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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: Wed, 27 Nov 2024 12:12:54 +0100
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On 27.11.2024 10:52, Mikko wrote:
> On 2024-11-26 11:05:30 +0000, WM said:
> 
>> On 26.11.2024 10:08, Mikko wrote:
>>> On 2024-11-25 13:55:57 +0000, WM said:
>>
>>>> But before touching a rational it will touch an irrational.
>>>
>>> Of course as the starting point is outside of all the intervals and
>>> every rational is in some of the intervals and therefore must be
>>> irrational. But when it has moved to another point it has already
>>> moved over both infinitely many irrationals
>>
>> This is true in every case. The intermediate numbers cannot be 
>> discerned. They are dark. This is so in fact between every pair of 
>> discernible real numbers: There are infinitely many dark numbers 
>> between them.
> 
> Some of the intermediate numbers can be expressed with a finite string.

But most cannot.

> In particular, every rational number can.

No. For every unit fraction there exist infinitely many smaller unit 
fractions, infinitely many of which cannot be expressed. They remain 
simply to be smaller.

Regards, WM