Definition A set is uncountable if it is not countableMathworldPlanetmath. In other words, a set S is uncountable, if there is no subset of (the set of natural numbers) with the same cardinality as S.

  1. 1.

    All uncountable sets are infiniteMathworldPlanetmath. However, the converseMathworldPlanetmath is not true, as is both infinite and countable.

  2. 2.

    The real numbers form an uncountable set. The famous proof of this result is based on Cantor’s diagonal argument.

Title uncountable
Canonical name Uncountable
Date of creation 2013-03-22 11:59:08
Last modified on 2013-03-22 11:59:08
Owner yark (2760)
Last modified by yark (2760)
Numerical id 9
Author yark (2760)
Entry type Definition
Classification msc 03E10
Synonym uncountable set
Related topic CardinalityOfTheContinuum