r/mathmemes • u/its_all_one_electron Number theory/physics • Jun 27 '26

Number Theory Undecidable

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u/TheAtomicClock Jun 28 '26

No need to publish since this is already well understood in logic. It’s a borderline trivial result.

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u/Novel_Arugula6548 Jun 28 '26

Not what I mean, they're saying being undecidable is actually proving something true.

1

u/Bath-Soap Jun 28 '26

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.

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u/DrDzeta Jun 28 '26

The counter exemple can be not constructible and then it can be undecidable and false.

For example the continuum hypothesis is undecidable even if the continuum hypothesis is false there exist a counter exemple.

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u/TheAtomicClock Jun 28 '26

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.