Uncountable set

An infinite set that is not countable, i.e. that is not equivalent to the set of natural numbers. For example, the set of real numbers is uncountable, whereas that of the rational numbers is countable.

Comments

In the literature the phrase "countable set" can mean either "a finite or countably infinite set" or "a countably infinite set" .

The uncountability of the set of real numbers is sometimes proved by the Cantor diagonalization principle (cf. Cantor theorem 1).

References

[a1]	K. Kuratowski, "Introduction to set theory and topology" , Pergamon (1961) pp. 62ff (Translated from French)