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A short time ago, I've been thinking about what statements in Number Theory are true but not provable. I've seen the proof of the Incompleteness Theorem (in the Gödel's works) and he gave an example of one statements which is true but not provable, but that statement is too weird (maybe because the proof was constructive, and Gödel constructed this statement, I think).My Logic professor said that, in past, mathematicians believe that Fermat's conjeture was other of those statements, but later Andrew Wiles gave a demonstration of the conjeture. So, I have two big questions.

  1. Could Goldbach's conjeture be true but not provable?
  2. What statements in Number Theory are true but not provable?

Thanks so much for all your ideas!

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  • $\begingroup$ I think there's already a thread about the construction of such sentences... Also see this question on Mathoverflow about Knuth's intuition that Goldbach might be unprovable. $\endgroup$
    – Hakim
    Commented Sep 27, 2014 at 22:05
  • $\begingroup$ @Hakim, yep, but we should put "intuition" in quotation marks, since its definition in this context is some approximation of "wild stab in the dark." $\endgroup$
    – goblin GONE
    Commented Sep 27, 2014 at 22:28
  • $\begingroup$ To the question asker: note that no particular statement of arithmetic is unprovable, in any absolute sense. Rather, once we've fixed our axioms, then certain theorems are not provable from those axioms. But we can always enlarge our axioms so as to prove any statement we wish. So when you ask: "Could Goldbach's conjeture be true but not provable?" you have to explain to us: provable with respect to which base system of axioms? $\endgroup$
    – goblin GONE
    Commented Sep 27, 2014 at 22:30
  • $\begingroup$ Well, if I strictly follow the Incompleteness Theorem, I would tell you provable with respect to Peano Axioms. $\endgroup$
    – Alexei0709
    Commented Sep 27, 2014 at 22:32
  • $\begingroup$ @goblin I referred to the content of the answers, but I do know that intuition can lamentably fail more than once. $\endgroup$
    – Hakim
    Commented Sep 27, 2014 at 22:45

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Firstly, by 'true' (and I wish people would stop using that word in connection with Godels theorem) we mean valid in the standard interpretation. Such statements are usually shown with the aid of set theory.

Second, Fermats last has not yet been shown to be a consequence of Peano's axioms. Can Wiles proof be formalized in PA ?

To answer your questions,

  1. Yes, at the time of this writing it could be but we dont know.

  2. There are a number of natural and elegant theorems known not to be provable in PA, Paris-Harrington, Goodstein, and Hercules and the Hydra.

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    $\begingroup$ I think you're using 'valid' in a non-standard way. Judging by what our friends say over at wikipedia have to say, a better choice would be satisfied. E.g. "By true, we mean satisfied by $\mathbb{N}$, where $\mathbb{N}$ is the standard interpretation of first-order arithmetic." $\endgroup$
    – goblin GONE
    Commented Sep 28, 2014 at 14:46
  • $\begingroup$ Wikapedia is not my frend. $\endgroup$
    – Rene Schipperus
    Commented Sep 28, 2014 at 14:50

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