Global square and mutual stationarity at the [formula omitted

Abstract

We give the proof of a theorem of Jensen and Zeman on the existence of a global □ sequence in the Core Model below a measurable cardinal κ of Mitchell order ( o M ( κ ) ) equal to κ + + , and use it to prove the following theorem on mutual stationarity at ℵ n . Let ω 1 denote the first uncountable cardinal of V and set Cof ( ω 1 ) to be the class of ordinals of cofinality ω 1 . Theorem If every sequence ( S n ) n < ω of stationary sets S n ⊆ Cof ( ω 1 ) ∩ ℵ n + 2 , is mutually stationary, then there is an inner model with infinitely many inaccessibles ( κ n ) n < ω so that for every m the class of measurables λ with o M ( λ ) ≥ κ m is, in V , stationary in κ n for all n > m . In particular, there is such a model in which for all sufficiently large m < ω , the class of measurables λ with o M ( λ ) ≥ ω m is, in V , stationary below ℵ m + 2 .