Path: news.eternal-september.org!eternal-september.org!.POSTED!not-for-mail
From: Richard Heathfield <rjh@cpax.org.uk>
Newsgroups: comp.theory
Subject: Re: Cantor Diagonal Proof
Date: Sun, 6 Apr 2025 21:35:21 +0100
Organization: Fix this later
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In-Reply-To: <vsungg$1redp$2@dont-email.me>

On 06/04/2025 21:17, Lawrence D'Oliveiro wrote:
> On Sun, 6 Apr 2025 08:05:42 +0100, Richard Heathfield wrote:
> 
>> On 06/04/2025 07:43, Lawrence D'Oliveiro wrote:
>>
>>> On Sun, 6 Apr 2025 07:27:43 +0100, Richard Heathfield wrote:
>>>
>>>> On 06/04/2025 06:40, Lawrence D'Oliveiro wrote:
>>>>
>>>>> On Sat, 5 Apr 2025 09:07:22 +0100, Richard Heathfield wrote:
>>>>>
>>>>>> But to be computable, numbers must be computed in a finite number of
>>>>>> steps.
>>>>>
>>>>> “Computable Number: A number which can be computed to any number of
>>>>> digits desired by a Turing machine.”
>>>>>
>>>>> <https://mathworld.wolfram.com/ComputableNumber.html>
>>>>
>>>> "The “computable” numbers may be described briefly as the real numbers
>>>> whose expressions as a decimal are calculable by finite means." - Alan
>>>> Turing.
>>>>
>>>> And therefore, to be computable, numbers must be computed in a finite
>>>> number of steps.
>>>
>>> I would say you are quoting Turing out of context.
>>
>> There is no context before his words because they are the paper's
>> opening words.
> 
> The paper is “On Computable Numbers, With An Application To The
> Entscheidungsproblem”
> <https://www.cs.virginia.edu/~robins/Turing_Paper_1936.pdf>.

Yes. That's a cleaner copy than the one I found, so I'm grateful 
to you.

> Go on, then. Show how his conclusions are at odds with the generally-
> accepted (and more concise) definition of “computable number” quoted
> above.

I see no disagreement between Wolfram and Turing's definitions, 
but Turing makes explicit the requirement for finitely many 
steps, a requirement that is merely implicit in Wolfram's 
wording, which says that the number /can/ be computed. A 
computation that takes infinitely many steps /cannot/ be 
completed. One may reason about the nature of the result of such 
a computation, but one cannot know what all the digits are.

-- 
Richard Heathfield
Email: rjh at cpax dot org dot uk
"Usenet is a strange place" - dmr 29 July 1999
Sig line 4 vacant - apply within