Axiom:Axiom of Pairing/Set Theory/Weak Form

Axiom

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

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

That is, let $a$ and $b$ be sets.

Then there exists a set $c$ such that $a \in c$ and $b \in c$.

Thus it is possible to create a set that contains as elements two sets that have already been created.

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

Sources

• 1960: Paul R. Halmos: Naive Set Theory ... (previous) ... (next): $\S 3$: Unordered Pairs
• 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: $\S 4$ The pairing axiom: Note $4$