axiom of power set

axiom of power set

1. (set theory) 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.
   • 1978, Thomas Jech, Set Theory, Academic Press, page 38,

     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).

   • 2003, Thomas Forster, Reasoning About Theoretical Entities, World Scientific, page 51,

     Verifying that the axiom of power set is in ${\displaystyle \{\phi \in {\mathcal {L}}^{*}:S\vdash {\mathcal {I}}(\phi )\}}$ relies on some rudimentary comprehension axioms.

   • 2011, Adam Rieger, 9: Paradox, ZF, and the Axiom of Foundation, David DeVidi, Michael Hallett, Peter Clark (editors), Logic, Mathematics, Philosophy: Vintage Enthusiasms: Essays in Honour of John L. Bell, Springer, page 183,

     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 ${\displaystyle \{\emptyset \}}$ to be a set.

• (axiom of set theory): power set axiom

• Zermelo–Fraenkel set theory on Wikipedia.Wikipedia