A set S is infiniteMathworldPlanetmath if it is not finite (; that is, there is no n for which there is a bijection between n and S.

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

Some examples of finite setsMathworldPlanetmath:

  • The empty setMathworldPlanetmath: {}.

  • {0,1}

  • {1,2,3,4,5}

  • {1,1.5,e,π}

Some examples of infinite sets:

  • {1,2,3,4,}.

  • The primes: {2,3,5,7,11,}.

  • The rational numbersPlanetmathPlanetmathPlanetmath: .

  • An interval of the reals: (0,1).

The first three examples are countableMathworldPlanetmath, 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