[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.