# Axiom:Axiom of Extension/Set Theory/Formulation 1

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## Axiom

Let $A$ and $B$ be sets.

The **axiom of extension** can be formulated as:

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

### Axiom of Extension for Classes

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**.

## 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

- 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?

- Weisstein, Eric W. "Zermelo-Fraenkel Axioms." From
*MathWorld*--A Wolfram Web Resource. https://mathworld.wolfram.com/Zermelo-FraenkelAxioms.html - Weisstein, Eric W. "Axiom of Extensionality." From
*MathWorld*--A Wolfram Web Resource. https://mathworld.wolfram.com/AxiomofExtensionality.html