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

1. Aug 27, 2014

### Demystifier

I am interested to know, possibly in non-technical lay terms, which axioms are really needed to prove the Goodstein's theorem:
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?

2. Aug 27, 2014

### Staff: Mentor

In the wiki article on Peano axioms they say that adding the multiply operation allows them to drop the second order axiom

http://en.wikipedia.org/wiki/Peano_axioms

3. Aug 28, 2014

### Demystifier

Thanks for the remark, but I don't see how is this related to the (im)possibility to prove the Goodstein theorem.

4. Aug 28, 2014

### Staff: Mentor

Perhaps others here can comment on your larger question.

5. Nov 12, 2014

### Demystifier

I have just found the answer to my question:
http://mathoverflow.net/questions/134590/goodsteins-theorem-without-transfinite-induction

So more-or-less as I expected, "Goodstein's theorem does not necessarily require transfinite induction for its proof, but it's not provable in elementary arithmetic. It can be proved by ordinary induction on a statement involving quantification over sets."

Which leads me to the next question:
Is there a known natural (non-self-referential) statement in second-order arithmetic which can neither be proved nor disproved in second-order arithmetic?

Last edited: Nov 13, 2014