""

The elements of the power set of the set {

*x*,

*y*,

*z*} "

ordered with respect to "

inclusion.

In "mathematics, the **axiom of power set** is one of the "Zermelo–Fraenkel axioms of "axiomatic set theory.

In the "formal language of the Zermelo–Fraenkel axioms, the axiom reads:

- $\forall x\,\exists y\,\forall z\,[z\in y\iff \forall w\,(w\in z\Rightarrow w\in x)]$

where *y* is the "Power set of *x*, ${\mathcal {P}}(x)$.

In English, this says:

- "Given any "set
*x*, "there is a set ${\mathcal {P}}(x)$ "such that, given any set *z*, this set *z* is a member of ${\mathcal {P}}(x)$ "if and only if every element of *z* is also an element of *x*.

More succinctly: *for every set $x$, there is a set ${\mathcal {P}}(x)$ consisting precisely of the subsets of $x$.*

Note the "subset relation $\subseteq$ is not used in the formal definition as subset is not a primitive relation in formal set theory; rather, subset is defined in terms of "set membership, $\in$. By the "axiom of extensionality, the set ${\mathcal {P}}(x)$ 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.

## Consequences[edit]

The Power Set Axiom allows a simple definition of the "Cartesian product of two sets $X$ and $Y$:

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

Notice that

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

and, for example, considering a model using the "Kuratowski ordered pair,

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

and thus the Cartesian product is a set since

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

One may define the Cartesian product of any "finite "collection of sets recursively:

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

Note that the existence of the Cartesian product can be proved without using the power set axiom, as in the case of the "Kripke–Platek set theory.

## References[edit]

- "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 on "PlanetMath, which is licensed under the "Creative Commons Attribution/Share-Alike License.*

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