Axiom of regularity

An axiom of Zermelo–Fraenkel set theory stating that every nonempty set contains an element disjoint from it. It rules out membership loops and infinitely descending membership chains.

Membership relation: The relation denoted by ∈, in which one set is an element of another. Regularity constrains this relation by requiring a minimal element in every nonempty set of sets.
Zermelo–Fraenkel set theory: An axiomatic foundation for mathematics built around sets and a collection of axioms governing their existence and equality. Regularity is one of its standard axioms.
Well-founded set: A set whose transitive closure has no infinite descending chain under membership. Regularity guarantees that every set is well-founded in this sense.
Non-well-founded set theory: Set theory that allows sets whose membership relation is not well-founded. It studies structures excluded by regularity.
Well-founded relation: A relation is well-founded when every nonempty subset of its domain has a minimal element. Regularity says membership is well-founded on every set.
Set: A collection of distinct objects treated as a single mathematical object. Regularity quantifies over sets and their elements.
Self-containing set: A set that is one of its own elements. Regularity implies that no set can contain itself as an element.
Quine atom: A non-well-founded set that is its own sole element. It directly violates regularity while illustrating a possible alternative universe of sets.
Transitive closure: The smallest transitive set containing a given set and all objects reachable from it by membership. Applying regularity to a transitive closure rules out infinite descending membership chains.
Empty set: The unique set with no elements, denoted ∅. It is disjoint from every set and serves as a simple example of a regular set.

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