A Bridge Between Q-Worlds

Andreas Döring, Benjamin Eva, Masanao Ozawa

Quantum set theory (QST) and topos quantum theory (TQT) are two long running
projects in the mathematical foundations of quantum mechanics that share a
great deal of conceptual and technical affinity. Most pertinently, 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 TQT and QST,
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.