Model Theory

This page concerns the course `Model Theory', taught at the University of Amsterdam from February - May 2019.
This course features in the local MSc Logic programme and the national Dutch Mastermath programme.

Contents of these pages

News

The 2018/2019 edition of this class has finished.
Next year, the standard 6 EC class Model Theory will be offered in the Master of Logic programme at the UvA.

Practicalities

Staff

Lecturer:
- Yde Venema, e-mail: y dot venema at uva dot nl
Teaching assistant (homework grading):
- Martijn den Besten, e-mail: martijndenb at gmail dot com

Dates/location:

Classes run from 8 February until 24 May. There are lectures and exercise sessions.
- There is one class every week, on Fridays from 10:00 - 12:45 in room SP F1.02 (Science Park, Amsterdam).
- This meeting takes the form of a lecture, but we will also discuss exercises.
The exam is on 7 June, from 10:00 - 13:00 in room OMHP D1.09 (Amsterdam Center).
The resit of the exam is on 28 June, from 10:00 - 13:00 in room NIKHEF F0.25 (Amsterdam Science Park).

Grading

The final grade will be determined by homework assignments and a final exam. See the special page on grading for more details.

Course material

The main text that we will be using is:
- van den Berg, Benno. Syllabus Model Theory, ILLC, Universiteit van Amsterdam, 2018/2019.
There is additional course material on
- ultraproducts
- ordinals and cardinal arithmetic (a scan of the appendix of Marker's book on model theory).
- Skolemisation
- Lindström's Theorem
- Morley's Theorem.
Other pertinent texts are
- Chang, Chen Chung; Keisler, H. Jerome. Model Theory,
  Studies in Logic and the Foundations of Mathematics (3rd ed.), Elsevier, ISBN 978-0-444-88054-3
- Hodges, Wilfrid, A shorter model theory,
  Cambridge University Press, 1997, ISBN-13: 978-0-521-58713-6, ISBN-10: 0521587131
- Marker, David. Model Theory: An Introduction.
  Graduate Texts in Mathematics 217. Springer, 2002. ISBN 0-387-98760-6.
- Katrin Tent and Martin Ziegler, A course in model theory,
  Cambridge University Press, 2012, ISBN-10: 052176324X, ISBN-13: 9780521763240
- Wilfrid Hodges, Model Theory,
  Cambridge University Press, ISBN-10: 0521304423, ISBN-13: 978-0521304429

Course Description

In (first-order) logic, the formal language of mathematical statements and their interpretation in mathematical structures is carefully identified.
Model theory, then, deals with questions such as:

What classes of structures can be captured by mathematical theories?
What are the pertinent constructions in mathematics to describe these classes?
On the other hand, what are the fundamental properties of mathematical theories?
And, how do these theories relate to properties of the classes of structures that they describe?

In this course we will give a general introduction to the methods and results of classical model theory.
More concretely, we will cover the following topics:

Basic notions: diagrams, compactness, Loewenheim-Skolem Theorem, games
Classical model theory: ultraproducts, preservation theorems, quantifier elimination
Types: type spaces, saturation, omitting types, omega-categoricity

The contents of the final part of the course have not been determined yet, but will consist of either

an introduction to modern model theory:: (stability: totally transcendental theories, Morley rank, indiscernibles, Morley's Theorem).
a special topic such as finite model theory or nonstandard analysis.

Prerequisites

We presuppose some background knowledge in formal logic; in particular familiarity with the syntax and semantics of first-order languages.
Some basic knowledge of topology will be good, and familiarity with algebraic structures (such as rings, fields, and vector spaces) will be useful in order to be able to appreciate the examples.
More importantly, we assume that participants in the course possess the mathematical maturity as can be expected from students in mathematics or logic at the MSc level.

Comments, complaints, questions: mail Yde Venema