TITLE: Goodstein's theorem without transfinite induction
QUESTION [18 upvotes]: Goodstein's theorem is an example of a theorem that is not provable from first order arithmetic. All proofs of the theorem seem to deploy transfinite induction and I've wondered if one could prove the theorem without transfinite induction. Some time ago I came across this old usenet post where Torkel Franzen writes


Goodstein's
    theorem does not necessarily require transfinite induction for its
    proof, but it's not provable in elementary arithmetic. It can be
    proved by ordinary induction on a statement involving quantification
    over sets...


What is the proof/statement Franzen is referring to?

REPLY [4 votes]: I am quite convinced that Goodstein's theorem can be proven with the constructive omega rule. Or "meta-logic". For every individual 'n' Goodstein's theorem can be proven in elementary arithmetic (the problem is the quantification). In a meta-proof you can prove that for every 'n' such valid proof can be generated. I believe that this "meta-proof" only requires elementary arithmetic (if the meta-proof would require second order logic, then we wouldn't have achieved anything).
Of course, I can only claim this by detailing this out, but given my understanding of the Goodstein theorem, I am quite convinced that it should be possible.