Descriptive Set Theory

摘要：这个项目的总体目标是发展波兰群的可定义作用理论，它们的轨道空间的结构和分类，以及与可定义等价关系密切相关的研究。这项工作是由基本的基本问题推动的，比如理解数学对象完全分类的本质，直到通过不变量的某种等价性概念，以及创建用于衡量此类分类问题的复杂性的数学框架。这一理论是在描述集合论的背景下发展起来的，它提供了基本的基本概念和方法。另一方面，鉴于这一理论的广泛范围，它与其他数学领域存在着自然的相互作用，如拓扑和变换群理论、拓扑动力学和遍历理论、模型理论和递归理论。在许多数学领域中出现的一个基本问题是对该领域研究的给定对象集合进行分类的问题。这相当于提供这些天体的“目录”或“清单”，原则上与生物学中的物种编目或天文学中的恒星和星系编目没有什么不同。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“完整”的理解。否则，或多或少会出现一种“混乱”的行为。因此，了解在什么情况下分类是可能的是非常重要的。由于什么构成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，这一困难的基本问题变得更加复杂，因此一个领域的“好”分类的标准可能不适用于另一个领域。在基本层面上，这个项目的目的是发展一种通用的量化理论，在许多情况下，它可以准确地衡量分类问题的复杂性，从而提供客观的手段，使人们能够在任何给定的领域中决定是否可能对所涉对象进行令人满意的分类。这是通过为每个要研究的对象集合关联一个适当的“大小”或“大小”概念来实现的，准确地说，这一概念衡量了其分类问题的难度。这个新的“大小”理论就是在这个项目中被研究的。

Abstract: The general aim of this project is the development of the theory of definable actions of Polish groups, the structure and classification of their orbit spaces, and the closely related study of definable equivalence relations. This work is motivated by basic foundational questions, like understanding the nature of complete classification of mathematical objects up to some notion of equivalence by invariants, and creating a mathematical framework for measuring the complexity of such classification problems. This theory is developed within the context of descriptive set theory, which provides the basic underlying concepts and methods. On the other hand, in view of the broad scope of this theory, there are natural interactions with other areas of mathematics, such as the theory of topological and transformation groups, topological dynamics and ergodic theory, model theory, and recursion theory. One of the fundamental questions that arises in many fields of mathematics is that of classifying a given collection of objects that has been studied in this field. This amounts to providing a "catalog" or "listing" of these objects, in principle not unlike that of cataloging species in biology or 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 very important to understand under what circumstances a classification is possible. This difficult foundational question is further complicated by the fact that what constitutes an acceptable classification is very much dependent on the particular field of mathematics studied, so the criteria for a "good" classification in one area might not be appropriate in another. At its basic level, this project aims to develop a general quantitative theory, which in many situations can precisely measure the complexity of a classification problem and thus provide objective means by which one can decide, in any given field, whether a satisfactory classification of the objects in question is possible. This is achieved by associating with each collection of objects to be studied an appropriate concept of "magnitude" or "size", which in a precise sense measures the difficulty of its classification problem. This new theory of "size" is investigated in this project.