Axiom of extensionality

The set-theoretic axiom stating that two sets are equal if and only if they have exactly the same elements. It treats sets as determined solely by their members.

Extensionality: A principle that objects are identical when they share all relevant properties or parts. The axiom applies this broader identity principle specifically to sets and their members.
Set theory: The mathematical study of sets, membership, and structures built from sets. The axiom is one of the foundational principles governing this subject.
Ernst Zermelo: A German mathematician who helped establish axiomatic set theory in the early twentieth century. His axiomatization made extensionality part of a systematic foundation for set theory.
Axiom of regularity: The set-theoretic axiom forbidding infinite descending chains of membership and sets that contain themselves. Together with extensionality, it supports the standard well-founded picture of sets.
Non-well-founded set theory: Set theories that permit membership structures excluded by the standard axiom of regularity. Some versions retain extensionality while allowing unusual self-referential membership structures.
Set equality: Equality between sets, holding when they contain precisely the same elements. The axiom gives the membership-based criterion for deciding when two sets are equal.
First-order logic: A formal system for reasoning with predicates, variables, quantifiers, and identity. The axiom is commonly stated as a universally quantified formula in this logic.
Zermelo–Fraenkel set theory: An axiomatic foundation for mathematics built around sets and their membership relation. Extensionality is one of its standard axioms.
Axiom of pairing: The set-theoretic axiom guaranteeing a set whose elements are any two given objects. Extensionality makes the set supplied by pairing unique by its specified members.
Quine atom: A set that is equal to the singleton containing itself, in a non-well-founded set theory. Extensionality can identify it with its singleton because both have the same sole member.

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