Fragility and indestructibility II

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 א2 is indestructible under א2-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 [18] 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 ω2 and is indestructible under ω2-directed closed forcing. Further, by proving a new preservation lemma we showed that the tree property at ω2 in a model of Mitchell [11] is indestructible under the forcing to add an arbitrary number of Cohen reals. It follows that the tree property at ω2 is consistent with 2 ω > ω2. The preservation lemma was Lemma 0.1. Let τ, η be cardinals with η regular and 2 ≥ η. Let P be τ-cc and R be τ-closed. Let Ṫ be a P-name for an η-tree. Then in V [P] forcing with R cannot add a branch through T . Date: June 23, 2015. 2010 Mathematics Subject Classification. 03E35,03E55.