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(Im)possibility to prove the Goodstein's theorem

  1. Aug 27, 2014 #1

    Demystifier

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    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. jcsd
  3. Aug 27, 2014 #2

    jedishrfu

    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

     
  4. Aug 28, 2014 #3

    Demystifier

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    Thanks for the remark, but I don't see how is this related to the (im)possibility to prove the Goodstein theorem.
     
  5. Aug 28, 2014 #4

    jedishrfu

    Staff: Mentor

    I was answering your simpler question regarding the induction principle.

    Perhaps others here can comment on your larger question.
     
  6. Nov 12, 2014 #5

    Demystifier

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    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
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