Alephzero

Aleph-zero, denoted aleph_0, is the smallest infinite cardinal in set theory. It represents the cardinality of the set of natural numbers and, more generally, the cardinality of any countably infinite set.

A set is countable if it is finite or has cardinality aleph_0. Examples include the natural numbers, the integers, and the rational numbers. Any set that can be put into a one-to-one correspondence with the natural numbers is countably infinite and has cardinality aleph_0. The first infinite ordinal, ω, has the same size as aleph_0.

Aleph_0 is the initial member of the sequence of aleph numbers, aleph_0 < aleph_1 < aleph_2, and so on, which index the infinite cardinals. It is a limit cardinal and serves as the baseline infinity from which larger infinities are defined.

In arithmetic of infinite cardinals, aleph_0 + aleph_0 = aleph_0 and aleph_0 × aleph_0 = aleph_0. The cardinality of the real numbers is 2^{aleph_0}, called the cardinality of the continuum. The relationship between 2^{aleph_0} and aleph_1 is central to the continuum hypothesis, which posits that 2^{aleph_0} = aleph_1; this statement is independent of the standard axioms of set theory (ZFC).

Historically introduced by Georg Cantor, aleph_0 is a foundational concept for distinguishing different sizes of infinity. It underpins the theory of countable versus uncountable sets and serves as the starting point for the hierarchy of infinite cardinals used in modern set theory.