This is a question related to ideas raised in http://arxiv.org/abs/1410.1224 and http://arxiv.org/pdf/1405.7456.pdf. Basically, the idea is the following:

Suppose I have a first-order theory $T$. Under what conditions are there "few" models of $T$ across all possible forcing extensions of the universe?

Maybe "few" is the wrong word - what I really mean is, when does the collection of "models of $T$ you can get by forcing" have some nontrivial structure?

There are a bunch of aspects of this question; the one I'm interested in right now is the following:

Say that a theory $T$ is

generically embeddableif - whenever $\mathbb{P}$ is a forcing notion and $\nu$ is a $\mathbb{P}$-name such that $\Vdash_\mathbb{P}``\nu\models T"$ - there is some forcing notion $\mathbb{Q}$ and $\mathbb{Q}$-name $\mu$ such that $$\Vdash_{\mathbb{P}\times\mathbb{Q}} ``\mbox{$(\mu[G_1]\models T)$ and there is a homomorphism $h\colon \nu[G_0]\rightarrow\mu[G_1]$}."$$ (Note that this homomorphism of course exists in $V[G_0\times G_1]$.)

To get a sense of what this means, let's look at a counterexample. Take $T$ to be the true theory of second-order arithmetic - that is, $T=Th(\omega\sqcup 2^\omega; +, \times, \in)$. Then models of $T$ code reals, and this is bad. Specifically, let $\mathbb{P}$ be Cohen forcing, and let $\nu$ be the $\mathbb{P}$-name picking out the "true" model $(\omega\sqcup 2^\omega; +, \times, \in)^{V[G]}$. Then if $G_0\times G_1$ is $\mathbb{P}\times\mathbb{Q}$ generic - for any $\mathbb{Q}$! - since the real $G_0$ won't be in $V[G_1]$, no model $N$ of $T$ in $V[G_1]$ will embed $\nu[G_0]$, since such a homomorphism would have to biject $\nu[G_0]$ with the well-founded segment of $N$, and from this we could recover $G_0$ in $V[G_1]$.

More generally, *theories which let you code sets are bad.*

My question is: **When is a theory generically embeddable?** Specifically, are there model-theoretic niceness properties which guarantee this? For instance, I can't come up with a *stable* example of a non-generically embeddable theory $T$, but I also can't prove there isn't one. I can prove that if $T$ is totally categorical, then $T$ is generically embeddable, but this isn't very interesting (in fact, it's hard not to prove this).

EDIT: Note that there are many natural strengthenings of this: e.g.

We may demand that $\mathbb{Q}=\mathbb{P}$, or even $\mathbb{Q}=\mathbb{P}$ and $\mu=\nu$!

We may ask for the homomorphism $h$ to be an elementary embedding, or satisfy some other strong property (e.g. the papers linked above looked at

*isomorphisms*, not homomorphisms).We may restrict attention to certain classes of forcing notions, or certain names (e.g. the papers linked above looked only at $\nu$ such that $\Vdash_\mathbb{P}$``$\nu[G]$ is countable").

For now, though, I'm interested in the question as it stands.