Topics in Model Theory

首席研究员建议继续研究数理逻辑的一个分支--模型理论的五个问题。前三个问题与推广和推广了谢拉的二分法定理有关。具体地说，PI建议研究可分类理论的模型分解树中可能出现的分支类型，将谢拉定理扩展到不可数语言的内在限制，以及那些其模型的特征是涉及模型的有限子集的不变量系统的理论。第一个问题源于对可数理论的不可数谱的分类的完成。最后两个问题涉及构建模型的新方法，这些方法让人联想到强迫建造。在与谢拉的合作中，PI定义了两种新的方法来构建具有无限重数类型的小型、稳定理论的模型，一种基于选择具有正向度量的类型的实现，另一种基于选择非贫乏实现。尽管在整个数学过程中，度量和范畴的概念通常是齐头并进的，但在这种情况下，这两个概念之间存在着本质上的不对称。最后一个问题是研究获得包含增长速度超过任何指数多项式的函数的实数的o-极小展开的可行性。似乎有可能利用Wilkie和Macintyre的新结果，以Cohen作用力的方式构造这样的展开。正如这里所用的术语，“模型理论”本质上是对代数结构的系统研究。一个理论只是一组句子，它的模型类由可能的宇宙组成，在这些宇宙中，理论的每个句子都是真的。事实证明，如果一个人问一个理论的模型的类别(如一定大小的模型的数量)的相当一般性的问题，答案只取决于一小部分元素的组合。也就是说，一定大小的模型的数量取决于这些组合中的哪些是理论允许的，哪些是禁止的。这项建议的大部分重点是为了更好地了解这些组合在不同环境中的情况。对其中一些组合的研究最近在PAC学习理论和神经网络结构的表达能力的研究中得到了应用。

The Principal Investigator proposes to continue his research on five problems of model theory, a branch of mathematical logic. The first three problems are connected with extending and generalizing Shelah's dichotomy theorem. Specifically, the PI proposes to study the possible types of branching that can occur in decomposition trees of models of classifiable theories, inherent limitations on extending Shelah's theorem to uncountable languages, and those theories whose models are characterized by systems of invariants involving finite subsets of the model. The first of these problems emanated from the completion of the classification of the uncountable spectra of countable theories. The last two problems deal with new methods of constructing models that are reminiscent of forcing constructions. In work with Shelah, the PI defined two new methods of constructing models of small, stable theories with a type of infinite multiplicity, one based on choosing realizations of the type with positive measure and the other based on choosing non-meager realizations. Whereas the notions of measure and category typically go hand in hand throughout mathematics, there is an essential asymmetry between the notions in this context. The final problem is to investigate the feasibility of obtaining an o-minimal expansion of the reals containing a function whose growth rate exceeds any exponential polynomial. It seems possible that by using new results of Wilkie and Macintyre, such an expansion could be constructed in the manner of Cohen forcing. As the term is used here, "model theory" is essentially the systematic study of algebraic structures. A theory is simply a set of sentences and its class of models consists of the possible universes in which each of the sentences of the theory is true. It turns out that if one asks rather general questions about the class of models of a theory (such as the number of models of a certain size) the answer depends only on a small set of combinations of elements. Tha t is, the number of models of a certain size is determined by which of these combinations are permitted by the theory and which are prohibited. Much of the focus of this proposal is to better understand what these sets of combinations look like in various contexts. The study of some of these sets of combinations has recently found applications in PAC learning theory and in the study of the expressive power of neural network architectures.