Re: Questions about Axiom of Choice

In article <4m5ne3F4l60dU1@xxxxxxxxxxxxxx>,
=?ISO-8859-1?Q?Jos=E9_Carlos_Santos?= <jcsantos@xxxxxxxx> wrote:
On 05-09-2006 17:25, agapito6314@xxxxxxx wrote:

My favorite is the Banach-Tarski paradox: it is possible to take
the unit ball (i.e. radius 1) in three dimensional euclidean space,
divide it into a finite number of pieces, move those pieces around
via rigid motions (i.e. translations and rotations), and re-assemble
them into a ball of radius 2.

Thank you and everyone who replied. This is a really dumb question but
is it not possible to prove or disprove this physically? That is we
can construct a solid unit radius ball, cut it up into a finite number
of pieces (according to some procedure) and produce the new (solid?)
bigger ball? Thanks again.

No, we can't do that. Each finite piece has a certain volume and the
sum of these volumes will be the volume of the original sphere.
Therefore, any new ball that we can build will have the same volume as
the original one; in particular, it will have the same size.

This assumes that the pieces have volumes. Before
Banach-Tarski, the question was open whether there was
a finitely additive extension of volume to all sets,
invariant under rotation and translation, just as there
is a finitely additive extension of area in the place
with those properties. As we now know, this is equivalent
to the group of rotations being amenable.
--
This address is for information only. I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Department of Statistics, Purdue University
hrubin@xxxxxxxxxxxxxxx Phone: (765)494-6054 FAX: (765)494-0558
.

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