Countable Set

In mathematics, a countable set is a set with the same cardinality (number of elements) as some subset of the set of natural numbers. A set that is not countable is called uncountable. The term was originated by Georg Cantor. The elements of a countable set can be counted one at a time—although the counting may never finish, every element of the set will eventually be associated with a natural number.

Some authors use countable set to mean a set with the same cardinality as the set of natural numbers. The difference between the two definitions is that under the former, finite sets are also considered to be countable, while under the latter definition, they are not considered to be countable. To resolve this ambiguity, the term at most countable is sometimes used for the former notion, and countably infinite for the latter. The term denumerable can also be used to mean countably infinite, or countable, in contrast with the term nondenumerable.