Recursive ordinal

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

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

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

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

See also

• Arithmetical hierarchy
• Large countable ordinal
• Ordinal notation

References

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