Axiom:Axiom of Unions/Set Theoretical and Class Theoretical

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Axiom of Unions: Difference between Formulations

Recall the two formulations of the axiom of unions:

Formulation 1

For every set of sets $A$, there exists a set $x$ (the union set) that contains all and only those elements that belong to at least one of the sets in the $A$:

$\forall A: \exists x: \forall y: \paren {y \in x \iff \exists z: \paren {z \in A \land y \in z} }$

Formulation 2

Let $x$ be a set (of sets).

Then its union $\bigcup x$ is also a set.

Equivalence of Formulations of Axiom of Unions notwithstanding, the two formulations have a subtle difference.

The purely set theoretical (formulation 1) version starts with a given set (of sets), and from it allows the creation of its union by providing a rule by which this may be done.

The class theoretical (formulation 2) version accepts that such a construct is already constructible in the context of the union of a class, and is itself a class.

What formulation 2 then goes on to state is that if $x$ is actually a set (of sets), then $\bigcup x$ is itself a set.

This is consistent with how:

the philosophy of axiomatic set theory defines the constructibility of sets from nothing

differs from

the class theoretical approach, in which classes may be considered to be already in existence, and it remains a matter of determining which of these classes are actually sets.