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{{Short description|Mathematical set that can be enumerated}}{{hatnote group|{{Redirect|Countable|the linguistic concept|Count noun|the statistical concept|Count data|the company|Countable (app)}}{{distinguish|text= (recursively) enumerable sets}}}}In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers.{{efn|name=ZeroN}} Equivalently, a set is countable if there exists an injective function from it into the natural numbers; this means that each element in the set may be associated to a unique natural number, or that the elements of the set can be counted one at a time, although the counting may never finish due to an infinite number of elements.In more technical terms, assuming the axiom of countable choice, a set is countable if its cardinality (the number of elements of the set) is not greater than that of the natural numbers. A countable set that is not finite is said to be countably infinite.The concept is attributed to Georg Cantor, who proved the existence of uncountable sets, that is, sets that are not countable; for example the set of the real numbers.

A note on terminology

Although the terms "countable" and "countably infinite" as defined here are quite common, the terminology is not universal.BOOK, Manetti, Marco, Topology, 19 June 2015, Springer, 978-3-319-16958-3, 26,weblink en, An alternative style uses countable to mean what is here called countably infinite, and at most countable to mean what is here called countable.{{Harvard citation no brackets|Rudin|1976|loc=Chapter 2}}{{harvnb|Tao|2016|p=181}}The terms enumerable{{Harvard citation no brackets|Kamke|1950|page=2}} and denumerable{{Harvard citation no brackets|Lang|1993|loc=§2 of Chapter I}}{{Harvard citation no brackets|Apostol|1969|loc=Chapter 1.14|p=23}} may also be used, e.g. referring to countable and countably infinite respectively,BOOK, Thierry, Vialar, Handbook of Mathematics, 4 April 2017, BoD - Books on Demand, 978-2-9551990-1-5, 24,weblink en, definitions vary and care is needed respecting the difference with recursively enumerable.BOOK, Mukherjee, Subir Kumar, First Course in Real Analysis, 2009, Academic Publishers, 978-81-89781-90-3, 22,weblink en,

Definition

A set S is countable if:

• Its cardinality |S| is less than or equal to aleph_0 (aleph-null), the cardinality of the set of natural numbers N.
• There exists an injective function from S to N.BOOK, Singh, Tej Bahadur, Introduction to Topology, 17 May 2019, Springer, 978-981-13-6954-4, 422,weblink en, BOOK, Katzourakis, Nikolaos, Varvaruca, Eugen, An Illustrative Introduction to Modern Analysis, 2 January 2018, CRC Press, 978-1-351-76532-9,weblink en,
• S is empty or there exists a surjective function from N to S.
• There exists a bijective mapping between S and a subset of N.{{harvnb|Halmos|1960|loc=p. 91}}
• S is either finite (|S|