Quantum set theory (QST) and topos quantum theory (TQT) are two long running projects in the mathematical foundations of quantum mechanics (QM) that share a great deal of conceptual and technical affinity. Most pertinently, both approaches attempt to resolve some of the conceptual difficulties surrounding QM by reformulating parts of the theory inside of nonclassical 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.

中文翻译：

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Q 世界之间的桥梁

量子集合论 (QST) 和拓扑量子理论 (TQT) 是量子力学 (QM) 数学基础中两个长期运行的项目，它们在概念和技术上具有很大的相似性。最相关的是，这两种方法都试图通过在非经典数学世界中重新表述部分理论来解决围绕 QM 的一些概念困难，尽管内部逻辑非常不同。我们将这样的数学宇宙，连同其中与物理解释相关的数学和逻辑结构，称为“Q 世界”。在这里，我们提供了一个统一的框架，使我们能够 (i) 更好地理解不同 Q 世界之间的关系，以及 (ii) 定义在 TQT 和 QST 之间传递概念和结果的通用方法，从而显着提高这两种方法的表达能力。在此过程中，我们开发了一种与副一致逻辑的新联系，并引入了一类新的结构，这些结构对最近关于副一致集理论的工作具有重要意义。