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infinite (Definition)

A set $ S$ is infinite if it is not finite; that is, there is no $ n \in \mathbb{N}$ for which there is a bijection between $ n$ and $ S$.

Assuming the Axiom of Choice (or the Axiom of Countable Choice), this definition of infinite sets is equivalent to that of Dedekind-infinite sets.

Some examples of finite sets:

Some examples of infinite sets:

The first three examples are countable, but the last is uncountable.



"infinite" is owned by yark. [ full author list (4) | owner history (3) ]
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See Also: finite, aleph numbers

Other names:  infinite set, infinite subset
Keywords:  infinite

Attachments:
if $A$ is infinite and $B$ is a finite subset of $A\,\!,$ then $A\setminus B$ is infinite (Theorem) by mathcam
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Cross-references: uncountable, countable, reals, interval, rational numbers, primes, empty set, finite sets, axiom of countable choice, bijection
There are 348 references to this entry.

This is version 14 of infinite, born on 2001-11-16, modified 2006-01-01.
Object id is 881, canonical name is Infinite.
Accessed 25982 times total.

Classification:
AMS MSC: 03E99 (Mathematical logic and foundations :: Set theory :: Miscellaneous)
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