Math 223M: Model Theory

# General Information

Time and Place: MWF 2-2:50pm, Mathematical Sciences Building 7608

Instructor: Matthias Aschenbrenner

Homepage: http://www.math.ucla.edu/~matthias

Office:
Mathematical Sciences Building 5614
Office Phone: (310) 206-8576
Office Hours: MWF 1-2pm, or by appointment.

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# Description

Model theory is a branch of mathematical logic which applies the methods of logic to the study of mathematical structures, and thus has impact on other parts of mathematics (e.g., number theory, analytic geometry). Since its beginnings in the early decades of the last century, the perception of what the subject is about has gone through various incarnations. Because many of the mathematical structures studied in model theory have an algebraic origin, Chang and Keisler (1973) simply decreed that universal algebra + logic = model theory, whereas Hodges (1993) defined model theory more broadly as the study of the construction and classification of structures within specified classes of structures. A modern view holds that model theory is the geography of tame mathematics (Hrushovski). Here, the emphasis is on identifying those classes of structures whose first-order theories can be understood (in some well-defined technical sense), and exploiting such an understanding as a tool in other parts of mathematics.

# Prerequisities

Basic knowledge of first-order logic (Math 220), especially the completeness theorem and elementary set theory, and abstract algebra (Math 210), especially field theory.

# Syllabus

Review of structures and theories. Quantifier elimination, model completeness. Types, saturation, omitting types. Totally transcendental theories, strong minimality, Morley's Theorem. Some o-minimality (time permitting).

# Course Text

I will follow my own notes, but the following book will be a good companion for this course: Model Theory: An Introduction by Dave Marker, Springer-Verlag, 2000.

Other texts on model theory that you might want to consult:
• A Course in Model Theory: An Introduction to Contemporary Mathematical Logic by Bruno Poizat, Springer-Verlag, 2000. (A Russian copy of Poizat's book may be downloaded and you can write (en français) to the author to buy a copy of the book in French.)
• A Shorter Model Theory by Wilfrid Hodges, Cambridge University Press, 1997. (See corrigenda.) An expanded version of this book is available under the title Model Theory.
• Introduction to Model Theory by Philipp Rothmaler, Gordon and Breach Science Publishers, 2000.
• Model Theory by C. C. Chang and H. J. Keisler, North-Holland, 1998.
• If you feel adventurous, check out the lecture notes (in German!) for a course in model theory taught by Volker Weispfenning which I wrote a long time ago.
A good general reference for mathematical logic is Mathematical Logic by Joseph R. Shoenfield, A K Peters, Ltd., 2000.

The classical works of Abraham RobinsonIntroduction to Model Theory and the Metamathematics of Algebra (1963), Complete Theories, (1956; new edition 1976), and On the Metamathematics of Algebra (1951) are still worth reading.

For a collection of recent survey articles on model theory see here.

# Homework

There will be an occasional problem set assigned every two weeks ago, which will be handed out in class, and will also posted on this website. Solutions are due in class on the date specified on the homework sheet.