Axiom of Regularity - Regularity and Russell's Paradox

Regularity and Russell's Paradox

Naive set theory (the axiom schema of comprehension and the axiom of extensionality) is inconsistent due to Russell's paradox. Set theorists have avoided that contradiction by replacing the axiom schema of comprehension with the much weaker axiom schema of separation. However, this makes set theory too weak. So some of the power of comprehension was added back via the other existence axioms of ZF set theory (pairing, union, powerset, replacement, and infinity) which may be regarded as special cases of comprehension. So far, these axioms do not seem to lead to any contradiction. Subsequently, the axiom of choice and the axiom of regularity were added to exclude models with some undesirable properties. These two axioms are known to be relatively consistent.

In the presence of the axiom schema of separation, Russell's paradox becomes a proof that there is no set of all sets. The axiom of regularity (with the axiom of pairing) also prohibits such a universal set, however this prohibition is redundant when added to the rest of ZF. If the ZF axioms without regularity were already inconsistent, then adding regularity would not make them consistent.

The existence of Quine atoms (sets that satisfy the formula equation x = {x}, i.e. have themselves as their only elements) is consistent with the theory obtained by removing the axiom of regularity from ZFC. Various non-wellfounded set theories allow "safe" circular sets, such as Quine atoms, without becoming inconsistent by means of Russell's paradox.(Rieger 2011, pp. 175,178)