Axiom of Power Set

The axiom asserting that for every set there is a set whose members are exactly its subsets. It guarantees that the power set of any set exists.

Power set: The set of all subsets of a given set. The axiom guarantees this collection is itself a set.
Set (mathematics): A collection of distinct objects treated as a single mathematical object. The axiom states what set must exist for each set already given.
Zermelo–Fraenkel set theory: An axiomatic foundation for mathematics whose axioms govern sets and their construction. Power Set is one of its standard axioms.
Kripke–Platek set theory: A set theory weaker than Zermelo–Fraenkel theory that omits the full Power Set axiom. Its omission shows that substantial set theory can be developed without full power sets.
Cantor's theorem: The theorem that every set has strictly smaller cardinality than its power set. It shows that repeated power-set formation creates ever-larger infinities.
Subset: A set whose every member is also a member of another set. Subsets are exactly the objects collected by the power-set axiom.
Axiom schema: A rule specifying an infinite family of axioms, one for each permitted formula or instance. Power Set is a single axiom, unlike schemas such as Separation and Replacement.
Constructive set theory: A family of set theories that uses intuitionistic logic and may weaken classical existence principles. Some constructive systems restrict or reject unrestricted power-set formation.
Cumulative hierarchy: A staged construction of sets, formed by iterating power sets through ordinal-indexed levels. The axiom supplies the successor-stage operation in this standard model of set theory.
Axiom of Extensionality: The axiom stating that sets with the same members are equal. It ensures that a set containing exactly the subsets is unique.

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