# Axiom:Axiom of Unions/Set Theoretical and Class Theoretical

## 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.