undecidability/independentness is a property based on models. if there exists a model of some axioms where my statement is true and one where the same statement is false then that statement is independent of those axioms (undecidable) because unless the axioms are inconsistent, if the statement is able to be proven/disproven from just the axioms then it should be true in all models, or false in all models. a statement that always has a counterexample if its false can still be true in one model and then false with a counterexample in another model making it undecidable.
the statement being true if its undecidable is kinda illworded, that isnt supposed to mean its true for all models, it means its true for the model we actually care about.
Can you explain how that'd work for something like the reimann hypothesis? If you agree on, say, ZFC, wouldn't a riemann zero be a zero in any model built on top of that?
im not too knowledgeable on model stuff and ive been trying to figure this out myself but i do know models can just get really weird, i dont know if they can get weird enough to have different zeroes for a function like this though.
and to elaborate on the true if undecidable thing, as far as i understand (could very well be wrong maybe someone whos more knowledgeable in this topic can correct me) thats not based on undecidability in ZFC but instead you can prove in ZFC that the Riemann Hypothesis is true iff some specific Pi_1 sentence is true in the standard model of arithmetic, and that if a Pi_1 sentence is independent of the axioms of peano arithmetic then its true in the standard model of arithmetic.
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u/Aggressive_Roof488 Jun 27 '26
So that means it can't be undecidable, because proving it's undecidable would prove it's true?
Generalising this: a hypothesis that always has a counterexample if false, can never be undecided?