Unrestricted comprehension + regularity

Question

I am learning set theory and I am curious if we could have unrestricted comprehension while blocking Russell's paradox using the axiom of regularity/foundation. To my very limited knowledge, axiom of regularity is not needed to block paradoxes since if ZF without regularity is inconsistent, then adding regularity would not make it consistent (and the restricted separation schema has already done the work). It seems to me that axiom of regularity also blocks the existence of a universal set and since it says no sets can be a member of itself, it should also block the construction of the Russell sets (since $\forall x (x\not\in x)$ is a description of the universal set under the Axiom of Regularity...?)

Answer 1

Adding an axiom can never make an inconsistent system consistent. Axioms can't block paradoxes. You should think of this the other way round: the axioms of regularity and replacement are designed to strengthen ZF in useful ways while avoiding inconsistency.

Answer 2

Rob wrote correctly, you cannot revive a dead person by giving him an extra leg to stand on.

But here is a direct proof: Consider the Russell class, $\{x\mid x\notin x\}$. By comprehension this is a set, and by regularity it is in fact the set of all sets. But now it must be a member of itself, and this is a contradiction to regularity.