### Pcf theory and Woodin cardinals

by Gitik and Schindler and Shelah. [GSSh:805]

Proc Logic Colloquium'2002 (ASL), 2006

We prove the following two results.
Theorem A: Let alpha be a limit ordinal. Suppose that
2^{| alpha |}< aleph_alpha and 2^{| alpha |^+}< aleph_{|
alpha |^+}, whereas aleph_alpha^{| alpha |}> aleph_{|
alpha |^+} . Then for all n< omega and for all bounded X subsetaleph_{| alpha |^+}, M_n^#(X) exists.
Theorem B: Let kappa be a singular cardinal of uncountable
cofinality. If {alpha < kappa | 2^alpha = alpha^+} is
stationary as well as co-stationary then for all n< omega and
for
all bounded X subset kappa, M_n^#(X) exists.
Theorem A answers a question of Gitik and Mitchell, and Theorem
B
yields a lower bound for an assertion discussed in Gitik, M.,
Introduction to Prikry type forcing notions, in: Handbook of set
theory, Foreman, Kanamori, Magidor (see Problem 4 there).
The proofs of these theorems combine pcf theory with core model
theory. Along the way we establish some ZFC results in cardinal
arithmetic, motivated by Silver's theorem and we obtain results of
core model theory, motivated by the task of building a ``stable core
model.'' Both sets of results are of independent interest.

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