FOM: determinacy and Friedman-style independence results

[Stephen G Simpson]

Benedikt Loewe 16 Mar 1998 11:38:59 writes:

 > The various Determinacy Axioms are reasonable generalizations from
 > theorems provable in ZF, and they are not a priori connected to
 > large cardinals, but there have been many equiconsistency
 > results. This research area had results twenty years ago that can
 > be compared (in terms of f.o.m. impact) to Harvey's result on
 > Greedy Ramsey Theory.

Loewe's idea of comparing and contrasting determinacy results to
Friedman-style combinatorial independence results is extremely
interesting.  I'd like to get into some details of this comparison.
Note that such comparisons need not be invidious in either direction.
Both sets of independence results are interesting and important in
their own ways.  We need to draw some distinctions.  My remarks below
are preliminary and incomplete and are intended to initiate discussion
on this worth-while topic.

Background: Friedman's independent statements involve insertion of
nodes in finite labeled trees.  The independence results to which
Loewe refers involve projective sets of reals and determinacy of
infinite games.

1.  Core math impact.  One point of comparison is in terms of actual
or potential impact on core mathematics.  I touched on this a little
in my posting of 16 Mar 1998 15:35:15.  Some people may view the
determinacy results as more appealing and natural and therefore more
mathematically applicable.  Another view is that reasonably natural
finitary statements such as those of Friedman are closer in spirit to
core mathematics.

2.  Absoluteness.  The Friedman-style combinatorial independent
statements, like all statements of finite mathematics, are absolute in
the sense of G"odel.  By contrast, the independent statements
concerning determinacy and projective sets, like the continuum
hypothesis and other well-known set-theoretic statements, are not
absolute in this sense.

3. Coherent specializations.  There is a coherent specialization of
the notion of set which settles most if not all determinacy and
projective set questions, namely V=L.  By contrast, it is known that
V=L does not settle the truth value of Friedman's statements.

Items 2 and 3 point to some obstacles that stand in the way of
obtaining Friedman-style finite combinatorial independence results.
When evaluating progress in this area, we need to beware of
underestimating the difficulties.

-- Steve