# 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:

where *y* is the Power set of *x*,
.

In English, this says:

- Given any set
*x*, there is a set such that, given any set*z*, this set*z*is a member of if and only if every element of*z*is also an element of*x*.

More succinctly: *for every set
, there is a set
consisting precisely of the subsets of
.*

Note the subset relation 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, . By the axiom of extensionality, the set 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 and :

Notice that

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

and thus the Cartesian product is a set since

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

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.*