Superstrong cardinal

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

Similarly, a cardinal κ is n-superstrong if and only if there exists an elementary embedding j : VM from V into a transitive inner model M with critical point κ and V_{j^n(\kappa)}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.

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