Mahlo cardinal: Difference between revisions

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In [[set theory]], a '''strong cardinal''' is a type of [[large cardinal]]. It is a weakening of the notion of a [[supercompact cardinal]].
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== Formal definition ==
 
If &lambda; is any [[ordinal number|ordinal]], &kappa; is '''&lambda;-strong''' means that &kappa; is a [[cardinal number]] and there exists an [[elementary embedding]] ''j'' from the universe ''V'' into a transitive [[inner model]] ''M'' with [[critical point (set theory)|critical point]] &kappa; and  
 
:<math>V_\lambda\subseteq M</math>
 
That is, ''M'' agrees with ''V'' on an initial segment. Then &kappa; is '''strong''' means that it is &lambda;-strong for all ordinals &lambda;.
 
== Relationship with other large cardinals ==
 
It is obvious from the definitions that strong cardinals lie below [[supercompact cardinal]]s and above [[measurable cardinal]]s in the consistency strength hierarchy.
 
They also lie below [[superstrong cardinal]]s and [[Woodin cardinal]]s. However, the least strong cardinal is larger than the least superstrong cardinal.
 
== References ==
 
* {{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]]
 
 
{{settheory-stub}}

Latest revision as of 20:40, 12 July 2014

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