Yes, it is very easy to check that [Gödel 1931] was for a

system for the foundation of mathematics.

If you figure this out correctly you can claim the third incompleteness theorem. (If I am right)

Results in [Gödel 1931] depend on existence of proposition

I'mUnprovable. Since, the proposition doesn't exist in

foundations, the results in [Gödel 1931] do not hold for foundations.