From ultrafilters to compactness

Por Pedro Filipe (Instituto Superior Técnico, Universidade de Lisboa).

Abstract: The compactness theorem is one of the key ingredients used in Lindstrom's Theorem that characterizes first-order logic and follows directly from Godel's completeness theorem, given the finite nature of proofs. In time, alternative proofs were found that don't require the usage of a formal proof system. In this seminar we will show one of these alternative proofs using ultrafilters and ultraproducts.

Decidability of first-order theories

Por Cristina Sernadas (Instituto Superior Técnico, Universidade de Lisboa).

Abstract: Some results and reduction techniques for proving decidability of mathematical theories and completeness of logics are presented. The crucial role of the theory of real closed ordered fields is explained. Selected illustrations from Euclidean Geometry to Quantum Logic are discussed.