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.