(1) Are there formulae $\varphi\_D(x)$ and $\varphi\_\in(x,y)$ defining an internal model $\mathcal{N}$ of $ZFC$ where $\mathcal{N}$ is not set-like and no definable, set-like, internal model $\mathcal{M}$ is elementary equivalent to $\mathcal{N}$?


(2) Are there formulae $\varphi\_D(x)$ and $\varphi\_\in(x,y)$ defining an internal model $\mathcal{N}$ of $ZFC$ where $\mathcal{N}$ is not set-like and no definable, set-like, $\it{well-founded}$, internal model $\mathcal{M}$ is elementary equivalent to $\mathcal{N}$?


A simple way to obtain non set-like models of $ZFC$ is to take a normal ultrafilter $U$ and take the iterated ultrapower of $V$ through length $Ord$. This gives us a non set-like model of $ZFC$, but it is elementary equivalent to $V$. (This approach uses parameters, but they are easily removed.)


In a vague nutshell, is there another way to define "long" models without some form of iteration?



Is it possible to define an internal model of ZFC which is not set-like and which is not elementary equivalent to any definable set-like model?
Let me describe another method for making long models definable.


First consider the case that $\kappa$ is inaccessible in $L$ and $\lambda$ is a worldly cardinal in $L$ above $\kappa$. Let $G$ be $L$-generic for the forcing to collapse $\lambda$ to $\kappa$. In the forcing extension, there is a set $E\subset\kappa$ that codes the structure $\langle L\_\lambda,\in\rangle$. Let $L[G][H]$ be the forcing extension that codes the set $E$ into the GCH pattern up to $\kappa$. This preserves the inaccessibility of $\kappa$. Let $M=(V\_\kappa)^{L[G][H]}$, which is a model of ZFC since $\kappa$ remained inaccessible. In $M$, we can define $E$ using the GCH pattern and therefore in $M$ we can define a copy of the structure $\langle L\_\lambda,\in\rangle$, which is a model of ZFC. This model is actually well-founded, and so the model $M$ will also look upon it as well-founded, and it is not set-like, since it has height $\lambda$, which is taller than $\kappa$.


So this method shows how a model of ZFC can define a much taller well-founded model of ZFC.


In your question, however, you had asked for more. You wanted the defined model to have a theory not realized in any set-like interpreted model of $M$. Let me modify the construction to achieve something closer to this. I will show how to arrange that $N$ satisfies a theory not satisfied by any set structure in $M$ nor any definable well-founded set-like class structure. (Thanks, Ali, for your comments about this.)


To do it, I will suppose not that $\kappa$ is actually inaccessible in $L$, but rather merely that it is inaccessible in $L\_\lambda$, which is itself a pointwise definable model. One can make this situation from the situation above simply by taking the Mostowski collapse of the definable elements of $L\_\lambda$. Now, these are countable ordinals in $L$, even though $L\_\lambda$ thinks $\kappa$ is inaccessible. But we can still do the forcing $G$ and $H$ and form the model $M=(V\_\kappa)^{L\_\lambda[G][H]}$ just as above, except that we take $G$ merely $L\_\lambda$-generic and $H$ is $\langle L\_\kappa,\in,E\rangle$-generic. In $M$, again we can define $L\_\lambda$ as a well-founded non-set-like model of ZFC.


The point of the pointwise definability assumption is that the theory $T$ of $L\_\lambda$ ensures that every model of it contains a copy of $L\_\lambda$. There can be no set model of this theory in $M$, because then $M$ would be able to take the definable elements of that model and thereby produce a copy of $L\_\lambda$ as a set, which is impossible in $M$ since $\kappa<\lambda$. Similarly, there can be no well-founded set-like class definable model of the theory in $M$, since any such model $N$ would have to have an ordinal with at least $\kappa$ many predecessors, and such an order relation would not be set-like in $M$.


What remains is the possibility that there could be a definable set-like class model in $M$, not necessarily well-founded, but which satisfies the theory. I don't at the moment know how to rule this out. The subtle point is that a set-like order in $M$ can contain a copy of $\lambda$ — after all, both $\kappa$ and $\lambda$ are countable ordinals and hence embed into $\mathbb{Q}$, which is set-like in $M$ — and the pointwise definability idea doesn't seem quite enough to rule out such ill-founded interpreted classs models.


Therefore, this answer doesn't quite fully answer the question.


