  If κ is a cardinal and η&gt;κ is an ordinal then κ is η-strong if there is a transitive class M and a non-trivial elementary embedding j: V → M such that crit(j)=κ, j(κ)&gt;η and Vη⊆ M.  A cardinal κ is strong iff it is η-strong for all η&gt;κ. One can also demand that the embedding preserve certain classes: If A is a class, κ is a cardinal, and η&gt;κ is an ordinal then κ is η-A-strong if there exists a j: V → M which witnesses that κ is η-strong and which has the additional feature that j(A ∩ Vκ) ∩ Vη = A ∩ Vη.  The following large cardinal notion plays a central role in the search for new axioms.