FOM: f.o.m. significance of Friedman's work

[Stephen G Simpson]

Joe Shoenfield is a preeminent figure in mathematical logic, so am
delighted that he joined in the discussion.  However, his comments of
26 Mar 1998 15:56:44 gloss over some crucial distinctions, and I very
much disagree with his lukewarm assesment of the significance of
Friedman's work.

Shoenfield correctly points out that the use of "large cardinals ...
to prove combinatorial principles not provable in ZFC" is an old
story.  However, there is vast gulf separating Friedman-style
combinatorics on finite labeled trees from Rowbottom-style
combinatorics on inaccessible cardinals.  Previous work on large
cardinal combinatorics (by Erd"os, Hajnal, Rowbottom, Silver, ...) did
not give rise to any finitary consequences; the combinatorics in
question was meaningful only for inaccessible cardinals (Ramsey
cardinals etc).  Moreover, the combinatorial properties were highly
non-absolute in the sense of G"odel.  Friedman's work is of a
completely different character.  It combines familiar large cardinal
techniques (indiscernibles, inner models, etc) with some very
delicate, peculiarly Friedmanesque techniques (parameterless set
theory, non-omega-models of set theory, etc) to obtain *finitary*
combinatorial statements that are independent of ZFC.  This marks a
dramatic revolution or sea change in f.o.m.  Shoenfield may not concur
with my use of the word "epochal", but I think that even Shoenfield
and other skeptics will eventually have to admit that the difference
is like night and day.

Let me try to present my personal perspective on the significance of
all this.  When I was a graduate student, independence results via
forcing and large cardinals were well known, but the holy grail of
f.o.m. was to import set-theoretic independence into the realm of
finite mathematics.  Without this, set theory would remain essentially
isolated from core mathematics, because the center of gravity of core
mathematics lies in the finitary realm.  In the late 1970's, important
work by Paris-Harrington and others (including Friedman) yielded
finite combinatorial statements independent of PA and other subsystems
of second order arithmetic.  But set-theoretic independence remained
out of reach; absoluteness presented a seemingly impenetrable barrier.
Now Friedman has broken through all the barriers to forge a direct
link between large cardinals and finite combinatorics.  This is
unprecedented, and it completely changes the possibilities for
f.o.m. for both the short term and the long term.

-- Steve