Axiom:Axiom of Powers/Set Theoretical and Class Theoretical

Axiom of Unions: Difference between Formulations

Recall the two formulations of the axiom of powers:

Formulation 1

For every set, there exists a set of sets whose elements are all the subsets of the given set.

$\forall x: \exists y: \paren {\forall z: \paren {z \in y \iff \paren {w \in z \implies w \in x} } }$

Formulation 2

Let $x$ be a set.

Then its power set $\powerset x$ is also a set.

Equivalence of Formulations of Axiom of Powers 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 power set 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 power set, and is itself a class.

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