# Math 512 Advanced Topics in Logic

# Model Theory and Infinitary Logic

# Fall 2013

Course Meeting: 9:00 MWF 310 AH

Call Number: 344413

Instructor: David Marker

Office: 411 SEO

Office Hours: TBA

phone: (312) 996-3069

e-mail: marker@uic.edu

course webpage:
http://www.math.uic.edu/~marker/math512-F13

### Description

This will be a course on the model theory of infinitary languages focusing on the model theory of L_{omega_1,omega}. The first part of the course will an introduction to the basics: back-and-forth systems, Scott's isomorphism theorem, Henkin arguments, omitting types,
prime models, up-Lowenheim Skolem Theorems and Hanf number considerations.

One major advanced topic is using infinitary methods to shed light on Vaught's Conjecture.

We will also look at issues about stability, existence of uncountable models, and categoricity questions. One goal will be to integrate recent work of Zilber, Baldwin, Larson, Laskowski, Koerwein and Friedman.

If time permits we may talk briefly about admissible sets and Barwise compactness.

There will be no homework.
### Prerequisites

The ideal preparation would include one semester each of model theory, descriptive set theory and set theory, but most of the material will be accessible to anyone with a modest understanding of graduate logic.
### References

- H. J. Keisler,
* Model Theory for Infinitary Logic*, North Holland 1971.
- J. Barwise,
* Admissible Sets and Structures*, Springer 1975.
- J. Barwise and S. Feferman,
*Model-Theoretic Logics*, Springer 1985. [Particularly the paper of M. Nadel]
- J. Baldwin, Categoricity, AMS University Lecture Notes 50, 2009.
- J. Steel, On Vaught's conjecture,
* Cabal Seminar 76-77*
Lecture Notes in Math., 689, Springer, 1978.

Comments: Keisler's book is sadly out of print. A copy should be on reserve at the library.
The Barwise and Barwise-Feferman books are available electronically through the library.
The article in Barwise-Feferman by Nadel is of particular interest.
Baldwin's book is easily available.
### Lecture Notes

The following are first drafts of the lecture notes. Beware of typos and small gaffs.