Axiom:Axiom of Pairing

Axiom

Set Theory

For any two sets, there exists a set to which only those two sets are elements:

$\forall a: \forall b: \exists c: \forall z: \paren {z = a \lor z = b \iff z \in c}$

Class Theory

Let $a$ and $b$ be sets.

Then the class $\set {a, b}$ is likewise a set.

Also known as

The Axiom of Pairing is also known as the Axiom of the Unordered Pair.

Some sources call it the Pairing Axiom.

Also see

• Definition:Doubleton
• Definition:Ordered Pair

• Results about the axiom of pairing can be found here.

Sources

• 2002: Thomas Jech: Set Theory (3rd ed.) ... (previous) ... (next): Chapter $1$: Pairing