FOM: cinq lettres, Banach-Tarski, Aristotle, Borel sets, Kanovei
Stephen G Simpson
simpson at math.psu.edu
Sat Dec 13 15:36:26 EST 1997
I just reread the "cinq lettres" exchange, and Walter Felscher is
absolutely correct: I went much too far in my interpretation of Borel,
Baire, and Lebesgue. These great mathematicians *did not* propose to
throw away all non-Borel sets of reals, as I had asserted. Indeed, so
far as I can see, Borel et al did not make any f.o.m. proposal that we
would now regard as inconsistent with full comprehension for sets of
reals. The "cinq lettres" exchange does include a severe critique of
the axiom of choice and of the use of mathematical objects that are
"not definable in a finite number of words". (The exact meaning of
this phrase is not clear.) Moreover, we know with hindsight that, in
a sense, the axiom of choice is needed to prove the existence of
non-Borel sets (see also the clarification below). However, these
facts in no way justify my misinterpretation of Borel et al.
Felscher also correctly points out that the Banach-Tarski paradox came
20 years after the 1905 "cinq lettres" exchange. On the other hand,
the related Hausdorff paradox (1/2 of the unit sphere being congruent
to 1/3 of it) came earlier, in 1914. (I am getting this from Gregory
Moore's book "Zermelo's Axiom of Choice".) In any case, I didn't
claim that "cinq lettres" was influenced by these particular
paradoxes, but only by pathology arising from the existence of a
well-ordering of the reals. As soon as the Hausdorff and
Banach-Tarski paradoxes appeared, they were viewed (with hindsight) as
being part of that same complex of pathological phenomena.
Obiously the history of mathematics is not among my strengths. But I
still think that the idea of studying the Borel universe, perhaps in
the framework of TBU_0, may be interesting and worthwhile. Walter
Felscher makes this point and then goes on to say:
> Being so, it should not need dubious historical references as
> crown's witnesses - just as NSA did not need the incomplete
> quotations from Leibniz for such purpose.
I accept this implicit criticism.
Speaking of dubious historical references, would anybody like to offer
some historical evidence for or against my "gut feeling" that reverse
mathematics goes back to Aristotle? (The history of philosophy is
also not one of my strengths.)
In another Banach-Tarski posting, I said:
> Your comments make it sound as if you [Steel] are unable to grasp
> the perspective of these great mathematicians. I know that you
> know better; I'm not sure about David Ross.
This was too harsh. I wish I had stated this better. Instead of "the
perspective of these great mathematicians", I should have said "the
above perspective". In context, this would have referred to the fact
that a well-ordering of the reals is a source of pathology. I know
that John Steel understands this point, even if he doesn't agree with
it. In David Ross's case, believe it or not, I misunderstood his
first Banach-Tarski posting as asserting that the free group on two
generators (rather than a well-ordering of the reals) may be the
source of the pathology. It turns out that he was saying no such
thing. His only point was that, in his opinion, Banach-Tarski is
overkill. He explained this in a subsequent Banach-Tarski posting.
A clarification on Borel sets:
Borel sets are usually defined as the smallest collection of sets
containing all intervals and closed under countable union and
intersection and complementation. Under this definition, it is
consistent with ZF that all sets of reals are Borel (e.g. the
Feferman-Levy model). In my view, this definition is almost useless
without the countable axiom of choice. A better approach is to define
Borel sets in terms of "Borel codes", i.e. countable well-founded
trees with intervals at the end nodes. Souslin and Lusin took a
similar approach to analytic sets; part of their motivation was to
avoid using any form of the axiom of choice. This is also the
approach that I take in my book (chapter V on ATR_0) and that I intend
for TBU_0. Souslin's theorem is valid in such contexts, so with
hindsight once again, Gregory Moore's discussion on pages 181-2 is a
little off base. Still, Moore's book is an excellent background
reference for a lot of this material.
I have proposed to restrict attention to Borel sets, but I also like
Kanovei's proposal to restrict attention to a wider class of sets, the
absolutely Delta^1_2 sets. Kanovei's reference to the
Schilling/Vaught paper on set operations is completely appropriate.
-- Steve
Stephen G. Simpson
Department of Mathematics, Pennsylvania State University
333 McAllister Building, University Park, State College PA 16802
Office 814-863-0775 Fax 814-865-3735
Email simpson at math.psu.edu Home 814-238-2274
World Wide Web http://www.math.psu.edu/simpson/
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