Mathematical Aspects of Recursive Function Theory

This chapter focuses on the mathematical aspects of recursive function theory. In mainstream recursion theory, the notions of computation and computability play a central role. There are several instances in mathematics in which recursion theoretic concepts arise naturally from concepts apparently having nothing to do with computability—for example, the Diophantine representability of recursively enumerable sets. Friedman has discovered a number of characterizations of recursion theoretic and logical notions in terms of concepts from mainstream mathematics. In a different direction, it has often been observed that certain structural property of recursive functions and their Gödel numberings, which again apparently have nothing to do with the notion of a computation, play a central role. The enumeration theorem and s-m-n theorems are the most noticeable examples of this. This observation suggests two areas of study. The axiomatizations of recursion theory and its generalizations can be studied, in which these structural properties, rather than the notion of a computation, occupy center stage. Then, the structures (models) naturally occurring in recursion theory, in terms of traditional mathematical and logical issues—such as definability theory, isomorphisms of structures, and automorphisms of a structure—can be studied. The chapter looks at two characterizations of recursion theoretic notions and also describes some work of Friedman showing how some concepts of recursion theory arise naturally in linear algebra and analysis.