<*mathematics*> The number of elements in a set;
intuitively, the set's "size" or "magnitude".
Notation: a double bar over the symbol denoting
the set; also "|S|" and "card S" when S is the symbol denoting
the set.
If two sets
have the same number of elements (i.e. there is a bijection
between them) then they have the same cardinality. A
cardinality is thus an isomorphism class in the category
of sets.

aleph 0 is defined as the cardinality of the first infinite ordinal, omega (the number of natural numbers).

[FOLDOC] and [Glossary of First-Order Logic]

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