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Is it possible to show that either a statement or its negation is non-independent of say, ZFC, without actually proving or disproving said statements? The reason I ask is because I've read of proofs of some number-theoretic statements which go by the strategy of first assuming the Riemann hypothesis and deriving the conclusion, and then assuming the negation of the Riemann hypothesis and deriving it again. It would seem to me that such proofs would be less meaningful if RH was shown to be independent of ZFC (however unlikely that may be). So I was wondering if, for the sake of completeness of these proofs one could show that RH is not independent of ZFC; but I don't know if that is doable without showing that RH or its negation introduces new contradictions to ZFC, which would be tantamount to proving or disproving it.

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    $\begingroup$ Provability and truth are different things. The Riemann Hypothesis, a statement about $\mathbb N$, either holds or fails, regardless of whether our best axiomatic systems can verify which of the two possibilities is the case. (By the way, if the Riemann Hypothesis is false, then this is provable in $\mathsf{ZFC}$ and in much weaker theories, see here.) $\endgroup$ – Andrés E. Caicedo Jul 27 '14 at 17:15
  • $\begingroup$ I'm not sure what exactly you mean by "it would be less meaningful". There is still a formal proof of the final result and since math these days is usually written from a formalist viewpoint (regardless of what your personal belief may be) I would say that it is as meaningful as anything else done in math for the most part. $\endgroup$ – UserB1234 Jul 28 '14 at 1:23

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