Re: Cantor Diagonal Proof

On 4/9/25 8:50 PM, Lawrence D'Oliveiro wrote:

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.

Sure it does, every digit you add gets it closer to its final value. What ever epsilon you ask for determines how many digits are needed.
Note, we RARELY actually REACH the limit, especially if the number doesn't have a finite exact representation.
The problem with the "Cantor Construction" is that one of the steps, "Compute to the Nth digit of the Nth number" isn't computable with a finite algorithm.