May 19th, 2017, 07:41 AM  #111  
Senior Member Joined: Aug 2012 Posts: 1,856 Thanks: 513  Quote:
 
May 19th, 2017, 08:51 PM  #112 
Banned Camp Joined: Mar 2017 From: . Posts: 338 Thanks: 8 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: 1,856 Thanks: 513  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 
Banned Camp Joined: Mar 2017 From: . Posts: 338 Thanks: 8 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: 1,856 Thanks: 513  
May 22nd, 2017, 10:08 AM  #116 
Banned Camp Joined: Mar 2017 From: . Posts: 338 Thanks: 8 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: 1,856 Thanks: 513  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 
Banned Camp Joined: Mar 2017 From: . Posts: 338 Thanks: 8 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: 1,856 Thanks: 513  
May 22nd, 2017, 02:42 PM  #120 
Senior Member Joined: Aug 2012 Posts: 1,856 Thanks: 513 
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? 

Tags 
difficulties, paradoxes, probability 
Thread Tools  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
Infinity just has too many paradoxes to be a real thing  uperkurk  Applied Math  11  September 30th, 2013 03:10 PM 
Euler's Paradoxes  unm  Number Theory  0  December 3rd, 2012 06:05 PM 
Greatest of all paradoxes  krausebj0  New Users  0  November 25th, 2011 03:50 PM 
Resolution of Russell's and Cantor's paradoxes  DaniilTeplitskiy  Applied Math  35  August 30th, 2011 11:58 PM 
Integration difficulties  JohnTan  Calculus  4  February 15th, 2010 03:03 PM 