Generalized Prikry forcing and iteration of generic ultrapowers

Moreover Bukovský [1] and Dehornoy [2] showed that the generic extension Mω[〈j0,n(κ) | n ∈ ω〉] is ⋂ n∈ω Mn in Theorem 1.1. (For the history of these results, read the introduction of Dehornoy [2] and pp.259-260 of Kanamori [6]. ) In Dehornoy [3], these results were generalized for the forcing of Magidor [7] which changes a measurable cardinal of higher Mitchell order into a singular cardinal of uncountable cofinality. In this paper we generalize Theorem 1.1 for normal filters which are not necessarily maximal. Of course, the above theorem can be restated using the dual ideal of U . In this paper we argue using ideals instead of filters. Assume κ is regular uncountable and I is a normal precipitous ideal on κ. Prikry forcing has two natural generalizations, PR∗ I and PR + I . PR ∗ I consists of all pairs 〈t, T 〉 such that t ∈ κ and T ⊆ κ is a tree in which every node has I-measure 1 immediate successors, i.e. {α < κ | s ̂ 〈α〉 ∈ T} is of I-measure 1 for every s ∈ T . PR I consists of all pairs 〈t, T 〉 such that t ∈ κ and T ⊆ κ is a tree in which every node has I-positive immediate successors. In both PR∗ I and PR + I , the order is defined as 〈t0, T0〉 ≤ 〈t1, T1〉 if and only if for every s0 ∈ T0, there is an s1 ∈ T1 such that t0 ̂ s0 = t1 ̂ s1. Note that if I is maximal then PR∗ I and PR + I coincide and this is the tree type Prikry forcing notion. Note also that if κ = ω2 and I is the ideal of bounded subsets of ω2 then PR I is the Namba forcing notion. So PR + I is often treated as a variant of Namba forcing. On the other hand, the iteration of ultrapowers has an obvious generalization, the iteration of generic ultrapowers. Unlike the iteration of ultrapowers, the iteration of generic ultrapowers is not uniquely determined by I. It depends on the choice of generic filters by which the ultrapowers are constructed.