Substructural approaches to logical consequence
In this mini-course, we will present a family of logical systems that do not take for granted all the structural features usually attributed to logical consequence, especially as conceived through the Tarskian tradition. Discussion of Montonicity, Contraction, and Exchange will be held, but special attention will be devoted to the slew of systems rejecting Reflexivity and Transitivity that were at the center of some vivid debates during the past decade. Particularly, we will analyze the families of three-valued valuations that, together with a non-transitive understanding of logical consequence, render the same valid inferences that Classical Logic. In connection with these, we will study different sequent calculi where the Cut rule is admissible, hoping to draw a connection between its underivability and the resulting system's substructurality.
Borel sets and reverse mathematics
Theorems about Borel sets are sometimes proved by recursing along the structure of the Borel sets, but more often they are proved via measure or category. It is natural to wonder if these are essentially different proofs. Reverse mathematics provides a way to formalize this kind of question. We analyze the statements "Every Borel set has the property of Baire" and "Every Borel set is measurable" to show that category arguments and measure arguments are strictly less powerful than arguments which recurse directly on the structure of a Borel set. This framework can then be applied to query the necessity of measure and category methods in various theorems about Borel sets, especially in descriptive combinatorics.
An introduction to construction squemes.
Construction/Capturing schemes are a powerful combinatorial tool introduced by Stevo Todorcevic. The point is to build uncountable structures by performing careful amalgamations on its finite structures. Using this schemes it is possible, for example, to build a Hausdorff gap or an Aronszajn tree in just countably many steps. The construction of some uncountable structures becomes easier when using construction schemes. This mini course will be a short introduction to Construction and Capturing schemes. We will survey some previously known results and present some new ones, which are part of a Joint work with Stevo Todorcevic and Jorge Cruz Chapital.