Tag Archives: abstract elementary classes

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.


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).

Model Theory of Abstract Elementary Classes: some recent trends. Tehran, 11/15

Here are (very sketchy) notes for a minicourse called Model Theory of Abstract Elementary Classes: some recent trends. I gave that minicourse at the IPM in November 2015. The four two-hour sessions were:

  • Day 1: The early days of AECs
  • Day 2: Stability Theory of AECs
  • Day 3: Stability Theory of AECs (II)
  • Day 4: Connections with Set Theory
  • Appendix: more examples