Re: Cantor Diagonal Proof

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:
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On Sun, 6 Apr 2025 07:27:43 +0100, Richard Heathfield wrote:
>

On 06/04/2025 06:40, Lawrence D'Oliveiro wrote:
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On Sat, 5 Apr 2025 09:07:22 +0100, Richard Heathfield wrote:
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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>

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"The “computable” numbers may be described briefly as the real numbers
whose expressions as a decimal are calculable by finite means." - Alan
Turing.
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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.

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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
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