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Path: ...!eternal-september.org!feeder3.eternal-september.org!news.eternal-september.org!.POSTED!not-for-mail
From: Mikko <mikko.levanto@iki.fi>
Newsgroups: sci.logic
Subject: Re: Incompleteness of Cantor's enumeration of the rational numbers
Date: Mon, 16 Dec 2024 11:55:10 +0200
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On 2024-12-15 11:08:45 +0000, WM said:

> On 15.12.2024 11:50, Mikko wrote:
>> On 2024-12-14 08:42:37 +0000, WM said:
>> 
>>> On 14.12.2024 09:30, Mikko wrote:
>>>> On 2024-12-13 10:28:44 +0000, WM said:
>>>> 
>>>>> On 13.12.2024 10:46, Mikko wrote:
>>>>> 
>>>>>> Between any two intervals there is space and that space contains other
>>>>>> intervals.
>>>>> 
>>>>> No. Starting from a point in the complement the cursor will hit a first 
>>>>> interval. This is true for all visible intervals.
>>>> 
>>>> False. From a point that is not a part of an interval no interval is the
>>>> nearest one because another interval is nearer.
>>> 
>>> IF ALL intervals and their endpoints are existing as invariable points 
>>> on the real line this cannot happen.
>> 
>> It can. Your { [q_n - sqrt(2)/2^n, q_n + sqrt(2)/2^n] | n = 1, 2, 3, ... }
>> is one such set.
> 
> That is nothing but an unfounded claim. In actual infinity of set 
> theory all intervals and their endpoints are existing as invariable 
> points from the beginning of the cursor's motion.

False. If that were true you could give an example of a possible starting
point and the first interval the cursor hits. But you can't.

-- 
Mikko