Axiom of Regularity

In mathematics, the axiom of regularity (also known as the axiom of foundation) is one of the axioms of Zermelo–Fraenkel set theory and was introduced by von Neumann (1925); it was adopted in a formulation closer to the one found in contemporary textbooks by Zermelo (1930). In first-order logic the axiom reads:

Or in prose:

Every non-empty set A contains an element B which is disjoint from A.

Two results which follow from the axiom are that "no set is an element of itself," and that "there is no infinite sequence (a_n) such that a_i+1 is an element of a_i 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, the axiom of regularity is equivalent, given the axiom of dependent choice, to the alternative axiom that there are no downward infinite membership chains.

The axiom of regularity is arguably the least useful ingredient of Zermelo–Fraenkel set theory, since virtually all results in the branches of mathematics based on set theory hold even in the absence of regularity; see chapter 3 of Kunen (1980). However, it is used extensively in establishing results about well-ordering and the ordinals in general. In addition to omitting the axiom of regularity, non-standard set theories have indeed postulated the existence of sets that are elements of themselves.

Given the other ZF axioms, the axiom of regularity is equivalent to the axiom of induction.