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From: Richard Heathfield <rjh@cpax.org.uk>
Newsgroups: comp.theory
Subject: Re: Cantor Diagonal Proof
Date: Wed, 9 Apr 2025 17:32:30 +0100
Organization: Fix this later
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On 09/04/2025 17:08, wij wrote:
> On Wed, 2025-04-09 at 15:48 +0100, Richard Heathfield wrote:
>> 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!
> 
> The correct equality is 1/3= 0.333... + nonzero_remainder.

Keep on dividing the remainder, and what do you get? Oh look! 
More 3s.

> If you use it to prove, that proof never finishes. Thus, invalid.

And Achilles never catches the tortoise. Yeah, right.

-- 
Richard Heathfield
Email: rjh at cpax dot org dot uk
"Usenet is a strange place" - dmr 29 July 1999
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