If $j : V \rightarrow M$ is an elementary embedding, what can we learn in $M$ from $j''ORD$? That is, what is $M[j''ORD]$? 


In particular, 



> Is it $M[j''ORD]$ equal to all of $V$?


If not, do we get a model intermediate between $M$ and $V$? If $\kappa$ is the critical point of $j$, is $\kappa$ still a large cardinal in $M[j''ORD]$? I am thinking of $j$ arising from a measure on $\kappa$, but I'm also interested in the more general situation.


My thinking goes like this: If we have the image of all of $V$, we can reconstruct $V$ itself by taking the Mostowski collapse of $j''V$ (and $j$ is the inverse of the Mostowski collapse). In $M[j''ORD]$, let's consider the class $W$ of sets with rank in $j''ORD$. Does the Mostowski collapse of $W$ yield all of $V$, and if not, what's missing? 


EDIT: formatting, clarification of question.



What can we learn about an elementary embedding from the image of the ordinals?
Nice question, Jonas!


Yes, in the case that $V$ satisfies ZFC and $M\subset V$, then indeed $M[j''\text{Ord}]=V$. To see this, consider first the case of a set of ordinals $A\subset\theta$ in $V$. Notice that from $j''\theta$ we may reconstruct $j\upharpoonright\theta$. Further, $j(A)$ is in $M$, and from $j(A)$ and $j\upharpoonright\theta$ we may easily reconstruct $A$ itself. So every set of ordinals in $V$ is in $M[j''\text{Ord}]$. If $V$ satisfies ZFC, then this suffices, since every set is coded by a set of ordinals. Namely, (as you know) if $a$ is any set, then $\langle \text{TC}(\{a\}),{\in}\rangle\cong\langle\theta,E\rangle$ for some cardinal $\theta$ and some binary relation $E$ on $\theta$, and then using a Gödel pairing function to code $E$ as a single set $A\subset\theta$. So for any set $a$, we find $E$ and then $A$, and by the previous argument $A$ is in $M[j''\text{Ord}]$, and so also $E$ and hence $a$ itself is there. Thus, $M[j''\text{Ord}]=V$, as desired. 


The argument relies on the axiom of choice, and in the most general case, I believe this is required. 


