Tag Archives: model theory

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.

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.