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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?
simple jack
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    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. – tryst with freedom Aug 20 '22 at 22:41
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    @Beautifullyirrational This question pertains to every book I have seen on elementary real analysis, not just Tao's. (It says so above.) I am just using that book as an example. – simple jack Aug 20 '22 at 22:43
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    I myself learned from Tao but I have checked out other books. I can say for sure that this is not the concern of most real analysis books at UG- to early graduate level. The reason I told specifically for Tao it is typicailly considered it is of level Tao is baby^2 Rudin. I have observed that many Analysis book also don't bother with construction of R and just take it as a field and work from there. I would recommend you check out an actual book whose focus is Set theory, try to see if you can get a partial answre and ask again. I am sure you'll be able to ask a great question then – tryst with freedom Aug 20 '22 at 22:44
  • @Beautifullyirrational I am confused because I am familiar with the model-theoretic definition of truth. The question would also pertain to most research papers in analysis, so the level of the textbook is not the issue. – simple jack Aug 20 '22 at 22:48
  • I think a very broad answer to this question is working mathematicians take only minimal axioms they need to move forward with a theory (and append as needed). One can play it safe by slapping ZFC on the preamble of a work--basically saying assume everything well-known to be true. This is like a release of liability in my opinion. Since any problems would point to hard Set Theory texts instead of your current source. If for example, you are studying Analysis, I think Peano's Axioms are a good place to start. When you get crazy counter examples, then you should worry about appending more. – Kevin Aug 20 '22 at 23:09
  • Is your question whether "truth" is the same as "provable", or specifically with respect to ZFC? – Joe Aug 20 '22 at 23:09
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    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. – Lee Mosher Aug 20 '22 at 23:11
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    @Joe No, I understand that by Gödel's first incompleteness theorem identifying truth with provability is not compatible with the law of excluded middle. – simple jack Aug 20 '22 at 23:13
  • @LeeMosher Since provability implies truth in every model? There is no concern for which model? – simple jack Aug 20 '22 at 23:14
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    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". – Andreas Blass Aug 21 '22 at 01:01
  • @AndreasBlass Thank you for pointing out that my proposal is nonsense. I will edit the question to reflect this. – simple jack Aug 21 '22 at 01:41
  • @AndreasBlass Also as mentioned in a previous comment (responding to Joe) I understand why logicians and set theorists do not identify truth with provability. At least I think I do. Please let me know if that response is incorrect, or if there is a problem with the updated question. – simple jack Aug 21 '22 at 01:49
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    Personally when I was a "working mathematician", I preferred not to use the words "true" or false, using, instead, "provable", "unproven", or "dis-provable". – George Ivey Aug 21 '22 at 01:56
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    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. – Alex Kruckman Aug 21 '22 at 03:22
  • @AlexKruckman Haha to me it just means “nonspecialist in logic/set theory.” Maybe Lee Mosher’s term “everyday mathematician” is better. If one were to interpret Tao’s statements about truth in model theory, is there a preferred model that would be used? Again I’m just using Tao as an example. Also, don’t you commit yourself just by making statements, since making a statement is the same as saying that the statement is true? – simple jack Aug 21 '22 at 03:37
  • Here is Tao's axiom of infinity: "Axiom 3.7 (Infinity). There exists a set N, whose elements are called natural numbers, as well as an object 0 in N, and an object n++ assigned to every natural number n ∈ N, such that the Peano axioms (Axioms 2.1 - 2.5) hold." from "Analysis I", p.44 Not much like ZFC. It is preferable to ZFC IMHO. – Dan Christensen Aug 21 '22 at 13:18
  • @DanChristensen Is it not equivalent? Either way that is not the point of the question. Using Tao as an example has caused distraction. – simple jack Aug 21 '22 at 16:47
  • This questions feels kind of off-topic to me. It appears to be asking people about a philosophy of mathematics, rather than mathematics itself. This question might be more appropriately asked on [philosophy.se]. I also fear that this is kind of an opinion-based question---such questions are generally off-topic here. – Xander Henderson Aug 21 '22 at 20:45
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    I tend to think of ZFC as being analogous to how a software engineer working in a high-level language might think of machine code: I'm glad this implementation of my high-level language exists and I really hope it's bug-free or else there will be some annoying downtime waiting for a hardware update, but the fact that it formally underpins the language I'm really speaking is only of academic interest to me. – hunter Aug 21 '22 at 04:10
  • @XanderHenderson I am asking about model theory, not philosophy. I thought this would be a straightforward question. – simple jack Aug 22 '22 at 00:26
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    Okay... but that still doesn't address the other part of my criticism, which is that this is an opinion-based question---what is an "everyday mathematician"? Does an "everyday mathematician" need a notion of model-theoretic proof (as you assert)? It is an interesting sociological or philosophical question (related to that which Kuhn tried to answer), but this question seems awfully "squishy" in defining its terms, and seems quite subjective. – Xander Henderson Aug 22 '22 at 03:09
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    @XanderHenderson I don't care if you close it. But I think it is pretty clear what is meant by "everyday mathematician." The question could be restricted to Tao's book, but the intention is not to single anyone out. – simple jack Aug 22 '22 at 03:11
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    @simplejack I clearly haven't closed this question, and I don't plan to. I am trying to ferret out the key mathematical idea. Thus far, it seems unclear---the variety of answers (which you seem to find inadequate) seems to confirm my claim that the question is a little muddled. – Xander Henderson Aug 22 '22 at 03:16
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    The following assumes we're working in classical first-order logic. By the Soundness and Completeness theorems, we have the following: $$\Gamma \vdash \varphi \ \text{ iff } \Gamma \models \varphi$$

    The symbol $\vdash$ means provability, so $\Gamma \vdash \varphi$ means $\Gamma$ proves $\varphi$. The symbol $\models$ means validity, so $\Gamma \models \varphi$ means for all models $\mathfrak{A}$, if $\Gamma$ is satisfied in $\mathfrak{A}$, then $\varphi$ must be satisfied in $\mathfrak{A}$....

     – ShyPerson Aug 22 '22 at 03:58
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    ...So to get to your questions, assuming "true" is a mushy way of saying "valid". Question 1. Yes, the model theorist would agree, because once the working mathematician produces a correct proof, we are assured by the Soundness theorem that the inference is valid. Question 2. If we know a statement is valid, we are assured by the Completeness theorem that there must be a proof out there somewhere. Even if we only have strong empirical evidence that a statement is true, that emboldens us to look for a proof (modulo incompleteness theorems, contradictions, etc.). – ShyPerson Aug 22 '22 at 03:59
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    @simplejack It is not just Tao who seems to be compelled by pedagogical or other concerns to develop their own versions of set theory in university math texts. He's just the most famous at this time. – Dan Christensen Aug 22 '22 at 04:14
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    @simplejack: Your comment about LEM (excluded middle) is incorrect. The generalized incompleteness theorems apply just as well to intuitionistic theories and other formal systems, as explained at the linked post. – user21820 Aug 22 '22 at 14:53
  • @user21820 How is that statement incorrect? I am aware of the general applicability of Gödel's theorems. If truth is defined as provability and falsity is defined as refutability, then Gödel's first incompleteness theorem says that some statements are neither true nor false, but undecidable. So the law of excluded middle would not hold. Do you disagree with this? – simple jack Aug 22 '22 at 21:53
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    Of course you're wrong. You have a basic and fundamental misconception! You cannot define truth as provability, in the same sense that you cannot define reality as your beliefs. No proper mathematician ever does that, and neither should you if you want to understand mathematics. This is not just a convention; "∀k∈ℕ ( You can prove Q(k) )" does not imply "You can prove ∀k∈ℕ ( Q(k) )", for any reasonable interpretation of "prove", so ∀intro fails if truth = provability, among many other things in ordinary mathematics. – user21820 Aug 23 '22 at 06:57
  • The law of the excluded middle is essential and of course it holds also if we do not know whether the given statement is true or false. For example, the Riemann hypothesis is either true or false, there is no other possibility. – Peter Aug 23 '22 at 11:13
  • There are actually proofs based on this, the statement follows if the Riemann hypothesis is true and also follows , if the Riemann hypothesis is false. So the statement must be true. This argumentation is even valid if the Riemann hypothesis cannot be proven. – Peter Aug 23 '22 at 11:13
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    @simplejack: I have granted you access to the Logic chat-room. I'm not sure when you requested, as the dumb SE system does not notify me about it. – user21820 Jul 04 '23 at 09:25

1 Answers1

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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.

Alex Kruckman
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    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. – Z. A. K. Aug 22 '22 at 17:29
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    @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. – Alex Kruckman Aug 22 '22 at 17:39
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    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 – Atticus Stonestrom Aug 22 '22 at 19:00
  • Thank you for the careful answer. You have answered my two questions but raised another. Why should the formalist not define truth as provability? This is the approach taken in Bourbaki. Indeed, it seems absurd to assert "$P$" without believing that "$P$ is true." And, as you have said, the formalist feels justified in asserting $P$ if and only if $P$ has been proved. – simple jack Aug 22 '22 at 21:57
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    @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. – Alex Kruckman Aug 22 '22 at 22:15
  • @AlexKruckman Got it, thanks. – simple jack Aug 22 '22 at 22:45
  • @Z.A.K.: On the other hand, formalists must essentially still accept the notion of truth for Σ1-sentences, whether over PA− or over TC (theory of concatenation). Otherwise even the notion of formal proofs is meaningless! Alex, that's my one quibble with your post, since it's just wrong to say that formalists do not accept any notion of truth at all. – user21820 Aug 23 '22 at 07:09
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    @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. – Alex Kruckman Aug 23 '22 at 12:41
  • @AlexKruckman: I know, but if one doesn't accept even TC, then one can't even do propositional logic, because ( A ⊢ A∨B ) is not valid! (We need closure under concatenation, among other things.) I've never seen a reasonable alternative that doesn't have severe issues in this aspect. – user21820 Aug 23 '22 at 17:05
  • @user21820 I don't think we actually disagree. I'm just saying that one could accept TC as being true about actual strings that we can physically write down, while simultaneously denying that "truth" makes any sense for abstract or idealized strings. – Alex Kruckman Aug 23 '22 at 17:15
  • @AlexKruckman: Oh then of course! – user21820 Aug 23 '22 at 18:10