Axiom of power set
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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:
- <math>\forall A, \exists\; {\mathcal{P}(A)}, \forall B: B \in {\mathcal{P}(A)} \iff (\forall C: C \in B \implies C \in A)</math>
Or in words:
- Given any set A, there is a set <math>\mathcal{P}(A)</math> such that, given any set B, B is a member of <math>\mathcal{P}(A)</math> if and only if B is a subset of A.
By the axiom of extensionality this set is unique. We call the set <math>\mathcal{P}(A)</math> 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.
[edit] Consequences
The Power Set Axiom allows the definition of the Cartesian product of two sets <math>X</math> and <math>Y</math>:
- <math> X \times Y = \{ (x, y) : x \in X \land y \in Y \}. </math>
The Cartesian product is a set since
- <math> X \times Y \subseteq \mathcal{P}(\mathcal{P}(X \cup Y)). </math>
One may define the Cartesian product of any finite collection of sets recursively:
- <math> X_1 \times \cdots \times X_n = (X_1 \times \cdots \times X_{n-1}) \times X_n. </math>
Note that the existence of the Cartesian product can be proved in Kripke–Platek set theory which does not contain the power set axiom.
This article incorporates material from Axiom of power set on PlanetMath, which is licensed under the GFDL.fr:Axiome de l'ensemble des parties it:Assioma dell'insieme potenza sv:Potensmängdsaxiomet zh:幂集公理
