Axiom of Extensionality

• name: Axiom of Extensionality

• aka : Axiom of Extension

• wikipedia: https://en.wikipedia.org/wiki/Axiom_of_extensionality

• found in: Zermelo-Fraenkel set theory

In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom of extensionality is one of the axioms of Zermelo-Fraenkel set theory.

• In the formal language of the Zermelo-Fraenkel axioms, the axiom reads (and assumes that equality is a primitive non-logical symbol):

∀A∀B. (∀X. X ∈ A ⟺ X ∈ B) ⟹ A = B

• in latex:

$\forall A\,\forall B\,(\forall X\,(X\in A\iff X\in B)\implies A=B)$

• in words:

Given any set A and any set B, if for every set X, X is a member of A if and only if X is a member of B, then A is equal to B (it is not really essential that X here be a set, but in ZF everything is).

• meaning: This axiom is about the equality of two sets. It is a roundabout way to say that two sets are equal when they have exactly the same elements.

• essentially: a set is identified by its members.

• rant: That's why all the empty sets are, in fact, one and the same (empty) set. There is just one empty set; it is unique and therefore deserving of the qualifier "the" (the empty set). Otherwise, we wouldn't be able to tell apart the empty sets from one another, because it is the elements that are important, that make a set distinguished. Without the elements, a set is nothing.