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”.