Re: Cantor Diagonal Proof

On 2025-04-11 00:29:56 +0000, Lawrence D'Oliveiro said:

On Thu, 10 Apr 2025 10:37:49 +0300, Mikko wrote:
 

On 2025-04-10 00:50:10 +0000, Lawrence D'Oliveiro said:
 

On Mon, 7 Apr 2025 20:48:27 -0400, Richard Damon wrote:
 

The paper clearly talks about the process continuing indefinitely.

 Note the key point about any computation of a computable number is that
the answer *converges* to the exact result in the limit. As you compute
more and more digits, the discrepancy between your approximation and
the correct answer can be made as close to zero as you like, just as
long as you don’t ask for it to be zero.
 The Cantor construction does not converge.

 If it is a computable number it does converge.

 That’s a key point of my proof: if it converges, then the number is
already in the list. The only way it can come up with a number not in the
list is by never converging.

That does not follow. Every sequence of digits specifies a convergent
sequence of numbers, whether computable or not. But none of the non-
computable numbers are in the list of computable numbers and no list
has all non-computable numbers.
--
Mikko