I am just getting acquainted with "very strong" large cardinal axioms, and it seems there is a consensus that among large cardinal axioms, the rank-into-rank cardinal axioms are at the threshold of consistency with $ZFC$. Axiom $I1$ (or $EE(1)$) states that there is a non-trivial elementary embedding $j$ from $V_{\lambda+1}$ into itself for some cardinal $\lambda$. Then $\kappa = crit(j)$ is a large cardinal satisfying various (other) large cardinal properties. Kunen Inconsistency Theorem tells us we cannot replace $\lambda+1$ with $\lambda+2$ in the above statement. My brief survey of the scene suggests that, besides Kunen's theorem, the tool shed for proving inconsistency of large cardinal axioms is rather spartan. Hence I am curious if the following somewhat natural strengthenings of Axiom $I1$ have been found to be inconsistent with $ZFC$.
- There is a cardinal $\kappa$ such that for all ordinal $\alpha$, there is a cardinal $\lambda > \kappa + \alpha$ and a non-trivial elementary embedding $j$ from $V_{\lambda+1}$ into itself with $crit(j) = \kappa$ and $j(\kappa) > \alpha$.
- There is a non-trivial elementary embedding $j$ from $V_{\lambda+1}$ into itself for some cardinal $\lambda$, and for all transitive class $M$ with $ORD \cup V_{\lambda+1} \subset M \not\models ZFC$ (or even just $V_{\lambda+1} \subset M \not\models ZFC$), $j$ can be extended to an elementary embedding from $M$ into itself.