Uncountable

Examples

The best known example of an uncountable set is the set ⁠${\displaystyle \mathbb {R} }$⁠ of all real numbers; Cantor's diagonal argument shows that this set is uncountable. The diagonalization proof technique can also be used to show that several other sets are uncountable, such as the set of all infinite sequences of natural numbers⁠${\displaystyle \mathbb {N} }$⁠ (see: (sequence A102288 in the OEIS)), and the set of all subsets of the set of natural numbers. The cardinality of ⁠${\displaystyle \mathbb {R} }$⁠ is often called the cardinality of the continuum, and denoted by ${\displaystyle {\mathfrak {c}}}$, or ${\displaystyle 2^{\aleph _{0}}}$, or ${\displaystyle \beth _{1}}$ (beth-one).