Preserving Preservation

Suslin树的保持与闭无界同构

Suslin tree preservation and club isomorphisms

本文构造了存在Suslin树且特定正规Aronszajn树闭无界同构的集合论模型，还证明自由正规Suslin树经c.c.c.力迫可实现n-自由且高维导出树特殊的结果。

Preserving Non-Null with Suslin+ forcing

We introduce the notion of effective Axiom A and use it to show that some popular tree forcings are Suslin+. We introduce transitive nep and present a simplified version of Shelah's "preserving a little implies preserving much": If I is a Suslin ccc ideal (e.g. Lebesgue-null or meager) and P is a transitive nep forcing (e.g. P is Suslin+) and P doesn't make any I-positive Borel set small, then P doesn't make any I-positive set small.

A model with Suslin trees but no minimal uncountable linear orders other than $ω_1$ and $-ω_1$

We show that the existence of a Suslin tree does not necessarily imply that there are uncountable minimal linear orders other than $ω_1$ and $-ω_1$, answering a question of J. Baumgartner. This is done by a Jensen-type iteration, proving that one can force CH together with a restricted form of ladder system uniformization on trees, all while preserving a rigid Suslin tree.

Transcendental pairs of generic extensions

We isolate a new preservation class of Suslin forcings and prove several associated consistency results in the choiceless theory ZF+DC regarding countable chromatic numbers of various Borel hypergraphs.

Preservation of a Souslin tree and side conditions

We show how to force, with finite conditions, the forcing axiom PFA(T), a relativization of PFA to proper forcing notions preserving a given Souslin tree T. The proof uses a Neeman style iteration with generalized side conditions consisting of models of two types, and a preservation theorem for such iterations. The consistency of this axiom was previously known by the standard countable support iteration, using a preservation theorem due to Miyamoto.

Cohen Preservation and Independence

We provide a general preservation theorem for preserving selective independent families along countable support iterations. The theorem gives a general framework for a number of results in the literature concerning models in which the independence number $\mathfrak{i}$ is strictly below $\mathfrak{c}$, including iterations of Sacks forcing, Miller partition forcing, $h$-perfect tree forcings, coding with perfect trees. Moreover, applying the theorem, we show that $\mathfrak{i} = \aleph_1$ in the Miller Lite model. An important aspect of the preservation theorem is the notion of "Cohen preserva…