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From: Mikko <mikko.levanto@iki.fi>
Newsgroups: comp.theory
Subject: Re: Cantor Diagonal Proof
Date: Sun, 6 Apr 2025 13:35:12 +0300
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On 2025-04-06 07:15:51 +0000, wij said:

> On Sun, 2025-04-06 at 06:43 +0000, 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. By your> 
>> (mis)interpretation of his words, even something like 1/3 is an> 
>> incomputable number, since its “expressions as a decimal are not> 
>> calculable by finite means”.
> 
> Simply put, repeating decimals are irrational.
> https://sourceforge.net/projects/cscall/files/MisFiles/RealNumber2-en.txt/download 
> 

Repeating decimals are rational. An irrational number has an infinite
non-repeating sequence of digits.

> All your doubt should be addressed in the file of the link.
> 
>> Turing would not have been so dumb as to have his definition of> 
>> computability depend on something as trivial as the choice of number 
>> base.


-- 
Mikko