Abstract: In the past couple of years I have been involved (joint work with Väänänen and independently with Shelah) with some logics in the vicinity of Shelah’s (a logic from 2012 that has Interpolation and a very weak notion of compactness, namely Strong Undefinability of Well-Orderings, and in some cases has a Lindström-type theorem for those two properties). Our work with Väänänen weakens the logic but keeps several properties. Our work with Shelah explores the connection with definability of AECs. These logics seem to have additional interesting properties under the further assumption of strong compactness of a cardinal, and this brings them close to recent work of Boney, Dimopoulos, Gitman and Magidor [BDGM]. During the first lecture, I plan to describe two games and a syntax of two logics: Shelah’s and my own logic (joint work with Väänänen) . I will stress some of the properties of these logics, with any use of large cardinal assumptions. During the second lecture, I plan to enter rather uncharted territory. I will describe some constructions done by Shelah (mostly) under the assumption of strong compactness, but I also plan to bring these logics to a territory closer to the work of [BDGM]. This second lecture will have more conjectures, ideas, and (hopefully interesting) discussions with some of the authors of that paper.
For the Carnegie Mellon Model Theory Seminar, I gave in March 2021 a series of two lectures with the title Around Shelah’s logic .
Abstract: In 2012, Shelah introduced a new logic called . 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 is a singular strong limit of countable cofinality): the logic 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 .
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 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).
Abstract: Two seemingly unrelated questions (the quest for natural logics of abstract elementary classes on the one hand, and the quest for logics adequate to model theory on the other hand) converge around the same combinatorial core: partition relations for scattered order types (due to Kómjath and Shelah). I will present recent results concerning the first question (and axiomatizing a.e.c.’s – joint work with Shelah) and the second question (joint work with Väänänen).
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.
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.
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.
Abstract: 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.
Two lectures on the same topic (Reconstruction Problem, SIP, quasiminimal classes), with different emphasis, given at Ben Gurion University in Beersheva and at the Hebrew University of Jerusalem, in November 2016.