en>Yobot: Tagging, added underlinked tag using AWB (10468)

2014-09-18T18:37:00Z

Tagging, added underlinked tag using AWB (10468)

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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~~.		The writer's title is Christy Brookins. I've usually loved living in Kentucky but now I'm considering other options. Distributing production has been his profession for some time. To climb is some thing I truly enjoy doing.<br><br>Also visit my blog - [http://hknews.classicmall.com.hk/groups/some-simple-tips-for-personal-development-progress/ clairvoyant psychic]

	~~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~~.

	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.~~

	~~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.~~

	~~==See also==~~
	*[[Large countable ordinals]]
	*[[Inaccessible cardinal]]
	*[[Constructible universe]]

	~~[[Category:Ordinal numbers]]~~

	~~{{settheory-stub}}~~

	~~{{unref\|date=December 2007}}~~

en>Bassel.shammoot at 12:52, 24 November 2010

2010-11-24T12:52:15Z

New page

In [[set theory]], an [[ordinal number]] α is an '''admissible ordinal''' if [[constructible universe|Lα]] 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 Lα⊧Σ0-collection.

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.

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.

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 Σ1(Lα) mapping from γ onto α. If M is a standard model of KP, then the set of ordinals in M is an admissible ordinal.

==See also==
*[[Large countable ordinals]]
*[[Inaccessible cardinal]]
*[[Constructible universe]]

[[Category:Ordinal numbers]]

{{settheory-stub}}

{{unref|date=December 2007}}

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en>Yobot: Tagging, added underlinked tag using AWB (10468)

en>Bassel.shammoot at 12:52, 24 November 2010