FOM: Reply to Steve's reply
Joseph Shoenfield
jrs at math.duke.edu
Mon Aug 31 11:32:27 EDT 1998
Lest this sequence of replies go to infinity, I will only reply to
a couple of points.
When I say that "understand" is vague, I mean that it can have many
meanings and degrees of meanings, even when carefully used. I understand
some things more and some things less. Different people can understand
the same thing in different ways.
Let me repeat why I think vague intuitive concepts are unsuitable for
fom. I think the important results in fom are mathematical; that is,
they consist of definitions, theorems, and proofs which are precise and
rigourous is the sense of mathematics. Thus we can use intuitive
concepts only by replacing them be precise concepts, or, as Harvey prefers
to say, by analysing them in a rigorous manner. This is clearly much
more difficult for vague concepts, since we must first decide which of the
meanings to use. I admit that we may succeed in this, and I tried to
indicate in another communication that I think this is what reverse
mathematics has accomplished. But it is a difficult matter.
I disagree totally with your statement that in normal mathematical
usage, the term "combinatorial" already has a component of
understandibility. You say I could find this out by trying to convince
the editors of the Journal of Combinatory Theorem that statements like
ConZFC are combinatorial and therefore would fit perfectly in their
journal. I believe they woud reply that ConZFC may be combinatorial but
that it does not fit in their journal.
I will agree that in some sense, Harvey's principal is an
understandable f.c.p and ConZFC is not. Whether this is serious progess
in f.o.m. depends on whether this notion of understandable f.c.p. is a
significant notion for f.o.m. It is not obvious to me (as it apparently
is to you and Harvey) that it is. In the absence of a consensus that it
is, the only way (in my opinion) to show that it is is to prove
mathematical results about it that we can agree are interesting and
informative about the nature of mathematics. This has not been done.
Is there some relevant qualitative difference between ConZFC and
Harvey's principle? Certainly. ConZFC is pi-0-1 and Harvey's
principle is not; and we know enough about pi-0-1 (limit lemma and so on)
to say that it is foundationally significant. I imagine this will not
please you and Harvey, since it will not contribute to your analysis of
the foundational significance of Harvey's principle which requires that it
be, in some sense, simpler than ConZFC. But this is not the only way to
do this analysis. If I were to try to do it, I would emphasize this
difference, and also emphasize Harvey's result that his principle can be
proved from certain large principles. This would lead me to hope that
there is a connection between the arithmetical hierarchy classification of
a proposition and the large cardinal assumptions needed to prove it. If
I could find such a connection, I think I would have a foundationally
significant result. <This is what I meant in a earlier communication,
where I suggested that Harvey's result could lead to a sort of analogue of
the Martin-Steel theorem.)
A few words on emotional words. I suppose "computability" is
emotional in the sense that some controversies about it have upset some of
the participants. I had something else in mind. The words I refer to
have strong meanings of approval or disapproval without indicating in any
way the reasons for approval or disapproval. A good example is the
golden in golden opportunity, which also seems to have upset Martin a
litle. Putiing it into your statement conveyed nothing except, perhaps,
your disapproval of those whoe might have done research in the field but
didn't. By the way, I think "despair" in my communication was an
emotional word; I apologize for it.
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