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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)


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