Davide Sutto

This dissertation explores the iterative conception of set from a twofold perspective. In the first part I run an extensive historical study of the conception divided into three papers. On a methodological standpoint, this inquiry employs the partition between pre-history and history---recently adopted in other historical inquires in the foundation of mathematics---to outline a continuum with the origins of the conception in the early developments of axiomatic set theory. Therefore, the first two papers present the pre-history of the conception, divided into two branches, and the third its history: Paper I describes the emergence of the notion of cumulative rank in set theory and the subsequent development of the cumulative hierarchy of sets; Paper II examines the process of adoption of the Axiom of Foundation, the crucial principle that frames the universe of sets as the cumulative hierarchy; Paper III investigates the different ways in which the iterative conception unfolds as \textit{the} axiomatic treatment of the hierarchy. The second part of the dissertation deals with contemporary approaches to the iterative conception. In Paper IV (Sutto, 2026) I sketch a taxonomy of set-theoretic potentialism, the view according to which the development of the cumulative hierarchy captured by the conception is characterized by an intrinsic open-endedness. In the last two papers I develop a new axiomatic approach to the conception that combines the reading of the process of set-formation as an abstraction from pluralities with Tim Button's recent work on theories of levels. In Paper V (Sutto, 2025) I propose a plural theory of levels that does not give up on the principle of unrestricted plural comprehension and preserves Gabriel Uzquiano's reading of pluralities as (proper) classes. In Paper VI I use the same tool to ground the critical view of pluralities and sets advocated by Salvatore Florio and Øystein Linnebo with its own plural iterative conception, which captures the idea that every plurality forms a set and exploits it to pin down the real object of contention between actualism and potentialism.

Set-theoretic potentialism is one of the most lively trends in the philosophy of mathematics. Modal accounts of sets have been developed in two different ways. The first, initiated by Charles Parsons, focuses on sets as objects. The second, dating back to Hilary Putnam and Geoffrey Hellman, investigates set-theoretic structures. The paper identifies two strands of open issues, technical and conceptual, to clarify these two different, yet often conflated, views and categorize the potentialist approaches that have emerged in the contemporary debate. The final outcome is a taxonomy that should help researchers navigate the rich landscape of modal set theories.

Georg Cantor informally distinguished between “consistent” and “inconsistent” multiplicities as those many things that, respectively, can and cannot be thought of as one, i.e., as a set. To clarify this distinction, the recent debate filtered the logic of plurals through two main approaches to the process of set-formation: limitation of size (Burgess) or set-theoretic potentialism (Linnebo). In this paper I propose a third route through the development of a plural iterative conception of set. Inspired by Tim Button’s Level Theory, I define and axiomatize the notion of a plural level, which explains Cantor’s multiplicities either as level-bound (consistent) or level unbound (inconsistent) pluralities. While this framework is clearly in contrast with the limitation of size view, it also revives a plausible actualist framework prematurely dismissed by the advocates of potentialism.

Set-theoretic potentialism is one of the most lively trends in the philosophy of mathematics. Modal accounts of sets have been developed in two different ways. The first, initiated by Charles Parsons, focuses on sets as objects. The second, dating back to Hilary Putnam and Geoffrey Hellman, investigates set-theoretic structures. The paper identifies two strands of open issues, technical and conceptual, to clarify these two different, yet often conflated, views and categorize the potentialist approaches that have emerged in the contemporary debate. The final outcome is a taxonomy that should help researchers navigate the rich landscape of modal set theories.