Editing Strong cardinal

In [[set theory]], a '''strong cardinal''' is a type of [[large cardinal]]. It is a weakening of the notion of a [[supercompact cardinal]].

== 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 ==

By definitions, strong cardinals lie below [[supercompact cardinal]]s and above [[measurable cardinal]]s in the consistency strength hierarchy.

&kappa; is &kappa;-strong if and only if it is measurable. If &kappa; is strong or &lambda;-strong for &lambda; ≥ &kappa;+2, then the [[ultrafilter]] ''U'' witnessing that &kappa; is measurable will be in ''V''<sub>&kappa;+2</sub> and thus in ''M''. So for any &alpha; < &kappa;, we have that there exist an ultrafilter ''U'' in ''j''(''V''<sub>&kappa;</sub>) &minus; ''j''(''V''<sub>&alpha;</sub>), remembering that ''j''(&alpha;) = &alpha;. Using the elementary embedding backwards, we get that there is an ultrafilter in ''V''<sub>&kappa;</sub> &minus; ''V''<sub>&alpha;</sub>. So there are arbitrarily large measurable cardinals below &kappa; which is regular, and thus &kappa; is a limit of &kappa;-many measurable cardinals.

Strong cardinals also lie below [[superstrong cardinal]]s and [[Woodin cardinal]]s in consistency strength.  However, the least strong cardinal is larger than the least superstrong cardinal.

Every strong cardinal is [[unfoldable cardinal|strongly unfoldable]] and therefore [[indescribable cardinal|totally indescribable]].

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

[[Category:Large cardinals]]


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