Axiom of Power Set

Set Theory · Axiom Academy

The power set of any set exists The Axiom of Power Set states: For any set A, there exists a set P(A) that contains exactly all the subsets of A. Let's compute the power set of A = 1, 2 : Therefore: P( 1, 2 ) = ∅, 1 , 2 , 1, 2 3. Connection to Naive Set Theory You may have seen power sets before in naive set theory. The axiom makes this rigorous: Naive approach: "P(A) is the set of all subsets of A" Axiomatic approach: The Power Set Axiom guarantees such a set exists Key properties remain the same: The Power Set Axiom has profound implications: Size explosion: P(A) is always strictly larger than A (Cantor's theorem) Infinite hierarchies: ∅, P(∅), P(P(∅)), ... creates infinitely many distinct infinite sets Real numbers: ℝ can be constructed as P(ℕ) in some models Boolean algebras: P(A) forms a complete Boolean algebra

This is the written version of the interactive lesson above. See the full Set Theory course.