Work in $ZF$. Let $j:V\to V$ be a non-trivial elementary embedding which is iterable, so that we can iterate it and form models $M\_\alpha, \alpha\in ON,$ with $M\_0=V,$ and elementary embeddings $j\_{\alpha, \beta},$ for $\alpha\leq \beta.$


Let $M=\bigcap\_\alpha M\_\alpha.$



> 
> What can we say about $M$?
> 
> 
> 


It seems it is not difficult to show that $M$ is also a model of $ZF$, since it suffices to show that it is closed under Godel operations.



> 
> Which large cardinals of $V$ are preserved in $M$? 
> 
> 
> 


Of course the answer seems to be trivial for some large cardinals, in particular for those below the critical point of $j$, where their existence require information in $V\_{crit(j)}$ So I am particularly interested in those large cardinals whose definitions require a proper class of information, like supercompact cardinals, ...



> 
> What interesting properties $M$ can have? In particular what can we say about the relation between $V$ and $M$?
> 
> 
> 


--


**Remark.** As it is stated in the comments by Hamkins, any such embedding is iterable, so we can remove the extra assumption of iterability of $j.$



Reinhardt cardinals and iterability
Note that $M=\bigcap\_{\alpha}M\_\alpha$ is just the $\mathrm{OR}^{\mathrm{th}}$ iterate $M\_{\mathrm{OR}}$ cut off at height $\mathrm{OR}$, so we have for example $V\_\kappa\preceq V\_\lambda\preceq M$ where $\kappa=\mathrm{crit}(j)$ and $\lambda=\kappa\_\omega(j)$, where $\kappa\_0(j)=\mathrm{crit}(j)$ and $\kappa\_{n+1}(j)=j(\kappa\_n(j))$ and $\kappa\_\omega(j)=\sup\_{n<\omega}\kappa\_n(j)$. So in terms of large cardinals with a first-order description, $M$ has the same kinds as does $V\_\kappa$. If there is an elementary $k:V\to V$ with $\kappa\_\omega(k)<\kappa$ then arguing like in [A weak reflection of Reinhardt by super Reinhardt cardinals](https://arxiv.org/abs/2005.11111), we get an embedding $\ell:M\to M$ witnessing that $\kappa\_n(k)$ is Reinhardt for $M$, for some $n<\omega$. (That is, for some $n<\omega$, we get $k\_n(\kappa)=\kappa$, and then we can just stretch out the structure $(V\_\kappa,{k\_n\upharpoonright V\_\kappa})$ to produce such an $\ell$.)


There are some facts established in [Reinhardt cardinals and iterates of V](https://arxiv.org/abs/2002.01215) on the relationship between the iterates $M\_\alpha$ and $V$. In particular, it is shown that for each $\alpha\geq\omega$, every set is set-generic over $M\_\alpha$, but $M\_\alpha$ is not a set-ground of $V$. And $\mathrm{HOD}\subseteq M\_\omega$ but $\mathrm{HOD}\not\subseteq M$.


