Uncountably

Uncountably is used to describe a property of sets or collections in mathematics. Specifically, a set is uncountable if it cannot be put into a one-to-one correspondence with the natural numbers. The term implies uncountably infinite cardinality, meaning its size exceeds that of any finite or countably infinite set. The adjective form is uncountable; the adverb form uncountably is used to describe an action or a class that has this property.

A set is countable if it is finite or countably infinite (in bijection with the natural numbers). If not, it is uncountable. Cantor's diagonal argument shows that the real numbers are uncountable, proving there exists no enumeration of the reals by natural numbers.

Examples include: the real numbers R are uncountable; the set of irrational numbers is uncountable; the power set of N is uncountable. Conversely, the set of algebraic numbers is countable. These examples illustrate the distinction between countable and uncountable infinities within standard set theory.

Notes: In common mathematical language, one speaks of uncountable sets or uncountably infinite sets, emphasizing the difference from countable infinity. The concept is central to set theory and cardinality, and relates to the continuum and Cantor's theorem. See also countable, continuum, Cantor's diagonal argument, cardinality.