I had some ideas regarding models of ZF. My ideas (phrased as questions) are:
- Given two models of ZF, what are the condition for a model containing both models (in the sense of embedding) to exist?
- What are the condition for the existence of minimal model (up to isomorphism) of this kind?
- Can this be generalized to a general collection of models, rather than two?
- Can this be generalized for other (first-order?) theories?
I am not looking for something specific, I am trying to find out more about these ideas. I will be grateful for any insights concerning these questions, their application or anything relevant.
I think you can embed any two ZF models into a bigger one as follows: Let the models be $M, N$. Let $T$ be the theory ZF plus the atomic diagram of $M$ and of $N$ using distinct constant symbols for each member of $M, N$ (notice that we do not add any statement involving constant symbols from both $M$ and $N$). This is finitely satisfiable because any finite graph (no cycles by foundation) can be $\in$-embedded into any model of ZF. Hence you can take a model of $T$ in which both $M, N$ are embedded.