Almost Strong Properness

A strong antidiamond principle compatible with CH

A strong antidiamond principle (*c) is shown to be consistent with CH. This principle can be stated as a "P-ideal dichotomy": every P-ideal on omega-1 (i.e. an ideal that is sigma-directed under inclusion modulo finite) either has a closed unbounded subset of omega-1 locally inside of it, or else has a stationary subset of omega-1 orthogonal to it. We rely on Shelah's theory of parameterized properness for NNR iterations, and make a contribution to the theory with a method of constructing the properness parameter simultaneously with the iteration. Our handling of the application of the NNR ite…

来自决定性模型的共尾强Chang猜想

The Cofinal Strong Chang Conjecture from Models of Determinacy

通过修改Woodin的证明，表明从决定性模型使用P_max力迫得到的模型满足更强的共尾版本的强Chang猜想，并由此证明Namba力迫是半正常的且Θ^{UB}=ω_3的一致性，以及Woodin (*)_{UB}公理蕴含该猜想。

How far is almost strong compactness from strong compactness

Bagaria and Magidor introduced the notion of almost strong compactness, which is very close to the notion of strong compactness. Boney and Brooke-Taylor asked whether the least almost strongly compact cardinal is strongly compact. Goldberg gives a positive answer in the case $\mathrm{SCH}$ holds from below and the least almost strongly compact cardinal has uncountable cofinality. In this paper, we give a negative answer for the general case. Our result also gives an affirmative answer to a question of Bagaria and Magidor.

一个具有强非饱和平方的几乎Kurepa Suslin树

An almost Kurepa Suslin tree with strongly non-saturated square

本文构造Knaster力迫，添加一棵带ω₂个强几乎不相交自同构族的Suslin树，使其平方强非饱和，并引入ρ-分离新方法。

Almost positively closed models

We introduce the notions of almost positively closed models and positive strong amalgamation property. We study the fundamental properties of these notions and develop some interactions between them.

The covering number of the strong measure zero ideal can be above almost everything else

We show that certain type of tree forcings, including Sacks forcing, increases the covering of the strong measure zero ideal $\mathcal{SN}$. As a consequence, in Sacks model, such covering number is equal to the size of the continuum, which indicates that this covering number is consistently larger than any other classical cardinal invariant of the continuum. Even more, Sacks forcing can be used to force that $\mathrm{non}(\mathcal{SN})<\mathrm{cov}(\mathcal{SN})<\mathrm{cof}(\mathcal{SN})$, which is the first consistency result where more than two cardinal invariants associated with $\mathcal…