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.

• 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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