# Cantor's diagonal argument may be false

Discussion in 'Physics & Math' started by phyti, Jul 17, 2021.

1. ### phytiRegistered Senior Member

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This is a trusted site (used over 1o yr).
app.box.com/s/6u5ydjoo8f97dnsord7b49dff4quqlnz
The counter example is a 3 page pdf using the 1891 example of a list of binary sequences.
It is not relative to any form of numbers, just sequences of symbols.

3. ### mathmanValued Senior Member

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Proof that reala are uncountable has other proofs as well. Using binary sequence is flawed, since most reals have infinite sequences, not listed.

5. ### James RJust this guy, you know?Staff Member

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I haven't looked at the pdf, but I don't see any "counter example" could be constructed.

What is the claimed flaw in Cantor's argument? Can you explain, briefly?

7. ### phytiRegistered Senior Member

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589
The binary tree shows the sequences occur in pairs via symmetry.
If Cantor can construct a 'new' sequence p from the diagonal d, then d its complement, is also new. Yet d is in the list, but can't be detected with one inspection.
Any sequence can be rotated 45 deg., but that doesn't change its identity.

8. ### mathmanValued Senior Member

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Symmetry doesn't work. For example first row .0000..., then all numbers starting with .1 won't be included.

9. ### phytiRegistered Senior Member

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I am only challenging the diagonal method, not its applications to numbers.
Follow Cantor's construction of p within the binary tree. He is just repeating an existing sequence.
His misdirection puts the readers focus on making the diagonal different.
We are aware that all sequences must be different in 1 or more positions.
The complement is different in all positions.

10. ### Sherlock HolmesRegistered Member

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The explanation of Cantor's method is (IMHO) a bit ambiguous, I'm not a mathematician but not a mathematical novice either.

The definition speaks of "...the set T of all infinite sequences of binary digits..." but does not state if the set T is itself infinite.

I can see that any member of T is an infinite sequence of bits, but is it speaking of a finite set of infinitely long members or an infinite set of infinitely long members?

It seems to be implied that T is infinite given it says "all infinite sequences" but I'm not totally clear in my mind, the very word "all" implies finite to me, countable.

One can't say "I have them all" or "I counted them all" unless "all" means finite - can one?

11. ### Sherlock HolmesRegistered Member

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Here's a more formal definition:

Cantor's Proof.

But it puzzles me, it says on page 2 "Now, this number can’t be in the table. Why not? Because..."

Well it can be in the table, we can literally just add it to the table as row number 6 say...

Hence my confusion...

12. ### mathmanValued Senior Member

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Adding to the table doesn't help. It was supposed to be complete in the first place. Also if you add it, then using Cantor's procedure, you could get another one, etc.

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14. ### phytiRegistered Senior Member

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589
Sherlock;

Each sequence is infinitely long.
The list is infinitely long.
Given the set N of natural/counting integers is infinite, it provides a means of ordering finite sets by size.
He visualizes transfinite sets as existing in a complete state, just as finite sets.
Cantor's idea of transfinite sets is similar in purpose, a means of ordering infinite sets by size. He uses the diagonal argument to show N is not sufficient to count the elements of a transfinite set, or make a 1 to 1 correspondence.
His method of swapping symbols on the diagonal d making it differ from each sequence in the list is true. His conclusion is false since he dismisses the possibility of duplication,
d being in the list, and not being detected.

Considering a finite sequence s of only 3 elements (0 or 1), there are 8 possible s.
There are 8! possible random lists. There would be duplication.
If any s is removed from the list and compared to the remaining s, it must differ from all those s. Thus being different does not mean not a member.

15. ### phytiRegistered Senior Member

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589
Here is more to consider.

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