Axiom of power set

File:Hasse diagram of powerset of 3.svg

The elements of the power set of the set {x, y, z} ordered with respect to inclusion.

In mathematics, the axiom of power set is one of the Zermelo–Fraenkel axioms of axiomatic set theory. It guarantees for every set Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} the existence of a set Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{P}(x)} , the power set of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} , consisting precisely of the subsets of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} . By the axiom of extensionality, the set Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{P}(x)} is unique.

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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \subseteq} 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \subseteq} is defined in terms of set membership, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \in} . Given this, in the formal language of the Zermelo–Fraenkel axioms, the axiom of power set reads:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \forall x \, \exists y \, \forall z \, [z \in y \iff \forall w \, (w \in z \Rightarrow w \in x)]}

where y is the power set of x, z is any element of y, w is any member of z.

In English, this says:

Given any set x, there is a set y such that, given any set z, this set z is a member of y if and only if every element of z is also an element of x.

Consequences

The power set axiom allows a simple definition of the Cartesian product of two sets Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Y} :

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X \times Y = \{ (x, y) : x \in X \land y \in Y \}. }

Notice that

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x, y \in X \cup Y }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{ x \}, \{ x, y \} \in \mathcal{P}(X \cup Y) }

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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (x, y) = \{ \{ x \}, \{ x, y \} \} \in \mathcal{P}(\mathcal{P}(X \cup Y)) }

and thus the Cartesian product is a set since

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X_1 \times \cdots \times X_n = (X_1 \times \cdots \times X_{n-1}) \times X_n. }

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. 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, the universe could contain sets that are not constructible.

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.