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October 4th, 2018, 10:42 AM   #1
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Cantor's Diagonal Argument is Wrong

This is a slight refinement (simplification and condensation) of my previous expositions which evolved in the quoted thread. I put it here because it was being buried, and because it addresses the subject directly.


Quote:
Originally Posted by zylo View Post
List of binary and decimal representation of ALL The Natural and Real [0,1) Numbers:

0-----0
1-----1
2-----01
3-----11
4-----001
........
........

Comments:
1) The list is endless and contains ALL imaginable strings of binary and decimal digits.
2) Adding a radix point before the binaries and after the natural numbers gives all real numbers in [0,1) in binary and decimal form.
5000 becomes 5000., or more commonly, .0005

Like all great ideas, it is beautifully simple, transparent, foundational, and can be understood by almost everyone, making it part of the world's cultural heritage.
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October 4th, 2018, 10:46 AM   #2
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As I stated before, it's unclear and incorrect.
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October 4th, 2018, 10:51 AM   #3
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I thought we'd agreed to close down this sort of idiocy.
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October 4th, 2018, 11:04 AM   #4
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Quote:
Originally Posted by skipjack View Post
As I stated before, it's unclear and incorrect.
It is also irrelevant. No one, particularly not Cantor, said that you could not represent all the natural numbers by an infinite list of infinite bit strings. What he proved was that, if you accept the idea of infinite strings and infinite lists at all, any infinite list that can be placed in one-to-one correspondence with the natural numbers will be incomplete.

Zylo's intuition is that this idea is ridiculous. Consequently, he assumes that such a list is complete. But an assumption is not a proof. He does everything in the world except address Cantor's argument.

There are two plausible responses to Cantor. One is to reject the idea that the infinite "exists" and therefore to adopt the position that theorems about infinity have the same epistemological status as theorems about the psychology of werewolves. The other is to accept that, at least in the human imagination, the infinite exists but follows rules that are completely alien to human intuition.
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October 4th, 2018, 11:05 AM   #5
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We don't need this as a separate thread for the time being, so I've closed it. I might ultimately reopen it by transferring all the related posts from the other thread.
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