Joel David Hamkins, W. Hugh Woodin
After small forcing, almost every strongness embedding is the lift of a strongness embedding in the ground model. Consequently, small forcing creates neither strong nor Woodin cardinals.
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Despite being an established notion in the large cardinal hierarchy, results about Woodin cardinals are sparse in the literature. Here we gather known results about the preservation of Woodin cardinals under certain forcing extensions, as well as giving a template for preserving Woodin cardinals through forcing. Using this template, we form an indestructibility result under certain Easton iterations.
Suppose that there is a measurable cardinal. If \aleph_ωis a strong limit cardinal, but the power of \aleph_ωis bigger than \aleph_{ω_1}, then there is an inner model with a Woodin cardinal. Modulo the need of the measurable cardinal this answers a question of Gitik and Mitchell.
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。
We give the definition of Woodin for strong compactness cardinals, the Woodinised version of strong compactness, and we prove an analogue of Magidor's identity crisis theorem for the first strongly compact cardinal.
It is shown that if there is a measurable cardinal above n Woodin cardinals and M_{n+1}^# doesn't exist then K exists. K is not fully iterable, though, but only iterable with respect to stacks of certain trees living between the Woodin cardinals. However, it is still true that if M is an omega-closed iterate of V then K^M is an iterate of K.
Under the assumption that $δ$ is a Woodin cardinal and $\GCH$ holds, I show that if $F$ is any class function from the regular cardinals to the cardinals such that (1) $κ<\cf(F(κ))$, (2) $κ<λ$ implies $F(κ)\leq F(λ)$, and (3) $δ$ is closed under $F$, then there is a cofinality-preserving forcing extension in which $2^γ= F(γ)$ for each regular cardinal $γ<δ$, and in which $δ$ remains Woodin. Unlike the analogous results for supercompact cardinals [Men76] and strong cardinals [FH08], there is no requirement that the function $F$ be locally definable.
The determinacy of lightface $Δ^1_{2n+2}$ and boldface $\boldsymbolΠ^1_{2n+1}$ sets implies the existence of an $(ω, ω_1)$-iterable $M_{2n+1}^{\#}$.
Power $Σ_1$ in Card with two Woodin cardinals
本文研究了在Card谓词下，两个Woodin基数与幂集操作$Σ_1$可定义的相容性问题。
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, …
The Virtual Large Cardinal Hierarchy
该研究分析超强、Woodin、Berkeley基数的虚拟版本，给出虚拟Woodin基数的等价刻画，引入有限语言虚拟Vopenka原理并证明其相关等价性，完善虚拟大基数层级的理论研究。
A synthetic overview on some known characterizations of Woodin cardinals
本文基于研讨会幻灯片，综合介绍了 Woodin 基数的多种已知刻画（含 Ernest Schimmerling 的成果），并力求结构自洽，明确展示通常省略的证明细节并给出准确文献参考。
We analyze the hereditarily ordinal definable sets $\operatorname{HOD}$ in $M_n(x)[g]$ for a Turing cone of reals $x$, where $M_n(x)$ is the canonical inner model with $n$ Woodin cardinals build over $x$ and $g$ is generic over $M_n(x)$ for the Lévy collapse up to its bottom inaccessible cardinal. We prove that assuming $\boldsymbolΠ^1_{n+2}$-determinacy, for a Turing cone of reals $x$, $\operatorname{HOD}^{M_n(x)[g]} = M_n(\mathcal{M}_{\infty} | κ_\infty, Λ),$ where $\mathcal{M}_\infty$ is a direct limit of iterates of $M_{n+1}$, $δ_\infty$ is the least Woodin cardinal in $\mathcal{M}_\infty$…
Mice with Woodin cardinals from a Reinhardt
本文证明若存在 Reinhardt 基数，则对任意传递集 X 和自然数 n，M_n(X) 存在且完全可迭代，且所有集合力迫扩张中射影决定性成立。
Chang models over derived models with supercompact measures
本文基于第三作者早期工作构造了带超紧测度的张型模型，扩张了特定hod鼠标的导出模型，证明其满足AD_R等性质，补充了Woodin的广义张模型结果。
Strong forcing axioms and the continuum problem (following Asperó's and Schindler's proof that $\mathbf{MM}^{++}$ implies Woodin's Axiom $(*)$)
本文基于Asperó和Schindler关于$\mathbf{MM}^{++}$蕴含Woodin公理$(*)$的证明，探讨了连续统问题及其与集合论近期众多进展的联系。
The study of inner models was initiated by Gödel's analysis of the constructible universe. Later, the study of canonical inner models with large cardinals, e.g., measurable cardinals, strong cardinals or Woodin cardinals, was pioneered by Jensen, Mitchell, Steel, and others. Around the same time, the study of infinite two-player games was driven forward by Martin's proof of analytic determinacy from a measurable cardinal, Borel determinacy from ZFC, and Martin and Steel's proof of levels of projective determinacy from Woodin cardinals with a measurable cardinal on top. First Woodin and later N…
Let GCH hold and let $j:V\longrightarrow M$ be a definable elementary embedding such that $crit(j)=κ$, $^κM\subseteq M$ and $κ^{++}=κ_{M}^{++}$. H. Woodin proved that there is a cofinality preserving generic extension in which $κ$ is measurable and GCH fails at it. This is done by using an Easton support iteration of Cohen forcings for blowing the power of every inaccessible $α\leqκ$ to $α^{++}$, and then adding another forcing on top of that. We show that it is enough to use the iterated forcing, and that the latter forcing is not needed. We will show this not only for the case where $κ^{++}=…
This paper explores several topics related to Woodin's HOD conjecture. We improve the large cardinal hypothesis of Woodin's HOD dichotomy theorem from an extendible cardinal to a strongly compact cardinal. We show that assuming there is a strongly compact cardinal and the HOD hypothesis holds, there is no elementary embedding from HOD to HOD, settling a question of Woodin. We show that the HOD hypothesis is equivalent to a uniqueness property of elementary embeddings of levels of the cumulative hierarchy. We prove that the HOD hypothesis holds if and only if every regular cardinal above the fi…
We prove the following result which is due to the third author. Let $n \geq 1$. If $\boldsymbolΠ^1_n$ determinacy and $Π^1_{n+1}$ determinacy both hold true and there is no $\boldsymbolΣ^1_{n+2}$-definable $ω_1$-sequence of pairwise distinct reals, then $M_n^\#$ exists and is $ω_1$-iterable. The proof yields that $\boldsymbolΠ^1_{n+1}$ determinacy implies that $M_n^\#(x)$ exists and is $ω_1$-iterable for all reals $x$. A consequence is the Determinacy Transfer Theorem for arbitrary $n \geq 1$, namely the statement that $\boldsymbolΠ^1_{n+1}$ determinacy implies $\Game^{(n)}(<ω^2 - \boldsymbolΠ…
We give Woodin's original proof that if there exists a $(κ+2)-$strong cardinal $κ,$ then there is a generic extension of the universe in which $κ=\aleph_ω,$ $GCH$ holds below $\aleph_ω$ and $2^{\aleph_ω}=\aleph_{ω+2}.$
We show that every Jonsson cardinal is Ramsey in the Steel core model, provided that this model exists and there is no model with a Woodin cardinal. This basic result is improved in two directions. First, we prove the same result for delta-Jonsson and delta-Erdos cardinals, where delta is any regular cardinal smaller than kappa. These notions correspond to Jonsson and Erdos cardinals, except that the submodel or set of indiscernibles is only required to have ordertype delta. The delta-Jonsson cardinals come up in the stationary tower forcing. Second, we weaken the assumption that the Steel cor…
Partial Tower Sealing
本文研究了部分塔密封的性质，证明其在特定模型中成立，并指出其一致性强度低于ZFC加上Woodin极限的存在。
Assuming the existence of a certain hod pair with a Woodin cardinal that is a limit of Woodin cardinals, we show that the Chang model satisfies $\mathsf{AD}^+$ in any set generic extensions.
I analyze the hierarchy of large cardinals between a supercompact cardinal and an almost-huge cardinal. Many of these cardinals are defined by modifying the definition of a high-jump cardinal. A high-jump cardinal is defined as the critical point of an elementary embedding $j: V \to M$ such that $M$ is closed under sequences of length $\sup\set{j(f)(κ) \st f: κ\to κ}$. Some of the other cardinals analyzed include the super-high-jump cardinals, almost-high-jump cardinals, Shelah-for-supercompactness cardinals, Woodin-for-supercompactness cardinals, \Vopenka\ cardinals, hypercompact cardinals, a…
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 …
Let $\mathsf{MM}^{++}(κ)$ state that the forcing axiom $\mathsf{MM}^{++}$ can be instantiated only for stationary set preserving posets of size at most $κ$. We give a detailed account of Asperò and Schindler's proof that $\mathsf{MM}^{++}(κ)+$there are class many Woodin cardinals implies Woodin's axiom $(*)$ if $\Diamond_κ$ holds and $κ>\aleph_2$. Our presentation takes advantage of the notion of consistency property: specifically we rephrase Asperò and Schindler's forcing as a specific instantiation of the notion of ``consistency property'' used by Makkai, Keisler, Mansfield and others in the…
More Derived Models in PFA
本文研究了在PFA下，Woodin基数极限处的衍生模型的性质，展示了其在不同条件下的行为，包括θ值的严格小于和AD_{ℝ}的满足情况。
Recognizable sets and Woodin cardinals: Computation beyond the constructible universe
本文研究了在构造宇宙外通过序数时间 Turing 机进行计算的可识别集，证明了可计算集属于 L，而无限时间计算的可识别对象可达 Woodin 数字的层次。
Given a Woodin cardinal $δ$, I show that if $F$ is any Easton function with $F"δ\subseteqδ$ and $\GCH$ holds, then there is a cofinality-preserving forcing extension in which $2^γ= F(γ)$ for each regular cardinal $γ<δ$, and in which $δ$ remains Woodin. I also present a new example in which forcing a certain behavior of the continuum function on the regular cardinals, while preserving a given large cardinal, requires large cardinal strength beyond that of the original large cardinal under consideration. Specifically, I prove that the existence of a $λ$-supercompact cardinal $κ$ such that $\GCH$…
Measures in Nairian models, I
在AD⁺ + V = L(P(ℝ))假设下，该研究证明超幂公理在HOD_s中成立，分析奈里安模型含大量可测基数，解决Henle等人的问题，还构造出ZF中每个不可数基数均为奇异的模型。
Long games just beyond fixed countable length
本文提出了一种变长可数长度的游戏，并证明其分析确定性与特定Woodin基数的内模型存在性等价。
Improving a result of Woodin, we identify some classes of individually consistent but mutually inconsistent generic large cardinal axioms.
We present several results relating the general theory of the stationary tower forcing developed by Woodin with forcing axioms. The main results is that the forcing axiom MM^{++} (also known as MM^{+ω_1}) decides the Π_2-theory of H_{ω_2} with respect to stationary set preserving forcings. We argue that this is a close to optimal generalization to H_{ω_2} of Woodin's absoluteness results for L(R).
A model of the Axiom of Determinacy in which every set of reals is universally Baire
本文基于导出模型构造，结合泛型迭代方法，证明了满足“每个实数集都是泛Baire集”的确定性公理模型的一致性强度对应存在武丁基数与强基数的极限基数的ZFC系统。
Woodin has shown that if there is a measurable Woodin cardinal then there is, in an appropriate sense, a sharp for the Chang model. We produce, in a weaker sense, a sharp for the Chang model using only the existence of a cardinal $κ$ having an extender of length $κ^{+ω_1}$.
We study $Σ_1(ω_1)$-definable sets (i.e. sets that are equal to the collection of all sets satisfying a certain $Σ_1$-formula with parameter $ω_1$) in the presence of large cardinals. Our results show that the existence of a Woodin cardinal and a measurable cardinal above it imply that no well-ordering of the reals is $Σ_1(ω_1)$-definable, the set of all stationary subsets of $ω_1$ is not $Σ_1(ω_1)$-definable and the complement of every $Σ_1(ω_1)$-definable Bernstein subset of ${}^{ω_1}ω_1$ is not $Σ_1(ω_1)$-definable. In contrast, we show that the existence of a Woodin cardinal is compatible …
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。
Nairian Models
本文引入了决定性公理的Nairian模型层级，通过施力构造出满足特定ZFC组合及超紧基数性质的模型，并据此证明了迭代性猜想的失败及回答了等一致性等关键问题，揭示了内模型理论方法的扩展障碍。
We determine the consistency strength of determinacy for projective games of length $ω^2$. Our main theorem is that $\boldsymbolΠ^1_{n+1}$-determinacy for games of length $ω^2$ implies the existence of a model of set theory with $ω+ n$ Woodin cardinals. In a first step, we show that this hypothesis implies that there is a countable set of reals $A$ such that $M_n(A)$, the canonical inner model for $n$ Woodin cardinals constructed over $A$, satisfies $A = \mathbb{R}$ and the Axiom of Determinacy. Then we argue how to obtain a model with $ω+ n$ Woodin cardinal from this. We also show how the pro…
The exact strength of generic absoluteness for the universally Baire sets
本文研究了泛Baire集的泛绝对性条件$\sf{Sealing}$的精确一致性强度，利用Woodin的$\sf{Core\\ Model\\ Induction}$技术证明其与$\sf{LSA-over-uB}$等一致，并分离出对应的精确大基数理论。
On a Question of Hamkins and Löwe on the modal logic of collapse forcing
本文回应Hamkins与Löwe关于集合论塌缩力迫模态逻辑的疑问，给出该性质两个弱版本的等一致性结果，并证明完整Hamkins-Löwe性质的一致性可由带上方不可达基数的Woodin基数的一致性推出。
Assuming that there is no inner model with a Woodin cardinal, we obtain a characterization of $λ$-tall cardinals in extender models that are iterable. In particular we prove that in such extender models, a cardinal $κ$ is a tall cardinal if and only if it is either a strong cardinal or a measurable limit of strong cardinals.
If kappa is any strongly unfoldable cardinal, then this is preserved in a forcing extension in which Diamond_kappa(REG) fails. This result continues the progression of the corresponding results for weakly compact cardinals, due to Woodin, and for indescribable cardinals, due to Hauser.
We prove in the theory "ZFC + there is no inner model with a Woodin cardinal" that there is a universal weasel. This shows in particular that one doesn't have to assume the existence of large cardinals in V to prove the existence of some such weasel.
The generalized continuum hypothesis above a strongly compact cardinal
本文证明若κ为强紧基数且GCH在κ之下成立，则GCH整体成立，解答了Woodin的长期开放问题。
Forcing Axioms and the Definabilty of the Nonstationary Ideal on $ω_1$
该数理逻辑研究探讨力迫公理下$\omega_1$非驻定理想的可定义性，证明$\text{BMM}$加武丁基数、$(*)$-公理下其不可被$\Sigma_1(\omega_1)$定义，同时构造出$\text{BPFA}$、$\text{MA}$等公理下其可定义甚至饱和的集合论宇宙。
W.H. Woodin showed that if $κ_1 < \cdots < κ_n$ are strong cardinals then two-step ${\bfΣ}^1_{n+3}$ generic absoluteness holds after collapsing $2^{2^{κ_n}}$ to be countable. We show that this number can be reduced to $2^{κ_n}$, and to $κ_n^+$ in the case $n = 1$, but cannot be further reduced to $κ_n$.
Laver, and Woodin independently, showed that models of ${\rm ZFC}$ are uniformly definable in their set-forcing extensions, using a ground model parameter. We investigate ground model definability for models of fragments of ${\rm ZFC}$, particularly of ${\rm ZF}+{\rm DC}_δ$ and of ${\rm ZFC}^-$, and we obtain both positive and negative results. Generalizing the results of Laver and Woodin, we show that models of ${\rm ZF}+{\rm DC}_δ$ are uniformly definable in their set-forcing extensions by posets admitting a gap at $δ$, using a ground model parameter. In particular, this means that models of…
The $\mathsf{HOD}$ Hypothesis and a supercompact cardinal
本文在超紧基数与HOD假设的前提下，证明V_κ中存在真类多在HOD中可测的正则基数，推导出Woodin局部普遍性定理，揭示了HOD假设下两类大基数的差异及V中大基数向HOD的局部反映特性。
We show that if the universe is self-iterable and $κ$ is an inaccessible limit of Woodin cardinal then $AD_R + "Θ$ is regular" holds in the derived model at $κ$. The proof is fine-structure free, and only assumes basic knowledge of iteration trees and iteration strategies. Our proof can be viewed as the fine-structure free version of the well-known fact that $AD_R + "Θ$ is regular" is true in the derived models of hod mice that have inaccessible limit of Woodin cardinals (see for example [6]). However, the proof uses a different set of ideas and is more general.
The consistency strength of determinacy when all sets are universally Baire
本文针对所有实数集合均为普遍 Baire 集时决定性公理的大基数强度，提出混合老鼠的新翻译程序，证明其恰等价于存在 Woodin 基数与强基数的共同极限。
We show that under the assumption of the existence of the canonical inner model with one Woodin cardinal $M_1$, there is a model of $\ZFC$ in which $\NS$ is $\aleph_2$-saturated and $Δ_1$-definable with $ω_1$ as a parameter which answers a question of Sy-David Friedman and Liuzhen Wu. We also show that starting from an arbitrary universe with a Woodin cardinal, there is a model with $\NS$ saturated and $Δ_1$-definable with a ladder system $\vec{C}$ and a full Suslin tree $T$ as parameters. Both results rely on a new coding technique whose presentation is the main goal of this article.
On $(Σ^2_1)^{uB}$ Absoluteness Between V and HOD
本文结合 Woodin 的 Σ²₁ 基定理与 Vopěnka 定理，证明在存在真类多 Woodin 基数时，V 中为真的 (Σ²₁)^uB 命题在 HOD 中也为真，并探讨了反向绝对性及其与 Ultimate L 公理的关系。
The Cofinal Strong Chang Conjecture from Models of Determinacy
通过修改Woodin的证明，表明从决定性模型使用P_max力迫得到的模型满足更强的共尾版本的强Chang猜想，并由此证明Namba力迫是半正常的且Θ^{UB}=ω_3的一致性，以及Woodin (*)_{UB}公理蕴含该猜想。
We present two ways in which the model $L({\mathbb R})$ is canonical assuming the existence of large cardinals. We show that the theory of this model, with {\em ordinal} parameters, cannot be changed by small forcing; we show further that a set of ordinals in $V$ cannot be added to $L({\mathbb R})$ by small forcing. The large cardinal needed corresponds to the consistency strength of $AD^{L({\mathbb R})}$; roughly $ω$ Woodin cardinals.
It is shown, from hypotheses in the region of $ω^2$ Woodin cardinals, that there is a transitive model of KP + AD$_\mathbb{R}$ containing all reals.
Local mantles of $L[x]$
本文在 Woodin 基数与真类多可测基数假设下，利用 Woodin 对 HOD 的分析方法研究 $L[x]$ 的局部 $κ$-mantle，证明其满足 ZFC+GCH 并含 Woodin 基数，且为完全可迭代策略 mouse。
The landmark Levy-Solovay Theorem limits the kind of large cardinal embeddings that can exist in a small forcing extension. Here I announce a generalization of this theorem to a broad new class of forcing notions. One consequence is that many of the forcing iterations most commonly found in the large cardinal literature create no new weakly compact cardinals, measurable cardinals, strong cardinals, Woodin cardinals, strongly compact cardinals, supercompact cardinals, almost huge cardinals, huge cardinals, and so on.
We describe an obstacle to the analysis of $\mathrm{HOD}^{L[x]}$ as a core model: Assuming sufficient large cardinals, for a Turing cone of reals $x$ there are premice $M,N$ in $\mathrm{HC}^{L[x]}$ such that the pseudo-comparison of $L[M]$ with $L[N]$ succeeds, is computed in $L[x]$, and lasts through $ω_1^{L[x]}$ stages. Moreover, we can take $M=M_1|(δ^+)^{M_1}$ where $M_1$ is the minimal iterable proper class inner model with a Woodin cardinal, and $δ$ is that Woodin. We can take $N$ such that $L[N]$ is $M_1$-like and short-tree-iterable.
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。
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