FOM: reverse math/recursion theory
Stephen G Simpson
simpson at math.psu.edu
Fri Aug 7 10:05:48 EDT 1998
Harvey Friedman writes:
> The Lempp/Simpson correspondence appeared on the FOM on 6/4/98
Thanks for your FOM follow-up to the Lempp/Simpson correspondence.
Unfortunately, Steffen Lempp probably didn't see your follow-up,
because he is not an FOM subscriber. In fact, most of the recursion
theorists are boycotting FOM. Only a few of them are subscribers, and
they have posted hardly anything. I wonder why. Do they explicitly,
consciously regard foundations of mathematics as irrelevant to what
they are doing? Would some recursion theorists please speak up? In
any case, I will forward your two follow-up messages to Lempp.
I like your account of the big picture in reverse mathematics. And,
your idea of positioning reverse mathematics as "the new recursion
theory" -- I love it!!
Reverse mathematics is full of open problems. Perhaps we should have
a public discussion here on FOM about how to choose specific problems
to work on. I have a lot of experience, having supervised 10
Ph.D. theses in this area. Recently Bishop-style constructive
analysis has influenced my thinking a lot, but I want to move into
You mentioned "the overdue Simpson book" on reverse mathematics.
Thanks for not saying "grotesquely overdue"! In the preface I
apologize to the many people whom I disappointed by not finishing the
By the way, the latest word from Springer-Verlag is that the book will
appear in November. In the meantime, chapter 1 is available at
> The problem with mathematical logic today is that there is too
> little variety.
I agree. What happened? How did a wonderful subject like
f.o.m. deteriorate in this way? My book is an attempt to rejuvenate
About your critique of recursion theory, I think your no-examples
point is very telling, as is your invidious comparison to "the new
descriptive set theory" (the fascinating work by Kechris, Hjorth,
... on Borel equivalence relations, etc). It remains a great mystery
why nobody knows of any specific nontrivial r.e. degrees. There are a
few interesting *classes* of r.e. degrees, but they don't seem to have
any applications outside r.e. degrees.
I have sometimes thought that instead of classifying r.e. degrees, it
might be more productive to classify Pi^0_1 subclasses of 2^omega up
to recursive homeomorphism. This project seems to have a better
prospect of applications. (And this remark is a blatant attempt to
draw Carl Jockusch into the discussion!)
More information about the FOM