Path: 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: Fri, 11 Apr 2025 13:24:48 +0100
Organization: Fix this later
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On 11/04/2025 13:11, wij wrote:
> On Fri, 2025-04-11 at 12:04 +0000, Mr Flibble wrote:
>> On Fri, 11 Apr 2025 19:52:20 +0800, wij wrote:
>>
>>> On Fri, 2025-04-11 at 04:21 -0700, Keith Thompson wrote:
>>>> wij <wyniijj5@gmail.com> writes:
>>>>> On Thu, 2025-04-10 at 17:23 -0700, Keith Thompson wrote:
>>>>>> wij <wyniijj5@gmail.com> writes:
>>>>>> [...]
>>>>>>> "lim(x->c) f(x)=L" means the limit of f approaching c is L, not
>>>>>>> f(c)=L 'eventually'.
>>>>>>> f at c is not defined (handled) in limit.
>>>>>>
>>>>>> Correct.
>>>>>>
>>>>>>> lim 0.333...=1/3    ... The *limit* is 1/3, not 0.333...=1/3
>>>>>>> 0.3+0.33+0.333+...  ... The sequence converges to 1/3 Σ(n=1,∞)
>>>>>>> 3/10^n     ... The sum converges to 1/3 (or you can use lim)
>>>>>>
>>>>>> The limit as the number of 3s increases without bound *is exactly
>>>>>> what we mean* by the notation "0.333...".  Once you understand
>>>>>> that, it's obvious that 0.333... is exactly equal to 1/3, and that
>>>>>> 0.333... is a rational number.
>>>>>
>>>>> You agree "f at c is not defined (handled) in limit", yet, on the
>>>>> other hand ASSERTING 0.333... is 'exactly' 1/3 from limit? Are you
>>>>> nut?
>>>>>
>>>>> As usual, you need to prove what you say. Or you are just showing
>>>>> yourself another olcott, just blink belief, nothing else.
>>>>
>>>> Keep the insults to yourself.  Last warning.
>>>
>>> I still think 'nut' is a common word, at least a terse word for people
>>> saying one thing and doing the other (or a liar more appropriate?)
>>>
>>>> My assertion is simply about what the "..." notation means.
>>>>
>>>> Do you agree that the limit of 0.3, 0.33, 0.333, as the number of 3s
>>>> increases without bound, is exactly 1/3?  (You said so above.)
>>>
>>> Increases without bound -> yes is exactly 1/3 -> no such logic
>>>
>>>> What exactly do you think the notation "0.333..." means?  I and many
>>>> others use that notation to mean the limit, which you agree is exactly
>>>> 1/3.
>>>
>>> Is this a lie? I have always consistently claiming "repeating decimals
>>> are irrational".
>>
>> The decimals only repeat in certain bases:
> 
> Agree

So is it your claim that 1/3 is irrational when represented in 
base 2, rational in base 3, irrational in base 4...? Seriously?

Numbers are what they are regardless of how they are expressed. 
1/3 is a ratio of two integers and is therefore rational by 
definition. The decimal expansion of 1/3 is 0.3r. Therefore, 0.3r 
is rational.

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