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.
Formal statement
The subset relation is not a primitive notion in formal set theory and is not used in the formal language of the Zermelo–Fraenkel axioms. Rather, the subset relation is defined in terms of set membership, . Given this, in the formal language of the Zermelo–Fraenkel axioms, the axiom of power set reads:
where y is the power set of x, z is any element of y, w is any member of z.
One may define the Cartesian product of any finitecollection of sets recursively:
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.
Limitations
The power set axiom does not specify what subsets of a set exist, only that there is a set containing all those that do.[2] Not all conceivable subsets are guaranteed to exist. In particular, the power set of an infinite set would contain only "constructible sets" if the universe is the constructible universe but in other models of ZF set theory could contain sets that are not constructible.
Paul Halmos, Naive set theory. Princeton, NJ: D. Van Nostrand Company, 1960. Reprinted by Springer-Verlag, New York, 1974. ISBN0-387-90092-6 (Springer-Verlag edition).
Jech, Thomas, 2003. Set Theory: The Third Millennium Edition, Revised and Expanded. Springer. ISBN3-540-44085-2.
Kunen, Kenneth, 1980. Set Theory: An Introduction to Independence Proofs. Elsevier. ISBN0-444-86839-9.