Church–Kleene ordinal
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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
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