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

Stephen G Simpson simpson at
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.

-- Steve

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