The best known example of an uncountable set is the set **R** of all real numbers; Cantor's diagonal argument shows that this set is uncountable. A modification of this argument can be used to show that several other sets are uncountable as well, for instance the set of all sequences of natural numbers (and even the set of all sequences consisting only of zeros and ones) and the set of all subsets of natural numbers.

Not all uncountable sets have the same size; the sizes of infinite sets are analyzed with the theory of cardinal numbers.
The statement that **R** is the smallest uncountable set (in the sense that its cardinal number is the smallest uncountable cardinal number) is the continuum hypothesis; this hypothesis is undecidable from the ordinary axioms of set theory.