Abstract: The main recent logic I will describe is Shelah’s infinitary logic (from 2012). I will describe some of the reasons for studying this logic (roughly, it is an infinitary logic that has interpolation and a weak form of compactness – therefore particularly well-adapted to model theory, as well as closure under chains) and some of the features lacking (mostly, a workable syntax). I will describe two other logics that have been created in order to capture better the syntax (one of these logics is my joint work with Väänänen, the other one is due originally to Karp and Cunningham and has recently been connected to by Dzamonja and Väänänen. Finally I will connect these logics with the problem of axiomatizing abstract elementary classes. In particular, I will describe canonical trees of models that enables one to build a sentence to test models for membership into aecs. This last part is joint work with Shelah.
(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.
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 .
A lecture on Reflection Principles & Abstract Elementary Classes given at the Centre de Recerca Matemàtica (Bellaterra, Catalunya), in the context of the Workshop on Set-Theoretical Aspects of the Model Theory of Strong Logics, in September of 2016.
A minicourse in Helsinki on Language, Logic and Nonelementary Classes: External/Internal Interactions, given at the University of Helsinki’s Mathematics Department in May of 2016.
The topics were
- Around The Presentation Theorem
- The Small Index Property for Homogeneous Classes
- Categories, AECs, Interpolation.
Here is the course material (presented on the blackboard).
For the UCLA Logic Seminar, in April 2016, I gave the lecture Categoricity in Non-Elementary contexts – The Role of Large Cardinals.
Here are (very sketchy) notes for a minicourse called Model Theory of Abstract Elementary Classes: some recent trends. I gave that minicourse at the IPM in November 2015. The four two-hour sessions were:
- Day 1: The early days of AECs
- Day 2: Stability Theory of AECs
- Day 3: Stability Theory of AECs (II)
- Day 4: Connections with Set Theory
- Appendix: more examples