Rami Grossberg (rami@cmu.edu)

MWF 10:30

12 Units

General: Model theory is one of the four major branches of mathematical logic, and has a number of applications to algebra (e.g. field theory, number theory, and group theory). 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.

Contents include: Similarity types, structures. 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: An undergraduate level course in logic.

Text: All the material (and much more) appears in the following books:

- C. C. Chang and H. J. Keisler, Model Theory, North-Holland 1990
- S. Shelah, Classification Theory North-Holland 1991.

Evaluation: Will be based on a final (3 hours written examination) and a 50 minutes midterm.

Suggested Exercises (from the handout):

- pages 18-19: all exercises (only #11 is really difficult).
- pages 33-36: 1-30 (especially 9, 11, 12, 15, 23, 29, 30) and #35
- pages 50-51: pay special attention to 4, 7, 11, 12, 15, 16,17, 18(difficult), 23,24,25.
- pages 62-63: 1,2,3,6,9,10,11,12,13,14
- pages 74-75: 2,3,6( non trivial), 7,11,16,23.