## 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 <= xIn 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

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