FOM: why is Martin-Steel good?

Stephen G Simpson simpson at
Mon Sep 7 21:53:35 EDT 1998

Joseph Shoenfield writes:

 > As to the Karp prize,
 > I mentioned it only to illustrate the large splash that the Martin-Steel
 > theorem has made in logic. ....
 > Harvey has had many successes
 > in logic; but I think it is clear that the particular result which has 
 > inspired this debate has not and will not create a similar splash.

Joe, this last assertion isn't clear, at least not to me.  It seems to
me that the program of which Harvey's result is a part, to extend the
incompleteness phenomenon into finite combinatorics, has much more
general intellectual appeal than the relationship between large
cardinals and projective sets.

However, rather than let you draw me into a probably unproductive
debate about the relative merits of these two research programs, let's
try to turn this into something useful and fruitful.  I propose to do
this by focusing on the Martin-Steel theorem as a case study of how to
evaluate f.o.m. research.

Joe, here is my challenge to you.  Obviously you have a high opinion
of the Martin-Steel theorem (as do I, as do many people).  Could you
please explain, as precisely and objectively as possible, *why* you
think the Martin-Steel theorem is a good theorem?  In other words,
what is it about the Martin-Steel theorem that makes it good?  After
that, maybe other people could answer the same question, and then all
of us could critique the answers, ....

It seems to me that by analyzing the Martin-Steel theorem from this
viewpoint (i.e. what makes it objectively good), we could shed a lot
of light on various issues: the relevance or non-relevance of informal
concepts in evaluating f.o.m. research, whether or not f.o.m. is
mathematics, whether or not it makes sense to evaluate f.o.m. research
as mathematical research, etc etc.

-- Steve

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