What does denumerable mean?
In mathematics, 'denumerable' describes a set whose elements can be placed in one-to-one correspondence with the natural numbers — meaning they can, at least in principle, be counted off in a single infinite list. A denumerable set may be finite or countably infinite; what matters is that no element is left unnumbered. The term belongs chiefly to set theory, where Georg Cantor's work made it central: surprisingly, infinite collections such as the integers and even the rational numbers are denumerable, while the real numbers are not. Because the concept hinges on the possibility of systematic listing rather than on actual counting, 'denumerable' carries a precise technical force that the everyday word 'countable' only loosely suggests. It appears almost exclusively in textbooks, papers, and lectures on logic and analysis, where its opposite is 'nondenumerable' or 'uncountable.'
adjectiveCapable of being counted or put into one-to-one correspondence with the natural numbers; in mathematics, describing a set that is either finite or countably infinite.
- (Mathematics) Of an infinite set, capable of being put into one-to-one correspondence with the natural numbers; countably infinite.
- (Rare/general) Capable of being counted or enumerated.
"The set of rational numbers is denumerable, even though it seems far larger than the set of natural numbers."
"The integers form a denumerable set because they can be listed as 0, 1, -1, 2, -2, and so on."
"Every denumerable subset of real numbers has measure zero."
Cantor stunned nineteenth-century mathematicians by proving the rationals are denumerable — infinitely many fractions, yet you can number them all.
Reviewed by Deb Chak, Editor. AI-assisted content curated by RJS Tech Solutions LLP.
Etymology of denumerable
'Denumerable' derives from Latin roots: the prefix 'de-' combined with 'numerare,' meaning to count, from 'numerus,' number. It entered English in the nineteenth century as mathematicians formalized ideas of infinity, likely influenced by French usage surrounding Cantor's set theory. It shares its root with words such as 'enumerate,' 'numeral,' and 'innumerable.'
Related word forms
How denumerable is actually used
A technical term of mathematical logic and set theory, essentially confined to formal academic writing; outside mathematics, 'countable' is the ordinary choice. British spelling and pronunciation follow the standard pattern; no significant regional variation exists.
Easily confused with denumerable
Both mean 'countable,' but 'denumerable' is used almost exclusively in mathematical set theory for sets that can be matched one-to-one with the natural numbers, while 'enumerable' has broader use, including computing and general contexts.