FOM: large cardinals and other areas of mathematics

Stephen G Simpson simpson at
Mon Mar 16 15:35:15 EST 1998

I wrote:
 > we don't know of any any coherent f.o.m. picture which implies the
 > 1-consistency of the large cardinals, other than the large cardinal
 > picture itself.

Benedikt Loewe replied:
 > What about Determinacy? 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.

Thanks for that comment.  

Perhaps your intention was to correct my remark by pointing out that
determinacy axioms prove the 1-consistency of large cardinals, yet
cannot themselves by characterized as large cardinal pictures.  That's
a valid point, which I probably should have mentioned.  On the other
hand, I don't know of any determinacy principle whose consistency
strength is in the same ball park as subtle cardinals, etc.  (Please
correct me if I'm behind the times here.)

 > 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.

That's a very interesting comparison which I hope will be discussed
extensively here on the FOM list.

Just to get the ball rolling, let me provocatively point out that
determinacy has not yet had much impact on what is sometimes called
"core mathematics".  The main impact has been with respect to
projective sets, Sigma^1_2 and above.  Sigma^1_1 sets are of interest
to some mathematicians in some corners of analysis, but ....

By contrast, Friedman's independence results refer to some fairly
elementary combinatorics of finite labeled trees.  Some people may
want to argue that this kind of combinatorics is also rather remote
from "the core".  But I would reply that it is arguably a lot closer
to "the core" than things like the continuum hypothesis and structural
properties of projective sets.  That's why I think this kind of work
by Friedman is of such tremendous f.o.m. importance.

Anyway, this is food for an interesting discussion.

-- Steve

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