Tag Archives: abstract elementary classes

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).

Axiomatizations of abstract elementary classes and natural logics for model theory: the role of partition relations – Toronto (v) 2/21

For the Toronto Set Theory Seminar, I gave the lecture Axiomatizations of abstract elementary classes and natural logics for model theory: the role of partition relations.

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).

Some New Infinitary Logics and a Canonical Tree – New York, 11/19

A lecture for the CUNY Graduate Center’s Model Theory Seminar.

Abstract: The main recent logic I will describe is Shelah’s infinitary logic L^1_\kappa (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 L^1_\kappa 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.

Interpolation and model theoretic forcing – some new perspectives (Campinas, Brazil, 12/18)

Interpolation and model theoretic forcing – some new perspectives.

(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 L^1_\kappa. In particular I will focus on some connections between model theoretic forcing and the model theory of abstract elementary classes.

Infinitary logic, large cardinals and AECs: some reflections (Montseny, Catalunya, 11/18)

Reflections on Set Theoretic Reflection – Montseny, Catalunya, nov. 2018.

Infinitary logic, large cardinals and AECs: some reflections.

Abstract:

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.

Logics underlying Abstract Elementary Classes (Warsaw, 4/18)

 
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 L^1_\kappa 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.
Logic Seminar, University of Warsaw.

Around the Small Index Property (on quasiminimal classes). Beersheva and Jerusalem, 11/16

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.

Slides and notes for blackboard lecture.

Language, Logic and Nonelementary Classes: External/Internal Interactions. Helsinki, 5/16.

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).