Axiom:Axiom of Powers/Set Theory

Axiom

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

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

Also known as

The axiom of powers is also known as:

the axiom of the power set

the power set axiom

Also see

• Definition:Power Set

Sources

• 1960: Paul R. Halmos: Naive Set Theory ... (previous) ... (next): $\S 5$: Complements and Powers
• 1982: Alan G. Hamilton: Numbers, Sets and Axioms ... (previous) ... (next): $\S 4$: Set Theory: $4.2$ The Zermelo-Fraenkel axioms: $\text {ZF5}$
• 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 $(1)$

• Weisstein, Eric W. "Zermelo-Fraenkel Axioms." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Zermelo-FraenkelAxioms.html
• Weisstein, Eric W. "Axiom of the Power Set." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/AxiomofthePowerSet.html