MATH 7830: Model Theory (Spring 2012)

Instructor: Richard Shore

Current plans are to cover the basic material for a first course in model theory. We will use Model Theory: An Introduction by David Marker. Another useful text is Hodges' Model Theory or his textbook version (Shorter Model Theory) as is the classic Model Theory by Chang and Keisler which is out of print but available electronically free to Cornell students.

Each book gives a two sentence definition/description of model theory.

• Chang and Keisler: Model theory is the branch of mathematical logic which deals with the connection between a formal language and its interpretations or models.....it can best be analyzed on the basis of a few general methods of constructing models.
• Marker: Model theory is a branch of mathematical logic where we study mathematical structures by considering the first-order sentences true in those structures and the sets definable by first order formulas.
• Hodges: Model theory is the study of the construction and classification of structures within specified classes of structures.
  

Thus, roughly in Marker’s words, we may begin with specific mathematical structures such as the field of real numbers and use model theoretic techniques to obtain new information about the structure and the sets definable in it. Or, we may begin by looking at theories and the class of structures they define, such as algebraically closed fields, and prove general structure theorems about the models of the theories.

We will follow both routes and also expect to bring in a viewpoint from computability by carrying along an analysis of the effectiveness (or usually the degree of noneffectiveness) of the model theoretic constructions developed such as for completeness, compactness and special models including atomic, prime, homogeneous and saturated as well as decidability issues for theories (quantifier elimination, Fraisse limits, indiscernibles, categoricity and interpretations). The viewpoint or material covered may be changed to suit the background and interests of participants so if you are interested please contact me.