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From: Mikko <mikko.levanto@iki.fi>
Newsgroups: comp.theory
Subject: Re: Cantor Diagonal Proof
Date: Mon, 14 Apr 2025 11:53:13 +0300
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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