# Axiom:Axiom of Powers

## Axiom

### 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**.

## Sources

- 2002: Thomas Jech:
*Set Theory*(3rd ed.) ... (previous) ... (next): Chapter $1$: Power Set