3. Axiom schema of specification (also called the axiom schema of separation or of restricted comprehension): If z is a set, and \phi\! is any property which may characterize the elements x of z, then there is a subset y of z containing those x in z which satisfy the property. The "restriction" to z is necessary to avoid Russell's paradox and its variant
now Russell's paradox is a famous example of an impredicative construction, namely the set of all sets which do not contain themselves
the axiom of separation is used to outlaw/block/ban impredicative statements like Russells paradox
but this axiom of separation is itself impredicative
"in ZF the fundamental source of impredicativity is the seperation axiom which asserts that for each well formed function p(x)of the language ZF the existence of the set x : x } a ^ p(x) for any set a Since the formular p may contain quantifiers ranging over the supposed "totality" of all the sets this is impredicativity according to the VCP this impredicativity is given teeth by the axiom of infinity "
thus it outlaws/blocks/bans itself thus ZFC contradicts itself and 1)ZFC is inconsistent 2) that the paradoxes it was meant to avoid are now still valid and thus mathematics is inconsistent Now we have paradoxes like Russells paradox Banach-Tarskin paradox Burili-Forti paradox Which are now still valid