July 13th, 2017, 10:12 AM  #1 
Senior Member Joined: Mar 2015 From: New Jersey Posts: 1,079 Thanks: 87  Cantors Diagonal Argument for n digits.
The number of binary sequences for n digits is always greater than n, for all n. Ex, n=2 10 01 11 00 11=00 is in the list. 00 01 10 11 01=10 is in the list. 
July 13th, 2017, 10:40 AM  #2 
Math Team Joined: Jan 2015 From: Alabama Posts: 2,524 Thanks: 643 
I am not clear why you have posted this. Do you have a question? And I don't understand what "11= 00" or 01 = 10" could mean. It is straight forward to prove that the set of all binary numeral with up to n digits (counting leading 0s) is . It is also easy to prove that for all positive integers n. 
July 13th, 2017, 11:47 AM  #3  
Global Moderator Joined: Oct 2008 From: London, Ontario, Canada  The Forest City Posts: 7,517 Thanks: 910 Math Focus: Elementary mathematics and beyond  Quote:
I hope we're not back to the nonsense of refuting Cantor's "diagonal argument"! It has been proven by a legitimate mathematical method and holds true. Whether or not one chooses to believe the concepts that it espouses is an entirely different matter.  
July 13th, 2017, 01:25 PM  #4 
Math Team Joined: Dec 2013 From: Colombia Posts: 6,857 Thanks: 2230 Math Focus: Mainly analysis and algebra 
More to the point Cantor's diagonal argument doesn't talk about finite sequences (those that cease after $n$ digits), it talks about infinite sequences (those that never cease).

July 13th, 2017, 03:19 PM  #5 
Member Joined: Oct 2009 Posts: 97 Thanks: 33 
This is interesting. As the OP shows, you can indeed extend Cantor's argument to show that $2^n > n$ for finite numbers $n$. Never thought of that, thanks!

July 13th, 2017, 03:55 PM  #6 
Math Team Joined: May 2013 From: The Astral plane Posts: 1,570 Thanks: 613 Math Focus: Wibbly wobbly timeywimey stuff. 
Again? Dan 
July 13th, 2017, 04:01 PM  #7 
Global Moderator Joined: Oct 2008 From: London, Ontario, Canada  The Forest City Posts: 7,517 Thanks: 910 Math Focus: Elementary mathematics and beyond  
July 13th, 2017, 06:12 PM  #8 
Senior Member Joined: Jun 2014 From: USA Posts: 299 Thanks: 21  
July 13th, 2017, 06:46 PM  #9 
Math Team Joined: Dec 2013 From: Colombia Posts: 6,857 Thanks: 2230 Math Focus: Mainly analysis and algebra 
Or, indeed any $n \not \in \mathbb N$.

July 14th, 2017, 10:14 AM  #10 
Global Moderator Joined: Dec 2006 Posts: 17,533 Thanks: 1322  

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