In mathematics, specifically computability and set theory, an ordinal is said to be computable or recursive if there is a computable well-ordering of a subset of the natural numbers having the order type .
The supremum of all computable ordinals is called the Church-Kleene ordinal, the first nonrecursive ordinal, and denoted by . The Church-Kleene ordinal is a limit ordinal. An ordinal is computable if and only if it is smaller than . Since there are only countably many computable relations, there are also only countably many computable ordinals. Thus, is countable.