FOM: Reply to Harvey

[Joseph Shoenfield]

    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.