MATH 7840 - Recursion Theory

Richard Shore, fall 2015.

MATH 7840 will be a first course in the theory of computability. We will assume some background in logic. MATH 6810 or CS 6820 should be more than sufficient.

The pace and content of the course will depend on the background of the students. Plausible outlines are as follows:

We will begin with a brief discussion of the basic properties of a reasonable model of computability: universal machines, the enumeration, s-m-n and recursion theorems, r.e. (effectively or computably enumerable) sets and the halting problem. Next will come the notions of relative computability, the Turing jump operator and the arithmetical hierarchy.

Then there will be some development of construction procedures for non-r.e. sets, in particular, the Kleene-Post finite extension method (really Cohen forcing in arithmetic). An example or two of other forcing type constructions such as with trees (perfect set forcing) to construct a minimal degree may also be presented later.

At this point there are two likely scenarios.

One will concentrate on the recursively (computably) enumerable sets and degrees. The primary text will then be some updated version of Recursively Enumerable Sets and Degrees by R. I. Soare. The heart of the course will be the development of various kinds of priority arguments for the construction of r.e. sets including finite and infinite injury as well as tree arguments. We will use these methods to analyze the structure of the (Turing) degrees of r.e. sets and something of their set theoretic structure as well.

The second scenario will instead study the structure of the Turing degrees of all sets and functions as well as important substructures such as the degrees below 0' (the Halting problem) and the degrees of the arithmetic sets (those definable in first order arithmetic or equivalently computable form some finite iteration of the Turing jump). The primary techniques will be forcing arguments in the setting of arithmetic rather than set theory. We begin with the development of construction procedures such as the Kleene-Post finite extension method (now seen as Cohen forcing in arithmetic) and minimal degree constrictions by forcing with trees (perfect set forcing), forcing with Pi-0-1 classes (closed sets) and others.

Relations with rates of growth and the jump hierarchy will be explored. We will prove the basic results about the complexity of theories of these structures such as that the theories of the degrees and the degrees below 0' are of the same complexity as second order arithmetic and first order arithmetic, respectively. We will also study the restrictions on possible automorphisms of the structures and definability results: which apparently external (but natural) relations on the structures can be defined internally. In particular, we may reach the proof that the Turing jump which captures quantification in arithmetic is definable in terms of relative computability alone.

In either case, connections between degree theoretic and other properties such as types of approximations, rates of growth and complexity of definition will be considered.