# 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.

## Historical notes

The concept and notation are due to Georg Cantor.

He defined the notion of cardinality.

He was the first to realize that infinite sets can have different cardinalities.

## Also see

A set has cardinality $\aleph_0$ if and only if 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.