March 20th, 2018, 07:21 AM  #21  
Senior Member Joined: Oct 2009 Posts: 430 Thanks: 144  Quote:
 
March 20th, 2018, 08:27 AM  #22  
Senior Member Joined: Mar 2015 From: New Jersey Posts: 1,392 Thanks: 100  Quote:
There are two possibilities: Yes or No.  
March 20th, 2018, 10:19 AM  #23 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,331 Thanks: 2457 Math Focus: Mainly analysis and algebra  
March 20th, 2018, 10:20 AM  #24 
Senior Member Joined: Oct 2009 Posts: 430 Thanks: 144  
March 20th, 2018, 06:55 PM  #25  
Senior Member Joined: Apr 2015 From: Planet Earth Posts: 129 Thanks: 25  Yes, it does. But... Quote:
For example, s(n)=mod(n,2) for all n in $\mathbb N$ is a pefectly valid definition of one of Cantor's strings. It does not end. Last edited by JeffJo; March 20th, 2018 at 06:59 PM.  
March 21st, 2018, 06:47 AM  #26 
Senior Member Joined: Mar 2015 From: New Jersey Posts: 1,392 Thanks: 100  
March 21st, 2018, 07:22 AM  #27 
Senior Member Joined: May 2016 From: USA Posts: 1,053 Thanks: 431  There we have it. Zylo is thinking about a finite list. There is not only no necessity that the list end, but it is explicitly an endless list because it is matched onetoone to the assumedly endless positive integers. There is nothing wrong with denying the acceptability of infinity, but doing so makes it ridiculous to discuss Cantor's diagonal theorem, which proves that, if infinity is accepted, then infinity is not a single, undifferentiated concept 
March 21st, 2018, 09:27 AM  #28 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,331 Thanks: 2457 Math Focus: Mainly analysis and algebra  
March 21st, 2018, 11:05 AM  #29 
Global Moderator Joined: Dec 2006 Posts: 19,190 Thanks: 1649  
March 21st, 2018, 06:02 PM  #30  
Senior Member Joined: Apr 2015 From: Planet Earth Posts: 129 Thanks: 25  Quote:
If "not," in what place do you think it fails? And why? Cantor's strings are not defined by a process that must end to define every character. Just like the Axiom of Infinity defines every natural number in $\mathbb N$  and establishes BY AXIOM that it is possible to do so  every character of that string is defined by a single statement. Without needing a process that ends.  

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argument, cantor, cantors, diagonal, infinity 
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