# Axiom:Axiom of Pairing/Set Theory/Weak Form

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## 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

## 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 $4$