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