Axiom of power set

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, 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 $x$, there is a set ${\displaystyle {\mathcal {P}}(x)}$ 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 ${\displaystyle {\mathcal {P}}(x)}$ 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 $X$ and $Y$:

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

Notice that

$x,y\in X\cup Y$

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

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

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:

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

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.

This article incorporates material from Axiom of power set on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.