My Math Forum Cantors Diag Argument Proves Reals Countable

 Topology Topology Math Forum

 March 4th, 2016, 04:11 PM #1 Banned Camp   Joined: Mar 2015 From: New Jersey Posts: 1,720 Thanks: 126 Cantors Diag Argument Proves Reals Countable Let T be the set of all infinite binary series. List all the elements of T and assume the list is uncountable. Use Cantor's Diagonal Argument to show there is a member of T not in the list. Contradiction. Therefore T is countable.
March 4th, 2016, 04:54 PM   #2
Senior Member

Joined: Feb 2016
From: Australia

Posts: 1,838
Thanks: 653

Math Focus: Yet to find out.
Quote:
 Originally Posted by zylo List all the elements of T and assume the list is uncountable.
How is this possible? Or am I just reading into the wording the wrong way.

 March 4th, 2016, 05:01 PM #3 Math Team   Joined: Dec 2013 From: Colombia Posts: 7,685 Thanks: 2665 Math Focus: Mainly analysis and algebra You aren't. Zylo doesn't understand that listing a set and counting a set are the same thing. He does have an irrational desperation to prove Cantor wrong though. Thanks from topsquark and Joppy
 March 4th, 2016, 05:51 PM #4 Math Team     Joined: May 2013 From: The Astral plane Posts: 2,273 Thanks: 943 Math Focus: Wibbly wobbly timey-wimey stuff. (sighs) Not again! How about this. Make a "list" of T and assume it is countably infinite. Therefore there exists a bijection f from the integers to T. Now use Cantor's diagonal argument to show there is an element of T that is not contained in the already listed elements of T. Therefore f is not onto and thus cannot be a bijection, a contradiction. Therefore T is not countably infinite. Why is this so difficult for you? -Dan Last edited by topsquark; March 4th, 2016 at 05:59 PM.
 March 4th, 2016, 06:19 PM #5 Math Team   Joined: Jan 2015 From: Alabama Posts: 3,264 Thanks: 902 I am starting to think that zylo is just posting these as a joke.
March 5th, 2016, 12:44 AM   #6
Senior Member

Joined: Jun 2015
From: England

Posts: 915
Thanks: 271

Quote:
 Let T be the set of all infinite binary series.
Pause right here.

How do you prove that T exists?

March 5th, 2016, 02:09 AM   #7
Global Moderator

Joined: Dec 2006

Posts: 20,975
Thanks: 2224

Quote:
 Originally Posted by zylo List all the elements of T and assume the list is uncountable. . . . Contradiction. Therefore T is countable.
You have assumed that T is uncountable and reached a contradiction, but you have also assumed that all the elements of T can be listed.

You've shown that at least one of your assumptions is untenable, but it may be your assumption that all the elements of T can be listed that is false, rather than your assumption that T is uncountable. That means that your argument is unsound (regardless of whether T is countable or uncountable).

 Tags argument, cantors, countable, diag, proves, reals

 Thread Tools Display Modes Linear Mode

 Similar Threads Thread Thread Starter Forum Replies Last Post zylo Topology 3 February 2nd, 2016 11:25 AM mathbalarka Number Theory 1 May 9th, 2013 05:51 AM HairOnABiscuit Abstract Algebra 11 September 11th, 2012 11:37 AM cernlife Real Analysis 5 May 30th, 2011 08:37 PM pi_314 Geometry 2 April 2nd, 2011 02:28 AM

 Contact - Home - Forums - Cryptocurrency Forum - Top