denumeration

Denumeration is the process of listing the elements of a set in a sequence that assigns to each element a unique natural number. In mathematics, to denumerate a set means to establish a bijection between the natural numbers N and the set. Such a bijection provides an explicit enumeration s0, s1, s2, ... of the elements.

A set that admits such a bijection is called denumerable, typically meaning countably infinite. Finite sets are countable as well, but the term denumerable is usually reserved for infinite sets.

Examples: The set of natural numbers N is denumerable by identity. The integers Z are denumerable, with an explicit listing such as 0, 1, -1, 2, -2, 3, -3, ... Also the rational numbers Q are denumerable, despite appearing dense. In contrast, the set of real numbers R is not denumerable; Cantor’s diagonal argument shows no bijection with N exists, hence R is uncountable.

Applications: Denumeration is a central notion in set theory and analysis for distinguishing countable from uncountable sets, and it underpins proofs about cardinality and convergence of sequences. It also informs discussions of computability and constructive mathematics, where explicit enumerations are often required.

See also: countable, countably infinite, Cantor's diagonal argument, enumerability.