Computable ordinal

In mathematics, specifically computability and set theory, an ordinal ${\displaystyle \alpha }$ is said to be computable or recursive if there is a computable well-ordering of a computable subset of the natural numbers having the order type ${\displaystyle \alpha }$.

It is easy to check that ${\displaystyle \omega }$ is computable. The successor of a computable ordinal is computable, and the set of all computable ordinals is closed downwards.

The supremum of all computable ordinals is called the Church–Kleene ordinal, the first nonrecursive ordinal, and denoted by ${\displaystyle \omega _{1}^{\mathsf {CK}}}$. The Church–Kleene ordinal is a limit ordinal. An ordinal is computable if and only if it is smaller than ${\displaystyle \omega _{1}^{\mathsf {CK}}}$. Since there are only countably many computable relations, there are also only countably many computable ordinals. Thus, ${\displaystyle \omega _{1}^{\mathsf {CK}}}$ is countable.

The computable ordinals are exactly the ordinals that have an ordinal notation in Kleene's ${\displaystyle {\mathcal {O}}}$.