Church–Kleene ordinal

In mathematics, the Church–Kleene ordinal, ωCK1, 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. The notation ωCK1 is in reference to ω₁, the first uncountable ordinal and the set of all countable (rather than recursive) ordinals.

References

• Church, Alonzo; Kleene, S. C. (1937), "Formal definitions in the theory of ordinal numbers.", Fundamenta mathematicae, Warszawa 28: 11–21 
• Church, Alonzo (1938), "The constructive second number class", Bull. Amer. Math. Soc. 44 (4): 224–232, doi:10.1090/S0002-9904-1938-06720-1, https://www.ams.org/bull/1938-44-04/S0002-9904-1938-06720-1/ 
• Kleene, S. C. (1938), "On Notation for Ordinal Numbers", The Journal of Symbolic Logic (The Journal of Symbolic Logic, Vol. 3, No. 4) 3 (4): 150–155, doi:10.2307/2267778 
• Rogers, Hartley (1987), The Theory of Recursive Functions and Effective Computability, First MIT press paperback edition, ISBN 978-0-262-68052-3