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| In [[set theory]], an [[ordinal number]] α is an '''admissible ordinal''' if [[constructible universe|L<sub>α</sub>]] is an [[admissible set]] (that is, a [[Inner model|transitive model]] of [[Kripke–Platek set theory]]); in other words, α is admissible when α is a limit ordinal and L<sub>α</sub>⊧Σ<sub>0</sub>-collection.
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| The first two admissible ordinals are ω and <math>\omega_1^{\mathrm{CK}}</math> (the least [[recursive ordinal|non-recursive ordinal]], also called the [[Church–Kleene ordinal]]). Any [[regular cardinal|regular]] uncountable cardinal is an admissible ordinal.
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| By a theorem of [[Gerald Sacks|Sacks]], the [[countable set|countable]] admissible ordinals are exactly those constructed in a manner similar to the Church-Kleene ordinal, but for Turing machines with [[Oracle machine|oracles]]. One sometimes writes <math>\omega_\alpha^{\mathrm{CK}}</math> for the <math>\alpha</math>-th ordinal which is either admissible or a limit of admissibles; an ordinal which is both is called ''recursively inaccessible'': there exists a theory of large ordinals in this manner that is highly parallel to that of (small) [[large cardinal property|large cardinals]] (one can define recursively [[Mahlo cardinal]]s, for example). But all these ordinals are still countable. Therefore, admissible ordinals seem to be the recursive analogue of regular [[cardinal number]]s.
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| Notice that α is an admissible ordinal if and only if α is a [[limit ordinal]] and there does not exist a γ<α for which there is a Σ<sub>1</sub>(L<sub>α</sub>) mapping from γ onto α. If M is a standard model of KP, then the set of ordinals in M is an admissible ordinal.
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| ==See also==
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| *[[Large countable ordinals]]
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| *[[Inaccessible cardinal]]
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| *[[Constructible universe]]
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| [[Category:Ordinal numbers]]
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| {{settheory-stub}}
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| {{unref|date=December 2007}}
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Latest revision as of 19:37, 18 September 2014
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