A BRIDGE BETWEEN Q-WORLDS. Issue 2 (2nd June 2021)

• Record Type: Journal Article
• Authors: DÖRING, ANDREAS; EVA, BENJAMIN; OZAWA, MASANAO
• Is Part Of: Review of symbolic logic. Volume 14:Issue 2(2021)
• Page Start: 447
• Page End: 486
• Publication Date: 2021-06-02
• Subjects: 81P10 -- 03C90 -- 18B25 -- 03B53 -- 03E70; quantum logic -- quantum set theory -- topos quantum theory -- quantum foundations -- paraconsistent logic; Logic, Symbolic and mathematical -- Periodicals; 511.3
• DOI: 10.1017/S1755020319000492
• Languages: English
• ISSNs: 1755-0203

Abstract

Abstract: 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.