Re: Visualization of unmeasurable sets and computability

On Jun 21, 1:35 am, Denis Feldmann <denis.feldmann.asuppri...@club-
internet.fr> wrote:
mike3a écrit :



On Jun 20, 10:54 pm, Rupert <rupertmccal...@xxxxxxxxx> wrote:
On Jun 20, 5:10 pm,mike3<mike4...@xxxxxxxxx> wrote:

On Jun 19, 8:15 pm, Rupert <rupertmccal...@xxxxxxxxx> wrote:
On Jun 20, 8:35 am,mike3<mike4...@xxxxxxxxx> wrote:
On Jun 19, 5:14 am, Denis Feldmann <denis.feldmann.asuppri...@club-
internet.fr> wrote:
mike3a écrit :> Hi.
Is it possible to visualize, even in a rough way (ie. an
approximation, it does not have to be exact! No visualizations are
exact anyway!), non-Lebesgue-measurable sets like those produced in
the ball-cutting of the Banach-Tarski Paradox?
No. And remember you are probably unable to wisualize perfectly good
easy to construct sets, like Cantor sets with positive measure, or even
just the set of all transcendant numbers (or their cartesian square, if
you are more atease with subsets of R^2 than with subsets of R
They cannot be visualized *exactly*, but this picture shows a
rough approximation of the postivie-measure Cantor set:
http://en.wikipedia.org/wiki/Image:Smith-Volterra_set.png
(white bars represent points in the set)
and this type of approximatevisualizationis what I mean by
"visualization". Of course it's "infinitely inexact", but
since you could not see the finer lines even if you were able
to display an hypothetical (and physically impossible)
infinitely-precise one simply because your eyes are finite in
their resolving power, it's good enough and gives you at least
some idea of what's going on. We can estimate a few elements
that are members of the set, and spit them out on the computer
screen, yielding a roughvisualization. It. Does. Not. Have.
To. Be. Exact. Heck, to throw more fuel on this fire, the
Mandelbrot set cannot be drawn exactly, yet it is possible to
draw a rough, approximatevisualizationof it that gives you
some idea of what it looks like.
So then the question comes out to: can we construct an algorithm
that spits out approximations of an arbitrary number of points to
any given degree of accuracy, that are inside a non-measurable
set?
If not, does this mean that non-measurable sets are also non-
computable (not sure if that is the right term, but what I mean by
"computable" is that you can construct an algorithm for a Turing
machine that will generate approximations of an arbitrary number of
points in the set to arbitrary accuracy?)? As if they are computable
then one could spit the estimated points out on a screen and generate
an approximatevisualization.
Try it already for one of the sets mentioned above, and good luck....
Anyway, your dreams of "seeing" the Banach-Tarski paradox with your own
eyes will never be fulfilled, I am afraid
Even in a rough, approximate way? I didn't say it needed to
be exact.
Therefore I direct you to the more general question, which
covers what I am trying to say even if you do not get my
drift:
***
Does non-measurability of a set imply that no algorithm can
exist that computes an arbitrary number of points of that
set to an arbitrary degree of accuracy?
***
Well, non-measurable sets are uncountable, and uncountable sets
contain non-computable reals....
But do they contain only a finite number of computable
reals (which would destroy any hope of approximating
the set to an arbitrary degree of accuracy and hence
of even creating a roughvisualization)?- Hide quoted text -
- Show quoted text -
I really think this idea of visualizing the sets is a non-starter. The
sets are dense, I tell you.

Does that mean that if I tried it the approximatevisualization
would be indistinguishable from an an approximatevisualization
of a solid ball?

Yes

Thanks.

I sort of figured it might look that way, but I did not
really *know* it for sure -- I didn't know they were dense,
for one.

.

Relevant Pages

  • Re: On writing negative zero - with or without sign
    ... of representing an "exact" zero... ... My definition of an "exact" zero is whatever internal representation ... *only* value associated with that approximation. ... A subset of floating point numbers can have an exact representation. ...
    (comp.lang.fortran)
  • Re: Visualization of unmeasurable sets and computability
    ... it does not have to be exact! ... rough approximation of the postivie-measure Cantor set: ... So then a "rough visualization" achieved by spitting out ... Does non-measurability of a set imply that no algorithm can ...
    (sci.math)
  • Re: The spinoza papers: design of the extra-precision number object 2
    ... approximation will be truncated to some arbitrary point (either a fixed ... But many "real" values are also exact. ... measuring, there comes a point (which varies depending on what we're ...
    (comp.programming)
  • Re: Visualization of unmeasurable sets and computability
    ... it does not have to be exact! ... rough approximation of the postivie-measure Cantor set: ... screen, yielding a rough visualization. ... So then the question comes out to: can we construct an algorithm ...
    (sci.math)
  • Re: relating controls to hcp
    ... [regarding exact values versus sims] ... >>>From hcp in the range 5 to 29, the best fit line for expected controls ... An approximation A * hcp - B cannot be accurate to two decimals. ... > the exact figures, to 2 dp, are shown below. ...
    (rec.games.bridge)