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 $S$

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

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)