Quantum Set Theory and Topos Quantum Theory

A Bridge Between Q-Worlds

Topos quantum theory and quantum set theory are two long running projects in the mathematical foundations of quantum mechanics that share a great deal of conceptual and technical affinity.Both approaches attempt to resolve some of the conceptual difficulties surrounding quantum mechanics by reformulating parts of the theory inside of non-classical mathematical universes, albeit with very different internal logics.We call such mathematical universes, together with those mathematical and logical structures within them that are pertinent to the physical interpretation, `q-worlds'.Here, we provide a unifying framework that allows us to (i) better understand the relationship between different q-worlds, and (ii) define a general method for transferring concepts and results between topos quantum theory and quantum set theory, thereby significantly increasing the expressive power of both approaches.Along the way, we develop a novel connection to paraconsistent logic and introduce a new class of structures that have significant implications for recent work on paraconsistent set theory.