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This is related to Are there statements that are undecidable but not provably undecidable and the links referred to there which addresses analagous questions, showing yes there are such statements.

This is an erroneous armchair argument to the contrary and I am wondering what is wrong with this line of reasoning. I have thought about it myself and asked others, but we aren't clear what's wrong. For a proposition P consider the statements

1) P can be proven.
2) P can be disproven.
3) Neither 1) or 2) can be proven.

Suppose 1), 2), and 3) cannot be proven. Then 3) is true.

Case 1: If we accept this as a proof of 3) we get a contradiction. So either 1), 2), or 3) can be proven.

Case 2: If we do not accept this as a proof of 3). Note 3) cannot be disproven or else 1) or 2) would be provable. So 3) must be undecidable. So 3) can be either true or false without contradiction since unprovability means independence. But this is absurd since if 3) were false 1) or 2) could be proven. So either 1), 2), or 3) was provable.

Stephen Harrison
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  • Why is this "absurd"? – Eric Wofsey Nov 27 '19 at 23:13
  • Thanks, I edited it. The 2 cases are there because I don't know what the exact conditions of being a proof of 3) should be – Stephen Harrison Nov 27 '19 at 23:17
  • @MCUrist If a statement is undecidable you can add it as an axiom, and still have a consistent system. Fix set theory as the logic and meta-logic, then sure. – Dole Nov 27 '19 at 23:33
  • Let me also remark that if you want your statements to be analogous to the question you linked and other similar questions, then 1) should simply be "P", not "P can be proven", and similarly for 2). – Eric Wofsey Nov 28 '19 at 00:59

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First of all, what you have written certainly is not a proof of 3). It is a proof of the statement "if 1), 2), and 3) cannot be proven, then 3)". So, if you can prove that 1), 2), and 3) cannot be proven, you would have a proof of 3). But it may be that 1), 2), and 3) cannot be proven, but you cannot prove that fact.

In your "Case 2", there are also errors. If 3) is disprovable, that does not mean that 1) or 2) is provable. It just means that it is provable that 1) or 2) is provable, but you cannot deduce the truth of a statement from its provability. (If you actually have a proof of a statement then you can conclude it is true, but if you just abstractly know it is provable as we do in this argument, you cannot conclude it is true.)

Moreover, even if 3) is not disprovable, there is no contradiction. Here you essentially just repeated your first error: you have shown that if 1) and 2) are not provable then 3) cannot be false. That does not prove 3) actually is true unless you first prove that 1) and 2) are not provable.

Eric Wofsey
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  • Where I don't follow is if we manage to prove something is provable in the abstract then why can't we conclude that thing is true without writing the proof down? This does not seem different than mathematical methods which say something definitely is possible in principle, but we don't have to run through the computations – Stephen Harrison Nov 28 '19 at 01:04
  • In this context, "concluding that something is true" means writing down a proof of it. That's the only way you can ever conclude something is true: by proving it. – Eric Wofsey Nov 28 '19 at 01:09
  • Sorry, I don't quite understand the difference between this and non-constructive methods. In numberphile's video on the Riemann Hypothesis a mathematician said an interesting way to prove the Riemann Hypothesis is to prove you can't disprove it – Stephen Harrison Nov 28 '19 at 01:10
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    That is a separate issue and has to do with the specific nature of the Riemann hypothesis. See https://math.stackexchange.com/questions/2305177/decidability-of-the-riemann-hypothesis-vs-the-goldbach-conjecture – Eric Wofsey Nov 28 '19 at 01:13
  • +1 I had this question too – Stephen Harrison Nov 28 '19 at 01:17
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    Just because you prove something does not mean it's true in the sense I believe you meant. It is possible that the logic is inconsistent also. – Dole Nov 28 '19 at 07:49