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:

:$forall\; A\; ,\; exists\; P\; ,\; forall\; B\; ,\; [B\; in\; P\; iff\; forall\; C\; ,\; (C\; in\; B\; Rightarrow\; C\; in\; A)]$

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

:Given any set "A", there is a set $mathcal\{P\}(A)$ such that, given any set "B", "B" is a member of $mathcal\{P\}(A)$ 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.)

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 is generally considered uncontroversial and it, or an equivalent axiom, appears in most alternative axiomatizations of set theory.

Consequences

The Power Set Axiom allows the definition of the Cartesian product of two sets $X$ and $Y$:

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

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

:$X\; imes\; Y\; subseteq\; mathcal\{P\}(mathcal\{P\}(X\; cup\; Y)).$

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

:$X\_1\; imes\; cdots\; imes\; X\_n\; =\; (X\_1\; imes\; cdots\; imes\; X\_\{n-1\})\; imes\; X\_n.$

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.

----