Editing Axiom of regularity (section)

{{short description|Axiom of set theory}}
In [[mathematics]], the '''axiom of regularity''' (also known as the '''axiom of foundation''') is an axiom of [[Zermelo–Fraenkel set theory]] that states that every [[Empty set|non-empty]] [[Set (mathematics)|set]] ''A'' contains an element that is [[Disjoint sets|disjoint]] from ''A''. In [[first-order logic]], the axiom reads:

<math display="block">\forall x\,(x \neq \varnothing \rightarrow (\exists y \in x) (y \cap x = \varnothing)).</math>

The axiom of regularity together with the [[axiom of pairing]] implies that [[Russell paradox|no set is an element of itself]], and that there is no infinite [[sequence]] (''a<sub>n</sub>'') such that ''a<sub>i+1</sub>'' is an element of ''a<sub>i</sub>'' for all ''i''. With the [[axiom of dependent choice]] (which is a weakened form of the [[axiom of choice]]), this result can be reversed: if there are no such infinite sequences, then the axiom of regularity is true. Hence, in this context the axiom of regularity is equivalent to the sentence that there are no downward infinite membership chains.

The axiom was originally formulated by von Neumann;{{sfn|von Neumann|1925}} it was adopted in a formulation closer to the one found in contemporary textbooks by Zermelo.{{sfn|Zermelo|1930}} Virtually all results in the branches of mathematics based on set theory hold even in the absence of regularity.{{sfn|Kunen|1980|loc=ch. 3}} However, regularity makes some properties of [[Ordinal number|ordinals]] easier to prove; and it not only allows induction to be done on [[well-ordering|well-ordered sets]] but also on proper classes that are [[well-founded relation|well-founded relational structures]] such as the [[lexicographical ordering]] on <math display="inline">\{ (n, \alpha) \mid n \in \omega \land \alpha \text{ is an ordinal } \} \,.</math>

Given the other axioms of Zermelo–Fraenkel set theory, the axiom of regularity is equivalent to the [[epsilon-induction|axiom of induction]]. The axiom of induction tends to be used in place of the axiom of regularity in [[intuitionism|intuitionistic]] theories (ones that do not accept the [[law of the excluded middle]]), where the two axioms are not equivalent.

In addition to omitting the axiom of regularity, [[Non-well-founded set theory|non-standard set theories]] have indeed postulated the existence of sets that are elements of themselves.