[Phys-L] Re: Zeno's Paradoxes

[Jack Uretsky <jlu@HEP.ANL.GOV>]

But why speculate?  The Wikkipedia article gives you the proof.  And the
first two hits on Google give the description and explanation for lay
persons.  It turns out that you only need to divide the sphere into five
subsets, in order to make two spheres.  The catch is that the subsets are
such that it is not possible to define a physically reasonable volume for
them.  See:

http://www.kuro5hin.org/story/2003/5/23/134430/275
http://www.math.hmc.edu/funfacts/ffiles/30001.1-3-8.shtml

or enter Google with "Banach-Tarski"





On Sat, 2 Apr 2005, Britton, Chuck wrote:

> If I remember correctly Banach & Tarsky were bothered by infinitesima=
> ls (and the associated infinities)
> because they allow one to subdivide a sphere the size of the sun=20
> into a (small) number of pieces -=20
> these pieces can then be reasembled into a sphere the size of a pea.
> No mathematically definable points left out and no overlapping.
>
> Or carving a cue ball into six mathematically connected subsets of th=
> e sphere=20
> and creating two identical cue balls each comprised of only three of =
> the oroginal six pieces.
> Again - with ALL mathematical points accounted for.
>
> I am a bit bothered by the fact that nobody seems to be bothered by t=
> hese shenanigans.
>
> Please enlight me as to these constructions aren't bothersome.
>
> The Axiom of Choice is tied up in it somehow.=20
> Accepting the axiom of Choice is equivalent to accepting infinitesima=
> ls/infinities,
> again if my poor rememberances are  correct.=20
> It WAS forty years ago in freshman calculus that my interest in this =
> was rekindled by the prof.
> SciAm had had an article on it in the earlier '60's.
>
> Eager minds are straining for further understanding.
>
>
> =46rom: Forum for Physics Educators on behalf of Jack Uretsky
> Sent: Sat 4/2/2005 1:29 PM
> To: PHYS-L@LISTS.NAU.EDU
> Subject: Re: Zeno's Paradoxes
> =20
> Who invited Banach and Tarski (and ignored Hausdorf) to Zeno's party?
> For those snowed by Chuck's posting, see Banach-Tarski in the Wikkipe=
> dia.
> Banach-Tarski does not present a paradox.
>                                 Regards,
>                                         Jack

--
"Trust me.  I have a lot of experience at this."
                General Custer's unremembered message to his men,
                just before leading them into the Little Big Horn Valley
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