Course info

Prerequisites

The student should have followed a course in mathematical analysis. No very specialized knowledge is required.

Aim of the course

Descriptive set theory, sometimes called: the study of definable subsets of the continuum, is a field of research started by French and Russian mathematicians like H. Lebesgue, É. Borel, M. Suslin and N. Lusin over a century ago. They mistrusted certain assumptions in Cantorian set theory like the existence of the power set of the set of the reals, the axiom of choice and the well-orderability of the set R of the reals. They defined the Borel subsets of R and the analytic and co-analytic subsets of R and they established the Borel hierarchy and the projective hierarchy. L.E.J. Brouwer shared some aspects of the critical attitude of the early descriptive set theorists but his criticism went further as he also attacked the use of classical logic. Brouwer sometimes called the early descriptive set theorists semi-intuitonists. In 1943, S.C. Kleene established, in recursion theory, the arithmetical hierarchy, a hierarchy of subsets of the set N of the natural numbers. He found this result on his own, but A. Mostowski, who also proved the theorem, was inspired by the old work in descriptive set theory. In 1955, J.W. Addison brought to light the analogy between descriptive set theory and recursion theory. Modern descriptive set theory is a blend of set theory and recursion theory. In the 1980's, intuitionistic descriptive set theory was developed. Infinite games, introduced into mathematics by the Polish mathematicians Banach and Mazur, play an important role in the subject. The course will stress the historical development of the subject. We will pay careful attention to the constructivity of our arguments. We will sketch some of the spectacular changes the subject undergoes when treated intuitionistically.

Lecturers

W. Veldman (RU)