Axiom of power set

The power set axiom allows a simple definition of the Cartesian product of two sets ${\displaystyle X}$ and ${\displaystyle Y}$:

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

Notice that

${\displaystyle x,y\in X\cup Y}$

${\displaystyle \{x\},\{x,y\}\in {\mathcal {P}}(X\cup Y)}$

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

${\displaystyle (x,y)=\{\{x\},\{x,y\}\}\in {\mathcal {P}}({\mathcal {P}}(X\cup Y))}$

and thus the Cartesian product is a set since

${\displaystyle X\times Y\subseteq {\mathcal {P}}({\mathcal {P}}(X\cup Y)).}$

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

${\displaystyle X_{1}\times \cdots \times X_{n}=(X_{1}\times \cdots \times X_{n-1})\times X_{n}.}$

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.

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. 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.

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