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.

描述集合论是与实数集合有关的数理逻辑的一个分支。 研究者打算研究应用描述性集合论中的三个主题，即描述性集合论与数学其他部分之间存在的一些联系，特别是（1）射影层次中点集的分类，（2）描述性集合理论和更精细的拓扑，（3）描述集合论和波兰群行动。 所有三个主题都与分析和拓扑相关。 第三个主题也与数理逻辑的其他部分相关，例如沃特猜想和可定义基数理论。 该项目中的一些问题将在强集合理论公理（例如确定性公理）的假设下考虑。 考虑到实数可以被认为是小学算术中的普通数轴，令人惊讶的是可以对它们施加如此多的结构以及可以针对该结构提出的问题是多么复杂。 描述性集合论是用现代数理逻辑的所有机制来解决这些问题的理论。 例如，该理论的标准构造是集合的 Borel 层次结构，由集合族的两个无限序列组成，即 Pi 集合和 Sigma 集合，它们是归纳定义的，因此复杂性不断增加。 该理论的一个明显应用是在该层次结构中精确定位分析中出现的特定集合，例如函数可微分的点集。 事实上，任何这样的可微性集合都属于 Borel 层次结构中的 Pi-one-one 系列，这是描述性集合论的一个典型且经典的定理（由 Mazurkiewicz 提出）。 研究者专注于诸如此类的问题，这些问题使逻辑与更广泛的数学世界相关。