Adding a club with finite conditions, Part II

We define a forcing poset which adds a club subset of a given fat stationary set S ⊆ ω2 with finite conditions, using S-adequate sets of models as side conditions. This construction, together with the general amalgamation results concerning S-adequate sets on which it is based, is substantially shorter and simpler than our original version in [3]. The theory of adequate sets introduced in [2] provides a framework for adding generic objects on ω2 with finite conditions using countable models as side conditions. Roughly speaking, an adequate set is a set of models A such that for all M and N in A, M and N are either equal or membership comparable below their comparison point βM,N . A technique which was central to the development of adequate sets in [2], as well as to our original forcing for adding a club to a fat stationary subset of ω2 in [3], involves taking an adequate set A and enlarging it to an adequate set which contains certain initial segments of models in A. In this paper we prove amalgamation results for adequate sets which avoid the method of adding initial segments of models. It turns out that these new results drastically simplify the amalgamation results from [3] for strongly adequate sets. As a result we are able to develop a forcing poset for adding a club to a given fat stationary subset of ω2 with finite conditions which is substantially shorter than our original argument in [3]. Forcing posets for adding a club to ω2 with finite conditions were originally developed by Friedman [1] and Mitchell [5], and then later by Neeman [6]. Adequate set forcing was introduced in [2] in an attempt to simplify and generalize the methods used by the first two authors. This new framework is also flexible as it admits useful variations. For example, in a subsequent paper [4] we show that the forcing poset for adding a club presented below can be modified to preserve CH, answering a problem of Friedman [1].