
Model Theory with nonstandard analysis
Credits 
8 credit points 
Instructors 
Gehrke, M. (Radboud Universiteit Nijmegen), Barmpalias, G. (Universiteit van Amsterdam), Venema, Y. (Universiteit van Amsterdam) 
Email 
m.gehrke@math.ru.nl, barmpalias@gmail.com, Y.Venema@uva.nl 
Description 
In (firstorder) 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 farreaching 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/modeltheoryfo/
In this course we will give a general introduction to the methods and results of classical model theory including compactness, the LowenheimSkolem 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 nonstandard 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). 
Examination 
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). 
Literature 
Main text: Hodges, Wilfrid (1997), A shorter model theory, Cambridge: Cambridge University Press, ISBN13: 9780521587136, ISBN10: 0521587131
Other pertinent texts: Chang, Chen Chung; Keisler, H. Jerome, Model Theory, Studies in Logic and the Foundations of Mathematics (3rd ed.), Elsevier, ISBN 9780444880543
Wilfrid Hodges, Model Theory, Cambridge University Press, ISBN10: 0521304423, ISBN13: 9780521304429
Marker, David (2002). Model Theory: An Introduction. Graduate Texts in Mathematics 217. Springer. ISBN 0387987606.
Robinson, Abraham (1996) [], NonStandard Analysis, Princeton University Press, ISBN13: 9780691044903
Berg, Imme van den; Neves, Vitor (Eds.) (2007), The Strength of Nonstandard Analysis, Springer, ISBN: 9783211499047
Ebbinghaus, HeinzDieter; Flum, Joergen (1991). Finite Model Theory, Springer, ISBN: 354060149X
Graedel et al. Finite Model Theory and Its Applications, Springer, ISBN 9783540004288 
Remarks 
See also: http://staff.science.uva.nl/~yde/teaching/mt/ 
