Is ZFC, in a pedantic sense, a different theory for every structure we're considering?

Question

(As a quick note, I'm using ZFC for concreteness, but this property applies to any theory with an axiom schema that uses parameters.)

If we look specifically at the axiom schema of separation and the axiom schema of replacement, we can see that they quantify over formulas with parameters $\varphi(\cdots)$ and not merely formulas.

However, one of the main things you can do with a theory is check whether a given structure satisfies it, $A \models T$.

This makes me wonder we can meaningfully call ZFC something other than a theory, like a theory-valued function or a parameterized theory or something and instead write $A \models \text{ZFC}(A)$ where $\text{ZFC}(A)$ tracks the fact that the parameters used to spell out the axiomatization of $\text{ZFC}$ are taken from $A$. (This also opens the door to potentially fun stuff like comparing $\text{ZFC}(A)$ and $\text{ZFC}(\Omega_A)$ where $\Omega_A$ contains exactly the ordinals inside of $A$.)

As is mentioned in this comment by Noah Schweber, among other places, this doesn't really cause problems in practice. My question is specifically not about potential foundational circularity caused by having parameters; it's merely about what to call ZFC to distinguish it from other theories that don't "change" depending on which structure we're inspecting and whether it makes sense to ever make this distinction in practice.

Answer 1

You're overlooking the universal quantifier(s) on the outside of the relevant sentences.

---

Let $S=\{\in\}$ be the language of set theory, and for $\mathcal{M}\models\mathsf{ZFC}$ let $S_\mathcal{M}$ be the language $S$ augmented by a constant $c_m$ for each $m\in \mathcal{M}$; we can conflate $\mathcal{M}$ with its expansion to an $S_\mathcal{M}$-structure in an obvious way.

Let's look at a single instance of separation for simplicity: given a formula $\varphi(x,y,z_1,...,z_n)$ in the language of set theory (without parameters), the corresponding instance of separation is $${\color{red}{\forall z_1,...,z_n}}\forall x\exists u\forall y[y\in u\leftrightarrow y\in x\wedge\varphi(x,y,z_1,...,z_n)].$$ This is a single sentences, but the red part lets the black part range over all $n$-tuples of parameters from whatever model we're working in. This individual sentence "does the same job" as the operation assigning, to each structure $\mathcal{M}$ in the language of set theory, the set of $S_\mathcal{M}$-sentences $$\{{}"\forall x\exists u\forall y[y\in u\leftrightarrow y\in x\wedge\varphi(x,y, c_{a_1},...,c_{a_n})]"{}: a_i\in\mathcal{M}\},$$ despite being a vastly simpler object.

---

Note that we can ask the same question about much simpler theories. Take (say) abelian group theory: is commutativity a single sentence in the language of group theory, or is it a structure-dependent set of sentences given by $$\{{}"a+b=b+a"{}: a,b\in\mathcal{X}\}$$ for each appropriate structure $\mathcal{X}$?

The culprit, I think, is our natural-language use of the term "parameters" to single out some variables in the separation/replacement schemes as auxiliary, together with a common (and unfortunate) suppression of outermost universal quantifiers. But there's fundamentally no issue here.

Answer 2

No: $\models$ is a relation between a structure (a semantic object) on the left and a theory, i.e., a set of sentences (a syntactic object) on the right. The definition of $\models$ captures what a sentence asserts about a structure. There is no sense in which ZFC changes depending on what structure we're inspecting. ZFC is defined purely syntactically as the deductive closure of a set of axioms and inference rules. In the remark by Noah Schweber that you cite, the parameter $p$ is just a syntactic variable and not an element of a structure (that's why Noah is able to form a sentence that quantifies over $p$).

PS: re the comments about schemata: it is $\varphi$ in $\varphi(...)$ that is the schematic variable in a presentation of an axiom schema, the variables that appear between the brackets are just ordinary logical variables. But this is just about our conventions for presenting an infinite number of axioms in a concise way.