[loicd]

> You can't claim that's it's even "widely accepted" that the axiom of choice is "true".

I have never claimed anything like that. The original comment was a reaction to the notion of "true but unprovable" which is wrong because what is true is precisely what is provable. You may have an intuitive notion of "true", but with logic, the devil is in the details. In my experience, it is better to stick to the mathematical definitions, especially when talking about things like the incompleteness theorem.

Now, the mathematical notions are as follows. First, you agree on some deduction rules, then some axioms (aka a theory), and by definition, what is true is what is satisfied by every model of the theory. A completeness theorem is then a theorem that states that what is true is precisely what is provable. (Proved by Gödel for classical logic.)

Of course, you may disagree with the choice of axioms. However, when introducing a new axiom, mathematicians don't argue whether it is "true" or not, they have to justify in one way or another that it is relatively consistent. The same thing is true for the deduction rules. In other words, consistency, not truth, is the right metric for axioms and deduction rules. Finally, observe that mathematicians who argue against the axiom of choice or the law of excluded middle do not claim that these are false, they claim that these are not constructive. Yet another notion not to be confused with truth.

[denotational]

> You may have an intuitive notion of "true", but with logic, the devil is in the details. In my experience, it is better to stick to the mathematical definitions, especially when talking about things like the incompleteness theorem.

> A completeness theorem is then a theorem that states that what is true is precisely what is provable. (Proved by Gödel for classical logic.)

As I pointed out in another comment, you are actually using a nonstandard definition of “true”/“truth” yourself; what you are calling “truth” is generally referred to as “validity”.

> However, when introducing a new axiom, mathematicians don't argue whether it is "true" or not

This is not representative of the historical development of mathematical logic and analytic philosophy at all.

[Y_Y]

I didn't claim you'd claimed it ;)

Where are you getting this definition of truth? I don't think things are a neat and simple as you're making out. Are you familiar with Tarski's work on (semantic) truth?