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