March 23rd, 2016, 08:00 AM  #1 
Senior Member Joined: Mar 2015 From: New Jersey Posts: 1,079 Thanks: 87  Cantor's Diagonal Argument Reconsidered
Let T be the list of all natural numbers, (1,2,3,4,5,........) Perform an operation which takes account of every member of the list to come up with 4. 4 is not a member of the list. Proof: 4 is different from every member of the list except itself. Every rearrangement of the list produces another unique member of the list which is not on the list. Therefore the list is empty. The point being an analogy with Cantor's diagonal argument: The fact that you produce a number different from every member of the list does not prove it is not on the list. Last edited by skipjack; March 23rd, 2016 at 02:57 PM. 
March 23rd, 2016, 09:14 AM  #2  
Senior Member Joined: Apr 2014 From: Glasgow Posts: 2,018 Thanks: 665 Math Focus: Physics, mathematical modelling, numerical and computational solutions  Quote:
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Last edited by skipjack; March 23rd, 2016 at 02:58 PM.  
March 23rd, 2016, 09:53 AM  #3  
Senior Member Joined: Mar 2015 From: New Jersey Posts: 1,079 Thanks: 87  Quote:
4 is in the list. EDIT The point is, showing that a number (sequence) is different than every member of a list of numbers (sequences), doesn't prove it is not in the list, 4 above, for ex. Last edited by zylo; March 23rd, 2016 at 10:05 AM.  
March 23rd, 2016, 01:18 PM  #4 
Math Team Joined: May 2013 From: The Astral plane Posts: 1,570 Thanks: 613 Math Focus: Wibbly wobbly timeywimey stuff.  Another thread? Really? Dan 
March 23rd, 2016, 03:02 PM  #5 
Senior Member Joined: Mar 2015 From: New Jersey Posts: 1,079 Thanks: 87 
To bring things a little closer to home. Assume T is the set of all countably infinite binary digits, i.e., the set of all natural numbers, which is countable. But Cantor's argument says T is uncountable? What's wrong? See OP. REF: Cantor's diagonal Argument: The set of all infinite binary sequences is uncountable. Let T be the set of all infinite binary sequences. Assume T is countable. Then all its elements can be enumerated: 1 0 0 1 1 0............... 0 0 1 0 1 1............... 0 1 0 1 0 0.............. ............................ Let s be the binary sequence consisting of the complemented underlined digits: s = 0 1 1................... s is different from every member of the list and s belongs to T. Contradiction. Therefore T isn't countable. ____________________________________ https://en.wikipedia.org/wiki/Cantor...gonal_argument _____________________________________ Each thread is a different seminal argument which would otherwise get buried. Considering all that has been written on the subject in the last 100+ years, and the assertion that it is fundamental to the foundations of mathematics, a few weeks and threads are not unreasonable. Last edited by skipjack; March 23rd, 2016 at 03:15 PM. 
March 23rd, 2016, 03:07 PM  #6 
Global Moderator Joined: Dec 2006 Posts: 17,445 Thanks: 1312  Adding this means that what you're describing isn't analogous to what Cantor did, as Cantor didn't use "except itself". Hence it's your conjectured scenario that doesn't come up to scratch, whilst Cantor's proof remains untarnished.

March 23rd, 2016, 03:14 PM  #7 
Global Moderator Joined: Dec 2006 Posts: 17,445 Thanks: 1312  It doesn't make sense to add "i.e., the set of all natural numbers" because no countably infinite binary sequences are natural numbers. As Cantor didn't do that, you're countering your own approach, not what Cantor did.

March 23rd, 2016, 03:43 PM  #8 
Senior Member Joined: Mar 2015 From: New Jersey Posts: 1,079 Thanks: 87 
Guess you missed my previous post: "Assume T is the set of all countably infinite binary digits, i.e., the set of all natural numbers, which is countable. But Cantor's argument says T is uncountable? What's wrong? See OP." "except itself" is obvious, but in any event doesn't change above proposition. It's part of the explanation in OP of why Cantor's argument doesn't work. 
March 23rd, 2016, 05:01 PM  #9 
Global Moderator Joined: Dec 2006 Posts: 17,445 Thanks: 1312 
What is the OP that you are referring to? I didn't miss your post with "Assume T" in its second sentence as that's what I quoted. It's not what Cantor does, but instead something that you've devised that doesn't make sense, as no element of T is a natural number.

March 23rd, 2016, 06:59 PM  #10  
Math Team Joined: Dec 2013 From: Colombia Posts: 6,854 Thanks: 2228 Math Focus: Mainly analysis and algebra 
Zylo: why does your making a host of assumptions that Cantor didn't (and which happen to be nonsense) make Cantor's proof wrong? If you wish to critique Cantor, you can start by learning what he did say, rather than making stuff up. Quote:
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This means that any enumeration E of elements of T is, not equal to T. Thus any countably infinite subset of T is not equal to T. The corollary to this statement is that T is not countably infinite. Why is this so difficult to grasp? Last edited by skipjack; March 23rd, 2016 at 10:05 PM.  

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