information:filtering, structural:expansion
History

The concept of well-foundedness and rank of a set were both introduced by Dmitry Mirimanoff (1917) cf. Lévy (2002, p. 68) and Hallett (1996, §4.4, esp. p. 186, 188). Mirimanoff called a set x "regular" (French: "ordinaire") if every descending chain x ∋ x ∋ x ∋... is finite. Mirimanoff however did not consider his notion of regularity (and well-foundedness) as an axiom to be observed by all sets; in later papers Mirimanoff also explored what are now called non-well-founded sets ("extraordinaire" in Mirimanoff's terminology).

Skolem (1923) and von Neumann (1925) pointed out that non-well-founded sets are superfluous (on p. 404 in van Heijenoort's translation) and in the same publication von Neumann gives an axiom (p. 412 in translation) which excludes some, but not all, non-well-founded sets. In a subsequent publication, von Neumann (1929, p. 231) gave an equivalent but more complex version of the Axiom of Class Foundation, cf. Suppes (1972, p. 53) and Lévy (2002, p. 72):

The contemporary and final form of the axiom is due to Zermelo (1930).