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

On 11/08/2024 08:55 AM, Ross Finlayson wrote:

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.)
>

That's part of usually what's called "measure theory",
and even today physicists are quite agog about "the measure problem".
When Vitali showed his geometric and analytic result,
then set theory arrived at "non-measurable" or "un-measurable".
Yet, it's still good, and, shows that it's a real thing
that doubling and halving measures and spaces are mathematical.
So, the quantum theorists have got plenty problems
about what's continuous again. ("The measure problem.")
Then as well, this of course gets into the multiple law(s)
of large numbers, what's called "non-standard" probability
theory, analysis, models of integers, and so on.
It's quite sensible with an approach like mine with
the three definitions of continuous domains the line-reals,
field-reals, and signal-reals, and the doubling line-reals
and halving signal-reals, after of course an apologetics in
the mathematical foundations to explain the slate of
uncountability and logical paradoxes so that there are none,
my "slates", that I've written since a decade and decades.
So, it probably helps if you know Riemann and Lebesgue
and Cauchy and Seidel and all these things after Weierstrass
that Dirichlet and Poincare make all out as with regards
to Vitali and Hausdorff and the "re-Vitali-ization",
measure theory.