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 schema, as we introduce a lot of similar axioms.

This axiom schema can be formally stated as follows:

Class Theory
The  in the context of class theory has a similar form:

Also see

 * Axiom:Comprehension Principle -- do not confuse that with this


 * Axiom of Specification from Replacement and Empty Set, proving that the  can be deduced from the and the