Cardinality

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

In mathematics, cardinality is a measure of a set's size, meaning the number of individual objects it contains, which may be infinite. The cardinality or cardinal number of a set ${\displaystyle A}$ and is written as ${\displaystyle |A|}$ between two vertical bars. The term derives from the grammatical term cardinal numeral, meaning, numbers used for counting. For finite sets, cardinality coincides with the natural number found by counting its elements. Infinite sets are encountered in nearly every area of mathematics, including mathematical analysis, abstract algebra, number theory, and so on.

Two sets are said to be equinumerous or have the same cardinality if there exists a one-to-one correspondence between them. That is, if their objects can be paired such that each object has a pair, and no object is paired more than once (see image). A set is countably infinite if it can be placed in one-to-one correspondence with the set of natural numbers ${\displaystyle \{1,2,3,4,\cdots \}.}$ For example, the set of even numbers ${\displaystyle \{2,4,6,..\}}$, the set of prime numbers ${\displaystyle \{2,3,5,\cdots \}}$, and the set of rational numbers. A set is uncountable if it is both infinite and cannot be put in correspondence with the set of natural numbers. For example, the set of real numbers or the powerset of the set of natural numbers.

Cardinal numbers extend the natural numbers as representatives of size. Most commonly, the aleph numbers are defined via ordinal numbers, and represent a large class of sets. The first aleph, aleph-null ${\displaystyle \aleph _{0}}$, represents the cardinality of the set of natural numbers and all other countable sets. The cardinality of the continuum is represented with the symbol ${\displaystyle {\mathfrak {c}}}$. Whether ${\displaystyle {\mathfrak {c}}}$ corresponds to the next largest aleph, aleph-one ${\displaystyle \aleph _{1}}$, is known as the continuum hypothesis, which has been shown to be unprovable in standard set theories such as Zermelo–Fraenkel set theory.