 
Summary: PROPER FORCING REMASTERED
BOBAN VELICKOVI´C AND GIORGIO VENTURI
Abstract. We present the method introduced by Neeman of gener
alized side conditions with two types of models. We then discuss some
applications: a variation of the FriedmanMitchell poset for adding a
club with finite conditions, the consistency of the existence of an 2 in
creasing chain in (1
1 ,
existence of a thin very tall superatomic Boolean algebra, originally
proved by BaumgartnerShelah. 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 
chain of countable elementary substructures of H(), for some cardinal
i.e. the structure consisting of sets whose transitive closure has cardinality
less than . The models in the side condition are used to control the
extension of the working part. This is crucial in showing some general
