Comment by bubblyworld

Thanks, I'm on the same page. Where I'm still confused - so you add in an axiom representing w-consistency that says "for all P, if for all n there exists a proof of P(n) then there exists a proof of forall n . P(n)". But does this really mean the same thing as w-consistency? We still have no link between our provability predicate and truth. For instance there could be models where the witnessing proofs are non-standard.

So I'm not convinced this is enough to conclude "forall x . P(x)" within PA itself, which is what you need to prove Goodstein's. We can merely conclude that there exists a (possibly non-standard) n coding for that statement.