FOM: Banach-Tarski and Borel sets; pathology
Stephen G Simpson
simpson at math.psu.edu
Thu Dec 11 19:07:06 EST 1997
John Steel said:
> If you restrict your view to Borel sets, you can see neither a
> wellorder nor a paradoxical decomposition, and if the sight of such
> offends you, then I suppose that's some advantage to you.
John, I myself am not necessarily offended by paradoxical
decompositions. However, I can well understand why thoughtful
mathematicians *might* be offended. Borel, Baire and Lebesgue were
not only offended but horrified by Zermelo's theorem on the existence
of a well-ordering of the reals, because (1) no specific well-ordering
was exhibited, (2) the existence of such a well-ordering gives rise to
pathology such as non-measurable sets and, later and more
dramatically, the Banach-Tarski paradox. That's why these great
mathematicians proposed to restrict attention to Borel sets: in order
to avoid such pathology. Your comments make it sound as if you are
unable to grasp the perspective of these great mathematicians. I know
that you know better; I'm not sure about David Ross.
For those who have no idea what I'm talking about, I would recommend
the historical discussion in Gregory Moore's book "Zermelo's Axiom of
Choice".
> you have made no progress whatsoever in taming the geometrical
> complexity of R^3 coming out of the fact that F_2 embeds in its
> isometry group.
John, I don't regard this as a fruitful problem to work on. Do you?
You and Ross have neglected to give reasons for your stated belief
that F_2 is a source of pathology. Beliefs such as this are unusual,
if not bizarre, and require explanation. After all, F_2 is a very
natural and easily defined mathematical object, and so is its
embedding in the rigid motions of R^3. (By contrast, a well-ordering
of the reals is not even provably definable in ZFC.) Nobody has
proposed or could propose any f.o.m. scenario under which F_2 would no
longer be embeddable in the rigid motions of R^3. Obviously any such
scenario would be much much much much more offensive than anything we
have discussed. (By contrast, Borel and Lebesgue proposed a perfectly
viable scenario with no well-ordering of the reals.)
Earlier in FOM there was a discussion of the fact that some pure
mathematicians don't appreciate f.o.m. because they "just don't get
it" when confronted with the distinction between basic mathematical
concepts and higher-level mathematical concepts. Now I want to remark
that there may be another class of people who "just don't get it". I
have in mind a class of logicians who have no appreciation for the
mathematical distinction between what is natural and what is
pathological. This distinction was spelled out at length in Harvey's
recent posting "FOM: 10:Pathology". Such logicians seem to have
somehow cut themselves off from the physical and geometrical roots of
mathematics. I don't really think that John Steel and David Ross
are of this ilk, but their comments on Banach-Tarski exhibit some
of the same obtuseness.
> There is a larger question here: how is TBU meant to be used?
John, you can set your mind at rest. I'm NOT going to petition the
regents of the University of California to make TBU the official
statewide axiom system for f.o.m., supplanting ZFC + a proper class of
Woodin cardinals. Does that make you breathe easier? :-)
Seriously, there are many possible uses for TBU_0. One of them is to
work out the consequences of the f.o.m. viewpoint expressed by Borel,
Baire and Lebesgue. Another might be to axiomatize certain classes of
theorems about Borel sets, to determine what follows from what. This
is valid f.o.m. activity. I'm not ready to claim that this is
earth-shaking stuff, but it's potentially interesting.
-- Steve
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