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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: Tue, 17 Dec 2024 15:08:43 +0200
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On 2024-12-16 11:04:17 +0000, WM said:

> On 16.12.2024 10:51, Mikko wrote:
>> On 2024-12-16 08:55:39 +0000, WM said:
>> 
>>> On 15.12.2024 22:14, Richard Damon wrote:
>>>> On 12/15/24 2:29 PM, WM wrote:
>>> 
>>>>> Next is a geometric property, in particular since the average distance 
>>>>> of intervals is infinitely larger than their sizes.
>>> 
>>>> Not sure where you get that the "average" distance of intervals is 
>>>> infinitely larger than ther sizes.
>>> 
>>> The accumulated size of all intervals is less than 3 over the infinite length.
>> 
>> True.
>> 
>>> Hence
>> 
>> False.
>> 
>>> there is at least one location with a ratio oo between distance to the 
>>> interval and length of the interval.
>> 
>> 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. That's what "dense" means.

-- 
Mikko