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Kunen showed that if $j:V \rightarrow M$ is a nontrivial elementary embedding from the von Neumann universe $V$ into a transitive class $M$, then $M \neq V$, or equivalently that there are no Reinhardt cardinals, assuming the axiom of choice.

Hamkins, Kirmayer, and Perlmutter (2012) cover a wide breadth of generalizations of Kunen's inconsistency result and cover the metamathematical results surrounding its proof ("a fuller power" of the theorem being revealed when understood in MK or NGBC set theory due to those theories' ability to talk about classes in addition to sets), including the proof's reliance on the axiom of choice. At the end it is stated that whether or not Kunen's full result and thus the nonexistence of a Reinhardt cardinal in ZF alone is an open problem:

Perhaps the principal open question is whether one can prove the Kunen inconsistency without using the axiom of choice.

My question is: What is the current progress on proving or refuting the existence of a Reinhardt cardinal in ZF? Are there any known conjectures that once resolved would imply the answer? Or, in lack of concrete results, what is the most current literature on this topic or any current literature that gives an overview as to the technical way that this problem should be approached?

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In "On the consistency of ZF with an elementary embedding from $V_{\lambda+2}$ into $V_{\lambda+2}$" (arXiv 2006.01077, 2020) Farmer Schlutzenberg showed that assuming a theory based on $I0$ and some dependent choice (specifically that there is an ordinal $\lambda$ such that $\lambda\text{-DC}$ holds and there is an $I0$ embedding $j : V_{\lambda+1} \to V_{\lambda+1}$), then it is consistent with $\mathrm{ZF}$ that there is a $\lambda$ with a nontrivial elementary embedding $k : V_{\lambda+2} \to V_{\lambda+2}$. About the significance of this result in showing inconsistency of $\mathrm{ZF}j+\mathrm{Reinhardt}$, the author says

Moreover, if $\mathrm{ZF}$ is in fact consistent with $k : V_{\lambda+2} \to V_{\lambda+2}$, this might suggest that defeating $j : V \to V$ without $\mathrm{AC}$ could demand significant modifications to Kunen's $\mathrm{ZFC}$ argument (and other arguments which have appeared since).

I am not aware of any previous relative consistency results showing, relative to large cardinals not known to be inconsistent with $\mathrm{ZFC}$, that some principle refuted in $\mathrm{ZFC}$ by Kunen's theorem is still consistent with $\mathrm{ZF}$.

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The problem seems to be open.

I would like to mention the following result of Woodin:

Theorem (ZF). Assume that ZFC proves the HOD Conjecture. Suppose $δ$ is an extendible cardinal. Then for all $λ > δ$ there is no non-trivial elementary embedding $j : V_{λ+2} → V_{λ+2}.$

From this, Woodin concludes the following: Thus (assuming that ZFC proves the HOD Conjecture) one nearly has a proof of Kunen’s Theorem without using the Axiom of Choice.

See Theorem 30 and the remarks after it of the following: The HOD dichotomy.

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