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Axiom of power set

## Summary

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

The elements of the power set of the set {x, y, z} ordered with respect to inclusion.

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.