PlanetMath: axiom of foundation

(more info)

Math for the people, by the people.

Main Menu

sections Encyclopædia
Papers
Books
Expositions

meta Requests (194)
Orphanage
Unclass'd
Unproven (498)
Corrections (25)
Classification

talkback Polls
Forums
Feedback
Bug Reports

downloads Snapshots
PM Book

information News
Docs
Wiki
ChangeLog
TODO List
Legalese
About

Entry average rating: No information on entry rating

axiom of foundation

(Definition)

The axiom of foundation (also called the axiom of regularity) is an axiom of ZF set theory prohibiting circular sets and sets with infinite levels of containment. Intuitively, it states that every set can be built up from the empty set. There are several equivalent formulations, for instance:

For any nonempty set there is some $y\in X$ such that $y\cap X=\emptyset$ .

For any set , there is no function from $\omega$ to the transitive closure of such that for every , $f(n+1)\in f(n)$ .

For any formula $\phi$ , if there is any set such that $\phi(x)$ then there is some such that $\phi(X)$ but there is no $y\in X$ such that $\phi(y)$ .

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.

"axiom of foundation" is owned by Henry.

(view preamble)

Other names:

foundation, regularity, axiom of regularity

Also defines:

artinian, artinian set, artinian sets

Attachments:: proof equivalence of formulation of foundation (Proof) by Henry

Log in to rate this entry.
(view current ratings)

Cross-references: consistent, satisfy, formula, transitive closure, function, empty set, levels, infinite, circular, set theory, ZF, axiom
There are 64 references to this entry.

This is version 6 of axiom of foundation, born on 2002-09-28, modified 2006-10-15.
Object id is 3485, canonical name is AxiomOfFoundation.
Accessed 14876 times total.

Classification:

AMS MSC:

03C99 (Mathematical logic and foundations :: Model theory :: Miscellaneous)

Pending Errata and Addenda

None.[ View all 4 ]

Discussion

forum policy

No messages.

Interact