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A BRIDGE BETWEEN Q-WORLDS

Part of: General logic Model theory Set theory Special categories

Published online by Cambridge University Press:  02 July 2020

ANDREAS DÖRING ,
BENJAMIN EVA  and
MASANAO OZAWA
ANDREAS DÖRING
Affiliation:
INDEPENDENT SCHOLAR E-mail: Andreas.doering@posteo.de
BENJAMIN EVA
Affiliation:
DEPARTMENT OF PHILOSOPHY DUKE UNIVERSITYDURHAM, NC27708, USAE-mail: Benjamin.eva@duke.edu
MASANAO OZAWA
Affiliation:
NAGOYA UNIVERSITY CHIKUSA-KU, NAGOYA464-8601, JAPAN and CHUBU UNIVERSITY 1200 MATSUMOTO-CHO, KASUGAI 487-8501, JAPANE-mail: ozawa@is.nagoya-u.ac.jp
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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.

Keywords

quantum logicquantum set theorytopos quantum theoryquantum foundationsparaconsistent logic

MSC classification

Primary: 81P10: Logical foundations of quantum mechanics; quantum logic
Secondary: 03C90: Nonclassical models (Boolean-valued, sheaf, etc.) 18B25: Topoi 03B53: Paraconsistent logics 03E70: Nonclassical and second-order set theories
Type
Research Article
Information
The Review of Symbolic Logic , Volume 14 , Issue 2 , June 2021 , pp. 447 - 486
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Copyright
© Association for Symbolic Logic, 2020

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