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

TL;DR

This paper explores the consistency of certain large cardinal axioms involving elementary embeddings in ZF set theory, extending known results from ZFC and constructing inner models with these properties.

Contribution

It proves the existence of inner models with elementary embeddings under ZF assuming $I_{0, ext{lambda}}$ and $ ext{lambda}$ even, without relying on the Axiom of Choice.

Findings

Inner models with elementary embeddings are constructed in ZF assuming $I_{0, ext{lambda}}$ and $ ext{lambda}$ even.

The existence of $A^ ext{ extasteriskcentered}$ for all $A ext{ in }V_{ ext{lambda+1}}$ is implied by the assumptions.

The theory does not necessarily imply the existence of $V_{ ext{lambda+1}}^ ext{ extasteriskcentered}$.

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.