Incidently, Russell found his famous paradox exactly a century ago in 1901 by studying Frege's axiomatic work on set theory with axiom V (1893) dealing with his axiom of abstraction. By introducing in 1908 the axiom schema of separation Zermelo was able to avoid Russell's paradox. Zermelo was able to show that the axiom schema of separation flowed naturally from the axiom of abstraction so there wasn't the problem of acting ad hoc (as many felt about Russell's theory of types).

Fraenkel (1922) strengthened Zermelo's axiom schema of separation with the axiom schema of replacement to handle transfinite induction and ordinal arithmetic which is why the axiomatic approach of Zermelo is called ZF set theory (Zermelo-Fraenkel) instead of just Zermelo set theory.

Warm regards, Harv