May 19th, 2017, 07:41 AM  #111  
Senior Member Joined: Aug 2012 Posts: 1,434 Thanks: 353  Quote:
 
May 19th, 2017, 08:51 PM  #112 
Senior Member Joined: Mar 2017 From: . Posts: 261 Thanks: 4 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: 1,434 Thanks: 353  Yes this is a problem for me. I would love to explain the wellordering theorem and I'm not sure how to get this conversation grounded to something sensible. Could you go over to the Wiki page on wellordering and just write down the exact definition of a wellordered set? 
May 22nd, 2017, 02:12 AM  #114 
Senior Member Joined: Mar 2017 From: . Posts: 261 Thanks: 4 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 Xa. This condition holds in all the orderings I have done. 
May 22nd, 2017, 08:46 AM  #115 
Senior Member Joined: Aug 2012 Posts: 1,434 Thanks: 353  
May 22nd, 2017, 10:08 AM  #116 
Senior Member Joined: Mar 2017 From: . Posts: 261 Thanks: 4 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: 1,434 Thanks: 353  Quote:
$\dots, 3, 2, 1, 0, 1, 2, 3, \dots$ 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: 261 Thanks: 4 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 
Senior Member Joined: Aug 2012 Posts: 1,434 Thanks: 353  
May 22nd, 2017, 02:42 PM  #120 
Senior Member Joined: Aug 2012 Posts: 1,434 Thanks: 353 
ps  Let me supply a little more detail. A totally ordered set is said to be wellordered 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 wellordered. Make sense? 

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