FOM: Combinatorics and Large Cardinals

[Joseph Shoenfield]

     I think my differences with Steve on the above topic are due to a
different notion of what fom is about.   I was rather shocked to see
him write some time ago that he prefers fom to mathematics.   In my
view, fom is a branch of mathematics whose subject is the structure
of mathematics.    Its object is to replace our intuitive notions
about this structure with precisely defined objects and then formulate
and prove statements about those objects.   (I think this is the moti-
vating idea behind Harvey's series of communications.)   From this
point of view, finite combinatorics becomes interesting only when it
has been precisely defined.   I suggested a syntactical definition
because it seemed to me that we should be able to identify a finite
combinatorial statement by looking at the sentence which expresses it.
     The fact that Harvey's combinatorial statement is pi-0-2 is
certainly significant.   It suggest to me that I should have com-
pared his result not to Erdos-Rowbottom but to Godel's result:
ConZFC is not provable in ZFC, but is provable if one assumes an
inaccessible cardinal.   ConZFC is pi-0-1, and I think it is clearly
a finite combinatorial statement.   Harvey's result is a natural
extension.   Perhaps his methods could lead to a series of results
on the relationship between combinatorial statements having certain
forms in the arithmetical hierarchy and large cardinals.   Possibly
one could find surprising and interesting connections similar to
those in the Martin-Steel Theorem.   This would be fine mathematics,
but perhaps not all that Steve claims for Harvey's result.   (Perhaps
my objection is due to my reluctance to use words like "epochal"
unless they are clearly called for.)