James Cummings, Sy David Friedman, Mohammad Golshani
Assuming that $GCH$ holds and $κ$ is $κ^{+3}$-supercompact, we construct a generic extension $W$ of $V$ in which $κ$ remains strongly inaccessible and $(α^+)^{HOD} < α^+$ for every infinite cardinal $α< κ$. In particular the rank-initial segment $W_κ$ is a model of ZFC in which $(α^+)^{HOD} < α^+$ for every infinite cardinal $α$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
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 that large cardinals need not generally exhibit their large cardinal nature in HOD. For example, a supercompact cardinal $κ$ need not be weakly compact in HOD, and there can be a proper class of supercompact cardinals in $V$, none of them weakly compact in HOD, with no supercompact cardinals in HOD. Similar results hold for many other types of large cardinals, such as measurable and strong cardinals.
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…
We deal with (< kappa)-supported iterated forcing notions which are (E_0,E_1)-complete, have in mind problems on Whitehead groups, uniformizations and the general problem. We deal mainly with the successor of a singular case. This continues [Sh:587]. We also deal with complimentary combinatorial results.
Assuming the existence of a supercompact cardinal and an inaccessible above it, we construct a model of ZFC, in which all uncountable regular cardinals are inaccessible in HOD.
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。
Given a forcing notion $P$ that forces certain values to several classical cardinal characteristics of the reals, we show how we can compose $P$ with a collapse (of a cardinal $λ>κ$ to $κ$) such that the composition still forces the previous values to these characteristics. We also show how to force distinct values to $\mathfrak m$, $\mathfrak p$ and $\mathfrak h$ and also keeping all the values in Cichoń's diagram distint, using the Boolean Ultrapower method of arXiv:1708.03691 . (In arXiv:2006.09826 , the same was done for the newer Cichoń's Maximum construction, which avoids large cardinals…
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 $\mathcal{M}_\infty$…
We deal with the problem of preserving various versions of completeness in (< kappa) --support iterations of forcing notions, generalizing the case ``S --complete proper is preserved by CS iterations for a stationary co-stationary S subseteq omega_1''. We give applications to Uniformization and the Whitehead problem. In particular, for a strongly inaccessible cardinal kappa and a stationary set S subseteq kappa with fat complement we can have uniformization for < A_delta : delta in S'>, A_delta subseteq delta = sup A_delta, cf(delta)=otp(A_delta) and a stationary non-reflecting set S' subseteq…
Suppose $κ$ is a singular strong limit cardinal of countable cofinality and let $\langle κ_{n}: n<ω\rangle$ be an incrasing sequence of regular cardinals cofinal in $κ$. We show that if $cf(2^κ)= κ^+$, then forcing with the full product $\prod_{n<ω}Add(κ_n,1)$ collapses $2^κ$ into $κ^+$. This result gives a consistent positive answer to a question of Sy Friedman. We also give a new proof of a result due to Shelah by showing that if the sequence carries a scale of length $κ^+,$ then forcing with $\prod_{n<ω}Add(κ_n,1)$ adds a generic filter for $Add(κ^+, 1)$, and indeed \[ \prod_{n<ω}Add(κ_n,1)…
Large cardinals beyond HOD
本文研究exacting与ultraexacting大基数，证明其与I0等一致，并给出其强度界及新刻画，表明它们与HOD假设相容。
On the optimality of the HOD dichotomy
本文研究HOD假设下大基数性质的最优性，证明可扩展基数在HOD中的紧致性与超紧性，并探讨弱覆盖失败及HOD正则基数等一致性结果。
Large cardinals, structural reflection, and the HOD Conjecture
本文引入了exacting cardinals与ultraexacting cardinals，证明其存在性挑战了大基数层谱的线性递增图景，并表明其超越现有与ZFC一致的大基数层级，能够反驳Woodin的HOD猜想与Ultimate-L猜想。
Supercompact extender based forcings are used to construct models with HOD cardinal structure different from those of V. In particular, a model with all regular uncountable cardinals measurable in HOD is constructed.
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。
We prove several consistency results concerning the notion of $ω$-strongly measurable cardinal in HOD. In particular, we show that is it consistent, relative to a large cardinal hypothesis weaker than $o(κ) = κ$, that every successor of a regular cardinal is $ω$-strongly measurable in HOD.
Starting from large cardinals we construct a model of $ZFC$ in which the $GCH$ fails everywhere, but such that $GCH$ holds in its $HOD$. The result answers a question of Sy Friedman. Also, relative to the existence of large cardinals, we produce a model of $ZFC+GCH$ such that $GCH$ fails everywhere in its $HOD$.
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 version of Woodin's HOD dichotomy is proved assuming the existence of just one strongly compact cardinal.
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。
Local Definability of $\mathrm{HOD}$ in $L(\mathbb{R})$
该研究在大基数假设下证明了$L(\mathbb{R})$中$\mathsf{HOD}$的局部可定义性，进一步阐释了Steel关于$\mathsf{HOD}^{L(\mathbb{R})}$是$Θ$之下核心模型的论断。
Compactness beyond choice and HOD
本文用逻辑和拓扑空间的紧致性刻画了多种Berkeley型大基数，表明紧致性假设可违反选择公理和V=HOD，从而提升已知一致性强度。
Compactness phenomena in HOD and the Optimality of Magidor's Covering theorem
该研究围绕集合论全域V与HOD的紧性现象，证明了奇异强极限基数的幂集分歧特性，回答了Hayut的问题，并揭示Magidor覆盖定理的最优性。
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。
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。
We construct a variety of inner models exhibiting features usually obtained by forcing over universes with large cardinals. For example, if there is a supercompact cardinal, then there is an inner model with a Laver indestructible supercompact cardinal. If there is a supercompact cardinal, then there is an inner model with a supercompact cardinal κfor which 2^κ=κ^+, another for which 2^κ=κ^++ and another in which the least strongly compact cardinal is supercompact. If there is a strongly compact cardinal, then there is an inner model with a strongly compact cardinal, for which the measurable c…
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.
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。
We introduce a category whose objects are stationary set preserving complete boolean algebras and whose arrows are complete homomorphisms with a stationary set preserving quotient. We show that the cut of this category at a rank initial segment of the universe of height a super compact which is a limit of super compact cardinals is a stationary set preserving partial order which forces $MM^{++}$ and collapses its size to become the second uncountable cardinal. Next we argue that any of the known methods to produce a model of $MM^{++}$ collapsing a superhuge cardinal to become the second uncoun…
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.
On $Σ_1$-Definable Closed Unbounded Sets
该文聚焦集合论中基数$κ$的$Σ_1$-可定义闭无界集，研究其对完全闭无界滤子的逼近程度及对定义复杂度的依赖，发现逼近能力差异大且依赖底层集合论宇宙性质。
Varsovian models $ω$
本文将Varsovian模型构造从有限Woodin与strong基数情形推广到N_ω；在星翻译可与P构造常规整合的假设下，证明其存在含ω个Woodin基数的可迭代策略鼠标内模型，并刻画其eventual generic HOD、mantle及核心模型K。
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。