Radin forcing and its iterations

We provide an exposition of supercompact Radin forcing and present several methods for iterating Radin forcing. In this paper we give an exposition of supercompact Radin forcing using coherent sequences of ultrafilters. This version of Radin forcing includes as special cases the Prikry forcing and Magidor forcing, both the measurable and supercompact versions. We also introduce some methods for iterating Radin forcing. First we show how to iterate Radin forcing over the same cardinal infinitely many times. Secondly we show that Magidor’s method of iterating Prikry forcing over different cardinals can be extended to iterate Radin forcing. Radin forcing was introduced in [8]. Mitchell [7] presented a version of Radin forcing which uses coherent sequences of ultrafilters in place of a measure sequence. Foreman and Woodin [5] developed a supercompact version of Radin forcing using measure sequences in the context of a proof that GCH can fail for every cardinal. See [2] for a more recent exposition of Radin forcing on a measurable cardinal. In Section 1 we review notation and prove some technical lemmas we need in the paper. Part I, consisting of Sections 2 to 7, is an exposition of supercompact Radin forcing using coherent sequences. Part II, consisting of Sections 8 and 9, presents two methods for iterating Radin forcing. Section 8 covers iterations of Radin forcing over the same cardinal. Section 9 extends Magidor’s method of iterating Prikry forcing over different cardinals to Radin forcing. 1. Notation and Background We assume that the reader is familiar with forcing, Prikry forcing, and supercompact cardinals; see [3] or [4]. For cardinals κ ≤ λ with κ regular, let Pκλ denote the set of a in [λ] such that a ∩ κ is an ordinal. Then Pκλ is a club subset of [λ]. For a, b in Pκλ, let a ⊂∼ b if a ⊆ b and |a| < b ∩ κ. In this paper, normal ultrafilter means a normal, fine, non-principal ultrafilter on some Pκλ. By fineness we mean that for all i < λ, the set {a ∈ Pκλ : i ∈ a} is in the ultrafilter. Normality is the property that for any function F : Pκλ → λ such that F (a) ∈ a for all a, there is β < λ such that the set {a : F (a) = β} is in the ultrafilter. If U is an ultrafilter on Pκλ and A ⊆ κ, we say that U concentrates on A if the set {a ∈ Pκλ : a ∩ κ ∈ A} is in U ; equivalently, κ ∈ j(A) where j : V → Ult(V,U).