Definition:Aleph Mapping

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

Let $\aleph : \left({ \operatorname{On}, \in }\right) \to \left({ \mathcal N' , \in }\right) $ be the unique order isomorphism between the two classes.

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