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Loosely speaking, there are three kinds of propositions.

  1. Those propositions which are true and can be proved to be true.

  2. Those propositions which are false and which can be proved to be false.

  3. Those propositions which are true, but it can't be proved that it is true.

Here the word "proof" is used in a strict mathematical sense.

My question simply is that under which category does Riemann Hypothesis belong? Or if a bit specification is more preferred, is RH ZFC-independent?

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if you prove something to be provable, then you proved it. –  Ittay Weiss Jun 22 at 7:28
    
I don't understand how proving that RH is provable is related with the position of the real part of zeros on the critical line. Can you be a bit elaborate? –  William Hilbert Jun 22 at 7:51
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I am not sure what this question is asking. Do you mean something along the lines: Is it possible that RH is ZFC-independent? –  Martin Sleziak Jun 22 at 8:08
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@MartinSleziak: My question actually is that: Has RH been proved to be formally decidable proposition? If yes, then I want some reference. –  William Hilbert Jun 22 at 12:20
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You may find mathoverflow.net/questions/79685/… helpful. –  Gerry Myerson Jul 17 at 6:41

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