Axiom:Axiom of Pairing/Set Theory

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Axiom

Formulation 1

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

$\forall A: \forall B: \exists x: \forall y: \paren {y \in x \iff y = A \lor y = B}$

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


Formulation 2

For any two sets, there exists a set containing those two sets as elements:

$\forall A: \forall B: \exists x: \forall y: \paren {y \in x \implies y = A \lor y = B}$

Thus it is possible to create a set that contains as elements at least 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


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.