Descriptive Set Theory

本项目的总目标是发展可定义等价关系理论，并对波兰群的可定义作用及其轨道空间的结构和分类进行密切相关的研究。这个理论发生在描述集合论的背景下，它提供了基本的概念和方法。另一方面，鉴于这一理论的广泛范围，与逻辑和数学的其他领域，如模型论，递归理论，拓扑和变换群理论，拓扑动力学，遍历理论，在数学的许多领域中出现的一个基本问题是，如何对在某些特定领域中被研究的给定对象集合进行分类。领域这相当于提供了这些物体的“目录”或“列表”，原则上与生物学中的物种或天文学中的恒星和星系的编目没有什么不同。如果这样的分类是可能的，那么人们就对所涉及的数学结构有了一个“完整”的理解，否则就可能会有或多或少的“混乱”行为。因此，了解在什么情况下分类是可能的是非常重要的。这一困难的基本问题进一步复杂化，因为什么是可接受的分类在很大程度上取决于所研究的特定数学领域，因此在一个领域中“好”分类的标准在另一个领域可能不合适。在其基本层面上，该项目旨在发展一个通用的定量理论，在许多情况下，它可以精确地衡量分类问题的复杂性，从而提供客观的手段，通过它，在任何给定的领域，是否有可能对所讨论的物体进行令人满意的分类。这是通过将每个待分类对象的集合与“大小“或“大小”的适当概念相关联来实现的，这在精确意义上衡量了其分类问题的难度。本课题研究的是这种新的“量级”理论。

DMS-998-7437ABSTRACTThis general aim of this project is the development of the theory ofdefinable equivalence relations, and the closely related study of definableactions of Polish groups, and the structure and classification of theirorbit spaces. This theory takes place within the context of descriptive settheory, which provides the basic underlying concepts and methods. On theother hand, in view of the broad scope of this theory, there are naturalinteractions with other areas of logic and mathematics, such as modeltheory, recursion theory, the theory of topological and transformationgroups, topological dynamics, ergodic theory, and operator algebras.One of the fundamental questions that arises in many areas of mathematics isthat of classifying a given collection of objects that is being studied insome particular field. This amounts to providing a "catalog" or "listing" ofthese objects, in principle not unlike that of cataloging species in biologyor stars and galaxies in astronomy. If such a classification is possible,one has a "complete" understanding of the mathematical structures involved.Otherwise a more or less "chaotic" behavior is expected. It is thus veryimportant to understand under what circumstances a classification ispossible. This difficult foundational question is further complicated by thefact that what constitutes an acceptable classification is very muchdependent on the particular field of mathematics studied, so the criteriafor a "good" classification in one area might not be appropriate in another.At its basic level, this project aims to develop a general quantitativetheory, which in many situations can precisely measure the complexity of aclassification problem and thus provide objective means by which one candecide, in any given field, whether a satisfactory classification of theobjects in question is possible. This is achieved by associating with eachcollection of objects to be classified an appropriate concept of "magnitude"or "size", which in a precise sense measures the difficulty of itsclassification problem. This new theory of "magnitude" is investigated inthis project.