Will Boney, Stamatis Dimopoulos, Victoria Gitman, Menachem Magidor
In [Bon20], model theoretic characterizations of several established large cardinal notions were given. We continue this work, by establishing such characterizations for Woodin cardinals (and variants), various virtual large cardinals, and subtle cardinals.
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This is the second combinatorial proof of the compactness theorem for singular from 1977. In fact it gives a somewhat stronger theorem.
We succeed to say something on the identities of (mu^+, mu) when mu>theta>cf(mu), mu strong limit theta--compact. This hopefully will help to prove the consistency of ``some pair (mu^+,mu) is not compact'', however, this has not been proved.
In a paper from 1997, Shelah asked whether $Pr_1(λ^+,λ^+,λ^+,λ)$ holds for every inaccessible cardinal $λ$. Here, we prove that an affirmative answer follows from $\square(λ^+)$. Furthermore, we establish that for every pair $χ<κ$ of regular uncountable cardinals, $\square(κ)$ implies $Pr_1(κ,κ,κ,χ)$.
We study uniqueness of limit models in abstract elementary classes (AECs) with no maximal models. We prove (assuming instances of diamonds) that categoricity in a cardinal of the form $μ^{+(n + 1)}$ implies the uniqueness of limit models of cardinality $μ^{+}, μ^{++}, \ldots, μ^{+n}$. This sheds light on a paper of Shelah and Villaveces, who were the first to consider uniqueness of limit models in this context. We also prove that (again assuming instances of diamonds) in an AEC with no maximal models, tameness (a locality property for types) together with categoricity in a proper class of card…
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.
Narrow systems revisited
本文引入并研究广义窄系统性质，证明了其足以蕴含$κ$以上的奇异基数假设，并在所有正则基数$κ\geqω_2$处一致成立，同时揭示了其与强无界次可加着色的联系。
We extend and improve the result of Makkai and Paré that the powerful image of any accessible functor F is accessible, assuming there exists a sufficiently large strongly compact cardinal. We reduce the required large cardinal assumption to the existence of $L_{μ,ω}$-compact cardinals for sufficiently large μ, and also show that under this assumption the λ-pure powerful image of F is accessible. From the first of these statements, we obtain that the tameness of every Abstract Elementary Class follows from a weaker large cardinal assumption than was previously known. We provide two ways of empl…
This is a revised version (of late 2020) of [Sh:700], which is arXiv:math/0012170 . First point is noting that the proof of Theorem 4.3 in [Sh:700], which says that the proof giving the consistency $ \mathfrak{b} = \mathfrak{d} = \mathfrak{u} < \mathfrak{a} $ also gives $ \mathfrak{s} = \mathfrak{d} $. The proof uses a measurable cardinal and a c.c.c. forcing so it gives large $ \mathfrak{d} $ and assumes a large cardinal. Second point is adding to the results of \S2,\S3 which say that (in \S3 with no large cardinals) we can force $ {\aleph_1} < \mathfrak{b} = \mathfrak{d} < \mathfrak{a}$. We …
Generic Absoluteness Revisited
本文研究递归公理与拉弗泛型大基数公理的关系，证明从特定递归公理片段和紧拉弗泛型巨大基数存在性可推出更一般的泛型绝对性结论，且相关假设在多数常见偏序集类下蕴含地基公理的否定，区分了两类公理的对应片段。
Infinite Combinatorics revisited in the absence of Axiom of Choice
该研究在ZF公理系统下考察无穷组合定理的可证性，通过组合论证与绝对性方法得到多个基数相关的组合结论，证明了若干陈述的等价性与不可证性，并明确了一致可数枚举存在性相关命题的等一致性强度。
We consider compactness characterizations of large cardinals. Based on results of Benda \cite{b-sccomp}, we study compactness for omitting types in various logics. In $\bL_{κ, κ}$, this allows us to characterize any large cardinal defined in terms of normal ultrafilters, and we also analyze second-order and sort logic. In particular, we give a compactness for omitting types characterization of huge cardinals, which have consistency strength beyond Vopěnka's Principle.
Let P be the direct product of countably many copies of the additive group Z of integers. We study, from a set-theoretic point of view, those subgroups of P for which all homomorphisms to Z annihilate all but finitely many of the standard unit vectors. Specifically, we relate the smallest possible size of such a subgroup to several of the standard cardinal characteristics of the continuum. We also study some related properties and cardinals, both group-theoretic and set-theoretic. One of the set-theoretic properties and the associated cardinal are combinatorially natural, independently of any …
Model-theoretic characterizations of large cardinals (Re)${}^2$visited
通过一阶逻辑扩展的模型论性质刻画多种大基数概念，包括Π_n-强基数、巨大基数等，并揭示Vopěnka原理与弱Vopěnka原理之间的模型论类比。
Game-theoretic variants of cardinal invariants
本文研究了连续统基数不变量的博弈论变体，分析了reaping、bounding、dominating数及null理想加法数的博弈性质，并确定了玩家的获胜策略。
We define a potentialist system of ZF-structures, that is, a collection of possible worlds in the language of ZF connected by a binary accessibility relation, achieving a potentialist account of the full background set-theoretic universe $V$. The definition involves Berkeley cardinals, the strongest known large cardinal axioms, inconsistent with the Axiom of Choice. In fact, as background theory we assume just ZF. It turns out that the propositional modal assertions which are valid at every world of our system are exactly those in the modal theory S4.2. Moreover, we characterize the worlds sat…
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.
We introduce a model-theoretic characterization of Magidor cardinals, from which we infer that Magidor filters are beyond ZFC-inconsistency
Inspired by Zermelo's quasi-categoricity result characterizing the models of second-order Zermelo-Fraenkel set theory $\text{ZFC}_2$, we investigate when those models are fully categorical, characterized by the addition to $\text{ZFC}_2$ either of a first-order sentence, a first-order theory, a second-order sentence or a second-order theory. Thus we mount an analysis of the categorical large cardinals. This mathematical analysis leads naturally to philosophical issues concerning structuralism and realism, including especially the tension between categoricity and reflection. Ultimately we ident…
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…
We study several cardinal characteristics of closed graphs G on compact metrizable spaces. In particular, we address the question when it is consistent for the bounding number to be strictly smaller than the smallest size of a set not covered by countably many compact G-anticliques. We also provide a descriptive set theoretic characterization of the class of analytic graphs with countable coloring number.
Through careful analysis of an argument of Brooke-Taylor and Rosicky, we show that the powerful image of any accessible functor is closed under colimits of $κ$-chains, $κ$ a sufficiently large almost measurable cardinal. This condition on powerful images, by methods resembling those of Lieberman and Rosicky, implies $κ$-locality of Galois types. As this, in turn, implies sufficient measurability of $κ$, via a paper of Boney and Unger, we obtain an equivalence: a purely category-theoretic characterization of almost measurable cardinals.
We give a new proof of a theorem of Becker that under AD+V=L(R), omega_2 is a kappa-supercompact for every kappa less than or equal to the supremum of all Suslin cardinals. Our proof uses inner model theory. It is still open whether one can prove, say under AD_R, that Suslin cardinals and their successors are <Theta-supercompact. That this is true is a theorem of Steve Jackson, who proved it using generic codes.
Supervaluation-Style Truth Revisited
针对Stern提出的超赋值风格真理论策略未完全实现的问题，研究者给出其核心论断的反例，同时提出该构想的不同语义实现与对应ℕ-范畴公理化，深化了两类真理论关系的认识。
Building models in small cardinals in local abstract elementary classes
本文证明了在局部抽象初等类中，假设Galois类型满足温和局部性条件，稳定性足以在小基数下构建更大模型，并显著改进了已有定理使其适用于部分不稳定类。
Noetherian type and Noetherian $π$-type are two cardinal functions which were introduced by Peregudov in 1997, capturing some properties studied earlier by the Russian School. Their behavior has been shown to be akin to that of the \emph{cellularity}, that is the supremum of the sizes of pairwise disjoint non-empty open sets in a topological space. Building on that analogy, we study the Noetherian $π$-type of $κ$-Suslin Lines, and we are able to determine it for every $κ$ up to the first singular cardinal. We then prove a consequence of Chang's Conjecture for $\aleph_ω$ regarding the Noetheria…
In the previous work, we have given a novel, game-semantic model of computation in an intrinsic, non-inductive and non-axiomatic manner, which is similar to Turing machines but beyond computation on natural numbers, e.g., higher-order computation. As the main theorem of the work, it has been shown that the game-semantic model may execute all the computation of the programming language PCF. The present paper revisits this result from an automata-theoretic perspective: It shows that deterministic non-erasing pushdown automata whose input tape is equipped with simple directed edges between cells …
Working in the context of $μ$-abstract elementary classes ($μ$-AECs) - or, equivalently, accessible categories with all morphisms monomorphisms - we examine the two natural notions of size that occur, namely cardinality of underlying sets and internal size. The latter, purely category-theoretic, notion generalizes e.g. density character in complete metric spaces and cardinality of orthogonal bases in Hilbert spaces. We consider the relationship between these notions under mild set-theoretic hypotheses, including weakenings of the singular cardinal hypothesis. We also establish preliminary resu…
We give a sharper version of a theorem of Rosicky, Trnkova and Adamek, and a new proof of a theorem of Rosicky, both about colimit preservation between categories of structures. Unlike the original proofs, which use category-theoretic methods, we use set-theoretic arguments involving elementary embeddings given by large cardinals such as alpha-strongly compact and C^(n)-extendible cardinals.
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