21-603 Model Theory I
Rami Grossberg (Rami@cmu.edu)
URL: www.math.cmu.edu/~rami
MWF 1:30-2:20PM, WeH 7201
Starting date: Monday, August 23, 2010
12 Units

General: Model theory is one of the four major branches of mathematical logic, and has applications to algebra (e.g.field theory, algebraic geometry, number theory, and group theory), analysis (non-standard analysis, complex manifolds and Banach spaces) 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.

Contents include: Similarity types, structures, Abstract Elemntary Classes. Downward Lowenheim-Skolem theorem. Construction of models from constants, applications of the compactness theorem, model completness, elementary decideability results, Henkin's omitting types theorem, prime models. Elementary chains of models, some basic two-cardinal theorems, saturated models (characterization and existence), 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 the majority of students where undergraduates, so 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 2009). 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, not to forward and/or share the contents without my explicit agreement.

The curent version on the web is outdated, a revision will be posted by August 23th.

Most of the material (and more) appears in the following books:

• C. C. Chang and H. J. Keisler, Model Theory, North-Holland 1990. price comparison.
The original version of this book appeared in 1973. Even more than 35 years later this is the most important comprehensive elementary introduction to model theory.

• Bruno Poizat, A course in Model Theory, Springer-Verlag 2000. price comparison.
This is a translation of Poizat's book that was published about 20 years ago by him in France. It is intelligently written, original in its approach. Unfortunately his treatment of forking is outdated. I recommend reading its introduction, it is quit entertaining (especially if you are not American).

• Saharon Shelah, Classification Theory North-Holland 1991. price comparison.
While parts of it are difficult to read this is the most important book in model theory.

Evaluation: Will be 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: Will be announced.

Model Theory homework.