MAGIC107: Computability Theory and algorithmic randomness

This is an introductory course on computability theory, the branch of mathematical logic concerned with classifying sets of natural numbers by how much extra information is needed to describe them by computer programs.  The computational strength of a set is intimately tied to the complexity of its definition, thus, more generally, computability theory is the study of definable sets of natural numbers.  Any countable mathematical object, such as a countable ring or a countable graph, can be coded by a set of natural numbers.  A continuous function on the reals, for example, can be coded by a set of natural numbers too.  Thus the techniques and results of computability theory can be applied to describe the complexities of all manner of mathematical objects.

The computational strength of a set is measured by its Turing degree.  The collection of all Turing degrees has a rich structure and is the central object of study in computability theory.  This course introduces the basic theory of the Turing degrees.  Time permitting, the course also surveys contemporary research directions that connect to other areas of mathematics, such as reverse mathematics and algorithmic randomness. 

The following prerequisites are preferable but not essential: