 Standard responses to the paradox attempt to limit in some way the conditions under which sets are formed. The goal is usually both to eliminate \(R\) (and similar contradictory sets) and, at the same time, to retain all other sets needed for mathematics. This is often done by replacing the unrestricted Comprehension Axiom with the more restrictive Separation Axiom, namely the axiom that given any (consistent) set \(S\) and any formula \(\phi(x)\) with \(x\) free, there will be a set \(\{x \in S: \phi(x)\}\) whose members are exactly those members of \(S\) that satisfy \(\phi(x)\). If we now let \(\phi(x)\) stand for the formula \(x \not\in x\), it turns out that the corresponding set, \(\{x \in S: x \not\in x\}\) will not be contradictory since it consists only of those members found within S that are not members of themselves. Hence the set fails to include itself. 