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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:

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

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

In English, this says:

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

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

Note the "subset relation ${\displaystyle \subseteq }$ 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, ${\displaystyle \in }$. By the "axiom 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, although "constructive set theory prefers a weaker version to resolve concerns about "predicativity.

## Consequences

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

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

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 the "Kuratowski 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 any "finite "collection of sets recursively:

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

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

• "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.