Axiom:Axiom of Extension

Axiom

The Axiom of Extension is the fundamental definition of the nature of a collection: it is completely determined by its elements.

Set Theory

Let $A$ and $B$ be sets.

The Axiom of Extension states that:

$A$ and $B$ are equal

if and only if:

they contain the same elements.

That is, if and only if:

every element of $A$ is also an element of $B$

and:

every element of $B$ is also an element of $A$.

This can be formulated as follows:

$\forall x: \paren {x \in A \iff x \in B} \iff A = B$

Class Theory

The axiom of extension in the context of class theory has the same form:

Let $A$ and $B$ be classes.

Then:

$\forall x: \paren {x \in A \iff x \in B} \iff A = B$

Also known as

The Axiom of Extension is also known as:

the Axiom of Extensionality

the Axiom of Extent.

Also see

• Leibniz's Law for Sets

• Definition:Set Equality
• Definition:Equals

Linguistic Note

The nature of the Axiom of Extension, or Axiom of Extensionality as it is frequently called, suggests that the Axiom of Extent ought in fact to be the preferred name, as it gives a precise definition of the extent of a collection.

However, the word extensionality is a term in logic which determines equality of objects by its external features, as opposed to intensionality, which is more concerned with internal structure.

Sources

• 2002: Thomas Jech: Set Theory (3rd ed.) ... (previous) ... (next): Chapter $1$: Extensionality