Cocovering and Set Forcing

1. Introduction Let V [G] be the result of adding a Cohen real and a Cohen subset of ω 2 to a model V of the GCH. Every infinite cardinal gets a new subset in V [G], namely, the Cohen real. Yet, in some sense, only ω and ω 2 get new subsets. One way to capture this is to say that a sort of dual to Covering holds between V and V [G], namely, " κ-cocovering " for κ other than ω and ω 2. We need some definitions. If κ is a regular cardinal in V , say that κ-cocovering holds between V and the outer model W if, given any unbounded a ⊆ κ that lies in W , there exists b ∈ V such that b ⊆ a and b is unbounded in κ. If V is a standard transitive model of ZFC, say that W ⊇ V is an outer model of V if W is also a standard transitive model of ZFC and V ∩ OR = W ∩ OR. In this paper we are only concerned with outer models such that (W ; V) satisfies ZFC (in a language with a predicate symbol for V). Ultimately, ZFC is our metatheory; talk of models can be understood as talk of set, or even countable set models. The reader will note that our results usually can be paraphrased in more traditional terminology, perhaps at the cost of some generality. This paper proves a sort of converse to the observation with which we began. The general case is not quite so simple as this example suggests. We show that if W is any outer model of V , and sufficient cocovering (and maybe a little covering) hold around κ, then every subset of κ that lies in W is set generic over V for a forcing of V-cardinality less than κ. To be precise, let us begin with a special case in which we assume a little of the GCH in the inner model V. Corollary 2.12 draws the following corollary of our main theorem. Theorem. Assume that • in V : κ is regular, κ <κ = κ, and 2 κ = κ +. Also assume that • both κ and κ +-cocovering hold between V and W. Then every subset of κ + that lies in W is set generic over V for a …