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 18:52:43 +0100
Organization: Fix this later
Lines: 52
Message-ID: <vt6c5b$10han$4@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$1@dont-email.me>
 <60cbb326c7d65b1bbd9451319bd07721c76d307f.camel@gmail.com>
 <vt61cc$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>
MIME-Version: 1.0
Content-Type: text/plain; charset=UTF-8; format=flowed
Content-Transfer-Encoding: 7bit
Injection-Date: Wed, 09 Apr 2025 19:52:44 +0200 (CEST)
Injection-Info: dont-email.me; posting-host="a0bfded38ff50eae03c6cf4454ca917e";
	logging-data="1066327"; mail-complaints-to="abuse@eternal-september.org";	posting-account="U2FsdGVkX191WGqUph/5aXIYVxXhgYQhN/NeCe9fXrLsSMNa5SAJ0Q=="
User-Agent: Mozilla Thunderbird
Cancel-Lock: sha1:u1MNZNa+6U1zu3juJSrJO/5IbdA=
In-Reply-To: <3e5a55b834962635ca7ecf428d074fba771a07f8.camel@gmail.com>
Content-Language: en-GB
Bytes: 3505

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:
>>
>> <snip>
>>
>>> Stick to the problem. Such puzzle won't prove the "0.999... problem".
>>
>> 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?
>>
> 
> 1/3 is rational "by definition".

I know.

> 0.333... (repeating decimal) is irrational.

If you say so, that's good enough for me. I'll let my old school 
know. They'll be appalled to hear they've been teaching it wrong 
all these years.

> 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)

You make it sound like Wij's Ant is busily calculating all the 
digits. Look up "pattern" when you get a second and give the ant 
a rest.

Achilles's race gives us /one/ model with /one/ solution that we 
can reach two different ways. We can use the equations of motion 
or we can use the sum of an infinite series. Both ways give the 
same answer - the precise moment when Achilles overtakes the 
tortoise and proves that infinite series can converge despite 
your objection.

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

-- 
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