infinite set

In set theory, an infinite set is a set that is not a finite set. Infinite sets may be countable or uncountable. Some examples are:the set of all integers, {..., -1, 0, 1, 2, ...}, is a countably infinite set; andthe set of all real numbers is an uncountably infinite set.
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<mathematics> A set with an infinite number of elements. There are several possible definitions, e.g.
(i) ("Dedekind infinite") A set X is infinite if there exists a bijection (one-to-one mapping) between X and some proper subset of X.
(ii) A set X is infinite if there exists an injection from N (the set of natural numbers) to X.
In the presence of the Axiom of Choice all such definitions are equivalent.
(1995-03-27)

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