Adding clubs with square

We present a technique for destroying stationary subsets of Pκκ using partial square sequences. We combine this method with Gitik’s poset for changing the cofinality of a cardinal without adding bounded sets to prove a variety of consistency results concerning saturated ideals and the set S(κ, κ+). In this paper we continue our study of consistency results concerning the set S(κ, κ) = {a ∈ Pκκ : o.t.(a) = (a ∩ κ)} from [5] and [6]. We present a method for destroying the stationarity of certain subsets of Pκκ, where κ is inaccessible, using partial square sequences. This method of destroying stationary sets has a variety of applications. We prove several results which support the general theme that the structure of S(κ, κ) can vary greatly depending on the particular model considered. For example, if κ is supercompact and μ < κ is regular, then there exists a generic extension in which S(κ, κ) is stationary and for almost all a in S(κ, κ), a∩κ is singular with cofinality μ. On the other hand, it is relatively consistent that for almost all a in S(κ, κ), a ∩ κ is measurable with Mitchell order (a ∩ κ). We also construct a model in which GCH holds and there is a stationary set S ⊆ Pκκ such that NS S is saturated and for all regular μ < κ, {a ∈ S : cf(a ∩ κ) = μ} is stationary. This material is related to our previous papers [5] and [6]. However, no prior acquaintance with these papers is required by the reader. Most of our consistency results use Easton support Prikry iterations. The theorems in this paper rely on Gitik’s technique from [3] for changing the cofinality of a cardinal without adding bounded sets. We present Gitik’s poset in complete detail. The contents of the paper are as follows. In Section 1 we outline preliminaries and notation. Section 2 provides additional background on forcing posets and large cardinals. Section 3 describes Easton support Prikry iterations. In Section 4 we present our method for destroying stationary sets using partial square sequences. In Section 5 we construct a model in which almost all a in S(κ, κ) satisfy that a∩κ is a measurable cardinal with Mitchell order (a∩ κ). Section 6 describes Gitik’s forcing poset for changing a cofinality without adding bounded sets. In Section 7 we show how to construct models in which we can control the cofinality of a∩κ for almost all a in S(κ, κ). In Section 8 we construct a model with a stationary set S ⊆ Pκκ such that NS S is saturated, GCH holds, and for every regular μ less than κ, the set {a ∈ S : cf(a ∩ κ) = μ} is stationary.