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Inmathematics, theaxiom of power set^[1] is one of theZermelo–Fraenkel axioms ofaxiomatic set theory. It guarantees for every set${\displaystyle x}$ the existence of a set${\displaystyle \mathcal{P}(x)}$, thepower set of${\displaystyle x}$, consisting precisely of thesubsets of${\displaystyle x}$. By theaxiom of extensionality, the set${\displaystyle \mathcal{P}(x)}$ is unique.

The axiom of power set appears in most axiomatizations of set theory. It is generally considered uncontroversial, althoughconstructive set theory prefers a weaker version to resolve concerns aboutpredicativity.

The subset relation${\displaystyle \subseteq }$ is not aprimitive notion informal set theory and is not used in the formal language of the Zermelo–Fraenkel axioms. Rather, the subset relation${\displaystyle \subseteq }$ is defined in terms ofset membership,${\displaystyle \in }$. Given this, in theformal language of the Zermelo–Fraenkel axioms, the axiom of power set reads:

${\displaystyle \mathrm{\forall }x\mathrm{\exists }y\mathrm{\forall }z[z\in y⟺\mathrm{\forall }w(w\in z\Rightarrow w\in x)]}$

wherey is the power set ofx,z is any element ofy,w is any member ofz.

In English, this says:

Given anysetx,there is a setysuch that, given any setz, this setz is a member ofyif and only if every element ofz is also an element ofx.

The power set axiom allows a simple definition of theCartesian product of two sets${\displaystyle X}$ and${\displaystyle Y}$:

${\displaystyle X\times Y=\{(x,y):x\in X\wedge y\in Y\}.}$

Notice that

${\displaystyle x,y\in X\cup Y}$

${\displaystyle \{x\},\{x,y\}\in \mathcal{P}(X\cup Y)}$

and, for example, considering a model using theKuratowski ordered pair,

${\displaystyle (x,y)=\{\{x\},\{x,y\}\}\in \mathcal{P}(\mathcal{P}(X\cup Y))}$

and thus the Cartesian product is a set since

${\displaystyle X\times Y\subseteq \mathcal{P}(\mathcal{P}(X\cup Y)).}$

One may define the Cartesian product of anyfinitecollection of sets recursively:

${\displaystyle X_1\times \cdots \times X_n=(X_1\times \cdots \times X_{n-1})\times X_n.}$

The existence of the Cartesian product can be proved without using the power set axiom, as in the case of theKripke–Platek set theory.

The power set axiom does not specify what subsets of a set exist, only that there is a set containing all those that do.^[2] Not all conceivable subsets are guaranteed to exist. In particular, the power set of an infinite set would contain only "constructible sets" if the universe is theconstructible universe but in other models of ZF set theory could contain sets that are not constructible.

• Paul Halmos,Naive set theory. Princeton, NJ: D. Van Nostrand Company, 1960. Reprinted by Springer-Verlag, New York, 1974.ISBN 0-387-90092-6 (Springer-Verlag edition).
• Jech, Thomas, 2003.Set Theory: The Third Millennium Edition, Revised and Expanded. Springer.ISBN 3-540-44085-2.
• Kunen, Kenneth, 1980.Set Theory: An Introduction to Independence Proofs. Elsevier.ISBN 0-444-86839-9.

This article incorporates material from Axiom of power set onPlanetMath, which is licensed under theCreative Commons Attribution/Share-Alike License.

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