Mathematical Sciences: Subsystems of Second Order Arithemtic

数学科学：二阶算术子系统

• 批准号: 9303478
• 负责人: Stephen Simpson
• 金额: $ 10.02万
• 依托单位: Pennsylvania State Univ University Park
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1993
• 资助国家: 美国
• 起止时间: 1993-07-01 至 1996-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9303478&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Subsystems; Second; Order

9303478 Simpson This project addresses a controversial question in the foundations of mathematics. The orthodox view (e.g. Bourbaki) holds that the ultimate axiomatic foundation for mathematics is set-theoretical. However, it is also clear that set theory itself is concerned mainly with objects (pathological subsets of the real line, large cardinals, etc.) which are rather far removed from ordinary mathematics (analysis, algebra, combinatorics, etc.). The question which drives this project is: Which set existence axioms are actually needed to prove specific theorems of ordinary mathematics? This question has been investigated in the context of subsystems of second order arithmetic, because: (1) the language of second order arithmetic is just rich enough to accommodate the bulk of ordinary mathematical practice; and (2) subsystems of second order arithmetic embody a detailed classification of set existence axioms according to "logical strength" as measured by proof theoretic ordinals. It turns out that, for many specific theorems of ordinary mathematics, one can precisely determine the weakest subsystem of second order arithmetic in which the given theorem is provable. Furthermore, the subsystems which arise in this way are few in number and correspond to certain alternative foundational programs such as recursive analysis (Myhill-Specker), predicativity (Weyl), predicative reductionism (Kreisel-Feferman-Friedman), and finitisic reductionism (Hilbert). The study of subsystems of second order arithmetic and their role in the foundations of mathematics has been vigorously pursued by Stephen G. Simpson, the Principal Investigator in this project, who is also in the final stages of writing a book which covers the entire subject. Some specific questions which Simpson is currently investigating are: Which set existence axioms are needed to prove basic theorems in countable combinatorics (e.g. the dual Ramsey theorem of Carlson and Simpson, and Laver's th eorem on embeddability of countable linear orderings) and in Borel combinatorics? Which set existence axioms are needed to prove basic theorems of dynamical systems theory? Which set existence axioms are needed to prove Lebesgue measurability of analytic sets? This project is concerned with the foundations and logical structure of mathematics. The orthodox view, promulgated by the Bourbaki school among others, is that the ultimate foundation of mathematics is set theoretical. This view was basic to the "New Math" experiment of the 1960's. The research in this project tends to show that, in certain respects, the orthodox view is incorrect and the foundational claims of set theory are unwarranted. Through systematic investigation of the relationships between axioms and theorems, it emerges that carefully chosen systems of second order arithmetic provide a logically more appropriate foundation for mathematics. ***

小行星9303478 这个项目解决了数学基础中一个有争议的问题。 正统观点（例如布尔巴基）认为数学的终极公理基础是集合论的。 然而，同样清楚的是，集合论本身主要关注对象（真实的线的病态子集，大基数等）。这是相当远离普通数学（分析，代数，组合数学等）。 推动这个项目的问题是： 哪些集合存在公理是证明普通数学中特定定理所必需的？ 这个问题已经在二阶算术的子系统的背景下进行了研究，因为：（1）二阶算术的语言足够丰富，可以容纳大量的普通数学实践;（2）二阶算术的子系统根据证明论序数测量的“逻辑强度”体现了集合存在公理的详细分类。 事实证明，对于普通数学的许多特定定理，人们可以精确地确定二阶算术的最弱子系统，其中给定的定理是可证明的。 此外，以这种方式出现的子系统数量很少，并且对应于某些替代的基础程序，如递归分析（Myhill-Specker），预测性（Weyl），预测还原论（Kreisel-Feferman-Friedman）和有限还原论（Hilbert）。 二阶算术的子系统及其在数学基础中的作用的研究一直是斯蒂芬G。辛普森，这个项目的主要研究者，谁也是在写一本书的最后阶段，其中涵盖了整个主题。 一些具体的问题，辛普森目前正在调查的是：哪些集存在公理需要证明基本定理在可数组合（如对偶拉姆齐定理的卡尔森和辛普森，和拉弗定理的嵌入性可数线性序）和博雷尔组合？ 证明动力系统理论的基本定理需要哪些集合存在公理？ 证明解析集的勒贝格可测性需要哪些集合存在公理？ 这个项目涉及数学的基础和逻辑结构。 布尔巴基学派和其他学派所宣扬的正统观点是，数学的最终基础是理论上的。 这种观点是20世纪60年代“新数学”实验的基础。 这个项目的研究表明，在某些方面，正统的观点是不正确的，集合论的基本主张是没有根据的。 通过对公理和定理之间的关系进行系统的研究，可以看出，精心选择的二阶算术系统为数学提供了逻辑上更合适的基础。 ***