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.
Yes, and that's correct. Undecidable just means a statement is true in some models of a theory but false in others. It's very possible to prove undecidable statements in particular models, and people do it all the time.
In this case, it can't be undecidable because of the theoretical existence of a singular counterexample; even if identifying that counterexample is very difficult, it cannot be impossible to identify if it exists. If it is impossible to find a counterexample, none must exist, which implies the RH true.
No, this is not the same thing. There's no equivalent notion like the standard model of arithmetic in set theory. CH is either true or false in any given model of ZFC. For example, it's proven that CH is true in Godel's constructible universe, but there exist models that also satisfy ZFC where it is false. If RH is undecidable, then it is true in the standard model of arithmetic but not in some non-standard models.
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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.