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May 19th, 2017, 07:41 AM   #111
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 Originally Posted by Mariga A well-order is a type of order. The example I have used is not even an order in mathematical sense. The set would be a well order if of course I consider the criteria for ordering and introduce a binary relation $r_n I'm wondering what stops you from clicking on the links I gave you. You aren't making any sense.  May 19th, 2017, 08:51 PM #112 Senior Member Joined: Mar 2017 From: . Posts: 210 Thanks: 2 Math Focus: Number theory I don't understand what doesn't make any sense. May 19th, 2017, 09:25 PM #113 Senior Member Joined: Aug 2012 Posts: 1,376 Thanks: 327 Quote:  Originally Posted by Mariga I don't understand what doesn't make any sense. Yes this is a problem for me. I would love to explain the well-ordering theorem and I'm not sure how to get this conversation grounded to something sensible. Could you go over to the Wiki page on well-ordering and just write down the exact definition of a well-ordered set?  May 22nd, 2017, 02:12 AM #114 Senior Member Joined: Mar 2017 From: . Posts: 210 Thanks: 2 Math Focus: Number theory A well ordered set is one which all it's subsets have a least element and also let's assume a is the least element in set X. then the next element after a, lets say b, is the least element in the subset X-a. This condition holds in all the orderings I have done. May 22nd, 2017, 08:46 AM #115 Senior Member Joined: Aug 2012 Posts: 1,376 Thanks: 327 Quote:  Originally Posted by Mariga A well ordered set is one which all it's subsets have a least element Every nonempty subset, right? So we agree that the natural numbers in their usual order are well-ordered, and the integers in their usual order aren't, agreed so far?  May 22nd, 2017, 10:08 AM #116 Senior Member Joined: Mar 2017 From: . Posts: 210 Thanks: 2 Math Focus: Number theory Yes. nonempty subset. Why are integers not well ordered in their usual order? May 22nd, 2017, 10:20 AM #117 Senior Member Joined: Aug 2012 Posts: 1,376 Thanks: 327 Quote:  Originally Posted by Mariga Yes. nonempty subset. Why are integers not well ordered in their usual order? The integers in their usual order are:$\dots, -3, -2, -1, 0, 1, 2, 3, \dots\$

Now, what is the smallest element of the integers?

Last edited by Maschke; May 22nd, 2017 at 10:27 AM.

 May 22nd, 2017, 10:59 AM #118 Senior Member   Joined: Mar 2017 From: . Posts: 210 Thanks: 2 Math Focus: Number theory Depending on our way of ordering, if it is a direct comparison of values of the integers, then the least element is undefined. and I don't know why you seem confused yet we agreed that this is counterintuitive.
May 22nd, 2017, 11:09 AM   #119
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 Originally Posted by Mariga Depending on our way of ordering, if it is a direct comparison of values of the integers, then the least element is undefined.
If the least element is undefined, then the set by definition is not well-ordered. Right?

 May 22nd, 2017, 02:42 PM #120 Senior Member   Joined: Aug 2012 Posts: 1,376 Thanks: 327 ps -- Let me supply a little more detail. A totally ordered set is said to be well-ordered if every nonempty subset has a smallest element. The integers are a nonempty subset of the integers. The integers have no first element. Therefore the integers are not well-ordered. Make sense?

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