# well-ordered set

## 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.
References in periodicals archive ?
And, in my defence, I keep a tight rein on the saucepan lid situation, and have an admirably well-ordered set of tea towels.
Each well-ordered set is isomorphic to a suitable [C.
Writing in an admirably succinct and clear style, he covers logic, sets, functions, counting infinite sets, infinite cardinals, well-ordered sets, inductions and numbers, prime numbers, and logic and meta-mathematics.
describes the uses of infinity as he explains elementary set theory, functions (including inverse functions), counting infinite sets (including Hilbert's Infinite Hotel), infinite Cardinals (including the two infinities), well-ordered sets (such as the arithmetic of ordinals and cardinals as ordinals), inductions and numbers, including number theory, prime numbers, including the Riemann Zeta function.
Now UN and HN are well-ordered sets, since 2, 1 are the first elements respectively of UN, and HN.

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