Axiom:Axiom of Extension/Set Theory/Formulation 1

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