Concerning multiple models of ZFC: I'm always confused by such statements about the foundations of set theory itself, they seem weirdly self-referential. ZFC certainly can't prove that it has multiple (or even any) models. Does such a statement need additional axioms, or is there a general theorem like "If a first order theory has any model, then it has multiple ones."?
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.
Edit: however, consistent second order theories don't always have a model. In first order, if S is an undecideable statement in theory T, then both T+S and T+!S have models, both of which are models for T, so undecideable statements always come from multiple models. But that does not need to be the case in second order, so your claim "it is independent from the axioms, i.e. there exist multiple models" is not neccessarily true, i.e. there may be cases when decideability of some statement fails in a theory with a unique model. Or is there another argument for your claim?
Forgive me for spamming questions, I just try to understand how these things fit together. But maybe we should just stick to first order, since everything else is too weird anyway.