TITLE: Models of set theory
QUESTION [11 upvotes]: How can one talk of a semantics or model of set theory (lets say ZF or ZFC) when the definition of a structure (and potential model) needs a carrier set in the first place (by its definition)?

REPLY [11 votes]: I already answered a similar question at length on MathOverflow so let me try a short answer, suitable for this forum.
You mistakenly assume that it is the business of model theory or logic to somehow "produce" set theory out of nothing. Logic and model theory are just two branches of ordinary mathematics -- they do not precede them, although they are a bit peculiar because their object of study is mathematics itself (its methods, its possiblities, its limitations). Therefore, model theorists and logicians are "allowed" to use all the usual tools of mathematics (numbers, sets, topological spaces, and so on).
When logicians speak of "foundations" of mathematics, they may give the impression that they are "building the cathedral" starting from its foundation. But it is much better to view what they are doing as a study of how the cathedral is built and how we can improve it. For instance, logicians have observed the fact that almost all of modern mathematics can be expressed in the language of set theory, but this does not mean that we need to "secure" set theory before the rest of mathematics can be done. History is my witness: geometry, algebra, and analysis existed before set theory and logic came along.
P.S. I will not deny that historically a primary objective of logic was in fact to secure foundations, especially the kind of logic that Bertrand Russell did in his Principia mathematica. However, at least since Gödel we have known that such an endeavor must fail. In any case, I am expressing here my personal view that attempts to secure absolute foundations of mathematics are a bit like attempts to prove there is god. Ultimately it comes down to an act of faith.