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).
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 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.