Axiom:Axiom of Extension/Class Theory

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

• Definition:Class Equality
• Definition:Equals

• Axiom:Axiom of Extension (Sets)

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]