I think saying that a theorem is true presumes the axioms from which it was proven and so the entire system is "true everywhere forever".

I often find it helpful to think of chess as my axiomatic system. When we say the king is in checkmate, it presumes that we accept all the underlying rules of chess. And these pieces that theoretically form a checkmate will always do so forever... Assuming the usual rules of chess, assuming they're unchanging, etc.

When you put things in terms of chess, these "deep" statements about "math" often become banal. And it works for any game that's a "formal system" (eg. most board games).

even if it’s true everywhere forever, it might still not be provable, because Gödel.

No. Gödel's completeness theorem says that if something is true in every model of a (first-order) theory, it must be provable. Gödel's incompleteness theorem says that for every sufficiently powerful theory, there exists statements that are true sometimes, and these can't be provable.

The key word is "everywhere".

Worse: If the chosen axioms are contradictory, then the theorem is effectively worthless.

And it is impossible to know whether axioms are consistent. You can only prove that they are not.

But that's math. And its proof is math. And that proof is true everywhere forever.

I see philosophy as a place to make nonrigorous arguments. Eventually, other fields advance enough to do away with many philosophical arguments, like whether matter is infinitely divisible or whether the physical brain or some metaphysical spirit determines our actions.

Since this is a question that math hasn't advanced enough to answer, we can have a philosophical argument about whether other fields will eventually advance enough to get rid of all philosophical arguments.