Generalized Recursion Theory

Provability, Computability and Reflection.

Fundamentals of Generalized Recursion Theory

Provability, Computability and Reflection

This monograph presents recursion theory from a generalized point of view centered on the computational aspects of definability. A major theme is the study of the structures of degrees arising from two key notions of reducibility, the Turing degrees and the hyperdegrees, using techniques and ideas from recursion theory, hyperarithmetic theory, and descriptive set theory. The emphasis is on the interplay between recursion theory and set theory, anchored on the notion of definability. The monograph covers a number of fundamental results in hyperarithmetic theory as well as some recent results on the structure theory of Turing and hyperdegrees. It also features a chapter on the applications of these investigations to higher randomness.

This volume presents a unified and coherent account of the many and various parts of general recursion theory.

This almost self-contained introduction to higher recursion theory is essential reading for all researchers in the field.