### Descriptive Set Theory - 8EC

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.

Rules about Homework / Exam

Part of the homework may be given in and will be graded.

This will be done every two weeks and will result in six grades.

The results of the homework  may influence the final grade, obtained via the written examination, only in a positive sense, up to a maximum weight of 40%.

Possibly,  10f=4h+6t, where f is the final grade, h is the homework grade, t is the grade for the written examination.  If only have only three homework grades that exceed the examination grade, the formula used will be: 10f=2h + 8t.

Lecture Notes / Literature

• A.S. Kechris, Classical Descriptive Set Theory, Springer, Berlin etc., 1995.
• Y.N. Moschovakis, Descriptive Set Theory, Second Edition, Mathematical Surveys and Monographs, vol. 155, American Mathematical Society, 2009.
• S.M. Srivastava, A Course on Borel Sets, Springer, 1998.
• W. Veldman, The Borel hierarchy theorem from Brouwer's intuitionistic perspective, The Journal of Symbolic Logic 73(2008)1-64.

Lecturers

W. Veldman (RU)