FOM: Martin-Steel Theorem
Joseph Shoenfield
jrs at math.duke.edu
Wed Sep 16 23:37:20 EDT 1998
Steve and Harvey have replied to my posting on the Martin-Steel
theorem. For both, the main point is that the problems of set theory
which I described are less important than certain other problems.
These are the problems which are of interest to core mathematicians;
interest here means more than curiosity, and implies a feeling that
the solution of the problem has significance for the kind of mathe-
matics which they do. This feature can also be described (in terms
which both Steve and Harvey have used) as: the problem has interactions
with core mathmatics.
What reasons do Steve and Harvey have for saying these problems
are particularly important? Steve says that this results from the
foundational perspective. If I have learned anything from the dis-
cussions on fom, it is that the various participants have quite
different ideas of where the fom perspective leads. I see no
reason to think that it leads to working on problems with interactions
with core mathematics. Harvey says that these problems have general
intellectual interest (gii). I am not sure what this term means;
I take it to mean intellectual interest which is not tied to any
particular field. I see no reason why problems of gii should be
more important than problems of special intellectual interest; I
think that in practice, the reverse is true. Problems with inter-
actions are of slightly more gii than problems without, since they
deal with 2 fields instead of 1; but I see no reason why interactions
with core mathematics have more gii that interactions with non-core
mathematics.
As the above shows, it is very diffucult to convince a person
that a criterion for the importance of a mathematical problem is
particularly significant if he does not alread believe this. I
think that this is fine, and that each person should choose the
problems he works on according to his own criteria of importance.
If (as seems unlikely) I convinced Steve and Harvey to stop working
on problems with interactions with core mathematics, the result would
be a loss of interesting results of the type which they have produced
in recent years. If (as seems even more unlikely) Steve and Harvey
convinced the leading set theorists to spend their time working on
such problems, the result would be a disaster for set theory.
Harvey also touches on a question which I barely mentioned: how
does one decide which new axioms of set theory are acceptable? As
he remarks, V=L solves more unsolved problems than any other new axiom
which has been considered. It is, as he said, rejected by set theor-
ists because it limits the sets which one can consider. If one
is only interested in constructible sets, one does not need such an
axiom; one simply decides to only prove theorems about constructible
sets.
Early large cardinals had a justification similar to that of the
axiom of infinity. The latter postulates a set containing 0 and
closed under the sucessor operation. Similarly, the existence of
an inacessible cardinal postulates the existence of a set closed
under certain set-theoretic operations. I once suggested to Solovay
in a casual conversation that the two axioms were equally justified.
He disagreed. It is easy to visualize the first set, say as a row of
telephone poles without end. But the structure of the second set is
very complicated. He suggested it would be an excellent project to
analyze this structure in a way that makes the existence of an
inacessible more evident. In any case, we do not have any such
closure definition for the interesting large cardinals, such as a
measurable cardinal.
Finally, let me repeat the last statement of my previous posting:
>What impresses me in the whole story is how the solutions of
problems leads to new concepts, which are then developed and inte-
grated with the old concepts. To me, this shows the folly of trying
to decide in advance of doing the mathematics what the fundamental
concepts of set theory are.
Both Steve and Harvey were unimpressed. Steve remarked that this
was not a distinguishing feature of set-theory. I couldn't agree
more. I made the same point in a previous posting on the fundamental
concepts of recursion theory. Steve felt the second statement was
a veiled attack on fom perspective. Well, there is certainly no veil.
It is an attack on a certain fom perspective, which is not mine but may
be close to Steve's. It is that one should first decide, using ones
intuition, what the fundamental concepts of the subject are and then
concentrate on problems concerning these. If the set theorists I
mentioned had adopted this perspective, set theory would have been
a much less interesting subject than it is today.
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