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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?

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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

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