This question probably doesn't make any sense, but I don't see why, so I ask it here hoping someone will illuminate the matter:
There is this whole area of study in Set Theory about the consistency, independence of axioms, etc. In some of these you use model theory (e.g. forcing) to prove results about set theory.
My question is: What is the foundation of this model theory we are using? We are certainly using sets to talk about the models, what some may call sets in the "meta"-mathematics, that is to say, the "real" mathematics.
But then, all these arguments in the end in are all about the theory of sets as a theory, and not the theory of sets as a foundation of math, since we are using these set sets in the meantime. So our set theory is not about the foundation of math.
Am I right?