Maria Beatrice Buonaguidi
A new foundation for mathematics. Naïve set theory in HYPE
Set theory, the theory studying collections in mathematics, is widely regarded to be the foundational framework for all of mathematical knowledge. However, the original formulation of the theory, despite having a strong philosophical justification, was susceptible to paradoxes, both philosophical and mathematical in nature, which can be traced back to the presence of intuitively justified mathematical entities behaving in a viciously circular way. A standard way to avoid the paradoxes is based on a theory which bans circularity: however, in many mathematical settings, this very circularity seems desirable, and an intrinsic feature of languages. The aim of this project is to build a theory which restores the lost circularity and philosophical intuitiveness, while being paradox-free. To do so, we will need to modify the system of reasoning which underlies the theory, and adopt a so-called non-classical logic. We choose to study non-classical logics with a strong conditional, which have the advantage of being relatively flexible to deal with paradoxes, while maintaining a certain mathematical strength. One example of such logic is HYPE, developed by Hannes Leitgeb in 2019. HYPE is presented as a hyperintensional logic, i.e. a system of reasoning which is suitable to deal with fine-grained logical contexts, such as contexts which deal with properties or belief. Using HYPE to develop a set theory is interesting for two reasons: firstly, it allows us to bring back the original, or naïve notion of set, which views collections as extensions of concepts, in a consistent way. Secondly, while having a relatively well-behaved consequence relation, it has a very flexible semantics, which allows us to model circular entities easily. The project will be devoted to finding new, mathematically viable solutions to the paradoxes of set theory by finding new alternative set theories with a sufficient mathematical content. The focus of the project will be mostly semantical in nature: we will study different theories based on their models, to investigate how they model circular entities, starting from a semantical treatment of a set theory based on HYPE. The aim of the project is to find a theory with a good balance between non-classicality and strength, and to investigate the advantages of employing logics with strong conditionals in set theory. This study, conducted with rigorous mathematical techniques, will shed light on longstanding and deep questions in the philosophy of mathematics, such as the status of circularity, and, if successful, will deliver a new framework which will be of interest to mathematicians, computer scientists and foundationally-minded scientists.