Re: Incompleteness of Cantor's enumeration of the rational numbers (doubling-spaces)

On 11/08/2024 02:18 AM, WM wrote:

On 08.11.2024 00:29, Jim Burns wrote:
 > On 11/7/2024 2:33 PM, WM wrote:
>
 >> It is impossible however to cover
 >> the real axis (even many times) by
 >> the intervals
 >> J(n) = [n - 1/10, n + 1/10].
 >
 > Those are not the cleverly.re.ordered intervals.
They are the intervals that we start with.
 >> No boundaries are involved because
 >> every interval of length 1/5 contains infinitely many rationals and
 >> therefore is essentially covered by infinitely many intervals of
 >> length 1/5
 >> - if Cantor is right.
 >
 > I haven't claimed anything at all about
 > your all.1/5.length intervals.
Then consider the two only alternatives: Either by reordering (one after
the other or simultaneously) the measure of these intervals can grow
from 1/10 of the real axis to infinitely many times the real axis, or not.
>
My understanding of mathematics and geometry is that reordering cannot
increase the measure (only reduce it by overlapping). This is a basic
axiom which will certainly be agreed to by everybody not conditioned by
matheology. But there is also an analytical proof: Every reordering of
any finite set of intervals does not increase their measure. The limit
of a constant sequence is this constant however.
>
This geometrical consequence of Cantor's theory has, to my knowledge,
never been discussed. By the way I got the idea after a posting of
yours: Each of {...,-3,-2,-1,0,1,2,3,...} is the midpoint of an interval.
>
Regards, WM
>

Perhaps you've never heard of Vitali's doubling-space,
the Vitali and Hausdorff's what became Banach-Tarski
the equi-decomposability, the doubling in signal theory
according to Shannon and Nyquist, and as with regards to
the quasi-invariant measure theory, where: taking a
continuum apart and putting it back together doubles things.
It's part of continuum mechanics and as with regards to infinity.
(Mathematical infinity.)