TITLE: Set theory and Model Theory
QUESTION [43 upvotes]: 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 sets in the meantime. So our set theory is not about the foundation of math. 
Am I right?

REPLY [8 votes]: First of all, I would like to say that your question is not only justified, but actually most welcome: it shows the King naked. Who is the King? You can call him Semantics,or the mathematical theory of Truth (notice the capital letter)  if you wish.
There is a widespread usage among logicians of sentences such as " the true universe of sets", the "set of natural numbers", and so forth, as if these entities were crystal clear and thus indisputable. They aren't, at least to me. But even if they are, even if there is such a thing as mathematical intuition which informs us on the world of platonic math (what my teacher aptly named PLATO's ATTIC), the fact is this: as soon as we SPEAK about them, they just become syntax, no more no less (ie part of a frame syntactic theory which grounds them). 
Model theory is a great branch of mathematics, no doubts about it. But it is formalized mathematics, and grounded in ZFC. Does that mean that all its results are empty, especially in the model theory of ZFC? I say absolutely no. Instead, it bespeaks of the uncanny capability of a powerful theory such as ZFC to "reflect" upon itself and the rest of mathematics. The king is naked, but a king nevertheless. Long live the king!
PS I just discovered the recent work of Joel D. Hamkins and a few others on the Multiverse: finally some people from the set theory ranks are moving away from a dogmatic  and monolitic notion of truth  (the ghost-like "true universe of sets" and its equally ghostly eternal properties) towards a new dynamic and contextual one. I think this is a beginning of a new era, if I am not mistaken....