infinite
A set $S$ is infinite^{} if it is not finite (http://planetmath.org/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^{} (http://planetmath.org/AxiomOfChoice) (or the Axiom of Countable Choice), this definition of infinite sets is equivalent^{} to that of Dedekindinfinite sets (http://planetmath.org/DedekindInfinite).
Some examples of finite sets^{}:

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The empty set^{}: $\{\}$.

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$\{0,1\}$

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$\{1,2,3,4,5\}$

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$\{1,1.5,e,\pi \}$
Some examples of infinite sets:

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$\{1,2,3,4,\mathrm{\dots}\}$.

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The primes: $\{2,3,5,7,11,\mathrm{\dots}\}$.

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The rational numbers^{}: $\mathbb{Q}$.

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An interval of the reals: $(0,1)$.
The first three examples are countable^{}, but the last is uncountable.
Title  infinite 

Canonical name  Infinite 
Date of creation  20130322 11:59:03 
Last modified on  20130322 11:59:03 
Owner  yark (2760) 
Last modified by  yark (2760) 
Numerical id  18 
Author  yark (2760) 
Entry type  Definition 
Classification  msc 03E99 
Synonym  infinite set 
Synonym  infinite subset 
Related topic  Finite 
Related topic  AlephNumbers 