  One can also obtain strong large cardinal axioms by placing closure conditions on the target model M.  For example, letting γ ≥ κ a cardinal κ is γ-supercompact if there is a transitive class M and a non-trivial elementary embedding j: V → M such that crit(j)=κ and γM⊆ M, that is, M is closed under γ-sequences.  (It is straightforward to see that if M is closed under γ-sequences then H(γ+)⊆ M; so this approach subsumes the previous approach.) A cardinal κ is supercompact if it is γ-supercompact for all γ ≥ κ.  Now, just as in the previous approach, one can strengthen these axioms by forging a link between the embedding j and the closure conditions on the target model.  A cardinal κ is n-huge if there is a transitive class M and a non-trivial elementary embedding j:V → M such that j n(κ)M ⊆ M, where κ=crit(j) and j i+1(κ) is defined to be j(j i(κ)). 