system has the notational advantage of not containing any explicitly typed variables, although in fact it can be seen as having an implicit type structure built into it, at least if the axiom of regularity is included. The details of this implicit typing are spelled out in [Zermelo 1930], and again in a well-known article of George Boolos [Boolos 1971]." went further and claimed that: In the same paper, Scott shows that an axiomatic system based on the inherent properties of the cumulative hierarchy turns out to be equivalent to ZF, including regularity. History The concept of well-foundedness and rank of a set were both introduced by Dmitry Mirimanoff (1917) cf. and . 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; in later papers Mirimanoff also explored what are now called non-well-founded sets ("extraordinaire" in Mirimanoff's terminology). and 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, gave the following axiom (rendered in modern notation by A. Rieger): . Regularity in the presence of urelements Urelements are objects that are not sets, but which can be elements of sets. In ZF set theory, there are no urelements, but in some other set theories such as ZFA, there are. In these theories, the axiom of regularity must be modified. The statement "" needs to be replaced with a statement that is not empty and is not an urelement. One suitable replacement is , which states that x is inhabited. See also Non-well-founded set theory Scott's trick Epsilon-induction References Sources reprinted in Reprinted in From Frege to Gödel, van Heijenoort, 1967, in English translation by Stefan Bauer-Mengelberg, pp. 291–301. ; translation in ; translation in External links Inhabited set and the axiom of foundation on nLab Axioms of set theory Wellfoundedness