Axiom:Axiom of Pairing/Set Theory

Axiom

Strong Form

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

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

Weak Form

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

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

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

• Equivalence of Formulations of Axiom of Pairing: Both forms of the axiom are equivalent, assuming the Axiom of Specification.
• Definition:Doubleton

• Results about Axiom of Pairing can be found here.

Relation to other axioms

The Axiom of Pairing can be deduced as a consequence of:

$(1): \quad$ The Axiom of Infinity and the Axiom of Replacement: see Axiom of Pairing from Infinity and Replacement

$(2): \quad$ The Axiom of Powers and the Axiom of Replacement: see Axiom of Pairing from Powers and Replacement.

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): pair set
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): pair set