Cardinality

It has been suggested that Equinumerosity be merged into this article. (Discuss) Proposed since May 2025.

In mathematics, the cardinality of a finite set is the number of its elements, and is therefore a measure of size of the set. Since the discovery by Georg Cantor, in the late 19th century, of different sizes of infinite sets, the term cardinality was coined for generalizing to infinite sets the concept of the number of elements.

More precisely, two sets have the same cardinality if there exists a one-to-one correspondence between them. In the case of finite sets, the common operation of counting consists of establishing a one-to-one correspondence between a given set and the set of the ⁠${\displaystyle n}$⁠ first natural numbers, for some natural number ⁠${\displaystyle n}$⁠. In this case, ⁠${\displaystyle n}$⁠ is the cardinality of the set.

A set is infinite if the counting operation never finishes, that is, if its cardinality is not a natural number. The basic example of an infinite set is the set of all natural numbers.

The great discovery of Cantor is that infinite sets of apparently different size may have the same cardinality, and, nevertheless, there are infinitely many different cardinalities. For example, the even numbers, the prime numbers and the polynomials whose coefficients are rational number have the same cardinality as the natural numbers. The set of the real numbers has a greater cardinality than the natural numbers, and has the same cardinality as the interval ⁠${\displaystyle [0,1]}$⁠ and as every Euclidean space of any dimension. For every set, its power set (the set of all its subsets) has a greater cardinality.

Cardinalities are represented with cardinal numbers, which are specific sets of a given cardinality, which have been chosen once for all. Some infinite cardinalities have been given a specific name, such as ⁠${\displaystyle \aleph _{0}}$⁠ for the cardinality of the natural numbers and ⁠${\displaystyle {\mathfrak {c}}}$⁠, the cardinality of the continuum, for the cardinality of the real numbers.