(forthcoming) Hume’s principle, bad company, and the axiom of choice (with Stewart Shapiro). The Review of Symbolic Logic. Penultimate draft: PDF.
This paper develops an account of pluralities based on the following simple claim: some things are nothing over and above the individual things they comprise. For some, this may seem like a mysterious statement, perhaps even meaningless; for others, like a truism, trivial and inferentially inert. I show that neither reaction is correct: the claim is both tractable and has important consequences for a number of debates in philosophy.
This note proves a new conservativity result for class theories. It tells us that as long as our set theory T contains an independently well-motivated reflection principle, anything provable about the sets in any reasonable class theory extending T is already provable in T itself.
This paper investigates the assumptions underlying modal structuralism, and looks at the prospects for supplementing them with a reflection principle. It shows that the viability of modal structuralism about set theory turns on a non-trivial assumption — the Stability principle — about the behaviour of structures across modal space. Once this assumption is accepted, however, I show that the modal structuralist can make sense of a significant fragment of set theory. The axiom schema of Replacement requires further assumptions, though, and I show that a recent proposal to use reflection principles to obtain it fails.
This paper introduces a new reflection principle. It says that whatever is true in all entities of some kind is also true in a small collection of them. When applied to sets and classes, it turns out to be remarkably strong (implying that there are so-called 1-extendible cardinals).
Potentialism is the view that the universe of mathematics is inherently potential. It comes in two main flavours: height-potentialism and width-potentialism. It is natural to think that height and width potentialism are just aspects of a broader phenomenon of potentialism, that they might both be true. The main result of this paper is that this is mistaken: height and width potentialism are jointly inconsistent. Indeed, I will argue that height potentialism is independently committed to an ultimate background universe of sets, an ultimate V, up to its height.