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From: Mikko <mikko.levanto@iki.fi>
Newsgroups: sci.logic
Subject: Re: Incompleteness of Cantor's enumeration of the rational numbers
Date: Thu, 19 Dec 2024 12:38:39 +0200
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On 2024-12-18 11:25:25 +0000, WM said:

> On 18.12.2024 11:16, Mikko wrote:
>> On 2024-12-17 19:29:52 +0000, WM said:
>> 
>>> On 17.12.2024 14:08, Mikko wrote:
>>>> On 2024-12-16 11:04:17 +0000, WM said:
>>>> 
>>>>>> False. Regardless which interval is "the" interval the distance to that
>>>>>> interval is finite and the length of the interval is non-zero so the
>>>>>> ratio is finite.
>>>>> 
>>>>> Well, it is finite but huge. Much larger than the interval and 
>>>>> therefore the finite intervals are not dense.
>>>> 
>>>> They are dense because there are other intervals between the point and the
>>>> interval.
>>> 
>>> The distance between intervals (in some location) is finite but much, 
>>> much larger than the finite length of the interval. This distance is 
>>> the distance between intervals which are next to each other. Therefore 
>>> there is nothing in between.
>> 
>> There is no next interval and therefore no distance to the next interval
>> as there are always other intervals nearer.
> 
> Mathematics says the covering by intervals is 3/oo. Therefore the ratio 
> between not covered part and covered part of the positive real axis is 
> oo/3. That implies an average of oo/3 which in some locations must be 
> realized.

Yes.

> That proves the existence of distances between next intervals much 
> larger than the finite lengths of the intervals.

No. There is no next interval because there are thore intervals nearer.

> It excludes density of intervals.

No, it does not.

-- 
Mikko