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 $\aleph \left({x}\right)$.