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 Dedekind-infinite sets (http://planetmath.org/DedekindInfinite).

Some examples of finite sets:

• The empty set: $\{\}$.

• $\{0,1\}$

• $\{1,2,3,4,5\}$

• $\{1,1.5,e,\pi \}$

Some examples of infinite sets:

• $\{1,2,3,4,\mathrm{\dots }\}$.

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

• The rational numbers: $\mathbb{Q}$.

• 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	2013-03-22 11:59:03
Last modified on	2013-03-22 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