Equivalence of Formulations of Axiom of Powers

Theorem

In the context of class theory, the following formulations of the Axiom of Powers are equivalent:

Formulation 1

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

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

Formulation 2

Let $x$ be a set.

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


Proof

It is assumed throughout that the Axiom of Extensionality and the Axiom of Specification both hold.


Formulation $1$ implies Formulation $2$

Let formulation $1$ be axiomatic:

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

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


Thus it is posited that for a given set $x$ the power set of $x$ exists:

$y := \powerset x := \set {z: z \subseteq x}$

and that this is a set.


Formulation $2$ asserts that given the existence of $\powerset x$, it is axiomatic that $\powerset x$ is itself a set.

Hence it follows that the truth of formulation $2$ follows from acceptance of the truth of formulation $1$.

From Power Set Exists and is Unique, which depends on:

the Axiom of Extensionality
the Axiom of Specification

$\powerset x$ is unique for a given $x$.

$\Box$


Formulation $2$ implies Formulation $1$

Let formulation $2$ be axiomatic:

Let $x$ be a set.

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


Let $x$ be a set.

Let us create the power set $y$ of x:

$\powerset x := \set {z: z \subseteq x}$

From Power Set Exists and is Unique, which depends on:

the Axiom of Extensionality
the Axiom of Specification

$\powerset x$ exists and is unique for a given $x$.


We have asserted the truth of formulation $2$.

That is, $\powerset x$ is a set.


As $x$ is arbitrary, it follows that $\powerset x$ exists and is unique for all sets $x$.

That is, the truth of formulation $1$ follows from acceptance of the truth of formulation $2$.

$\blacksquare$


Sources

  • 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $2$: Some Basics of Class-Set Theory: $\S 6$ The power axiom: Remarks $(1)$
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