Abstract: I will provide a general overview of AECs with emphasis on their connection with various abstract notions of “existential closure”. I will present a recent joint result of myself together with Shelah (2019) on axiomatizability of AECs in infinitary logic, and examples of their interaction with closure notions. As an example, I will present aspects of a construction of a canonical “existential closure” for locally finite groups due to Shelah, and their connection with more general stability-theoretical questions.
Abstract: in 2003, Komjath and Shelah proved a partition theorem on scattered order types; these in turn could be understood as partition relations for classes of well-founded trees. Recently, two different kinds of applications of the same partition relation have been used in infinitary logic and in model theory: one by Väänänen and Velickovic on games related to Shelah’s logic , another by Shelah and myself on the “canonical tree” of an AEC (a generalization of the Scott sentence for an abstract elementary class). I will describe the Komjath-Shelah result in the first part and then narrow in the applications (with more details on the second one, from some recent joint work with Shelah).
Abstract: I will describe two recent lines of interplay between Abstract Elementary Classes and Categorical Logic: the problem of building the “Galois group” of an AEC (building on Lascar and Poizat’s work on the “Galois theory of model theory”, and on the role of the Small Index Property – joint work of mine with Ghadernezhad) and interpreting -categoricity in terms of properties of classifying topoi (recent work of Espíndola, connected to his ground-breaking work on Shelah’s eventual categoricity conjecture). My talk will stress the way these connections appear and the opening of new lines of possibility.
(A lecture during the Latin American Symposium of Mathematical Logic, Concepción, Chile, December 2019.)
Abstract: The first logic (called for a singular strong limit cardinal) I will speak about was introduced by Saharon Shelah in 2012. The logic has many properties that make it very well adapted to model theory, despite being stronger than. However, it also lacks a good syntactic definition. With Väänänen, we introduced the second logic (called ,) as a variant of with a transparent syntax and many of the strong properties of Shelah’s logic. The third logic (called Chain Logic), while not new (it is due to Karp), has been revisited recently by Dzamonja and Väänänen) also in relation to Shelah’s and the Interpolation property.
I will provide a description of these three logics, with emphasis on their relevance to model theory.
Abstract: We study interpretations in Abstract Elementary Classes. We recast notions of interpretability and internality (following Hrushovski and Kamensky who based their work on earlier results due to Makkai and Reyes) and their version of a model theoretic Galois theory. We present results for AECs that satisfy a Small Index Property (enabling one to recover the biinterpretability class of theories) and on the Galois groups of those AECs. We stress the role of the Presentation Theorem and of a recent logic due to Shelah that enables us to define AECs.