Layering Everything and More. A Compact Study of the Iterative Conception of Set

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.