On Modal Set Theory. Three Routes to the Iterative Conception ​ (original) (raw)

The thesis presents an assessment of the three main theories available in the contemporary debate over the modal profile intrinsic to the Iterative Conception of Set. The outcome of the research offers a prospect of what is, according to my analysis, the best route to the potential hierarchy of sets. The work starts with a preliminary historical overview of the path the Iterative Conception (Ch.0), before turning to the contemporary discussion . Each theory is then presented according to three criteria: the informal profile, the formal regimentation, and the cost-benefit analysis. The first proposal considered is Øystein Linnebo’s modal set theory, the very first contemporary view that combines set theory and modal logic in a satisfactory way, via a framework that resorts to plural resources (Ch. 1). The second account is due to James Studd and it basically amounts to an improvement over Linnebo’s theory by capturing the potential hierarchy of sets via a bimodal approach (Ch. 2). The third and most recent account is due to Tim Button, who champions a view, originally stated by Hilary Putnam, according to which the modal and the non-modal accounts of set theory are, in fact, equivalent (Ch. 3). Finally, I offer my own general assessment, taking a stand, contra Button, in favor of the ultimate irreducibility of modal set theory to its non-modal counterpart, and, contra Studd, in favor of Linnebo’s approach (Ch. 4). However, some space for an improvement towards a more viable bimodal enhancement is left over among the possible extensions of the debate. ​