Given a concrete mathematical statement, such as BSD conjecture(https://en.wikipedia.org/wiki/Birch_and_Swinnerton-Dyer_conjecture), do we know if it is provable?

  • $\begingroup$ I suspect the answer, in the vast majority of specific cases, is going to be, quite simply, "We don't." I've never heard of a non-independence result that doesn't itself discern whether the statement is true or false. I would be interested in finding out if such a thing exists - by the consistency theorem, you could start with two models, one of the statement and one of its negation, and try and derive a contradiction. $\endgroup$ – Dustan Levenstein Feb 7 '16 at 16:21
  • $\begingroup$ What do you mean by provable? If you mean does a proof exist - then it is just as hard as to prove conjecture. Only provably correct conjecture provably exist a proof. If you mean if it is possible to have a proof, however, then it is easy. The only thing that you cannot write a proof are "non-statements". For example, one cannot write a proof to "Good Morning", or "How are you" $\endgroup$ – Andrew Au Feb 7 '16 at 16:27
  • $\begingroup$ @AndrewAu I was thinking people are trying to prove BSD conjecture, but is it possible that the conjecture is not provable by Godel's incompleteness theorem? $\endgroup$ – Qixiao Feb 7 '16 at 17:46
  • $\begingroup$ A statement is not "provable" in and of itself. It is only provable relative to a particular axiom system. The most common way to show an axiom system doesn't prove a statement is to build a model of the system that doesn't satisfy the statement. For BSD there seems to be no specific reason to suspect it is unprovable from ZFC set theory. $\endgroup$ – Carl Mummert Feb 13 '16 at 13:34
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    $\begingroup$ In general, however, there is no algorithm that can decide whether arbitrary statements are provable from ZFC. They have to be considered on a case by case basis. $\endgroup$ – Carl Mummert Feb 13 '16 at 13:35

You're using the wrong term. You mean to ask whether we can tell if a conjecture is decidable, meaning that it is either provable or disprovable. But no we cannot tell whether a statement is decidable if the quantifier complexity is too high. Furthermore, it may be possible that even the decidability of a statement is itself undecidable! (See below for an example.)

First read https://math.stackexchange.com/a/1643073/21820, to ensure that you fully understand the import of Godel's incompleteness theorem. After that, consider the following. $\def\imp{\rightarrow}$

[We work in a meta-system and assume that $PA$ is omega-consistent.]

Let $φ = \square_{PA} Con(PA) \lor \square_{PA} \neg Con(PA)$. [So $φ$ expresses "Con(PA) is decidable over $PA$".]

If $PA \vdash φ$:

  Within $PA$:

    $\square Con(PA) \lor \square \neg Con(PA)$.

    If $\square Con(PA)$:

      $\neg Con(PA)$. [by the internal incompleteness theorem]

      $\square \bot$.

      $\square \neg Con(PA)$. [by (D1),(D2)]

    $\square \neg Con(PA)$. [by basic logic]

    $\neg Con(PA)$. [because $PA$ is omega-consistent]

  Contradiction. [with the external incompleteness theorem]

Therefore $PA \nvdash φ$.

If $PA \vdash \neg φ$:

  Within $PA$:

    $\neg \square Con(PA)$. [by basic logic]

    If $\square \bot$:

      $\square Con(PA)$. [by (D1),(D2)]


    $\neg \square \bot$.


  Contradiction. [with the external incompleteness theorem]

Therefore $PA \nvdash \neg φ$.

Thus $φ$ is independent of $PA$.


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