Definition:Cardinal

Definition
Let $S$ be a set.

Associated with $S$ there exists a set $\operatorname{Card} \left({S}\right)$ called the cardinal of $S$.

It has the properties:
 * $\operatorname{Card} \left({S}\right) \sim S$, i.e. $\operatorname{Card} \left({S}\right)$ is (set) equivalent to $S$
 * $S \sim T \iff \operatorname{Card} \left({S}\right) = \operatorname{Card} \left({T}\right)$.

Also see

 * Cardinal Number, an approach to this concept using ordinals.