首页/论文/1908.02442

arXiv 1908.024422019-08-07math.LO

A game-theoretic proof of Shelah's theorem on labeled trees

Trevor M. Wilson

英文摘要：

We give a new proof of a theorem of Shelah which states that for every family of labeled trees, if the cardinality $κ$ of the family is much larger (in the sense of large cardinals) than the cardinality $λ$ of the set of labels, more precisely if the partition relation $κ\to (ω)^{\mathord{<}ω}_λ$ holds, then there is a homomorphism from one labeled tree in the family to another. Our proof uses a characterization of such homomorphisms in terms of games.

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 lazy model theoretician's guide to Shelah's eventual categoricity conjecture in universal classes

We give a short overview of the proof of Shelah's eventual categoricity conjecture in universal classes with amalgamation in arXiv:1506.07024 .

2015-11-30math.LO

Remarks on Shelah's classification theory and Quillen's negation

We give category-theoretic reformulations of stability, NIP, NTP, and non-dividing by observing that their characterisations in terms of indiscernible sequences are naturally expressed as Quillen lifting properties %(negation) of certain morphisms associated with linear orders, in a certain category extending the categories of topological spaces and of simplicial sets. This suggests an approach to a homotopy theory for model theory.

2020-10-17math.LO

On the number of independent orders

We investigate a model theoretic invariant $κ_{srd}^m(T)$, which was introduced by Shelah in his famous book, and prove that $κ_{srd}^m(T)$ is sub-additive. When $κ_{srd}^m(T)$ is infinite, this gives the equality $κ^m_{srd}(T)=κ^1_{srd}(T)$, answering a question by Shelah. We apply the same proof method to analyze another invariant $κ^m_{ird}(T)$, and show that it is also sub-additive, improving a result in the book.

2020-04-05math.LO

轨迹可定义性II：模型论的线性性

Trace definability II: model-theoretic linearity

研究展示了NIP结构中Shelah完备化出现新代数结构的例子，特别是构造了一个弱弱有序结构，其Shelah完备化解释了无限域。

2026-05-12math.LO

A short proof of Shelah's eventual categoricity conjecture for AEC's with interpolation, under $GCH$

We provide a short proof of Shelah's eventual categoricity conjecture, assuming the Generalized Continuum Hypothesis ($GCH$), for abstract elementary classes (AEC's) with interpolation, a strengthening of amalgamation which is a necessary and sufficient condition for an AEC categorical in a high enough cardinal to satisfy eventual categoricity. The proof builds on an earlier topos-theoretic argument which was syntactic in nature and recurred to $κ$-classifying toposes. We carry out here the same proof idea but from the semantic perspective, making use of a connection between $κ$-classifying to…

2020-12-28math.CTmath.LO

Invariants Related to the Tree Property

We consider global analogues of model-theoretic tree properties. The main objects of study are the invariants related to Shelah's tree property $κ_{\text{cdt}}(T)$, $κ_{\text{sct}}(T)$, and $κ_{\text{inp}}(T)$ and the relations that obtain between them. From strong colorings, we construct theories $T$ with $κ_{\text{cdt}}(T) > κ_{\text{sct}}(T) + κ_{\text{inp}}(T)$. We show that these invariants have distinct structural consequences, by investigating the decay of saturation in ultrapowers of models of $T$, where $T$ is some theory with $κ_{\text{cdt}}(T)$, $κ_{\text{sct}}(T)$, or $κ_{\text{inp…

2019-08-12math.LO

更多相关论文收起更多论文

论超滤构造

On ultrafilter construction

本文从模型论视角回顾了20世纪至今的正则超滤构造，并解释了Malliaris和Shelah近期发展的“规范布尔代数”。

2025-09-17math.LO

Eberlein-Smulian compactness and Kolmogorov extension theorems; a model theoretic approach

This paper has two parts. First, we complete the proof of the Kolmogorov extension theorem for unbounded random variables using compactness theorem of integral logic which was proved for bounded case in [8]. Second, we give a proof of the Eberlein-Smulian compactness theorem by Ramsey's theorem and point out the correspondence between this theorem and a result in Shelah's classification theory.

2015-09-20math.LO

On model-theoretic tree properties

We study model theoretic tree properties ($\text{TP}, \text{TP}_1, \text{TP}_2$) and their associated cardinal invariants ($κ_{\text{cdt}}, κ_{\text{sct}}, κ_{\text{inp}}$, respectively). In particular, we obtain a quantitative refinement of Shelah's theorem ($\text{TP} \Rightarrow \text{TP}_1 \lor \text{TP}_2$) for countable theories, show that $\text{TP}_1$ is always witnessed by a formula in a single variable (partially answering a question of Shelah) and that weak $k-\text{TP}_1$ is equivalent to $\text{TP}_1$ (answering a question of Kim and Kim). Besides, we give a characterization of $\…

2016-10-21math.COmath.LO

定量解析稳定正则性

Quantitative analytic stable regularity

University of Illinois Chicago（伊利诺伊大学芝加哥分校）

研究二元实值函数的定量稳定正则性，通过“解析对称引理”等工具证明相关引理，并对随机抽样方法进行函数理论处理，扩展了Malliaris和Shelah关于稳定图的工作。

2026-07-27math.COmath.LO

Some applications of the ultrapower theorem to the theory of compacta

The ultrapower theorem of Keisler-Shelah allows such model-theoretic notions as elementary equivalence, elementary embedding and existential embedding to be couched in the language of categories (limits, morphism diagrams). This in turn allows analogs of these (and related) notions to be transported into unusual settings, chiefly those of Banach spaces and of compacta. Our interest here is the enrichment of the theory of compacta, especially the theory of continua, brought about by the immigration of model-theoretic ideas and techniques.

1996-10-15math.LO

Model theoretic connected components of groups

We give a general exposition of model theoretic connected components of groups. We show that if a group G has NIP, then there exists the smallest invariant (over some small set) subgroup of G with bounded index (Theorem 5.3). This result extends theorem of Shelah. We consider also in this context the multiplicative and the additive groups of some rings (including infite fields).

2010-02-09math.GRmath.LO

Universal classes near $\aleph_1$

Shelah has provided sufficient conditions for an $L_{ω_1, ω}$-sentence $ψ$ to have arbitrarily large models and for a Morley-like theorem to hold of $ψ$. These conditions involve structural and set-theoretic assumptions on all the $\aleph_n$'s. Using tools of Boney, Shelah, and the second author, we give assumptions on $\aleph_0$ and $\aleph_1$ which suffice when $ψ$ is restricted to be universal: $\mathbf{Theorem}$ Assume $2^{\aleph_{0}} < 2 ^{\aleph_{1}}$. Let $ψ$ be a universal $L_{ω_{1}, ω}$-sentence. - If $ψ$ is categorical in $\aleph_{0}$ and $1 \leq I(ψ, \aleph_{1}) < 2 ^{\aleph_{1}}$, …

2018-06-04math.LO

Stability, NIP, and NSOP; Model Theoretic Properties of Formulas via Topological Properties of Function Spaces

We study and characterize stability, NIP and NSOP in terms of topological and measure theoretical properties of classes of functions. We study a measure theoretic property, `Talagrand's stability', and explain the relationship between this property and NIP in continuous logic. Using a result of Bourgain, Fremlin and Talagrand, we prove the `almost definability' and `Baire~1 definability' of coheirs assuming NIP. We show that a formula $ϕ(x,y)$ has the strict order property if and only if there is a convergent sequence of continuous functions on the space of $ϕ$-types such that its limit is not…

2019-02-15math.LO

Model theoretic stability and definability of types, after A. Grothendieck

We point out how the "Fundamental Theorem of Stability Theory", namely the equivalence between the "non order property" and definability of types, proved by Shelah in the 1970s, is in fact an immediate consequence of Grothendieck's "Crit{è}res de compacit{é}" from 1952. The familiar forms for the defining formulae then follow using Mazur's Lemma regarding weak convergence in Banach spaces.

2015-01-09math.FAmath.LO

合并与Keisler序

Amalgamation and Keisler's Order

本文统一了超滤子构造，证明了在可数低理论满足独立合并条件时，Keisler序中$T_{k,k-1}$不优于$T$，从而分离了所有$T_{k+1,k}$类。

2024-09-20math.LO

Kurepa trees and spectra of $\mathcal{L}_{ω_1,ω}$-sentences

We use set-theoretic tools to make a model-theoretic contribution. In particular, we construct a \emph{single} $\mathcal{L}_{ω_1,ω}$-sentence $ψ$ that codes Kurepa trees to prove the consistency of the following: (1) The spectrum of $ψ$ is consistently equal to $[\aleph_0,\aleph_{ω_1}]$ and also consistently equal to $[\aleph_0,2^{\aleph_1})$, where $2^{\aleph_1}$ is weakly inaccessible. (2) The amalgamation spectrum of $ψ$ is consistently equal to $[\aleph_1,\aleph_{ω_1}]$ and $[\aleph_1,2^{\aleph_1})$, where again $2^{\aleph_1}$ is weakly inaccessible. This is the first example of an $\mathc…

2020-03-19math.LO

关于连续模型论中$\mathrm{C}^*$-代数超积的同构问题

On the isomorphism problem for ultraproducts of $\mathrm{C}^*$-algebras in continuous model theory

Chiba University（千叶大学）

本文探讨了$\mathrm{C}^*$-代数超积的同构问题，证明在否定连续假说下，存在两个元素等价的无限维单位$\mathrm{C}^*$-代数，其超积不相等，展示了模型论与连续假说的关系。

2026-05-13math.LOmath.OA