21-603 Model Theory I
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:

I will not use a text, but will distribute my own notes on a weekly basis.

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

Date for midterm: A 50 minute test was given on Monday March 8 instead of a regular lecture.
Suggested Exercises (from the handout):