Time and Place: MWF 2-2:50pm, Mathematical Sciences
E-mail address: matthiasmath.ucla.edu
Office: Mathematical Sciences Building 5614
Office Phone: (310) 206-8576
Office Hours: MWF 1-2pm, or by appointment.
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original course announcement.
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
simply decreed that universal
algebra + logic = model theory
(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
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.
Basic knowledge of first-order logic (Math 220), especially the
completeness theorem and elementary set theory, and abstract algebra
(Math 210), especially field theory.
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
I will follow my own notes, but the
following book will be a good companion for this course: Model
Theory: An Introduction
, Springer-Verlag, 2000.
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
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
- Introduction to Model
by Philipp Rothmaler, Gordon and Breach Science Publishers, 2000.
- Model Theory by C.
Chang and H. J.
- 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
logic is Mathematical
A K Peters, Ltd., 2000.
The classical works of Abraham
to Model Theory and the Metamathematics of Algebra
, (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.
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.
Back to my home
modified March 30, 2009.