Cardinality

Cardinality is a concept in set theory that measures the size of a set. Two sets have the same cardinality when there exists a bijection between them, and this equivalence partitions all sets into cardinality classes. The cardinality of a set A is denoted |A|. For finite sets, |A| is the number of elements; this agrees with the intuitive notion of size.

Infinite sets may also have a cardinal number. Sets such as the natural numbers N, integers Z, and rational numbers Q are all countably infinite, meaning they have cardinality aleph-null (aleph_0). The real numbers R form a larger cardinal, called the cardinality of the continuum, denoted c, with c = 2^{aleph_0}.

Cantor's diagonal argument shows that R is uncountable, so no bijection exists between N and R. In general, if there are injections both ways between two sets, Cantor-Bernstein-Schroeder implies a bijection and equal cardinalities.

Cardinal arithmetic studies the sum, product, and exponentiation of cardinals. For infinite cardinals κ and λ with at least one infinite, κ + λ = max{κ, λ} and κ · λ = max{κ, λ}. In particular, aleph_0 + aleph_0 = aleph_0 and aleph_0 · aleph_0 = aleph_0. The continuum is c = 2^{aleph_0}, and exponentiation yields further infinities.

The continuum hypothesis asserts that there is no cardinal strictly between aleph_0 and c. It is known to be independent of the standard axioms of set theory (ZFC): Gödel showed CH cannot be disproved from ZFC, and Cohen showed it cannot be proved from ZFC.