| Sysop: | Amessyroom |
|---|---|
| Location: | Fayetteville, NC |
| Users: | 27 |
| Nodes: | 6 (0 / 6) |
| Uptime: | 38:01:07 |
| Calls: | 631 |
| Calls today: | 2 |
| Files: | 1,187 |
| D/L today: |
22 files (29,767K bytes) |
| Messages: | 173,681 |
On Tuesday, September 4, 2018 at 1:13:06 PM UTC-7, Ross A. Finlayson wrote:
On Tuesday, September 4, 2018 at 8:51:32 AM UTC-7, Ross A. Finlayson wrote: >>> On Monday, September 3, 2018 at 10:41:40 PM UTC-7, peteolcott wrote:
On 9/3/2018 1:03 PM, Ross A. Finlayson wrote:
On Sunday, September 2, 2018 at 6:03:35 AM UTC-7, Peter Percival wrote: >>>>>> peteolcott wrote:
On 8/31/2018 10:04 PM, exflaso.quodlibet@gmail.com wrote:
From your own video, the number of points magically triples
The number of points in the original sphere is equal to the sum of the >>>>>> number of points in the five pieces is equal to the sum of the number of >>>>>> points in the two spheres. That number is 2^aleph_0.
by simply rotating this set of points. That is categorically
impossible. My system of categorically exhaustive (thus
infallible) reasoning caught that mistake.
Remember it's not just "there's a bijective function however
deranged it may be between sets of equal cardinality", but
"using only rigid translations, Vitali splits all the points
on [0,1] and would get a measure of 2 instead of 1, and where
splitting as otherwise summatory or recursive preserves measure".
I boil it down to much simpler than all that.
If you remove a single point from a sphere and
do not put it back it ceases to be a sphere.
Copyright 2018 Pete Olcott
Vitali and Hausdorff and later Banach and Tarski
found some interesting "paradoxical" decompositions
of "the points of the line", "paradoxical" meaning
"established two ways not agreeing". Then the goal
of logic for paradox is to resolve the paradox, to
remove what must be some ambiguity or find what must
be some truth about the objects that it is so.
(Otherwise it's just erroneous not a "paradox".)
So, talking about a "point in the line" or a "point
on the line" is talking about two different things,
two different "models" (or constructed entities) of
what this "unit line segment" is. "Splitting all the
points" is not the same thing as "partitioning a
segment".
I agree that the pointed (punctured) line is not
the same as the line. Then that it's a "removable
discontinuity" here is about Vitali making "removable
discontinuities" all over one "continuous domain",
that then the resulting "measure" in terms of a
measure and a "measure 1.0" in particular, has that
the measure of a resulting translation of the
"removable discontinuities" preserves the measure.
This then is usually that "these sets are non-measurable",
or instead here that they illustrate features of points
"in the line" (i.e., "all" the points) and "on the line"
(i.e., "each" point).
That this involves infinitesimals in the real numbers
and about Veronese and Peano and standard infinitesimals,
which are like Newton and Leibniz' differentials but that's
only part of the story, this richer story of the infinitesimals
finds more important features of their analytical character
(real analysis).
So, geometrically the punctured line looks different,
but "splitting" all the points in the line is making
two copies that fit exactly on the line (doubled).
This is also phrased for a usual expectation
that "measure is invariant" about what results
as about a "quasi-invariant measure"
https://en.wikipedia.org/wiki/Quasi-invariant_measure
and here as from the examples about Lebesgue for the
field continuity and Hausdorff for the line continuity
(Lebesgue measure vis-a-vis Hausdorff measure) than
in the quasi-invariant is in the example of Gaussian
measure (as about usual centralizing tendencies in
contrast to usual dispersive tendencies) and what
conditions hold (or, don't) in the maintenance of
invariant measure.
That's an example of what's going on in "dynamical"
measure theory and as discussed above about Brian and
Oprocha and so on, and after Bergelson, about the ergodic
and the Markov and non-Markov (but ergodic) basically
gets into geometry having framed below, cardinals having
framed above, now building that _in the various ways it has_
to the middle.
From Dr. Oprocha's publications, http://home.agh.edu.pl/~oprocha/?p=2 ,
you can read a lot about what's going on the the research
that's about analysis (and then what's for foundations to
accommodate, i.e. the needful).
This includes for example "re-Vitali-izing" the measure theory.
The Wikipedia's quite grown these years and these days
there's a page about the "Vitali covering lemma", and
about Hausdorff and Lebesgue measure, and here in this
context about differences between the areal and volumetric
and what is density.
https://en.wikipedia.org/wiki/Vitali_covering_lemma
Here this usual notion of "removable discontinuity"
is reflected in "negligeable collection" (or, negligible
quantity).
(Some tagged posts of T.Tao about measure and multiple
recurrence:
https://terrytao.wordpress.com/tag/lebesgue-measure/ https://terrytao.wordpress.com/tag/multiple-recurrence/ )
Then, there is room for dispute of "The Vitali covering
theorem is not valid in infinite-dimensional settings",
as where it's framed as about the Gaussian instead of
as we discuss here other relevant measures, in the
infinite-dimensional setting. (And about a spiral-
space-filling curve as finite- or infinite-dimensional.)
There is much of this type of reasoning put to the
van der Waerden theorem about arithmetic progressions
much as an analog to Dirichlet/pigeonhole principle,
https://en.wikipedia.org/wiki/Van_der_Waerden%27s_theorem
except as a complement of how they disperse besides
how they nest.
"Van der Waerden's theorem is a theorem in the branch
of mathematics called Ramsey theory."
So, you can see that "re-Vitali-ization" is ongoing in
mathematics writ large, and quite more accessibly these
days then that as the dust arises of the construction,
it all has to settle to the foundations again. Then,
looking to "re-Vitali-ize" measure as from some quite
fundamental notions as the spiral-space-filling curve
and unique properties of the equivalency function (in
continuity, analysis, differential analysis, singularity
theory, and probability theory) makes a neat course for
then laying out what are building up as these quite
immense structures in categorical algebra in descriptive
set theory, as somewhat neatly primitive (and accordingly
fundamental).
Op 19/09/2018 om 14:57 schreef Peter Percival:
peteolcott wrote:
It is categorically impossible and analogous to
the circle with a missing point which is itself
analogous to BanachrCoTarski.
I once compared you unfavourably to Nam, and George Greene disagreed. I stand by my appraisal. At least Nam knows that proof counts in mathematics. He can't recognize a proof as being valid, and he couldn't produce one himself to save his life, but at least he knows that they
are the gold standard. You, on the other hand, faced with a counter-intuitive result (in this thread Banach-Tarski) announce that it
is categorically impossible. That it has been proved doesn't count for you, you just repeat "categorically impossible" like a parrot. Analogies furnish neither proofs nor refutations. (That analogies might guide someone towards a proof or a refutation is true, but that doesn't make analogies proofs or refutations.)
And another thing - none of the newsgroups you are posting to is appropriate for a discussion about the Banach-Tarski paradox.
This thread, if needs be, is a demonstration of BT, in the "sphere" of
ideas: a statement (po's) even if infertile, can multiply itself out of nothing to as many clones as one wishes (or not wishes) :-o)
So I don't see the end of it, unless people decide they've done their
last bit :o
--
guido wugi