Axiom:Axiom of Powers

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Set Theory

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

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

Class Theory

Let $x$ be a set.

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

Also known as

The axiom of powers is also known as:

the axiom of the power set
the power set axiom

Set Theoretical and Class Theoretical Formulations

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.

Also see

  • Results about the axiom of powers can be found here.