FOM: Feferman's draft paper "Intuition vs Monsters"

Stephen G Simpson simpson at
Sun Jul 26 17:03:59 EDT 1998

To: Sol Feferman

Dear Sol,

Thanks very much for your draft paper "Intuition vs Monsters", which I
view as an interesting continuation of the discussion of pathology,
the Banach-Tarski paradox, etc. which occurred on the FOM list last
December, in which you participated prominently.  I would urge you to
throw "Intuition vs Monsters" open to discussion on the FOM list.

I am intrigued by your thesis that "monsters" (i.e. pathological
objects such as the Banach/Tarski decomposition of the unit sphere)
are best viewed simply as counterexamples showing that certain
hypotheses cannot be omitted from certain theorems, in the context of
rigorous mathematics.  If this thesis is correct, then it gives a
large class of examples of why it's a very good thing to have high
standards of rigor.  I wonder how the quasi-empricists (Tymoczko,
Hersh, et al) view these monsters, since they apparently don't fully
acknowledge current standards of mathematical rigor.

However, I'm not entirely convinced of your thesis.  After reading
your paper, I still cling to the idea of an intuitive distinction
between "pathological objects" and "natural objects" in mathematics,
or perhaps rather a series of such intuitions.  I think that these
intuitive distinctions provide important guidance in executing and
evaluating various mathematical and foundational research programs.
See also Harvey Friedman's long FOM posting of 8 Dec 1997 00:37:46,
entitled "Pathology".

In "Intuition vs Monsters", you asked a technical question about the
strength of a form of the axiom of choice which you called ACE.
Unfortunately, I had trouble extracting a plain text file from your
Microsoft Word file, so I'm not sure of some of your mathematical
formulas.  Is ACE the statement that a choice function exists for
every partial arithmetical equivalence relation on the reals?  Off
hand I don't know the strength of this statement.  Maybe Joe Shipman
would know, since he is interested in the strength of various
statements related to the existence of non-measurable sets.

Your FOM posting of 14 Dec 1997 08:54:57 mentioned what you called
OBT, the Foreman/Dougherty variant of the Banach/Tarski paradox,
involving only open sets.  You asked a couple of questions, to which I
may still owe you answers.

 > I'd like to hear what people think of OBT and how it might affect
 > what we think of restricting to the mathematics of Borel sets.

Here you were referring to my proposal for a theory of the Borel
universe, in my FOM posting of 6 Dec 1997 17:46:34.  I think that OBT
doesn't really affect my proposal, because there is still a sense in
which Borel sets of reals are much less pathological than arbitrary
sets of reals (as detailed in my posting), and therefore there still
exist good reasons for wanting to study the consequences of
restricting the set-theoretic universe to Borel sets.

 > A technical question is what part of ZF is needed for OBT.  Does
 > Simpson's TBU_0 suffice?

I imagine that RCA_0 suffices.  But I haven't looked up the
Foreman/Dougherty paper to check the details.

-- Steve Simpson

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