Axiom of power set
||This article includes a list of references, related reading or external links, but its sources remain unclear because it lacks inline citations. (March 2013)|
In the formal language of the Zermelo–Fraenkel axioms, the axiom reads:
where P stands for the power set of A, . In English, this says:
- Given any set A, there is a set such that, given any set B, B is a member of if and only if B is a subset of A. (Subset is not used in the formal definition above because the axiom of power set is an axiom that may need to be stated without reference to the concept of subset.)
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.
The Power Set Axiom allows a simple definition of the Cartesian product of two sets and :
and thus the Cartesian product is a set since
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.
- 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.