Meaning of axiom of power set | Babel Free

Definitions

The axiom that the power set of any set exists and is a valid set, which appears in the standard axiomatisation of set theory, ZFC.

Equivalents

Examples

“The axiom of choice differs from other axioms of ZF by stating existence of a set (i.e., a choice function) without defining it (unlike, for instance, the axiom of pairing or the axiom of power set).”

“Verifying that the axiom of power set is in #92;#123;#92;phi#92;in#92;mathcal#123;L#125;#42;#58;S#92;vdash#92;mathcal#123;I#125;(#92;phi)#92;#125; relies on some rudimentary comprehension axioms.”

“But the ZF axioms of which the hierarchy is an intuitive model involve impredicative quantifications. Most striking is the axiom of power set in tandem with the axiom of separation.”

“2012, A. H. Lightstone, H. B. Enderton (editor), Mathematical Logic: An Introduction to Model Theory, Plenum Press, Softcover, page 292, The Axiom of Power Set asserts that the collection of all subsets of a set is a set. […] Adding the Axiom of Power Set compels the collection empty to be a set.”

CEFR level