Rami Grossberg (Rami@cmu.edu)

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

MWF 4:00-34:50PM, ZOOM

Starting date: Monday, August 31, 2020

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), 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, Basics of infinitary logics and Abstract Elementary Classes, 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.

Prerequisites: This is a graduate level course, while at the beginning the pace will be slow in order to accommodate everybody, eventually the course will speeds up. In the past many, in fact at times the majority of students were undergraduates. As I am interested to have undergraduate students 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 2020). 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:

- 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 35 years ago by him in French. 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 willing to be lectured by a French Chauvinist about his views of Americans).

- 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%), two 50 minutes midterms (20% each) and a 3 hours in class comprehensive final written examination (40%).

Model Theory homework.

Accommodations for Students with Disabilities:

If you have a disability and are registered with the Office of Disability Resources, I encourage you to use their online system to notify me of your accommodations and discuss your needs with me as early in the semester as possible. I will work with you to ensure that accommodations are provided as appropriate. If you suspect that you may have a disability and would benefit from accommodations but are not yet registered with the Office of Disability Resources, I encourage you to contact them at access@andrew.cmu.edu.

Statement of Support for Studentsâ€™ Health & Well-being

Take care of yourself. Do your best to maintain a healthy lifestyle this semester by eating well, exercising, avoiding drugs and alcohol, getting enough sleep and taking some time to relax. This will help you achieve your goals and cope with stress.

If you or anyone you know experiences any academic stress, difficult life events, or feelings like anxiety or depression, we strongly encourage you to seek support. Counseling and Psychological Services (CaPS) is here to help: call 412-268-2922 and visit http://www.cmu.edu/counseling/. Consider reaching out to a friend, faculty or family member you trust for help getting connected to the support that can help.

If you or someone you know is feeling suicidal or in danger of self-harm, call someone immediately, day or night:

CaPS: 412-268-2922 Re:solve Crisis Network: 888-796-8226 If the situation is life threatening, call the police On campus: CMU Police: 412-268-2323 Off campus: 911

If you have questions about this or your coursework, please let me know. Thank you, and have a great semester.

Rami's home page.

Last modified:
August 15 ^{th}, 2019 |