Applications of Set Theory to Topology and Topology to Model Theory

这项研究将一个数学领域(集合论)应用到另一个领域(拓扑学)，并将第二个领域应用到第三个领域(模型理论)。集合论是对数学宇宙的基本构件(集合)的研究；拓扑学是对空间几何性质的抽象的研究；模型理论是对语言及其所描述的结构的关系的研究。根据定义，纯数学，如这些领域，本身并不关心现实世界的应用。尽管如此，这些基础研究一次又一次地带来了重要的技术进步，例如计算机的发展，这是这种研究的早期先驱所没有预见到的。因此，明智的政府会为基础研究提供资金。*集合论在拓扑学中的应用是由作者和几位同事在近50年前开创的，它通过显示其对潜在集合论的依赖，彻底改变了后一领域中更抽象的部分，有可能为同一问题产生定理和反例，这取决于人们采用的不同于数学中通常使用的集合论原理，因为那些通常的原理都不能以任何方式解决这些问题。*最近，我们一直在使用新的集合论方法来解决拓扑学中长期悬而未决的问题，我们提出的程序继续沿着这些路线进行。这类工作的消费者是拓扑学家和/或集合论者。*拓扑学在模型理论中的应用有很长的历史，但研究人员注意到，除了在初级拓扑学教科书中可以找到的应用之外，这种应用几乎没有什么用处。因此，他倡导在模型理论中使用更复杂的拓扑学方法，并在他的博士生C.Eagle用这种方法发表了一篇重要论文时证明了这一点。在拟议的研究计划中，Eagle和研究人员将继续这些拓扑应用。贝尔范畴定理被广泛应用于数学的许多领域；它允许一个人断言对一个正在构造的对象的无限多个要求的满足。拓扑学家已经极大地扩展了这一定理可以应用的情况，但模型理论家到目前为止还没有利用这些扩展。Eagle和研究人员将使用这些扩展来在模型理论及其应用中的基本问题上取得进展，这些问题涉及哪些性质可以用哪些语言描述，以便例如分析(处理极限的数学领域，扩展微积分)的研究人员可以使用它们。**

This research applies one field of mathematics (Set Theory) to another (Topology), as well as the second to a third (Model Theory). Set Theory is the study of the fundamental building blocks (sets) of the mathematical universe; Topology is the study of abstractions of geometric properties of space; Model Theory is the study of languages and how languages relate to the structures they describe. By definition, Pure Mathematics, such as these fields, does not concern itself with real-world applications. Nonetheless, time after time, such basic research has led to important technological advances, such as the development of the computer, which were not foreseen by the early pioneers in such research. Therefore, wise governments fund basic research. ***Applications of Set Theory to Topology, which were pioneered by the author and a few colleagues almost 50 years ago, have revolutionized the more abstract portions of the latter field by showing its dependence on underlying Set Theory, with the possibility of producing both theorems and counterexamples for the same question, depending on what set-theoretic principles beyond the usual ones used in Mathematics one adopts, because those usual principles do not settle such questions either way.***Recently, we have been using new set-theoretic methods to settle longstanding open questions in Topology, and our proposed program continues along these lines. Consumers of such work are topologists and/or set-theorists. ***Applications of Topology to Model Theory have a long history, but the investigator noticed that such applications used little beyond what could be found in elementary topology textbooks. He has therefore championed the use of more sophisticated topological methods in Model Theory, and was vindicated when his doctoral student, C. Eagle, produced an important thesis using such methods. The proposed research program will see Eagle and the investigator continue these topological applications. The Baire Category Theorem is widely used in many areas of Mathematics; it allows one to assert the satisfaction of infinitely many requirements on an object one is constructing. Topologists have greatly extended the kinds of situations in which this theorem can be applied, but Model Theorists have until now not made use of these extensions. Eagle and the investigator will be using these extensions to make progress on fundamental questions in Model Theory and its applications, concerning what properties can be described in which languages, so that researchers in e.g. Analysis (the field of Mathematics that deals with limits, extending Calculus), can make use of them.**