Definition:Aleph Number

Definition
The aleph numbers are a sequence of numbers used to represent the cardinality (or size) of infinite sets that can be well-ordered.

The cardinality of the natural numbers is $\aleph_0$.

The next larger cardinality is $\aleph_1$.

Then $\aleph_2$ and so on.

Also see
A set has cardinality $\aleph_0$ it is countably infinite.

It is possible to define a cardinal number $\aleph_\alpha$ for every ordinal number $\alpha$.

The aleph numbers are best known for their relevance to the Continuum Hypothesis.

This hypothesis states that the cardinality of the set of real numbers (cardinality of the continuum) is $2^{\aleph_0}$.

See Continuum Hypothesis for more details.