Club isomorphisms on higher Aronszajn trees

Club Stationary Reflection and the Special Aronszajn Tree Property

We prove that it is consistent that Club Stationary Reflection and the Special Aronszajn Tree Property simultaneously hold on $ω_2$, thereby contributing to the study of the tension between compactness and incompactness in set theory. The poset which produces the final model follows the collapse of an ineffable cardinal first with an iteration of club adding (with anticipation) and second with an iteration specializing Aronszajn trees. In the first part of the paper, we prove a general theorem about specializing Aronszajn trees on $ω_2$ after forcing with what we call $\mathcal{F}$-Strongly Pr…

A Large Pairwise Far Family of Aronszajn Trees

We construct a large family of normal $κ$-complete $\mathbb{R}_κ$-embeddable non-special $κ^+$-Aronszajn trees which have no club isomorphic subtrees using an instance of the proxy principle of Brodsky-Rinot.

力迫公理与Aronszajn树的结构理论

Forcing axioms and the structure theory of Aronszajn trees

本文研究Aronszajn树的结构定理对力迫公理的需求，证明MA(ℵ₁)足以保证存在2^ℵ₁棵两两非同构的相干A-树。

Distributive Aronszajn trees

Ben-David and Shelah proved that if $λ$ is a singular strong-limit cardinal and $2^λ=λ^+$, then $\square^*_λ$ entails the existence of a normal $λ$-distributive $λ^+$-Aronszajn tree. Here, it is proved that the same conclusion remains valid after replacing the hypothesis $\square^*_λ$ by $\square(λ^+,{<}λ)$. As $\square(λ^+,{<}λ)$ does not impose a bound on the order-type of the witnessing clubs, our construction is necessarily different from that of Ben-David and Shelah, and instead uses walks on ordinals augmented with club guessing. A major component of this work is the study of postprocess…

A forcing axiom for a non-special Aronszajn tree

Suppose that $T^*$ is an $ω_1$-Aronszajn tree with no stationary antichain. We introduce a forcing axiom PFA($T^*$) for proper forcings which preserve these properties of $T^*$. We prove that PFA($T^*$) implies many of the strong consequences of PFA, such as the failure of very weak club guessing, that all of the cardinal characteristics of the continuum are greater than $ω_1$, and the $P$-ideal dichotomy. On the other hand, PFA($T^*$) implies some of the consequences of diamond principles, such as the existence of Knaster forcings which are not stationarily Knaster.

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。