\begin{definition}[Definition:Cardinal]
Let $S$ be a set.
Associated with $S$ there exists a set $\map \Card S$ called the '''cardinal of $S$'''.
It has the properties:
:$(1): \quad \map \Card S \sim S$
that is, $\map \Card S$ is (set) equivalent to $S$
:$(2): \quad S \sim T \iff \map \Card S = \map \Card T$
\end{definition}