January 11th, 2017, 03:29 PM  #21  
Member Joined: Jun 2014 From: Alberta Posts: 56 Thanks: 2  Quote:
Or is "going to infinity" and "exhausting infinity" different things? If so, then that would make sense to me about why my method wouldn't work.  
January 11th, 2017, 03:34 PM  #22  
Senior Member Joined: Aug 2012 Posts: 2,410 Thanks: 754  Quote:
I apologize if my explanation added confusion.  
January 11th, 2017, 03:38 PM  #23  
Math Team Joined: Dec 2013 From: Colombia Posts: 7,690 Thanks: 2669 Math Focus: Mainly analysis and algebra  Quote:
Every number on the list has a finite (natural) number of digits followed by infinite zeros. Specifically, it contains no decimals that do not end in infinite zeros such as $(\pi3)$, $(\sqrt2 1)$ and $\frac13$. The only way your list could move onto numbers that do not end in infinite zeros would be for the process of listing all decimals with $N$ digits before the infinite zeros to halt (where $N$ is some natural number). This means that no number having $(N+1)$ digits followed by infinite zeros would appear on the list. So it's still not a complete list.  
January 11th, 2017, 03:44 PM  #24 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,690 Thanks: 2669 Math Focus: Mainly analysis and algebra  "Going to infinity" has no meaning in this context. The number of digits before the infinite zeros follows the sequence 1, 2, 3,... This sequence never reaches "infinity" because "infinity" is not a number. So each number on your list has a finite number of digits before the infinite (that is "never ending") sequence of zeroes.

January 11th, 2017, 03:54 PM  #25 
Senior Member Joined: May 2016 From: USA Posts: 1,310 Thanks: 552  This is a debatable assertion. Real numbers "exist" as a mathematical construct. They cannot be demonstrated to "exist" in the physicotemporal world. Nor can any transfinite number be shown to have a physicotemporal existence. It would, I think. be preferable to say something like "if you accept that real numbers and at least one transfinite numbers 'exist' as mathematical constructs, then it follows that the (transfinite) number of the real numbers exceeds the (transfinite) number of the natural numbers." I suspect that the psychological barriers to grasping Cantor include the refusal to entertain seriously the "existence" of transfinite numbers and the failure to realize that the "existence" of transcendental numbers entails the "existence" of transfinite numbers. Cantor's argument is elegant and relatively simple once you accept its ontological premises. But those premises can never be "proved" physically. 
January 11th, 2017, 04:02 PM  #26  
Senior Member Joined: Aug 2012 Posts: 2,410 Thanks: 754  Quote:
Conceptually it's like a lookup table or array in a computer program; except that the array has infinitely many cells, one for each natural number; and we can look up any cell of the array immediately. That's the conceptual model of standard math. Everything exists at once and is always available for use. That's one of the basic rules of the game. As you can see this is a highly abstract conceptual world we live in when we do math. That para was a replacement for an extended discussion of the role of the Axiom of Infinity, which I wrote in an earlier draft of #11 and then deleted as being inappropriate to the OP's question. But for the record, Cantor's great breakthrough was to go past Aristotle and posit the existence of a completed infinity of natural numbers. Without that, there's no transfinite theory at all. Last edited by Maschke; January 11th, 2017 at 04:09 PM.  
January 11th, 2017, 05:30 PM  #27 
Member Joined: Jun 2014 From: Alberta Posts: 56 Thanks: 2 
Thanks everyone, I think I get it now.


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