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 $ZFC$.

  1. 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 $j(\kappa) > \alpha$.
  2. 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.
  • $\begingroup$ What is the role of $\kappa$ in (1)? Clearly it can't be the critical point of all these embeddings. $\endgroup$ – Asaf Karagila Jun 16 '16 at 17:26
  • $\begingroup$ Also, for (2) the term "Inner model" usually refers to class-models of $\sf ZFC$, but in some contexts of $\sf ZF$. It's not clear what you mean by inner model here. $\endgroup$ – Asaf Karagila Jun 16 '16 at 17:27
  • $\begingroup$ @Asaf: the $\kappa$ in (1) is exactly the critical point of all those embeddings. I am thinking of it as some kind of upward reflection property. For (2) I shall remove the part in parenthesis so as not to confuse. I might have interpreted the term "inner model" wrongly. Of course this whole question may sound dumb, but I am trying to push things as far as I can think of to see where and how things break down w.r.t. consistency with $ZFC$. $\endgroup$ – John Toh Jun 16 '16 at 17:48
  • $\begingroup$ If for every $\alpha$, the critical point is above $\alpha$, that makes no sense. $\kappa$ should be fixed. $\endgroup$ – Asaf Karagila Jun 16 '16 at 17:53
  • $\begingroup$ @JohnToh How does that work exactly for (1)? What if $\alpha>\kappa$ - then $crit(j)>\alpha$ so $crit(j)\not=\kappa$. $\endgroup$ – Noah Schweber Jun 16 '16 at 17:53

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