# How often are forcing extensions of countable computably saturated models of $\mathsf{ZFC}$ computably saturated?

Recall that given a finite language $$\mathcal{L}$$, we say that an $$\mathcal{L}$$-structure is computably saturated (or recursively saturated) if for any computable set $$\Sigma(\bar{x},y)$$ of $$\mathcal{L}$$-formulas in the variables $$\bar{x}y$$ and any $$\bar{a} \in M^{\bar{x}}$$, if $$\Sigma(\bar{a},y)$$ is finitely satisfiable in $$M$$, then it is satisfied in $$M$$. An easy inductive argument shows that every countable $$\mathcal{L}$$-structure has a countable computably saturated elementary extension. Another easy argument shows that any expansion of a computably saturated structure by finitely many constants is still computably saturated.

One important property of countable computably saturated structures is resplendence, which for our purposes can be defined like this: An $$\mathcal{L}$$-structure $$M$$ is resplendent if for any finite extensions $$\mathcal{L}' \supseteq \mathcal{L}$$ and any $$\mathcal{L}'$$-sentence $$\varphi$$ that is consistent with $$\mathrm{Th}(M)$$, there is an expansion $$M'$$ of $$M$$ that is a computably saturated model of $$\mathrm{Th}(M)\cup\{\varphi\}$$.

All countable computably saturated structures are resplendent. Moreover, the same is true after any expansion by any finite number of constants.

Forcing is typically defined in terms of well-founded models of $$\mathsf{ZFC}$$, but, as discussed in the answers to this question, forcing ultimately makes sense over arbitrary models of $$\mathsf{ZFC}$$: Given an model $$M$$ of $$\mathsf{ZFC}$$ and a forcing poset $$\mathbb{P} \in M$$, we can consider the language that contains $$\in$$, $$\mathbb{P}$$, and a unary predicate $$G$$. There is a single sentence $$\chi(\mathbb{P},G)$$ in this language that says that $$\mathbb{P}$$ is a forcing poset and $$G$$ is a generic filter on $$\mathbb{P}$$. As Emil Jeřábek points out in his answer to that question, $$\mathsf{ZFC} \cup \{\chi(\mathbb{P},G)\}$$ is a conservative extension of $$\mathsf{ZFC}$$. (The rest of the work of a forcing argument is showing that $$(M,\mathbb{P},G)$$ interprets an end extension of $$M$$ modeling $$\mathsf{ZFC}$$ in which $$G$$ is a set. We don't really need to worry about that for the sake of this question.)

We have, furthermore, that the theory $$\mathrm{eldiag}(M) \cup \{\chi(\mathbb{P},G)\}$$ is consistent. If $$M$$ is a countable computably saturated model of $$\mathsf{ZFC}$$, then, by resplendence, we have that for any forcing poset $$\mathbb{P}\in M$$, there is a $$G \subseteq \mathbb{P}$$ which is $$M$$-generic such that $$(M,\mathbb{P},G)$$ is computably saturated. (In particular, this means that the actual forcing extension $$M[G]$$ will be computably saturated as well.)

My question is about whether this always happens, 'usually' happens, or 'usually' doesn't happen for various choices of $$G$$.

Question 1: Let $$M$$ be a countable computably saturated model of $$\mathsf{ZFC}$$, and let $$\mathbb{P} \in M$$ be a forcing poset. If $$G \subseteq \mathbb{P}$$ is an $$M$$-generic filter, does it follow that $$(M,\mathbb{P},G)$$ is computably saturated?

I doubt that this does in fact always hold, but I don't know how to build a counterexample. What's less certain to me is whether a 'sufficiently generic' $$M$$-generic filter results in a computably saturated expansion.

To state this carefully, we need the following observation: Fix a countable computably saturated model $$M$$ of $$\mathsf{ZFC}$$ and a forcing poset $$\mathbb{P} \in M$$. Let $$A$$ be the set of all elements $$\alpha$$ of $$\mathbb{P}^\omega$$ such that for each $$i<\omega$$, $$\alpha(i+1) \leq \alpha(i)$$. $$\mathbb{P}^\omega$$ can easily be identified with Baire space, and $$A$$ is a $$G_\delta$$ subset of $$\mathbb{P}^\omega$$ and therefore itself a Polish space. It's easy to see that the set $$B = \{\alpha \in A : \alpha\text{ generates an }M\text{-generic filter}\}$$ is comeager in $$A$$. (This is essentially the Rasiowa–Sikorski lemma.) For any $$\alpha \in A$$, let $$G_\alpha = \{p \in \mathbb{P} : p \geq \alpha(i)\text{ for some }i\}$$. (This is what we mean by 'generating' a filter.)

Question 2: Is the set $$\{\alpha \in B : (M,\mathbb{P},G_\alpha)\text{ is computably saturated}\}$$ always comeager in $$A$$?

I would also be interested in answers involving a weaker set theory than $$\mathsf{ZFC}$$.

The answer to Question 1 is positive (thus the answer to Question 2 is also positive). More explicitly, the positive answer to Question 1 follows from the following well-known facts:

Lemma 1. $$(M,\mathbb{P},G)$$ is parametrically definable in $$M[G]$$.

Lemma 2. $$M[G]$$ is recursively saturated.

Lemma 3. Any structure that is parametrically definable in a computably saturated structure is also computably saturated.

Lemma 1 is due independently to Laver and Woodin, who proved that $$M$$ is parametrically definable in $$M[G]$$; see this MO post of Hamkins for more detail.

Lemma 2 was proved as part of the proof of Theorem 2.6 of this paper of mine. You can also find a proof of Lemma 2 in this blogpost of Kameryn Williams (see the third proposition). The same blogpost also includes a reference to another result of mine which shows that Lemma 2 can fail if $$\mathbb{P}$$ is a proper class notion of forcing in $$M$$.

Lemma 3 follows from the definitions involved.

Finally, let me point out that Lemma 2 is true for all models $$M$$ of ZF, but I do not know the status of Lemma 1 for a model of ZF in which AC fails.