  Nevertheless, there is a lot of room below the above upper bound.  For example, a very strong axiom is the statement that there is a non-trivial elementary embedding j:Vλ+1→ Vλ+1.  The strongest large cardinal axiom in the current literature is the axiom asserting that there is a non-trivial elementary embedding j: L(Vλ+1)→ L(Vλ+1) such that crit(j)&lt;λ.  In recent work, Woodin has discovered axioms much stronger than this.