# 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

- they contain the same elements.

That is, if and only if:

and:

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

## 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

- 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**axiom of extensionality** - 2002: Thomas Jech:
*Set Theory*(3rd ed.) ... (previous) ... (next): Chapter $1$: Extensionality - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**axiom of extensionality**