Mathematical Sciences: Subsystems of Second Order Arithemtic

点击翻译按钮获取中文摘要

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. ***