This is more of a question about philosophy of mathematics. There is an apparent circularity in set theory and logic, in the means that we use set theory to define model theory, and we use results from model theory for proofs in set theory. And, analogously, we use set theory to prove facts from logic (unique readability, for example) and we define set theory using logic. I feel that we can reduce all this notions to a single syntactical idea of logical consequence, and from there we can define things in a non-circular way. However, I can't seem to make this notions all fit together in a logical, foundational manner. Does anyone know of an article or book that addresses this issue?
I wonder about the foundations of set theory and my question can be stated in some related forms:   If we base Zermelo–Fraenkel set theory on first order logic, does that mean first order logic is not allowed to contain the notion of sets? The axioms of Zermelo–Fraenkel set theory seem to already expect the notion of a set to be defined. Is there are pre-definition of what we are dealing with? And where? In set theory, if a function is defined as a set using tuples, why or how does first order logic and the axioms of Zermelo–Fraenkel set theory contain parameter dependend properties $\psi(u_1,u_2,q,...)$, which basically are functions?
I know they have their adherents, but do more or less esoteric branches of mathematics such as Category Theory and/or Quantum Logic provide powerful tools for new theory development or are they just occasionally-useful presentational frameworks?  More particularly, is it worth investing the time to study these formalisms?