Cardinal number

In mathematics, a cardinal number, or cardinal for short, is a kind of number that measures the cardinality of a set, i.e., how many elements there are in a set. The cardinal number associated with a set ⁠${\displaystyle A}$⁠ is generally denoted by ⁠${\displaystyle \vert A\vert }$⁠, with a vertical bar on each side, though it may also be denoted by ${\displaystyle A}$, ${\displaystyle \operatorname {card} (A),}$ or ${\displaystyle \#A.}$

Cardinality is defined in terms of bijective functions. Two sets have the same cardinality if, and only if, there is a one-to-one correspondence (bijection) between the elements of the two sets. The cardinality of a finite set can be identified with a natural number, which can be found simply by counting its elements. For example, the sets ⁠${\displaystyle \{1,2,3\}}$⁠ and ⁠${\displaystyle \{4,5,6\}}$⁠ both have the same cardinality 3, as evidenced by the bijection ⁠${\displaystyle \{1\mapsto 4,2\mapsto 5,3\mapsto 6\}}$⁠.

The behavior of cardinalities of infinite sets is more complex. For example, there exists a bijection between the set of all natural numbers ⁠${\displaystyle \mathbb {N} }$⁠ and the set of all rational numbers ⁠${\displaystyle \mathbb {Q} }$⁠, and thus ⁠${\displaystyle \vert \mathbb {N} \vert =\vert \mathbb {Q} \vert }$⁠ even though ⁠${\displaystyle \mathbb {N} }$⁠ is a proper subset of ⁠${\displaystyle \mathbb {Q} }$⁠—something that cannot happen with proper subsets of finite sets. However, a fundamental theorem due to Georg Cantor shows that it is possible for two infinite sets to have different cardinalities, and in particular the cardinality of the set of real numbers ⁠${\displaystyle \mathbb {R} }$⁠ is greater than the cardinality of ⁠${\displaystyle \mathbb {N} }$⁠.

The cardinality of ⁠${\displaystyle \mathbb {N} }$⁠ is usually denoted by ⁠${\displaystyle \aleph _{0}}$⁠ (aleph-null), since it is the smallest aleph number. The properties of other aleph numbers and of infinite cardinal numbers in general depend on statements independent of Zermelo–Fraenkel set theory, such as the axiom of choice and the continuum hypothesis. For example, all infinite cardinal numbers are aleph numbers if and only if the axiom of choice is true.

Cardinality is studied for its own sake as part of set theory. It is also a tool used in branches of mathematics including model theory, combinatorics, abstract algebra and mathematical analysis. In category theory, the cardinal numbers form a skeleton of the category of sets.