Model Theory: a view towards
intuitionistic and categorical logic
PhD level course*, 7.5 hp, Fall 2015
Classical model theory is sometimes explained by the equation
Model theory = universal algebra + logic.
In this course we consider a more general setting where the logic can not only be classical, but intuitionistic or categorical. The universal algebra is also understood from the more general perspective of category theory, giving direct applicability to structures found in algebraic geometry and topology, e.g. operads and sheaves, as well as structures in constructivism and computability theory.
Equational logic and Birkhoff’s completeness and variety theorem. Algebraic theories in sets and arbitrary categories (Lawvere theories). Operads. Generalized algebraic theories, essentially algebraic theories and Cartesian logic. Gabriel-Ulmer duality. Substitution, relations and quantifiers, pullbacks, subobjects and adjoints in categories. General syntax-semantic duality. Boolean and Heyting algebra valued models. Topological models of intuitionistic logic. Kripke and sheaf models. Independence results, especially in arithmetic and analysis. Results about model classes. Comparison with classical model theory. (The content can be adapted to the interests of the participants, to some degree. )
Course literature: to be announced
Lecturer Prof. Erik Palmgren (08-16 45 32, palmgren[at]math.su.se)
Schedule Lectures Thursday 10.15-12 in room 306, building 6, Kräftriket. First lecture 15 October. Final lecture February 4. No lectures in the period 18 December - 13 January.
No lecture December 3. Instead attend one of the lectures of Categorical Logic Workshop 3-4 December.
Programme is available here:
(*)The course can also be taken as the advanced level course Topics in mathematics: mathematical logic. A good course in logic and knowledge of abstract algebra and basic topology are assumed as background.