Comment by codeflo

I also like the recursion. In essence, you're making a meta-proof about what PA proves, and given that you trust PA, you also trust this meta-proof.

> I think just PA+"PA is consistent" is enough?

It's not clear to me how. I believe PA+"PA is consistent" would allow a model where Goodstein's theorem is true for the standard natural numbers, but that also contains some nonstandard integer N for which Goodstein's theorem is false. I think that's exactly the case that's ruled out by the stronger statement of ω-consistency.