We presuppose some background knowledge in formal logic; in particular familiarity with the syntax and semantics of first-order languages (the first two chapters from "Logic and Structure" by Van Dalen should be sufficient). Some basic knowledge of set theory (ordinals, cardinals), topology (compact Hausdorff spaces, isolated points) will be good, and familiarity with algebraic structures (such as rings, fields, and vector spaces) will be useful in order to be able to appreciate some of the examples. More importantly, we assume that participants in the course possess the mathematical maturity as can be expected from students in mathematics or logic at the MSc level.
The main aim of the course is to provide the students with an overview of classical model theory; additionally, the course will give either an introduction to modern model theory (leading up to Morley's Theorem) or treat a special topic (for instance, finite model theory or nonstandard analysis).
In this course we will first give a general introduction to the methods and results of classical model theory. This part will be divided into three blocks:
(1) Basic notions: diagrams, compactness, Löwenheim-Skolem Theorem, games
(2) Classical model theory: ultraproducts, preservation theorems, quantifier elimination
(3) Types: type spaces, omitting types, saturation, omega-categoricity
The final part of the course will either be devoted to an introduction to modern model theory::
(4a) Stability: totally transcendental theories, Morley rank, indiscernibles, Morley's Theorem.
Alternatively, in the final classes of the course we will cover a special topic such as finite model theory or nonstandard analysis.