July 13th, 2017, 11:12 AM  #1 
Senior Member Joined: Mar 2015 From: New Jersey Posts: 1,227 Thanks: 93  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, 11:40 AM  #2 
Math Team Joined: Jan 2015 From: Alabama Posts: 2,953 Thanks: 800 
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, 12:47 PM  #3  
Global Moderator Joined: Oct 2008 From: London, Ontario, Canada  The Forest City Posts: 7,739 Thanks: 1000 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, 02:25 PM  #4 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,146 Thanks: 2386 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, 04:19 PM  #5 
Senior Member Joined: Oct 2009 Posts: 232 Thanks: 84 
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, 04:55 PM  #6 
Math Team Joined: May 2013 From: The Astral plane Posts: 1,668 Thanks: 657 Math Focus: Wibbly wobbly timeywimey stuff. 
Again? Dan 
July 13th, 2017, 05:01 PM  #7 
Global Moderator Joined: Oct 2008 From: London, Ontario, Canada  The Forest City Posts: 7,739 Thanks: 1000 Math Focus: Elementary mathematics and beyond  
July 13th, 2017, 07:12 PM  #8 
Senior Member Joined: Jun 2014 From: USA Posts: 320 Thanks: 23  
July 13th, 2017, 07:46 PM  #9 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,146 Thanks: 2386 Math Focus: Mainly analysis and algebra 
Or, indeed any $n \not \in \mathbb N$.

July 14th, 2017, 11:14 AM  #10 
Global Moderator Joined: Dec 2006 Posts: 18,586 Thanks: 1489  

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