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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: Tue, 26 Nov 2024 11:09:23 +0200
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On 2024-11-25 14:38:13 +0000, WM said:

> On 25.11.2024 09:43, Mikko wrote:
>> 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"?
> 
> The simple example contradicts a bijection between the two sets 
> described above.

What does "contradicts a bijection" mean?

-- 
Mikko