319015 Introduction to mathematical logic and model theory (V) (WiSe 2008/2009)

This lecture course is aimed primarily at graduate students of mathematics and mathematical economics (after the Vordiplom or Bachelor). Its purpose is to acquaint students with widely-used methods from mathematical logic which are applied in both pure mathematics (e.g. algebra, functional analysis) and mathematical economics (e.g. social choice theory).

The first part of this lecture course covers Boolean algebras, filters and ultrafilters. These notions capture much of the algebraic reasoning involved in mathematical logic. Subsequently, the syntax and semantics of both propositional logic and first-order predicate logic is introduced, and deductive systems for both propositional logic and predicate logic will be studied. In particular, Gödel's completeness theorem will be proven. The last part of the lecture course introduces a basic technique of model theory, viz. the ultraproduct method. Time permitting, some applications of the ultraproduct method will be presented: (1) a rigorous development of elementary infinitesimal ("nonstandard") analysis; (2) the characterisation of social aggregation functions as Arrowian dictatorships.

Bibliography

J. L. Bell, A. B. Slomson. Models and Ultraproducts: An Introduction. 2nd revised printing.
Amsterdam: North-Holland Publishing Company, 1971.

Additional references will be given over the course of the lectures.