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.

Talk: Relativism and Metalogic, or; Are the Natural Numbers Indeterminate?

This will be a talk in the Logik Café at the University of Vienna Department of Philosophy. Slides here.

Abstract: It is commonplace to assume that if our discourse involving natural numbers and arithmetic is coherent at all, then it determines a unique structure (up to isomorphism). Recently, however, some mathematicians have challenged this assumption (especially the “New York School” of set theorists), arguing that the independence phenomenon in mathematics suggests that our talk about natural numbers is, in fact, indeterminate. In particular, an assumption of this view is that only statements that can be given first-order formulation have determinate content. In this paper we argue that such a view runs the risk of being self-undermining. We will suggest that modifying the view to include a schematic commitment to *feasibly checkable* fragments of set theory and mathematics as interpreted in these fragments yields a coherent conception of mathematics. We will, however, suggest that the view comes at the price of significant metamathematical reinterpretation, with several open philosophical and technical questions. This is joint work with Daniel Waxman.