Axiom:Axiom of Specification/Set Theory

Axiom
The axiom of specification is an axiom schema which can be formally stated as follows:

For any propositional function $\map P y$, we introduce the axiom:


 * $\forall z: \exists x: \forall y: \paren {y \in x \iff \paren {y \in z \land \map P y} }$

where each of $x$, $y$ and $z$ range over arbitrary sets.

This means that if you have a set, you can create a set that contains some of the elements of that set, where those elements are specified by stipulating that they satisfy some (arbitrary) condition.

Also see

 * Axiom:Axiom of Specification/Class Theory