well-ordered set

from The Free On-line Dictionary of Computing (8 July 2008)

well-ordered set

   <mathematics> 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.

   (1995-04-19)