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In Terrence Tao's book Analysis I, the axioms of ZFC are considered to be true statements, and every other true statement in the book is proved from these axioms. Model theory is not mentioned.

This approach to truth is not peculiar to Tao's book or to its subject. As far as I can tell, this is the standard approach for books and research papers outside of mathematical logic and set theory.

I have two questions:

  1. When the everyday mathematician says that a certain statement is true, does the model theorist agree? Why or why not?
  2. Why does the everyday mathematician need the model-theoretic definition of truth, if only provability is ever used?
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    $\begingroup$ Heh, I would say you are trying to squeeze out too much from Tao's book. I suggest reading an actual book on Set theory. His book is not that concerned with foundational issues. $\endgroup$ Aug 20, 2022 at 22:41
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    $\begingroup$ While I might agree that everyday mathematicians agree that provability in ZFC is a convenient way to establish truth, perhaps even sufficient for most of what they do, I doubt much that many of then identify truth with provability in ZFC. Most everyday mathematicians probably will suspend judgement on such philosophical issues until the problem they are working on "forces" them to make a choice. $\endgroup$
    – Lee Mosher
    Aug 20, 2022 at 23:11
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    $\begingroup$ You seem to conflate several proposals. "True in the von Neumann universe $V_\kappa$ where $\kappa$ is an inaccessible cardinal" has different meanings for different inaccessible $\kappa$'s. And none of those meanings is the same as "provable in ZFC". $\endgroup$ Aug 21, 2022 at 1:01
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    $\begingroup$ Personally when I was a "working mathematician", I preferred not to use the words "true" or false, using, instead, "provable", "unproven", or "dis-provable". $\endgroup$ Aug 21, 2022 at 1:56
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    $\begingroup$ Your question seems to suggest that logicians are not working mathematicians. I know many counterexamples! Joking aside, the meaning of "true" in a mathematical context (unqualified by "true in a structure/model") is really a question of philosophy of mathematics, and while logicians may be particularly disposed to have thought deeply (or to care at all) about the philosophical question, both logicians and other mathematicians hold a wide variety of views on this issue. George Ivey's comment above is typical. Most mathematicians, and also many logicians, would prefer not to commit themselves. $\endgroup$ Aug 21, 2022 at 3:22

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It's important to distinguish truth in a structure (e.g. "$\varphi$ is true in $M$") from unqualified truth (e.g. "$P$ is true").

The former expresses a mathematical relationship, precisely defined by the model-theoretic definition of truth, between a sentence $\varphi$ of a logic and a structure $M$, which is a set possibly with extra functions and relations defined on it. It's important here that both $\varphi$ and $M$ are mathematical objects.

The latter is a linguistic construction that we sometimes use at the meta-level, i.e., when we talk about mathematics. What we mean when we say "$P$ is true" is really a question of philosophy of mathematics.

Model theorists, of course, may use both senses of "truth" and mean different things by them. I think this answers your two questions:

  1. When the everyday mathematician says that a certain statement is true, does the model theorist agree?

This depends on whether their views on philosophy of mathematics align. No one has mentioned truth in a structure in this scenario, so there is nothing special about model theorists here - the same holds between two "everyday mathematicians" or between two logicians.

  1. Why does the everyday mathematician need the model-theoretic definition of truth, if only provability is ever used?

They don't. The model-theoretic definition applies in a very specific context where we are considering logical sentences as mathematical objects. That is, it is only relevant when we are doing mathematical logic.

But! That's not to say that "provability is all that's used". An earlier version of your question asked whether everyday mathematicians identify truth with provability. I would say that most of them do not. To explain a bit more, I'll sketch two common views on philosophy of mathematics (there are many more, and these two views also have many variations). Let's contrast the meanings of the assertions "$P$", "$P$ is true", and "$P$ is provable" (implicitly: from a background foundational system, say ZFC), for people holding these views.

A Platonistic View: There is a canonical "real" universe of mathematical objects. To address a comment in an earlier version of your question: the mathematical universe is not $V_\kappa$ for some inaccessible cardinal $\kappa$, because $\kappa\notin V_\kappa$, and the universe contains all mathematical objects.

For the platonist, asserting "$P$" is the same as asserting "$P$ is true", and it's about what holds in the real mathematical universe. It is different from asserting "$P$ is provable". For example, there is a fact of the matter about whether there is a bijection between the real numbers and the first uncountable ordinal. That is, either the Continuum Hypothesis ($CH$) is true, or $\lnot CH$ is true, but we know that neither is provable from ZFC.

Unfortunately, proof is the only means for us mere mortals to be sure of what is the truth. So the platonist is unlikely to assert "$P$ is true" unless they have a proof of $P$ (or they are feeling reckless or provocative). For example, the platonist is likely to say "I believe ZFC is consistent", rather than "ZFC is consistent", because this statement is not provable in ZFC.

A Formalist View: There is no such thing as mathematical truth, because ideal mathematical objects do not actually exist in the world. For the formalist, asserting "$P$" is the same as asserting "$P$ is provable". It is not the same as asserting "$P$ is true", because "$P$ is true" has no meaning.

For the formalist, the fact that neither $CH$ nor $\lnot CH$ is provable settles the matter. But they might be interested in conditional statements like "such and such additional axiom of set theory implies $CH$", and they do not need to be convinced that the additional axiom is "true".

Also, the formalist might say something like "I believe that it's true that ZFC is consistent" and mean this as a statement about our physical world in the sense that it would be impossible to actually write down a proof of a contradiction from the axioms of ZFC.

As I said above, there are a wide variety of views on philosophy of mathematics and the nature of truth. I'm not suggesting that all platonists or all formalists agree with everything I wrote in my caricatures of these views.

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    $\begingroup$ Lovely! This is pretty much exactly the answer I wanted to write earlier today. I'd like to add one thing to Point 2: everyday mathematicians may also need the model-theoretic notion of truth, purely as a technical tool. E.g. an everyday mathematician may still want to prove that an injective polynomial map $\mathbb{C}^n \rightarrow \mathbb{C}^n$ is surjective. Of course, when used this way, it doesn't really matter that the notion is called truth in a structure, which sounds similar to the colloquial true: it'd perform the same technical role even if it was called sploop in a looplab. $\endgroup$
    – Z. A. K.
    Aug 22, 2022 at 17:29
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    $\begingroup$ @Z.A.K. Right. An "everyday mathematician" might find the tools of model theory useful for proving theorems, in which case they need to know at least the basic definitions of model theory. Just the same as how I need to know the basic definitions of group theory if I want to use automorphism groups as a tool to prove something in model theory. I feel like I'm always making the point that model theory is just a field of mathematics like any other. Its definitions and objects of study don't have some special extra-mathematical character just because they use the same language as our foundations. $\endgroup$ Aug 22, 2022 at 17:39
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    $\begingroup$ great answer; every time I see these philosophical questions around logic and foundations come up I've struggled to articulate my thinking, and I think you've hit nail on the head by pointing out the difference between truth in a structure and "truth" abstractly. I really agree that's the fundamental core behind many of these questions, and once it's identified imo it eliminates a lot of common confusions $\endgroup$ Aug 22, 2022 at 19:00
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    $\begingroup$ @simplejack Indeed, provability should establish truth, but the converse is not so clear. Also, the predicate "is provable" does not behave in ways we intuitively expect "is true" to behave. For example, $CH\lor \lnot CH$ is provable, but we can't conclude that $CH$ is provable or $\lnot CH$ is provable. $\endgroup$ Aug 22, 2022 at 22:15
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    $\begingroup$ @user21820 Yes, of course we have to start somewhere. But I don't think it's incoherent to reject the notion of mathematical truth entirely. Proofs, formal and otherwise, are objects of our physical world once we write them down, and we can accept truths about them without agreeing to identify these truths with $\Sigma_1$ sentences about abstract natural numbers. I hinted at this when I talked about consistency of ZFC as a statement about our physical world. $\endgroup$ Aug 23, 2022 at 12:41

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