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.
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 (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 and my own logic (joint work with Väänänen) . 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.
Model Theoretic Forcing has been interweaved with interpolation theorems in infinitary logic since the early work of Mostowski, Vaught, Harnik and others. I will present some of these historical connections and their effect on Shelah’s much more recent logic . In particular I will focus on some connections between model theoretic forcing and the model theory of abstract elementary classes.
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.
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.
Some connections first between categoricity in model theory and the role of large cardinals in pinning down tameness (work of Boney and Unger), with a slight reframing of Boney’s proof. Then, more model theory and set theory connections, around the combinatorics of pcf structures, problems of absoluteness and tree properties.
1st Mexico-USA Logicfest – ITAM, Mexico City, January 2018.