Recursive ordinal

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

It is easy to check that 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 . The Church–Kleene ordinal is a limit ordinal. An ordinal is recursive if and only if it is smaller than . Since there are only countably many recursive relations, there are also only countably many recursive ordinals. Thus, is countable.

The recursive ordinals are exactly the ordinals that have an ordinal notation in Kleene's .

See also[edit]


  • 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