Editing Axiom of regularity (section)

== History ==

The concept of well-foundedness and [[Von Neumann universe|rank]] of a set were both introduced by [[Dmitry Mirimanoff]].{{sfn|Mirimanoff|1917}}<ref>cf. {{harvnb|Lévy|2002|p=68}} and {{harvnb|Hallett|1996|loc=§4.4, esp. p. 186, 188}}.</ref> Mirimanoff called a set ''x'' "regular" ({{langx|fr|ordinaire}}) if every descending chain ''x'' ∋ ''x''<sub>1</sub> ∋ ''x''<sub>2</sub>  ∋ ... is finite. Mirimanoff however did not consider his notion of regularity (and well-foundedness) as an axiom to be observed by all sets;{{sfn|Halbeisen|2012|pp=62–63}} in later papers Mirimanoff also explored what are now called [[non-well-founded set theory|non-well-founded sets]] ({{lang|fr|extraordinaire}} in Mirimanoff's terminology).{{sfn|Sangiorgi|2011|pp=17–19, 26}}

Skolem{{sfn|Skolem|1923}} and von Neumann{{sfn|von Neumann|1925}} pointed out that non-well-founded sets are superfluous{{sfn|van Heijenoort|1967|p=404}} and in the same publication von Neumann gives an axiom{{sfn|van Heijenoort|1967|p=412}} which excludes some, but not all, non-well-founded sets.{{sfn|Rieger|2011|p=179}} In a subsequent publication, von Neumann{{sfn|von Neumann|1929|p=231}} gave an equivalent but more complex version of the axiom of class foundation:<ref>cf. {{harvnb|Suppes|1972|p=53}} and {{harvnb|Lévy|2002|p=72}}</ref>

<math display="block"> A \neq \emptyset \rightarrow \exists x \in A\,(x \cap A = \emptyset).</math>

The contemporary and final form of the axiom is due to Zermelo.{{sfn|Zermelo|1930}}