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

On 11/11/2024 12:09 PM, Jim Burns wrote:

On 11/11/2024 2:04 PM, Ross Finlayson wrote:

On 11/11/2024 11:00 AM, Ross Finlayson wrote:

On 11/11/2024 10:38 AM, Jim Burns wrote:

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Our sets do not change.
Everybody who believes that
  intervals could grow in length or number
is deeply mistaken about
  what our whole project is.

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How about Banach-Tarski equi-decomposability?

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The parts do not change.
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(We had a great long thread over on sci.logic
about Banach-Tarski and Vitali-Hausdorff, there's
quite a bit about the historical and technical arrival,
including references and links to Hausdorff's original.
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Vitali's doubling-space reflects on "Zeno's graduation
course", where Zeno also has a doubling-space or
doubling-measure argument, since about 2300 years ago.
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These are considered part of "mathematics", if
your project is wider than "bumbing- or dumbing-down W.M.".)

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My project is potentially (dare I say it?) much wider
than explanations that sets do not change.
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But what do I accomplish by talking about more,
if I say one thing, and a different thing is heard?
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Well, you see, the story goes, that a mathematical ball,
with a unit volume, is according to measure, that the
theory of measure, indicates that any manner of partitioning
said ball or its decomposition, would result in whatever
re-composition, a volume, the same.
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Yet, according to the theory, the Banach-Tarski "equi-decomposability"
arrives instead at making two identical copies.
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This is quite after Hausdorff established this geometrically,
and quite after Vitali made a unit line segment, partitioned,
by "not-equi-decomposed", re-composed, to a length L where 1 < L < 3,
or that exactly, L = 2, which is exactly twice, double, the L = 1.
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Then, it's said that von Neumann in one of his phases
established quite a variety of similar results on the plane.
Though, I imagine he had his influences or who really
came up with these ideas.
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Yet, it's to Vitali to whom this great and important aspect
of mathematics is assigned, that broke standard measure theory
and made "non-measurable" out of things.
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While that is so, we can point at "Zeno's graduation course"
or the many runners doing wind-sprints on the field, and
how it's so that he shows it add up to twice, when otherwise
these days the bumblebee flying back and forth between
the oncoming trains is simply derived the distance from the time.
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So, "mathematics" is rather an inclusive endeavor,
and it includes these things.
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Then, given that WM is a nuisance soft-ball straw-man,
then, what are we to make of this observation that
"intervals could grow in length", under what transformations?
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It's called doubling-spaces and doubling-measure and
with regards to halving-spaces and halving-measure,
and it's a feature of continuum and, as it's called,
infinitesimal analysis.
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The, ..., "re-Vitali-ization", or, "quasi-invariant",
measure theory, ....
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It's a thing, ....
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Look back on the discussion on sci.logic "Banach-Tarski",
you'll find learnings.
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