well-founded set

In mathematics, the axiom of regularity (also known as the axiom of foundation) is an axiom of Zermelo–Fraenkel set theory that states that every non-empty set A contains an element that is disjoint from A. In first-order logic, the axiom reads:

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The axiom implies 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.

In mathematics, a binary relation, R, is well-founded (or wellfounded) on a class X if and only if every non-empty subset S ⊆ X has a minimal element; that is, some element m of any S is not related by sRm (for instance, "m is not smaller than") for any s ∈ S.

(Some authors include an extra condition that R is set-like, i.e., that the elements less than any given element form a set.)