Abstract

We present the method introduced by Neeman of generalized side conditions with two types of models. We then discuss some applications: a variation of the Friedman-Mitchell poset for adding a club with finite conditions, the consistency of the existence of an ω2 increasing chain in, originally proved by Koszmider, and the existence of a thin very tall superatomic Boolean algebra, originally proved by Baumgartner-Shelah. We expect that the present method will have many more applications.

Introduction

We present a generalization of the method of model as side conditions. Generally speaking a poset that uses models as side conditions is a notion of forcing whose elements are pairs, consisting of a working part which is some partial information about the object we wish to add and a finite ϵ-chain of countable elementary substructures of H(θ), for some cardinal θ i.e. the structure consisting of sets whose transitive closure has cardinality less than θ. The models in the side condition are used to control the extension of the working part. This is crucial in showing some general property of the forcing such as properness.

The generalization we now present amounts to allowing also certain uncountable models in the side conditions.