# Generic two-cardinal behavior of first-order sentences

This is a hopefully improved version of a question I asked before and then deleted because it was based on some fundamentally incorrect assumptions.

Some first-order theories are able to control the relative sizes of definable sets, i.e. for some designated unary predicate $P$ and $Q$, there is some cardinal function $f$ such that $|P(\mathfrak{A})|\leq f(|Q(\mathfrak{A}|)$ for all models $\mathfrak{A}$.

It follows from Vaught's theorem on cardinals far apart and a compactness argument that a theory controls the relative size of $P$ and $Q$ if and only if some sentence does, so we can focus our attention on sentences.* If $\varphi$ is a sentence in any language containing $P$ and $Q$, then for any model $V$ of $ZFC$, let

$$f^V_\varphi (\kappa)=\sup\{ |P(\mathfrak{A})|^V:V \ni \mathfrak{A}\models \varphi,|Q(\mathfrak{A})|^V=\kappa \}$$

We really only care about this for infinite cardinals because the behavior with finite cardinals is easy to classify, so assume that $\varphi$ is consistent with both $P$ and $Q$ being infinite (although don't assume that it necessarily implies that they are infinite). Given this assumption $f^V_\varphi(\kappa)\geq\kappa$ always. By Vaught's theory on cardinals far apart we have a dichotomy: either $f^V_\varphi (\kappa)=\infty$ for all infinite $\kappa$ or $f^V_\varphi(\kappa)\leq\beth_n^V (\kappa)$ for some $n<\omega$. And this is an absolute property since it can be determined by the consistency of ${\varphi}\cup T$ for some fixed countable theory $T$ in an expanded language (although I don't know if the specific $n$ is absolute, but I heavily suspect there's always an absolute upper bound on $n$). Vaught's two-cardinal theorem also shows that $f^V_\varphi (\kappa)=\kappa$ is an absolute property. There's a very precise syntactic characterization of when $f^V_\varphi (\kappa)=\kappa$ given in Hodges' Model Theory in terms of 'layerings' and broadly speaking I'm interested in precise syntactic characterizations of particular $f^V_\varphi (\kappa)$, but the question needs to be phrased correctly and there are more preliminary questions that are already difficult.

Certain cardinal functions can be exhibited. For example, $f^V_\varphi (\kappa)=\kappa,\,\kappa^+,\,\text{ded}(\kappa),\,2^\kappa$ are all possible, and the set of such functions is closed under composition and cardinal addition and exponentiation, but depending on $V$, $\kappa^+$ might equal $2^\kappa$, so there's no hope of a syntactic characterization of $f^V_\varphi (\kappa)=\kappa^+$ as opposed to $f^V_\varphi (\kappa)=2^\kappa$ without thinking about different models of set theory.

It's well known that in general two-cardinal questions are very difficult and sensitive to set theory, but one might hope that forcing generic versions of such questions would be tamer. So given two sentences $\varphi$ and $\psi$, write $\varphi \sim \psi$ if $f_\varphi^{V[G]}=f_\psi^{V[G]}$ for all forcing extensions $V[G]$. So then we arrive at some natural questions:

Question 1: How absolute is $\sim$?

Question 2: Is there a simpler description of $\sim$ not in terms of forcing?

Question 3: What are the equivalence classes of $\sim$?

Question 4: Is there a simple list of cardinal functions that generate precisely the equivalence classes of $\sim$?

*I realized that there might be a theory $T$ such that $T \not\sim \varphi$ for any $\varphi$ such that $T\vdash \varphi$ (or even any $\varphi$ at all) and it may even be that $f_T^V(\kappa) \neq \inf \{f_\varphi^V(\kappa):T \vdash \varphi\}$, but the absoluteness question probably becomes a lot more subtle in this case since $T$ isn't a finite object, so I want to ignore it for now.