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)