Computable Ordinal
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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 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 ${\displaystyle \omega _{1}^{CK}}$. The Church-Kleene ordinal is a limit ordinal. An ordinal is computable if and only if it is smaller than ${\displaystyle \omega _{1}^{CK}}$. Since there are only countably many computable relations, there are also only countably many computable ordinals. Thus, ${\displaystyle \omega _{1}^{CK}}$ is countable.

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

## References

• Hartley Rogers Jr. The Theory of Recursive Functions and Effective Computability, 1967. Reprinted 1987, MIT Press, ISBN 0-262-68052-1 (paperback), ISBN 0-07-053522-1
• Gerald Sacks Higher Recursion Theory. Perspectives in mathematical logic, Springer-Verlag, 1990. ISBN 0-387-19305-7