# Definition:Aleph Mapping

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## Definition

Let $\NN'$ denote the class of infinite cardinals.

Then $\aleph$ (that is: **aleph**) is defined as the unique order isomorphism between the two ordered structures $\struct {\On, \in}$ and $\struct {\NN', \in}$

where $\On$ denotes the class of all ordinals.

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## Definition by Transfinite Recursion

$\aleph$ may also be defined via the Second Principle of Transfinite Recursion:

- $\aleph_0 = \omega$

- $\ds \aleph_{x^+} = \bigcap \set {y \in \NN' : x < y}$

- $\ds \aleph_y = \bigcup_{x \mathop \in y} \aleph_x$ where $y$ is a limit ordinal.

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## Also see

An explicit construction for the $\aleph$ function is given by Order Isomorphism between Ordinals and Proper Class/Corollary where $F = \aleph$ and $A = \NN'$.

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## Notation

The value of the aleph mapping at an ordinal $x$ is denoted $\aleph_x$ instead of $\map \aleph x$.

## 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

- 1971: Gaisi Takeuti and Wilson M. Zaring:
*Introduction to Axiomatic Set Theory*: $\S 10.44$, $\S 10.45$