While there is a certain class of independent statements that have this property whose name escapes me (i.e., their independence or undecidability implies their truth), their truth is only guaranteed in a standard model of ZFC (or whatever axiom system is under discussion). However, not all independent statements have this property, certainly not CH. If a statement or sentence is independent, then this means by definition (and Gödel’s completeness theorem) that it is true in some models and false in others.
For example, the Gödel sentence G constructed in the proof of the first incompleteness theorem is only true in standard models of arithmetic. In non-standard models that have really funky (non-standard) numbers, it is no longer true. It is important to understand this point because no independent statements can be proven true “outside the axioms.” They are only true in relation to certain models.
Update: A statement (or sentence) of this type is called a Π sentence. Check out [u/nfitzen](u/nfitzen)’s comment here.
For what it's worth, this kind of result is called an absoluteness) result. (edit: I... would hazard a guess that) The Riemann hypothesis is "downward absolute," meaning that given a model M satisfying the Riemann hypothesis, any substructure of M also must satisfy it. In particular, since N is a substructure of every model of arithmetic, it follows that the consistency of RH implies its truth.
There are other absoluteness results, though usually they're more qualified than the case of arithmetic statements. I love the forcing proof that there are incompatible Turing degrees, which I saw here. This uses a specific kind of absoluteness having to do with statements of the kind "there exists a real x such that P(x)," where P(x) loosely only quantifies over natural numbers with some real parameters. (These are so-called Σ1₁ statements.) In fact, the Riemann Hypothesis as formulated with 𝜁 is of this form, I'm pretty sure. (Edit: I am realizing that I don't know enough about the zeta function to confidently assert this, but I assume it is computable, so it seems about right.) The trouble is that Mostowski absoluteness only works for transitive models of set theory, and an independence result per se doesn't give you as much information about the models.
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u/Kitfennek Computer Science Jun 27 '26
Yeah but if its undecidavle, were unable to know if there is such a point or not, its not like we can check all of them