Hyperarithmetic theory is the first step beyond classical recursion theory. It is the primary source of ideas and examples in higher recursion theory. It is also a crossroad for several areas of mathematical logic: in set theory it is an initial segment of Godel's L; in model theory, the least admissible set after ; in descriptive set theory, the setting for effective arguments. In this book, hyperarithmetic theory is developed at length and used to lift classical recursion theory from integers to recursive ordinals (metarecursion). Two further liftings are then made, first ordinals ( -recursion) and then to sets (E-recursion). Techniques such as finite and infinite injury, forcing and fine structure and extended and combined Dynamic and syntactical methods are contrasted. Several notions of reducibility and computation are compared. Post's problem is answere affirmatively in all three settings. This long-awaited volume of the -series will be a "Must" for all working in the field.

E-recursively enumerable degrees

Higher recursion theory

We discuss the conceptual problem of identifying the natural notions of computability at higher types (over the natural numbers). We argue for an eclectic approach, in which one considers a wide range of possible approaches to defining higher type computability and then looks for regularities. As a first step in this programme, we give an extended survey of the different strands of research on higher type computability to date, bringing together material from recursion theory, constructive logic and computer science. The paper thus serves as a reasonably complete overview of the literature on higher type computability. Two sequel papers will be devoted to developing a more systematic account of the material reviewed here.

Notions of computability at higher types I

This chapter discusses E-recursion with Kleene's notion of recursion in objects of finite type in the so-called normal case. The objects of type 0 are the natural numbers. An object of type n + 1 is a set of objects of type n. Kleene defines {e} (x) for x an object of finite type by means of schemes. Normality allows each computation to invoke the appropriate equality predicate.

Chapter 10 E-recursion

A central issue in E-recursion theory is the relative status of E-recursive enumerability and Σ1-definability in an E-closed structure. In most initial segments of L these two are not the same. However, as is shown here, every E-closed initial segment of L is canonically represented as the union of Π1-absolute admissible sets with gaps, in which sets the two notions are identical. This representation is used to prove a meta-theorem which provides a translation of Friedberg-style finite injury constructions from classical recursion theory into successful E-recursion theoretic constructions for initial segments of L. The key to the method is to construe an E-recursive enumeration as a direct limit of Σ1-enumerations, in each of which, requirements are satisfied using techniques from α-recursion theory. Suppose that L k is E-closed; a subset of κ is said to be scattered if its order type is less than ρ k1 , the E-recursively enumerable projectum of L k . As an application of this method, it is shown that there is an E-recursively enumerable degree on L k which is not scattered.

E-recursively enumerable degrees

Citations

Higher recursion theory

Notions of computability at higher types I

Chapter 10 E-recursion

Reflection and the priority method in E-recursion theory

Related Papers (5)

On degrees of recursively enumerable sets

Recursively enumerable degrees and the conjugacy problem

Recursively Enumerable L‐Sets

The recursively enumerable α-degrees are dense

The Recursively Enumerable Degrees are Dense