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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?
The guarantee that such a set can't exist is already given by the argument of Russell's paradox: its existence leads to a contradiction therefore it can't exist.
The problem with unrestricted comprehension was that it guaranteed the set does exist, which causes a problem because of the conflicting guarantees.
I don't think the axiom scheme of separation "resolves" Russell's paradox at all, but restricts the way of using predicates to determine sets.
The paradox is nothing but a proof that there is no one-to-one correspondence between predicates and classes: there are predicates that not defines a class. When writing sets as $\{x|p(x)\}$ that one-to-one correspondence is understood. Therefore axiomatic set theory is to show that sets exists and show rules how to create sets. If $A$ is a set, then the set $\{x\in A|p(x)\}$ never will cause any trouble, due to the theories.
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 $B\notin A$.
You are correct.
Russell's Paradox exposed the inconsistency of a proposed formal set theory of his day. This so-called naive set theory allowed you to derive theorems of the form:
$\exists S: \forall a:[a \in S \iff P(a)]$ for any unary predicate $P$.
This was called the axiom schema of unrestricted comprehension.
For $P(a)\equiv a\notin a$, however, we could then prove to the contrary that:
$\neg\exists S:\forall a:[a\in S \iff a\notin a]$
The moral of the story: Just because you can "define" a set doesn't mean it actually exists. Some restrictions must apply.
These restrictions were described, at least in part, in the so-called axiom schema of restricted comprehension that replaced the above axiom:
$\forall X:\exists S: \forall a:[a \in S \iff a\in X \land P(a)]$ for any unary predicate $P$ where $S$ is not referred to in $P(a)$.
This eliminated the direct route to Russell's Paradox and plugged one hole in naive set theory. Various ways including effectively banning self-reference altogether (e.g. $x\in x$) were also introduced into set theory. This, it was hoped, would banish anything resembling Russell's Paradox for good.
