This term we will cover basics of model theory, first order languages, ultrafilters, Gödel's completeness theorem and the compactness theorem, Löwenheim-Skolem theorems, Fraissé theory and the back-and-forth method, quantifier elimination, and if time permits some applications, such as the solution to Hilbert's 17th problem.
28.09: Definition of first order language, formulas, and structures.
30.09: Substructures, elementary equivalence and elementary substructures, Tarski-Vaught.
05.10: Examples of theories, definability.
07.10: Characterization of definability.
12.10: Interpretability.
14.10: Ultrafilters, definition and applications.
19.10: Ultraproducts, Los Lemma.
21.10: The compactness theorem.
26.10: Applications of compactness, nonstandard analysis.
28.10: Godel's completeness theorem.
2.11: Model existence theorem, Henkin construction.
4.11: Extension to Henkin systems.
9.11: Lowenheim-Skolem theorems.
11.11: Lowenheim-Skolem theorems.
16.11: Back-and-forth, DLO's and their properties, the random graph (online).
18.11: Fraisse classes and limits (online).
23.11: Fraisse limits; quantifier elimination (online).
30.11: Quantifier elimination, algebraic criterion.
02.12: Presburger arithmetic.