axiom schema of separation

Let ϕ(u,p) be a formulaMathworldPlanetmathPlanetmath. For any X and p, there exists a set Y={uX:ϕ(u,p)}.

The Axiom Schema of Separation is an axiom schemaMathworldPlanetmath of Zermelo-Fraenkel set theoryMathworldPlanetmath. Note that it represents infinitely many individual axioms, one for each formula ϕ. In symbols, it reads:


By Extensionality, the set Y is unique.

The Axiom Schema of Separation implies that ϕ may depend on more than one parameter p.

We may show by inductionMathworldPlanetmath that if ϕ(u,p1,,pn) is a formula, then


holds, using the Axiom Schema of Separation and the Axiom of Pairing.

Another consequence of the Axiom Schema of Separation is that a subclass of any set is a set. To see this, let 𝐂 be the class 𝐂={u:ϕ(u,p1,,pn)}. Then


holds, which means that the intersectionMathworldPlanetmathPlanetmath of 𝐂 with any set is a set. Therefore, in particular, the intersection of two sets XY={xX:xY} is a set. Furthermore the difference of two sets X-Y={xX:xY} is a set and, provided there exists at least one set, which is guaranteed by the Axiom of InfinityMathworldPlanetmath, the empty setMathworldPlanetmath is a set. For if X is a set, then ={xX:xx} is a set.

Moreover, if 𝐂 is a nonempty class, then 𝐂 is a set, by Separation. 𝐂 is a subset of every X𝐂.

Lastly, we may use Separation to show that the class of all sets, V, is not a set, i.e., V is a proper class. For example, suppose V is a set. Then by Separation


is a set and we have reached a Russell paradox.

Title axiom schema of separation
Canonical name AxiomSchemaOfSeparation
Date of creation 2013-03-22 13:42:46
Last modified on 2013-03-22 13:42:46
Owner Sabean (2546)
Last modified by Sabean (2546)
Numerical id 18
Author Sabean (2546)
Entry type Axiom
Classification msc 03E30
Synonym separation schema
Synonym separation