I am interested to know, possibly in non-technical lay terms, which axioms are really needed to prove the Goodstein's theorem:(adsbygoogle = window.adsbygoogle || []).push({});

http://en.wikipedia.org/wiki/Goodstein's_theorem

and/or other theorems of that type.

It is claimed that it cannot be proved in first order arithmetic. My first question is: What exactly is missing in first order arithmetic which is needed to prove the theorem? Is it perhaps the induction axiom?

The theorem has been proved by using trans-finite numbers. But that's very strange, given that the theorem talks only about finite numbers. My second question is: Can the theorem be proved without using trans-finite numbers?

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# (Im)possibility to prove the Goodstein's theorem

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