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March 30th, 2016, 02:16 PM  #1 
Senior Member Joined: Mar 2015 From: New Jersey Posts: 1,603 Thanks: 115  Cantor's diagonal proof and the real numbers
At this point we have two issues: 1) Cantor's proof. Wrong in my opinion, see: Cantor's Diagonal Argument. Infinity is Not a Number 2) Cantor's proof has nothing to do with the real numbers. A real irrational number is not an infinite binary digit. It is the limit of binary digits. (sqrt2). WRONG!! See EDIT below. A natural number can be assigned to every binary fraction for all n (the binary fractions can be counted): .1011... > 1011... . But lim .1011... exists lim 1011 doesn't exist.* The question remains, can a natural or pos rational number be assigned to a lim? You have to answer this before you can decide if the reals are countable. For example: sqrt2 > 2 sqrtn > n Square roots of natural numbers are countable. nth roots are countable. *skipjack, see link EDIT HOWEVER If you can show that infinite binary sequences are uncountable, then it follows that the reals are uncountable because, by the above, there are more reals than infinite binary sequences. But note that all infinite binary fractions are countable by: .1011... > 1011... for all n. Last edited by zylo; March 30th, 2016 at 02:57 PM. 
March 30th, 2016, 02:49 PM  #2 
Global Moderator Joined: Dec 2006 Posts: 19,963 Thanks: 1849  That seems to be an admission that you are merely giving your opinions, not discovering genuine mathematical or logical errors that shouldn't need to be a matter of opinion. Let's put this particular issue on hold briefly, as I would like you to reply first to my recent posts in the thread that you linked to above. 
March 30th, 2016, 04:14 PM  #3 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,511 Thanks: 2514 Math Focus: Mainly analysis and algebra 
Actually, there is only one issue: you have no idea what you are talking about, but persist in making nonsensical claims instead of attempting to understand.

March 30th, 2016, 05:36 PM  #4  
Senior Member Joined: Mar 2015 From: New Jersey Posts: 1,603 Thanks: 115  Quote:
But the limit of an infinite binary digit is not always an irrational number: Lim .3333333333.....=1/3 The question still remains, unanswered by Cantor, Are limits of countably infinite binary numbers countable. Countably infinite binary digits are natural numbers. 101101..... , n digits, is a natural number for all n. There is not an "infinite" binary digit, so Cantors set of "infinite" binary digits is meaningless right off the bat. Last edited by zylo; March 30th, 2016 at 05:57 PM.  
March 30th, 2016, 06:55 PM  #5  
Math Team Joined: Dec 2013 From: Colombia Posts: 7,511 Thanks: 2514 Math Focus: Mainly analysis and algebra  Quote:
They are. Quote:
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This is more nonsense. Unfortunately, it can't be rescued: it's just wrong. It's a product of your complete lack of understanding of the subject and your utter refusal to learn anything about it. Quote:
Perhaps one day you will realise just how little you know. I know that when I did that, it freed up my mind to understand many more marvellous things. Please, by whatever you hold sacred, stop writing meaningless, confused nonsense and try to learn something. Last edited by skipjack; March 30th, 2016 at 08:58 PM.  
March 30th, 2016, 09:04 PM  #6 
Global Moderator Joined: Dec 2006 Posts: 19,963 Thanks: 1849  
March 31st, 2016, 08:54 AM  #7 
Senior Member Joined: Mar 2015 From: New Jersey Posts: 1,603 Thanks: 115 
Convergence of an infinite binary fraction is elementary calculus. A binary fraction can be associated with a natural number as follows. .101.... to n places can be associated with the natural number 101....but lim.101.... can't because lim101.... doesn't exist. The fractional part of pi for any finite number of decimal places can be associated with a natural number, but not the limit. .1415 > 1415 but limit .1415...... = pi3 I personally find it interesting that any countably infinite fraction can be associated with a natural number. But there is a deeper significance in that all countably infinite binary (or decimal) fractions can be counted. But that doesn't imply that all reals in [0,1)can be counted, because some are the limit of a countably infinite binary fraction. A basic principle has been developed here: All binary fractions can be counted. Some real numbers are limits of binary fractions and the above association doesn't work; counting these is open for discussion. binary fraction: .1011000001......., countably infinite places. lim of binary fraction: lim.1011000001...as n approaches infinity 
March 31st, 2016, 10:24 AM  #8  
Math Team Joined: Dec 2013 From: Colombia Posts: 7,511 Thanks: 2514 Math Focus: Mainly analysis and algebra 
Stop trying to prove things, you aren't competent to do so. Everything your write is nonsense because you don't understand the first thing about this stuff. Quote:
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binary fraction: 0.1011000001... terminating after a finite number of digits.But even after correcting these definitions nothing "has been developed here" this is all centuries old stuff. And the counting of the nonterminating binary representations is not up for discussion  they are not countable (although some subsets of them are countable).  
March 31st, 2016, 03:00 PM  #9  
Global Moderator Joined: Dec 2006 Posts: 19,963 Thanks: 1849  Quote:
 

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