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From: wij <wyniijj5@gmail.com>
Newsgroups: comp.theory
Subject: Re: Cantor Diagonal Proof
Date: Fri, 11 Apr 2025 23:43:21 +0800
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On Fri, 2025-04-11 at 09:07 -0400, Richard Damon wrote:
> On 4/11/25 7:32 AM, wij wrote:
> > On Fri, 2025-04-11 at 09:50 +0000, Alan Mackenzie wrote:
> > > wij <wyniijj5@gmail.com> wrote:
> > > > On Thu, 2025-04-10 at 17:23 -0700, Keith Thompson wrote:
> > > > > wij <wyniijj5@gmail.com> writes:
> > > > > [...]
> > > > > > "lim(x->c) f(x)=3DL" means the limit of f approaching c is L, n=
ot
> > > > > > f(c)=3DL 'eventually'.=C2=A0 f at c is not defined (handled) in=
 limit.
> > >=20
> > > > > Correct.
> > >=20
> > > > > > lim 0.333...=3D1/3=C2=A0=C2=A0=C2=A0 ... The *limit* is 1/3, no=
t 0.333...=3D1/3
> > > > > > 0.3+0.33+0.333+...=C2=A0 ... The sequence converges to 1/3
> > > > > > =CE=A3(n=3D1,=E2=88=9E) 3/10^n=C2=A0=C2=A0=C2=A0=C2=A0 ... The =
sum converges to 1/3 (or you can use lim)
> > >=20
> > > > > The limit as the number of 3s increases without bound *is exactly=
 what
> > > > > we mean* by the notation "0.333...".=C2=A0 Once you understand th=
at, it's
> > > > > obvious that 0.333... is exactly equal to 1/3, and that 0.333... =
is a
> > > > > rational number.
> > >=20
> > > > You agree "f at c is not defined (handled) in limit", yet, on the o=
ther hand
> > > > ASSERTING 0.333... is 'exactly' 1/3 from limit? Are you nut?
> > >=20
> > > No, Keith Thompson is simply correct, here.=C2=A0 It is you who are n=
uts,
> > > making unfounded claims about mathematics without having studied it.
> > >=20
> > > The sentence ....
> > > > > The limit as the number of 3s increases without bound *is exactly=
 what
> > > > > we mean* by the notation "0.333...".
> > > .... is entirely correct.
> > >=20
> > > > As usual, you need to prove what you say. Or you are just showing y=
ourself
> > > > another olcott, just blink belief, nothing else.
> > >=20
> > > No, one doesn't need continually to prove standard mathematical
> > > definitions and results.=C2=A0 One could spend the whole day, every d=
ay, doing
> > > nothing else.
> > >=20
> > > It is _you_ who needs to prove your remarkable assertions.=C2=A0 You =
can't, of
> > > course, because they're false.=C2=A0 What you could do, of course, is=
 to show
> > > a bit of respect for those who have studied and learnt mathematics.
> >=20
> > I am not interesting to blind beliefs.
> > As I may guess from your posts, your knowledge is essentially 'what peo=
ple say'
> > without knowing the meaning of words.
> > You may say it is 'standard', 'mainstream'...,etc. But whatever it is, =
simply
> > no logical proof.
> >=20
> > Remind you, the so called 'standard', 'mainstream'=C2=A0is on the side =
of logical proof.
> > They may evolve/change from errors. It is not a static thing and not th=
e source of fact.
> >=20
> > To save garbage talks, provide your logical proof (as usual, I believe =
NONE).
> >=20
> >=20
>=20
> Remmeber, the claim is that 0.33333... is 1/3 in the limit, i.e. that=20
> for any possible epsilon, no matter how small, but still positive, there=
=20
> is a point in the sequence of generatation of 0.3333... that all points=
=20
> after that will be closer to the limit then epsilon.
>=20
> We can compute that point, and thus show the limit is that value.
>=20
> We do that by taking the log base 10 of epsilon, taking its floor (the=
=20
> largest integer that is less than or equal to the value), negate it, and=
=20
> use that many 3's (but at least one if we start with a big epsilon).
>=20
> For instance, an epsilon of 0.001 has a log base 10 of -3, so we say=20
> that all number of the pattern with at least 3 3's are that close.
>=20
> we can show the example as 1/3 - 0.333 will be 0.0003333... which is=20
> less than 0.0004 which is less than 0.001, and adding more 3s to the=20
> number just makes us closer.

Firstly, we are now talking about limit, nothing to do with "repeating deci=
mals
are irrational".
Your statement above is sloppy, cannot be verified or refuted. It contains =
too
many concepts to be defined. So, you just jump to the conclusion (or assert=
ion)
you like. So, no valid proof is taken.

limit only says the *limit* is 1/3, all others are your wishes (or most peo=
ple,
'standard', 'mainstream',... whatever you like, it does not matter).
1. No one disagree that the sequence 0.3,0,33..... 0.3333 can go on forever=
..
2. No one disagree that we can choose an arbitrary epsolon/delta whatever,
   to make the error arbitrarily close to the limit (i.e L or 1/3).
So, don't make implications that I disagrees with these basics. (your are j=
ust
slight, others may even imply I claim 1/3 is irrational. Smear as proof?).

In logic language and point of view, the premise (i.e. the sequence 0.3, 0.=
33...)=C2=A0
does not contain 1/3 (also whatever epsilon/delta you like), therefore,=C2=
=A0
no possibility a valid logical proof can yield the conclusion 1/3 (QED).

> Thus we have the limit, and thus the proof by the definition.
>=20
> It seems your problem is you don't actually believe in the concept of=20
> limit as a rigerous mathematical process,=C2=A0which just means you aren'=
t=20
> following the defintions.