Forcing closed unbounded subsets of א

1. Introduction Suppressing some important technical hypotheses for the moment, the project under-taken in this paper is this: Given a tree T of height and cardinality ω 1 , construct (uniformly in T) a subset S T ⊆ ℵ ω 1 +1 such that, if W is an outer model of V having the same ω 1 , ℵ ω 1 , and ℵ ω 1 +1 in which S T has a closed unbounded (club) subset, then T has a cofinal branch in W. Conversely, if T has a cofinal branch in W , then there exists a cardinal preserving set forcing extension of W in which S T has a club subset. This reduces the " branch problem " for trees of height and cardinality ω 1 to the " club subset problem " for subsets of ℵ ω 1 +1. The point is that there is no parameter-free first-order characterization of those trees of height and cardinality ω 1 that have a branch in some ω 1-preserving outer model [S1]. It follows that there is no parameter-free first-order characterization of the subsets of ℵ ω 1 +1 that have a club subset in some ω 1 , ℵ ω 1 , and ℵ ω 1 +1 preserving outer model. The reason that we require these three cardinals to be preserved is so that ℵ ω 1 +1 of the inner model is still ℵ ω 1 +1 in the outer model. That is, it maintains its status as the successor of the least limit cardinal of uncountable cofinality. These rules allow that unboundedly many cardinals less than ℵ ω 1 may be collapsed. If we require all cardinals less than or equal to ℵ ω 1 +1 to be preserved, then uncharacterizability follows from work in [S1]. Properties of the form ". .. has. .. in some. .. outer model " are not first-order. They are best understood in terms of standard transitive models. Say that W is an outer model of the standard transitive model V of ZFC, if W ⊇ V , (W ; V) is a 1. INTRODUCTION model of ZFC in a language with a predicate symbol for V , and the models W and V contain the same ordinals. A first-order characterization (relative to ZFC) of a family delineated by a non-first-order property is a first-order formula that defines that family in …