# Definition:Cardinal

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## 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 $S$

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

### Informal Definition

A natural number considered as an indication of the size of a set is referred to as a **cardinal number**.

## Notes

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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 $\SS / \sim$, where $\SS$ is the set of all sets, which leads to a contradiction.

See Cardinality for further discussion on the subject.

## Also see

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

- Results about
**cardinals**can be found**here**.

## 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): $\S 8$ - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**cardinal number (cardinality)** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**cardinal number (cardinality)**