Gabriel Goldberg
While many inner model theoretic combinatorial principles are incompatible with large cardinal axioms, on some rare occasions, large cardinals actually imply that the structure of the universe of sets is analogous to the canonical inner models. This note provides two new instances of this phenomenon.
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We prove that the Generalized Continuum Hypothesis holds above a supercompact cardinal assuming the Ultrapower Axiom, an abstract comparison principle motivated by inner model theory at the level of supercompact cardinals.
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.
In this paper, we characterize the possible cofinalities of the least $λ$-strongly compact cardinal. We show that, on the one hand, for any regular cardinal, $δ$, that carries a $λ$-complete uniform ultrafilter, it is consistent, relative to the existence of a supercompact cardinal above $δ$, that the least $λ$-strongly compact cardinal has cofinality $δ$. On the other hand, provably the cofinality of the least $λ$-strongly compact cardinal always carries a $λ$-complete uniform ultrafilter.
We summarize the known methods of producing a non-supercompact strongly compact cardinal and describe some new variants. Our Main Theorem shows how to apply these methods to many cardinals simultaneously and exactly control which cardinals are supercompact and which are only strongly compact in a forcing extension. Depending upon the method, the surviving non-supercompact strongly compact cardinals can be strong cardinals, have trivial Mitchell rank or even contain a club disjoint from the set of measurable cardinals. These results improve and unify previous results of the first author.
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 study the general problem of the behaviour of the continuum function in the presence of non-supercompact strongly compact cardinals.
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 show that, assuming GCH, if $κ$ is a Ramsey or a strongly Ramsey cardinal and $F$ is a class function on the regular cardinals having a closure point at $κ$ and obeying the constraints of Easton's theorem, namely, $F(α)\leq F(β)$ for $α\leqβ$ and $α<\cf(F(α))$, then there is a cofinality preserving forcing extension in which $κ$ remains Ramsey or strongly Ramsey respectively and $2^δ=F(δ)$ for every regular cardinal $δ$.
If kappa is strongly compact, lambda > kappa is regular, then (2^{< lambda})^+ --> (lambda+eta)^2_theta holds for eta,theta<kappa.
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。
We introduce the strongly uplifting cardinals, which are equivalently characterized, we prove, as the superstrongly unfoldable cardinals and also as the almost hugely unfoldable cardinals, and we show that their existence is equiconsistent over ZFC with natural instances of the boldface resurrection axiom, such as the boldface resurrection axiom for proper forcing.
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 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 $λ=λ^{<κ}$.
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…
We prove some consistency results about b(lambda) and d(lambda), which are natural generalisations of the cardinal invariants of the continuum b and d. We also define invariants b_cl(lambda) and d_cl(lambda), and prove that almost always b(lambda) = b_cl(lambda) and d(lambda)=d_cl(lambda)
This paper presents the main results in my Ph.D. thesis. In what follows several proofs of SCH are presented introducing a family of covering properties which implies both SCH and the failure of various forms of square. These covering properties are also applied to investigate models of strongly compact cardinals or of strong forcing axioms like MM or PFA.
Assuming the existence of a strong cardinal and a measurable cardinal above it, we construct a model of $ZFC$ in which for every singular cardinal $δ$, $δ$ is strong limit, $2^δ=δ^{+3}$ and the tree property at $δ^{++}$ holds. This answers a question of Friedman, Honzik and Stejskalova [8]. We also produce, relative to the existence of a strong cardinal and two measurable cardinals above it, a model of $ZFC$ in which the tree property holds at all regular even cardinals. The result answers questions of Friedman-Halilovic [5] and Friedman-Honzik [6].
Using Koszmider's strongly unbounded functions, we show the following consistency result: Suppose that $κ,λ$ are infinite cardinals such that $κ^{+++} \leq λ$, $κ^{<κ}=κ$ and $2^κ= κ^+$, and $η$ is an ordinal with $κ^+\leq η<κ^{++}$ and $cf(η) = κ^+$. Then, in some cardinal-preserving generic extension there is a superatomic Boolean algebra $B$ such that - $ht(B) = η+ 1$, - the cardinality of the $α$th level of $B$ is $κ$ for every $α<η$, - and the cardinality of the $η$th level of $B$ is $λ$ Especially, $\<ω\>_{ω_1}\concatenation \<ω_3\>$ and $\<ω_1\>_{ω_2}\concatenation \<ω_4\>$ can be cardi…
We show relative to strong hypotheses that patterns of compact cardinals in the universe, where a compact cardinal is one which is either strongly compact or supercompact, can be virtually arbitrary. Specifically, we prove if V is a model of ``ZFC + Omega is the least inaccessible limit of measurable limits of supercompact cardinals + f :Omega-->2 is a function'', then there is a partial ordering P in V so that for Vbar = V^P, Vbar_Omega models ``ZFC + There is a proper class of compact cardinals + If f(alpha) = 0, then the alpha-th compact cardinal isn't supercompact + If f(alpha) = 1, then t…
Patterns of Structural Reflection in the large-cardinal hierarchy
该研究揭示首个可测基数以下大基数层级的结构反射新模式，刻画强不可展开与微妙基数，引入$C^{(n)}$-强不可展开基数并证明其形成自然层级，证实大基数低区域与更高区域结构完全类似。
Large cardinal characterizations via compactness for list colourings
该研究引入含知名大基数概念的紧性基数分层，通过其刻画弱紧等大基数，并得到新紧性基数大小的下界。
We show that Shelah's Eventual Categoricity Conjecture follows from the existence of class many strongly compact cardinals. This is the first time the consistency of this conjecture has been proven. We do so by showing that every AEC with $LS(K)$ below a strongly compact cardinal $κ$ is $< κ$ tame and applying the categoricity transfer of Grossberg and VanDieren. These techniques also apply to measurable and weakly compact cardinals and we prove similar tameness results under those hypotheses. We isolate a dual property to tameness, called \emph{type shortness}, and show that it follows simila…
Weak compactness cardinals for strong logics and subtlety properties of the class of ordinals
该研究通过序数类组合性质刻画抽象逻辑弱紧基数存在性，证明其不蕴含强不可达基数存在，并给出其与真类微妙基数的一致性等价条件及HOD内模型中的相关推论。
In this paper we investigate more characterizations and applications of $δ$-strongly compact cardinals. We show that, for a cardinal $κ$ the following are equivalent: (1) $κ$ is $δ$-strongly compact, (2) For every regular $λ\ge κ$ there is a $δ$-complete uniform ultrafilter over $λ$, and (3) Every product space of $δ$-Lindelöf spaces is $κ$-Lindelöf. We also prove that in the Cohen forcing extension, the least $ω_1$-strongly compact cardinal is a precise upper bound on the tightness of the products of two countably tight spaces.
This paper establishes a number of constraints on the structure of large cardinals under strong compactness assumptions. These constraints coincide with those imposed by the Ultrapower Axiom, a principle that is expected to hold in Woodin's hypothesized Ultimate \(L\), providing some evidence for the Ultimate \(L\) Conjecture. We show that every regular cardinal above the first strongly compact that carries an indecomposable ultrafilter is measurable, answering a question of Silver for large enough cardinals. We show that any successor almost strongly compact cardinal of uncountable cofinality…
We introduce the notion of weakly extendible cardinals and show that these cardinals are characterized in terms of weak compactness of second order logic. The consistency strength and largeness of weakly extendible cardinals are located strictly between that of strongly unfoldable (i.e. shrewd) cardinals, and strongly uplifting cardinals. Weak compactness of many other logics can be connected to certain variants of the notion of weakly extendible cardinals. We also show that, under V=L, a cardinal $κ$ is the weak compactness number of ${\cal L}^{\aleph_0,II}_{stat,κ,ω}$ if and only if it is th…
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.
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 prove the consistency of the theory ZFC + there is a strongly compact cardinal from the existence of a cardinal preserving embedding from the universe into an inner model. The proof almost shows that under SCH, every cardinal preserving embedding preserves the cofinality and continuum functions.
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。
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。
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 $κ^{++}$.
We prove two ZFC theorems about cardinal invariants above the continuum which are in sharp contrast to well-known facts about these same invariants at the continuum. It is shown that for an uncountable regular cardinal $κ$, $\mathfrak{b}(κ) = κ^{+}$ implies $\mathfrak{a}(κ) = κ^{+}$. This improves an earlier result of Blass, Hyttinen, and Zhang. It is also shown that if $κ\geq {\beth}_ω$ is an uncountable regular cardinal, then $\mathfrak{d}(κ) \leq \mathfrak{r}(κ)$. This result partially dualizes an earlier theorem of the authors.
A Strongly Non-Saturated Aronszajn Tree Without Weak Kurepa Trees
本文基于 Chang 猜想的否定与不可达基数等大基数假设，通过构造 c.c.c. 和 Y-proper 等力迫偏序集，证明了强非饱和 Aronszajn 树的存在性与弱 Kurepa 树的不存在性及相关双基数树性质保持一致。
If an extension Vbar of V satisfies the delta approximation and cover properties for classes and V is a class in Vbar, then every suitably closed embedding j:Vbar to Nbar in Vbar with critical point above delta restricts to an embedding j|V:V to N amenable to the ground model V. In such extensions, therefore, there are no new large cardinals above delta. This result extends work in math.LO/9808011.
In this article we prove three main theorems: (1) guessing models are internally unbounded, (2) for any regular cardinal $κ\ge ω_2$, $\textsf{ISP}(κ)$ implies that $\textsf{SCH}$ holds above $κ$, and (3) forcing posets which have the $ω_1$-approximation property also have the countable covering property. These results solve open problems of Viale and Hachtman-Sinapova.
We show that the notions of "strongly unfoldable cardinals", introduced by Villaveces in his model-theoretic studies of models of set theory, and "shrewd cardinals", introduced by Rathjen in a proof-theoretic context, coincide. We then proceed by using ideas from the proof of this equivalence to establish the existence of "ordinal anticipating Laver functions" for strong unfoldability. With the help of these functions, we show that the principle $\Diamond_κ(\mathrm{Reg})$ holds at every strongly unfoldable cardinal $κ$ with the property that there exists a subset $z$ of $κ$ such that every sub…
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。
Assuming an abstract comparison principle called the Ultrapower Axiom, which is motivated by the comparison process of inner model theory and generalizes the statement that the Mitchell order is linear on normal ultrafilters, we characterize strongly compact cardinals in terms of supercompactness: a strongly compact cardinal is either supercompact or a limit of supercompact cardinals. Assuming the Ultrapower Axiom and the GCH, we also prove a local result that roughly states that every countably complete ultrafilter factors as a finite iteration of supercompact ultrafilters.
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.
In the context of large cardinals, the classical diamond principle Diamond_kappa is easily strengthened in natural ways. When kappa is a measurable cardinal, for example, one might ask that a Diamond_kappa sequence anticipate every subset of kappa not merely on a stationary set, but on a set of normal measure one. This is equivalent to the existence of a function l:kappa-->V_kappa such that for any A in H(kappa+) there is an embedding j:V-->M having critical point kappa with j(l)(kappa)=A. This and similar principles formulated for many other large cardinal notions, including weakly compact, i…
We observe that the nonstandard finite cardinality of a definable set in a strongly minimal pseudofinite structure D is a polynomial over the integers in the nonstandard finite cardinality of D. We conclude that D is unimodular, hence also locally modular. We also deduce a regularity lemma for graphs definable in strongly minimal pseudofinite structures. The paper is elementary, and the only surprising thing about it is that the results were not explicitly noted before. In the new version we add a comment on relations to work of Macpherson and Steinhorn, as well as on the limited nature of the…
This paper continues a line of investigation of the Halpern--Läuchli Theorem at uncountable cardinals. We prove in ZFC that the Halpern--Läuchli Theorem for one tree of height $κ$ holds whenever $κ$ is strongly inaccessible and the coloring takes less than $κ$ colors. We prove consistency of the Halpern--Läuchli Theorem for finitely many trees of height $κ$, where $κ$ is a strong limit cardinal of countable cofinality. On the other hand, we prove failure of weak forms of Halpern--\Lauchli\ for trees of height $κ$, whenever $κ$ is a strongly inaccessible, non-Mahlo cardinal or a singular strong…
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 prove the consistency of the failure of the weak diamond $Φ_λ$ at strongly inaccessible cardinals. On the other hand, we show that the very weak diamond $Ψ_λ$ is equivalent to the statement $2^{<λ}<2^λ$ and hence holds at every strongly inaccessible cardinal.
We prove from suitable large cardinal hypotheses that the least weakly compact cardinal can be unfoldable, weakly measurable and even nearly $θ$-supercompact, for any desired $θ$. In addition, we prove several global results showing how the entire class of weakly compact cardinals, a proper class, can be made to coincide with the class of unfoldable cardinals, with the class of weakly measurable cardinals or with the class of nearly $θ_κ$-supercompact cardinals $κ$, for nearly any desired function $κ\mapstoθ_κ$. These results answer several questions that had been open in the literature and ex…
Low level definability above large cardinals
研究大基数与可定义性之间的关系，证明了某些基数下的良序和族的存在性，以及内模型的构造。
We investigate the consistency strength of the statement: $κ$ is weakly compact and there is no tree on $κ$ with exactly $κ^{+}$ many branches. We show that this statement fails strongly (in the sense that there is a sealed tree with exactly $κ^{+}$ many branches) if there is no inner model with a Woodin cardinal. Moreover, we show that for a weakly compact cardinal $κ$ the nonexistence of a tree on $κ$ with exactly $κ^{+}$ many branches and, in particular, the Perfect Subtree Property for $κ$, implies the consistency of $AD_{\mathbb{R}} + DC$.
Recently the second author introduced combinatorial principles that characterize supercompactness for inaccessible cardinals but can also hold true for small cardinals. We prove that the proper forcing axiom PFA implies these principles hold for $ω_2$. Using this, we argue to show that any of the known methods for forcing models of PFA from a large cardinal assumption requires a strongly compact cardinal. If one forces PFA using a proper forcing, then we get the optimal result that a supercompact cardinal is necessary.
Assuming three strongly compact cardinals, it is consistent that \[ \aleph_1 < \mathrm{add}(\mathrm{null}) < \mathrm{cov}(\mathrm{null}) < \mathfrak{b} < \mathfrak{d} < \mathrm{non}(\mathrm{null}) < \mathrm{cof}(\mathrm{null}) < 2^{\aleph_0}.\] Under the same assumption, it is consistent that \[ \aleph_1 < \mathrm{add}(\mathrm{null}) < \mathrm{cov}(\mathrm{null}) < \mathrm{non}(\mathrm{meager}) < \mathrm{cov}(\mathrm{meager}) < \mathrm{non}(\mathrm{null}) < \mathrm{cof}(\mathrm{null}) < 2^{\aleph_0}.\]
For a topological space $X$, let $X_δ$ be the space $X$ with $G_δ$-topology of $X$. For an uncountable cardinal $κ$, we prove that the following are equivalent: (1) $κ$ is $ω_1$-strongly compact. (2) For every compact Hausdorff space $X$, the Lindelöf degree of $X_δ$ is $\le κ$. (3) For every compact Hausdorff space $X$, the weak Lindelöf degree of $X_δ$ is $\le κ$. This shows that the least $ω_1$-strongly compact cardinal is the supremum of the Lindelöf and the weak Lindelöf degrees of compact Hausdorff spaces with $G_δ$-topology. We also prove the least measurable cardinal is the supremum of…
Assuming $κ$ is a supercompact cardinal and $λ$ is an inaccessible cardinal above it, we present an idea due to Magidor, to find a generic extension in which $κ=\aleph_ω$ and $λ=\aleph_{ω+1}.$
Bagaria and Magidor introduced the notion of almost strong compactness, which is very close to the notion of strong compactness. Boney and Brooke-Taylor asked whether the least almost strongly compact cardinal is strongly compact. Goldberg gives a positive answer in the case $\mathrm{SCH}$ holds from below and the least almost strongly compact cardinal has uncountable cofinality. In this paper, we give a negative answer for the general case. Our result also gives an affirmative answer to a question of Bagaria and Magidor.
In this paper we introduce a new hierarchy of large cardinals between I3 and I2, the iterability hierarchy, and we prove that every step of it strongly implies the ones below.
Extendible cardinals, and Laver-generic large cardinal axioms for extendibility
引入用于可扩展性的Laver-生成大基数公理，证明其能推导出先前已知的大多数推论，并探讨了其一致性、反射性质及相关分离结果。
The $κ$-Strongly Proper Forcing Axiom
研究高阶力迫公理的相容性，通过超紧基数力迫得到$\kappa$-强正常且$\kappa$-格力迫公理的一致性，消除大基数需求，并构造$2^\kappa$任意大的模型，同时证明其某些自然强化的不相容性。
In earlier work of the second and third author the equivalence of a finite square principle square^fin_{lambda,D} with various model theoretic properties of structures of size lambda and regular ultrafilters was established. In this paper we investigate the principle square^fin_{lambda,D}, and thereby the above model theoretic properties, at a regular cardinal. By Chang's Two-Cardinal Theorem, square^fin_{lambda,D} holds at regular cardinals for all regular filters D if we assume GCH. In this paper we prove in ZFC that for certain regular filters that we call "doubly^+ regular", square^fin_{la…
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。
Aronszajn trees and maximality
在弱康托尔基数存在条件下，证明了宽μ⁺-Aronszajn树的一致性，该树能容纳所有宽μ⁺-Aronszajn树的嵌入。
In [13] the authors show that if $μ$ is a strongly compact cardinal, $K$ is an Abstract Elementary Class (AEC) with $LS(K)<μ$, and $K$ satisfies joint embedding (amalgamation) cofinally below $μ$, then $K$ satisfies joint embedding (amalgamation) in all cardinals $\ge μ$. The question was raised if the strongly compact upper bound was optimal. In this paper we prove the existence of an AEC $K$ that can be axiomatized by an $\mathcal{L}_{ω_1,ω}$-sentence in a countable vocabulary, so that if $μ$ is the first measurable cardinal, then (1) $K$ satisfies joint embedding cofinally below $μ$ ; (2) $…
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