General: Model theory is one of the major branches of mathematical logic, has applications to algebra (e.g.field theory, algebraic geometry, number theory, and group theory), analysis (non-standard analysis, complex manifolds and Banach spaces), combinatorics and theoretical computer science (via finite model theory) as well as to set theory and set-theoretic topology. This course is the first in a sequence of three courses. The purpose of this course is to present the basic concepts and techniques of model theory with an emphasis on pure model theory. The main theorem of the course is Morley's theorem. It will be presented in a way that permits several powerful extensions.
Saharon Shelah the most prominent logician of our time, wrote a short article about his views of model theory. I will be taking a similar approach in this course. To find out what Shelah thinks of model theory and the subject's the major problems: Saharon Shelah's short article
Contents include: Similarity types, structures, Abstract Elemntary Classes. Lowenheim-Skolem-Tarski theorems. Construction of models from constants, applications of the compactness theorem, model completness, elementary decideability results, cardinal transfer theorems, Henkin's omitting types theorem, prime models. Elementary chains of models, some basic two-cardinal theorems, saturated models (characterization and existence), special models, the monster model, basic results on countable models including Ryll-Nardzewski's theorem. Indiscernible sequences, and connections with Ramsey theory, Ehrenfeucht-Mostowski models. Introduction to stability (including the equivalence of the order-property to instability), chain conditions in group theory corresponding to stability/superstablity/omega-stability, strongly minimal sets, various rank functions, primary models, and a proof of Morley's categoricity theorem. Basic facts about infinitary languages, computation of Hanf-Morley numbers.
Prerequisites: This is a graduate level course, while at the beginning the pace will be slow in order to accommodate everybody, the course speeds up in the second half. In the past many, in fact at times the majority of students were undergraduates. As I am interested to have undergraduate students attending this course attending my courses, I decided to keep the prerequisites to the minimum of "an undergraduate level" course in logic.
Text: Rami Grossberg, A course in model theory I: An introduction,
a book in preperation.
Table of contents (as of August 2017). This is the first volume in a three volume book series to be published by Cambridge University Press. The full text is available to registered students from a protected directory here. If you use this link, you agree not to publish it and not to share the contents.
Most of the material (and more) appears in the following books:
Evaluation: Based on weekly homework assignments (20%), a 50 minutes midterm (20%) and a 3 hours in class comprehensive final written examination (60%).Date for the midterm: Wednesday 10/18/2017.
|Last modified: October 11 th, 2017|