Why does the axiom of separation prevent the contradictions caused by unrestricted comprehension?

Unrestricted comprehension leads to Russell’s paradox because it allows the formation of a set from any condition, such as the set of all sets that are not members of themselves ($R=\{x|x\notin x\}$). This creates a logical contradiction: if $R$ is a member of itself, it must not be; if it is not, it must be.1

The axiom schema of separation prevents this by restricting the principle of comprehension in two critical ways:

• Preexisting Sets: It does not allow the formation of sets from the entire universe of discourse. Instead, it asserts that given a preexisting set $A$ and a condition $S(x)$, one can only form a new set consisting of the elements of $A$ that satisfy $S(x)$.2
• No Universal Set: By requiring that new sets be subsets of existing ones, the axiom prevents the construction of "too large" collections, such as the set of all sets. Consequently, there is no universal set, which effectively breaks the "vicious circle" that generates the paradox.1