Equivalence of Formulations of Axiom of Pairing for Classes

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Theorem

The following formulations of the Axiom of Pairing in the context of class theory are equivalent:

Formulation 1

Let $a$ and $b$ be sets.

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

Formulation 2

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

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


Proof

It is assumed that all classes are subclasses of a basic universe $V$.


$(1)$ implies $(2)$

Let formulation $1$ of the Axiom of Pairing be assumed:

Let $a$ and $b$ be sets.

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


Thus we have that $c = \set {a, b}$ is a set such that both $a \in c$ and $b \in c$.

Thus formulation $2$ of the Axiom of Pairing is seen to hold.

$\Box$


$(2)$ implies $(1)$

Let formulation $2$ of the axiom of pairing be assumed:

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

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


Then the class $\set {a, b}$ is a subclass of $c$.

We have by hypothesis that $c$ is a subclass of a basic universe $V$.

Hence by the Axiom of Swelledness, every subclass of $c$ is a set.

That is, $\set {a, b}$ is a set.

Thus formulation $1$ of the Axiom of Pairing is seen to hold.

$\blacksquare$


Sources

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