Recently, a friend of mine introduced me to Goodstein's theorem, which I found to be very interesting and mind-blowing. The theorem basically says that every Goodstein sequence (the wikipedia article does a good job of explaining it) terminates at 0. What actually surprises me most is that this theorem can't be proven using the 'peano axioms', which to a layman like me seems to be just the 'usual' axioms I've been working with since I was introduced to arithmetic as a child.
I'm really interested in this theorem, and I want to be able to follow the proof and understand what is going on, but I have no idea what subjects I need to know more about in order to do this. Can anyone here guide me in the correct direction?