Zero sharp - Revision history

en>Trovatore: Undid revision 614402926 by Blackbombchu (talk) this article uses Harvard-style cites

2014-06-25T19:33:18Z

Undid revision 614402926 by Blackbombchu (talk) this article uses Harvard-style cites

← Older revision		Revision as of 21:33, 25 June 2014
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	~~The author~~ is known the ~~particular name of Gabrielle Lattimer though she doesn't tremendously like being called like this~~. ~~To bake is something that this lady has been doing for months~~ and ~~months~~. ~~Her job~~ is ~~probably~~ a ~~cashier but inside~~ the ~~her~~ [http://~~Photobucket~~.com/~~images/husband husband~~] and ~~your~~ will ~~start their own home office~~. ~~She~~'~~s always loved living into South Carolina~~. ~~She is running and examining~~ a ~~blog here:~~ http://~~prometeu~~.net<br><br>~~Look into my homepage;~~ [http://prometeu.net clash of clans hack 2014]		It begins by consuming in a pair of a lovely island where your peaceful village is experiencing beaches and woods till the enemies known because BlackGuard led by Lieutenant Hammerman invades your tropical island. After managing to guard against a smallish invasion force, he offers to avenge his loss appearing in battle.<br><br>Keep a video game champion. These can be a lot amongst fun for you together gaming friends. You may either do this online, in the house or at a colleguerrrs place. Serve a good time snacks and get as a number of people as you can involved. This is a great way to enjoy your game playing with friends.<br><br>Gallstones are known as typically the [http://www.thefreedictionary.com/games+primary games primary] forex. The Jewels are previously used to purchase resources along among speeding up numerous imperative tasks. The Diamond rings can also be used to buy bonus items. Apart from that, it can possibly let the leader noticeable any undesired debris with the intention to obtain a much more gems. Players can easily obtain Gems through concluding numerous tasks or certainly using the clash of clans compromise available online.<br><br>Seen the evaluations and see the trailers before buying a relevant video game. Help it become one thing you will be looking at before you get in which. These video games aren't low-cost, and also you will get nearly as drastically cash whenever you commerce inside a employed game which you have few utilized several times.<br><br>Sensei Wars, the feudal Japan-themed Clash of Clans Cheats attacker from 2K, has aloof accustomed its aboriginal agreeable amend again there barrage on iOS aftermost 12 ,.<br><br>If you are the proud owner of an ANY easily transportable device that runs located on iOS or android as a touchscreen tablet computer or a smart phone, then you definitely may possibly have already been mindful of the [http://Revolution.net/ revolution] using place right now a world of mobile computer game "The Clash Associated with Clans", and you would be in demand pointing to conflict of families free jewels compromise because a lot of gems, elixir and valuable metal are needed seriously to acquire every battle.<br><br>Now that you have read this composition, you need to have an easier time locating as well as , loving video games in your lifetime. Notwithstanding your favored platform, from your cellphone in your own own computer, playing in addition , enjoying video gaming enables you to take the benefit of the worries of the particular busy week get advise If you liked this posting and you would like to get extra info concerning [http://prometeu.net clash of clans hack 2014] kindly check out our site. .

en>JRSpriggs: changed 3 ""s to ""s to match the associated "^"s

2014-02-26T09:53:41Z

changed 3 ""s to ""s to match the associated ""s

← Older revision		Revision as of 11:53, 26 February 2014
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	~~In [[set theory]], a '''Woodin cardinal''' (named for [[W. Hugh Woodin]])~~ is ~~a [[cardinal number]] λ such that for all functions~~		The author is known the particular name of Gabrielle Lattimer though she doesn't tremendously like being called like this. To bake is something that this lady has been doing for months and months. Her job is probably a cashier but inside the her [http://Photobucket.com/images/husband husband] and your will start their own home office. She's always loved living into South Carolina. She is running and examining a blog here: http://prometeu.net<br><br>Look into my homepage; [http://prometeu.net clash of clans hack 2014]

	~~:''f'' : λ → λ~~

	~~there exists a cardinal κ < λ with~~

	~~: {''f''(β)\|β < κ} &sube; κ~~

	~~and an [[elementary embedding]]~~

	~~:''j'' : ''V'' → ''M''~~

	~~from~~ the ~~[[Von Neumann universe]] ''V'' into a transitive [[inner model]] ''M'' with [[critical point (set theory)\|critical point]] κ and~~

	~~:V<sub>j(f)(κ)</sub> &sube; ''M'~~'.

	~~An equivalent definition~~ is this~~: λ is Woodin [[if and only if]] λ is [[inaccessible cardinal\|strongly inaccessible]] and~~ for ~~all <math>A \subseteq V_\lambda</math> there exists a <math>\lambda_A</math> < λ which is <math><\lambda</math>-<math>A</math>-strong.~~

	<math>\lambda _A</math> being <math><\lambda</math>-<math>A</math>-strong means that for all [[ordinal number\|ordinals]] α < λ, there exist a <math>j: V \to M</math> which is an [[elementary embedding]] with [[critical point (set theory)\|critical point]] <math>\lambda _A</math>, <math>j(\lambda _A) > \alpha</math>, <math>V_\alpha \subseteq M</math> and ~~<math>j(A) \cap V_\alpha = A \cap V_\alpha</math>. (See also [[strong cardinal]]~~.)

	~~A Woodin cardinal~~ is ~~preceded by~~ a ~~[[stationary set]] of [[measurable cardinal]]s, and thus it is a [[Mahlo cardinal]]. However,~~ the ~~first Woodin cardinal is not even [[Weakly compact cardinal\|weakly compact]].~~

	~~== Consequences ==~~

	~~Woodin cardinals are important in [[descriptive set theory]]. By a result<ref>~~[http://~~www~~.~~jstor.org~~/~~stable~~/~~1990913 A Proof of Projective Determinacy]</ref> of [[Donald A. Martin\|Martin]~~] and ~~[[John R~~. Steel\|Steel]], existence of infinitely many Woodin cardinals implies [[projective determinacy]], which in turn implies that every projective set is [[measurable]], has the [[Baire property]] (differs from an open set by a [[meagre set\|meager set]], that is, a set which is a countable union of nowhere dense sets), and the [[perfect set property]] (is either countable or contains a [[Perfect set\|perfect]] subset).

	The consistency of the existence of Woodin cardinals can be proved using determinacy hypotheses. Working in [[Zermelo–Fraenkel set theory\|ZF]]+[[axiom of determinacy\|AD]]+[[axiom of dependent choice\|DC]] one can prove that <math>\Theta _0</math> is Woodin in the class of hereditarily ordinal-definable sets. <math>\Theta _0</math> is the first ordinal onto which the continuum cannot be mapped by an ordinal-definable surjection (see [[Θ (set theory)]]).

	~~[[Saharon Shelah\|Shelah]] proved that if the existence of a Woodin cardinal is consistent then it is consistent that the nonstationary ideal on ω<sub>1</sub> is <math>\aleph_2</math>-saturated.~~
	~~Woodin also proved the equiconsistency of the existence of infinitely many Woodin cardinals and the existence of an <math>\aleph_1</math>-dense ideal over <math>\aleph_1</math>.~~

	~~==Hyper-Woodin cardinals==~~
	~~A [[cardinal number\|cardinal]] κ is called hyper-Woodin if there exists a [[normal measure]] ''U'' on κ such that for every set ''S'', the set~~

	~~:{λ < κ \| λ is <κ-''S''-[[strong cardinal\|strong]]}~~

	~~is in ''U'~~'.

	λ is ~~<κ-S-strong if~~ and ~~only if for each δ < κ there is~~ a ~~[[transitive class]] ''N'' and an [[elementary embedding]]~~

	:j : ~~V → N~~

	~~with~~

	~~:λ = crit(j),~~
	~~:j(λ)≥ δ, and~~

	~~:<math>j(S) \cap H_\delta = S \cap H_\delta<~~/~~math>~~.

	~~The name alludes to the classical result that a cardinal is Woodin if and only if for every set ''S'', the set~~

	~~:{λ~~ < ~~κ \| λ is <κ-''S''-[[strong cardinal\|strong]]}~~

	~~is a [[stationary set]]~~

	~~The measure ''U'' will contain the set of all [[Shelah cardinal]]s below κ.~~

	~~==Weakly hyper-Woodin cardinals==~~
	A [[cardinal number\|cardinal]] κ is called weakly hyper-Woodin if for every set ''S'' there exists a [[normal measure]] ''U'' on κ such that the set {λ < κ \| λ is <κ-''S''-strong} is in ''U''. λ is <κ-S-strong if and only if for each δ < κ there is a transitive class N and an elementary
	~~embedding j : V → N with λ = crit(j), j(λ)~~ >~~= δ, and~~ <~~math~~>~~j(S) \cap H_\delta = S \cap H_\delta.</math>~~

	~~The name alludes to the classic result that a cardinal is Woodin if for every set ''S'', the set {λ < κ \| λ is <κ-''S''-[[strong cardinal\|strong]]} is stationary.~~

	~~The difference between hyper-Woodin cardinals and weakly hyper-Woodin cardinals is that the choice of ''U'' does not depend on the choice of the set ''S'' for hyper-Woodin cardinals.~~

	~~== Notes and references ==~~
	~~<references/>~~

	~~== Further reading ==~~
	* {{cite book\|last=Kanamori\|first=Akihiro\|year=2003\|authorlink=Akihiro Kanamori\|publisher=Springer\|title=The Higher Infinite: Large Cardinals in Set Theory from Their Beginnings\|edition=2nd\|isbn=3-540-00384-3}}

	* For proofs of the two results listed in consequences see ''Handbook of Set Theory'' (Eds. Foreman, Kanamori, Magidor) (to appear). [http://handbook.assafrinot.com/ Drafts] of some chapters are available.
	* Ernest Schimmerling, ''Woodin cardinals, Shelah cardinals and the Mitchell-Steel core model'', Proceedings of the American Mathematical Society 130/11, pp. ~~3385–3391, 2002,~~ [http://~~www.math.cmu.edu/~eschimme/papers/hyperwoodin~~.~~pdf online]~~

	* {{cite journal \| last = Steel \| first = John R. \| authorlink = John R. Steel \|date=October 2007 \| title = What is a Woodin Cardinal? \| journal = [[Notices of ~~the American Mathematical Society]] \| volume = 54 \| issue = 9 \| pages = 1146–7 \| url = http://www.ams.org/notices/200709/tx070901146p.pdf \| format = [[PDF]] \| accessdate = 2008-01-15 }}~~

	~~{{DEFAULTSORT:Woodin Cardinal}}~~
	~~[[Category:Large cardinals]]~~
	~~[[Category:Determinacy]~~]

en>Zeimusu at 22:21, 5 January 2014

2014-01-05T22:21:07Z

← Older revision		Revision as of 00:21, 6 January 2014
Line 1:		Line 1:
			In [[set theory]], a '''Woodin cardinal''' (named for [[W. Hugh Woodin]]) is a [[cardinal number]] λ such that for all functions

			:''f'' : λ → λ

	~~My name~~ is ~~Reagan Edden but everybody calls me Reagan~~. ~~I'm from Austria~~. ~~I'm studying at the university~~ (~~3rd year~~) and ~~I play~~ the ~~Xylophone for 8 years~~. ~~Usually~~ [http://~~search~~.~~Huffingtonpost~~.~~com~~/~~search?q=I~~+~~choose&s_it=header_form_v1 I choose~~] ~~music from~~ the ~~famous films :~~). <br>~~I have two sister~~. ~~I like Hunting~~, ~~watching TV~~ (~~The Big Bang Theory~~) and ~~[http~~:/~~/www~~.~~google~~.~~de/search?q~~=~~Amateur+astronomy Amateur astronomy~~].<br><br>~~Check out my page~~ - ~~fifa 15 coin generator~~ ([http://~~www~~.~~thiscutie~~.com/~~members~~/~~margopkd~~/~~activity~~/~~27655~~/ ~~Http~~://www.~~thiscutie~~.~~com~~/~~members~~/~~margopkd~~/~~activity/27655~~])		there exists a cardinal κ < λ with

			: {''f''(β)\|β < κ} &sube; κ

			and an [[elementary embedding]]

			:''j'' : ''V'' → ''M''

			from the [[Von Neumann universe]] ''V'' into a transitive [[inner model]] ''M'' with [[critical point (set theory)\|critical point]] κ and

			:V<sub>j(f)(κ)</sub> &sube; ''M''.

			An equivalent definition is this: λ is Woodin [[if and only if]] λ is [[inaccessible cardinal\|strongly inaccessible]] and for all <math>A \subseteq V_\lambda</math> there exists a <math>\lambda_A</math> < λ which is <math><\lambda</math>-<math>A</math>-strong.

			<math>\lambda _A</math> being <math><\lambda</math>-<math>A</math>-strong means that for all [[ordinal number\|ordinals]] α < λ, there exist a <math>j: V \to M</math> which is an [[elementary embedding]] with [[critical point (set theory)\|critical point]] <math>\lambda _A</math>, <math>j(\lambda _A) > \alpha</math>, <math>V_\alpha \subseteq M</math> and <math>j(A) \cap V_\alpha = A \cap V_\alpha</math>. (See also [[strong cardinal]].)

			A Woodin cardinal is preceded by a [[stationary set]] of [[measurable cardinal]]s, and thus it is a [[Mahlo cardinal]]. However, the first Woodin cardinal is not even [[Weakly compact cardinal\|weakly compact]].

			== Consequences ==

			Woodin cardinals are important in [[descriptive set theory]]. By a result<ref>[http://www.jstor.org/stable/1990913 A Proof of Projective Determinacy]</ref> of [[Donald A. Martin\|Martin]] and [[John R. Steel\|Steel]], existence of infinitely many Woodin cardinals implies [[projective determinacy]], which in turn implies that every projective set is [[measurable]], has the [[Baire property]] (differs from an open set by a [[meagre set\|meager set]], that is, a set which is a countable union of nowhere dense sets), and the [[perfect set property]] (is either countable or contains a [[Perfect set\|perfect]] subset).

			The consistency of the existence of Woodin cardinals can be proved using determinacy hypotheses. Working in [[Zermelo–Fraenkel set theory\|ZF]]+[[axiom of determinacy\|AD]]+[[axiom of dependent choice\|DC]] one can prove that <math>\Theta _0</math> is Woodin in the class of hereditarily ordinal-definable sets. <math>\Theta _0</math> is the first ordinal onto which the continuum cannot be mapped by an ordinal-definable surjection (see [[Θ (set theory)]]).

			[[Saharon Shelah\|Shelah]] proved that if the existence of a Woodin cardinal is consistent then it is consistent that the nonstationary ideal on ω<sub>1</sub> is <math>\aleph_2</math>-saturated.
			Woodin also proved the equiconsistency of the existence of infinitely many Woodin cardinals and the existence of an <math>\aleph_1</math>-dense ideal over <math>\aleph_1</math>.

			==Hyper-Woodin cardinals==
			A [[cardinal number\|cardinal]] κ is called hyper-Woodin if there exists a [[normal measure]] ''U'' on κ such that for every set ''S'', the set

			:{λ < κ \| λ is <κ-''S''-[[strong cardinal\|strong]]}

			is in ''U''.

			λ is <κ-S-strong if and only if for each δ < κ there is a [[transitive class]] ''N'' and an [[elementary embedding]]

			:j : V → N

			with

			:λ = crit(j),
			:j(λ)≥ δ, and

			:<math>j(S) \cap H_\delta = S \cap H_\delta</math>.

			The name alludes to the classical result that a cardinal is Woodin if and only if for every set ''S'', the set

			:{λ < κ \| λ is <κ-''S''-[[strong cardinal\|strong]]}

			is a [[stationary set]]

			The measure ''U'' will contain the set of all [[Shelah cardinal]]s below κ.

			==Weakly hyper-Woodin cardinals==
			A [[cardinal number\|cardinal]] κ is called weakly hyper-Woodin if for every set ''S'' there exists a [[normal measure]] ''U'' on κ such that the set {λ < κ \| λ is <κ-''S''-strong} is in ''U''. λ is <κ-S-strong if and only if for each δ < κ there is a transitive class N and an elementary
			embedding j : V → N with λ = crit(j), j(λ) >= δ, and <math>j(S) \cap H_\delta = S \cap H_\delta.</math>

			The name alludes to the classic result that a cardinal is Woodin if for every set ''S'', the set {λ < κ \| λ is <κ-''S''-[[strong cardinal\|strong]]} is stationary.

			The difference between hyper-Woodin cardinals and weakly hyper-Woodin cardinals is that the choice of ''U'' does not depend on the choice of the set ''S'' for hyper-Woodin cardinals.

			== Notes and references ==
			<references/>

			== Further reading ==
			* {{cite book\|last=Kanamori\|first=Akihiro\|year=2003\|authorlink=Akihiro Kanamori\|publisher=Springer\|title=The Higher Infinite: Large Cardinals in Set Theory from Their Beginnings\|edition=2nd\|isbn=3-540-00384-3}}

			* For proofs of the two results listed in consequences see ''Handbook of Set Theory'' (Eds. Foreman, Kanamori, Magidor) (to appear). [http://handbook.assafrinot.com/ Drafts] of some chapters are available.
			* Ernest Schimmerling, ''Woodin cardinals, Shelah cardinals and the Mitchell-Steel core model'', Proceedings of the American Mathematical Society 130/11, pp. 3385–3391, 2002, [http://www.math.cmu.edu/~eschimme/papers/hyperwoodin.pdf online]

			* {{cite journal \| last = Steel \| first = John R. \| authorlink = John R. Steel \|date=October 2007 \| title = What is a Woodin Cardinal? \| journal = [[Notices of the American Mathematical Society]] \| volume = 54 \| issue = 9 \| pages = 1146–7 \| url = http://www.ams.org/notices/200709/tx070901146p.pdf \| format = [[PDF]] \| accessdate = 2008-01-15 }}

			{{DEFAULTSORT:Woodin Cardinal}}
			[[Category:Large cardinals]]
			[[Category:Determinacy]]

en>R.e.b.: O not 0

2011-10-19T17:10:55Z

O not 0

New page

My name is Reagan Edden but everybody calls me Reagan. I'm from Austria. I'm studying at the university (3rd year) and I play the Xylophone for 8 years. Usually [http://search.Huffingtonpost.com/search?q=I+choose&s_it=header_form_v1 I choose] music from the famous films :). I have two sister. I like Hunting, watching TV (The Big Bang Theory) and [http://www.google.de/search?q=Amateur+astronomy Amateur astronomy]. Check out my page - fifa 15 coin generator ([http://www.thiscutie.com/members/margopkd/activity/27655/ Http://www.thiscutie.com/members/margopkd/activity/27655])

Zero sharp - Revision history

en>Trovatore: Undid revision 614402926 by Blackbombchu (talk) this article uses Harvard-style cites

en>JRSpriggs: changed 3 ""s to ""s to match the associated ""s

en>Zeimusu at 22:21, 5 January 2014

en>R.e.b.: O not 0

en>JRSpriggs: changed 3 ""s to ""s to match the associated "^"s