Tag Archives: infinitary logic

On Some New Infinitary Logics . . . and (their) Model Theory SLALM – Concepción, Chile 12/19

Integral del silencio. Roberto Matta.

On Some New Infinitary Logics . . . and (their) Model Theory

(A lecture during the Latin American Symposium of Mathematical Logic, Concepción, Chile, December 2019.)

Abstract: The first logic (called L^1_\kappa for \kappa a singular strong limit cardinal) I will speak about was introduced by Saharon Shelah in 2012. The logic L^1_\kappa has many properties that make it very well adapted to model theory, despite being stronger thanL_{\kappa,\omega}. However, it also lacks a good syntactic definition.
With Väänänen, we introduced the second logic (called L^{1,c}_\kappa,) as a variant of L^1\kappa 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 L^1_\kappa and the Interpolation property.

I will provide a description of these three logics, with emphasis on their relevance to model theory.

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.

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.