# Definition:Aleph Number

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## 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.

### Aleph-Null: $\aleph_0$

**Aleph-null** is the cardinal number of a set which is in one-to-one correspondence with the natural numbers $\N$.

## Also see

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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.

- Results about
**aleph numbers**can be found**here**.

## Historical Note

The concept and notation of **aleph numbers** are due to Georg Cantor.

He defined the notion of cardinality.

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

## Linguistic Note

Aleph, $\aleph$, is the first letter of the Hebrew alphabet.

It is pronounced ** al-eph**, with the stress on the first syllable.

## Sources

- 1989: Ephraim J. Borowski and Jonathan M. Borwein:
*Dictionary of Mathematics*... (previous) ... (next):**aleph** - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next):**aleph**