Rami Grossberg (Rami@cmu.edu)

URL: www.math.cmu.edu/~rami

MWF 1:30-2:20PM, WeH 7201

Starting date: Monday, August 29, 2016

12 Units

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) 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 the major problems: Saharon Shelah's short article

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, 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, 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 2012). 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 and
not to share the contents without my explicit agreement.

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

- C. C. Chang and H. J. Keisler, Model Theory, Third Edition (Dover Books on Mathematics)
Paperback.

The original version of this book appeared in 1973. More than 40 years later, this is the most important comprehensive elementary introduction to model theory. Its republication by Dover makes it the best buy in the category of logic books that I know.

- Bruno Poizat, A course in Model Theory, Springer-Verlag 2000.

This is a translation of Poizat's book that was published about 30 years ago by him in France. It is intelligently written and 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 1990. You may buy it for lots of money (if you can find a copy) or get a free online copy.
Recently Shelah placed the entire book on the web, it is available from
here.

While parts of the book are impossible to read, this is the most important book in model theory.

Evaluation: Based on weekly homework assignments (20%), a 50 minutes midterm (20%) and a 3 hours in class comprehensive final written examination (60%).

Model Theory homework.

Rami's home page.

Last modified:
August 29 ^{th}, 2016 |