Strongly adequate sets and adding a club with finite conditions

We continue the study of adequate sets which we began in [2] by introducing the idea of a strongly adequate set, which has an additional requirement on the overlap of two models past their comparison point. We present a forcing poset for adding a club to a fat stationary subset of ω2 with finite conditions, thereby showing that a version of the forcing posets of Friedman [1] and Mitchell [3] for adding a club on ω2 can be developed in the context of adequate sets. The idea of an adequate set of models was introduced by the author in [2]. Roughly speaking, an adequate set is a set consisting of countable models which are pairwise membership comparable below a particular ordinal called their comparison point. The relevance of the comparison point is that the two models have only a finite overlap past this ordinal. We presented a general framework in [2] for using adequate sets of models as side conditions in forcing on ω2 with finite conditions. Examples of forcings which fit into this framework include adding a generic function on ω2, forcing a nonreflecting stationary subset of ω2 ∩ cof(ω), and adding an ω1Kurepa tree. In earlier work Friedman [1] and Mitchell [3] separately introduced forcing posets which add a club to a fat stationary subset of ω2 with finite conditions, using countable models as side conditions. In this paper we develop an analogue of these forcings in the context of adequate sets. To achieve this, we introduce the idea of a strongly adequate set of models, which differs from an adequate set by obeying an additional requirement on the overlap of models past their comparison point. This paper is a sequel to [2]. We assume that the reader is familiar with the material in Sections 1–3 of that paper. Our forcing poset is similar to the FriedmanMitchell posets in the sense that we approximate a generic club using intervals. Neeman’s method [4] for adding a club is somewhat different; he adds a generic sequence of models with finite conditions for which the suprema of models appearing on the sequence form a club. 1. Background Assumptions and Notation For easy reference, we review here the notation, concepts, and results of Sections 1–3 of [2]. Assumption 1: Assume 21 = ω2. 2010 Mathematics Subject Classification. 03E40.