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well ordered set (Definition)

A well-ordered set is a totally ordered set in which every nonempty subset has a least member.

An example of well-ordered set is the set of positive integers with the standard order relation $(\mathbbmss{Z}^+,<)$, because any nonempty subset of it has least member. However, $\mathbbmss{R}^+$ (the positive reals) is not a well-ordered set with the usual order, because $(0,1)=\{x:0<x<1\}$ is a nonempty subset but it doesn't contain a least number.

A well-ordering of a set $X$ is the result of defining a binary relation $\leq$ on $X$ to itself in such a way that $X$ becomes well-ordered with respect to $\leq$.



"well ordered set" is owned by drini. [ full author list (2) | owner history (2) ]
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See Also: well-ordering principle for natural numbers

Other names:  well-ordered, well-ordered set
Also defines:  well-ordering

Attachments:
properties of well-ordered sets (Theorem) by GrafZahl
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Cross-references: binary relation, least number, contain, reals, relation, order, integers, positive, subset, totally ordered set
There are 34 references to this entry.

This is version 11 of well ordered set, born on 2001-10-17, modified 2005-07-26.
Object id is 271, canonical name is WellOrderedSet.
Accessed 17549 times total.

Classification:
AMS MSC: 06A05 (Order, lattices, ordered algebraic structures :: Ordered sets :: Total order)
 03E25 (Mathematical logic and foundations :: Set theory :: Axiom of choice and related propositions)
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Well-ordered set? by Schneemann on 2006-02-15 11:05:34
Hi!
I am browsing the articles about well-ordered sets and tree (set theory). I am just asking myself what's the difference between well-ordered sets and the natural numbers.

I imagine the following set, with the natural "<" relation. (NN_0 = the natural numbers, with zero)

M = { m + k/(k+1) | m, k in NN_0 }

Is (M, <) a well-ordered set? Is it a tree, in the set-theoretic sense? Every subset has a smallest member, right?

If so, it would be helpful as an example, to show how the concept of well ordering does not imply an ordering like the natural numbers have..

Regards, Schneemann..
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