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.

Talk/Slides: Intensional Classes and Intuitionistic Topoi

This will be a talk at UNICAMP on 8th December. 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.

Talk: Countabilism and Maximality

This was a talk at Speaking the Unspeakable: Paradoxes between Truth and Proof at the University of Campinas, Brazil. You can find the slides here.

Abstract:

It is standard in set theory to assume that Cantor’s Theorem establishes that there are uncountable sets. In this paper, we present versions of set theory with classes that imply that every set is countable, and the continuum is a proper class. Within these theories we show how standard set theories (including ZFC with large cardinals added) can be incorporated. We discuss some properties of the theories, in particular that they provide a radically new perspective on the notion of maximality. We conclude that the systems considered raise questions concerning the foundational purpose of set theory.