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”.
You can go deeper. To prove anything, including the consistency or inconsistency of a theory, you need to work within a different system of axioms, and assume that it is consistent, etc.