MATH 7840 - Recursion Theory

Richard Shore, fall 2014.

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.

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.

We will then 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 (to be made available). 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. Connections between degree theoretic and other properties such as types of approximations, rates of growth and complexity of definition will be considered.