arXiv math/98091961998-09-15math.LO

On Hanf numbers of the infinitary order property

Rami Grossberg, Saharon Shelah

英文摘要：

We study several cardinal, and ordinal--valued functions that are relatives of Hanf numbers. Let kappa be an infinite cardinal, and let T subseteq L_{kappa^+, omega} be a theory of cardinality <= kappa, and let gamma be an ordinal >= kappa^+. For example we look at (1) mu_{T}^*(gamma, kappa):= min {mu^* for all phi in L_{infinity, omega}, with rk(phi)< gamma, if T has the (phi, mu^*)-order property then there exists a formula phi'(x;y) in L_{kappa^+, omega}, such that for every chi >= kappa, T has the (phi', chi)-order property}; and (2) mu^*(gamma, kappa):= sup{mu_T^*(gamma, kappa)| T in L_{kappa^+,omega}}.

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。

阅读原文

阅读 PDF ↗

arXiv 页面 ↗HTML 原文 ↗

本文内容

英文摘要相关论文

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

相关论文

The Hanf number for amalgamation of coloring classes

We study amalgamation properties in a family of abstract elementary classes that we call coloring classes. The family includes the examples previously studied in previous work of Baldwin, Kolesnikov, and Shelah. We establish that the amalgamation property is equivalent to the disjoint amalgamation property in all coloring classes; find the Hanf number for the amalgamation property for coloring classes; and improve the results of Baldwin, Kolesnikov, and Shelah by showing, in ZFC, that the (disjoint) amalgamation property for classes $K_α$ studied in that paper must hold up to $\beth_α$ (only a…

2015-07-13math.LO

Hanf number of the first stability cardinal in AECs

We show that $\beth_{(2^{\operatorname{LS}({\bf K})})^+}$ is the lower bound to the Hanf numbers for the length of the order property and for stability in stable abstract elementary classes (AECs). Our examples satisfy the joint embedding property, no maximal model, $(<\aleph_0)$-tameness but not necessarily the amalgamation property. We also define variations on the order and syntactic order properties by allowing the index set to be linearly ordered rather than well-ordered. Combining with Shelah's stability theorem, we deduce that our examples can have the order property up to any $μ<\beth_…

2021-10-08math.LO

Hanf number for the strictly stable cases

Suppose t = (T,T_1, p) is a triple of two theories T subset T_1 in vocabularies tau subset tau_1 (respectively) of cardinality lambda and a tau_1-type p over the empty set; in the main case here is with T stable. We show the Hanf number for the property: "there is a model M_1 of T_1 which omits p, but M_1 restricted to tau is saturated" is larger than the Hanf number of L_{lambda^+, kappa} but smaller than the Hanf number of L_{(2^lambda)^+, kappa} when T is stable with kappa = kappa(T). In fact, we characterize the Hanf number of t when we fix (T, lambda) where T is a first order complete, la…

2019-02-06math.LO

Polyadic-like algebras without the amalgamation property

Usually when we have polyadic-like algebras, meaning that we have infinitary substitutions (that is substitutions moving infinitely many points) in the similarity type, then we get the superamalgamation property especially if this class of algebras happen to be a variety. This for example happens for (full) polyadic algebras, full Heyting algebras and reducts of those using only finitely many infinitary substitutions, like Sains Boolean and Heyting algebras. (The last is studied by Sayed Ahmed) . In cylindric-like algebras (like quasi-polyadic algebras) when we do not have infinitary substitut…

2013-04-04math.LO

A Note on Extensions of Infinitary Logic

We show that a strong form of the so called Lindstrom's Theorem fails to generalize to extensions of L_{kappa,omega} and L_{kappa,kappa}: For weakly compact kappa there is no strongest extension of L_{kappa,omega} with the (kappa,kappa)-compactness property and the Lowenheim-Skolem theorem down to kappa. With an additional set-theoretic assumption, there is no strongest extension of L_{kappa,kappa} with the (kappa,kappa)-compactness property and the Lowenheim-Skolem theorem down to <kappa.

2000-09-07math.LO

Uniform Versions of Infinitary Properties in Banach Spaces

In functional analysis it is of interest to study the following general question: Is the uniform version of a property that holds in all Banach spaces also valid in all Banach spaces? Examples of affirmative answers to the above question are the host of proofs of almost-isometric versions of well known isometric theorems. Another example is Rosenthal's uniform version of Krivine's Theorem. Using an extended version of Henson's Compactness result for positive bounded formulas in normed structures, we show that the answer of the above question is in fact yes for every property that can be expres…

1999-05-11math.FAmath.LO

更多相关论文收起更多论文

Galois-stability for Tame Abstract Elementary Classes

We introduce tame abstract elementary classes as a generalization of all cases of abstract elementary classes that are known to permit development of stability-like theory. In this paper we explore stability results in this context. We assume that $\K$ is a tame abstract elementary class satisfying the amalgamation property with no maximal model. The main results include: (1) Galois-stability above the Hanf number implies that κ(K) is less than the Hanf number. Where κ(K) is the parallel of \kapppa(T) for f.o. T. (2) We use (1) to construct Morley sequences (for non-splitting) improving previo…

2005-09-23math.LO

抽象逻辑的上向Löwenheim-Skolem-Tarski数

Upward Löwenheim-Skolem-Tarski Numbers for Abstract Logics

该论文研究抽象逻辑的上向Löwenheim-Skolem-Tarski数，证明二阶逻辑的ULST数是最小可扩展基数，并引入强ULST数及新大基数概念刻画多种强逻辑。

2024-04-18math.LO

The Joint Embedding Property and Maximal Models

We introduce the notion of a `pure` Abstract Elementary Class to block trivial counterexamples. We study classes of models of bipartite graphs and show: Main Theorem (cf. Theorem 3.5.2 and Corollary 3.5.6): If $(λ_i : i \le α<\aleph_1)$ is a strictly increasing sequence of characterizable cardinals (Definition 2.1) whose models satisfy JEP$(<λ_0)$, there is an $L_{ω_1,ω}$ -sentence $ψ$ whose models form a pure AEC and (1) The models of $ψ$ satisfy JEP$(<λ_0)$, while JEP fails for all larger cardinals and AP fails in all infinite cardinals. (2) There exist $2^{λ_i^+}$ non-isomorphic maximal mod…

2015-02-19math.LO

Infinitary generalizations of Deligne's completeness theorem

Given a regular cardinal $κ$ such that $κ^{<κ}=κ$, we study a class of toposes with enough points, the $κ$-separable toposes. These are equivalent to sheaf toposes over a site with $κ$-small limits that has at most $κ$ many objects and morphisms, the (basis for the) topology being generated by at most $κ$ many covering families, and that satisfy a further exactness property $T$. We prove that these toposes have enough $κ$-points, that is, points whose inverse image preserve all $κ$-small limits. This generalizes the separable toposes of Makkai and Reyes, that are a particular case when $κ=ω$, …

2017-09-06math.CTmath.LO

Shelah's Categoricity Conjecture from a successor for Tame Abstract Elementary Classes

Let K be an Abstract Elemenetary Class satisfying the amalgamation and the joint embedding property, let μbe the Hanf number of K. Suppose K is tame. MAIN COROLLARY: (ZFC) If K is categorical in a successor cardinal bigger than \beth_{(2^μ)^+} then K is categorical in all cardinals greater than \beth_{(2^μ)^+}. This is an improvment of a Theorem of Makkai and Shelah ([Sh285] who used a strongly compact cardinal for the same conclusion) and Shelah's downward categoricity theorem for AECs with amalgamation (from [Sh394]).

2005-09-19math.LO

Quillen Model Structures-Based Notions of Locality of Logics over Finite Models

Locality is a property of logics, based on Hanf's and Gaifman's theorems, and that was shown to be very useful in the context of finite model theory. In this paper I present a homotopic variation for locality, namely a Quillen model category-based framework for locality under k-logical equivalence, for every primitive-positive sentence of quantifier-rank k.

2020-05-18math.CTmath.LO

偏序游戏的汉弗数

Hanf numbers for poset games

研究偏序游戏中的汉弗数性质，通过分析偏序类的结构和序数上界，确定获胜策略的序数特征。

2026-01-05math.COmath.LO

Shelah's eventual categoricity conjecture in universal classes: part I

We prove: $\mathbf{Theorem}$ Let $K$ be a universal class. If $K$ is categorical in cardinals of arbitrarily high cofinality, then $K$ is categorical on a tail of cardinals. The proof stems from ideas of Adi Jarden and Will Boney, and also relies on a deep result of Shelah. As opposed to previous works, the argument is in ZFC and does not use the assumption of categoricity in a successor cardinal. The argument generalizes to abstract elementary classes (AECs) that satisfy a locality property and where certain prime models exist. Moreover assuming amalgamation we can give an explicit bound on t…

2017-03-05math.LO