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| In [[mathematics]], specifically [[set theory]], an [[ordinal number|ordinal]] <math>\alpha</math> is said to be '''recursive''' if there is a [[recursive set|recursive]] [[well-order]]ing of a [[subset]] of the [[natural numbers]] having the [[order type]] <math>\alpha</math>.
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| It is trivial to check that <math>\omega</math> is recursive, the [[successor ordinal|successor]] of a recursive ordinal is recursive, and the [[Set (mathematics)|set]] of all recursive ordinals is [[closure (mathematics)|closed]] downwards. The [[supremum]] of all recursive ordinals is called the [[Church-Kleene ordinal]] and denoted by <math>\omega^{CK}_1</math>. Indeed, an ordinal is recursive if and only if it is smaller than <math>\omega^{CK}_1</math>. Since there are only countably many recursive relations, there are also only [[countable|countably]] many recursive ordinals. Thus, <math>\omega^{CK}_1</math> is countable.
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| The recursive ordinals are exactly the ordinals that have an [[ordinal notation]] in [[Kleene's O|Kleene's <math>\mathcal{O}</math>]].
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| ==See also==
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| *[[Arithmetical hierarchy]]
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| *[[Large countable ordinals]]
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| *[[Ordinal notation]]
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| == References ==
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| * 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
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| * Sacks, G. ''Higher Recursion Theory''. Perspectives in mathematical logic, Springer-Verlag, 1990. ISBN 0-387-19305-7
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| [[Category:Set theory]]
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| [[Category:Computability theory]]
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| [[Category:Ordinal numbers]]
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| {{settheory-stub}}
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Latest revision as of 17:10, 26 September 2014
Nice to meet you, I am Marvella Shryock. Minnesota has always been his home std test but his wife wants them to move. One of the extremely very best issues in the globe for him is to gather badges but he is struggling to discover time for it. Hiring is his profession.