Comment by dwohnitmok

I haven't had the time to go through your argument in more detail, but it's important to note (because the natural language version of what you've presented is ambiguous) that you haven't shown "PA proves 'Provable(forall n, G(n))'" in which case it would be the case that in fact "PA proves 'forall n, G(n)'", but rather "PA proves 'forall n, Provable(G(n))'".

My logic is very rusty at this point, but if someone could give me an argument that you cannot move the 'Provable' outside the 'forall', I would really appreciate that, without making reference to Goodstein sequences. In other words, that in general for propositions 'P' it is not true that proving "forall n, Provable(P(n))" implies you can prove "Provable(forall n, P(n))".