Tag Archives: logic

A partition relation for well-founded trees by Komjáth and Shelah. . . and two applications to model theory – Virtual – 5/21

For the Virtual Logic Seminar, I gave in May 2021 the lecture A partition relation for well-founded trees by Komjáth and Shelah. . . and two applications to model theory.

Abstract: in 2003, Komjáth and Shelah proved a partition theorem on scattered order types; these in turn could be understood as partition relations for classes of well-founded trees. Recently, two different kinds of applications of the same partition relation have been used in infinitary logic and in model theory: one by Väänänen and Velickovic on games related to Shelah’s logic L^1_\kappa, another by Shelah and myself on the “canonical tree” of an AEC (a generalization of the Scott sentence for an abstract elementary class). I will describe the Komjáth-Shelah result in the first part and then narrow in the applications (with more details on the second one, from some recent joint work with Shelah). Time permitting, I will also address a third interaction between partition relations and model theoretic issues.

ZFC vs HoTT – a possible crisis in foundations of mathematics? – Torino (v) – 4/21

As part of my Torino Lectures in Set Theory, Special Lecture 1 was devoted to ZFC vs HoTT – a possible crisis in foundations of mathematiccs?

Abstract: The lecture will address the oft-heard issue on “New Foundations” provided by HoTT, primarily for a group of students who are learning for the first time set theory. The students have just learned the intricate details of Gödel’s beautiful argument of consistency of AC + GCH with ZFC and are now entering the big step: learning forcing for the first time in their lives. It will be followed by conversation with special invited guests Mirna Džamonja, Matteo Viale and Fernando Zalamea.

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.

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).

Lógica(s) y Topología(s) – desde Stone hasta Lurie / Bogotá, 2/20

Para el III Simposio Carlos Ruiz en la Universidad Nacional (Bogotá) di la charla Lógica(s) y Topología(s) – desde Stone hasta Lurie.

Resumen: En esta charla hago un recuento de tres teoremas de dualidad: Stone (1936), Makkai (1988 – generalizando resultados anteriores con Reyes) y Lurie (preprint, 2019) también llamados “Teoremas de Completitud Conceptual” en algunos contextos. Señalo el rol central de la topología como herramienta de reconstrucción (de la sintaxis a partir de la semántica) y planteo otros dos escenarios de reconstrucción análogos (trabajos conjuntos con Ghadernezhad – 2017 y con Shelah – en proceso).

Mirar hacia fuera desde la lógica matemática (Bucaramanga, 10/18)

Días de la Matemática – CEMAT – UIS

Bucaramanga, 17 a 19 de octubre de 2018.

Mirar hacia fuera desde la lógica matemática (un paseo por preguntas de la química matemática)

Resumen – Modelar fenómenos externos a nuestra disciplina nunca es fácil. Requiere “cruzar puentes” a veces difíciles, requiere escuchar preguntas de investigadores de otros campos. La lógica matemática, y en particular su sub-área llamada “teoría de modelos” permiten asomarse a entender un poco a descifrar algunas preguntas planteadas por la física y por la química.
En esta charla miraremos algunas preguntas que han hecho algunos químicos – preguntas sobre la estructura de las sustancias químicas, sobre la tensión entre cálculos cada vez más complejos y (según algunos de ellos) un escaso entendimiento real de preguntas fundamentales. Y usaremos algo de teoría de modelos como “prisma” para abordar algunas de esas preguntas (principalmente debidas al químico suizo de finales del siglo pasado, Hans Primas).
La teoría de modelos recientemente ha indagado las llamadas “estructuras de aproximación” (o estructuras límite) en teoría de números y en física matemática, y también ha permitido entender fenómenos de no-localidad en física matemática. La no-localidad es central en el desarrollo de una versión “interna” de la química matemática, según Primas entre otros autores ya clásicos de esta disciplina.
La conferencia tendrá una estructura de “paseo” por diversas preguntas.

Entre ZFC y HoTT – sobre posibles crisis … (Medellín, 10/18)

Coloquio – Escuela de Matemáticas – Universidad Nacional de Colombia – Medellín

1 de octubre de 2018

Entre ZFC y HoTT – sobre posibles crisis de fundamentos en la matemática

Resumen: Se ha hablado recientemente de una nueva crisis en los fundamentos de la matemática, en relación con la propuesta originada en trabajos de Voevodsky sobre la “teoría homotópica de tipos” (Homotopy Type Theory, mejor conocida por su acrónimo HoTT) y una posible re-fundamentación de la matemática basada en esta. Hace poco más de un siglo hubo otra crisis que finalmente se decantó en la axiomatización de Zermelo y Fraenkel. Daré un panorama de lo qué está pasando realmente en HoTT y con el nuevo Axioma UF (Univalent Foundations), y trataré de poner en perspectiva la pregunta sobre la crisis. Esta charla se ubicará entre los dos extremos conjuntista y “tipo-teórico-homotópico”: el debate sobre el tema ha tenido contribuciones interesantes de Dzamonja (del lado conjuntístico) y Lurie (crítico, desde el lado categórico). Mostraré algo del debate reciente.

Membre du jury of Doctoral Thesis defense (Vienna, 7/18)

Mr. Fernando Gálvez (Univ. de Paris) will defend his doctoral thesis Le débat sur les notions d’objet et de structure mathématiques au sein du structuralisme contemporain : les travaux de Shapiro, Parsons et Hellman. (Director: Marco Panza).

The public defense will be at the University of Vienna (Philosophy Department) on 31 July 2018.

I will participate, as member of the Committee.

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.