Axiom of extensionality

The axiom of extensionality, also known as the axiom of extent, appears in many versions of axiomatic set theory, including Zermelo–Fraenkel set theory. It essentially defines what a set is: informally, two sets are equal precisely when they contain the same members.

The term "extensionality" comes from logic. An intensional definition gives the necessary and sufficient conditions for a term to apply to an object—for instance, "an even number is an integer divisible by 2." An extensional definition, by contrast, lists every object the term applies to—for example, "an even number is any of the integers 0, 2, 4, 6, 8..., -2, -4, -6, -8..." In logic, the extension of a predicate is the set of all things for which that predicate holds true.

This logical concept was introduced into set theory in 1893 by Gottlob Frege in his Basic Laws of Arithmetic. He formally used the idea of an extension: for a predicate F, its extension (in German, Umfang) is the set of all objects satisfying F. For instance, if F(x) is "x is even," then its extension is the set {..., -4, -2, 0, 2, 4, ...}. Frege defined his infamous Basic Law V as: the extension of F equals the extension of G if and only if for all x, F(x) is equivalent to G(x). This law states that if two predicates have the same extension—meaning they are satisfied by the same objects—then they are logically equivalent. However, it was later discovered that this axiom leads to Russell's paradox.

The first clear statement of the modern axiom of extensionality appeared in 1908, in a paper by Ernst Zermelo on the well-ordering theorem. There, Zermelo presented the first axiomatic set theory, now called Zermelo set theory, which became the foundation for modern set theories. The term Zermelo used for "extensionality" was Bestimmtheit. The English word "extensionality" only became common in mathematical and logical texts during the 1920s and 1930s, especially with the formalization of logic and set theory by figures such as Alfred Tarski and John von Neumann.

In the formal language of Zermelo–Fraenkel set theory, the axiom is written as: ∀x∀y [∀z (z ∈ x ↔ z ∈ y) → x = y]. In words: if the sets x and y have the same members, then they are the same set. In pure set theory, every member of a set is itself a set, though this is not the case in set theories that include urelements.