Modal Structuralism and Reflection

Abstract

Modal structuralism promises an interpretation of set theory that avoids commitment to abstracta. This article investigates its underlying assumptions. In the first part, I start by highlighting some shortcomings of the standard axiomatisation of modal structuralism, and propose a new axiomatisation I call MSST (for Modal Structural Set Theory). The main theorem is that MSST interprets exactly Zermelo set theory plus the claim that every set is in some inaccessible rank of the cumulative hierarchy. In the second part of the article, I look at the prospects for supplementing MSST with a modal structural reflection principle, as suggested in Hellman (2015). I show that Hellman’s principle is inconsistent (Theorem 5.32), and argue that modal structural reflection principles in general are either incompatible with modal structuralism or extremely weak.

Iterative Conceptions of Set

“…Forcing axioms can pop in various ways, either by admitting too many parameters, allowing too many kinds of forcing, or not keeping a tight enough control on the sentences allowed (see [Bagaria, 2005]). Standard reflection principles blow up at the level of third-order reflection (see [Reinhardt, 1974] and [Koellner, 2009]) and modal reflection principles are pretty flammable too (see [Roberts, 2019]).…”