[loicd]

> This is very far away from my original point

Yes, the discussion has deviated, and I don't think we will resolve the disagreement, but I wanted to make my position clearer w.r.t to the claim that "most mathematicians are Platonists [...] and they believe the objects they work with are real".

> It’s not clear to me which definition of (non-technical, unqualified/alone) “true” you are using.

I may be elliptic and not very clear, but I have not changed my definition. We can't do mathematics in a vacuum. There is always a context, which consists of a language, i.e. a fixed set of constant, function and relation symbols, and a theory, which is a fixed set of statements of the language. Typical theories are ZF, ZFC, PA, etc. For me, "true" (alone) means satisfied in all models of the theory, and equivalently by completeness, provable from the theory. (And by the way, your notion of "true" (alone) as "satisfied in the standard model" is equivalent to requiring that the theory be complete.) That would be your definition 1, except for the "non-technical" part. Now, the discussion deviated towards set theory because to compare my idea of "true" (alone) with yours, I used your comment:

> “True in the standard model” is generally what most working mathematicians who are not logicians mean by “true”.

which lacked context and seemed to me to be especially problematic in the context of set theory. And also, "most working mathematicians who are not logicians" implies a context of set theory. So the "non-technical" definition would be your definition 2 although I think ZF+DC (the axiom of dependent choice) is closer to what most mathematicians won't have a problem with than ZFC (depends on the discipline I suppose). Probably a mistake to talk about "most mathematicians" though.

> If you think it isn’t true then you are saying that we don’t really understand the naturals intuitively and we can only understand them by axiomatisation.

I mean something more subtle. I think we understand the naturals intuitively but only to some extent. Enough to write some axioms, but not enough to reliably answer many seemingly simple questions about them. I also think that our intuitive understanding is not static but grows as we study mathematics.

> we can ever say that proof is what determines truth given we know from Gödel that proof is fundamentally limited.

This is perhaps where the disagreement is? I don't have a problem with the fact that proofs are fundamentally limited.

[denotational]

I think we're understanding the phrase "true in all models" in different ways.

I understand it to mean: "true in all structures compatible with the language".

On the other hand, I think you understand it to mean: "true in all models of some latent theory left implicit", where the theory may be ZF(C) or something else depending on context?

I'm basing that on your comment:

> For me, "true" (alone) means satisfied in all models of the theory, and equivalently by completeness, provable from the theory. [...] That would be your definition 1 [...]

That isn't my definition 1, because I'm refering to all structures compatible with the language, not all models of some theory. This is probably an abuse of terminology on my part because usually we reserve the term model for structures that model a theory [1], sorry for the confusion!

> And by the way, your notion of "true" (alone) as "satisfied in the standard model" is equivalent to requiring that the theory be complete.

Please can you explain this? I don't think that's what I mean. We know that PA isn't complete, but when I say the Gödel sentence is true I mean that it's true in the standard model of the naturals.

> There is always a context, which consists of a language, i.e. a fixed set of constant, function and relation symbols, and a theory, which is a fixed set of statements of the language

I completely disagree with the idea that this context always existed, it's too Formalist. There is a rich history of mathematics before the concept of a formal language and a formal theory existed; if you were to ask Gauss if he worked in ZF or ZFC or TG I don't think he would have an answer, but clearly he had some concept of mathematical truth.

[1] : Although all structures are vacuously models of the empty theory, so technically they are models, but that's not very convincing or useful...

[loicd]

> On the other hand, I think you understand it to mean: "true in all models of some latent theory left implicit", where the theory may be ZF(C) or something else depending on context?

Yes, that's what I mean. (For me, "structure" is preferred to "model" when nothing is implied.)

> The standard model that most set theorists have in mind is something like the Von Neumann Universe, V.

Now I am getting confused. Isn't that equivalent to requiring the axiom of regularity? I have a book on set theory by JL Krivine with the theorem: "V is the whole universe iff the axiom of regularity holds". This book also proves that if "U is a universe (i.e. a model of ZF) then the collection V inside U satisfes ZF+axiom of regularity" (which proves the relative consistence of the axiom of regularity).

To talk about the Von Neumann Universe, you must assume some "surrounding" universe which is a fixed but arbitrary model of ZF. Thus, X is true in the Von Neumann Universe if and only if X is satisfied in all models of ZF+axiom of regularity. That certainly matches my idea of "true", albeit with a weaker set of axioms... (I proposed ZF+DC as a least common denominator because a large part of analysis can't be done without some form of axiom of choice.)

> Please can you explain this?

Let us call S your standard model of PA. I understood your idea of "X is true" as "S satisfies X". Now, let T be the set of all statements satisfied by S. Then T is a complete, consistent theory that extends PA and "X is true" if and only if "T proves X". (Of course, T is much larger than PA, and in fact, by incompleteness, there are no recursively enumerable theories equivalent to T.) This correspondence between complete consistent theories and models is not one-to-one though, a complete consistent theory may have infinitely many models.

> if you were to ask Gauss if he worked in ZF or ZFC or TG [...]

Fair enough, but I think he was familiar with Euclid's elements, and would have agreed on the fact that there are things that are assumed to be true because they are intuitive and things that are proved to be true. In my view, ZF is the culmination of an effort to minimize that intuitive part. By constrast, the notion of model (and Tarski's notion of truth) are more modern.