Let $\sigma(x)$ be a formula (in one free variable $x$) in the language of set theory such that $\mathsf{ZFC} \models \forall x (x \text{ countable } \Rightarrow \sigma(x))$. (E.g., $\sigma(x) =$ "$x$ is countable".)
Let $V$ be a model of $\mathsf{ZFC}$ and let $M \in V$. We add to the language of set theory a constant symbol $c$ whose interpretation in $V$ should be $M$. Then, in this new language, we have a sentence $\sigma(c)$ that says, when interpretated in $V$, that $M$ satisfies $\sigma$.
By forcing, we find a generic extension $V[G]$ of $V$ in which $M$ is countable, see for instance this post. Hence $V[G] \models \mathsf{ZFC} \cup {\sigma(c)}$.
I have three questions:
(1) Is it true from that $\mathsf{ZFC} \models \sigma(M)$? (or might there be some issue that in $V[G]$ the constant symbol $c$ has an interpretation different from $M$?)
(2) If (1) has a positive answer, is it really sensible to say that it is consistent with $\mathsf{ZFC}$ that $M$ satisfies $\sigma$ or does this not make sense because $M$ is some fixed set that may not be definable via set-theoretical language?
(3) What if we replace $M$ by a whole subclass of $S \subseteq V$ (for instance, instead of the singleton class containing $M$, we do the above process for the class $S$ of all partially ordered sets). Using the Compactness Theorem, can we say that it is consistent with $\mathsf{ZFC}$ that every $M \in S$ satisfies $\sigma$?
