FOM: David Ross's comment on the Banach-Tarski paradox
Stephen G Simpson
simpson at math.psu.edu
Wed Dec 10 19:26:27 EST 1997
Commenting on the CH thread, Robert Tragesser said:
> if Steve Simpson feels that there is too much spontaneity, that too
> much is being sent that was not sufficiently thought out in
> advance, perhaps the better thing to do (instead of a summary)
> would be to offer some guidelines (if only in the form of good
> examples and bad examples)?
Let me offer an example from a different thread.
(1) The Banach-Tarski paradox is the following theorem of ZFC: There
is a decomposition of the 3-dimensional sphere of radius 1 into seven
pieces which can be moved and reassembled by rigid motions to form the
sphere of radius 2. The point of this paradox is that the volume of
the sphere is, in a sense, not uniquely defined. The seven pieces are
necessarily not Lebesgue measurable, hence not Borel.
(2) I mentioned the Banach-Tarski paradox as an example of the kind of
pathology that the "cinq lettres" group (Borel, Baire, Lebesgue, ...)
wanted to avoid by restricting attention to Borel sets. I think that
this comment was clear enough in context.
(3) The free group on two generators x and y is, by definition, the
group generated by x and y subject to no relations except those
implied by the group-theoretic axioms. This well-known group, call it
F_2, has infinitely many elements. It can be shown that the group of
rigid motions in 3-dimensional space contains a subgroup isomorphic to
F_2. This fact may be tangentially relevant to one of the well-known
proofs of the Banach-Tarski paradox (via amenability).
After I made my comment on the Banach-Tarski paradox, David Ross
responded with the following mysterious, elliptical comment: What's
nasty about the Banach-Tarski paradox isn't the Banach-Tarski paradox
itself, but rather the fact that the group of rigid motions in
3-dimensional Euclidean space contains F_2.
I'm not sure whether David Ross was trying to support or undercut my
comment about the "cinq lettres" group. Let's disregard that aspect
and focus instead on the substance of Ross's comment.
It's hard to understand Ross's comment. I don't know how Ross
expected anyone to understand his comment. I don't know what was in
Ross's mind when he made his comment. Ross's comment makes no sense.
The only explanation of Ross's comment that I can think of is the
following: Some very specialized mathematicians might regard F_2 as
"nasty", because it isn't the kind of group they are accustomed to
studying. For instance, specialists in the theory of finite groups
might grumble that F_2 is "nasty", because F_2 isn't a finite group.
However, this doesn't imply that all or even most mathematicians would
regard F_2 as "nasty". In fact, most mathematicians do not regard F_2
as "nasty" or pathological, nor should they, since F_2 is perfectly
natural from the viewpoint of group theory. Therefore, in the light
of this proposed explanation, Ross's comment still makes no sense.
If Ross expects anyone to understand his comment, he needs to add some
clarification. And he had better include enough relevant and coherent
reasoning to make it understandable.
More information about the FOM