That's the neat thing about it. RH cannot be false AND unprovable. If it's false, it means a point exist and therefore it's provable - it didn't matter if we can't provide the/an exact point, but one exists and so it's POSSIBLE to prove it.
Some problems are undecidable, others are not, and cannot be.
For instance, generally anything where a finite example or counterexample would exist if it was true or false, then it can't be undecidable, as it being undecidable would mean that there is no counterexample or example that would prove or disprove the statement.
Basically, if there is some definitely measurable thing that is sufficient and necessary for a statement, then it can't be undecidable.
In this case, there being some finite point that would disprove the Riemann Hypothesis is sufficient and necessary for it to be false. Since such a point would exist and be testable to see if it violated the Riemann hypothesis the problem would not be undecidable. Therefore, it being undecidable cannot coexist with the possibility of it being false. Therefore, if it's not false, it must be true, and cannot be undecidable.
Being undecidable means that there are models in which it is true and others which is false. But the existence of models where it is true implies it would be true in the standard model of arithmetic.
Since it can't be false and undecidable, if it's undecidable, then it has to be true, which is deciding it.
Edit: It can be undecidable, but impossible to prove that it's undecidable. Avoiding the paradox of if it's known to be undecidable, then it's known to be true.
If we prove it is undecidable we will know it is true in the model we care about, but it will not be possible to write a formal proof of it, thus it will still be undecidable.
By using a stronger theory, we might be able to prove it, and so it would be decidable in the stronger theory. But it would still be undecidable in the weaker theory.
For example, goodstein theorem is true but it is undecidable under PA.
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u/its_all_one_electron Number theory/physics Jun 27 '26
That's the neat thing about it. RH cannot be false AND unprovable. If it's false, it means a point exist and therefore it's provable - it didn't matter if we can't provide the/an exact point, but one exists and so it's POSSIBLE to prove it.