 In mathematics, a countable set is a set with the same cardinality number of elements as some subset of a set of natural numbers. A countable set is either a finite set or a countably infinite set. Whether finite or infinite, the elements of a countable set can always be counted one at a time and, although the counting may never finish, every element of the set is associated with a unique natural number. Some authors use countable set to mean countably infinite alone. 1. To avoid this ambiguity, the term that most countable may be used when finite sets are included and countably infinite, enumerable 2 or denumerable 3 otherwise. George Cantor introduced the term countable set, contrasting sets that are countable with those that are uncountable i.e., non-denumerable or non-denumerable 4. Today, countable sets formed the foundation of a branch of mathematics called discrete mathematics.