Superstrong cardinal

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 &kappa; and $$V_{j(\kappa)}$$ &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 &kappa; and $$V_{j^n(\kappa)}$$ &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.