well-ordered set


A set with a total ordering and no infinite descending chains. A total ordering "<=" satisfies

 x <= x

 x <= y <= z  =>  x <= z

 x <= y <= x  =>  x = y

 for all x, y: x <= y or y <= x

In addition, if a set W is well-ordered then all non-empty subsets A of W have a least element, i.e. there exists x in A such that for all y in A, x <= y.

Ordinals are isomorphism classes of well-ordered sets, just as integers are isomorphism classes of finite sets.

Last updated: 1995-04-19

Nearby terms:

well-connectedwell-known portwell-ordered setWEPWesley Clark

Try this search on Wikipedia, Wiktionary, Google, OneLook.