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?
Everything in such questions is based on ordinals. To understand what it is, you need to understand what is a well order.
http://en.wikipedia.org/wiki/Well-order
Then Ordinal numbers
http://en.wikipedia.org/wiki/Ordinal_number
And how they are introduced in usual fundation of mathematics.
http://en.wikipedia.org/wiki/Axiom_of_infinity