A descriptive Main Gap Theorem

IF 0.9 1区 数学 Q1 LOGIC
Francesco Mangraviti, L. Ros
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引用次数: 3

Abstract

Answering one of the main questions of [S.-D. Friedman, T. Hyttinen and V. Kulikov, Generalized descriptive set theory and classification theory, Mem. Amer. Math. Soc. 230(1081) (2014) 80, Chap. 7], we show that there is a tight connection between the depth of a classifiable shallow theory [Formula: see text] and the Borel rank of the isomorphism relation [Formula: see text] on its models of size [Formula: see text], for [Formula: see text] any cardinal satisfying [Formula: see text]. This is achieved by establishing a link between said rank and the [Formula: see text]-Scott height of the [Formula: see text]-sized models of [Formula: see text], and yields to the following descriptive set-theoretical analog of Shelah’s Main Gap Theorem: Given a countable complete first-order theory [Formula: see text], either [Formula: see text] is Borel with a countable Borel rank (i.e. very simple, given that the length of the relevant Borel hierarchy is [Formula: see text]), or it is not Borel at all. The dividing line between the two situations is the same as in Shelah’s theorem, namely that of classifiable shallow theories. We also provide a Borel reducibility version of the above theorem, discuss some limitations to the possible (Borel) complexities of [Formula: see text], and provide a characterization of categoricity of [Formula: see text] in terms of the descriptive set-theoretical complexity of [Formula: see text].
一个描述性的主间隙定理
回答[s . d .]傅利民,《广义描述集理论与分类理论》,《中国科学》。阿米尔。数学。Soc. 230(1081)(2014) 80,第7章),我们证明了在可分类的浅层理论[公式:见文]的深度与其大小[公式:见文]模型上同构关系[公式:见文]的Borel秩之间存在紧密联系,对于[公式:见文]任何满足[公式:见文]的基数。这是通过在所述秩和[公式:见文]-[公式:见文]的[公式:见文]-大小模型的[公式:见文]之间建立联系来实现的,并产生以下描述集理论类似于Shelah的主要间隙定理:给定一个可数的完全一阶理论[公式:见文],要么[公式:见文]是具有可数Borel秩的Borel(即非常简单,给定相关Borel层次的长度为[公式:(见原文),或者根本就不是Borel。这两种情况之间的分界线与希拉定理相同,即可分类的浅层理论的分界线。我们还提供了上述定理的Borel可约性版本,讨论了[公式:见文]的可能(Borel)复杂性的一些限制,并根据[公式:见文]的描述性集合理论复杂性提供了[公式:见文]的范畴性的表征。
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来源期刊
Journal of Mathematical Logic
Journal of Mathematical Logic MATHEMATICS-LOGIC
CiteScore
1.60
自引率
11.10%
发文量
23
审稿时长
>12 weeks
期刊介绍: The Journal of Mathematical Logic (JML) provides an important forum for the communication of original contributions in all areas of mathematical logic and its applications. It aims at publishing papers at the highest level of mathematical creativity and sophistication. JML intends to represent the most important and innovative developments in the subject.
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