# Two logics, and their connections with large cardinals / Questions for BDGM – New York (v) – 4/21

For the CUNY Graduate Center Set Theory Seminar I gave a series of two lectures with title Two logics, and their connections with large cardinals / Questions for BDGM.

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 $L^1_\kappa$ (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 $L^1_\kappa$ and my own logic (joint work with Väänänen) $L^{1,c}_\kappa$. 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.

# 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).

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

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 than$L_{\kappa,\omega}$. 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.

# Tree Partition Properties / natural logics for AEC (a tale of two cities) – Bogotá 11/19

During DiPriscoFest I gave the lecture Tree Partition Properties / natural logics for AEC (a tale of two cities) as homage to Carlos Di Prisco, the Caracas Logic Group and the Intertwined History of two Logic Groups: Caracas and Bogotá.

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

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.

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.

# Between Infinitary Logic and Abstract Elementary Classes (Barranquilla, 6/18)

(slides)

Encuentro de Sociedad Matemática Mexicana y Sociedad Colombiana de Matemáticas, Universidad del Norte, Barranquilla, mayo y junio de 2018.

A lecture on the role of the problem of capturing abstract elementary classes and Shelah’s logic $L^1_\kappa$.

# 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.