Fragility and indestructibility II
Annals of Pure and Applied Logic
In this paper we continue work from a previous paper on the fragility and indestructibility of the tree property. We present the following: (1) A preservation lemma implicit in Mitchell's PhD thesis, which generalizes all previous versions of Hamkins' Key lemma. (2) A new proof of theorems the 'superdestructibility' theorems of Hamkins and Shelah. (3) An answer to a question from our previous paper on the apparent consistency strength of the assertion "The tree property at ℵ 2 is indestructible
... under ℵ 2 -directed closed forcing". (4) Two models for successive failures of weak square on long intervals of cardinals. Techniques for preserving the tree property are central to a growing literature of consistency results obtaining the tree property as successive regular cardinals (see for example [11, 1, 3, 15, 20]). These techniques can be viewed abstractly as indestructibility results, which typically arise from either integration of preparation forcing or preservation lemmas. In our original paper  we proved results using both of these methods. In particular, using methods of Abraham and Cummings and Foreman [3, 1] we showed that modulo the existence of a supercompact cardinal it is consistent that the tree property holds at ω 2 and is indestructible under ω 2 -directed closed forcing. Further, by proving a new preservation lemma we showed that the tree property at ω 2 in a model of Mitchell  is indestructible under the forcing to add an arbitrary number of Cohen reals. It follows that the tree property at ω 2 is consistent with 2 ω > ω 2 . The preservation lemma was Lemma 0.1. Let τ, η be cardinals with η regular and 2 τ ≥ η. Let P be τ + -cc and R be τ + -closed. LetṪ be a P-name for an η-tree. Then in V [P] forcing with R cannot add a branch through T .