The Tree Property Below

An important remark is that in the model for the main theorem אω is not strong limit, in particular 21 = אω+2. In doing so we avoid a difficult question of Woodin’s which asks whether it is consistent to have the failure of both SCH and weak square at אω. The main theorem of this paper should be compared with a theorem in [9] where a similar result is obtained for successive failures of weak square. The result makes use of the poset and a some of the main lemmas from Neeman’s paper [5]. The reader is advised to have a copy of it on hand. Throughout the paper we have attempted to keep the notation very close to Neeman’s. One notable convention is the use of τ -closed to mean every decreasing sequence of length τ has a lower bound. The paper is organized into sections based on these topics: preliminaries, definition of the main forcing, cardinal structure, the tree property below אω, the tree property at אω+1, the tree property at אω+2, and the tree property above אω+2.