Mathematical Sciences: Subsystems of Second Order Arithemtic

9303478辛普森这个项目解决了数学基础中一个有争议的问题。正统观点(如布尔巴基)认为，数学的最终公理基础是集合论。然而，同样清楚的是，集合论本身主要关注对象(实线的病理性子集、大基数等)。它们与普通数学(分析、代数、组合学等)相去甚远。推动这一项目的问题是：证明普通数学的特定定理实际上需要哪些集合存在公理？这个问题是在二阶算术的子系统的背景下研究的，因为：(1)二阶算术的语言刚刚丰富到足以容纳大部分普通的数学实践；(2)二阶算术的子系统体现了根据证明论序数衡量的“逻辑强度”对集合存在公理的详细分类。结果表明，对于普通数学中的许多具体定理，可以精确地确定所给定理可证明的二阶算术中最弱的子系统。此外，以这种方式产生的子系统在数量上很少，并且对应于某些可供选择的基础程序，例如递归分析(MyHill-specker)、预测性(Weyl)、谓词还原(Kreisel-Feferman-Friedman)和有限还原(Hilbert)。这个项目的首席研究员史蒂芬·G·辛普森一直在积极研究二阶算术的子系统及其在数学基础中的作用，他也正在撰写一本涵盖整个主题的书的最后阶段。Simpson目前正在研究的一些具体问题是：在可数组合学(例如，Carlson和Simpson的对偶Ramsey定理，以及关于可数线性序可嵌入性的Laver定理)和Borel组合学中，需要使用哪些集合存在公理来证明基本定理？证明动力系统理论的基本定理需要哪些集合存在公理？需要哪种集合存在公理来证明解析集的勒贝格可测性？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，数学的最终基础是既定的理论。这一观点是1960年S《新数学》实验的基础。该项目的研究表明，在某些方面，正统观点是不正确的，集合论的基本主张是站不住脚的。通过系统地研究公理和定理之间的关系，发现精心选择的二阶算术系统为数学提供了一个在逻辑上更合适的基础。***

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. ***