My Math Forum  

Go Back   My Math Forum > College Math Forum > Topology

Topology Topology Math Forum


Reply
 
LinkBack Thread Tools Display Modes
March 6th, 2016, 08:58 PM   #1
Banned Camp
 
Joined: Mar 2015
From: New Jersey

Posts: 1,720
Thanks: 125

Cantors Diagonal Argument and Epimenides

Let S be the set of all infinite binary sequences. By Cantor's Diagonal Argument:

S countable -> S uncountable -> S countable -> ...... ,

which is circular, which is Epimenides Liar Paradox:

I am a liar T -> I am a liar F -> I am a liar T .........

Last edited by skipjack; March 6th, 2016 at 09:07 PM.
zylo is offline  
 
March 6th, 2016, 09:07 PM   #2
Global Moderator
 
Joined: Dec 2006

Posts: 20,746
Thanks: 2133

S uncountable -> S countable isn't shown by Cantor's (or any other) argument.
skipjack is offline  
March 6th, 2016, 09:37 PM   #3
Math Team
 
Joined: Dec 2013
From: Colombia

Posts: 7,663
Thanks: 2643

Math Focus: Mainly analysis and algebra
Zylo, what is your motivation behind all of these threads? Are you channelling Leopold Kronecker?

You have the appearance of somebody who is desperate to achieve something, but it's not clear if that something has a clear definition or not. What is clear is that you don't have enough understanding to be making any claims. If you don't understand a particular result, you should admit that and then work to understand it, rather than simply claiming it to be false. I would go so far as to suggest that if you disagree with any result, you should do the same.

In this case, you have yet again ignored the responses to your previous thread. In that thread you claimed that the diagonal argument could be used, with an assumption of the uncountability of a set, to prove the countability of that set. Now you use the same result, despite every response to the first thread pointing out how your argument was mistaken. Why do you refuse to listen to the arguments of others?

Last edited by skipjack; March 6th, 2016 at 11:17 PM.
v8archie is offline  
March 8th, 2016, 05:37 AM   #4
Math Team
 
Joined: Jan 2015
From: Alabama

Posts: 3,264
Thanks: 902

What Cantor's proof says is that "if S is the set of all infinite binary sequences then S--> there exist an infinite binary sequence that is not in S", a contradiction.
Country Boy is offline  
March 8th, 2016, 06:08 AM   #5
Math Team
 
Joined: Dec 2013
From: Colombia

Posts: 7,663
Thanks: 2643

Math Focus: Mainly analysis and algebra
Quote:
Originally Posted by Country Boy View Post
What Cantor's proof says is that "if S is the set of all infinite binary sequences then S--> there exist an infinite binary sequence that is not in S", a contradiction.
No, Cantor's proof says that, given any countable set S of infinite binary sequences, there exists an infinite binary sequence that is not in S. This result then provides a contradiction if we assume that the set of infinite binary sequences is countable.

If we assume that the set of infinite binary sequences is uncountable, Cantor's argument doesn't tell us anything interesting. In fact it doesn't apply, because the argument requires that the set be listed and no uncountable set can be listed.
v8archie is offline  
Reply

  My Math Forum > College Math Forum > Topology

Tags
argument, cantors, diagonal, epimenides



Thread Tools
Display Modes


Similar Threads
Thread Thread Starter Forum Replies Last Post
Cantors Diag Argument Proves Reals Countable zylo Topology 6 March 5th, 2016 02:09 AM
Cantor's Diagonal Argument zylo Math 22 January 26th, 2016 08:05 PM
The Super Diagonal Argument AplanisTophet Number Theory 0 October 24th, 2014 08:59 PM
Help! Cantor's Diagonal Argument mjcguest Applied Math 9 July 25th, 2013 07:22 AM
Cantorīs diagonal argument netzweltler Applied Math 191 November 7th, 2010 01:39 PM





Copyright © 2019 My Math Forum. All rights reserved.