Is it possible to prove Goodstein's theorem without transfinite induction? Is there such a proof?
- Anybody can ask a question
- Anybody can answer
- The best answers are voted up and rise to the top
I'm pretty sure the short answer is "no" (if you mean, can you prove Goodstein's theorem without invoking apparatus as strong as a transfinite induction which can't be reduced to an ordinary induction). For if I recall correctly, Goodstein's theorem is actually equivalent (over a weak base theory) to transfinite induction up to $\varepsilon_0$. My first port of call to check this would be Kirby and Paris's paper on 'Accessible independence results for Peano Arithmetic'.
I found the following paper which seems fairly relevant: