Model theory, Keisler measures, and groups - Ehud Hrushovski, Ya’acov Peterzil and Anand Pillay, Groups, measures, and the NIP. Journal of the American Mathematical Society, vol. 21 (2008), no. 2, pp. 563–596. - Ehud Hrushovski and Anand Pillay, On NIP an
Model theory, Keisler measures, and groups - Ehud Hrushovski, Ya’acov Peterzil and Anand Pillay, Groups, measures, and the NIP. Journal of the American Mathematical Society, vol. 21 (2008), no. 2, pp. 563–596. - Ehud Hrushovski and Anand Pillay, On NIP an
复制标题
模型理论、凯斯勒测度和群 - Ehud Hrushovski、Yaacov Peterzil 和 Anand Pillay，群、测度和 NIP。
DOI：
10.1017/bsl.2018.68
复制
发表时间：
2018
期刊：
影响因子：
--
通讯作者：
Chernikov, Artem
中科院分区：
文献类型：
--
作者：
Chernikov, Artem
传统的模型论研究一阶理论模型中的可定义集合。为了这个目的，它往往是更有益的研究类型，即超过滤器的布尔代数的可定义集。虽然类型的紧致空间包含相同的Stone对偶信息，但使用类型并在更大的饱和模型中实现它们允许使用可定义集合的“通用”点（例如，非标准分析），并提供了一种重要的方法，无论是纯模型理论还是应用模型理论。类型可以被视为凯斯勒测度的特例，或者可定义集合的布尔代数上的可加概率测度（因此类型是在{0，1}中取值的凯斯勒测度）。鉴于测度论在数学上的重要性，值得注意的是，这些测度直到最近才在模型论中受到关注。他们首先被认为是在HJ Keisler，“措施和分叉”，纯和应用逻辑年鉴45（1987年），非常有见地地概括方面的希拉的分叉从类型在稳定的理论措施在NIP理论。一些后续工作在分类设置出现在90年代初，由阿尔伯特和恩斯利，但似乎去很大程度上被忽视。Karpinski和Macyntire，关注应用到神经网络，研究了可定义性性质的某些Keisler措施在o-最小结构和p-adics在“连接模型理论和代数和解析几何”，149-177，四。垫，6、部门数学、那不勒斯第二大学，卡塞塔，2000年。然而，近几十年来，模型理论中的分析对象，特别是凯斯勒措施，受到几个相互交织的研究路线的推动。我们回顾了在这些发展中发挥关键作用的三篇连续文章。
Traditionally model theory investigates definable sets in models of first-order theories. Towards this purpose, it is often more instructive to study types, ie, ultrafilters on the boolean algebra of definable sets. While the compact space of types contains the same information by Stone duality, working with types and realizing them in larger saturated models allows to work with “generic” points of definable sets (eg, nonstandard analysis), and provides an important method at the core of both pure and applied model theory. Types can be viewed as a special case of Keisler measures, or finitely additive probability measures on the Boolean algebra of definable sets (so a type is a Keisler measure taking values in {0, 1}). Given the general mathematical importance of measure theory, it is remarkable in retrospective that these measures have received little attention in model theory until recently. They were first considered in HJ Keisler,“Measures and forking”, Annals of Pure and Applied Logic 45 (1987), very insightfully generalizing aspects of Shelah’s forking from types in stable theories to measures in NIP theories. Some follow up work in the-categorical setting appeared in the early 90’s, by Albert and Ensley, but seemed to go largely unnoticed. Karpinski and Macyntire, concerned with applications to neural networks, studied definability properties of certain Keisler measures in o-minimal structures and in the p-adics in “Connections between model theory and algebraic and analytic geometry”, 149–177, Quad. Mat., 6, Dept. Math., Seconda Univ. Napoli, Caserta, 2000. Recent decades, however, witnessed resurgence of analytic objects in model theory, and Keisler measures in particular, motivated by several intertwined lines of research. We review three consecutive articles that played a crucial role in these developments.