Kenta Tsukuura
It is known that the set of possible cofinalities $\mathrm{pcf}(A)$ has good properties if $A$ is a progressive interval of regular cardinals. In this paper, we give an interval of regular cardinals $A$ such that $\mathrm{pcf}(A)$ has no good properties in the presense of a measurable cardinal, or in generic extensions by Prikry-type forcing.
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In this paper we investigate some properties of forcing which can be considered "nice" in the context of singularizing regular cardinals to have an uncountable cofinality. We show that such forcing which changes cofinality of a regular cardinal, cannot be too nice and must cause some "damage" to the structure of cardinals and stationary sets. As a consequence there is no analogue to the Prikry forcing, in terms of "nice" properties, when changing cofinalities to be uncountable.
In this paper, we answer a question asked in "A minimal Prikry type forcing for singularizing a measurable cardinal" regarding a Mathias criteria for Tree-Prikry forcing. Also we will investigate Prikry forcing using various filters. For completeness and self inclusion reasons, we will give proofs of many known theorems.
We define a version of Gitik-Sharon diagonal Prikry forcing using a strongly compact cardinal, and prove its basic properties.
Let $D$ be an infinite discrete set of measurable cardinals. It is shown that generalized Prikry forcing to add a countable sequence to each cardinal in $D$ is subcomplete. To do this it is shown that a simplified version of generalized Prikry forcing which adds a point below each cardinal in $D$, called generalized diagonal Prikry forcing, is subcomplete. Moreover, the generalized diagonal Prikry forcing associated to $D$ is subcomplete above $μ$, where $μ$ is any regular cardinal below the first limit point of $D$.
Prikry type forcings and the Bukovský-Dehornoy phenomena
本文系统梳理了Bukovský和Dehornoy独立发现的常见力迫现象，先给出Prikry型力迫的基本结果，再将其推广到Gitik-Sharon力迫、基于Extender的Prikry力迫、带交错坍缩的Prikry力迫及Radin力迫。
We work with symmetric inner models of forcing extensions based on strongly compact Prikry forcing to extend some known results.
We consider here Easton support iterations of Prikry type forcing notions. New ways of constructing normal ultrafilters in extensions are presented. It turns out that, in contrast with other supports, seemingly unrelated measures or extenders can be involved here.
We study which $κ$-distributive forcing notions of size $κ$ can be embedded into tree Prikry forcing notions with $κ$-complete ultrafilters under various large cardinal assumptions. An alternative formulation -- can the filter of dense open subsets of a $κ$-distributive forcing notion of size $κ$ be extended to a $κ$-complete ultrafilter.
We show that the cofinalities of both the Miller ideal m^0 (the sigma-ideal naturally related to Miller forcing) and the Laver ideal ell^0 (related to Laver forcing) are larger than the size of the continuum in ZFC.
We study the Mathias--Prikry and the Laver type forcings associated with filters and coideals. We isolate a crucial combinatorial property of Mathias reals, and prove that Mathias--Prikry forcings with summable ideals are all mutually bi-embeddable. We show that Mathias forcing associated with the complement of an analytic ideal always adds a dominating real. We also characterize filters for which the associated Mathias--Prikry forcing does not add eventually different reals, and show that they are countably generated provided they are Borel. We give a characterization of $ω$-hitting and $ω$-s…
We introduce a class of notions of forcing which we call $Σ$-Prikry, and show that many of the known Prikry-type notions of forcing that centers around singular cardinals of countable cofinality are $Σ$-Prikry. We show that given a $Σ$-Prikry poset $\mathbb P$ and a name for a non-reflecting stationary set $T$, there exists a corresponding $Σ$-Prikry poset that projects to $\mathbb P$ and kills the stationarity of $T$. Then, in a sequel to this paper, we develop an iteration scheme for $Σ$-Prikry posets. Putting the two works together, we obtain a proof of the following. Theorem. If $κ$ is the…
In Part I of this series, we introduced a class of notions of forcing which we call Sigma-Prikry, and showed that many of the known Prikry-type notions of forcing that center around singular cardinals of countable cofinality are Sigma-Prikry. We showed that given a Sigma-Prikry poset P and a P-name for a non-reflecting stationary set T, there exists a corresponding Sigma-Prikry poset that projects to P and kills the stationarity of T. In this paper, we develop a general scheme for iterating Sigma-Prikry posets and, as an application, we blow up the power of a countable limit of Laver-indestruc…
On the Intermediate Models of Strongly Compact Prikry Forcing
通过分析强紧Prikry力迫的中间模型，利用κ-完备细测度刻画其所有投影，并推广了Benhamou-Hayut-Gitik等人的结果。
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…
On fresh sets in iterations of Prikry type forcing notions
该研究考察 Prikry 型力迫迭代中 fresh sets 的存在与不存在性，推广已有结果并回答相关审稿问题，同时讨论驻集保持。
We prove the consistency of the failure of the singular cardinals hypothesis at $\aleph_ω$ together with the reflection of all stationary subsets of $\aleph_{ω+1}$. This shows that two classic results of Magidor (from 1977 and 1982) can hold simultaneously.
On semiproperness of Namba forcings and Ideals in Prikry extensions
本文研究了$\mathcal{P}_κλ$上Namba力迫变体的半正则性，证明其蕴含$\mathrm{SSR}([λ]^ω,{<}κ)$，并指出Prikry型力迫扩张中不存在半正则饱和理想。
The purpose of this paper is to present some results which suggest that the Singular Cardinals Hypothesis follows from the Proper Forcing Axiom. What will be proved is that a form of simultaneous reflection follows from the Set Mapping Reflection Principle, a consequence of PFA. While the results fall short of showing that MRP implies SCH, it will be shown that MRP implies that if SCH fails first at kappa then every stationary subset of S_{kappa^+}^omega = {a < kappa^+ : cf(a) = omega} reflects. It will also be demonstrated that MRP always fails in a generic extension by Prikry forcing.
We continue Gitik, Kanovei and Koepke's work and study sets in generic extensions by the Magidor forcing and by the Prikry forcing with non-normal ultrafilters.
On the strength of ultrafilters above choiceless large cardinals and their Prikry forcings
本文研究无选择大基数上超滤子强度与普利克里力迫，证明超滤子的大基数强度界，拓展普利克里力迫理论并限制可坍缩/奇异化基数，提出张量普利克里系统以构造无选择集合论的一致性结果（如所有不可数基数奇异的模型）。
We show: There are pairs of universes V_1 subseteq V_2 and there is a notion of forcing P in V_1 such that the change mentioned in the title occurs when going from V_1[G] to V_2[G] for a P-generic filter G over V_2. We use forcing iterations with partial memories. Moreover, we implement highly transitive automorphism groups into the forcing orders.
We study the nonstationary-support iteration of Prikry forcings below a measurable cardinal κ, characterizing all the normal measures it carries in the generic extension. We then analyze the restriction of ultrapower embeddings, taken with such a normal measure in the generic extension, to the ground model. We prove that every such restriction is an iterated ultrapwer of the ground model, and provide a sufficient condition for its definability there. This is done without core-model theoretic arguments: the assumption that GCH holds in the ground model up to κsuffices.
We present a modification to the Prikry on Extenders forcing notion allowing the blow up of the power set of a large cardinal, change its cofinality to omega without adding bounded subsets, working directly from arbitrary extender (e.g., n-huge extender). Using this forcing, starting from a superstrong cardinal, we construct a model in which the added Prikry sequences are a scale in the normal Prikry sequence.
I isolate a combinatorial property of a poset $\mathbb{P}$ that I call the strong Prikry property, which implies the existence of an ultrafilter on the complete Boolean algebra $\mathbb{B}$ of $\mathbb{P}$ such that one inclusion of the Boolean ultrapower version of the so-called \Bukovsky-Dehornoy phenomenon holds with respect to $\mathbb{B}$ and $U$. I show that in all cases that were previously studied, and for which it was shown that they come with a canonical iterated ultrapower construction whose limit can be described as a single Boolean ultrapower, the posets in question satisfy this p…
On Cohen and Prikry Forcing Notions
本文证明用 κ 上的 Prikry 力迫可添加 κ⁺ 个 Cohen 子集，并引入非 Galvin 性质的强化，仅需单个可测基数即证明其相容性，同时讨论基于 extender 的 Prikry 力迫。
We generalize results of Gitik, Dzamonja-Shelah, and Magidor-Sinapova on the existence of pseudo-Prikry sequences, which are sequences that approximate the behavior of the generic objects introduced by Prikry-type forcings, in outer models of set theory. Such sequences play an important role in the study of singular cardinal combinatorics by placing restrictions on the type of behavior that can consistently be obtained in outer models. In addition, we provide results about the existence of diagonal pseudo-Prikry sequences, which approximate the behavior of the generic objects introduced by dia…
In this paper, we introduce several classes of Prikry-type forcing notions, two of which are used to produce minimal generic extensions, and the third is applied in $α$-recursion theory to produce minimal covers. The first forcing as a warm up yields a minimal generic extension at a measurable cardinal (in $V$), the second at an $ω$-limit of measurable cardinals $\langleγ_n\colon n<ω\rangle$ such that each $γ_n$ ($n>0$) carries $γ_{n-1}$-many normal measures. Via a notion of $V_γ$-degree (see Definition \ref{def:vgammadegree}), we transfer the second Prikry-type construction for minimal generi…
We define extender sequences, generelizing measure sequences from Radin Forcing. Using the extender sequences we combine Gitik-Magidor forcing for adding many Prikry sequences with Radin forcing. This forcing satisfies Prikry like condition, destroys no cardinals, and has a kind of properness. Depending on the large cardinal we start with, this forcing can blow the power of a cardinal together with changing its' cofinality to a prescribe value. If this prescribed value is the cardinal itself then it remains regular and even can remain measurable.
We continue the work done by the authors and before that by the second author, Kanovei and koepke. We prove that for every set of ordinals $A$ in a Magidor-Radin generic extension using a coherent sequence such that $o^{\vec{U}}(κ)<κ^+$, there is $C'\subseteq C_G$, such that $V[A]=V[C']$. Also we prove that the supremum of a fresh set in a Prikry, tree Prikry, Magidor, Radin-Magidor and Radin forcing, changes cofinality to $ω$.
Starting from large cardinals we construct a pair $V_1\subseteq V_2$ of models of $ZFC$ with the same cardinals and cofinalities such that $GCH$ holds in $V_1$ and fails everywhere in $V_2$.
We give a detailed proof of the properties of the usual Prikry type forcing notion for turning a measurable cardinal into $\aleph_ω$.
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。
Consider $(κ^{+++},κ^{++}) \twoheadrightarrow (κ^+,κ)$ where $κ$ is an uncountable regular cardinal. By a result of Shelah's we have $\operatorname{cof}(X \cap κ^{++}) = κ$ for almost all $X \subset κ^{+++}$ witnessing this. Here we consider the question if there could be a similar result for $X \cap κ^+$. We use this discussion to give an interesting example of a pseudo Prikry forcing answering a question of Sinapova.
Dobrinen, Hathaway and Prikry studied a forcing $\mathbb{P}_κ$ consisting of perfect trees of height $λ$ and width $κ$ where $κ$ is a singular $ω$-strong limit of cofinality $λ$. They showed that if $κ$ is singular of countable cofinality, then $\mathbb{P}_κ$ is minimal for $ω$-sequences assuming that $κ$ is a supremum of a sequence of measurable cardinals. We obtain this result without the measurability assumption. Prikry proved that $\mathbb{P}_κ$ is $(ω,ν)$-distributive for all $ν<κ$ given a singular $ω$-strong limit cardinal $κ$ of countable cofinality, and Dobrinen et al$.$ asked whether …
I survey an array of topics in set theory in the context of a novel class of forcing notions: subcomplete forcing. Subcompleteness was originally defined by Ronald Jensen. I have attempted to make the subject somewhat more approachable to set theorists, while showing various properties of subcomplete forcing which one might desire of a forcing class, drawing comparisons between subcomplete forcing and countably closed forcing. In particular, I look at the interaction between subcomplete forcing and $ω_1$-trees, preservation properties of subcomplete forcing, the subcomplete maximality principl…
We investigate forcing properties of perfect tree forcings defined by Prikry to answer a question of Solovay in the late 1960's regarding first failures of distributivity. Given a strictly increasing sequence of regular cardinals $\langle κ_n: n< ω\rangle$, Prikry defined the forcing $\mathbb{P}$ all perfect subtrees of $\prod_{n<ω}κ_n$, and proved that for $κ=\sup_{n<ω}κ_n$, assuming the necessary cardinal arithmetic, the Boolean completion $\mathbb{B}$ of $\mathbb{P}$ is $(ω,μ)$-distributive for all $μ<κ$ but $(ω,κ,δ)$-distributivity fails for all $δ<κ$, implying failure of the $(ω,κ)$-d.l. …
Two questions on $E_0$-like generic equivalence
本文解答了关于$E_0$-类泛型等价的两个问题，纠正了原条件表述，并构造了满足修正条件的Prikry型及更广泛例子，同时证明条件蕴含不可达基数。
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.
Non-Normal Magidor-Radin Types of Forcings
本文发展 Magidor 和 Radin 力迫的非正规变体，证明其可添加 Cohen 泛型函数，但仅在非改变共尾性情形；并证明扩展器基力迫不添加新子集，且 Gitik 力迫变体可改变可测基数共尾性至 ω₁ 并引入 Cohen 子集。
We show that while the length $ω$ iterated ultrapower by a normal ultrafilter is a Boolean ultrapower by the Boolean algebra of Prikry forcing, it is consistent that no iteration of length greater than $ω$ (of the same ultrafilter and its images) is a Boolean ultrapower. For longer iterations, where different ultrafilters are used, this is possible, though, and we give Magidor forcing and a generalization of Prikry forcing as examples. We refer to the discovery that the intersection of the finite iterates of the universe by a normal measure is the same as the generic extension of the direct li…
The lottery preparation, a new general kind of Laver preparation, works uniformly with supercompact cardinals, strongly compact cardinals, strong cardinals, measurable cardinals, or what have you. And like the Laver preparation, the lottery preparation makes these cardinals indestructible by various kinds of further forcing. A supercompact cardinal kappa, for example, becomes fully indestructible by kappa-directed closed forcing; a strong cardinal kappa becomes indestructible by less-than-or-equal-kappa-strategically closed forcing; and a strongly compact cardinal kappa becomes indestructible …
Generalizing random real forcing for inaccessible cardinals
本文针对不可达基数$\u03bb$对应的$\u03bb^\u03bb$空间，在附加假设下构造了具备随机实数力迫性质的推广力迫，证明其保持基数与共尾性且不添加无支配实数，填补了随机实数力迫在高阶无穷空间的推广空白。
An optimal extension of the Jensen covering lemma, within the limits imposed by Prikry forcing, is proved. If L[E] is an "iterable" weasel with no measurable cardinals, then either L[E] has "indiscernibles", or every uncountable set of ordinals is contained in a set in L[E] of the same cardinality. (The terms "iterable" and "indiscernibles" are made precise in the paper.) Most importantly, there is no hypothesis explicitly limiting the large cardinals which are consistent in L[E].
Of Mice and Machetes
本文证明非超不可达正则基数类相关的内部模型可由迭代“大砍刀”老鼠并取超类 Magidor 迭代 Prikry 力迫扩张生成，且这些简单老鼠本身属于 $L[\mathsf{Reg}]$。
The effects of forcing on exacting and ultraexacting cardinals
本文研究集合论强制对exacting与ultraexacting基数的影响，证明I2嵌入产生具有真类exacting基数的ZFC模型，并探讨HOD猜想下相关性质及强制扩张。
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。
In this paper, we investigate the relationship between the Tukey order and PCF theory, as applied to sets of regular cardinals. We show that it is consistent that for all sets $A$ of regular cardinals that the Tukey spectrum of $A$, denoted $\operatorname{spec}(A)$, is equal to the set of possible cofinalities of $A$, denoted $\operatorname{pcf}(A)$; this is to be read in light of the $\mathsf{ZFC}$ fact that $\operatorname{pcf}(A)\subseteq\operatorname{spec}(A)$ holds for all $A$. We also prove results about when regular limit cardinals must be in the Tukey spectrum or must be out of the Tuke…
In this issue we celebrate the appearance of the proceedings of the first SPM Workshop, announce several mathematical breakthroughs, have two extended contributions by Babinkostova, and a new open problem by Kalenda. Contents: Editor's note; Proceedings of the first workshop on Coverings, Selections, and Games in Topology; A five element basis for the uncountable linear orders; Set mapping reflection; The Proper Forcing Axiom, Prikry forcing, and the Singular Cardinals Hypothesis; A solution to the L space problem and related ZFC constructions; Countable Tightness, Elementary Submodels and Hom…
We prove the following two results. Theorem A: Let alpha be a limit ordinal. Suppose that 2^{|alpha|}<aleph_alpha and 2^{|alpha|^+}<aleph_{|alpha|^+}, whereas aleph_alpha^{|alpha|}>aleph_{|alpha|^+}. Then for all n< omega and for all bounded X subset aleph_{|alpha|^+}, M_n^#(X) exists. Theorem B: Let kappa be a singular cardinal of uncountable cofinality. If {alpha<kappa| 2^alpha=alpha^+} is stationary as well as co-stationary then for all n< omega and for all bounded X subset kappa, M_n^#(X) exists. Theorem A answers a question of Gitik and Mitchell, and Theorem B yields a lower bound for an …
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