(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.