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In [[mathematics]], a [[Set (mathematics)|set]] is '''countable''' if either it is [[finite set|finite]] or it can be made in [[one to one correspondence]] with the set of [[natural number]]s.{{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 set]]s, that is, sets that are not countable; for example the set of the [[real number]]s.
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