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?
(1) This is actually not a problem in the form you have stated it — the rules of what is a valid proof in first-order logic can be stated without any reference to sets, such as by speaking purely about operations on concrete strings of symbols, or by arithmetization with Gödel numbers.
However, if you want to do model theory on your first-order theory you need sets. And even if you take the syntactical viewpoint and say that it is all just strings, that just pushes the fundamental problem down a level, because how can we then formalize reasoning about natural numbers (or symbol strings) if first-order logic itself “depends on” natural numbers (or symbol strings)?
The answer to that is that is just how it is — the formalization of first-order logic is not really the ultimate basis for all of mathematics, but a mathematical model of mathematical reasoning itself. The model is not the thing, and mathematical reasoning is ultimately not really a formal theory, but something we do because we intuitively believe that it works.
(2) This is a misunderstanding. In axiomatic set theory, the axioms themselves are the definition of the notion of a set: A set is whatever behaves like the axioms say sets behave.
(3) What you quote is how functions usually are modeled in set theory. Again, the model is not the thing, and just because we can create a model of our abstract concept of functional relation in set theory, it doesn’t mean that our abstract concept an sich is necessarily a creature of set theory. Logic has its own way of modeling functional relations, namely by writing down syntactic rules for how they must behave — this is less expressive but sufficient for logic’s need, and is no less valid as a model of functional relations than the set-theoretic model is.