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 ${\displaystyle \aleph _{0}}$ (read aleph-nought, aleph-zero, or aleph-null); the next larger cardinality of a well-ordered set is ${\displaystyle \aleph _{1},}$ then ${\displaystyle \aleph _{2},}$ then ${\displaystyle \aleph _{3},}$ and so on. Continuing in this manner, it is possible to define an infinite cardinal number ${\displaystyle \aleph _{\alpha }}$ for every ordinal number ${\displaystyle \alpha ,}$ 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.

${\displaystyle \aleph _{0}}$ (aleph-nought, aleph-zero, or aleph-null) is the cardinality of the set of all natural numbers, and is an infinite cardinal. The set of all finite ordinals, called ${\displaystyle \omega }$ or ${\displaystyle \omega _{0}}$ (where ${\displaystyle \omega }$ is the lowercase Greek letter omega), also has cardinality ${\displaystyle \aleph _{0}}$. A set has cardinality ${\displaystyle \aleph _{0}}$ if and only if it is countably infinite, that is, there is a bijection (one-to-one correspondence) between it and the natural numbers. Examples of such sets are:

Among the countably infinite sets are certain infinite ordinals, including for example ${\displaystyle \omega }$, ${\displaystyle \omega +1}$, ${\displaystyle \omega \cdot 2}$, ${\displaystyle \omega ^{2}}$, ${\displaystyle \omega ^{\omega }}$, and ${\displaystyle \varepsilon _{0}}$. For example, the sequence (with order type ${\displaystyle \omega \cdot 2}$) of all positive odd integers followed by all positive even integers ${\displaystyle \{1,3,5,7,9,\cdots ;2,4,6,8,10,\cdots \}}$ is an ordering of the set (with cardinality ${\displaystyle \aleph _{0}}$) of positive integers.

If the axiom of countable choice (a weaker version of the axiom of choice) holds, then ${\displaystyle \aleph _{0}}$ is smaller than any other infinite cardinal, and is therefore the (unique) least infinite ordinal.

${\displaystyle \aleph _{1}}$ is the cardinality of the set of all countable ordinal numbers. This set is denoted by ${\displaystyle \omega _{1}}$ (or sometimes Ω). The set ${\displaystyle \omega _{1}}$ is itself an ordinal number larger than all countable ones, so it is an uncountable set. Therefore, ${\displaystyle \aleph _{1}}$ is the smallest cardinality that is larger than ${\displaystyle \aleph _{0},}$ the smallest infinite cardinality.