# Axiom:Axiom of Extension/Class Theory

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## Axiom

Let $A$ and $B$ be classes.

Then:

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

Hence the order in which the elements are listed in the classes is immaterial.

## 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

- 2010: Raymond M. Smullyan and Melvin Fitting:
*Set Theory and the Continuum Problem*(revised ed.) ... (previous) ... (next): Chapter $2$: Some Basics of Class-Set Theory: $\S 1$ Extensionality and separation: $P_1$**[Axiom of extensionality]**