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From: wij <wyniijj5@gmail.com>
Newsgroups: comp.theory
Subject: Re: Cantor Diagonal Proof --- PLO
Date: Sun, 06 Apr 2025 15:00:16 +0800
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On Sat, 2025-04-05 at 14:40 -0700, Keith Thompson wrote:
> Lawrence D'Oliveiro <ldo@nz.invalid> writes:
> [...]
> > The cardinality of the set of integers (and therefore also the set of=
=20
> > computer programs, and of the set of computable numbers) is conventiona=
lly=20
> > written as =E2=84=B5=E2=82=80. The cardinality of the set of reals is w=
ritten as =E2=84=B5=E2=82=81. Both=20
> > are infinite, but =E2=84=B5=E2=82=81 is supposed to be a larger infinit=
y than =E2=84=B5=E2=82=80 -- at=20
> > least, that=E2=80=99s what the Cantor diagonal construction is supposed=
 to prove.
> >=20
> > In this thread I am trying to point out why the proof doesn=E2=80=99t w=
ork. For a=20
> > start, in general, the diagonal construction never converges to an answ=
er.
>=20
> Which is more likely, that you've found a flaw in a proof that's
> been accepted by mathematicians for over a century, or that you've
> reached an incorrect conclusion?
>=20
> There's nothing wrong with trying to find flaws in established
> proofs.=C2=A0 It can be a great way to understand the proof more deeply.
> But please consider the possibility that you're mistaken and everyone
> else is right.
>=20
> Cantor's construction proves that, given a list of all real numbers,
> there is a number that is not in the list -- and therefore, by
> contradiction, that no such list is possible.
>=20
> https://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument
>=20

IMO, Cantor Diagonal Proof (along with Cantor set theory) is a trick of mag=
ic for convenience.

.... Snippet from https://sourceforge.net/projects/cscall/files/MisFiles/Rea=
lNumber2-en.txt/download
Theorem 5: The set of elements composed of finite discrete symbols and the =
set
         of elements composed of infinite discrete symbols cannot form 1-1
         correspond.
  Proof: According to the meaning of finite/infinite, during the correspond=
ence
         process, the elements of the finite set will be exhausted (accordi=
ng to
         the definition), while the elements of the infinite set won't.

Theorem 6: There is no 1-1 correspondence procedure between the sets =E2=84=
=9D and =E2=84=95.
  Proof: It can be proved according to Theorem 5.
.....=C2=A0

The wording looks not good, but you should get the idea.
All is that simple. Just infinite and finite.