Cantor's diagonal argument may be false
- Thread starter phyti
- Start date
Proof that reala are uncountable has other proofs as well. Using binary sequence is flawed, since most reals have infinite sequences, not listed.
I haven't looked at the pdf, but I don't see any "counter example" could be constructed.
How was this supposedly done?
What is the claimed flaw in Cantor's argument? Can you explain, briefly?
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.
Symmetry doesn't work. For example first row .0000..., then all numbers starting with .1 won't be included.
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.
Sherlock Holmes
Registered Member
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?
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?
Sherlock Holmes
Registered Member
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...
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...
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.
Sherlock Holmes
Registered Member
Well I will continue to think about this, it is quite subtle I know, I last read about this kind of thing years ago and I know intuition is not helpful !
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.
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.