Making sense of formulas of non-standard length in set theory?

Question

I’m trying to understand the implications of this question and the posted answers/comments. What I think I understand so far is the following:

Suppose we have a metalanguage and a $\mathsf{ZFC}$ universe $V$ whose $\omega$ is isomorphic the whole numbers of the metalanguage. Within $V$ we can define an $\omega$-nonstandard model of $\mathsf{ZFC}$ (call it $\mathcal{M}$) which has its own smallest inductive set (call it $\omega^{\mathcal{M}}$). And within $\mathcal{M}$ we can imagine an encoding where the first few elements of $\omega^{\mathcal{M}}$ are interpreted as logical symbols ($\exists$, $\land$, etc.) and the rest are interpreted as “variables”. And so if $\alpha$ is any nonstandard element of $\omega^{\mathcal{M}}$, then according to this encoding, many functions $\alpha \mapsto \omega^{\mathcal{M}}$ could be interpreted as “formulas” of nonstandard length.

So far so good. But why do we care about these essentially fake nonstandard-length “formulas” (which we invented by imposing an interpretation on certain functions within $\mathcal{M}$)? How are they at all relevant? My understanding was that when we “do set theory” we are assuming up front a fixed metalanguage (with a fixed notion of metalanguage numbers), and so the possible lengths of formulas (being defined by the fixed metalanguage) is the same regardless of which model the formulas are talking about. Is this understanding wrong? If so, then how do we know/decide what each model’s unique notion of metalanguage number is supposed to be?

Update (to add a bit more detail to what I’m wondering about):

For brevity, I’ll refer to the notion of formula defined by the metalanguage as “metalanguage formulas” and the notion of formula defined by encoding functions $\alpha \mapsto \omega^{\mathcal{M}}$ within $\mathcal{M}$ as “internal formulas”. Then:

1. When we speak of something like “the proper classes of model $\mathcal{M}$”, which notion of “formula” is being used to define what a proper class is? Given that the possible “lengths” of metalanguage formulas may differ from the possible “lengths” of internal formulas (due to the nonstandardness of $\omega^{\mathcal{M}}$), I don’t see any reason to assume that we end up with the same proper classes regardless of which notion of “formula” we are employing. Is it perhaps assumed by default (unless otherwise specified) that “formula” means “metalanguage formula”?
2. Similarly, due to (for example) the possible nonstandardness of $\omega^{\mathcal{M}}$, “ZFC” may mean different things depending on whether we’re talking in terms of metalanguage formulas or internal formulas (e.g. the formulas comprising the Axiom Schema of Replacement won’t be identical). If model $\mathcal{M}$ satisfies all axioms of ZFC in the sense of metalanguage formulas, does that necessarily mean it also satisfies all axioms of ZFC in the sense of internal formulas? It seems not at all obvious that this would necessarily be true.

Update 2:

For what it’s worth, the original motivation that led to this question was this paper, which on page 3 says that absoluteness of the satisfaction relation fails only for formulas that are not of “standard-finite length”

Answer 1

I think it's not so much that nonstandard-length objects are "useful" but that it's important to remember that they exist. For example, the fact that PA+¬Con(PA) is consistent might seem paradoxical, since a model of PA+¬Con(PA) must include a witness for ¬Con(PA), i.e. a number $w$ encoding a proof of contradiction from PA, which (this is the error) ought to correspond to a real proof that PA is inconsistent. The paradox is avoided by noticing that $w$ must be nonstandard.

As to your question about satisfaction of axiom schema instances, satisfaction is only defined for metalanguage formulas. If $\phi$ is an object in $\mathcal M$ representing a nonstandard-length formula, it doesn't make sense to ask whether $\mathcal M$ satisfies $\phi$. In the metalanguage, $\phi$ is not a real formula, and internally we have no "$\mathcal M$". Similarly, I think the question about proper classes needs to be more carefully formulated. (How would you define the class associated with a given internal formula?)

Answer 2

The OP wrote:

"I understand how such nonstandard length, sort of suspicious (fake?) “formulas” can be said to exist. What I don’t understand is whether/how such “formulas” are used for anything."

One simple-minded illustration of the usefulness of such formulas is infinitesimal calculus as done in axiomatic approaches to nonstandard analysis. Here you have unlimited ("infinite") numbers $H$ in $\mathbb N$. This enables one, for instance, to consider a uniform partition of the interval $[0,1])$ into subintervals of infinitesimal length $\frac{1}{H}$. One can then build an infinite Riemann sum over such a partition. The formula expressing such a sum is essentially a formula of unlimited length $H$. If one has an integrable function $f$ on $[0,1]$, the integral of $f$ is then the standard part of such a Riemann sum.