FOM: Informal concepts in fom
Joseph Shoenfield
jrs at math.duke.edu
Fri Sep 4 15:50:41 EDT 1998
This is a rebuttal to Steve's Aug 31 reply to an earlier posting of
mine. Steve and I certainly have some differences, but I think some of
them are unimportant. For example, the relative merits of the terms fcp
and understandable fcp. This started with a dispute over whether ComZFC
is an fcp. It soon became apparent that our difference was over the
meaning of fcp. Steve isolated the difference by pointing out that his
notion of an fcp included a component of understanding; and I agreed that
this was a good explanation of our difference. Under these conditions,
I would have thought he would have approved of using understandable fcp
for his notion; but he is clearly uncomfortable with this terminology, and
uses it only with the caveat that it is my terminology. Therefore I will
agree to use fcp for his notion, at least in this posting, since my notion
of an fcp is no longer relevant to our discussions.
Now to a more important point. In my original posting I said:
>Whether this [the fact that Harvey's principle is an fpc] is serious
progress in fom depends on whether this notion of fcp is a significant
notion for fom.
Steve replied:
>The notion of fcp is one of those intuitive informal notions which
are of tremendous significance for evaluating progress in fom.
It seems to me that Steve is agreeing with my statement, but saying
that it is irrelevant here because fcp is a significant notion for fom.
As should be clear from my having made the statement, I am not convinced
that fcp is a significant notion for fom.
What should Steve do in such a case? On possibility would be to
repeat the question posed by Richter in a different case: why no wait and
see what happens? But I think Steve is unwilling to say anything which
would indicate that the question of whether fcp is a significant notion
for fom is still open. He might also try to find arguments which which
might convince me that his position is correct. Instead of this, he
jacks up the stakes by say that fcp is TREMENDOUSLY important for fom.
But all this does is make his position more difficult to defend.
I have suggested that Steve should defend his position by proving
mathematical theorems which clearly show the importance of fcp in fom.
This, of course, is very difficult. He must first (in Harvey's
felicitous phrase) formally analyse fcp in order to obtain a mathematical
concept which corresponds to it. Then he must think of some theorems
which would demonstrate the important of that mathematical concepts for
fom. Finally, and most difficult, he must prove these theorems. There
may be a third alternative (besides doing all this and posing Richter's
question) which will contribute something to our discussion; but I do not
know what it is.
Now to some other points. The Martin-Steel Theorem says that if
there are n Woodin cardinal and a measurable cardinal above all of them,
then ever pi-1-n+1 is determinate. (A Woodin cardinal is a certain type
of large cardinal, larger than a measurable but smaller than a
supercompact.) They also have results which show that this result cannot
be improved in an obvious way. Thus they have solved the problem of
finding the relation between determinacy and large cardinal hypotheses, a
problem which has received much attention from some of the best set
theorists over a period of many years. The connection between this and
the sort of results which I conjectured might be obtainable by extending
Harvey's work is very tenuous; both relate certain undecidable statements
to large cardinals, and both say something about pi-1-n sentences for each
n. I guess my secret reason for mentioning it was to encourage Harvey to
believe that if he proved such a result, his might receive some of the
accolade which Martin and Steel have received (e.g., the Karp prize).
The main purpose of my entire discussion was to show that the analysis of
the foundational significance of a theorem is often a difficult matter,
and can sometimes be performed in quite different ways by competent
researchers.
I won't say any more about "golden"; if I did, it would be much like
what I said about "tremendous".
I just remembered that I failed to respond to an earlier question of
yours about "Mathematical Logic". A couple of years ago, there was a
movement in the ASL to reprint useful books which have gone out of print;
"Mathematical Logic" was to be one of them. This was stalled for various
reasons, including the death of the president of the ASL. I do not know
if it is presently active. I suggest that anyone who feels such reprints
would be useful should write to the new president, Tony Martin, giving the
reasons why they support the proposal.
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