Forcing Closed Unbounded Subsets of N_w+1

Summary

Introduction

This paper addresses a special case of the following problem: Suppose that k is an infinite cardinal. Characterize those X ⊆ k⁺ such that X contains a closed unbounded (club) subset of k⁺ in some k and k⁺ preserving outer model.

If V is a transitive standard model of ZFC, say that W is an outer model of V if W ⊇ V is also a standard transitive model of ZFC and V ∪ OR = W ∪ OR.

Assume, as we shall everywhere, that the GCH holds in the inner model V.

If k = ω, then this problem has a well known solution. There exists a club subset of ω₁ contained in X in some ω₁-preserving outer model iff X is stationary in ω₁. The paper [S] to which this is a sequel shows that for regular k ≥ ω₁ this characterization problem is generally unsolvable and never uniformly solvable in the way it is for k = ω. (See §6 for this statement in more detail.)

This paper addresses a special case of this problem when k is singular, namely, the case of “bounded pattern width” subsets of N_w+1. The ideas employed can be applied more generally, but further ideas are needed to settle the general problem. Consequently, here we shall be content to restrict ourselves to this case.