Strong cardinal: Difference between revisions

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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
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:<math>{ }^\lambda M\subseteq M \,.</math>
 
That is, ''M'' contains all of its &lambda;-sequences.  Then &kappa; is '''supercompact''' means that it is &lambda;-supercompact for all ordinals &lambda;.
 
Alternatively, an uncountable cardinal &kappa; is '''supercompact''' if for every ''A'' such that |''A''| ≥ &kappa; there exists a [[normal measure]] over [''A'']<sup>< &kappa;</sup>.
 
[''A'']<sup>< &kappa;</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 &kappa;, then a cardinal with that property exists below &kappa;. For example, if &kappa; is supercompact and the [[Generalized Continuum Hypothesis]] holds below &kappa; then it holds everywhere because a bijection between the powerset of &nu; and a cardinal at least &nu;<sup>++</sup> would be a witness of limited rank for the failure of GCH at &nu; so it would also have to exist below &kappa;.
 
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]]

Latest revision as of 19:54, 26 July 2014


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