Cornell Math - MATH 787, Fall 2000

MATH 787 — Fall 2000
Set Theory

There are two possible syllabi for this course. It may either be the usual basic introduction to axiomatic set theory (I) or a course in descriptive set theory (II). We give the corresponding course descriptions below and ask that anyone interested in either version contact us.

I. Basic introduction to axiomatic set theory: The standard course begins with the axioms for Zermelo-Fraenkel Set theory and the elementary theory of ordinal and cardinal numbers. We develop enough of the structure of Goedel's constructible universe  L  to prove the consistency of the general continuum hypothesis, the axiom of choice and various combinatorial principles useful for establishing consistency results in topology and algebra (e. g. the Souslin and Whitehead problems). We also investigate some of the forcing constructions of Cohen, Martin, Solovay and others to construct models of set theory in which the continuum hypothesis fails and various problems of combinatorics, topology and algebra have different solutions than they do in Goedel's universe. There may also be some discussion of combinatorial properties of some of what are now considered to be the smaller of the large cardinals.

Prerequisites: A familiarity with predicate logic and naive set theory.

Text: Set Theory, An Introduction to Independence Proofs, K. Kunen

II. Descriptive set theory: Descriptive set theory concentrates on the analysis of sets of reals (or more generally of Polish spaces). In particular, analyzes the structure of definable sets beginning with the hierarchy of Borel sets determined by beginning with the open sets and then iterating complementation and countable union. It then continues into the projective hierarchy which is generated from the Borel sets by projection and complementation. Typical issues considered are uniformization (of relations by functions of the same class in the hierarchy), measurability, perfect set property (which uncountable sets must conation perfect subsets) and the property of Baire (differing from an open set by a meager set). We will study and use the determinacy of infinite games as well as effective notions and approaches of recursion theory to analyze the sets in these hierarchies.

Prerequisites: A familiarity with predicate logic, naive set theory and the basics of elementary point set topology.

Probable Text: Descriptive Set Theory, Yiannis M. Moschovakis and/or Classical Descriptive Set Theory, Alexander S. Kechris.