Definition:Cardinal

Definition
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 $S$
 * $(2): \quad S \sim T \iff \map \Card S = \map \Card T$

Also see

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