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From: wij <wyniijj5@gmail.com>
Newsgroups: comp.theory
Subject: Re: Cantor Diagonal Proof
Date: Thu, 10 Apr 2025 19:35:38 +0800
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On Wed, 2025-04-09 at 18:52 +0100, Richard Heathfield wrote:
> On 09/04/2025 18:30, wij wrote:
> > On Wed, 2025-04-09 at 18:10 +0100, Richard Heathfield wrote:
> > > On 09/04/2025 17:45, wij wrote:
> > >=20
> > > <snip>
> > >=20
> > > > Stick to the problem. Such puzzle won't prove the "0.999... problem=
".
> > >=20
> > > You know what? You're right. 1/3 is irrational, 1 is infinite, 3
> > > is unnatural, and Achilles never did catch the tortoise. I don't
> > > know what I was thinking, claiming that rationals are rational!
> > > Clearly the arithmetic I learned at school was deeply flawed, and
> > > I was naif to think it could be trusted. How happy I am, now that
> > > I have learned the truth from some faceless bloke on the
> > > Internet. Well done you, eh? 1/3 is irrational; who knew?
> > >=20
> >=20
> > 1/3 is rational "by definition".
>=20
> I know.
>=20
> > 0.333... (repeating decimal) is irrational.
>=20
> If you say so, that's good enough for me. I'll let my old school=20
> know. They'll be appalled to hear they've been teaching it wrong=20
> all these years.
>=20
> > Achilles puzzle gives us two models. One can catch the tortoise, one
> > cannot. Since we believe Achilles can, so it make? people to believe
> > 0.99.... will finish (false from the other model. Also the term
> > 'repeating' implies (define) never terminate)
>=20
> You make it sound like Wij's Ant is busily calculating all the=20
> digits. Look up "pattern" when you get a second and give the ant=20
> a rest.
>=20
> Achilles's race gives us /one/ model with /one/ solution that we=20
> can reach two different ways. We can use the equations of motion=20
> or we can use the sum of an infinite series. Both ways give the=20
> same answer - the precise moment when Achilles overtakes the=20
> tortoise and proves that infinite series can converge despite=20
> your objection.

"lim(x->c) f(x)=3DL" means the limit of f approaching c is L, not f(c)=3DL =
'eventually'.
f at c is not defined (handled) in limit.

lim 0.333...=3D1/3    ... The *limit* is 1/3, not 0.333...=3D1/3
0.3+0.33+0.333+...  ... The sequence converges to 1/3
=CE=A3(n=3D1,=E2=88=9E) 3/10^n     ... The sum converges to 1/3 (or you can=
 use lim)

lim =CE=A3(n=3D1,=E2=88=9E) 3/10^n =3D 1/3  ... use lim
=CE=A3(n=3D1,=E2=88=9E) 3/10^n =E2=89=92 1/3      ... use '=E2=89=92' ('=3D=
' can be confusing)
 =20
'=E2=88=9E' is (or can be) merely a symbol like the unit of weight/length,.=
..., then,
everything is simple and clear.

Symbol '=E2=88=9E'::=3D A never-ending counting loop procedure with the fol=
lowing
              algebraic rules:
         1. =E2=88=80n=E2=88=88=E2=84=95, n<=E2=88=9E
         2. The multiplicative inverse of =E2=88=9E is 1/=E2=88=9E, and the=
 additive inverse of
            =E2=88=9E is -=E2=88=9E

         Note: 'never-ending counting loop procedure' is to be defined, but=
, as
               long as algebraic principles are followed, '=E2=88=9E' is co=
mputationally
               safe.

=E2=84=9D contains infinity naturally (method of exhaustion is everywhere).=
 You don't=20
need all sort of NSA, Cantor's cardinality theory to handle infinity, becau=
se=20
they are at most (if consistent) subset of =E2=84=9D.

> But hey! Maybe the equations of motion are wrong too. Why stop at=20
> maths when you can tear down physics at the same time?

Yes, the world is a strange thing, every logical is possible.