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

[Solomon Feferman]

Dear Steve,
  Thanks for your quick and interesting comments.  A few replies below.  

Best,
Sol

On Sun, 26 Jul 1998, Stephen G Simpson wrote:

> 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.

This *is* what I say about the older pathologies (nowhere differentiable
continuous function, space-filling curve, etc.), and I duly consider
whether it is also applicable to the B-T theorem, but my conclusion p. 11
is that there is no obvious way to do so.

> 
> 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".

Well, it seems to me if we try to take this line, there are a whole series
of questions to be asked:  Is the real number system a "natural object",
like (presumably) the "natural" continuum, are curves natural objects (if
so what are they), are pieces of solids natural objects (if so just what
are they), etc.?

> 
> 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.  

Maybe try reading it in binary format for that.

Is ACE the statement that a choice function exists for
> every partial arithmetical equivalence relation on the reals?  

That's exactly it.  Of course it can be expressed precisely in different 
formal frameworks, set-theoretical or type-theoretical.  But since it is
essentially 3d order, I don't see any way of formulating it in the
language of subsystems of 2nd order analysis.  I have chosen the
finite type theory formalism of my old Handbook article in this note. 
Obviously, using a constructibility interpretation, in proof theoretic
strength it can be reduced to a not too strong subsystem of analysis.  But
I conjecture that the statement is actually much weaker than what one
would get that way.  

>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.

I'd be interested in any sharp results on ACE, or even first rough (lower
and upper) bounds on this.

> 
> 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.

I'm clearer now on these things than I was then.  OBT is pretty clearly
predicative, surely provable in a conservative extension of PA and
probably already in WKL_0.  Someone should check though.