Friedman's Research on Subsystems of Second Order Arithmetic*

Publisher Summary This chapter focuses on Friedman's research on subsystems of second-order arithmetic, which can be examined from two related but essentially disparate points of view. On the one hand, such systems have many interesting meta-mathematical properties, which can be investigated using proof-theoretic and model-theoretic tools. On the other hand, subsystems of second-order arithmetic are a natural vehicle for the formal axiomatic study of ordinary mathematics. Ordinary mathematics comprises branches of mathematics—such as geometry, number theory, differential equations, algebra, and functional analysis. The chapter is concerned mainly with formal systems for the hyperarithmetical sets and represents a vast extension of the program of Reverse Mathematics. Reductionist programs, where a reductionist is anyone who proposes to reduce a large part of mathematics to some restricted set of “acceptable” principles, are also discussed.

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