The overall goal of this project is to reach a better understanding of higher order concepts in mathematics, with applications to computer science, physics, and philosophy. The main innovation is to use so-called team semantics, developed by the PI in his earlier work. A team is essentially a relation, a collection of tuples, on a fixed domain. A tuple can be thought of as a function from a fixed finite set of attributes to the domain. If the attributes are e.g. “name”, “rank” and “salary”, then examples of possible tuples are e.g. (Jones, typist, 50.000) (Brown, director, 120.000) and (Taylor, secretary, 50.000). By studying dependences and independence in such teams of tuples a surprising amount of information can be unearthed. Technically speaking anything that is existential second order, e.g. functional dependence, can be expressed. This turns out to be useful in applications to database theory. On the other hand, teams offer a technical tool to address many questions involving plurality, e.g. non-locality phenomena in Quantum Mechanics, or independence phenomena in the multiverse of set theory. More exactly the goal is to axiomatize and find tractable fragments of existential second order, which itself is seriously non-axiomatizable (Pi-3-complete) and intractable (NO-complete). On the plurality side the goal is to develop a new understanding of inner models of set theory. The importance of this project to society stems from the ubiquity of the concepts of dependence and independence in society. In this project we develop a philosophical and mathematical theory of dependence and independence concepts that covers applications from database theory to Quantum Foundations and to the multiverse of set theory.
