How does Zermelo-Fraenkel set theory avoid Russell’s paradox?

Zermelo-Fraenkel (ZF) set theory avoids Russell’s paradox by replacing the unrestricted comprehension principle with more restrictive axioms that prevent the formation of paradoxical sets, such as the set of all sets that do not contain themselves.12

In earlier systems, the unrestricted comprehension principle allowed for the creation of a set from any property, which led to the "vicious circle" of self-reference. ZF avoids this by utilizing the axiom schema of separation, which dictates that new sets can only be constructed as subsets of existing, previously defined sets. By requiring that elements be separated from an already established set, ZF ensures that one cannot construct a set as large as the "universe of all sets," thereby eliminating the possibility of forming the contradictory set $R=\{x\mid x\notin x\}$.123

Furthermore, the iterative conception of sets in ZF, often supplemented by the foundation axiom, explicitly prohibits sets from containing themselves as members, further reinforcing the system's consistency against such paradoxes.2