Talk: Maximality and Countability (or `Some systems of set theory on which every set is countable’)

Handout here. This will be a talk at the New York Set Theory Seminar.

Abstract: It is standard in set theory to assume that Cantor’s Theorem establishes that the continuum is an uncountable set. A challenge for this position comes from the observation that through forcing one can collapse any cardinal to the countable and that the continuum can be made arbitrarily large. In this paper, we present a different take on the relationship between Cantor’s Theorem and extensions of universes, arguing that they can be seen as showing that every set is countable and that the continuum is a proper class. We examine several theories based on maximality considerations in this framework (in particular countabilist analogues of reflection principles) and show how standard set theories (including ZFC with large cardinals added) can be incorporated. We conclude that the systems considered raise questions concerning the foundational purposes of set theory.

Talk: Intensional Classes and Intuitionistic Topoi

This will be a talk at the OLOFOS seminar “Formal analysis of modality and pragmatics in science and mathematics”. You can get the slides here.

Abstract: A popular view in the philosophy of set theory is that of *potentialism*: the position that the set-theoretic universe unfolds as more sets come into existence or become accessible to us. A difficult question for the potentialist is to explain how *classes* (understood as intensional entities) behave on this framework, and in particular what logic governs them. In this talk we’ll see how category-theoretic resources can be brought to bear on this issue. I’ll first give a brief introduction to topos theory, and then I’ll explain how (drawing on work of Lawvere) we can think of intensional classes for the potentialist as given by a functor category. I’ll suggest some tentative directions for research here, including the possibility that this representation indicates that the logic of intentional classes should be intuitionistic rather than classical, and that the strength of the intuitionistic logic is dependent upon the partial order on the worlds.

Talk/Slides: `Intensional Classes and Intuitionistic Topoi’ Barcelona Set Theory Seminar 9 December 2020

This will be a talk at the Barcelona Set Theory seminar on 9th December 2020. You can find the slides here.

Abstract: A popular view in the philosophy of set theory is that of potentialism: the position that the set theoretic universe unfolds as more sets come into existence. A difficult question for the potentialist is to explain how *classes* (understood as intensional entities) behave on this framework, and in particular what logic governs them. In this talk we’ll see how category-theoretic resources can be brought to bear on this issue. I’ll first give a brief introduction to topos theory, and then I’ll explain how (drawing on work of Lawvere) we can think of intensional classes for the potentialist as given by a functor category. I’ll suggest some tentative directions for research here, including the possibility that this representation indicates that the logic of intentional classes should be intuitionistic rather than classical, and that the strength of the intuitionistic logic is dependent upon the partial order on the worlds.