# Definition:Cardinal

## Contents

## 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.

## Also see

- Results about
**cardinals**can be found here. - Cardinal Number, an approach to this concept using ordinals.

## Sources

- 1972: A.G. Howson:
*A Handbook of Terms used in Algebra and Analysis*... (previous) ... (next): $\S 4$: Number systems $\text{I}$: A set-theoretic approach - 1975: T.S. Blyth:
*Set Theory and Abstract Algebra*... (previous) ... (next): $\S 8$