axiom of power set

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axiom of power set

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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”, in 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