FOM: reverse math/recursion theory

Stephen G Simpson simpson at
Fri Aug 7 10:05:48 EDT 1998

Harvey Friedman writes:
 > The Lempp/Simpson correspondence appeared on the FOM on 6/4/98
 > 5:51PM.

Dear Harvey,

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
other areas.

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
book sooner.

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!)

Best regards,
-- Steve

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