# Church–Kleene ordinal

In mathematics, the Church–Kleene ordinal, ${\displaystyle \omega _{1}^{\mathrm {CK} }}$, named after Alonzo Church and S. C. Kleene, is a large countable ordinal. It is the set of all recursive ordinals and consequently the smallest non-recursive ordinal. Since the successor of a recursive ordinal is recursive, the Church–Kleene ordinal is a limit ordinal. It is also the first ordinal which is not hyperarithmetical, and the first admissible ordinal after ω.