June 24th, 2016, 10:32 AM  #41  
Senior Member Joined: May 2016 From: USA Posts: 594 Thanks: 251  Quote:
He thinks it is a contradiction to say that each member of the set of natural numbers is finite, but the number of members of the set is infinite. Bizarre. Last edited by skipjack; June 25th, 2016 at 08:37 PM.  
June 24th, 2016, 12:38 PM  #42 
Senior Member Joined: Jan 2014 From: The backwoods of Northern Ontario Posts: 367 Thanks: 68 
But he indicated that to say, "The natural numbers collectively are finite" is false. Isn't that another way of saying that the number of elements in the set of natural numbers is infinite? Perhaps to be clear, it would have been better for him to have said, "To say, 'The natural numbers collectively are finite in number' is false." 
June 24th, 2016, 02:05 PM  #43  
Math Team Joined: Dec 2013 From: Colombia Posts: 6,700 Thanks: 2177 Math Focus: Mainly analysis and algebra  Quote:
This is why he highlights the two possible meanings. At some point he is going to reach the sentence with one meaning and then interpret it using the other. Or perhaps I'm doing him a disservice and he really is incapable of distinguishing between a set and its elements despite having had the difference highlighted to him many times.  
June 24th, 2016, 10:28 PM  #44 
Global Moderator Joined: Dec 2006 Posts: 16,944 Thanks: 1255  
June 25th, 2016, 06:17 AM  #45 
Senior Member Joined: Mar 2015 From: New Jersey Posts: 1,010 Thanks: 83 
n binary digits define 2^n natural numbers for every natural number n. Cantor's proof fails. The reals are countable. Last edited by skipjack; June 25th, 2016 at 08:49 PM. 
June 25th, 2016, 06:46 AM  #46 
Math Team Joined: Dec 2013 From: Colombia Posts: 6,700 Thanks: 2177 Math Focus: Mainly analysis and algebra 
Only for natural numbers $n$ and thus only for finite sequences. Cantor doesn't talk about finite sequences. Even if the diagonal argument were flawed, there are other proofs that the reals are uncountable, so you are completely wrong. You are obviously incapable of rational thought on this subject. You have done exactly as I predicted in using the artificial "ambiguity" that you set up in exactly the way I predicted that you would. 
June 25th, 2016, 06:56 AM  #47  
Math Team Joined: Jan 2015 From: Alabama Posts: 2,413 Thanks: 613  Quote:
For any finite number n, n binary digits define 2^n natural numbers. You are immediately extending that to a (countably) infinite set of digits without justifying the jump. Last edited by skipjack; June 25th, 2016 at 08:44 PM.  
June 25th, 2016, 05:33 PM  #48  
Senior Member Joined: May 2016 From: USA Posts: 594 Thanks: 251  Quote:
And this is totally irrelevant to real numbers. I feel a singularity in the derp field approaching. Question: how many binary digits does it take to define all natural numbers? Let's say it is p. So how many binary digits does it take to define $2^p + 10.$ EDIT: And what is magical about binary digits anyway? Last edited by skipjack; June 25th, 2016 at 08:45 PM.  
June 25th, 2016, 08:47 PM  #49 
Global Moderator Joined: Dec 2006 Posts: 16,944 Thanks: 1255 
derp?

June 25th, 2016, 08:52 PM  #50 
Newbie Joined: Jun 2016 From: ha noi Posts: 2 Thanks: 2 
Same applies for binary digits. 

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