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
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