21-703 Model Theory II - Spring 2008 MWF 2:30-3:20PM, PH A19A

Instructor: Rami Grossberg
Office: WEH 7204
Phone: x8482 (268-8482 from external lines), messages at x2545
Email: Rami@cmu.edu
URL: www.math.cmu.edu/~rami
Office Hours: By appointment or whenever else you can find me.

Purpose. This is a second course in model theory. The main topic of discussion will be classification theory for Abstract Elementary Classes (AECs).

Course description.

The subject started 30 years ago by Shelah, the goal is to discover the concepts and tools necessary for the development of model theory for infinatry logic and ultimately to have a complete theory of invariants of models up to isomorphism whenever this is possible and also establish the reason for inexistence of a theory of invariants. Shelah also proposed a conjecture as a test for the development of the theory: Shelah's categoricity conjecture, it is a parallel to Morley's categoriicty theorem for Lw1,w. Despite the existence of about a thousand pages of partial results the conjecture is still open.

Shelah in his list of open problems in model theory [Sh 702] writes: I see this [classification of Abstract Elementary Classes] as the major problem of model theory.''

Until 2001 virtually nobody besides Shelah published work on AECs. Interest and progress in the field by others materialized from two directions: Boris Zilber managed to construct a function over the complex numbers sharing many formal properties with exponentiation and as well satisfying Schanuel's conjecture over the complex numbers (this is a far reaching conjecture in transcendetal number theory implying solutions to many difficult long standing problems, e.g. it implies that \pi + \e is a transcendetal number). Zilber's construction uses a combination of methods from number theory with abstract model-theoretic concepts of Shelah. In parallel Grosberg and VanDieren introduced the notion of tamness and proved new cases of Shelah's categoricity conjecture. Both directions influenced many new people to enter the field.

In the near future fast progress is expected in the pure theory as well as in applications to complicated mathematical structures like homological algebra and quantum groups. Hopefully some of the techniques will turn to be usefull also in the study of classes of finite models, but we will concentrate at uncountable models.

I will focus on the basic parts of the theory that may eventually converge to a proof cases of Shelah's categoricity conjecture.
I will start the ocurse with some fundamental theorems concerning first-order theories (omitting types and computing Hanf-Morley numbers, characterizations of stability) as well with basic set-theoretic machinary manipulating models of weak set theories and the interplay between stationary sets of ordinals and elementary chains of models. Topics to be covered: Chang and Shelah's presentation theorems, Undefineability of well-ordering, Galois-types, Galois-stability and Galois-saturation. The role of the amalgamation property and its connections with the number of pairwise non-isomorphic models, the connection between properties of models of cardinality \lambda and \lambda^+.

Evaluation: Weekly homework assignments (20%) 30%, Midterm 20% and an inclass 3 hour final 50%.

Prerequisites: The contents of a basic graduate course in model theory like 21-603 or permission of the instructor.

Textbook: A course in Model Theory III: Classification Theory for Abstract Elementary Classes, by Rami Grossberg.

Material for students in this course, if you use this link you agree not to forward and/or share the contents without my explicit agreement.