Abstract
One consequence of Shelah's research in Set Theory
is the completely unexpected power of the axioms
of ZFC; notable examples are Shelah's theorem that
ZFC implies both diamond on
P
1(2)
and that cofinality of
P1()
is less than
4.
The latter result is
a single instance of a broad family of results obtained from
PCF theory.
Earlier results demonstrated the analogous power of
large cardinal axioms, by Laver's theorem if
kappa is supercompact then diamond holds
on Pkappa(lambda) for all lambda
and by Solovay's theorem
the Singular Cardinals Hypothesis holds above
a supercompact cardinal. In fact above supercompact cardinal
Shelah's PCF theory looks trivial.
Is there a nontrivial PCF theory above a supercompact
cardinal? We discuss emerging evidence that there is such
a theory and the connections
to the inner model problem for large cardinals.
H. Woodin