Tutor: Meini Cristina
Progetto: As a reaction to Russellâ€™s Paradox, set theorists have discarded the assumption that there is a domain containing all sets over which sentences of set theory are interpreted. On the other hand, set-theoretical sentences such as "all sets are self-identical" seem to presuppose a quantifier that rages over absolutely all sets there are. In my research project, I try to connect both aspects in a single, non-standard semantic framework for set theory. The idea, roughly, is to bisect the sentences of the language of set theory in problematic and unproblematic sentences, where a paradigmatic case for a problematic sentence is one postulating a Russell set, and a paradigmatic case for an unproblematic sentence is the one mentioned above, postulating the self-identity of all sets. In the next step, the unproblematic sentences are interpreted over an all-inclusive domain, whereas the problematic sentences are interpreted over an increasing sequence of relativized domains, similar to the cumulative hierarchy. The project relies on upcoming work of Lorenzo Rossi, as well as the paper "Towards a Theory of Second-Order Consequence" by AgustĂn Rayo and Garbiel Uzquiano.