From Newsgroup: sci.physics.relativity

On 09/05/2018 09:04 AM, Ross A. Finlayson wrote:

On Wednesday, September 5, 2018 at 8:35:29 AM UTC-7, Jim Burns wrote:

On 9/5/2018 9:43 AM, Peter Percival wrote:

Jim Burns wrote:

So rotating M just the right amount, an amount which is
about 1/1000 of a degree, makes over a hundred thousand
points distributed around the circle blink into and out
of existence.

And there are even more impressive examples. In fact, there
will be rotations arbitrarily close to zero degrees in which
an arbitrarily large number of points blink in and out.

----
I strongly suspect that this is unhelpful to Pete Olcott,
who is having trouble imagining one point blinking
in and out.

But it is very nice, if one likes that sort of thing.

Yes. This may interest people:
https://en.wikipedia.org/wiki/Non-measurable_set#Example.

Thank you.
I think this example -- though only for a circle -- demonstrates
the general plan to generate the non-measurable sets for
Banach-Tarski. Nice.

The next section
https://en.wikipedia.org/wiki/Non-measurable_set#Consistent_definitions_of_measure_and_probability
is relevant to Pete Olcott's categorically correct analysis.
The assumptions that ground the Banach-Tarski result are
laid out neatly.

<wiki>
The BanachrCoTarski paradox shows that there is no way to
define volume in three dimensions unless one of the
following four concessions is made:

1. The volume of a set might change when it is rotated.
2. The volume of the union of two disjoint sets might be
different from the sum of their volumes.
3. Some sets might be tagged "non-measurable", and one would
need to check whether a set is "measurable" before talking
about its volume.
4. The axioms of ZFC (ZermelorCoFraenkel set theory with the
axiom of Choice) might have to be altered.

Standard measure theory takes the third option.
</wiki>

Banach-Tarski was originally geometric, after Hausdorff
for circles after Vitali for line segments with respect
to measures and also about "doubling measures".

https://en.wikipedia.org/wiki/Doubling_space

Working it out from algebra under rotation groups,
maybe would more naturally be for two boxes instead
of two balls.

For topologists these are are being some same kind
of surfaces, balls and boxes, but let's not be
squaring the circle with classical constructions
without unbounded exhaustion.

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