On the Role of O' in Recursion Theory

Publisher Summary This chapter shows some applications of π 0 1 classes to the study of the structure of Turing degrees ordered by Turing reducibility. The chapter first explains that 0' is definable in the structure resulting from the structure of Turing degrees by adding the relation “to be a degree of a complete extension of Peano Arithmetic (PA).” In this connection the chapter exhibits a kind of priority-free construction of degrees having a self-referential flavor. It is noted that a π 0 1 class of functions can be thought of as just the class of all infinite branches of a recursive tree. A class C of functions is called recursively bounded if there is a recursive function that bounds every f Є C on all arguments. A class of sets is a π 0 1 class just if the corresponding class of characteristic functions is a π 0 1 class.