FOM: Large Cardinals and other areas of mathematics

Benedikt Loewe loewe at
Mon Mar 16 14:38:59 EST 1998

On Sun, 15 Mar 1998, Stephen G Simpson wrote:

> to obtain the combinatorial theorems in question.  Moreover: (2) 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.

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. 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

Cf. Ronald Bjoern Jensen, Inner Models and Large Cardinals, Bulletin of
Symbolic Logic 1 (1995), p. 393 - 407, especially section 4 and 5
Akihiro Kanamori, Menachem Magidor, The evolution of large cardinal axioms
in set theory, in: Gert H.Mueller, Dana S.Scott (eds.), Higher Set Theory,
Springer Lecture Notes in Mathematics 669, Berlin 1978, p. 99 - 276
and many, many other places in the literature.

                                                   Benedikt Loewe
                                                      UC Berkeley
                                        Department of Mathematics
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                                                Berkeley CA 94720
                                                 (510) 848 - 8512

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