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From: Mikko <mikko.levanto@iki.fi>
Newsgroups: sci.logic
Subject: Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary)
Date: Mon, 25 Nov 2024 10:43:32 +0200
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On 2024-11-24 14:01:15 +0000, WM said:

> On 24.11.2024 13:38, Mikko wrote:
>> On 2024-11-23 08:49:18 +0000, WM said:
>> 
>>>>> It is relevant by three reasons:
>>>>> 1) The limit of the sequence f(n) of relative coverings in (0, n] is 
>>>>> 1/10, not 1. Therefore the relative covering 1 would contradict 
>>>>> analysis.
>>>>> 2) Since for all intervals (0, n] the relative covering is 1/10, the 
>>>>> additional blackies must be taken from the nowhere.
>>>>> 3) Since a shifted blacky leaves a white unit interval where it has 
>>>>> left, the white must remain such that the whole real axis can never 
>>>>> become black.
>>>> 
>>>> You say that it is relevant but you don't show how that is relevant
>>>> to the fact that there is no real number between the intervals (n/2, n/2+1)
>>>> that is not a part of at least one of those intervals.
>>> 
>>> Because that has nothing to do with the topic under discussion. See 
>>> points 1, 2, and 3. They are to be discussed.
>> 
>> The subject line specifies that the discussion should be about Cantor's
>> enumeration of the rational numbers.
>> 
>> OP specifies that the discussion shall be baout the sequence of
>> itnrevals
> 
> That is a mistake. Should read:
> [q_n - ε*sqrt(2)/2^n, q_n + ε*sqrt(2)/2^n].

OK but the following applies to that, too:

>> The 1, 2, and 3 above are not relevant to the topic sepcified by the
>> subject line and OP.

> My last example contradicts a simpler bijection, namely that between 
> all natural numbers and all natural numbers divisible by 10: Let every 
> unit interval on the real axis after a number 10n carry a black hat. 
> Then it should be possible to cover all intervals with black hats.

What does "contradicts" in "contradicts a simpler bijection"?

-- 
Mikko