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axiom of power set (Axiom)

The axiom of power set is an axiom of Zermelo-Fraenkel set theory which postulates that for any set $ X$ there exists a set $ \mathcal{P}(X)$, called the power set of $ X$, consisting of all subsets of $ X$. In symbols, it reads:

$\displaystyle \forall X \exists \mathcal{P}(X) \forall u (u \in \mathcal{P}(X) \leftrightarrow u \subseteq X). $
In the above, $ u \subseteq X$ is defined as $ \forall z(z \in u \rightarrow z \in X)$. By the extensionality axiom, the set $ \mathcal{P}(X)$ is unique.

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

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

The Cartesian product is a set since

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

We 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. $



"axiom of power set" is owned by mathcam. [ full author list (3) | owner history (3) ]
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Other names:  power set axiom, powerset axiom, axiom of powerset
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Cross-references: collection, finite, Cartesian product, extensionality, subsets, power set, postulates, Zermelo-Fraenkel set theory, axiom
There are 3 references to this entry.

This is version 8 of axiom of power set, born on 2003-06-26, modified 2006-12-12.
Object id is 4399, canonical name is AxiomOfPowerSet.
Accessed 8825 times total.

Classification:
AMS MSC: 03E30 (Mathematical logic and foundations :: Set theory :: Axiomatics of classical set theory and its fragments)
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