Truth in a different universe of sets?

Question

I understand that provability and truth as different concepts. Provability is syntactic, it only concerns whether the given sentence can be derived by reiterating the inference rules over a collection of axioms. Truth, on the other hand, is a model theoretic notion. Quoted from section 1.3 of [1],

We say that $A$ is a model of $\phi$, or that $\phi$ is true in $A$, if $A \vDash \phi$.

where $A$ is a structure, $\phi$ is an atomic formula in the signature of $A$ without variables, and $\vDash$ defined as usual. And by saying that $\phi$ is true, or $ \vDash \phi$, whenever $A \vDash \phi$ for all structures $A$ (with the same signature).

This question is really about re-examining the definition of truth, with a specific focus on the definition of structure since it depends on what a set means. So again according to section 1.1 of [1], a structure $A$ is a collection of four sets: a set of domain, a set of constant elements, a set of relation symbols, and a set of function symbols.

This definition is the one that confuses me the most. In particular, I wonder what a set means here. In usual mathematics, we talk about sets based on the ZFC set theory. But there are other systems such as the ZF set theory, and any other topoi (in regard that the category of ZFC sets forms a topos)!

Different theory provides a different underlying universe of sets. Should we change the underlying universe, would we get a different definition of a structure, and thus a different definition of truth? (Please note that I'm not asking about taking generalized truth values in some topos.)

Reference

[1] A shorter model theory-[Wilfrid Hodges]

Answer 1

Yes, there are a variety of mathematical systems that are able to serve as a foundation of mathematics, whether one uses ZFC set theory, ZFC plus large cardinals, or ZF, or PA, or category theory, type theory, and so forth.

Each of these theories is able to provide notions of model, satisfaction, truth, provability (along with all the rest of mathematics, number theory, topology, and so on). In this sense, one can undertake model theory and metamathematics in any of a variety of foundational background theories.

In some cases, different metatheoretic background theories are interpretable within one another, and in these cases, there will be a basic translation of theorems from one system to the other. In other cases, there is at least a relevant conservativity result. For example, ZFC and ZF have exactly the same arithmetic consequences, and this will imply that they have the same proof-theoretic features for any arithmetically definable first-order language. Basically, they will have the same consistency and validity consequences for the usual theories we consider.

Meanwhile, it can also happen that different choices for the background theory can have substantive consequences in the metamathematical features that one finds.

This occurs even just within the various set-theoretic systems. The large cardinal set theories, for example, imply outright that certain theories are consistent, whereas these consistency assertions are not provable in the weaker systems. Not all the systems have exactly the same consistency strength.

In ZFC, for example, the theory PA does not entail Con(PA), but this is not provable in PA alone, since there are models of PA in which PA is thought to be inconsistent, in which case it entails every assertion.

So the different foundational background theories can even disagree on whether a given theory has a given consequence. Thus, they would disagree on whether a given statement is true in all models of the theory.

This can happen even when it isn't just that one system thinks the theory is consistent and the other does not. For example, if one works in $\text{ZFC}+\neg\operatorname{Con}(\text{ZFC})+\rho$ or $\text{ZFC}+\neg\rho$, where $\rho$ is the Rosser sentence for ZFC (or stronger theories, whatever), then these two background theories will satisfy different incompatible $\Sigma_1$ arithmetic assertions. In one theory, the first number with a certain property is of one sort, and in the other, it is of another incompatible sort. So these two theories will both think the standard model $\mathbb{N}$ exists, but they will disagree on what is true in that model.

More generally, in light of the incompleteness theorem, none of the usual systems can be complete for arithmetic truth, and so it is consistent with any of them that they disagree on what is true in the standard model $\mathbb{N}$. Different models of set theory, when used as a metamathematical background context, can have fundamental disagreements on the model theory of our familiar theories.

In summary, yes, the metatheoretic background theory can have consequences as to what is true, what is valid, what is consistent, even for comparatively simple definable models and assertions.

Answer 2

In contrast to Joel's answer, I would like to point out that the notion of truth-in-a-structure, or whether $A \models \phi$ for a given $A,\phi$, is not so wild and capricious. Although the question of whether $\phi$ is true in all models of a given theory $T$ of course depends on what models of $T$ exist, given a particular one $A$, the satisfaction relation is relatively concrete. One just needs to establish the notion recursively on the complexity of formulas.

For any particular formula $\phi$, one can write down a step-by-step procedure for checking whether a given model $A$ satisfies $\phi$. If $H$ is a larger structure (of set theory) with $A \in H$, and $H$ satisfies a very basic set theory like Kripke-Platek, so that it can do basic operations like pairing and union and taking subsets defined by simple ($\Delta_0$) formulas, then $H$ will reach the same conclusion about whether $A \models \phi$ as any other $H'$ with the same properties.

Even the notion of whether $A \models T$, for theories $T$ possessing infinitely many formulas, is highly absolute. One just needs the machinery to carry out the definition of satisfaction for all formulas in the language of $T$, which involves a recursion of length $\omega$. So any two mathematical universes possessing $A$ will agree on whether $A \models T$ as long as they have standard $\omega$ and can carry out recursions of length $\omega$ where each step involves constructing a subset defined by a simple formula (like taking complements, intersections, cartesian projections).

(Someone more familiar with weak set theories might want to improve this answer by getting more specific and dropping some keywords...)

EDIT: I should acknowledge the interesting paper of Hamkins and Yang, where they show that you can have two models with the same (nonstandard) natural numbers that disagree on whether some (nonstandard) sentence is satisfied by the natural number structure. The point, as I understand it, is that the inductive notion of satisfaction is not completely determined by the starting point when you get into nonstandard numbers. Thus I should have specified above in the first version that satisfaction of a standard theory by a model is absolute between universes with standard $\omega$.