Definition:Aleph Mapping

Definition
Let $\NN'$ 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 $\struct {\On, \in}$ and $\struct {\NN', \in}$

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

Definition by Transfinite Recursion
$\aleph$ may also be defined via 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.

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

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