FOM: set theory-2
Joseph Shoenfield
jrs at math.duke.edu
Fri Sep 18 22:07:17 EDT 1998
Steve and Harvey have replied to my posting of Sep 16. Again,
the main issue is the importance of what Steve calls set-theory-2.
(As Steve points out, he did not say that set-theory 2 was more
important than set-thory 1; but all of the points he made concerned
the importance of set-theory 2.) I have suggested that for this
and similar issues, argument is futile after both sides have
explained their positions clearly. I will try to explain why I
think this is so. I think it would be most informative if I
make the point only for this particular issue; but I think my
reasons are quite general. In particular, they do not (as Steve
seems to think) depend on who is involved in the argument.
In the posting of Sep 16, I summarized as well as I could in a
short space the arguments which Steve and Harvey gave for the
importance of set-theory 2 and stated very briefly why they did not
convince me of this importance. For example, I said that I did not
see that a foundational perspective on set theory led one to problems
having interaction with core mathematics. Steve has given no reason
why it should lead to such problems. If he did, we would perhaps
clarify the difference between our foundational perspectives; but
I cannot imagine either of us convincing the other that his perspective
is better. I said that I did not find general intellectual interests
more important than other intellectual interests. Harvey replied that
the biggest advances in science surround investigations of gii. Of
course, I disagree. We could both give many examples which we thought
supported our view. If we did, I am certain that the result would
be constant disagreement on what are the biggest advances and whether
they have gii. I see no way to settle such disagreements. In trying
to show that problems in set-theory 2 have gii, he described a par-
ticular problem and stated that it has the most profound implications
for the future of set theory. If gii means what I think it does,
he should have instead considered implications for science as a whole.
In evaluating the importance of a mathematical result, Harvey puts
great value on the number and variety of people who find the result
important. I don't. If I proved a result in set theory, I would
rather have Bob Solovay say it was important than any 20 Fields
Medalists. I don't see how any person could make a rational judgment
about which attitude is better.
Harvey asks if I am working on a certain project (roughly, analyz-
ing the notion of an inacessible cardinal with the object of making
the acceptance of such cardinals seem more rational). I don't think
it was a very serious question, but I will give two reasons why I am
not because they reflect differences between Harvey and me over what
one should work on. First, none of the techniques which I know and
feel at home with seem to have a chance of making progress on the
project. Second, even if completed, I don't think it would be too
important, because the existence of an inacessible is not now considered
a useful new axiom compared, say, to the existence of a measurable.
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