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July 13th, 2017, 10:12 AM   #1
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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.
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July 13th, 2017, 10:40 AM   #2
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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.
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July 13th, 2017, 11:47 AM   #3
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Quote:
Originally Posted by zylo View Post
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.
What, exactly, are you trying to say?

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.
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July 13th, 2017, 01:25 PM   #4
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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).
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July 13th, 2017, 03:19 PM   #5
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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!
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July 13th, 2017, 03:55 PM   #6
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Again?

-Dan
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July 13th, 2017, 04:01 PM   #7
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Again?
Not for long, if at all.
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July 13th, 2017, 06:12 PM   #8
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Quote:
Originally Posted by zylo View Post
The number of binary sequences for n digits is always greater than n, for all n.
What about n = 0?

...couldn't help myself.
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July 13th, 2017, 06:46 PM   #9
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Or, indeed any $n \not \in \mathbb N$.
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July 14th, 2017, 10:14 AM   #10
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Quote:
Originally Posted by zylo View Post
The number of binary sequences for n digits is always greater than n, for all n.
Cantor used non-terminating binary sequences, not binary sequences n digits long.
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