It is known that the Goodstein theorem http://en.wikipedia.org/wiki/Goodstein's_theorem which is a theorem about natural numbers, cannot be proved from the standard axioms of natural numbers, that is Peano axioms http://en.wikipedia.org/wiki/Peano_axioms . It is also known that Goodstein theorem can be proved from a more powerful axiom system which includes transfinite numbers. My question is: Can Goodstein theorem be proved from a natural axiom system more powerful than Peano axioms, but without transfinite numbers? I expect that it can. More precisely, I suspect that Peano axioms cannot prove the Goodstein theorem because these axioms do not contain an axiomatization of powers (but only of addition and multiplication). If one would add appropriate natural axioms for powers (similar to those for addition and multiplication), I expect that then one could prove the Goodstein theorem without transfinite numbers. Can someone confirm or reject my expectations?