Path: news.eternal-september.org!eternal-september.org!.POSTED!not-for-mail
From: wij <wyniijj5@gmail.com>
Newsgroups: comp.theory
Subject: Re: Cantor Diagonal Proof
Date: Sat, 12 Apr 2025 01:35:08 +0800
Organization: A noiseless patient Spider
Lines: 207
Message-ID: <213bd514554b2c2277c389f45afe874757e3b6cb.camel@gmail.com>
References: <vt3dg5$1qj4p$1@dont-email.me> <vt61cc$putp$1@dont-email.me>
	 <a3088f983cc8deed93d9cef50aaaaeb0f0be0aa3.camel@gmail.com>
	 <vt67eu$10han$2@dont-email.me>
	 <ebc8d3cda53aa225977faf7bd5e209c23a19c27f.camel@gmail.com>
	 <vt69ln$10han$3@dont-email.me>
	 <3e5a55b834962635ca7ecf428d074fba771a07f8.camel@gmail.com>
	 <vt6c5b$10han$4@dont-email.me>
	 <ff91dc05893d54c73ff17c4b4ecf1b18d0554084.camel@gmail.com>
	 <878qo74kbl.fsf@nosuchdomain.example.com>
	 <b6d3a579ffa0cb0f197e7972d984f5134c1ef466.camel@gmail.com>
	 <vtaolr$2cq5$1@news.muc.de>
	 <aa4c50cc28f5d0a6e3ac1d33b89b6a8e2cc0005b.camel@gmail.com>
	 <9344a1a25b0c3859ac75c481222d8e13082426f3@i2pn2.org>
	 <7657577c486deca73d3bc371e70c4d5d1455f606.camel@gmail.com>
	 <e481c06316cef04b40329e23ba6a972fbe282893@i2pn2.org>
MIME-Version: 1.0
Content-Type: text/plain; charset="UTF-8"
Content-Transfer-Encoding: quoted-printable
Injection-Date: Fri, 11 Apr 2025 19:35:10 +0200 (CEST)
Injection-Info: dont-email.me; posting-host="05950bb31754b8261a44aa641f85531a";
	logging-data="2307992"; mail-complaints-to="abuse@eternal-september.org";	posting-account="U2FsdGVkX190yTjfg9yERxho+lGXYZNm"
User-Agent: Evolution 3.54.3 (3.54.3-1.fc41)
Cancel-Lock: sha1:gagPqhN2XsDZYGVGzF5Gw525ZQc=
In-Reply-To: <e481c06316cef04b40329e23ba6a972fbe282893@i2pn2.org>

On Fri, 2025-04-11 at 12:42 -0400, Richard Damon wrote:
> On 4/11/25 11:43 AM, wij wrote:
> > 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, not
> > > > > > > > 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=
, not 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 exa=
ctly what
> > > > > > > we mean* by the notation "0.333...".=C2=A0 Once you understan=
d that, 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 t=
he other 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 a=
re nuts,
> > > > > making unfounded claims about mathematics without having studied =
it.
> > > > >=20
> > > > > The sentence ....
> > > > > > > The limit as the number of 3s increases without bound *is exa=
ctly 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 showi=
ng yourself
> > > > > > 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, eve=
ry day, 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 mathematic=
s.
> > > >=20
> > > > I am not interesting to blind beliefs.
> > > > As I may guess from your posts, your knowledge is essentially 'what=
 people 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 s=
ide of logical proof.
> > > > They may evolve/change from errors. It is not a static thing and no=
t the source of fact.
> > > >=20
> > > > To save garbage talks, provide your logical proof (as usual, I beli=
eve NONE).
> > > >=20
> > > >=20
> > >=20
> > > Remmeber, the claim is that 0.33333... is 1/3 in the limit, i.e. that
> > > for any possible epsilon, no matter how small, but still positive, th=
ere
> > > is a point in the sequence of generatation of 0.3333... that all poin=
ts
> > > 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 (th=
e
> > > largest integer that is less than or equal to the value), negate it, =
and
> > > 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
> > > 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
> > > less than 0.0004 which is less than 0.001, and adding more 3s to the
> > > number just makes us closer.
> >=20
> > Firstly, we are now talking about limit, nothing to do with "repeating =
decimals
> > are irrational".
>=20
> But the limit is what DEFINES what a repeating decimal represents, at=20
> least within the normal Real Number System.
>=20
> > Your statement above is sloppy, cannot be verified or refuted. It conta=
ins too
> > many concepts to be defined. So, you just jump to the conclusion (or as=
sertion)
> > you like. So, no valid proof is taken.
>=20
> Of course it can be verified and thus refuted if it was wrong.
>=20
> What concepts used were not defined in normal mathematics?
>=20
> >=20
> > limit only says the *limit* is 1/3, all others are your wishes (or most=
 people,
> > '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 for=
ever.
>=20
> Right, and the question is, what value does that sequence, when taken=20
> "to the end" become. Since we can't actually do the infinite operation=
=20
> to the end, we define it "in the limit".
>=20
> Of course, in some hyper-mathematics which uses trans-finite values,=20
> like the infintesimals, we might be able to come up with other=20
> definitions, but then we are not working with what are called the "Real=
=20
> Numbers", but some Hyper-Real number system, which you claim not to be=
=20
> doing,
>=20
> > 2. No one disagree that we can choose an arbitrary epsolon/delta whatev=
er,
> > =C2=A0=C2=A0=C2=A0 to make the error arbitrarily close to the limit (i.=
e L or 1/3).
>=20
> Which means that the limit of the sequence, which is the definition of=
=20
> what that notation means, is defined and found.
>=20
> > So, don't make implications that I disagrees with these basics. (your a=
re just
> > slight, others may even imply I claim 1/3 is irrational. Smear as proof=
?).
>=20
> But if the value of 0.333... but the definition of the representation=20
> *IS* 1/3, and you claim that 0.333... is irrational, you are claiming=20
> that 1/3 is irrational. That or=C2=A0 your logic doesn't support the axio=
m of=20
> equivalence (if A =3D B and B =3D C, thus A =3D C)
>=20
> >=20
> > In logic language and point of view, the premise (i.e. the sequence 0.3=
, 0.33...)
> > does not contain 1/3 (also whatever epsilon/delta you like), therefore,
> > no possibility a valid logical proof can yield the conclusion 1/3 (QED)=
..
>=20
> But no one says that the series contains the limit, just that the=20
> "value" of the series is that limit.
========== REMAINDER OF ARTICLE TRUNCATED ==========