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 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 ℵ2ℵ2 is indestructible under ℵ2ℵ2-directed closed forcing”.(4)Two models for successive failures of weak square on long intervals of cardinals.