Cardinal number

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.

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

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