Definition:Aleph Mapping

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

Define $\aleph$ as the unique order isomorphism between the two classes $\left({ \operatorname{On}, \in }\right)$ 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.

Notation
The value of the aleph function at an ordinal $x$ shall be denoted $\aleph_x$ instead of $\aleph \left({x}\right)$.