Can a statement like Goodstein's theorem (which is about finite objects) be proven by finitist mathematics? It cannot be proven by Peano arithmetic, which is bi

Can a statement like Goodstein's theorem (which is about finite objects) be proven by finitist mathematics? It cannot be proven by Peano arithmetic, which is bi-interpretable with ZFC with the axiom of infinity replaced by its negation [1], or any weaker theory. [1] https://math.stackexchange.com/questions/315399/how-does-zfc-infinitythere-is-no-infinite-set-compare-with-pa https://math.stackexchange.com/questions/315399/how-does-zfc...