Path: ...!eternal-september.org!feeder3.eternal-september.org!news.eternal-september.org!eternal-september.org!.POSTED!not-for-mail
From: Richard Heathfield <rjh@cpax.org.uk>
Newsgroups: comp.theory
Subject: Re: Cantor Diagonal Proof
Date: Wed, 9 Apr 2025 15:48:44 +0100
Organization: Fix this later
Lines: 79
Message-ID: <vt61cc$putp$1@dont-email.me>
References: <vt3dg5$1qj4p$1@dont-email.me> <vt3eme$2bi5g$2@dont-email.me>
 <vt3qqn$1qj4q$1@dont-email.me>
 <1ab7fe6b234496769adde06995790eebb827756e.camel@gmail.com>
 <vt5qac$j4kv$1@dont-email.me>
 <60cbb326c7d65b1bbd9451319bd07721c76d307f.camel@gmail.com>
MIME-Version: 1.0
Content-Type: text/plain; charset=UTF-8; format=flowed
Content-Transfer-Encoding: 8bit
Injection-Date: Wed, 09 Apr 2025 16:48:46 +0200 (CEST)
Injection-Info: dont-email.me; posting-host="a0bfded38ff50eae03c6cf4454ca917e";
	logging-data="850873"; mail-complaints-to="abuse@eternal-september.org";	posting-account="U2FsdGVkX19AvX89QgsnarsEK0oRioX/OJDoI3KbcicU0Os0uAJbng=="
User-Agent: Mozilla Thunderbird
Cancel-Lock: sha1:Vq7fYfX9aMj3WycaLYDus7JG4Jw=
Content-Language: en-GB
In-Reply-To: <60cbb326c7d65b1bbd9451319bd07721c76d307f.camel@gmail.com>
Bytes: 5018

On 09/04/2025 15:31, wij wrote:
> On Wed, 2025-04-09 at 13:48 +0100, Richard Heathfield wrote:
>> On 09/04/2025 13:25, wij wrote:
>>> On Tue, 2025-04-08 at 19:44 +0100, Andy Walker wrote:
>>>> On 08/04/2025 16:17, Richard Heathfield wrote:
>>>>> It will, however, take me some extraordinarily convincing
>>>>> mathematics before I'll be ready to accept that 1/3 is irrational.
>>>>
>>>> 	I don't think that's quite what Wij is claiming.  He thinks,
>>>> rather, that 0.333... is different from 1/3.  No matter how far you
>>>> pursue that sequence, you have a number that is slightly less than
>>>> 1/3.  In real analysis, the limit is 1/3 exactly.  In Wij-analysis,
>>>> limits don't exist [as I understand it], because he doesn't accept
>>>> that there are no infinitesimals.  It's like those who dispute that
>>>> 0.999... == 1 [exactly], and when challenged to produce a number
>>>> between 0.999... and 1, produce 0.999...5.  They have a point, as
>>>> the Archimedean axiom is not one of the things that gets mentioned
>>>> much at school or in many undergrad courses, and it seems like an
>>>> arbitrary and unnecessary addition to the rules.  But we have no good
>>>> and widely-known notation for what can follow a "...", so the Wijs of
>>>> this world get mocked.  He doesn't help himself by refusing to learn
>>>> about the existing non-standard systems.
>>>
>>> Lots of excuses like POOH. You cannot hide the fact that you don't have a
>>> valid proof in those kinds of argument.
>>> If you propose a proof, be sure you checked against the file I provided.
>>> I have no no time for garbage talk.
>>
>> I have read that document, about which I have a simple question.
>>
>>   From Theorem 2 and Axiom 2, if x can be expressed in the form of
>> p/q, then p and q will be infinite numbers (non-natural numbers).
>> Therefore, x is not a rational number. And since a non-rational
>> number is an irrational number, the proposition is proved.
>>
>> Let p = 1
>> Let q = 3
>>
>> Is it or is it not your contention that p and q are "infinite"
>> (non-natural) numbers?
> 
> The audience of the file was originally intended to include 12 years old kids.
> Wordings in the file wont' be precise enough to meet rigorous requirements.
> The mentioned paragraph was revised (along with several others):
> 
> Theorem 2: ℚ+ℚ=ℚ (the sum of a rational number and a rational number is still a
>          rational number), but it is only true for finite addition steps.
>    Proof: Let Q'={p/q| p,q∈ℕ, q≠0 and p/q>0}, then Q'⊂ℚ. Since the sum of any two
>           terms in Q' is greater than the individual terms, the sum q of the
>           infinite terms (q=q₁+q₂+q₃...) is not a fixed number.
> 
> What I intended to mean is: 0.999...= 999.../1000... (in p/q form)
> Since p,q will be infinitely long to denote/define 0.999..., p,q won't be
> natural numbers. Thus, "ℚ+ℚ=ℚ" is conditionally true (so false).
> 
> But I still think your English is worse than olcott's (and mine).

Charmed, I'm sure.

>> Prediction: you will evade the question. Why not surprise me?
> Ok, I evade more clarification.

I deduce from what you intended to mean (and that's very classy 
English, so well done you) that you didn't intend to mean that 1 
and 3 are "infinite".

And you're right. 1 and 3 are both integers. Natural numbers. 
Whole numbers. Finite numbers. Not infinite.

Let us calculate the ratio of these two integers, 1/3. Oh look, 
it's 0.3r. So 0.3r is the ratio of two integers (i.e. rational) 
after all. Quelle surprise!

-- 
Richard Heathfield
Email: rjh at cpax dot org dot uk
"Usenet is a strange place" - dmr 29 July 1999
Sig line 4 vacant - apply within