棍棒

Club-guessing, stationary reflection, and coloring theorems

We obtain strong coloring theorems at successors of singular cardinals from failures of certain instances of simultaneous reflection of stationary sets. Along the way, we establish new results in club-guessing and in the general theory of ideals.

Club guessing 工具箱 I

A club guessing toolbox I

本文综述文献中各种 club-guessing 形式，并系统研究 club-guessing 序列的改进方式，特别是猜测频率的校准，为不熟悉该领域的读者提供独立说明章节。

无顶端扩展子的基于扩展子的Magidor-Radin迫迫

Extender-based Magidor-Radin forcings without top extenders

本文发展了无顶端扩展子的基于扩展子的Magidor-Radin迫迫，用于在不可达基数的club子集上实现SCH失败，并构造基数算术行为在stationary类上不同的模型。

Club Chang's Conjecture

Chang's Conjecture (CC) asserts that for every $F:[ω_2]^{<ω} \to ω_2$, there exists an $X$ that is closed under $F$ such that $|X|=ω_1$ and $|X \cap ω_1| =ω$. By classic results of Silver and Donder, CC is equiconsistent with an $ω_1$-Erdos cardinal. Using stronger large cardinal assumptions (between $o(κ) = κ^+$ and $o(κ) = κ^{++}$), we prove that it is consistent to also require that $X$ contains a closed unbounded set of ordinals in $\text{sup}(X \cap ω_2)$. We denote this stronger principle \textbf{Club-CC}, and also show that, unlike CC, Club-CC implies failure of certain weak square prin…

Towers and clubs

We revisit several results concerning club principles and nonsaturation of the nonstationary ideal, attempting to improve them in various ways. So we typically deal with a (non necessarily normal) ideal $J$ extending the nonstationary ideal on a regular uncountable (non necessarily successor) cardinal $κ$, our goal being to witness the nonsaturation of $J$ by the existence of towers (of length possibly greater than $κ^+$).

Towers and gaps at uncountable cardinals

Our goal is to study the pseudo-intersection and tower numbers on uncountable regular cardinals, whether these two cardinal characteristics are necessarily equal, and related problems on the existence of gaps. First, we prove that either $\mathfrak p(κ)=\mathfrak t(κ)$ or there is a $(\mathfrak p(κ),λ)$-gap of club-supported slaloms for some $λ< \mathfrak p(κ)$. While the existence of such gaps is unclear, this is a promising step to lift Malliaris and Shelah's proof of $\mathfrak p=\mathfrak t$ to uncountable cardinals. We do analyze gaps of slaloms and, in particular, show that $\mathfrak p(…

AD$^+$ 蕴含 $ω_1$ 是 $Θ$-Berkeley 基数

AD$^+$ implies that $ω_1$ is a $Θ$-Berkeley cardinal

本文在 AD$^+$ 下证明正则 Suslin 基数、$ω_1$ 及 ${\tildeδ}^1_{2n}$ 等具有不同强度的 $Θ$-Berkeley 性质，并说明 $ω_1$ 非 $Θ^+$-Berkeley 从而结果最优。

$P_κ(λ)$的$μ$-俱乐部：天堂中的乐园

$μ$-Clubs of $P_κ(λ)$ : Paradise in heaven

该研究在无大基数内模型等条件下，证明了$P_κ(λ)$与$P_κ(u(κ,λ))$上的$μ$-俱乐部滤子同构，且其对偶理想不满足特定饱和性，推进了无穷基数组合相关研究。

Measuring club sequences, together with the Continuum Hypothesis

We answer a question of Moore by building a forcing extension satisfying measuring together with CH. The construction works over any model of ZFC and can be described as a forcing iteration with countable structures as side conditions and with symmetry constraints. Also, we show that a small variation of this construction produces a model of measuring together with the continuum being larger than the second uncountable cardinal.

Forcing a $\square(κ)$-like principle to hold at a weakly compact cardinal

Hellsten \cite{MR2026390} proved that when $κ$ is $Π^1_n$-indescribable, the \emph{$n$-club} subsets of $κ$ provide a filter base for the $Π^1_n$-indescribability ideal, and hence can also be used to give a characterization of $Π^1_n$-indescribable sets which resembles the definition of stationarity: a set $S\subseteqκ$ is $Π^1_n$-indescribable if and only if $S\cap C\neq\emptyset$ for every $n$-club $C\subseteqκ$. By replacing clubs with $n$-clubs in the definition of $\Box(κ)$, one obtains a $\Box(κ)$-like principle $\Box_n(κ)$, a version of which was first considered by Brickhill and Welch …

Another method to add a closed unbounded set of former regulars

A club consisting of former regulars is added to an inaccessible cardinal, without changing cofinalities outside it. The initial assumption is optimal. A variation of the Radin forcing without a top measurable cardinal is introduced for this.

Cardinal characteristics of the continuum and partitions

We prove combinatorial theorems concerning the stick principle and cardinal characteristics.

Approximating diamond principles on products at an inaccessible cardinal

We isolate \emph{the approximating diamond principles}, which are consequences of the diamond principle at an inaccessible cardinal. We use these principles to find new methods for negating the diamond principle at large cardinals. Most notably, we demonstrate, using Gitik's overlapping extenders forcing, a new method to get the consistency of the failure of the diamond principle at a large cardinal $θ$ without changing cofinalities or adding fast clubs to $θ$. In addition, we show that the approximating diamond principles necessarily hold at a weakly compact cardinal. This result, combined wi…

$P_κ(λ)$上的$μ$-俱乐部：人间天堂

$μ$-Clubs OF $P_κ(λ)$ : Paradise on earth

该研究在ZFC公理框架下，针对无穷基数三元组$(μ, κ, λ)$，证明存在大量满足$u(κ,λ)>λ$的情形，使得$P_κ(λ)$与$P_κ(u(κ,λ))$上的$μ$-俱乐部滤子同构，拓展了$V=L$下的相关结论。

来自扩展逻辑的内模型：第二部分

Inner Models from Extended Logics: Part 2

本文引入由平稳逻辑生成的内模型 $C(aa)$，在 Woodin 基数类或 $MM^{++}$ 假设下证明其正则不可数基数可测、理论力迫绝对且满足 CH，并借助 club 确定性与 aa-鼠标完成构造与证明。

Characterizations of the weakly compact ideal on $P_κλ$

Hellsten \cite{MR2026390} gave a characterization of $Π^1_n$-indescribable subsets of a $Π^1_n$-indescribable cardinal in terms of a natural filter base: when $κ$ is a $Π^1_n$-indescribable cardinal, a set $S\subseteqκ$ is $Π^1_n$-indescribable if and only if $S\cap C\neq\emptyset$ for every $n$-club $C\subseteq κ$. We generalize Hellsten's characterization to $Π^1_n$-indescribable subsets of $P_κλ$, which were first defined by Baumgartner. After showing that under reasonable assumptions the $Π^1_0$-indescribability ideal on $P_κλ$ equals the minimal \emph{strongly} normal ideal $\text{NSS}_{κ…

Forcing Diamond 及其在可迭代性中的应用

Forcing Diamond and Applications to Iterability

本文证明高阶Sacks力迫与club Miller力迫可添加diamond序列，并肯定回答了κ-Sacks力迫的κ-支撑迭代不坍缩κ+且为κ-proper的长期问题，结果适用于其他高阶树力迫。