Editing Zermelo–Fraenkel set theory (section)

=== Axiom of power set ===
{{Main|Axiom of power set}}

By definition, a set <math>z</math> is a [[subset]] of a set <math>x</math> if and only if every element of <math>z</math> is also an element of <math>x</math>:

<div style="margin-left:1.6em;"><math>(z \subseteq x) \Leftrightarrow ( \forall q (q \in z \Rightarrow q \in x)).</math></div>

The Axiom of power set states that for any set <math>x</math>, there is a set <math>y</math> that contains every subset of <math>x</math>:

<div style="margin-left:1.6em;"><math>\forall x \exists y \forall z (z \subseteq x \Rightarrow z \in y).</math></div>

The axiom schema of specification is then used to define the [[power set]] <math>\mathcal{P}(x)</math> as the subset of such a <math>y</math> containing the subsets of <math>x</math> exactly:

<div style="margin-left:1.6em;"><math>\mathcal{P}(x) = \{ z \in y: z \subseteq x \}.</math></div>

Axioms ''1–8'' define ZF. Alternative forms of these axioms are often encountered, some of which are listed in {{harvtxt|Jech|2003}}. Some ZF axiomatizations include an axiom asserting that the [[axiom of empty set|empty set exists]]. The axioms of pairing, union, replacement, and power set are often stated so that the members of the set <math>x</math> whose existence is being asserted are just those sets which the axiom asserts <math>x</math> must contain.

The following axiom is added to turn ZF into ZFC: