Power Set can be Derived using Axiom of Abstraction
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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