Reverse Mathematics

Pennsylvania State Univ University Park, University Park PA

Investigators

Abstract

The investigator believes that mathematical logic needs to return to its roots in Foundations of Mathematics, in the great tradition of Frege, Russell, Hilbert, Turing, and G del. A basic question in Foundations of Mathematics is: Which set-existence axioms are needed to prove specific theorems of core mathematics? Here ``core mathematics'' comprises standard topics in analysis, algebra, topology/geometry, etc. The investigator and his colleagues study this question in terms of Subsystems of Second Order Arithmetic, as exposited in the investigator's recently published research monograph of that title. An extensive series of case studies reveals that (i) many core mathematical theorems are logically equivalent to the set-existence axioms needed to prove them, (ii) only a handful of set-existence axioms arise in this way, (iii) the corresponding subsystems are linearly ordered by logical implication. This gives a far-reaching classification of hundreds of core mathematical theorems into a small number of classes, the classes being defined in terms of logical equivalence over a weak base system. This ongoing classification project is known as Reverse Mathematics. It has many implications for philosophically motivated foundational programs such as constructivism (Bishop), computable mathematics (Pour-El/Richards), finitistic reductionism (Hilbert), predicativism (Weyl/Feferman), and predicative reductionism. The Reverse Mathematics classification project is also a rich source of challenging technical problems. The investigator and his colleagues are pursuing several of these, with emphasis on analysis, geometry, and countable combinatorics. A direction for the future is to weaken the base system, in order to greatly broaden the scope of the classification project, providing significant points of contact with other parts of mathematics such as number theory and computational complexity. Foundations of Mathematics is the study of the most basic concepts and logical structure of mathematics, with an eye to the unity of human knowledge. This line of research deals fruitfully with fundamental questions such as: What is the nature of mathematical proof? What is the logical structure of mathematics? What are the appropriate axioms for mathematics? Foundations of Mathematics is a rich subject with a long history, going back to Aristotle and Euclid and continuing in the hands of outstanding modern figures such Frege, Russell, Hilbert, Turing, and G del. The investigator and his colleagues continue in this foundational tradition by pursuing a far-reaching classification project known as Reverse Mathematics. Specific mathematical theorems are classified according to the axioms needed to prove them. This reveals a remarkably simple logical structure within mathematics. The existence of such a structure has many profound implications. The ongoing research contributes to clarification of the role of the infinite in mathematics, the nature of mathematical constructions, the role of impredicative definitions in mathematics, and related issues.