What does countably mean?
'Countably' is an adverb used almost entirely within mathematics to describe something that can be counted — that is, enumerated by pairing its elements with the natural numbers 1, 2, 3, and so on. Its most famous appearance is in the phrase 'countably infinite', which describes sets, such as the integers or the rational numbers, that are infinite yet can still be listed in a systematic sequence. This contrasts with uncountable sets, like the real numbers, which provably cannot be enumerated in any such list — a distinction established in Georg Cantor's groundbreaking work on set theory in the late nineteenth century. The word also appears in phrases like 'countably many' and 'countably additive', the latter central to measure theory. Though obscure outside technical contexts, 'countably' encodes one of the most surprising ideas in mathematics: that infinity itself comes in different sizes.
adverbIn a way that can be counted; especially in mathematics, in such a manner that the elements of a set can be put into one-to-one correspondence with the natural numbers.
- (Mathematics) In a manner that can be counted or put into one-to-one correspondence with the natural numbers.
"The rationals are countably infinite, even though they are densely packed along the number line."
"The set of integers is countably infinite."
"A measure is said to be countably additive if the measure of a union of disjoint sets equals the sum of their measures."
There are exactly as many even numbers as whole numbers — that's what 'countably infinite' lets mathematicians prove, and it still breaks people's brains.
Reviewed by Deb Chak, Editor. AI-assisted content curated by RJS Tech Solutions LLP.
Etymology of countably
'Countably' derives from the verb 'count', which came into Middle English from the Old French 'conter' (to count or tell), ultimately from the Latin 'computare', meaning to calculate or sum up — the same root that gives us 'compute' and 'computer'. The adjective 'countable' formed from 'count' with the suffix '-able', and the adverb 'countably' arose by adding '-ly'. Its technical sense crystallised in the nineteenth century with the rise of set theory, when Cantor introduced 'countable' (German 'abzählbar') as a precise mathematical term for sets that can be enumerated.
Related word forms
How countably is actually used
'Countably' is almost exclusively a technical term used in mathematics, particularly set theory and measure theory. It rarely appears in everyday language. It most commonly modifies adjectives such as 'infinite' or 'many'.
Easily confused with countably
Though similar in rhythm and ending, 'comfortably' means 'in a comfortable manner', whereas 'countably' is a technical term meaning 'in a countable (enumerable) way'.