Talk:Well-order
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I think this would be clearer with an example. Does a well-order require the definitions of mathematical sets?
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[edit] my example stinks
I added an example of how to well order the integers (all the integers), but my reference book is at home, I didn't write it well and it needs a proof it well orders the integers. Somebody please improve it. (marked for cleanup) RJFJR 21:46, Feb 7, 2005 (UTC)
[edit] definition?
The definition of a well ordered set says that every subset of a set has a least element. However, since the relational operator must be anti-symmetric, I'm having a hard time imagining a totally ordered set which itself has a least element but has a subset without a least element.
Can anyone give an example of why this definition is necessary? It might be something nice to include on the page. 129.110.240.1 05:18, 25 Mar 2005 (UTC)
Easy. The set of real numbers [0,1]. It has a least number, 0. What is the least element of the subset consisting of the interval (0,1] ? It is the smallest number greater than zero. But what is the smallest real number greater than zero? [Actually, if the Axiom of Choice is true then there is a way to well order the reals, but the proof, in addition to requiring AC, is non constructive so no one knows how to well order the reals. Unlike well ordering integers where there is a way to well order the negative numbers). RJFJR 15:22, Mar 25, 2005 (UTC)
I removed the cleanup template from here since it has been cleanedup. RJFJR 13:46, Jun 24, 2005 (UTC)
[edit] New version
The new version looks great! RJFJR 13:43, Jun 24, 2005 (UTC)
[edit] Strict or nonstrict?
The article is not very specific about whether it's discussing strict or nonstrict partial orders. The link to total order uses the nonstrict notion, but the examples given seem to be strict. Among adepts this is one of those things you don't worry too much about because it's usually clear from context, but in an article like this maybe we should be a little more careful. --Trovatore 8 July 2005 00:04 (UTC)
[edit] intro para
The intro para was stated in such a way that the linearity of the ordering was a consequence of the proposed definition (a partial order in which every subset has a least element). The alternative way of defining a well ordering is as a linear order which is well founded in the sense that every nonempty subset has a minimal element. The third possibility, namely a poset in which every nonempty subset has a minimal element, is not the same. To prevent confusion between least and minimal, I think it is better to be explicit that a well ordering is defined to be a well founded linear order. CMummert 13:10, 1 July 2006 (UTC)
