Degrees of Unsolvability Structure and Theory

This book presents the theory of degrees of unsolvability in textbook form. It is accessible to any student with a slight background in logic and recursive function theory. Degrees are defined and their basic properties established, accompanied by a number of exercises. The structure of the degrees is studied and a new proof is given that every countable distributive lattice is isomorphic to an initial segMent of degrees. The relationship between these initial segments and the jump operator is studied. The significance of this work for the first-order theory of degrees is analyzed: it is shown that degree theory is equivalent to second-order arithmetic. Sufficient con- ditions are established for the degrees above a given degree to be not isomorphic to and have different first-order theory than the degrees, with or without jump. The degrees below the halting problem are introduced and surveyed. Priority arguments are presented. The theory of these degrees is shown to be undecidable. The history of the subject is traced in the notes and annotated bibliography.