Re: Questions about Axiom of Choice

In article <1157473528.401566.308170@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
<agapito6314@xxxxxxx> wrote:

Daniel Grubb wrote:
1.- It is a "pure existence" statement, that is it tells something
exists but not what it is. Is math not full of these kinds of
statements?

It is now. Until relatively recently, an existence proof was required
to give a method of construction.

2.- Some of its implications are weird or counterintuitive. Can
someone please explain some of these?

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.

--Dan Grubb

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.

The "pieces" are weird indeed, and cannot be obtained
by any sort of "measurable" cuts. This is because the
group of all rotations in 3-space, as a discrete group,
is not amenable, which means that there exists functions
which have the property that one cannot define an
"average" over the group with the property that the
action of the group preserves the average. I believe
that one does not need the axiom of choice to prove this.





--
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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