  One can obtain stronger large cardinal axioms by forging a link between the embedding j and the amount of resemblance between M and V.  For example, a cardinal κ is superstrong if there is a transitive class M and a non-trivial elementary embedding j: V → M such that crit(j)=κ and   Vj(κ)⊆ M.  If κ is superstrong then κ is a Woodin cardinal and there are arbitrarily large Woodin cardinals below κ. 