"What I meant to say is that multiple models are not the only reason for something to be true but unprovable, the incompleteness theorem also holds in more general conditions." I can't give a clean rebuttal for this, but I believe this to be profoundly mistaken. It might be technically correct though, depending on how you'd formalize this statement.
To formalize math you need a logic that has some properties: it should be decidable whether a proof is correct, you should be able to write it down, it should not be contradictory. If you take these together the only way a statement is unprovable, is if it is independent from the axioms, i.e. there exist multiple models.
Löwenheim-Skolem implies the existence of a countable model of ZFC. https://en.m.wikipedia.org/wiki/L%C3%B6wenheim%E2%80%93Skole...
"What I meant to say is that multiple models are not the only reason for something to be true but unprovable, the incompleteness theorem also holds in more general conditions." I can't give a clean rebuttal for this, but I believe this to be profoundly mistaken. It might be technically correct though, depending on how you'd formalize this statement. To formalize math you need a logic that has some properties: it should be decidable whether a proof is correct, you should be able to write it down, it should not be contradictory. If you take these together the only way a statement is unprovable, is if it is independent from the axioms, i.e. there exist multiple models.