Interpolation and model theoretic forcing – some new perspectives.
(A lecture in Cantor Meets Robinson – Set theory, model theory and their philosophy. University of Campinas, Brazil, December 2018.)
Model Theoretic Forcing has been interweaved with interpolation theorems in infinitary logic since the early work of Mostowski, Vaught, Harnik and others. I will present some of these historical connections and their effect on Shelah’s much more recent logic . In particular I will focus on some connections between model theoretic forcing and the model theory of abstract elementary classes.
Reflections on Set Theoretic Reflection – Montseny, Catalunya, nov. 2018.
Infinitary logic, large cardinals and AECs: some reflections.
The interaction between infinitary logic and the model theory of abstract elementary classes has had a serious imprint of large cardinals since the inception of AECs. Although later developments in AECs have emphasized a more purely model theoretic treatment, capturing independence-like relations, there are various fundamental questions on the relation between various logics and AECs — and, in some of these, large cardinals are central.
I will discuss some work by Boney on these connections, as well as some recent joint work by Väänänen and myself.
Encuentro de Sociedad Matemática Mexicana y Sociedad Colombiana de Matemáticas, Universidad del Norte, Barranquilla, mayo y junio de 2018.
A lecture on the role of the problem of capturing abstract elementary classes and Shelah’s logic .
: I will first describe Abstract
Elementary Classes as a global generalization of Infinitary Logic. I will emphasize constructions such as Galois types, the Representation Theorem and various open problems. In the second half, I will focus on some recent research on logics underlying AECs – with special emphasis on Shelah’s
logic (satisfying Interpolation and weak remnants of compactness) and the role it plays in controlling Abstract
Elementary Classes. This second part contains recent results of research and several open questions.
Logic Seminar, University of Warsaw.