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January 28th, 2016, 12:22 PM  #21  
Math Team Joined: Dec 2013 From: Colombia Posts: 7,685 Thanks: 2665 Math Focus: Mainly analysis and algebra  Quote:
Having said that we can add more real numbers to the list if we like. There's always a diagonal.  
January 28th, 2016, 12:33 PM  #22 
Senior Member Joined: Dec 2015 From: France Posts: 103 Thanks: 1  Do you mean that once an infinite list is created, no new member can be added?

January 28th, 2016, 12:55 PM  #23 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,685 Thanks: 2665 Math Focus: Mainly analysis and algebra 
No. Bit it doesn't make the list any longer. It's still an infinite list.

January 28th, 2016, 01:04 PM  #24  
Senior Member Joined: Dec 2015 From: France Posts: 103 Thanks: 1  Quote:
First you say We do not change the list, then you say even it is changed, it is not any longer. If the list cannot be changed, it leads to a thing, if it change, it leads to another thing. What is your mind. It changes or not?  
January 28th, 2016, 06:55 PM  #25 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,685 Thanks: 2665 Math Focus: Mainly analysis and algebra 
The diagonal proof is not about building an infinite list, it just says that we should assume that we have one. But if you want to add entries to an infinite list, you can. But it doesn't make the list any longer. 
January 28th, 2016, 07:27 PM  #26  
Senior Member Joined: Aug 2012 Posts: 2,395 Thanks: 749  Quote:
It changes but it still has the same cardinality. Suppose I have an infinite set X = 1, 2, 3, 4, ... Now suppose I "add" an element to the end to get Y = 1, 2, 3, 4, ..., 0. But this second set has the same cardinality as the first one, and the bijection is to just map 0 to 1, 1 to 2, 2 to 3, etc. Y  X 0  1 1  2 2  3 etc. So you see we've added a new element to X to get Y, yet we have not changed the size.  
January 29th, 2016, 06:48 AM  #27 
Senior Member Joined: Dec 2015 From: France Posts: 103 Thanks: 1  So, the length of the list is infinity. When added one, it is still infinity. The number of digits is nonechanged infinity. And is the diagonal unchanged, or not? THe added sequence is in the diagonal or not?

January 29th, 2016, 06:49 AM  #28  
Senior Member Joined: Dec 2015 From: France Posts: 103 Thanks: 1  Quote:
 
January 29th, 2016, 07:21 AM  #29  
Math Team Joined: Dec 2013 From: Colombia Posts: 7,685 Thanks: 2665 Math Focus: Mainly analysis and algebra  Quote:
Quote:
But we don't run out of columns with which to build the diagonal, because the horizontal list of columns is also nonterminating. Last edited by v8archie; January 29th, 2016 at 07:27 AM.  

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argument, assumption, cardinality, diagonal, diagonal argument, hidden, real 
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