# Axiom:Axiom of Extension/Set Theory

## Axiom

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:

### Formulation 1

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

### Formulation 2

In set theories that define $=$ instead of admitting it as a primitive, the **axiom of extension** can be formulated as:

- $\forall x: \paren {\paren {A = B \land A \in x} \implies B \in x}$

The order of the elements in the sets is immaterial.

Hence a set is completely and uniquely determined by its elements.

## 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

- 1960: Paul R. Halmos:
*Naive Set Theory*... (previous) ... (next): $\S 1$: The Axiom of Extension - 1964: W.E. Deskins:
*Abstract Algebra*... (previous) ... (next): Chapter $1$: A Common Language: $\S 1.1$ Sets - 1966: Richard A. Dean:
*Elements of Abstract Algebra*... (previous) ... (next): $\S 0.2$. Sets - 1993: Keith Devlin:
*The Joy of Sets: Fundamentals of Contemporary Set Theory*(2nd ed.) ... (previous) ... (next): $\S 1$: Naive Set Theory: $\S 1.1$: What is a Set? - 1996: Winfried Just and Martin Weese:
*Discovering Modern Set Theory. I: The Basics*... (previous) ... (next): Part $1$: Not Entirely Naive Set Theory: Chapter $1$: Pairs, Relations, and Functions