FOM: Banach-Tarski; we must strive for clarity!

Stephen G Simpson simpson at
Thu Dec 11 20:27:25 EST 1997

David Ross writes:
 > I think Balwin and Steel made my meaning clear (as I apparently
 > hadn't),

In my opinion, they didn't.  I hope you are willing to try again.

 > I'd hoped readers of this list would be familiar with the result
 > (due to Stan Wagon, early 80s) that Day's conjecture holds for
 > rigid motions in R^n, n>2.

I would be surprised if more than a handful of the 213 FOM subscribers
are familiar with this technical result.  I myself am not familiar
with it, though I've been through a proof of the Banach-Tarski paradox
and I understand the relationship to amenability.

The FOM subscriber base includes people with a variety of backgrounds.
People who post to FOM need to exercise care and strive for clarity.

 > While the one volume paradox - most troubling to barbers - might go
 > away if we restricted our attention to Borel sets, most of the
 > others would not.

What is the "one volume paradox"?  What paradoxes are you talking
about that would not go away if we restricted attention to Borel sets?
Do these other paradoxes have anything to do with the Banach-Tarski
paradox?  Could you please explain yourself?  If you are too tired or
busy to explain yourself, could you please give a reference to the

 > In fact, as the passage (in the proof of BT)
 > from existence of F2 in this group to the Hausdorff paradox is

What is the "Hausdorff paradox"?  What is its relevance to this
discussion?  Could you please explain yourself or give a reference to
the literature?

 > BTW, I never meant this as an indictment of the interesting
 > programme of restricting our atention to Borel sets, merely as an
 > indictment of the choice of one of the motivating examples.

But you didn't explain why you think the Banach-Tarski paradox is a
bad motivating example.  Could you please have another go at it?

 > (I also never meant this as an indictment of the free group on two
 > generators, which is one of my favorite mathematical objects.)

Thank you for that partial clarification.  In your previous posting on
this subject, it sounded for all the world as if you were saying that
F_2 is "nasty".

-- Steve

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