Re: Cantor Diagonal Proof

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

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..