\begin{definition}[Definition:Aleph Mapping]
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 ordinal numbers.
{{explain|A link is needed to a proof that such a unique order isomorphism exists.}}
\end{definition}