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Concept
Countable set
Summary
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. 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.
Although the terms "countable" and "countably infinite" as defined here are quite common, the terminology is not universal. An alternative style uses countable to mean what is here called countably infinite, and at most countable to mean what is here called countable. To avoid ambiguity, one may limit oneself to the terms "at most countable" and "countably infinite", although with respect to concision this is the worst of both worlds. The reader is advised to check the definition in use when encountering the term "countable" in the literature.
The terms enumerable and denumerable may also be used, e.g. referring to countable and countably infinite respectively, but as definitions vary the reader is once again advised to check the definition in use.
A set is countable if:
Its cardinality is less than or equal to (aleph-null), the cardinality of the set of natural numbers .
There exists an injective function from to .
is empty or there exists a surjective function from to .
There exists a bijective mapping between and a subset of .
is either finite () or countably infinite.
All of these definitions are equivalent.
Official source
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