Measurable cardinal: Difference between revisions

Revision as of 16:43, 2 December 2013

In mathematics, a cardinal number κ is called superstrong if and only if there exists an elementary embedding j : V → M from V into a transitive inner model M with critical point κ and $V_{j (κ)}$ ⊆ M.

Similarly, a cardinal κ is n-superstrong if and only if there exists an elementary embedding j : V → M from V into a transitive inner model M with critical point κ and $V_{j^{n} (κ)}$ ⊆ M. Akihiro Kanamori has shown that the consistency strength of an n+1-superstrong cardinal exceeds that of an n-huge cardinal for each n > 0.

References

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+In [[mathematics]], a [[cardinal number]] &kappa; is called '''superstrong''' [[if and only if]] there exists an [[elementary embedding]] ''j'' : ''V'' &rarr; ''M'' from ''V'' into a transitive inner model ''M'' with [[critical point (set theory)|critical point]] &kappa; and <math>V_{j(\kappa)}</math> &sube; ''M''.
+Similarly, a cardinal κ is '''n-superstrong''' if and only if there exists an [[elementary embedding]] ''j'' : ''V'' &rarr; ''M'' from ''V'' into a transitive inner model ''M'' with [[critical point (set theory)|critical point]] &kappa; and <math>V_{j^n(\kappa)}</math> &sube; ''M''. [[Akihiro Kanamori]] has shown that the consistency strength of an n+1-superstrong cardinal exceeds that of an [[n-huge cardinal]] for each n > 0.
+== References ==
+* {{cite book|last=Kanamori|first=Akihiro|authorlink=Akihiro Kanamori|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:Set theory]]
+[[Category:Large cardinals]]
+{{settheory-stub}}

Measurable cardinal: Difference between revisions

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