Forcing with adequate sets of models as side conditions

We present a general framework for forcing on ω2 with finite conditions using countable models as side conditions. This framework is based on a method of comparing countable models as being membership related up to a large initial segment. We give several examples of this type of forcing, including adding a function on ω2, adding a nonreflecting stationary subset of ω2 ∩ cof(ω), and adding an ω1-Kurepa tree. The method of forcing with countable models as side conditions was introduced by Todorčević ([13]). The original method is useful for forcing with finite conditions to add a generic object of size ω1. The preservation of ω1 is achieved by including finitely many countable elementary substructures as a part of a forcing condition. The models which appear in a condition are related by membership. So a condition in such a forcing poset includes a finite approximation of the object to be added, together with a finite ∈-increasing chain of models, with some relationship specified between the finite fragment and the models. Friedman ([3]) and Mitchell ([10], [11]) independently lifted this method up to ω2 by showing how to add a club subset of ω2 with finite conditions. In the process of going from ω1 to ω2, they gave up the requirement that models appearing in a forcing condition are membership related, replacing it with a more complicated relationship between the models. Later Neeman ([12]) developed a general approach to the subject of forcing with finite conditions on ω2. A major feature of Neeman’s approach is that a condition in his type of forcing poset includes a finite ∈-increasing chain of models, similar to Todorčević’s original idea, but he includes both countable and uncountable models in his conditions, rather than just countable models. Other recent papers in which side conditions are used to add objects of size ω2 include [1], [2], [5], and [14]. In this paper we present a general framework for forcing a generic object on ω2 with finite conditions, using countable models as side conditions. This framework is based on a method for comparing elementary substructures which, while not as simple as comparing by membership, is still natural. Namely, the countable models appearing in a condition will be membership comparable up to a large initial segment. The largeness of the initial segment is measured by the fact that above the point of comparison, the models have only a finite amount of disjoint overlap. We give several examples of this kind of forcing poset, including adding a generic function on ω2, adding a nonreflecting stationary subset of ω2∩ cof(ω), and adding an ω1-Kurepa tree. Since these three kinds of objects can be forced using classical methods, the purpose of these examples is to illustrate the method, rather than proving new consistency results.