Mathematische Logik / Mathematical Logic (V3A5/F4A1)

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Lecturers

• Prof. Dr. Philipp Hieronymi
• Leon Chini (e-mail: lchini at uni-bonn.de, office: N1.015)

Lectures

Mondays 14:15-16:00, Wednesdays 12:15-14:00, Wegelerstr. 10 - Zeichensaal

Tutorials:

• Gruppe 1: Thursdays 12.15 - 14, Endicher Allee 60, 0.011
• Gruppe 2: Fridays 12.15 - 14, Endicher Allee 60, 0.011
• Gruppe 3: Fridays 14.15 - 16, Endicher Allee 60, 0.011

The first tutorials will be during the Week October 16-20.

Content

Model theory, a part of mathematical logic, studies mathematical structures, such as groups, graphs and fields, by understanding which logical sentences are true in these structures, and by determining which sets are definable by logical formulas in these structures. So broadly speaking, we investigate mathematical objects by analyzing how they are defined. This is a common theme in mathematics. For example, algebraic geometry studies geometric objects (varieties over a field) using their syntatical description (the fact that they are zero sets of polynomial equations). In model theory, we replace varieties by arbitrary definable sets in mathematical structures and replace the polynomial equations by arbitrary first-order formulas. Indeed, as Hodges wrote aptly, one can argue that

model theory = algebraic geometry − fields.
Model theory has recently seen striking applications to other areas of mathematics, where it has been used as a tool to attack and solve formerly intractable problems. Examples are Hrushovski’s proof of the function field Modell-Lang conjecture in all characteristics, Pila’s proof of the André-Oort conjecture for products of modular curves and the applications of o-minimality to Hodge theory by Bakker, Brunebarbe, Klingler, and Tsimerman.

This is a first course in model theory. We cover basic notions and results of the area, while focusing on applications to and interactions with other areas of mathematics. Given the striking applications of model theory, this course should be of interest not only to students planning to study logic, but to all students interested in algebra, geometry and number theory.

Prerequisites

A prerequisite for this course is basic knowledge of the syntax and sematics of first-order logic, and some experience with expressing mathematical statements in first-order logic. Students who have taken Einführung in die Mathematische Logik (V2A2) or an equivalent course, easily satisfy this prerequiste. This (and further) material can be reviewed in Part I of Prof. Koepke's script.

Topics

1. Compactness via ultraproducts
2. Definability
3. Theories and types
4. Elementary maps
5. Saturated models
6. Quantifier eliminiation
7. Algebraic examples
8. Model completeness
9. Indiscrenible sequences
10. ...

Recommend books

1. David Marker, "Model Theory: An Introduction"
2. Katrin Tent and Martin Ziegler, "A Course in Model Theory"