FOM: Reply to Harvey
Joseph Shoenfield
jrs at math.duke.edu
Fri Aug 14 11:22:18 EDT 1998
In a recent communication, Harvey commented on my remarks on the
significance of a result of his. In this result, he formulated a
combinatorial statement about trees and proved: (a) the result is not
provable in ZFC; (b) the result is provable from an additional assumption
about the existence of large cardinals. This result was praised by
Steve Simpson as an "epochal advance in fom". This comment led to
several replies, including one from me suggesting that the praise was
excessive.
The original comment by Steve did not make clear what in Harvey's
result was so significant foundationally about Harvey's results. Sub-
sequent comments by Steve, Harvey, and others seem to make it clear that
the significance is that the tree principle is a finite combinatorial
statement and hence is in some sense simpler (or more basic or fundamental)
than other statements undecidable in ZFC.
I maintained that ConZFC is a finite combinatorial statement
concerning certain finite sequences of sentences of the language of set
theory. Harvey disagrees because without an understanding of infinite
sets, ConZFC is "an incomprehensible pile of ad hoc unitelligible
nonsense". I think Harvey is replacing the notion of finite
combinatorial statement by the notion of an interesting finite combinatorial
statement. This is a much less satisfactory notion, since its meaning
depends on questions of interest and taste, which cannot be settled by
the tools of mathematics.
Harvey quotes some comments of mine on what matters in fom with some
approval. He then suggest that my argument applies here, since ConZFC
does not matter to finite combinatorialists. I think this is a
completely different situation. I suggested that a certain discussion
of Lagrange's theorem did not matter to people who used that therorem or
wished to understand it. ConZFC matters to people who want to use set
theory or understand it. I don't think the opinions of finite
combinatorialists is relevant to the foundational significance of
Harvey's result, since they are not knowledgable about fom.
Of course, it might happen that consideration of the notion of an
interesting finite combinatorial statement would lead to a precisely
defined notion which differentiates the tree principle from ConZFC and
which is foundationally signficant. If so, one could hope to prove
non-trivial mathemtical results about the precise notion which all of us
would find significant foundationally. I therefore strongly urge Harvey
to pursue his idea of showing that the tree principle is in some simple
language (almost) the shortest independent sentence.
I have avoided any discussion of "epochal" and other (to me)
excessive adjectives. I would have objected much less to Steve's
original statement if he had used "important" instead of "epochal", or at
least showed why he considered epochal more appropriate. To me, epochal
suggests the beginning of a new epoch, in which many good logicians would
utilize the concepts and methods of the epochal result to prove important
mathematical results. An example of such a result is Cohen's proof of
the independence of CH. History shows that even the best qualified
people cannot decide whether a result is epochal when it is first
proved. The solution of Hilbert's tenth problem was admitted by everyone
to be good mathematics and foundationally significant; but it has
certainly not been epochal in my sense.
Harvey also disagreed (at least partly) with a couple of comments of
mine on the nature of fom; I will defent them in a future communication.
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