Fragility and indestructibility II

Abstract

Abstract. 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 [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 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 [11] 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 .

Trees and Stationary Reflection at Double Successors of Regular Cardinals

“…Fact 2.12. [6,24] Let λ be regular, and let P be square-λ-cc. If T is a tree of height λ, then forcing with P does not add cofinal branches to T .…”

The tree property and the continuum function below

“…Again, we leave it as an exercise for the reader to verify that the cardinals in A are preserved, , and the continuum function below is controlled by f . The proof is basically the same as in using the usual Easton‐style analysis, the product analysis of the forcing in and Lemma . Let be fixed.…”

“…(cf. [, Lemma 4.7]; cf. also [, Theorem 16.30]; it is easy to see that the forcing is actually ‐Knaster by the same argument).…”

Set-theoretic geology

“…In particular, since the second factor Q 2 can be trivial, it follows that every forcing extension V ⊆ V [G], with V -generic filter G ⊆ Q ∈ V , exhibits the δapproximation and δ-covering property for δ = |Q| + , and this is why theorem 6 generalizes theorem 5. Lemma 8 has an improved proof in [HJ10, lemma 12], following Mitchell's treatment [Mit06], which really goes all the way back to [Mit73], and see also the generalization in [Ung,§1].…”