FOM: f.o.m. significance of Friedman's work (corrections)
Stephen G Simpson
simpson at math.psu.edu
Fri Mar 27 14:55:08 EST 1998
In my previous posting I wrote:
> 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
a vast gulf
> 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. ....
The last sentence quoted above is not quite correct. The properties
of an inaccessible cardinal being Rowbottom, Ramsey, etc are
non-absolute, but some related large cardinal properties are well
known to be downward absolute. For instance, it is well known that if
kappa is n-subtle in V then it is n-subtle in L.
This correction does not affect the point that I was making. The
point was that there is a vast gulf between finitary combinatorics on
the one hand, and large cardinal combinatorics on the other.
More information about the FOM