Axiom:Comprehension Principle

Definition
The comprehension principle states:
 * Given any property $P$, there exists a unique set which consists of all and only those objects which have property $P$:


 * $\set {x: \map P x}$

Formally, let $P$ be an arbitrary predicate where $S$ is not free. Then the following is an instance of the comprehension principle:
 * $\exists S : \forall x : \paren {x \in S \iff P(x)}$

From the definition of a set:


 * A set is any aggregation of objects, called elements, which can be precisely defined in some way or other.

Also known as
The comprehension principle can also be referred to as:
 * the abstraction principle
 * the axiom of abstraction
 * the unlimited abstraction principle

Also see

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


 * Definition:Set Definition by Predicate