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From: Richard Damon <richard@damon-family.org>
Newsgroups: sci.logic
Subject: Re: Incompleteness of Cantor's enumeration of the rational numbers
Date: Wed, 6 Nov 2024 19:51:03 -0500
Organization: i2pn2 (i2pn.org)
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On 11/6/24 1:22 PM, WM wrote:
> On 06.11.2024 12:46, Richard Damon wrote:
>> On 11/5/24 7:34 AM, WM wrote:
>>> On 05.11.2024 13:03, Richard Damon wrote:
>>>> On 11/5/24 3:45 AM, WM wrote:
>>>
>>>>> I describe that a point between two finite intervals has two finite 
>>>>> intervals around it. Even this simple conclusion must be denied by 
>>>>> the believers in matheology.
>>>
>>>> If you mean that between to finite length intervals, which include 
>>>> there endpoints, and a point that is between those two intervals, 
>>>> then YES there exist two other intervals between the point and those 
>>>> two intervals.
>>>
>>> These other intervals also have irrational endpoints. Every point 
>>> outside of an interval is next to some endpoint which is irrational.
>>
>> Or rational endpoints.
> 
> No, the rationals are centres of their intervals.

They can also be endpoints of intervals.

>>
>> There is no "next to" on the dense line
> 
> Every positive point is nearer to zero than to any negative point.
> Of -x and 0 the latter is next to any positive x.
> 

But that isn't "next to".

There is no point that is NEXT TO any other point, as if X and Y were 
"next to" each other, then the point (X+Y)/2 would be between them, and 
thus they are not next to each other.

By the rules of arithmetic of Rational and Reals, that expresion is 
ALWAYS a value, and will ALWAYS be between the two numbers.

> Regards, WM
>