My question is essentially in the title. I'm reading some notes about the proof of consistency of ZFC assuming the consistency of ZF. The author first assumes theres is a model for ZF-Foundation and proves that the von Neumann universe is then a model for ZF.
Subsequently the author constructs the constructible universe L and proves that it is a model for the axiom of constructibility V=L and subsequently for AC and GCH.
As far as I understand why L believes in V=L, it's not clear to me whether V is a model of V=L or not.
To be a model for a sentence means that the sentence is true or false under an interpretation, so it seems to me that there should be a definite answer to that, i.e. either V is a model of V=L or it is not.
I was unable to find any reference for that in the literature. Would anyone care to help me with that?
(In particular if V believed in V=L then it would already be a model for ZFC. Or is the answer that we simply cannot deduce $\langle V,\in \rangle\models V=L$ nor V doesn't model V=L? But if so, is it a theorem than we cannot prove any of the above (from ZFC) or we simply don't know?)