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From: wij <wyniijj5@gmail.com>
Newsgroups: comp.theory
Subject: Re: Cantor Diagonal Proof
Date: Sun, 06 Apr 2025 15:15:51 +0800
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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:
>=20
> > On 06/04/2025 06:40, Lawrence D'Oliveiro wrote:
> >=20
> > > On Sat, 5 Apr 2025 09:07:22 +0100, Richard Heathfield wrote:
> > >=20
> > > > But to be computable, numbers must be computed in a finite number o=
f
> > > > steps.
> > >=20
> > > =E2=80=9CComputable Number: A number which can be computed to any num=
ber of
> > > digits desired by a Turing machine.=E2=80=9D
> > >=20
> > > <https://mathworld.wolfram.com/ComputableNumber.html>
> >=20
> > "The =E2=80=9Ccomputable=E2=80=9D numbers may be described briefly as t=
he real numbers
> > whose expressions as a decimal are calculable by finite means." - Alan
> > Turing.
> >=20
> > And therefore, to be computable, numbers must be computed in a finite
> > number of steps.
>=20
> I would say you are quoting Turing out of context. By your=20
> (mis)interpretation of his words, even something like 1/3 is an=20
> incomputable number, since its =E2=80=9Cexpressions as a decimal are not=
=20
> calculable by finite means=E2=80=9D.

Simply put, repeating decimals are irrational.
https://sourceforge.net/projects/cscall/files/MisFiles/RealNumber2-en.txt/d=
ownload

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=20
> computability depend on something as trivial as the choice of number base=
..