Model Theory with non-standard analysis
||8 credit points
||Gehrke, M. (Radboud Universiteit Nijmegen), Barmpalias, G. (Universiteit van Amsterdam), Venema, Y. (Universiteit van Amsterdam)
||email@example.com, firstname.lastname@example.org, Y.Venema@uva.nl
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 they relate to properties of the classes of structures that they describe?
These questions are fundamental to the whole enterprise of mathematics and the insights and methods of model theory have far-reaching consequences in many branches of mathematics. For a nice introduction and overview of the field, please see the Stanford Encyclopedia of Philosophy website: http://plato.stanford.edu/entries/modeltheory-fo/
In this course we will give a general introduction to the methods and results of classical model theory including compactness, the Lowenheim-Skolem theorems, and various preservation theorems illustrated by examples and applications in algebra, analysis, and discrete mathematics. Various model theoretic techniques for constructing models will be introduced and applied, such as unions of elementary chains, omitting types construction, ultraproducts and saturated models. In addtion, we will cover a few more advanced topics including non-standard analysis (with applications both in analysis and algebra), and elements of finite model theory (in which the model theory of finite structures is the focus).
The final grade will be determined by biweekly homework
assignments and an exam (oral or written; the precise structure may depend on the class size).
Hodges, Wilfrid (1997), A shorter model theory, Cambridge: Cambridge University Press, ISBN-13: 978-0-521-58713-6, ISBN-10: 0521587131
Other pertinent texts:
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
Wilfrid Hodges, Model Theory, Cambridge University Press, ISBN-10: 0521304423, ISBN-13: 978-0521304429
Marker, David (2002). Model Theory: An Introduction. Graduate Texts in Mathematics 217. Springer. ISBN 0-387-98760-6.
Robinson, Abraham (1996) , Non-Standard Analysis, Princeton University Press, ISBN13: 978-0-691-04490-3
Berg, Imme van den; Neves, Vitor (Eds.) (2007), The Strength of
Nonstandard Analysis, Springer, ISBN: 978-3-211-49904-7
Ebbinghaus, Heinz-Dieter; Flum, Joergen (1991). Finite Model Theory, Springer, ISBN: 3-540-60149-X
Graedel et al. Finite Model Theory and Its Applications, Springer, ISBN 978-3-540-00428-8
||See also: http://staff.science.uva.nl/~yde/teaching/mt/