# 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 $S$
$S \sim T \iff \operatorname{Card} \left({S}\right) = \operatorname{Card} \left({T}\right)$.

## Notes

A cardinal is an equivalence class of sets of the same cardinality, so each cardinal has no specific nature as a set.

The cardinals do not form a set, since this would be $\mathcal S / \sim$, where $\mathcal S$ is the set of all sets, which by Set of all Sets leads to a contradiction.

See Cardinality for further discussion on the subject.