Editing Axiom of power set (section)

{{short description|Concept in axiomatic set theory}}
{{More footnotes|date=May 2020}}
[[Image:Hasse diagram of powerset of 3.svg|thumb|250px|The elements of the power set of the set {{nowrap|{{mset|''x'', ''y'', ''z''}}}} [[order theory|ordered]] with respect to [[Inclusion (set theory)|inclusion]].]]

In [[mathematics]], the '''axiom of power set'''<ref>{{Cite web|url=https://www.britannica.com/science/axiom-of-power-set|title=Axiom of power set {{!}} set theory {{!}} Britannica|website=www.britannica.com|language=en|accessdate=2023-08-06}}</ref> is one of the [[Zermelo–Fraenkel axioms]] of [[axiomatic set theory]]. It guarantees for every set <math>x</math> the existence of a set <math>\mathcal{P}(x)</math>, the [[power set]] of <math>x</math>, consisting precisely of the [[subset]]s of <math>x</math>. By the [[axiom of extensionality]], the set <math>\mathcal{P}(x)</math> is unique.

The axiom of power set appears in most axiomatizations of set theory. It is generally considered uncontroversial, although [[constructive set theory]] prefers a weaker version to resolve concerns about [[predicativity]].