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From: wij <wyniijj5@gmail.com>
Newsgroups: comp.theory
Subject: Re: Cantor Diagonal Proof
Date: Sun, 06 Apr 2025 18:42:05 +0800
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Message-ID: <809a03d4b71e717fca7bf77bd900c4e977844c2d.camel@gmail.com>
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On Sun, 2025-04-06 at 13:35 +0300, Mikko wrote:
> On 2025-04-06 07:15:51 +0000, wij said:
>=20
> > 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 numb=
er of
> > > > > > steps.
> > > > >=20
> > > > > =E2=80=9CComputable Number: A number which can be computed to any=
 number 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 the real numbers
> > > > whose expressions as a decimal are calculable by finite means." - A=
lan
> > > > Turing.
> > > >=20
> > > > And therefore, to be computable, numbers must be computed in a fini=
te
> > > > 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.
> >=20
> > Simply put, repeating decimals are irrational.
> > https://sourceforge.net/projects/cscall/files/MisFiles/RealNumber2-en.t=
xt/download=C2=A0
> >=20
>=20
> Repeating decimals are rational.

Prove it (be sure not to make mistakes shown in the link above)

>  An irrational number has an infinite
> non-repeating sequence of digits.
>=20
> > All your doubt should be addressed in the file of the link.
> >=20
> > > 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=
=20
> > > base.
>=20
>=20