-measurable ('''Ulam’s Theorem'''), and the first measurable cardinal number is certainly strongly inaccessible. However, the first measurable cardinal number is considerably larger than the first uncountable strongly-inaccessible cardinal number ('''Tarski’s Theorem'''). It is not known (1987) whether the hypothesis that measurable cardinal numbers exist contradicts the axioms of set theory.
  
A cardinal number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020370/c020370108.png" /> is called a limit cardinal number if for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020370/c020370109.png" /> there exists a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020370/c020370110.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020370/c020370111.png" />. Examples of limit cardinal numbers are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020370/c020370112.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020370/c020370113.png" />, while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020370/c020370114.png" /> is a non-limit cardinal number. A regular limit cardinal number is called weakly inaccessible. A cardinal number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020370/c020370115.png" /> is said to be a strong limit cardinal if for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020370/c020370116.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020370/c020370117.png" />. A strong regular limit cardinal number is called strongly inaccessible. It follows from the generalized [[Continuum hypothesis|continuum hypothesis]] that the classes of strongly- (or weakly-) inaccessible cardinal numbers coincide. The classes of inaccessible cardinal numbers can be further classified (the so-called Malo scheme), which leads to the definition of hyper-inaccessible cardinal numbers. The assertion that strongly- (or weakly-) inaccessible cardinal numbers exist happens to be independent of the usual axioms of axiomatic set theory.
+
====References====
  
A cardinal number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020370/c020370118.png" /> is said to be measurable (more precisely, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020370/c020370120.png" />-measurable), if there exists a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020370/c020370121.png" /> of cardinality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020370/c020370122.png" /> and a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020370/c020370123.png" /> defined on all elements of the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020370/c020370124.png" />, taking the values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020370/c020370125.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020370/c020370126.png" /> and such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020370/c020370127.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020370/c020370128.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020370/c020370129.png" /> and such that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020370/c020370130.png" /> is a sequence of pairwise disjoint subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020370/c020370131.png" />, then
+
<table>
 +
<TR><TD valign="top">[1]</TD><TD valign="top">
 +
P.S. [P.S. Aleksandrov] Alexandroff, “Einführung in die Mengenlehre und die allgemeine Topologie”, Deutsch. Verlag Wissenschaft. (1984) (Translated from Russian)</TD></TR>
 +
<TR><TD valign="top">[2]</TD><TD valign="top">
 +
G. Cantor, “New ideas in mathematics”, ''Handbook Math. Libraries'', '''6''' (1914), pp. 90–184. (In Russian)</TD></TR>
 +
<TR><TD valign="top">[3]</TD><TD valign="top">
 +
F. Hausdorff, “Grundzüge der Mengenlehre”, Leipzig (1914). (Reprinted (incomplete) English translation: Set theory, Chelsea (1978))</TD></TR>
 +
<TR><TD valign="top">[4]</TD><TD valign="top">
 +
K. Kuratowski, A. Mostowski, “Set theory”, North-Holland (1968).</TD></TR>
 +
<TR><TD valign="top">[5]</TD><TD valign="top">
 +
W. Sierpiński, “Cardinal and ordinal numbers”, PWN (1965). (Translated from Polish)</TD></TR>
 +
</table>
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020370/c020370132.png" /></td> </tr></table>
+
====Comments====
  
Every cardinal number less than the first uncountable strongly-inaccessible cardinal number is non-measurable (Ulam's theorem), and the first measurable cardinal number is certainly strongly inaccessible. However, the first measurable cardinal number is considerably larger than the first uncountable strongly-inaccessible cardinal number (Tarski's theorem). It is not known (1987) whether the hypothesis that measurable cardinal numbers exist contradicts the axioms of set theory.
+
König’s theorem stated above is usually called the '''König–Zermelo Theorem'''.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  P.S. [P.S. Aleksandrov] Alexandroff,  "Einführung in die Mengenlehre und die allgemeine Topologie" , Deutsch. Verlag Wissenschaft.  (1984)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  G. Cantor,  , ''New ideas in mathematics'' , ''Handbook Math. Libraries'' , '''6'''  (1914)  pp. 90–184  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  F. Hausdorff,  "Grundzüge der Mengenlehre" , Leipzig  (1914)  (Reprinted (incomplete) English translation: Set theory, Chelsea (1978))</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  K. Kuratowski,  A. Mostowski,  "Set theory" , North-Holland  (1968)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  W. Sierpiński,  "Cardinal and ordinal numbers" , PWN  (1965)  (Translated from Polish)</TD></TR></table>
 
 
  
 
+
<table>
====Comments====
+
<TR><TD valign="top">[a1]</TD><TD valign="top">
König's theorem stated above is usually called the König–Zermelo theorem.
+
T.J. Jech, “Set theory”, Acad. Press (1978), Chapt. 7. (Translated from German)</TD></TR>
 
+
<TR><TD valign="top">[a2]</TD> <TD valign="top">
====References====
+
A. Levy, “Basic set theory”, Springer (1979).</TD></TR>
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> T.J. Jech,   "Set theory" , Acad. Press (1978) pp. Chapt. 7 (Translated from German)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> A. Levy,   "Basic set theory" , Springer (1979)</TD></TR></table>
+
</table>

Latest revision as of 11:57, 10 April 2018

2020 Mathematics Subject Classification: Primary: 03E10 [MSN][ZBL]

transfinite number, power in the sense of Cantor, cardinality of a set $ A $

That property of $ A $ that is intrinsic to any set $ B $ with the same cardinality as $ A $. In this connection, two sets $ A $ and $ B $ are said to have the same cardinality (or to be equivalent) if and only if there is a bijective function $ f: A \to B $ with domain $ A $ and range $ B $. G. Cantor defined the cardinal number of a set as that property of it that remains after abstracting the qualitative nature of its elements and their ordering. By way of stressing the double act of abstraction, Cantor introduced the symbol $ \overline{\overline{A}} $ to denote the cardinal number of $ A $. The most commonly used from among the various notations for a cardinal number are the symbols $ \mathsf{card}(A) $ and $ A $. If $ A $ is a finite set containing $ n $ elements, then $ \mathsf{card}(A) = n $. If $ \mathbf{Z}^{+} $ denotes the set of natural numbers, then $ \mathsf{card}(\mathbf{Z}^{+}) = \aleph_{0} $ (see Aleph). If $ \mathbf{R} $ denotes the set of real numbers, then $ \mathsf{card}(\mathbf{R}) = \mathfrak{c} $, the power of the continuum. The set $ 2^{A} $ of all subsets of $ A $ is not equivalent to $ A $ or to any subset of $ A $ (Cantor’s Theorem). In particular, no two members of the sequence $$ A, ~ 2^{A}, ~ 2^{2^{A}}, ~ 2^{2^{2^{A}}}, ~ \ldots \qquad (1) $$ are equivalent. When $ A = \mathbf{Z}^{+} $, the above sequence gives rise to infinitely many distinct infinite cardinal numbers. Further cardinal numbers are obtained by taking the union $ Q $ of the sets in (1) and constructing the analogous sequence, setting $ A = Q $. This process can be continued infinitely often. The scale (class) of all infinite cardinal numbers is much richer than the scale (class) of finite cardinals. Furthermore, there are so many of them that it is not possible to form a set containing at least one of each cardinal number.

One can define for cardinal numbers the operations of addition, multiplication, raising to a power, as well as taking the logarithm and extracting a root.

  • The cardinal number $ \delta $ is the sum of $ \alpha $ and $ \beta $, written as $ \delta = \alpha + \beta $, if and only if every set of cardinality $ \delta $ is equivalent to the disjoint union $ A \sqcup B $ of sets $ A $ and $ B $, where $ \mathsf{card}(A) = \alpha $ and $ \mathsf{card}(B) = \beta $.
  • The cardinal number $ \gamma $ is the product of $ \alpha $ and $ \beta $, written as $ \gamma = \alpha \beta $, if and only if every set of cardinality $ \gamma $ is equivalent to the Cartesian product $ A \times B $ of sets $ A $ and $ B $, where $ \mathsf{card}(A) = \alpha $ and $ \mathsf{card}(B) = \beta $.
  • Addition and multiplication of cardinal numbers is commutative and associative, and multiplication is distributive with respect to addition.
  • The cardinal number $ \kappa $ is the power with base $ \alpha $ and exponent $ \beta $, written as $ \kappa = \alpha^{\beta} $, if and only if every set of cardinality $ \kappa $ is equivalent to the set $ A^{B} $ of all functions $ f: B \to A $, where $ \mathsf{card}(A) = \alpha $ and $ \mathsf{card}(B) = \beta $.
  • The cardinal number $ \alpha $ is said to be smaller than or equal to the cardinal number $ \beta $, written as $ \alpha \leq \beta $, if and only if every set of cardinality $ \alpha $ is equivalent to some subset of a set of cardinality $ \beta $.
  • If $ \alpha \leq \beta $ and $ \beta \leq \alpha $, then $ \alpha = \beta $ (the so-called Cantor–Bernstein theorem), so that the scale of cardinal numbers is totally ordered.
  • For each cardinal number $ \beta $, the set $ \{ \alpha \mid \alpha < \beta \} $ is totally ordered, which enables one to define the logarithm $ {\log_{\alpha}}(\beta) $ of $ \beta $ to the base $ \alpha $, with $ \alpha \leq \beta $, as the smallest cardinal number $ \gamma $ such that $ \alpha^{\gamma} \geq \beta $.
  • Similarly, the $ \alpha $-th root $ \beta^{1 / \alpha} $ of the cardinal number $ \beta $ is defined as the smallest cardinal number $ \delta $ such that $ \delta^{\alpha} \geq \beta $.

Any cardinal number $ \alpha $ can be identified with the smallest ordinal number of cardinality $ \alpha $. In particular, $ \aleph_{0} $ corresponds to the ordinal number $ \omega_{0} $, $ \aleph_{1} $ to the ordinal $ \omega_{1} $, etc. Thus, the scale of cardinal numbers is a sub-scale of the scale of ordinal numbers. A number of properties of ordinal numbers carry over to cardinal numbers; however, these same properties can also be defined ‘intrinsically’.

If $ \alpha_{t} < \beta_{t} $ for every $ t \in T $ and if $ \mathsf{card}(T) \geq \omega_{0} $, then König’s Theorem states that $$ \sum_{t \in T} \alpha_{t} < \prod_{t \in T} \beta_{t}. \qquad (2) $$ If, in (2), one sets $ T = \mathbf{Z}^{+} $ and $ 1 < \alpha_{n} < \alpha_{n + 1} $, then $$ \alpha_{\omega_{0}} = \alpha_{1} + \alpha_{2} + \cdots < \alpha_{1} \alpha_{2} \cdots. \qquad (3) $$ In particular, for any $ \alpha \geq 2 $, it is impossible to express the power $ \alpha^{\omega_{0}} $ as the sum of an infinite increasing sequence of length $ \omega_{0} $, all terms of which are less than $ \alpha^{\omega_{0}} $.

For each cardinal number $ \alpha $, the cofinal character of $ \alpha $, denoted by $ \mathsf{cf}(\alpha) $, is defined as the smallest cardinal number $ \gamma $ such that $ \alpha $ can be written as $ \displaystyle \sum_{t \in T} \beta_{t} $ for suitable $ \beta_{t} < \alpha $ and $ \mathsf{card}(T) = \gamma $.

If $ \mathsf{cf}(\alpha) = \alpha $, then $ \alpha $ is called regular, otherwise it is called singular. For each cardinal number $ \alpha $, the smallest cardinal number $ \alpha^{+} $ greater than $ \alpha $ is regular (granted the axiom of choice). An example of a singular cardinal number is the cardinal number $ \alpha_{\omega_{0}} $ on the left-hand side of (3) under the condition that $ \omega_{0} < \alpha_{1} $. In this case, $$ \mathsf{cf}(\alpha_{\omega_{0}}) = \omega_{0} < \alpha_{\omega_{0}}. $$

A cardinal number $ \alpha $ is called a limit cardinal number if and only if for any $ \beta < \alpha $, there exists a $ \gamma $ such that $ \beta < \gamma < \alpha $. Examples of limit cardinal numbers are $ \omega_{0} $ and $ \alpha_{\omega_{0}} $, while $ \omega_{1} $ is a non-limit cardinal number. A regular limit cardinal number is called weakly inaccessible. A cardinal number $ \alpha $ is said to be a strong limit cardinal if and only if for any $ \beta < \alpha $, we have $ 2^{\beta} < \alpha $. A strong regular limit cardinal number is called strongly inaccessible. It follows from the generalized continuum hypothesis ($ \mathsf{GCH} $) that the classes of strongly- (or weakly-) inaccessible cardinal numbers coincide. The classes of inaccessible cardinal numbers can be classified further (the so-called Mahlo Scheme), which leads to the definition of hyper-inaccessible cardinal numbers. The assertion that strongly- (or weakly-) inaccessible cardinal numbers exist happens to be independent of the usual axioms of axiomatic set theory.

A cardinal number $ \alpha $ is said to be measurable (more precisely, $ \{ 0,1 \} $-measurable) if and only if there exists a set $ A $ of cardinality $ \alpha $ and a function $ \mu: 2^{A} \to \{ 0,1 \} $ with the following properties:

  • $ \mu(A) = 1 $.
  • $ \mu(\{ a \}) = 0 $ for any $ a \in A $.
  • If $ (X_{n})_{n \in \omega_{0}} $ is a sequence of pairwise disjoint subsets of $ A $, then $ \displaystyle \mu \! \left( \bigcup_{n \in \omega_{0}} X_{n} \right) = \sum_{n \in \omega_{0}} \mu(X_{n}) $.

Every cardinal number less than the first uncountable strongly-inaccessible cardinal number is non-measurable (Ulam’s Theorem), and the first measurable cardinal number is certainly strongly inaccessible. However, the first measurable cardinal number is considerably larger than the first uncountable strongly-inaccessible cardinal number (Tarski’s Theorem). It is not known (1987) whether the hypothesis that measurable cardinal numbers exist contradicts the axioms of set theory.

References

[1] P.S. [P.S. Aleksandrov] Alexandroff, “Einführung in die Mengenlehre und die allgemeine Topologie”, Deutsch. Verlag Wissenschaft. (1984) (Translated from Russian)
[2] G. Cantor, “New ideas in mathematics”, Handbook Math. Libraries, 6 (1914), pp. 90–184. (In Russian)
[3] F. Hausdorff, “Grundzüge der Mengenlehre”, Leipzig (1914). (Reprinted (incomplete) English translation: Set theory, Chelsea (1978))
[4] K. Kuratowski, A. Mostowski, “Set theory”, North-Holland (1968).
[5] W. Sierpiński, “Cardinal and ordinal numbers”, PWN (1965). (Translated from Polish)

Comments

König’s theorem stated above is usually called the König–Zermelo Theorem.

References

[a1] T.J. Jech, “Set theory”, Acad. Press (1978), Chapt. 7. (Translated from German)
[a2] A. Levy, “Basic set theory”, Springer (1979).
How to Cite This Entry:
Cardinal number. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Cardinal_number&oldid=12770
This article was adapted from an original article by B.A. Efimov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article