Axiom:Axiom of Specification

Axiom
For every set and every condition, there corresponds a set whose elements are exactly the same as those elements of the original set for which the condition is true.

Because we cannot quantify over functions, we need an axiom for every condition we can express. Therefore, this axiom is sometimes called an axiom scheme, as we introduce a lot of similar axioms.

This axiom scheme is formally stated as follows:

For any function of propositional logic $P \left({y}\right)$, we introduce the axiom:


 * $\forall z: \exists x: \forall y: \left({y \in x \iff \left({y \in z \land P \left({y}\right)}\right)}\right)$

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.

Otherwise known as:
 * The Axiom of Specification;
 * The Axiom of Comprehension;
 * The Axiom of Selection;
 * The Axiom of Separation (although this can be confused with the separation axioms of Hausdorff, which arise in topology, so it's not often used).

This can be deduced from the Axiom of Replacement.

Alternative specification
Alternatively, this axiom can be specified as follows:


 * If $\phi$ is a property (with parameter $p$), then for any $X$ and $p$ there exists a set $Y = \left( {u \in X: \phi \left({u, p}\right)}\right)$ that contains all those $u \in X$ that have the property $\phi$.


 * $\forall X: \forall p: \exists Y: \forall u: \left({u \in Y \iff \left({u \in X \land \phi \left({u, p}\right)}\right)}\right)$