FOM: Urbana thoughts; Soare; Pillay

[Stephen G Simpson]

Matthew Frank Sun, 11 Jun 2000 10:11:16 -0500 (CDT) writes:

 > Nabutovsky and Weinberger had needed some new results in
 > computability (which Soare then provided) for their proofs.

Ummm, well, ....  My understanding of it (I have been reading up on
the Nabutovsky/Weinberger stuff) is that Nabutovsky and Weinberger
originally cited an old theorem about r.e. degrees, the Sacks Density
Theorem.  But then, after discussing the matter with Soare, they found
that they could get a slightly better geometrical result by applying a
much easier, tailor-made theorem about r.e. sets, which Soare
provided.  The easier theorem, due to Soare (see also Soare's ASL 2000
abstract), is that there exists a sequence of r.e. sets such that any
settling-down function for any one of the sets in the sequence
dominates any recursive function composed with any settling-down
function for the next set in the sequence.

This theorem of Soare is much easier because, according to some
recursion theorists that I asked at ASL 2000, the proof does not need
a priority argument.  Another vindication of the so-called Simpson's
Thesis!  [ This is a reference to the discussion of the role of
priority arguments in applied recursion theory, FOM,
July/August/September 1999.  See for instance my posting of Wed, 4 Aug
1999 19:18:00 -0400 (EDT). ]

Matthew Frank continues:

 > Soare mentioned a theorem that a compact Riemannian manifold with
 > an unsolvable word problem has only finitely many closed
 > contractible geodesics [...]

According to my notes taken at Soare's ASL 2000 talk on the
Nabutovsky/Weinberger stuff, this is a statement that Soare attributed
to Nabutovsky.  [ During the talk Soare promised to put his
transparencies up on the web, but they are not there yet. ]  However,
the statement seems to be running the wrong way.  Shouldn't "only
finitely many" be replaced by "infinitely many"?  This way it seems to
make a lot more sense.  [ Unfortunately I am away from home right now
and don't have access to my copies of Nabutovsky's papers. ]

Let me say for the record that this recent talk by Soare was very good
in that it included a fair amount of detail, both geometrical and
recursion-theoretic, much more than when Soare spoke on the same topic
at the Boulder CTA meeting a year ago.  Also, Soare is now much more
careful in describing his own role as a consultant for the
Nabutovsky/Weinberger "Fractal" paper, which finally became available
in March 2000.

Regarding Pillay's talk at ASL 2000, Matthew Frank says:

 > He [...] exhorted logicians to be intellectually responsible,
 > although the details of that are somewhat unclear (e.g. for Pillay
 > that primarily meant a sensitivity to the concerns of
 > mathematicians, while for others it might mean a sensitivity to
 > foundational issues).

Yes.  In fact, Pillay often goes out of his way to reject and
obliterate foundational issues.  For example, in his talk he stated,
proudly and incorrectly, that the G"odel incompleteness phenomenon is
nowhere to be seen ("not part of the picture") in the kind of model
theory that he likes.  (See also Harvey's "Urbana Thoughts" posting of
Sat, 10 Jun 2000 00:08:46 -0400.  See also the dialog with Pillay on
FOM in October 1997.)

-- Steve