axiom of foundation

The axiom of foundationMathworldPlanetmath (also called the axiom of regularity) is an axiom of ZF set theoryMathworldPlanetmath prohibiting circular sets and sets with infiniteMathworldPlanetmath levels of containment. Intuitively, it that every set can be built up from the empty setMathworldPlanetmath. There are several equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath formulations, for instance:

For any nonempty set X there is some yX such that yX=.

For any set X, there is no function f from ω to the transitive closureMathworldPlanetmathPlanetmath of X such that for every n, f(n+1)f(n).

For any formulaMathworldPlanetmathPlanetmath ϕ, if there is any set x such that ϕ(x) then there is some X such that ϕ(X) but there is no yX such that ϕ(y).

Sets which satisfy this axiom are called artinian. It is known that, if ZF without this axiom is consistent, then this axiom does not add any inconsistencies.

One important consequence of this property is that no set can contain itself. For instance, if there were a set X such that XX then we could define a function f(n)=X for all n, which would then have the property that f(n+1)f(n) for all n.

Title axiom of foundation
Canonical name AxiomOfFoundation
Date of creation 2013-03-22 13:04:31
Last modified on 2013-03-22 13:04:31
Owner Henry (455)
Last modified by Henry (455)
Numerical id 10
Author Henry (455)
Entry type Definition
Classification msc 03C99
Synonym foundation
Synonym regularity
Synonym axiom of regularity
Defines artinian
Defines artinian set
Defines artinian sets