Editing Axiom of regularity (section)

== Regularity and Russell's paradox ==
[[Naive set theory]] (the axiom schema of [[unrestricted comprehension]] and the [[axiom of extensionality]]) is inconsistent due to [[Russell's paradox]]. In early formalizations of sets, mathematicians and logicians have avoided that contradiction by replacing the axiom schema of comprehension with the much weaker [[axiom schema of separation]]. However, this step alone takes one to theories of sets which are considered too weak.{{clarification needed|date=January 2023|reason=This seems to have in mind a specific result or interpretation, however what that might be is not stated. Ideally that would be given, along with a citation of at least one reference stating/explaining the corresponding result/interpretation.}}{{citation needed|date=January 2023}} 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.{{citation needed|date=January 2023}}{{clarification needed|date=January 2023|reason=Interpreting this literally, if these axioms were all special cases of the particular comprehension axiom, then adding them back would neither strengthen nor weaken the theory. So clearly something slightly different is what the author had in mind, and expressed it this way heuristically. Fine. But a reference to a more precise explanation/result could still be helpful.}} 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 [[universal set|set of all sets]]. The axiom of regularity together with the axiom of pairing also prohibit such a universal set. However, Russell's paradox yields a proof that there is no "set of all sets" using the axiom schema of separation alone, without any additional axioms. In particular, ZF without the axiom of regularity already prohibits such a universal set.

If a theory is extended by adding an axiom or axioms, then any (possibly undesirable) consequences of the original theory remain consequences of the extended theory. In particular, if ZF without regularity is extended by adding regularity to get ZF, then any contradiction (such as Russell's paradox) which followed from the original theory would still follow in the extended theory.

The existence of [[Quine atom]]s (sets that satisfy the formula equation ''x''&nbsp;=&nbsp;{''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-well-founded set theory|non-wellfounded set theories]] allow "safe" circular sets, such as Quine atoms, without becoming inconsistent by means of Russell's paradox.{{sfn|Rieger|2011|pp=175,178}}