April 4th, 2016, 07:53 AM  #1 
Senior Member Joined: Mar 2015 From: New Jersey Posts: 1,555 Thanks: 113  Summary of Cantor's Diagonal Argument is Wrong
1) Cantor's Diagonal Argument is wrong because countably infinite binary sequences are natural numbers. 2) Cantor's Diagonal Argument fails because there is no natural number greater than all natural numbers. 3) Cantor's Diagonal Argument is not applicable for infinite binary sequences because countably infinite is a subset of infinite. 4) Without the point, every real number in [0,1) can be uniquely associated with a countably infinite binary sequence which is a natural number. .00110....... > 00110....... 0x2^{1} + 0x2^{2} + 1x2^{3}+.... > 0x2^{0} + 0x2^{1} + 1x2^{2} + .... The reals are countable. This is summarized here because it's buried in: Cantor's Diagonal Argument. Infinity is Not a Number Last edited by skipjack; April 4th, 2016 at 12:22 PM. 
April 4th, 2016, 08:25 AM  #2  
Global Moderator Joined: Dec 2006 Posts: 19,733 Thanks: 1810  Quote:
You haven't proved your opening assertion (numbered 1), or given any examples to support it. You haven't explained why your second and third points have any bearing on Cantor's diagonal argument. Also, you haven't explained which particular part of Cantor's argument is affected. Do your points all occur in the thread you linked? I can't find them.  
April 4th, 2016, 09:01 AM  #3 
Senior Member Joined: Mar 2015 From: New Jersey Posts: 1,555 Thanks: 113  
April 4th, 2016, 09:27 AM  #4 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,468 Thanks: 2499 Math Focus: Mainly analysis and algebra 
So this is yet another thread for you to post the same old rubbish with exactly the same stupid mistakes. This thread is pointless and utterly without value.

April 4th, 2016, 12:12 PM  #5 
Senior Member Joined: Mar 2015 From: New Jersey Posts: 1,555 Thanks: 113 
Let L be the list of ALL countably infinite binary sequences. S is not in L S is larger than any element of L because, whatever Sn is, S has "1" digits past the last "1" digit of Sn. (There is no natural number greater than all natural numbers). If you don't accept the sentence in parentheses, you will never get it. 
April 4th, 2016, 12:19 PM  #6 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,468 Thanks: 2499 Math Focus: Mainly analysis and algebra  I realise that this is quite a groundbreaking discovery for you, but for most of us it is entirely obvious. Unfortunately, you have no idea how to use this fact correctly.

April 4th, 2016, 12:31 PM  #7 
Global Moderator Joined: Dec 2006 Posts: 19,733 Thanks: 1810  You originally gave this as a generalization of the corresponding statement for finite binary sequences, but you never justified making the generalization. Not every generalization from finite to infinite is correct. You haven't proved that such a list exists, so that is not a valid definition. 
April 4th, 2016, 12:52 PM  #8 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,468 Thanks: 2499 Math Focus: Mainly analysis and algebra 
It's worth pointing out that even if such a list existed (which it doesn't) it would not be a unique list. This is a very trivial fact, but isn't reflected by zylo's statements.

April 4th, 2016, 02:12 PM  #9  
Senior Member Joined: Mar 2015 From: New Jersey Posts: 1,555 Thanks: 113  Quote:
https://en.wikipedia.org/wiki/Cantor...gonal_argument "The diagonal argument was not Cantor's first proof of the uncountability of the real numbers; it was actually published much later than his first proof, which appeared in 1874. Cantor's diagonal argument, was published in 1891 by Georg Cantor." _____________________________________ In over 100yrs, the mathematicians still haven't gotten it right. So it took me a few posts and a couple of weeks. Cut me some slack. ============================ "You haven't proved that such a list exists, so that is not a valid definition:" skipjack As you well know, that's Cantor's assumption from which he tries (unsuccessfully) to draw a contradiction. Perhaps you should review Cantor's proof. There you will find: https://en.wikipedia.org/wiki/Cantor...gonal_argument "He assumes for contradiction that T was countable. Then all its elements could be written as an enumeration s1, s2, â€¦ , sn, â€¦ ." Last edited by skipjack; April 4th, 2016 at 03:18 PM.  
April 4th, 2016, 02:41 PM  #10 
Math Team Joined: May 2013 From: The Astral plane Posts: 1,891 Thanks: 769 Math Focus: Wibbly wobbly timeywimey stuff.  Wow. Just. Wow. Dan Last edited by skipjack; April 4th, 2016 at 03:18 PM. 

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