# Definition:Aleph Mapping

## Definition

Let $\mathcal N'$ denote the class of all infinite cardinal numbers.

Then $\aleph$ (that is: **aleph**) is defined as the unique order isomorphism between the two ordered structures $\left({\operatorname{On}, \in}\right)$ and $\left({\mathcal N', \in}\right)$

where $\operatorname{On}$ denotes the class of ordinal numbers.

## Definition by Transfinite Recursion

$\aleph$ may also be defined via transfinite recursion:

- $\aleph_0 = \omega$

- $\displaystyle \aleph_{x^+} = \bigcap \left\{ y \in \mathcal N' : x < y \right\}$

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

## Also see

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

## Notation

The value of the aleph mapping at an ordinal $x$ is denoted $\aleph_x$ instead of $\aleph \left({x}\right)$.

## 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*: $\S 10.44$, $\S 10.45$