comprehension axiom

The axiom of comprehensionPlanetmathPlanetmath (CA) states that every formulaMathworldPlanetmathPlanetmath defines a set. That is,

Xx(xXϕ(x))for any formulaϕwhereXdoes not occur free inϕ

The specification and separation are sometimes used in place of comprehension, particularly for weakened forms of the axiom (see below).

In theories which make no distinction between objects and sets (such as ZF), this formulation leads to Russell’s paradoxMathworldPlanetmath, however in stratified theories this is not a problem (for example second order arithmetic includes the axiom of comprehension).

This axiom can be restricted in various ways. One possibility is to restrict it to forming subsets of sets:

YXx(xXxYϕ(x)) for any formula ϕ where X does not occur free in ϕ

This formulation (used in ZF set theoryMathworldPlanetmath) is sometimes called the Aussonderungsaxiom.

Another way is to restrict ϕ to some family F, giving the axiom F-CA. For instance the axiom Σ10 -CA is:

Xx(xXϕ(x)) where ϕ is Σ10 and X does not occur free in ϕ

A third form (usually called separation) uses two formulas, and guarantees only that those satisfying one are included while those satisfying the other are excluded. The unrestricted form is the same as unrestricted collectionMathworldPlanetmath, but, for instance, Σ10 separation:

 where ϕ and ψ are Σ10 and X does not occur free in ϕ or ψ

is weaker than Σ10 -CA.

Title comprehension axiom
Canonical name ComprehensionAxiom
Date of creation 2013-03-22 12:56:54
Last modified on 2013-03-22 12:56:54
Owner Henry (455)
Last modified by Henry (455)
Numerical id 10
Author Henry (455)
Entry type Definition
Classification msc 03F35
Synonym CA
Synonym -CA
Synonym comprehension
Synonym comprehension axiom
Synonym axiom of comprehension
Synonym separation
Synonym separation axiomMathworldPlanetmath
Synonym axiom of separation
Synonym specification
Synonym specification axiom
Synonym axiom of specification
Synonym Aussonderungsaxiom