Aleph number

In mathematics, particularly in set theory, the aleph numbers are a sequence of numbers used to represent the cardinality (or size) of infinite sets. They were introduced by the mathematician Georg Cantor and are named after the symbol he used to denote them, the Hebrew letter aleph (ℵ). The smallest cardinality of an infinite set is that of the natural numbers, denoted by ℵ 0 {\displaystyle \aleph _{0}} (read aleph-nought, aleph-zero, or aleph-null); the next larger cardinality of a well-ordered set is ℵ 1 , {\displaystyle \aleph _{1},} then ℵ 2 , {\displaystyle \aleph _{2},} then ℵ 3 , {\displaystyle \aleph _{3},} and so on. Continuing in this manner, it is possible to define an infinite cardinal number ℵ α {\displaystyle \aleph _{\alpha }} for every ordinal number α , {\displaystyle \alpha ,} as described below. The concept and notation are due to Georg Cantor, who defined the notion of cardinality and realized that infinite sets can have different cardinalities. The aleph numbers differ from the infinity ( ∞ {\displaystyle \infty } ) commonly found in algebra and calculus, in that the alephs measure the sizes of sets, while infinity is commonly defined either as an extreme limit of the real number line (applied to a function or sequence that "diverges to infinity" or "increases without bound"), or as an extreme point of the extended real number line.