Tra teoria dei modelli e teoria degli insiemi – Torino (v) – 5/21

As part of my Torino Lectures in Set Theory, Special Lecture 3 was given in Italian. The title was Tra teoria dei modelli e teoria degli insiemi.

La terza lezione speciale (parte del corso Teoria degli insieme) sarà questo prossimo giovedì (16:30 ore a Torino). Darò io stesso la lezione stavolta, su connessioni tra teoria dei modelli e teoria degli insiemi. La lezione sarà diretta principalmente a studenti che iniziano il loro apprendimento delle tecniche di forcing e grandi cardinali (la lezione 2, che Vika Gitman ha dato due settimane fa, anche aveva un mix di questi due temi).
Sarà anche la prima volta che parlerò in italiano in pubblico online (forse soltanto per una parte della lezione; dipende di come andrà tutto). Ho invitato gente che in principio può capire la lezione così.

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).

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).

Partitions of well-founded trees: Three connections with model theory – Tehran (v) – 2/21

For the IPM (Tehran), the lecture Partitions of well-founded trees: Three connections with 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 other interactions between partition relations and model theoretic issues.