en>Michael Devore: /* References */ fix ref to not have double "ed" edition

2014-07-26T18:54:53Z

References: fix ref to not have double "ed" edition

← Older revision		Revision as of 20:54, 26 July 2014
Line 1:		Line 1:
	~~In [[set theory]], a '''supercompact cardinal''' is a type of [[large cardinal]]. They display a variety of reflection properties.~~

	~~==Formal definition==~~

	If &~~lambda; is any [[ordinal number\|ordinal]~~], ~~κ is '''λ-supercompact''' means that there exists an elementary embedding ''j'' from~~ the ~~universe ''V'' into a transitive~~ [~~[inner model]~~] ~~''M'' with [[critical point (set theory~~)~~\|critical point]] κ~~, ~~''j''(κ)~~>~~λ and~~		Once a association struggle begins, you will see The specific particular War Map, one particular map of this conflict area area [https://Www.google.com/search?hl=en&gl=us&tbm=nws&q=association+wars association wars] booty place. Warm and friendly territories will consistently often be on the left, while having the adversary association at intervals the right. If you have any inquiries with regards to where in addition to tips on how to utilize clash of clans cheats ([http://prometeu.net Full Piece of writing]), you can contact us with our own internet site. Any boondocks anteroom on these war map represents some type of war base.<br><br>Yet unfortunately Supercell, by allowing typically the illusion on the multiplayer game, taps into the actual instinctual male drive [http://Www.Sharkbayte.com/keyword/returning returning] to from the status hierarchy, and even though it''s unattainable to the the surface of your hierarchy if you don't need to been logging in each day because the game became available plus you invested physical money in extra builders, the drive for obtaining a small bit further obliges enough visitors to use up a real income on virtual 'gems'" that game could be the top-grossing app within the App Store.<br><br>For anybody who is getting a online round for your little one, look for one who enables numerous customers to do with each other. Video gaming can be a solitary action. Nevertheless, it is important for you to motivate your youngster really social, and multi-player clash of clans hack is capable to complete that. They give sisters and brothers and buddies to all among take a moment with laugh and compete with each other.<br><br>Investigate evaluations and see each of our trailers before buying a relevant video game. Allow it to become one thing you are thinking about before you get it. These video games aren't low-cost, and also you won't get nearly as substantially cash whenever you industry inside a employed cd which you have simply just utilized several times.<br><br>Exclusive some online games provde the comfort of establishing a true-entire world time clock accessible in the playstation game itself. Is definitely usually a downside in full-monitor game titles. You don't want each of them using up even a good deal more of your time and even energy than within any budget place a point clock of your close by to your display exhibit screen to be able if you want to monitor just how long you've been enjoying.<br><br>By borer on a boondocks anteroom you possibly is going to appearance added advice that play, scout, entente troops, or attack. Of course, these positive results will rely on appearance of the competition you might be found in.<br><br>Really a nice technique. Breaking the appraisement bottomward into chunks of their precious time that accomplish college to be able to bodies (hour/day/week) causes this particular to be accessible so as to visualize. Everybody comprehends what it appears this kind of to accept to delay a day. Is usually additionally actual accessible for you to tune. If a change your current apperception after and adjudge those one day should wholesale more, all you claim to complete is amend 1 value.

	:<~~math~~>~~{ }^\lambda M\subseteq M \~~,.</~~math>~~

	~~That is~~, ''~~M'' contains all~~ of ~~its λ-sequences. Then κ is~~ '''~~supercompact''' means~~ that it is ~~λ-supercompact~~ for ~~all ordinals λ~~.

	~~Alternatively~~, ~~an uncountable cardinal κ~~ is ~~'''supercompact''' if~~ for ~~every ''A'' such~~ that ~~\|''A''\| ≥ κ there exists~~ a ~~[[normal measure]] over [''A'']~~<~~sup~~>< ~~κ</sup~~>.

	~~[''A~~''~~]<sup>< κ</sup> is defined~~ as ~~follows:~~

	:<~~math~~>~~[A]^{< \kappa} := \{X \subseteq A\| \|X\| < \kappa\} \,.~~<~~/math~~>

	~~==Properties==~~
	~~Supercompact cardinals have reflection properties~~. If a ~~cardinal with some property (say a 3~~-~~[[huge cardinal]]) that is witnessed by~~ a ~~structure~~ of ~~limited rank exists above~~ a ~~supercompact cardinal κ, then~~ a ~~cardinal with~~ that ~~property exists below κ~~. ~~For example~~, ~~if κ is supercompact and~~ the ~~[[Generalized Continuum Hypothesis]] holds below κ then it holds everywhere because a bijection between the powerset of ν and a cardinal at least ν~~<~~sup~~>++<~~/sup~~> ~~would be~~ a ~~witness of limited rank for~~ the ~~failure~~ of ~~GCH at ν~~ so it ~~would also have~~ to ~~exist below κ~~.

	~~Finding~~ a ~~canonical inner model for supercompact cardinals is~~ one ~~of the major problems of [[inner model theory]].~~

	~~==References==~~

	* {{cite book\|author=Drake, ~~F. R.\|title=Set Theory: An Introduction~~ to ~~Large Cardinals (Studies in Logic and the Foundations of Mathematics ; V~~. ~~76)\|publisher=Elsevier Science Ltd\|year=1974\|isbn=0-444-10535-2}}~~
	* {{cite book\|author=Jech, Thomas\|title=Set theory, third millennium edition (revised and expanded)\|publisher=Springer\|year=2002\|isbn=3-540-44085-2\|authorlink=Thomas Jech}}
	* {{cite book\|last=Kanamori\|first=Akihiro\|year=2003\|publisher=Springer\|title=The Higher Infinite : Large Cardinals in Set Theory from Their Beginnings\|edition=2nd ed\|isbn=3-540-00384-3}}

	~~[[Category:Large cardinals]]~~

en>Hans Adler: refine stub

2010-04-26T13:58:18Z

refine stub

New page

In [[set theory]], a '''supercompact cardinal''' is a type of [[large cardinal]]. They display a variety of reflection properties.

==Formal definition==

If λ is any [[ordinal number|ordinal]], κ is '''λ-supercompact''' means that there exists an elementary embedding ''j'' from the universe ''V'' into a transitive [[inner model]] ''M'' with [[critical point (set theory)|critical point]] κ, ''j''(κ)>λ and

:<math>{ }^\lambda M\subseteq M \,.</math>

That is, ''M'' contains all of its λ-sequences. Then κ is '''supercompact''' means that it is λ-supercompact for all ordinals λ.

Alternatively, an uncountable cardinal κ is '''supercompact''' if for every ''A'' such that |''A''| ≥ κ there exists a [[normal measure]] over [''A'']< κ.

[''A'']< κ is defined as follows:

:<math>[A]^{< \kappa} := \{X \subseteq A| |X| < \kappa\} \,.</math>

==Properties==
Supercompact cardinals have reflection properties. If a cardinal with some property (say a 3-[[huge cardinal]]) that is witnessed by a structure of limited rank exists above a supercompact cardinal κ, then a cardinal with that property exists below κ. For example, if κ is supercompact and the [[Generalized Continuum Hypothesis]] holds below κ then it holds everywhere because a bijection between the powerset of ν and a cardinal at least ν++ would be a witness of limited rank for the failure of GCH at ν so it would also have to exist below κ.

Finding a canonical inner model for supercompact cardinals is one of the major problems of [[inner model theory]].

==References==

* {{cite book|author=Drake, F. R.|title=Set Theory: An Introduction to Large Cardinals (Studies in Logic and the Foundations of Mathematics ; V. 76)|publisher=Elsevier Science Ltd|year=1974|isbn=0-444-10535-2}}
* {{cite book|author=Jech, Thomas|title=Set theory, third millennium edition (revised and expanded)|publisher=Springer|year=2002|isbn=3-540-44085-2|authorlink=Thomas Jech}}
* {{cite book|last=Kanamori|first=Akihiro|year=2003|publisher=Springer|title=The Higher Infinite : Large Cardinals in Set Theory from Their Beginnings|edition=2nd ed|isbn=3-540-00384-3}}

[[Category:Large cardinals]]

Strong cardinal - Revision history

en>Michael Devore: /* References */ fix ref to not have double "ed" edition

en>Hans Adler: refine stub