I’m not too knowledgeable on the topic, but Gödel’s proof uses a smaller logic system — on which a meta-language can be used to prove consistency/completeness. It is precisely about not being able to prove these properties “from within”.
for the foundation of mathematics.
I'mUnprovable. Since, the proposition doesn't exist in
foundations, the results in [Gödel 1931] do not hold for foundations.
system for the foundation of mathematics.
claimed to prove incompleteness for a system for the
foundations of mathematics.
1st-order systems such a PA were introduced later and
quickly shown to be inadequate for the foundations of mathematics.