On the consistency of ZF with an elementary embedding from $V_{\lambda+2}$ into $V_{\lambda+2}$

Abstract

According to a theorem due to Kenneth Kunen, under ZFC, there is no ordinal $\lambda$ and non-trivial elementary embedding $j:V_{\lambda+2}\to V_{\lambda+2}$. His proof relied on the Axiom of Choice (AC), and no proof from ZF alone has been discovered. $I_{0,\lambda}$ is the assertion, introduced by W. Hugh Woodin, that $\lambda$ is an ordinal and there is an elementary embedding $j:L(V_{\lambda+1})\to L(V_{\lambda+1})$ with critical point ${<\lambda}$. And $I_0$ asserts that $I_{0,\lambda}$ holds for some $\lambda$. The axiom $I_0$ is one of the strongest large cardinals not known to be inconsistent with AC. It is usually studied assuming ZFC in the full universe $V$ (in which case $\lambda$ must be a limit ordinal), but we assume only ZF. We prove, assuming ZF + $I_{0,\lambda}$ + "$\lambda$ is an even ordinal", that there is a proper class transitive inner model $M$ containing $V_{\lambda+1}$ and satisfying ZF + $I_{0,\lambda}$ + "there is an elementary embedding $k:V_{\lambda+2}\to V_{\lambda+2}$"; in fact we will have $k\subseteq j$, where $j$ witnesses $I_{0,\lambda}$ in $M$. This result was first proved by the author under the added assumption that $V_{\lambda+1}^\#$ exists; Gabe Goldberg noticed that this extra assumption was unnecessary. If also $\lambda$ is a limit ordinal and $\lambda$-DC holds in $V$, then the model $M$ will also satisfy $\lambda$-DC. We show that ZFC + "$\lambda$ is even" + $I_{0,\lambda}$ implies $A^\#$ exists for every $A\in V_{\lambda+1}$, but if consistent, this theory does not imply $V_{\lambda+1}^\#$ exists.