Topics in Model Theory

首席研究员建议继续研究数理逻辑分支模型论的五个问题。前三个问题与推广和推广希拉二分定理有关。具体来说，PI建议研究可分类理论模型分解树中可能出现的分支类型，将Shelah定理扩展到不可数语言的固有限制，以及那些模型以涉及模型的有限子集的不变量系统为特征的理论。第一个问题产生于可数理论的不可数谱的分类的完成。最后两个问题涉及构造模型的新方法，这使人想起强制构造。在与Shelah的合作中，PI定义了两种新的方法来构建具有无限多重类型的小稳定理论模型，一种基于选择具有正测度类型的实现，另一种基于选择非贫实现。尽管度量和范畴的概念在数学中通常是齐头并进的，但在这种情况下，这两个概念之间存在着本质上的不对称。最后的问题是研究得到包含增长率超过任何指数多项式的函数的实数的0极小展开式的可行性。通过使用Wilkie和Macintyre的新结果，似乎可以用Cohen强迫的方式构建这样的展开式。正如这里使用的术语，“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，理论的每一个句子都是真的。事实证明，如果一个人对一个理论的模型类别提出相当普遍的问题（比如一定规模的模型的数量），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是由这些组合中哪些是理论允许的，哪些是被禁止的。这个建议的重点是更好地理解这些组合在不同的上下文中是什么样子。对其中一些组合的研究最近在PAC学习理论和神经网络架构的表达能力研究中得到了应用。

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.