On the consistency of ZF with an elementary embedding from Vλ+2 into Vλ+2

According to a theorem due to Kenneth Kunen, under ZFC, there is no ordinal $\lambda$ and nontrivial 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 is has been discovered.

$I_{0,\lambda }$ is the assertion, introduced by 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 $k$ ⊆$j$, 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.