Power Set can be Derived using Axiom of Abstraction

Theorem

Let $a$ be a set.

By application of the Axiom of Abstraction, the power set $\powerset a$ can be formed.

Hence the power set $\powerset a$ can be derived as a valid object in Frege set theory.

Proof

Let $P$ be the property defined as:

$\forall x: \map P x := \paren {x \subseteq a}$

where $\lor$ is the disjunction operator.

Hence, using the Axiom of Abstraction, we form the set:

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

$\blacksquare$

Sources

• 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $1$: General Background: $\S 7$ Frege set theory