Countable set
In mathematics, a
countable set is a set with the same cardinality 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 and 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 senses of
countable set is in how they handle finite sets. Under the first definition finite sets are considered to be countable, while under the second definition they are not. To resolve this ambiguity, the term
at most countable is sometimes used for the first definition, and
countably infinite for the second. The term
denumerable can also be used to mean countably infinite, or countable, in contrast with the term
nondenumerable.