My Math Forum  

Go Back   My Math Forum > High School Math Forum > Probability and Statistics

Probability and Statistics Basic Probability and Statistics Math Forum


Thanks Tree17Thanks
Reply
 
LinkBack Thread Tools Display Modes
May 19th, 2017, 07:41 AM   #111
Senior Member
 
Joined: Aug 2012

Posts: 1,434
Thanks: 353

Quote:
Originally Posted by Mariga View Post
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 $r_n<r_{n+1}$ or $r_n\leq r_{n+1}$.

In all cases, the elements the sets begin with could be considered the least element. And the criteria I have used in all my orderings is a direct comparison of the values of the elements.
I'm wondering what stops you from clicking on the links I gave you. You aren't making any sense.
Maschke is online now  
 
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.
Mariga is offline  
May 19th, 2017, 09:25 PM   #113
Senior Member
 
Joined: Aug 2012

Posts: 1,434
Thanks: 353

Quote:
Originally Posted by Mariga View Post
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?
Maschke is online now  
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 X-a.

This condition holds in all the orderings I have done.
Mariga is offline  
May 22nd, 2017, 08:46 AM   #115
Senior Member
 
Joined: Aug 2012

Posts: 1,434
Thanks: 353

Quote:
Originally Posted by Mariga View Post
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?
Maschke is online now  
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?
Mariga is offline  
May 22nd, 2017, 10:20 AM   #117
Senior Member
 
Joined: Aug 2012

Posts: 1,434
Thanks: 353

Quote:
Originally Posted by Mariga View Post
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.
Maschke is online now  
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.
Mariga is offline  
May 22nd, 2017, 11:09 AM   #119
Senior Member
 
Joined: Aug 2012

Posts: 1,434
Thanks: 353

Quote:
Originally Posted by Mariga View Post
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?
Maschke is online now  
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 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?
Maschke is online now  
Reply

  My Math Forum > High School Math Forum > Probability and Statistics

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





Copyright © 2017 My Math Forum. All rights reserved.