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From: WM <wolfgang.mueckenheim@tha.de>
Newsgroups: sci.math
Subject: Re: Incompleteness of Cantor's enumeration of the rational numbers
Date: Mon, 4 Nov 2024 18:32:10 +0100
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On 04.11.2024 15:37, Jim Burns wrote:
> On 11/4/2024 6:26 AM, WM wrote:
>> On 03.11.2024 23:18, Jim Burns wrote:
>>> On 11/3/2024 3:38 AM, WM wrote:
> 
>>>> Further there are never
>>>> two irrational numbers
>>>> without a rational number between them. 
> 
>>>> (Even the existence of neighbouring intervals
>>>> is problematic.) 
>>>
>>> There aren't any neighboring intervals.
>>> Any two intervals have intervals between them.
>>
>> That is wrong in geometry.
>> The measure outside of the intervals is infinite.
>> Hence there exists at least one point outside.
>> This point has two nearest intervals
> 
> This point,
> which is on the boundary of two intervals,
> is not two irrational points.

You are wrong. The intervals together cover a length of less than 3. The 
whole length is infinite. Therefore there is plenty of space for a point 
not in contact with any interval.
> 
> Further there are never
> two irrational numbers
> without an interval between them.

Not in reality. But in the used model.
The rationals are dense but the intervals are not. This proves that the 
rationals are not countable.

Regards, WM
> 
>