**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 L_{w1,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.

Rami's home page.

Last modified:
January 10^{th}, 2008 |