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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 '14 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 '14 at 7:51
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 '14 at 8:08
@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 '14 at 12:20
You may find… helpful. – Gerry Myerson Jul 17 '14 at 6:41
  1. Since it is still a conjecture (a "Millenium Prize Problem") we don't know if it is true or false
  2. Therefore we cannot even know whether it is independent from $ZFC$ because it would imply that we know that it is also true (chek the replies here).
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This isn't quite accurate - in principle, we could prove that RH was decidable without deciding it (e.g. prove "RH iff R(5, 5) is odd"), although I don't think anyone believes this is a real possibility. You are right, though, that if we proved that RH was independent that would imply that RH is true. Of course, that's all relative to the consistency of the system in which independence is proved - if ZFC proved that PA did not decide RH, all that would mean is that RH is true or ZFC is inconsistent. Of course, most of us strongly doubt that latter possibility, but it's worth mentioning. – Noah Schweber Feb 29 at 15:35
@NoahSchweber What is R? – Dustan Levenstein Apr 6 at 14:36
Nevermind, I got it. – Dustan Levenstein Apr 6 at 14:41

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