Russell's paradox and axiom of separation I don't quite understand how the axiom of separation resolves Russell's paradox in an entirely satisfactory way (without relying on other axioms).
I see that it removes the immediate contradiction that is produced by unrestricted comprehension, but it seems that we still need further axioms to guarantee that a well-formed set $S$ will never contain the set of all given elements (of $S$) which do not contain themselves.
Is that correct? 

Axioms are used to show that under some circumstances, some sets exist.
Russell's paradox shows that not every definable collection can be a set. So we have to restrict what sort of definable collections we allow. The separation axiom schema essentially say that if $A$ is already a set, then every definable subcollection of $A$ is also a set.
If we repeat the Russell paradox over a set $A$ using Separation, then we don't get a contradiction. We get an interesting theorem:

For every set $A$ there is a set $B$ such that $B\notin A$.

