Some applications of mixed support iterations

Abstract. We give some applications of mixed support forcing iterations to the topics of disjoint stationary sequences and internally approachable sets. In the first half of the paper we study the combinatorial content of the idea of a disjoint stationary sequence, including its relation to adding clubs by forcing, the approachability ideal, canonical structure, the proper forcing axiom, and properties related to internal approachability. In the second half of the paper we present some consistency results related to these ideas. We construct a model in which a disjoint stationary sequence exists at the successor of an arbitrary regular uncountable cardinal. We also construct models in which the properties of being internally stationary, internally club, and internally approachable are distinct. In [16] we worked out a general schema for a mixed support forcing iteration. In this paper we present some applications of this forcing iteration to the topics of disjoint stationary sequences and internally approachable sets. Let κ be a regular uncountable cardinal. A disjoint stationary sequence on κ+ is a sequence 〈Sα: α