well-ordered set

(mathematics)

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.