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 $M$?

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 $M$?

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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 $j.$

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 $M$?

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 $M$?