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From: WM <wolfgang.mueckenheim@tha.de>
Newsgroups: sci.logic
Subject: Re: Incompleteness of Cantor's enumeration of the rational numbers
Date: Tue, 17 Dec 2024 20:29:52 +0100
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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.

> That's what "dense" means.

Yes that is the meaning of "dense". There is no finite distance between 
next points, e.g. rationals. Therefore the intervals are not dense. 
Therefore the intervals do not cover all rational points.

Regards, WM