# well-ordered set

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## well-ordered set

[′wel ¦ȯr·dərd ′set] (mathematics)

A linearly ordered set where every subset has a least element.

## 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.

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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