numberscardinal

Numberscardinal, commonly called cardinal numbers, are numbers that express the size or count of a set. They differ from ordinal numbers, which describe position in a sequence. In set theory, cardinal numbers are formalized as the equivalence classes of sets under the relation of equipotence (having the same size). A representative of a cardinal is associated with a least ordinal of that size, reflecting the connection between cardinals and ordinals.

For finite sets, cardinal numbers correspond to natural numbers: the cardinality of a finite set is a nonnegative integer. For infinite sets, cardinal numbers describe infinite sizes, with the smallest infinite cardinal denoted aleph-null (aleph0). Notation often uses vertical bars, |A|, to denote the cardinality of a set A. Arithmetic with cardinals differs from ordinary arithmetic, especially for infinite cardinals. For finite cardinals, addition and multiplication behave as usual, but for infinite cardinals kappa, kappa + kappa = kappa and kappa × 2 = kappa. The power set operation yields a strictly larger cardinal: for any set A, |P(A)| = 2^|A|, and 2^aleph0 is the cardinality of the continuum, denoted c. The continuum hypothesis posits no cardinal strictly between aleph0 and c, a statement independent of standard set theory (ZFC).

Cardinal numbers underpin cardinal arithmetic and have central roles in mathematics, including model theory, algebra, and analysis. They provide a rigorous means to compare sizes of infinite sets and to formulate questions about infinity, such as the existence of well-ordered representations and the structure of the ordinals and cardinals.