# Around Shelah’s logic L^1_\kappa – Pittsburgh (v) – 3/21

For the Carnegie Mellon Model Theory Seminar, I gave in March 2021 a series of two lectures with the title Around Shelah’s logic $L^1_\kappa$.

Abstract: In 2012, Shelah introduced a new logic called $L^1_\kappa$. The declared intention was to solve the problem of finding a logic with interpolation and maximality properties with respect to weak forms of compactness. This logic satisfies the declared intention (when the parameter $\kappa$ is a singular strong limit of countable cofinality): the logic $L^1_\kappa$ has interpolation (in itself) and undefinability of well-order. However, the logic’s syntax is given only through a variant of the classical Ehrenfeucht-Fraïssé game and is not built recursively. In recent joint work with Väänänen, we have provided an approximation (from below) to Shelah’s logic that indeed has a recursively built syntax and clarifies some questions on Shelah’s $L^1_\kappa$.

In the first lecture, I will give a presentation of Shelah’s logic, with one proof (of a long string of theorems leading to a Lindström-like characterization). In the second lecture, I will describe our logic (from our joint work with Väänänen) that approximates $L^1_\kappa$ and allows us to understand in a different way some of the properties of Shelah’s logic. If time allows, I will also mention other two approaches (joint work of Väänanen with Dzamonja for one of these approaches and with Velickovic for the other one).