Consider a first-order axiomization of ZFC. The quantifiers range over all the sets. However, we can prove that (in ZFC) there is no set which contains all sets. Soooo.........how can we make a _model_ for ZFC? The first thing you do when you make a model for a set of axioms is specify a domain, which is a set of things which the quantifiers range over......this seems to be exactly what you can't do with ZFC.
So what am I missing?