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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: Thu, 7 Nov 2024 09:46:05 +0100
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On 06.11.2024 21:20, Jim Burns wrote:
> On 11/6/2024 2:50 PM, WM wrote:

>> From every positive point we know that
>> it is not 0 and
>> not in contact with (-oo, 0].
>> Same for every point not in an interval.
> 
> Is 0 "not in contact with" [-1,0) ⊆ ℝ

0 is not a positive point.
> A point can be not.in the closure of each
> of infinitely.many sets
> and also in the closure of their union.

These ideas are irrelevant because we can use the following estimation 
that should convince everyone:

Use the intervals I(n) = [n - sqrt(2)/2^n, n + sqrt(2)/2^n]. Since n and 
q_n can be in bijection, these intervals are sufficient to cover all 
q_n. That means by clever reordering them you can cover the whole 
positive axis except "boundaries".

And an even more suggestive approximation:
Replace the I(n) by intervals J(n) = [n - 1/10, n + 1/10] (as soon as 
the I(n) are smaller than 2/10).
These intervals (without splitting or modifying them) can be reordered, 
to cover the whole positive axis except boundaries.
Reordering them again in an even cleverer way, they can be used to cover 
the whole positive and negative real axes except boundaries. And 
reordering them again, they can be used to cover 100 real axes in parallel.

Even using only intervals J(P) = [p - 1/10, p + 1/10] where p is a prime 
number can accomplish the same.

Is this the power of infinity?
Or is it only the inertia of brains conquered by matheology?

Regards, WM