well-ordered set

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well-ordered set

[′wel ¦ȯr·dərd ′set]
A linearly ordered set where every subset has a least element.
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.

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.
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