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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, 28 Nov 2024 13:46:20 +0200
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On 2024-11-27 11:12:54 +0000, WM said:

> 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.

The number 0 can be expressed with a finite string. The successor of an
expresssible number can be expressed. The ratio of two expressible numbers
can be expressed. Nothing else is a rational number.

-- 
Mikko