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 (1986, §4.4, esp. p. 186, 188). Mirimanoff called a set x "regular" (French: "ordinaire") if every descending chain x ∋ x1 ∋ x2 ∋ ... is finite. Mirimanoff however did not consider his notion of regularity (and well-foundedness) as an axiom to be observed by all sets (Halbeisen 2012, pp. 62–63); in later papers Mirimanoff also explored what are now called non-well-founded sets ("extraordinaire" in Mirimanoff's terminology) (Sangiorgi 2011, pp. 17–19, 26).
According to Adam Rieger, von Neumann (1925) describes non-well-founded sets as "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 (Rieger 2011, p. 179). In a subsequent publication, von Neumann (1928) gave the following axiom (rendered in modern notation by A. Rieger):
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Read more about this topic: Axiom Of Regularity
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