# When does the set enter set theory?

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?