Topics in Model Theory

模型理论主题

• 批准号: 9704364
• 负责人: Michael Chris Laskowski
• 金额: $ 6.57万
• 依托单位: University of Maryland, College Park
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1997
• 资助国家: 美国
• 起止时间: 1997-06-01 至 2000-11-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9704364&HistoricalAwards=false
• 关键词: Topics; Model; Theory

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.

主要研究者建议继续研究数理逻辑的分支模型论的五个问题。 前三个问题是与延伸和推广希拉的二分法定理。 具体来说，PI建议研究可能发生在可分类理论模型的分解树中的分支类型，将Shelah定理扩展到不可数语言的固有限制，以及其模型的特征在于涉及模型有限子集的不变量系统的理论。 第一个这些问题产生的完成分类的不可数谱可数理论。 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构建。 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多重性的小型稳定理论的模型，一种是基于选择具有正测度的类型的实现，另一种是基于选择非贫乏实现。 尽管测度和范畴的概念在整个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。 最后一个问题是调查的可行性，获得一个O-最小的扩展包含一个函数的增长率超过任何指数多项式的实数。 利用Wilkie和Macintyre的新结果，这样的展开似乎可以用Cohen强迫的方式来构造。 正如这里所使用的术语，“模型论”本质上是对代数结构的系统研究。 一个理论仅仅是一组句子，它的模型类由理论的每个句子为真的可能的宇宙组成。事实证明，如果一个人问关于一个理论的模型类的相当一般的问题（例如一定大小的模型的数量），答案只取决于元素组合的一小部分。 也就是说，一定规模的模型的数量取决于这些组合中哪些是理论允许的，哪些是禁止的。 这个建议的重点是更好地理解这些组合在各种上下文中的样子。 对这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的研究中找到了应用。