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.

Chi Tat Chong, National University of Singapore; Liang Yu, Nanjing University, Jiangsu, China.

Preliminaries

1. ¹₁-uniformization and Applications to Turing Degrees

2. Rigidity of Hyperdegrees

3. Basis Theorems and ¹₁-Hyperarithmetic

4. The Jump Operator

5. Independence Results in the Turing Degrees

6. Higher Randomness

References

Index