What does denumerably mean?
"Denumerably" is an adverb used almost entirely within mathematics and logic to describe sets that can be counted — that is, whose elements can be paired one-to-one with the natural numbers 1, 2, 3, and so on. A set described as "denumerably infinite" may have endlessly many members, yet they can in principle be listed in a single sequence: the natural numbers themselves, the integers, and the rational numbers are classic examples. The term derives from the verb "denumerate," meaning to enumerate or count out, and functions as the adverbial form of the adjective "denumerable." It contrasts sharply with "uncountably," which applies to sets like the real numbers, too vast to be enumerated even in principle. Because Georg Cantor's work revealed this surprising divide between sizes of infinity, "denumerably" carries a certain conceptual elegance for mathematicians. Outside formal writing it is rare; general readers will more often meet its synonym "countably."
adverbIn a manner that can be counted or enumerated; in mathematics, in such a way that a set's elements can be put into one-to-one correspondence with the natural numbers.
- In mathematics: in a way that permits one-to-one correspondence with the natural numbers (countably).
- Generally: in a manner that can be counted or enumerated.
"The rational numbers are denumerably infinite, even though they appear densely packed on the number line."
"Cantor showed that while the reals are uncountable, the algebraic numbers are denumerably infinite."
"The set contains denumerably many elements, so each can be assigned a unique positive integer."
The rationals seem infinitely crowded between any two whole numbers — yet they're still only denumerably infinite, meaning you could, in principle, list them all one by one.
Reviewed by Deb Chak, Editor. AI-assisted content curated by RJS Tech Solutions LLP.
Etymology of denumerably
Denumerably traces to the Latin root numerus, meaning "number," via the verb denumerare, "to count out fully." English acquired "denumerate" in the sense of enumerating, and mathematicians formed "denumerable" and its adverb in the nineteenth century, when Cantor's set theory made precise the notion of a countably infinite set. Cognates sharing the Latin root include "enumerate," "numeral," and "innumerable."
Related word forms
How denumerably is actually used
Strictly technical vocabulary, found almost exclusively in mathematics, logic, and theoretical computer science. 'Denumerably infinite' is the standard fixed phrase. Some mathematicians prefer 'countably' as the more common synonym.
Easily confused with denumerably
'Numerably' means simply capable of being counted, while 'denumerably' is the technical mathematical term for countable via one-to-one correspondence with the natural numbers.