One of the

Zermelo-Fraenkel axioms of

set theory, and the most obviously

fundamental. It simply says that two

sets are

equal to each other if they have the same

elements.

This is not entirely trivial. Two sets could be defined by different properties, for example the set of primes less than 5, and the set of solutions of x^2 - 5x + 6 = 0. But a set is an elementary entity defined purely by what members it has, so these are both the set {2, 3}.

Zermelo restricted this principle to sets but it may also be applied to classes generally, in which case it is not an axiom, but is called the Principle of Extensionality.

Formally: **A**x **A**y (x = y <=> **A**z (z **in** x <=> z **in** y))

ariels points out another one that having the same members would not necessarily define the same set if we didn't have this axiom. If sets came in flavours or colours, so that 2 could be red or blue, while still being 2. Then {red 2, red 3} and {blue 2, blue 3} might be different. This axiom says that if 2 is the same thing regardless of its colour, then these are both {2, 3} and the same set. Extensionality says that membership is the only way sets can be distinguished.