# Axiom of power set

 related topics {math, number, function}

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 P stands for the power set, $\mathcal{P}(A)$, of A. In English, this says:

By the axiom of extensionality this set is unique. We call the set $\mathcal{P}(A)$ the power set of A. Thus, the essence of the axiom is that every set has a power set.

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 the definition of the Cartesian product of two sets X and Y:

Notice that

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 in Kripke–Platek set theory which does not contain the power set axiom.

### 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.