Uncountable set

Infinite set that is not countable In mathematics, an uncountable set, informally, is an infinite set that contains too many elements to be countable. The uncountability of a set is closely related to its cardinal number: a set is uncountable if its cardinal number is larger than aleph-null, the cardinality of the natural numbers. Examples of uncountable sets include the set ⁠ R {\displaystyle \mathbb {R} } ⁠ of all real numbers and the set of all subsets of the natural numbers. Characterizations There are many equivalent characterizations of uncountability. A set X is uncountable if and only if any of the following conditions hold: There is no injective function (hence no bijection) from X to the set of natural numbers. X is nonempty and for every ω-sequence of elements of X, there exists at least one element of X not included in it. That is, X is nonempty and there is no surjective function from the natural numbers to X. The cardinality of X is neither finite nor equal to ℵ 0 {\displaystyle \aleph _{0}} (aleph-null). The set X has cardinality strictly greater than ℵ 0 {\displaystyle \aleph _{0}} . The first three of these characterizations can be proved equivalent in Zer