Set theory - the study of collections of objects, which in turn are treated as mathematical objects of their own, can serve as a foundation for all of mathematics. Using the basic axioms governing the behavior of sets, we can prove the existence of practically any known mathematical object. A model of these axioms is a "universe" of sets. However, there is no unique universe of sets - various distinct universes can be constructed. This project examines the ways we can use logic, and in particular "non-standard" logic, to investigate these universes. We focus on two paths, shedding light on the topic from different directions: First, we use modal logic, used to model notions of possibility and necessity, to learn about the interactions between universes which are constructed from one another via the method of "forcing". Second, we use strong logics - extensions of first order logic - to construct smaller universes, and utilize the logic to investigate the results.

2026

2030