Consider the three following large cardinal axioms:


1. there exists a nontrivial elementary embedding $j:V\to V$.
2. there exists a n.e.e. $j:V\to M$ such that $M^{j^\omega(crit(j))}\subseteq M$.
3. there exists a n.e.e. $j:V\_{\lambda+2}\to V\_{\lambda+2}$ for some $\lambda$.


Kunen's inconsistency theorem states that, in ZFC (with replacement extended to formulas containing $j$), these three statements are false. However there is no such (known) result within ZF (with replacement sufficiently extended) alone.


Clearly (1) implies (2). I remember seeing somewhere that (2) implies (1), but I don't remember where, so I might be mistaken. Does anyone knows how these three statements relates to each other?


EDIT: [this question](https://mathoverflow.net/questions/87222/what-are-the-known-implications-of-there-exists-a-reinhardt-cardinal-in-the-th?rq=1) already deals with the case (1)=>(3) 



Are these large cardinals properties equivalent?
No.


This is a very recent work in progress of myself with Juan Aguilera.



> 
> **Definition.** We say that $\kappa$ is a *Kunen cardinal* if there is a nontrivial elementary embedding $j\colon V\_{\lambda+2}\to V\_{\lambda+2}$ with $\lambda=\sup j^n(\kappa)$.
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(Sometimes it is easier to talk about the critical points of the sequence, and sometimes on $\lambda$, like in the case of $I0$ and such.)



> 
> **Theorem.** If $\kappa$ is a Reinhardt cardinal, i.e. the critical point of an elementary embedding $j\colon V\to V$, then $\kappa$ is the limit of Kunen cardinals.
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*Proof.* Note that being a Kunen cardinal is a first-order property in the language of set theory. And note that a Reinhardt cardinal itself is Kunen by a fairly easy verification. 


Let $j\colon V\to V$ be a nontrivial elementary embedding, and $\kappa$ its critical point. Now, it is easy to see that the set $A\subseteq\kappa$ defined as $\{\mu<\kappa\mid\mu\text{ is a Kunen cardinal}\}$ satisfies $\kappa\in j(A)$. Therefore it follows that $A$ is in the normal measure derived from $j$, and therefore it is stationary. 


In particular, a Reinhardt cardinal is the limit of Kunen cardinals. $\square$


