Countably

Countably is an adverb used in set theory to describe the size of a set. A set is countable if it is finite or countably infinite; that is, there exists a bijection between the set and the natural numbers.

A set that is infinite but countable is said to be countably infinite, and there are countably many elements. The natural numbers N are the canonical example, and many familiar sets are countable, including the integers Z and the rational numbers Q. By contrast, the real numbers R are uncountable, meaning there is no bijection with N.

In practice, countable is used to indicate that elements can be listed in a sequence indexed by natural numbers. Subsets of countable sets are countable, and the union of countably many countable sets is countable.

Examples and consequences: the set of prime numbers is countable; the algebraic numbers are countable; the set of finite strings over any finite alphabet is countable. The term countably is often used in statements like “there are countably many primes” or “the set is countably infinite.” The concept helps distinguish finite, countably infinite, and uncountable sets in cardinality theory.