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.


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.

Paper: Inner-Model Reflection Principles

Last year I had pleasure of working on a project with some fantastic mathematicians (Andrés Eduardo Caicedo, Gunter Fuchs, Joel Hamkins, Jonas Reitz, and Ralf Schindler) concerning certain kinds of principles which state that formulas in set theory hold in substructures of the universe. (For the cognoscenti: Analogues of Lévy-Montague Reflection, but for inner models.) I think it’s quite interesting, since the notion of how absoluteness between V and its substructures plays out (both with respect to height and width) is something that needs more work, and this represented a good step on here. The paper is now out, and can be found here.


We introduce and consider the inner-model reflection principle, which asserts that whenever a statement \phi(a) in the first-order language of set theory is true in the set-theoretic universe V, then it is also true in a proper inner model W \subsetneq V. A stronger principle, the ground-model reflection principle, asserts that any such \phi(a) true in V is also true in some non-trivial ground model of the universe with respect to set forcing. These principles each express a form of width reflection in contrast to the usual height reflection of the Lévy–Montague reflection theorem. They are each equiconsistent with ZFC and indeed \Pi_2-conservative over ZFC, being forceable by class forcing while preserving any desired rank-initial segment of the universe. Furthermore, the inner-model reflection principle is a consequence of the existence of sufficient large cardinals, and lightface formulations of the reflection principles follow from the maximality principle MP and from the inner-model hypothesis IMH. We also consider some questions concerning the expressibility of the principles.