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From: wij <wyniijj5@gmail.com>
Newsgroups: comp.theory
Subject: Re: Another proof: The Halting Problem Is Undecidable.
Date: Mon, 14 Oct 2024 08:40:37 +0800
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On Sun, 2024-10-13 at 22:01 +0100, Ben Bacarisse wrote:
> wij <wyniijj5@gmail.com> writes:
>=20
> > If 0.999..=3D1, you have to explain your arithmetic system.
>=20
> Almost.=C2=A0 First you have to explain the notation.=C2=A0 That's easy (=
but
> relatively advanced) as 0.999... denotes the limit of a sequence of
> partial sums (sometimes called a series limit).=C2=A0 The arithmetic syst=
em
> (the reals, where all such sums converge) comes after saying what the
> ... denotes.
>=20
> When *you* say that 0.999... =3D/=3D 1 you always avoid saying what the
> notation (specifically the ...) means in formal terms.
>=20

What I would say now is probably not different from
https://sourceforge.net/projects/cscall/files/MisFiles/RealNumber2-en.txt/d=
ownload=20

"..." conventionally means "so on,..,etc.", likely infinitely. I use it lik=
e that.

As indicated in the above file:
"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."

(did your formal system say that? I guess no)

That means whatever a formal system can do is limited. Almost everything=
=20
formally stated is impossible.

Take Archimedes' Axiom for example: It did not mention anything about infin=
ity.
How can it *formally* deny the existence of infinity/infinitesimal? And, th=
e
Archimedes' Axiom argument is likely suffering from circular reasoning as a=
n
axiom. It needs to say what the R it is addressing. (The Axiom is new, it l=
acks
the source, and likely a 'popular' rumor to me if it is also 'formal'. E.g.
I just saw another one https://proofwiki.org/wiki/Axiom_of_Archimedes
=E2=88=80x=E2=88=88=E2=84=9D: =E2=88=83n=E2=88=88=E2=84=95, n>x
The problem is what exactly had Archimedes said?=C2=A0).