Mirna Džamonja
We prove that for regular $λ$ above a strong limit singular $μ$ certain guessing principles follow just from cardinal arithmetic assumptions. The main result is that for such $λ$ and $μ$ there are coboundedly many regular $κ<μ$ such that $\clubsuit^-(S^λ_κ)$ holds whenever $λ=λ^{<κ}$.
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We obtain results on the condensation principle called local club condensation. We prove that in extender models an equivalence between the failure of local club condensation and subcompact cardinals holds. This gives a characterization of $\square_κ$ in terms of local club condensation in extender models. Assuming $\gch$, given an interval of ordinals $I$ we verify that iterating the forcing defined by Holy-Welch-Wu, we can preserve $\gch$, cardinals and cofinalities and obtain a model where local club condensation holds for every ordinal in $I$ modulo those ordinals which cardinality is a si…
Strong failures of club guessing at the successor of a regular cardinal
该研究在正则基数κ的后继处，通过Krueger的带模型边条件的力迫迭代技术，力迫使极弱俱乐部猜测与℧原则同时失效，且ω₁处的失效是PFA的已知结果。
We prove the consistency, assuming an ineffable cardinal, that any two normal countably closed $ω_2$-Aronszajn trees are club isomorphic. This work generalizes to higher cardinals the property of Abraham-Shelah that any two normal $ω_1$-Aronszajn trees are club isomorphic, which follows from $\textsf{PFA}$. The statement that any two normal countably closed $ω_2$-Aronszajn trees are club isomorphic implies that there are no $ω_2$-Suslin trees, so our proof also expands on the method of Laver-Shelah for obtaining the $ω_2$-Suslin hypothesis.
Zaba: a case for clubs
该研究源于范德弗勒格特工作，对不可达基数κ的广义贝尔空间相关基数系数提出看法，认为合适系数由限制在俱乐部上的二元关系确定，主张系统使用俱乐部。
Suppose that lambda is the successor of a singular cardinal mu whose cofinality is an uncountable cardinal kappa. We give a sufficient condition that the club filter of lambda concentrating on the points of cofinality kappa is not lambda^+-saturated. The condition is phrased in terms of a notion that we call weak reflection. We discuss various properties of weak reflection
We give a direct, detailed and relatively short proof of Shelah's theorem on club guessing sequences on $S^{μ^+}_μ$ (for any regular, uncountable cardinal $μ$).
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.
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。
We give two results on guessing unbounded subsets of lambda^+. The first is a positive result and applies to the situation of lambda regular and at least equal to aleph_3, while the second is a negative consistency result which applies to the situation of lambda a singular strong limit with 2^lambda>lambda^+. The first result shows that in ZFC there is a guessing of unbounded subsets of S^{lambda^+}_lambda. The second result is a consistency result (assuming a supercompact cardinal exists) showing that a natural guessing fails. A result of Shelah in math.LO/9808140 shows that if 2^lambda=lambd…
We study combinatorial principles known as stick and club. Several variants of these principles and cardinal invariants connected to them are also considered. We introduce a new kind of side-by-side product of partial orders which we call pseudo-product. Using such products, we give several generic extensions where some of these principles hold together with not CH and Martin's Axiom for countable p.o.-sets. An iterative version of the pseudo-product is used under an inaccessible cardinal to show the consistency of the club principle for every stationary subset of limits of omega_1 together wi…
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。
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…
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 $\mathfrak p(…
AD$^+$ 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。
We investigate negative square-brackets partition relations at successors of singular cardinals of countable cofinality. Along the way we prove some club-guessing results.
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…
$μ$-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。
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 …
The failure of square at all uncountable cardinals is weaker than a Woodin limit of Woodin cardinals
该研究通过强制选择公理，在满足ZF的最小初始段的Nairian模型中展示所有不可数基数上平方失败，并表明正则基数在HOD中为omega-强可测，从而揭示了HOD假设在ZFC中的不可证性。
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.
We analyze a natural function definable from a scale at a singular cardinal, and using this function we are able to obtain quite strong negative square-brackets partition relations at successors of singular cardinals. The proof of our main result makes use of club-guessing, and as a corollary we obtain a fairly easy proof of a difficult result of Shelah connecting weak saturation of a certain club-guessing ideal with strong failures of square-brackets partition relations. We then investigate the strength of weak saturation of such ideals and obtain some results on stationary reflection.
We investigate, in ZFC, the behavior of abstract elementary classes (AECs) categorical in many successive small cardinals. We prove for example that a universal $\mathbb{L}_{ω_1, ω}$ sentence categorical on an end segment of cardinals below $\beth_ω$ must be categorical also everywhere above $\beth_ω$. This is done without any additional model-theoretic hypotheses (such as amalgamation or arbitrarily large models) and generalizes to the much broader framework of tame AECs with weak amalgamation and coherent sequences.
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…
We prove that a large class of presaturated ideals at inaccessible cardinals can be de-saturated while preserving their presaturation, answering both a question of Foreman and of Cox and Eskew. We do so by iterating a generalized version of Baumgartner and Taylor's forcing to add a club with finite conditions along an inaccessible cardinal, and invoking Foreman's Duality Theorem.
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…
$μ$-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。
On $ω$-strongly measurable cardinals in $\mathbb{P}_{\max}$ extensions
本文证明了在特定Chang型决定性模型的$\mathbb{P}_{max}$扩张中，特定基数上的club filter限制到HOD上构成超滤，从而回答了Ben-Neria和Hayut提出的问题4.11。
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。
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…
We show that the tree property, stationary reflection and the failure of approachability at $κ^{++}$ are consistent with $\mathfrak{u}(κ) = κ^+ < 2^κ$, where $κ$ is a singular strong limit cardinal with the countable or uncountable cofinality. As a by-product, we show that if $λ$ is a regular cardinal, then stationary reflection at $λ^+$ is indestructible under all $λ$-cc forcings (out of general interest, we also state a related result for the preservation of club stationary reflection).
We continue the work of [KlSh:362] and prove that for lambda successor, a lambda-categorical theory T in L_{kappa^*, omega} is mu-categorical for every mu, mu <= lambda which is above the (2^{LS(T)})^+-beth cardinal.
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]).
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。
We show that $\beth_{(2^{\operatorname{LS}({\bf K})})^+}$ is the lower bound to the Hanf numbers for the length of the order property and for stability in stable abstract elementary classes (AECs). Our examples satisfy the joint embedding property, no maximal model, $(<\aleph_0)$-tameness but not necessarily the amalgamation property. We also define variations on the order and syntactic order properties by allowing the index set to be linearly ordered rather than well-ordered. Combining with Shelah's stability theorem, we deduce that our examples can have the order property up to any $μ<\beth_…
The stable core, an inner model of the form $\langle L[S],\in, S\rangle$ for a simply definable predicate $S$, was introduced by the first author in [Fri12], where he showed that $V$ is a class forcing extension of its stable core. We study the structural properties of the stable core and its interactions with large cardinals. We show that the $\operatorname{GCH}$ can fail at all regular cardinals in the stable core, that the stable core can have a discrete proper class of measurable cardinals, but that measurable cardinals need not be downward absolute to the stable core. Moreover, we show th…
We prove that for any superatomic Boolean Algebra of cardinality >beth_omega there is an automorphism moving uncountably many atoms. Similarly for larger cardinals. Any of those results are essentially best possible.
We show in this note that in the forcing extension by $Add(ω,\beth_ω)$, the following Ramsey property holds: for any $r\in ω$ and any $f: \mathbb{R}\to r$, there exists an infinite $X\subset \mathbb{R}$ such that $X+X$ is monochromatic under $f$. We also show the Ramsey statement above is true in $\mathrm{ZFC}$ when $r=2$. This answers two questions by Komjáth, Leader, Russell, Shelah, Soukup and Vidnyánszky.
We prove that the statement "for every infinite cardinal nu, every graph with list chromatic nu has coloring number at most beth_omega (nu)" proved by Kojman [6] using the RGCH theorem [11] implies the RGCG theorem via a short forcing argument. Similarly, a better upper bound than beth_omega (nu) in this statement implies stronger forms of the RGCH theorem hold, whose consistency and the consistency of their negations are wide open. Thus, the optimality of Kojman's upper bound is a purely cardinal arithmetic problem, and, as discussed below, is hard to decide.
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。
There are several examples in the literature showing that compactness-like properties of a cardinal $κ$ cause poor behavior of some generic ultrapowers which have critical point $κ$ (Burke \cite{MR1472122} when $κ$ is a supercompact cardinal; Foreman-Magidor \cite{MR1359154} when $κ= ω_2$ in the presence of strong forcing axioms). We prove more instances of this phenomenon. First, the Reflection Principle (RP) implies that if $\vec{\mathcal{I}}$ is a tower of ideals which concentrates on the class $GIC_{ω_1}$ of $ω_1$-guessing, internally club sets, then $\vec{\mathcal{I}}$ is not presaturated…
For $K$ an abstract elementary class with amalgamation and no maximal models, we show that categoricity in a high-enough cardinal implies structural properties such as the uniqueness of limit models and the existence of good frames. This improves several classical results of Shelah. $\mathbf{Theorem}$ Let $μ\ge \text{LS} (K)$. If $K$ is categorical in a $λ\ge \beth_{\left(2^μ\right)^+}$, then: 1) Whenever $M_0, M_1, M_2 \in K_μ$ are such that $M_1$ and $M_2$ are limit over $M_0$, we have $M_1 \cong_{M_0} M_2$. 2) If $μ> \text{LS} (K)$, the model of size $λ$ is $μ$-saturated. 3) If $μ\ge \beth_…
A question dating to Sibe Mardešić and Andrei Prasolov's 1988 work Strong homology is not additive, and motivating a considerable amount of set theoretic work in the ensuing years, is that of whether it is consistent with the ZFC axioms for the higher derived limits $\mathrm{lim}^n$ $(n>0)$ of a certain inverse system $\mathbf{A}$ indexed by ${^ω}ω$ to simultaneously vanish. An equivalent formulation of this question is that of whether it is consistent for all $n$-coherent families of functions indexed by ${^ω}ω$ to be trivial. In this paper, we prove that, in any forcing extension given by ad…
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