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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: Sun, 24 Nov 2024 14:56:42 +0200
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On 2024-11-22 11:05:58 +0000, WM said:

> On 22.11.2024 09:52, Mikko wrote:
>> On 2024-11-21 19:22:39 +0000, WM said:
> 
>>>> True but irrelevant because it may be even closer to the end of
>>>> another interval.
>>> 
>>> Every end of any interval is irrational.
>> 
>> Irrelevant to the claim that a point outside of all intervals has two
>> nearest intervals
> 
> What else should it have?

For every point that is neither an endpoint nor interior point of any
of the intervals there is no nearest interval because between that point
and any interval there is another interval. That is true whether the
intervals have rational or irrational endpoints. Therefore the above
statement that every end of any interval is irrational is irrelevant.

>> and also to the claim that there is a point outside
>> of all intervals
> 
> That is proven by the measure of the intervals being less than 3.

Which proof method proves the same whether the the endpoints are
rational or irrational. Therefore the above statement that every
end of any interval is irrational is irrelevant.

>> and also to the topic specified on the subjec line.
> 
> That is concerned by the fact that in the complement of measure oo - 3 
> rational numbers must exist.

As every rational number is the midpoint of one of the intervals it is
obvious that no rational is outside of all intervals. Whether the end
points are rational or irrational is irrelevant.

>> All those claims are true for some sets of intervals and false for others.
> 
> Let me know what else than an irrational number could be next to a 
> point in the complement, if there were no rational numbers.

No point is next to any other point because there are other points between
the two.

-- 
Mikko