Menachem Magidor, Saharon Shelah
Assuming some large cardinals, a model of ZFC is obtained in which aleph_{omega+1} carries no Aronszajn trees. It is also shown that if lambda is a singular limit of strongly compact cardinals, then lambda^+ carries no Aronszajn trees.
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May the successor of a singular cardinal be Jonsson?
该研究为数学逻辑领域的综述工作，汇总了奇异基数的后继成为Jonsson基数的必要条件，为相关大基数问题研究提供参考。
Starting from the existence of many supercompact cardinals, we construct a model of ZFC in which the tree property holds at a countable segment of successor of singular cardinals.
We prove that successors of singular limits of strongly compact cardinals have the strong tree property. We also prove that aleph_{omega+1} can consistently satisfy the strong tree property.
We investigate negative square-brackets partition relations at successors of singular cardinals of countable cofinality. Along the way we prove some club-guessing results.
It is proved that the consistency strength of having definable tree property for successors of all regular cardinals is the consistency strength of having proper class many small large cardinals which are defined very similar to indescribables but are much weaker in consistency strength. Also the consistency strength of definable tree property for successor of a singular cardinal is reduced to the existence of a supercompact cardinal and a measurable above it.
We present a weak sufficient condition for the existence of Souslin trees at successor of regular cardinals. The result is optimal and simultaneously improves an old theorem of Gregory and a more recent theorem of the author.
Assuming the existence of a strong cardinal $κ$ and a measurable cardinal above it, we force a generic extension in which $κ$ is a singular strong limit cardinal of any prescribed cofinality, and such that the tree property holds at $κ^{++}$.
We investigate the existence of strong colorings on successors of singular cardinals. This work continues Section 2 of [Sh:413] (math.LO/9809199), but now our emphasis is on finding colorings of pairs of ordinals, rather than colorings of finite sets of ordinals.
Assuming the existence of a strong cardinal $κ$, a weakly compact cardinal $λ$ above it and $γ> λ,$ we force a generic extension in which $κ$ is a singular strong limit cardinal of any given cofinality $δ$, $2^κ\geq γ$ and such that the tree property holds at $κ^{++}$.
Assuming that $0^\dagger$ does not exist, we prove that if there is a partition of $\mathbb R$ into $\aleph_ω$ Borel sets, then there is also a partition of $\mathbb R$ into $\aleph_{ω+1}$ Borel sets.
Assume $λ$ is a singular limit of $η$ supercompact cardinals, where $η\leq λ$ is a limit ordinal. We present two forcing methods for making $λ^+$ the successor of the limit of the first $η$ measurable cardinals while the tree property holding at $λ^+.$ The first method is then used to get, from the same assumptions, tree property at $\aleph_{η^2+1}$ with the failure of $SCH$ at $\aleph_{η^2}$. This extends results of Neeman and Sinapova. The second method is also used to get tree property at successor of an arbitrary singular cardinal, which extends some results of Magidor-Shelah, Neeman and S…
We describe a framework for proving consistency results about singular cardinals of arbitrary cofinality and their successors. This framework allows the construction of models in which the Singular Cardinals Hypothesis fails at a singular cardinal of uncountable cofinality, while its successor enjoys various combinatorial properties. As a sample application, we prove the consistency (relative to that of ZFC plus a supercompact cardinal) of there being a strong limit singular cardinal $κ$ of uncountable cofinality where SCH fails and for which there is a collection of graphs on $κ^+$ whose size…
For an inaccessible cardinal $κ$, the super tree property (ITP) at $κ$ holds if and only if $κ$ is supercomact. However, just like the tree property, it can hold at successor cardinals. We show that ITP holds at the successor of the limit of $ω$ many supercompact cardinals. Then we show that it can consistently hold at $\aleph_{ω+1}$. We also consider a stronger principle, ISP, and certain weaker variations of it. We determine which level of ISP can hold at a successor of a singular. These results fit in the broad program of testing how much compactness can exist in the universe, and obtaining…
If $κ$ is regular and $2^{<κ}\leqκ^+$, then the existence of a weakly presaturated ideal on $κ^+$ implies $\square^*_κ$. This partially answers a question of Foreman and Magidor about the approachability ideal on $ω_2$. As a corollary, we show that if there is a presaturated ideal $I$ on $ω_2$ such that $\mathcal{P}(ω_2)/I$ is semiproper, then CH holds. We also show some barriers to getting the tree property and a saturated ideal simultaneously on a successor cardinal from conventional forcing methods.
The paper is concerned with the existence of a universal graph at the successor of a strong limit singular mu of cofinality aleph_0. Starting from the assumption of the existence of a supercompact cardinal, a model is built in which for some such mu there are mu^{++} graphs on mu^+ that taken jointly are universal for the graphs on mu^+, while 2^{mu^+}>> mu^{++} . The paper also addresses the general problem of obtaining a framework for consistency results at the successor of a singular strong limit starting from the assumption that a supercompact cardinal kappa exists. The result on the exist…
We investigate the problem of when $\leqλ$--support iterations of $<λ$--complete notions of forcing preserve $λ^+$. We isolate a property -- {\em properness over diamonds} -- that implies $λ^+$ is preserved and show that this property is preserved by $λ$--support iterations. We close with an application of our technology by presenting a consistency result on uniformizing colorings of ladder systems on $\{δ<λ^+:\cf(δ)=λ\}$ that complements a theorem of Shelah.
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。
Let K be an abstract elementary classes which has arbitrarily large models and satisfies the amalgamation and joint embedding properties. Theorem 1. Suppose K is χ-tame. If K is categorical in some λ^+ >LS(K) then it is categorical in all μ\geq (λ+χ)^+. Theorem 2. If K is LS(K)-tame and is categorical both in LS(K) and in LS(K)^+ then K is categorical in all μ\geq LS(K).
Let $\mathrm{cof}(μ)=μ$ and $κ$ be a supercompact cardinal with $μ<κ$. Assume that there is an increasing and continuous sequence of cardinals $\langleκ_ξ\mid ξ<μ\rangle$ with $κ_0:=κ$ and such that, for each $ξ<μ$, $κ_{ξ+1}$ is supercompact. Besides, assume that $λ$ is a weakly compact cardinal with $\sup_{ξ<μ}κ_ξ<λ$. Let $Θ\geqλ$ be a cardinal with $\mathrm{cof}(Θ)>κ$. Assuming the $\mathrm{GCH}_{\geqκ}$, we construct a generic extension where $κ$ is strong limit, $\mathrm{cof}(κ)=μ$, $2^κ= Θ$ and both $\mathrm{TP}(κ^+)$ and $\mathrm{TP}(κ^{++})$ hold. Further, in this model there is a very …
We prove that $α_M(λ)$ can be successor of a supercompact cardinal, when $λ$ is a Magidor cardinal. From this result we obtain the consistency of $α_M(λ)$ being a successor of a singular cardinal with uncountable cofinality.
We introduce a combinatorial notion of measures called Rudin-Keisler capturing and use it to give a new construction of elementary substructures around singular cardinals. The new construction is used to establish mutual stationary results at the first successor of singular cardinals $\langle \aleph_{ωn + 1}\rangle_{n< ω}$.
We obtain an array of consistency results concerning trees and stationary reflection at double successors of regular cardinals $κ$, updating some classical constructions in the process. This includes models of $\mathsf{CSR}(κ^{++})\wedge \mathsf{TP}(κ^{++})$ (both with and without $\mathsf{AP}(κ^{++})$) and models of the conjunctions $\mathsf{SR}(κ^{++}) \wedge \mathsf{wTP}(κ^{++}) \wedge \mathsf{AP}(κ^{++})$ and $\neg \mathsf{AP}(κ^{++}) \wedge \mathsf{SR}(κ^{++})$ (the latter was originally obtained in joint work by Krueger and the first author \cite{GilKru:8fold}, and is here given using di…
Strong Tree Properties Along Many Segments of Successors of Singulars
该研究从大基数出发，通过力迫构造证明了ITP可在奇异基数的可数初始段后继处同时成立，并推广得到奇异基数多共尾度长段后继处的强树性质的一致性。
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]).
The Strong and Super Tree Property at Successors of Singular Cardinals
该研究聚焦奇异基数后继的强树性质与ITP（超树性质），构造通用力迫类，推广Neeman的结果，证明多种基数组合下两类性质同时成立的一致性，推进了同时在多个相继基数上获得两类树性质的问题研究。
We strengthen non-structure theorems for almost free Abelian groups by studying long Ehrenfeucht-Fraisse games between a fixed group of cardinality lambda and a free Abelian group. A group is called epsilon-game-free if the isomorphism player has a winning strategy in the game (of the described form) of length epsilon in lambda. We prove for a large set of successor cardinals lambda=mu^+ existence of nonfree (mu*omega_1)-game-free groups of cardinality lambda. We concentrate on successors of singular cardinals.
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
Total Failure of Approachability at Successors of Singulars of Countable Cofinality
在可数余数的后继点中，研究了非可接近点的存在性，解决了Mitchell和Foreman、Shelah的问题。
Some models of combinatorial principles have been obtained by collapsing a huge cardinal in the case of the successors of regular cardinals. For example, saturated ideals, Chang's conjecture, polarized partition relations, and transfer principles for chromatic numbers of graphs. In this paper, we study these in the case of the successors of singular cardinals. In particular, we show that Prikry forcing preserves the centeredness of ideals but kills the layeredness. We also study $\binom{μ^{++}}{μ^{+}}\to\binomκ{μ^{+}}_μ$ and $\mathrm{Tr}_{\mathrm{Chr}}(μ^{+++},μ^{+})$ in the extension by Prikr…
A cardinal kappa is countably closed if mu^omega < kappa whenever mu < kappa. Assume that there is no inner model with a Woodin cardinal and that every set has a sharp. Let K be the core model. Assume that kappa is a countably closed cardinal and that alpha is a successor cardinal of K with kappa < alpha < kappa^+. Then cf( alpha ) = kappa. In particular, K computes successors of countably closed singular cardinals correctly. (The hypothesis of countable closure is not required; see "Weak covering without countable closure", W. J. Mitchell and E. Schimmerling, Math. Res. Lett., Vol. 2, No. 5, …
We get a quite maximal version of the colouring property $Pr_1$ by proving $Pr_1(λ,λ,λ,θ)$ when $λ= \partial^+, \partial > θ$ are regular cardinals.
The notion of stationary reflection is one of the most important notions of combinatorial set theory. We investigate weak reflection, which is, as the name suggests, a weak version of stationary reflection. This sort of reflection was introduced in [DjSh:545] (math.LO/9601219), where it was shown that weak reflection has applications to various guessing principles, in the sense that if there is no weak reflection, than a guessing principle holds, and an application dealing with the saturation of normal filters. Further investigations of weak reflection were carried in [CDSh:571] (math.LO/95042…
Tree Properties at Successors of Singulars of Many Cofinalities
本研究利用多个超紧基数，证明树性质可在多个不同共尾性的奇异基数小后继上相容成立，构造了特定基数上树性质成立的模型，并将结果推广到强树性质及不可数长共尾序列情形。
We extend prior results of Cody-Eskew, showing the consistency of GCH with the statement that for all regular cardinals $κ\leq λ$, where $κ$ is the successor of a regular cardinal, there is a rigid saturated ideal on $\mathcal{P}_κλ$. We also show the consistency of some instances of rigid saturated ideals on $\mathcal{P}_κλ$ where $κ$ is the successor of a singular cardinal.
Superclub, splitting, separating statements
本文证明 superclub 蕴含分裂数等于最小不可数基数，并在弱紧、超紧及正则、奇异基数后继处利用 Galvin 性质分离 tiltan 与 superclub。
Ultrafilters over Successor Cardinals and the Tukey Order
该研究聚焦正则不可数基数（尤其$\omega_1$）上的超滤与Tukey序，证明了相关断言的ZFC独立性，给出新型超滤构造，分析了特定公理下超滤的序性质并得到多个相关结果。
We formulate and prove (in {\sf ZFC}) a strong coloring theorem which holds at successors of singular cardinals, and use it to answer several questions concerning Shelah's principle $Pr_1(μ^+,μ^+,μ^+,\cf(μ))$ for singular $μ$.
We prove that on many inaccessible there is a Jonsson algebra, so e.g. the first regular Jonsson cardinal lambda which is lambda x omega-Mahlo. We give further restrictions on successor of singulars which are Jonsson cardinals. E.g. there is a Jonsson algebra on the successor of beth_omega. Lastly, we give further information on guessing of clubs.
Negating the Galvin Property
本文证明了 Galvin 性质在强极限奇异基数及一般奇异基数的后继处可一致地不成立，并分析了 Prikry 型力迫对其强失败的影响及大基数背景下的更强形式。
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 $κ$,$λ$ be regular uncountable cardinals such that $λ> κ^+$ is not a successor of a singular cardinal of low cofinality. We construct a generic extension with $s(κ) = λ$ starting from a ground model in which $o(κ) = λ$ and prove that assuming $\neg 0^¶$, $s(κ) = λ$ implies that $o(κ) \geq λ$ in the core model.
Erdos, Hajnal and Rado asked whether $\binom{\aleph_{ω_1}}{\aleph_2}\rightarrow\binom{\aleph_{ω_1}}{\aleph_0}_2$ and whether $\binom{\aleph_{ω_1}}{\aleph_2}\rightarrow\binom{\aleph_{ω_1}}{\aleph_1}_2$. We prove that both relations are independent over ZFC. We shall also prove that $\binomμ{\aleph_2}\rightarrow\binomμ{\aleph_2}_2$ is independent over ZF for some $μ$ of cofinality $ω_1$.
1. For many regular cardinals lambda (in particular, for all successors of singular strong limit cardinals, and for all successors of singular omega-limits), for all n in {2,3,4, ...} : There is a linear order L such that L^n has no (incomparability-)antichain of cardinality lambda, while L^{n+1} has an antichain of cardinality lambda . 2. For any nondecreasing sequence (lambda2,lambda3, ...) of infinite cardinals it is consistent that there is a linear order L such that L^n has an antichain of cardinality lambda_n, but not one of cardinality lambda_n^+ .
We establish a coloring theorem for successors of a singular cardinals, and use it prove that for any such cardinal $μ$, we have $μ^+\nrightarrow[μ^+]^2_{μ^+}$ if and only if $μ^+\nrightarrow[μ^+]^2_θ$ for arbitrarily large $θ<μ$.
We build a supercompact version of the forcing defined in \cite{gitik2019}. For each singular cardinal in the ground model with any fixed cofinality, which is a limit of supercompact cardinals, it is possible to force so that the size of the powerset of the singular cardinal is arbitrarily large, while preserving the singular cardinal. An important feature of this forcing is that it is possible to define the forcing so that the successor of the singular cardinal is collapsed, but all the cardinals above it are preserved.
Splendid extensions
研究在特定条件下，后继基数κ相关的PA模型扩展问题，通过强制构造全域的方法，使得大小为κ的PA模型扩展为M且无辉煌扩张，若有不可表达基数，在ZFC中该结论在其下某基数处成立。
Failure of Approachability at the Successor of the first Singular for any Cofinality
该研究解决了两个关于ℵ_{ω+1}组合学的长期公开问题，证明对任意n≥1，GCH成立且存在共尾性ℵ_n的驻集不在可逼近性理想中，并得到良态性与可逼近性在共尾性ℵ_1的驻点上不同的模型。
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 the influence of the existence of large cardinals on the existence of wellorderings of power sets of infinite cardinals $κ$ with the property that the collection of all initial segments of the wellordering is definable by a $Σ_1$-formula with parameter $κ$. A short argument shows that the existence of a measurable cardinal $δ$ implies that such wellorderings do not exist at $δ$-inaccessible cardinals of cofinality not equal to $δ$ and their successors. In contrast, our main result shows that these wellorderings exist at all other uncountable cardinals in the minimal model containing a…
We prove several results giving lower bounds for the large cardinal strength of a failure of the singular cardinal hypothesis. The main result is the following theorem: Theorem: Suppose $κ$ is a singular strong limit cardinal and $2^κ>= λ$ where $λ$ is not the successor of a cardinal of cofinality at most $κ$. (i) If $\cofinality(κ)>\gw$ then $o(κ)\geλ$. (ii) If $\cofinality(κ)=\gw$ then either $o(κ)\geλ$ or $\set{\ga:K\sat o(\ga)\ge\ga^{+n}}$ is cofinal in $κ$ for each $n\in\gw$. In order to prove this theorem we give a detailed analysis of the sequences of indiscernibles which come from appl…
A cardinal $λ$ satisfies a property P robustly if, whenever $\mathbb{Q}$ is a forcing poset and $|\mathbb{Q}|^+ < λ$, $λ$ satisfies P in $V^{\mathbb{Q}}$. We study the extent to which certain reflection properties of large cardinals can be satisfied robustly by small cardinals. We focus in particular on stationary reflection and the tree property, both of which can consistently hold but fail to be robust at small cardinals. We introduce natural strengthenings of these principles which are always robust and which hold at sufficiently large cardinals, consider the extent to which these strengthe…
We prove the consistency of a strong polarized relation for a cardinal and its successor, using pcf and forcing
Quadruples and cubes
本文在ZFC中证明阶梯立方体与四元组关系的失败条件，并利用奇妙理想建立后继基数上的正极化关系，同时否定连续基数上的此类理想存在。
Stationary reflection using ancestrally forced conditions
该研究提出新方法，从κ⁺-超紧基数出发，构建奇异基数后继处平稳反射模型，得到ℵ_{ω₁+1}处可数同时反射的模型。
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.
Let kappa be a regular uncountable cardinal and lambda >=kappa^+ . The principle of stationary reflection for P_kappa lambda has been successful in settling problems of infinite combinatorics in the case kappa=omega_1. For a greater kappa the principle is known to fail at some lambda. This note shows that it fails at every lambda if kappa is the successor of a regular uncountable cardinal or kappa is countably closed.
We survey some recent results on the validity of Jensen's diamond principle at successor cardinals. We also discuss weakening of this principle such as club guessing, and anti-diamond principles such as uniformization. A collection of open problems is included.
Two-cardinal Kurepa Hypotheses
本文研究双基数 Kurepa 假设 $\mathsf{KH}(κ,λ)$ 的蕴含关系，证明其唯一的 ZFC 约束，并讨论奇异基数情形及其与 Chang 猜想、scales 的联系，同时推广 Cummings 的独立性结果。
We investigate the possibilities of global versions of Chang's Conjecture that involve singular cardinals. We show some $\mathrm{ZFC}$ limitations on such principles, and prove relative to large cardinals that Chang's Conjecture can consistently hold between all pairs of limit cardinals below $\aleph_{ω^ω}$.
We extend a transitive model V of ZFC + GCH cardinal preservingly to a model N of ZF + "GCH holds below Alef_omega" + "there is a surjection from the power set of Alef_omega onto lambda" where lambda is an arbitrarily high fixed cardinal in V.
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