Church–Kleene ordinal

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In mathematics, the Church–Kleene ordinal, ωCK
, named after Alonzo Church and S. C. Kleene, is a large countable ordinal. Its order type is the set of all recursive ordinals and consequently it is 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 that is not hyperarithmetical, and the first admissible ordinal after ω.

The notation ωCK
is in reference to ω
, the first uncountable ordinal and the set of all countable (rather than recursive) ordinals.


  • Church, Alonzo; Kleene, S. C. (1937), "Formal definitions in the theory of ordinal numbers.", Fundamenta mathematicae, Warszawa, 28: 11–21, JFM 63.0029.02
  • Church, Alonzo (1938), "The constructive second number class", Bull. Amer. Math. Soc., 44 (4): 224–232, doi:10.1090/S0002-9904-1938-06720-1
  • Kleene, S. C. (1938), "On Notation for Ordinal Numbers", Journal of Symbolic Logic, Vol. 3, No. 4, 3 (4): 150–155, doi:10.2307/2267778, JSTOR 2267778
  • Rogers, Hartley (1987) [1967], The Theory of Recursive Functions and Effective Computability, First MIT press paperback edition, ISBN 978-0-262-68052-3