On a conjecture of Tarski on products of cardinals

Suslin cardinals and cutpoints in mouse limits

We obtain a partial result on the following conjecture. Conjecture. Let (P, Σ) be a projectum stable mouse pair, and let κbe a cardinal of V such that κ< o(M_\infty(P, Σ)); then the following are equivalent: (1) κis a Suslin cardinal, (2) κis a cutpoint of M_\infty(P, Σ).

On the consistency of local and global versions of Chang's Conjecture

We show that for many pairs of infinite cardinals $κ> μ^+ > μ$, $(κ^{+}, κ)\twoheadrightarrow (μ^+, μ)$ is consistent relative to the consistency of a supercompact cardinal. We also show that it is consistent, relative to a huge cardinal that $(κ^{+}, κ)\twoheadrightarrow (μ^+, μ)$ for every successor cardinal $κ$ and every $μ< κ$, answering a question of Foreman.

Galvin猜想与弱沉淀理想

Galvin's Conjecture and Weakly Precipitous Ideals

本文通过组合博弈证明温和的大基数假设使ω₁上所有正规理想满足弱沉淀性，并将Galvin猜想的证明推广到更弱的大基数假设下。

Tameness from Large Cardinal Axioms

We show that Shelah's Eventual Categoricity Conjecture follows from the existence of class many strongly compact cardinals. This is the first time the consistency of this conjecture has been proven. We do so by showing that every AEC with $LS(K)$ below a strongly compact cardinal $κ$ is $< κ$ tame and applying the categoricity transfer of Grossberg and VanDieren. These techniques also apply to measurable and weakly compact cardinals and we prove similar tameness results under those hypotheses. We isolate a dual property to tameness, called \emph{type shortness}, and show that it follows simila…

Todorcevic关于Rado猜想的难题

Todorcevic's Problem on Rado's Conjecture

本文证明了Rado猜想在所有正则基数上一致成立，从而回答了Todorčević在2024年Mostowski讲座中提出的关于两个连续基数上Rado猜想一致性的问题。

Singular cofinality conjecture and a question of Gorelic

We give an affirmative answer to a question of Gorelic \cite{Gorelic}, by showing it is consistent, relative to the existence of large cardinals, that there is a proper class of cardinals $α$ with $cf(α)=ω_1$ and $α^ω> α.$

Shelah's Categoricity Conjecture from a successor for Tame Abstract Elementary Classes

Let K be an Abstract Elemenetary Class satisfying the amalgamation and the joint embedding property, let μbe the Hanf number of K. Suppose K is tame. MAIN COROLLARY: (ZFC) If K is categorical in a successor cardinal bigger than \beth_{(2^μ)^+} then K is categorical in all cardinals greater than \beth_{(2^μ)^+}. This is an improvment of a Theorem of Makkai and Shelah ([Sh285] who used a strongly compact cardinal for the same conclusion) and Shelah's downward categoricity theorem for AECs with amalgamation (from [Sh394]).

Categoricity and multidimensional diagrams

We study multidimensional diagrams in independent amalgamation in the framework of abstract elementary classes (AECs). We use them to prove the eventual categoricity conjecture for AECs, assuming a large cardinal axiom. More precisely, we show assuming the existence of a proper class of strongly compact cardinals that an AEC which has a single model of some high-enough cardinality will have a single model in any high-enough cardinal. Assuming a weak version of the generalized continuum hypothesis, we also establish the eventual categoricity conjecture for AECs with amalgamation.

关于Weasel的Mouse序的一个猜想

On a Conjecture Regarding the Mouse Order for Weasels

研究Steel关于Ω+1-可迭代1-小型weasels的Mouse序猜想，证明其在存在满足KP且有Σ₁-Woodin基数的可迭代premouse时不成立，并解决相关开放问题。

Shelah's eventual categoricity conjecture in universal classes. Part II

We prove that a universal class categorical in a high-enough cardinal is categorical on a tail of cardinals. As opposed to other results in the literature, we work in ZFC, do not require the categoricity cardinal to be a successor, do not assume amalgamation, and do not use large cardinals. Moreover we give an explicit bound on the "high-enough" threshold: $\mathbf{Theorem}$ Let $ψ$ be a universal $\mathbb{L}_{ω_1, ω}$ sentence. If $ψ$ is categorical in some $λ\ge \beth_{\beth_{ω_1}}$, then $ψ$ is categorical in all $λ' \ge \beth_{\beth_{ω_1}}$. As a byproduct of the proof, we show that a conj…

数学中小基数的紧致性：原理、后果与局限性

Compactness for small cardinals in mathematics: principles, consequences, and limitations

本文讨论小正则基数（如ω_n、ℵ_{ω+1}等）的紧致性原理，分为逻辑原理（树性质）和数学原理（图、群、拓扑空间的紧致性），重点研究这些原理在力迫扩张中的不可破坏性，并观察传统问题（如Suslin假设、Whitehead猜想）与ω_2处最强紧致性形式的独立性。

Borel's Conjecture in Topological Groups

We introduce a natural generalization of Borel's Conjecture. For each infinite cardinal number $κ$, let {\sf BC}$_κ$ denote this generalization. Then ${\sf BC}_{\aleph_0}$ is equivalent to the classical Borel conjecture. Assuming the classical Borel conjecture, $\neg{\sf BC}_{\aleph_1}$ is equivalent to the existence of a Kurepa tree of height $\aleph_1$. Using the connection of ${\sf BC}_κ$ with a generalization of Kurepa's Hypothesis, we obtain the following consistency results: (1)If it is consistent that there is a 1-inaccessible cardinal then it is consistent that ${\sf BC}_{\aleph_1}$. (…

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…

Fallen Cardinals

We prove that for every singular cardinal mu of cofinality omega, the complete Boolean algebra compP_mu(mu) contains as a complete subalgebra an isomorphic copy of the collapse algebra Comp Col(omega_1,mu^{aleph_0}). Consequently, adding a generic filter to the quotient algebra P_mu(mu)=P(mu)/[mu]^{<mu} collapses mu^{aleph_0} to aleph_1. Another corollary is that the Baire number of the space U(mu) of all uniform ultrafilters over mu is equal to omega_2. The corollaries affirm two conjectures by Balcar and Simon. The proof uses pcf theory.

Shelah's eventual categoricity conjecture in universal classes: part I

We prove: $\mathbf{Theorem}$ Let $K$ be a universal class. If $K$ is categorical in cardinals of arbitrarily high cofinality, then $K$ is categorical on a tail of cardinals. The proof stems from ideas of Adi Jarden and Will Boney, and also relies on a deep result of Shelah. As opposed to previous works, the argument is in ZFC and does not use the assumption of categoricity in a successor cardinal. The argument generalizes to abstract elementary classes (AECs) that satisfy a locality property and where certain prime models exist. Moreover assuming amalgamation we can give an explicit bound on t…

关于Halin端度猜想的评论

Remarks on Halin's end-degree Conjecture

本文在ZFC框架下证明了Halin端度猜想的新实例，并探讨了其与奇异基数假设的关系，揭示了该猜想在特定模型中的失败情况，提供了新的独立性结果。

The generic pair conjecture for dependent finite diagrams

This paper generalizes Shelah's generic pair conjecture (now theorem) for the measurable cardinal case from first order theories to finite diagrams. We use homogeneous models in the place of saturated models.

线性序的加法算术

The Additive Arithmetic of Linear Orders

该研究系统发展线性序的加法算术，通过欧几里得算法解决塔尔斯基1956年提出的线性序加法交换问题，刻画加法交换对及可在有序和中表示的交换半群。

The Strength of Menger's Conjecture

Menger conjectured that subsets of R with the Menger property must be $σ$-compact. While this is false when there is no restriction on the subsets of R, for projective subsets it is known to follow from the Axiom of Projective Determinacy, which has considerable large cardinal consistency strength. We note that in fact, Menger's conjecture for projective sets has consistency strength of only an inaccessible cardinal.

Rado猜想与随机代数

Rado's Conjecture and the random algebra

本文研究Rado猜想与随机代数的一致性，证明Rado猜想与$2^\omega=\omega_2$及Cichon图中随机代数对应的基数不变量一致，通过观察随机代数不特化非特殊树并构造新的Mitchell力迫得到结果。

The Open Graph Axiom and Menger's Conjecture

Menger conjectured that subsets of $\mathbb R$ with the Menger property must be $σ$-compact. While this is false when there is no restriction on the subsets of $\mathbb R$, for projective subsets it is known to follow from the Axiom of Projective Determinacy, which has considerable large cardinal consistency strength. We show that the perfect set version of the Open Graph Axiom for projective sets of reals, with consistency strength only an inaccessible cardinal, also implies Menger's conjecture restricted to this family of subsets of $\mathbb R$.

A proof of Shelah's eventual categoricity conjecture and an extension to accessible categories with directed colimits

We provide a proof, in $ZFC$, of Shelah's eventual categoricity conjecture for abstract elementary classes (AEC's). Moreover, assuming in addition the Singular Cardinal Hypothesis ($SCH$), we prove a direct generalization to the more general context of accessible categories with directed colimits. If $\mathcal{K}$ is such a category, we show that there is a cardinal $μ$ such that if $\mathcal{K}$ is $λ$-categorical for some $λ\geq μ$ (i.e., it has only one object of internal size $λ$ up to isomorphism), then $\mathcal{K}$ is eventually categorical (i.e., it is $λ'$-categorical for every $λ' \g…

On a conjecture for $\aleph_0$-bounded groups

We show that it is consistent, relative to the consistency of a strongly inaccessible cardinal, that an instance of the generalized Borel Conjecture introduced in [8] holds while the classical Borel Conjecture fails.

不同基数的Jónsson群

Jónsson groups of various cardinalities

本文利用Obraztsov嵌入定理，证明无穷基数κ存在Jónsson群当且仅当存在同基数Jónsson代数，将Jónsson群的存在性问题归约为已研究的Jónsson基数问题，推出任意大基数Jónsson群存在且Babai无穷边轨道猜想成立。

由二阶逻辑得到的新内模型

New inner models from second order logics

本文定义由二阶逻辑可数子集片段产生的新内模型 C2(omega)，比较其与 C(aa) 的大基数强度，并研究 C2(omega, aa) 及 HOD1 与 HOD 同一性的独立性。

在 $V=L$ 下 Enochs 猜想的一个反例

A counterexample to Enochs' conjecture under $V=L$

在假设不存在不可测基数下，构造了一个 $\mathbb F_2$-代数及右模覆盖类，该类不封闭于可数链余极限，从而在 $V=L$ 下给出 Enochs 猜想的反例。