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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 22:06:39 +0100
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On 07.11.2024 21:32, Jim Burns wrote:
> On 11/7/2024 12:59 PM, WM wrote:

>> When we cover the real axis by intervals
>> --------_1_--------_2_--------_3_--------_4_--------_5_--------_...
>> J(n) = [n - √2/10, n + √2/10]
>> and shuffle them in a clever way,
>> then all rational numbers are midpoints of intervals
>> and no irrational number is outside of all intervals.
> 
> No irrational is not in contact with
> the union of intervals.

On the contrary, every irrational (and every rational) is in contact 
with infinitely many intervals, because in the length of √2/5 there are 
infinitely many rationals and therefore infinitely many midpoints of 
intervals.

> The first is true because,
> for each irrational x,
> each interval of which x is in its interior
> holds rationals, and
> rationals are points in the union of intervals.
> 
> There is an enumeration of ℚ⁺

If it is true, then the measure of the intervals of 3/10 of the real 
axis grows to infinitely many times the real axis.

> The infinite sum of measures = 2³ᐟ²/9
> d is _not in contact with_ each interval.

The intervals are shifted such that every rational number is the 
midpoint of an interval. Every point p on the real axis is covered by 
infinitely many intervals, namely by all intervals having midpoint 
rationals q with
d - √2/10 < q < d + √2/10.
> 
> However,
> each interval of which d is in its interior
> holds rationals, which are points in ⋃(ε.cover)
> Thus, d is _in contact with_ ⋃(ε.cover)
> but not with any of its intervals.

The intervals have length √2/5. There is no ε.
> 
>> Do you believe this???
> 
> Don't you believe this???

No, it violates mathematics and logic when by reordering the measure of 
a set of disjunct intervals grows.

Regards, WM