Re: Visualization of unmeasurable sets and computability

On Jun 20, 2:53 am, Denis Feldmann <denis.feldmann.asuppri...@club-
internet.fr> wrote:
mike3a écrit :



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)

You know too much for your own good, so I will assume you are trolling.
But see below.


This is not a troll: I really want to know, as maybe I don't
know as much as I think I do. In which case I would like to
fill some of that ignorance with knowledge.





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?

Yes (see below)


Yes. So then a "rough visualization" achieved by spitting out
such approximations on a computer screen is possible.



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.

But in fact, you *hint* at it, expecting the approximation to somehow
let you understand what is happening...


But even if you _could_ create an exact value, the discreteness
of visualization methods like computer screens are going to
prevent the visualization from being exact.

What would you get, if you had a program that randomly spit out
estimated points in one of the non-measurable Banach-Tarski
paradox sets on a computer screen? That's what I am trying to
figure out, and therefore the first question would be if such a
program is even possible in the first place. Hence the question
of computability of the elements in the set and/or the set
itself.

It does not have to help understand much, I'm just curious to
see if there would be some sort of interesting patterns visible.
Like how the rough visualization of the Cantor set shows a
pattern even though the visualization is far from exact
(infinitely far, in fact.)



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?
***

As this question is incredibly stupid, given what has already be said to
you, just one easy answer : take any non-measureble set. Union it with
the rationals (or *any* dense resursive set of measure 0). plot the
points in the added set. Is not that a perfectly good algorithm
answering your question? If not, why?



(aside: How can a question be "stupid" if I know as
much as you claim? Especially since what has been
said to me before did not seem to be understandable
as completely answering the question -- since it
did not seem to be based on exactly what I was
asking because I didn't know how to phrase it.)

It would. But can you compute the union, which would
require an algorithm for deciding if a given rational
is in the set? Basically, does an algorithm exist
that can decide whether or not some element is in the
set?



.

Relevant Pages

  • Re: Visualization of unmeasurable sets and computability
    ... it does not have to be exact! ... screen, yielding a rough visualization. ... So then the question comes out to: can we construct an algorithm ...
    (sci.math)
  • 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: Visualization of unmeasurable sets and computability
    ... it does not have to be exact! ... rough approximation of the postivie-measure Cantor set: ... your dreams of "seeing" the Banach-Tarski paradox with your own ... Does that mean that if I tried it the approximate visualization ...
    (sci.math)
  • Re: Visualization of unmeasurable sets and computability
    ... it does not have to be exact! ... rough approximation of the postivie-measure Cantor set: ... So then the question comes out to: can we construct an algorithm ... then one could spit the estimated points out on a screen and generate ...
    (sci.math)
  • Minimum Dominating Set
    ... Is there an algorithm for finding Minimum Dominating Set of a graph? ... This algorithm should find the exact answer, not the approximation one. ...
    (comp.theory)
Loading