# A Bridge Between Q-Worlds

## TL;DR

This paper unifies quantum set theory and topos quantum theory within a common framework, enhancing understanding of their relationship and enabling transfer of concepts, with implications for paraconsistent logic and set theory.

## Contribution

It introduces a unifying framework connecting QST and TQT, allowing transfer of concepts and results, and explores new structures related to paraconsistent logic.

## Key findings

- Established a formal connection between QST and TQT
- Developed a method for transferring concepts between the two approaches
- Introduced new structures impacting paraconsistent set theory

## Abstract

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.