Cardinal number

In mathematics, a cardinal number, or cardinal, is a measure of the size of a set, generalizing the concept of counting from finite collections to infinite ones. Finite cardinal numbers are precisely the natural numbers: a set with exactly \(n\) elements has cardinality \(n\). For infinite sets, cardinal numbers describe different magnitudes of infinity, providing a formal way to say, for instance, that there are more real numbers than natural numbers. The theory of cardinals was pioneered by Georg Cantor in the late nineteenth century and is a cornerstone of modern set theory. The cardinality of a set \(A\) is denoted by \(|A|\), and two sets have the same cardinality if and only if there exists a bijection (a one-to-one correspondence) between them.

Definition and basic properties

In the presence of the Axiom of Choice, every set can be well-ordered, and the cardinal number of a set \(A\) can be defined as the least ordinal number that is equipotent to \(A\)—that is, the smallest ordinal that can be put into bijection with \(A\). Such an ordinal is called an initial ordinal. For a finite set with \(n\) elements, this initial ordinal is the natural number \(n\). For infinite sets, the initial ordinals are exactly the aleph numbers introduced by Cantor. Without the Axiom of Choice, not every set is equipotent to an ordinal, and cardinals are defined using Scott’s trick as the class of all sets of minimal rank that are equipotent to a given set, a definition that does not rely on well-orderability.

The fundamental equivalence relation for cardinals is equinumerosity: two sets \(A\) and \(B\) have the same cardinality, written \(|A| = |B|\), if there exists a bijection \(f: A \to B\). Cardinal numbers are ordered by the relation \(|A| \le |B|\), which holds if and only if there exists an injective function from \(A\) to \(B\). The Schröder–Bernstein theorem asserts that if \(|A| \le |B|\) and \(|B| \le |A|\), then \(|A| = |B|\), establishing that the ordering is a partial order (and, with the Axiom of Choice, a well-ordering of the cardinals).

Finite cardinals

The finite cardinals are the natural numbers \(0, 1, 2, \dots\). The cardinal number \(0\) corresponds to the empty set; \(1\) to any singleton set; and in general, \(n\) to any set with exactly \(n\) elements. The usual arithmetic of natural numbers—addition, multiplication, and exponentiation—coincides with the corresponding operations on finite cardinals defined via set-theoretic constructions. Within set theory, the finite cardinals are usually identified with the set-theoretic natural numbers, i.e., the finite von Neumann ordinals.

Infinite cardinals

An infinite set is one that is not finite, and its cardinal is called an infinite cardinal. The smallest infinite cardinal is \(\aleph_0\) (aleph-null), the cardinality of the set of natural numbers \(\mathbb{N}\). A set with cardinality \(\aleph_0\) is called countably infinite. Examples include the integers, the rational numbers, and the set of all finite sequences of natural numbers.

Cantor showed, via the diagonal argument, that the power set of any set—the set of all its subsets—has a strictly larger cardinality. Thus, the cardinality of the real numbers, denoted by \(\mathfrak{c}\) (or \(2^{\aleph_0}\)), is strictly greater than \(\aleph_0\). The sequence of infinite cardinals is traditionally laid out using the aleph hierarchy: let \(\aleph_1\) be the smallest cardinal greater than \(\aleph_0\), \(\aleph_2\) the smallest greater than \(\aleph_1\), and so on. In general, for any ordinal \(\alpha\), \(\aleph_{\alpha+1}\) is the successor of \(\aleph_\alpha\), and for limit ordinals \(\lambda\), \(\aleph_\lambda\) is the supremum of \(\{\aleph_\beta \mid \beta < \lambda\}\). Under the Axiom of Choice, every infinite cardinal is some \(\aleph_\alpha\); without Choice, there can be infinite sets whose cardinals are not comparable with the alephs.

Cardinal arithmetic

Operations on cardinal numbers are defined by constructions on sets. Let \(\kappa\) and \(\lambda\) be cardinals, and let \(A, B\) be disjoint sets with \(|A| = \kappa\), \(|B| = \lambda\).

• Addition: \(\kappa + \lambda = |A \cup B|\).
• Multiplication: \(\kappa \cdot \lambda = |A \times B|\).
• Exponentiation: \(\kappa^\lambda = |A^B|\), where \(A^B\) is the set of all functions from \(B\) to \(A\).

For finite cardinals, these operations agree with the usual arithmetic on natural numbers. For infinite cardinals, with the Axiom of Choice, the sums and products simplify dramatically: if \(\kappa\) and \(\lambda\) are infinite, then \(\kappa + \lambda = \kappa \cdot \lambda = \max(\kappa, \lambda)\). Exponentiation, however, is more complex. Cantor’s theorem states that for any cardinal \(\kappa\), \(\kappa < 2^\kappa\). In particular, \(2^{\aleph_0} > \aleph_0\). Determining the precise value of \(2^{\aleph_0}\) in terms of the aleph sequence is the subject of the continuum hypothesis.

Relationship with ordinal numbers

Cardinal numbers and ordinal numbers serve different purposes in set theory. An ordinal number describes the order type of a well-ordered set—its length and the relative positions of its elements. A cardinal number measures only the size of the set, abstracting away any order structure. Many distinct ordinals can have the same cardinality: for example, \(\omega\), \(\omega+1\), \(\omega+2\), and \(\omega^2\) are all countable ordinals, sharing the cardinal \(\aleph_0\). Every well-ordered set can be associated with a unique ordinal that is its order type and a unique cardinal that is its cardinality. Under the standard von Neumann assignment, a cardinal is identified with the smallest ordinal of that cardinality, making the set of all cardinals a subclass of the ordinals.

The continuum hypothesis

The continuum hypothesis (CH) is the statement that there is no cardinal number strictly between \(\aleph_0\) and \(\mathfrak{c} = 2^{\aleph_0}\); more precisely, \(2^{\aleph_0} = \aleph_1\). Cantor himself believed CH to be true and spent many years attempting to prove it. In 1940, Kurt Gödel showed that CH cannot be disproved from the standard Zermelo–Fraenkel set theory with the Axiom of Choice (ZFC). In 1963, Paul Cohen demonstrated that CH also cannot be proved from ZFC, thereby establishing its independence. The generalized continuum hypothesis (GCH) asserts that for every infinite cardinal \(\kappa\), the next cardinal in the power-set order is the successor cardinal: \(2^\kappa = \kappa^+\). GCH is also independent of ZFC and has far-reaching consequences in many areas of set theory.

Large cardinals

Beyond the aleph hierarchy built by the successor and limit operations, set theorists study large cardinals—cardinal numbers whose existence cannot be proved in ZFC but which appear as essential hypotheses in many consistency and independence proofs. Examples include inaccessible cardinals, Mahlo cardinals, measurable cardinals, and Woodin cardinals. A cardinal \(\kappa\) is strongly inaccessible if it is uncountable, regular (its cofinality is itself), and a strong limit (\(\lambda < \kappa\) implies \(2^\lambda < \kappa\)). The existence of such cardinals implies the consistency of ZFC and gives rise to a hierarchy of stronger and stronger axioms of infinity. The theory of large cardinals is a central area of modern set-theoretic research.