FOM: Goading Harvey
Joseph Shoenfield
jrs at math.duke.edu
Mon Sep 7 11:24:02 EDT 1998
Steve has completely misinterpreted my motives in my discussion of
the possibility of Harvey's work leading to a theorem somewhat resembling
the Martin-Steel Theorem. I did want to goad Harvey into working on
such a result. This is because Harvey is the only person (in my opinion)
with both the knowledge and the mathematical ability needed to prove such
a result, and because I think such a result would fit in very well with
his program. I had no intention of comparing him unfavorably with Martin
or Steel. They are certainly three of the outstanding logicians of today.
I see no reason for me to try to compare them; if I did, I would not disguise
this comparison in the way that Steve suggests I have. As to the Karp prize,
I mentioned it only to illustrate the large splash that the Martin-Steel
theorem has made in logic. I certainly did not think Harvey would be
tempted by the meager financial rewards. 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.
I haven't thought nearly enough about the matter to make a conjecture;
but I will state a possible theorem which I hope will clarify what I have
in mind: There is a sequence {An} of sentences such that An is strictly
pi-1-n+1, and An can be proved from ZFC and the existence of n large
cardinals of a certain sort but not (under obvious consistency hypotheses)
from n-1 such cardinals. The kind of large cardinal remains to be
determined. (Recall that the definition of a Woodin cardinal was an
essential preliminary to the Martin-Steel theorem; this was part of the
reason that Woodin shared in the prize.) One would of course hope to
show that one could not replace An by a pi-1-n sentence. It would be
nice if An was simple in some of the ways that undecidable sentences
considered by Harvey and others are simple; but I do not think this
essential for the significance of such a result.
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