Mathematical Sciences: Applied Descriptive Set Theory

数学科学：应用描述集合论

• 批准号: 9206922
• 负责人: Howard Becker
• 金额: $ 3.92万
• 依托单位: University of South Carolina at Columbia
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1992
• 资助国家: 美国
• 起止时间: 1992-09-01 至 1995-02-28
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9206922&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Applied; Descriptive; Set

Descriptive set theory is the branch of mathematical logic concerned with sets of real numbers. The investigator intends to work on three topics in applied descriptive set theory, that is, on some connections which exist between descriptive set theory and some other parts of mathematics, specifically (1) classification of pointsets in the projective hierarchy, (2) descriptive set theory and finer topologies, (3) descriptive set theory and Polish group actions. All three topics are related to analysis and topology. The third topic is also related to other parts of mathematical logic, such as Vaught's Conjecture and the theory of definable cardinality. Some of the problems in the project will be considered under the assumption of strong set theoretic axioms such as the axiom of determinacy. Considering that the real numbers can be thought of as the ordinary number line of grade school arithmetic, it is surprising how much structure can be imposed upon them and how intricate the questions that can be asked about this structure. Descriptive set theory is the theory that addresses these questions with all the machinery of modern mathematical logic. For example, a standard construct of this theory is the Borel hierarchy of sets, consisting of two infinite sequences of families of sets, the Pi sets and the Sigma sets, defined inductively, and hence of increasing complexity. One of the obvious applications of the theory is to locate precisely in this hierarchy a particular set which arises in analysis, say the set of points at which a function is differentiable. In fact, it is a typical and classical theorem of descriptive set theory (due to Mazurkiewicz) that any such set of differentiability belongs to the family Pi-one-one in the Borel hierarchy. The investigator is focussed on questions such as these, which make logic relevant to the wider world of mathematics.

描述性集理论是与真实数字集有关的数学逻辑的分支。 研究人员打算在应用描述性集理论中研究三个主题，也就是说，在描述集理论与数学的其他部分之间存在的某些联系上，特别是（1）投射层次结构中的点分类，（2）描述性集合的理论和较细性的拓扑和较大的拓扑，（3）描述性集合理论和波兰人组动作。 这三个主题均与分析和拓扑有关。 第三个主题也与数学逻辑的其他部分有关，例如Vaught的猜想和可定义的基数理论。 在强大的设定理论公理（例如确定性公理）的假设下，该项目中的某些问题将被考虑。 考虑到实数可以被认为是小学算术的普通数字线，这令人惊讶的是，可以将多少结构施加到它们上，以及有关该结构的问题。 描述性集理论是通过现代数学逻辑的所有机械来解决这些问题的理论。 例如，该理论的标准构建体是集合的Borel层次结构，由两个无限的集合家族，PI集和Sigma集序列组成，这些序列是归纳定义的，因此复杂性的增加。 该理论的明显应用之一是精确地定位在该层次结构中，该集合在分析中出现的特定集合，例如函数可区分的点集。 实际上，这是描述性集理论的典型和古典定理（由于Mazurkiewicz的原因），任何此类可区分性都属于Borel层次结构中的家族。 研究者专注于这样的问题，这些问题使逻辑与更广泛的数学世界有关。