An Smax Variation for One Souslin Tree

We present a variation of the forcing Smax as presented in Woodin [4]. Our forcing is a Pmax-style construction where each model condition selects one Souslin tree. In the extension there is a Souslin tree TG which is the direct limit of the selected Souslin trees in the models of the generic. In some sense, the generic extension is a maximal model of "there exists a minimal Souslin tree," with TG being this minimal tree. In particular, in the extension this Souslin tree has the property that forcing with it gives a model of Souslin's Hypothesis. ?1. Preliminaries. This paper relies on the P~max forcing machinery developed in Woodin [4]. All proofs which are essentially identical to the corresponding proofs in that book have been omitted or sketched. In almost all cases in this paper, co -trees are considered to be partial orders on co, satisfying the usual definitions, along with the condition that for tree T and ordinals ae, fl, levT(ae) > levT (fl) -* ar > fl (in the ordinal ordering, where levT (fl) = y means that fl is on level y of T). So, for instance, 0 is the least node in any tree with a unique root, and the remaining ordinals of co are all on the first level. A trivial subtree of a tree T is the subtree above some node ar, denoted T., All of the trees we construct in this paper will be normal, in the terminology of Jech [1], and so will have a unique root and co immediate successors for every node. We use lower case letters, usually p or q, to indicate paths through trees. The term chain is used to refer to a subset of a tree linearly ordered by its relation. The terms path and branch are used interchangeably to refer to chains p which contain the root and satisfy the property that if a <?T Pl <?T y, and ar and y are in p, then so is fi. Thus, an cw,-path, or co,-branch, of an co,-tree is a chain which extends through all levels of the tree, and the path derived from a particular chain is just the smallest path which contains it. By a path through a tree, we mean a path which extends through the entire height of the tree. We use the notation that T 8 6 is the tree derived from taking the first 8 levels of T. INS denotes the nonstationary ideal on co,. Our forcing construction is called STWax in which we preserve the Sousliness of one particular Souslin tree. This tree will satisfy a particular homogeneity property, as defined below. Received September 30, 1996: revised March 31, 1997 ? 1999. Association for Symbolic Logic 0022-4812/99/6401-0008/$2.80