Review of Symbolic Logic 10 (4):782-807 (2017)

Abstract
In 1981, Takeuti introduced quantum set theory by constructing a model of set theory based on quantum logic represented by the lattice of closed linear subspaces of a Hilbert space in a manner analogous to Boolean-valued models of set theory, and showed that appropriate counterparts of the axioms of Zermelo–Fraenkel set theory with the axiom of choice hold in the model. In this paper, we aim at unifying Takeuti’s model with Boolean-valued models by constructing models based on general complete orthomodular lattices, and generalizing the transfer principle in Boolean-valued models, which asserts that every theorem in ZFC set theory holds in the models, to a general form holding in every orthomodular-valued model. One of the central problems in this program is the well-known arbitrariness in choosing a binary operation for implication. To clarify what properties are required to obtain the generalized transfer principle, we introduce a class of binary operations extending the implication on Boolean logic, called generalized implications, including even nonpolynomially definable operations. We study the properties of those operations in detail and show that all of them admit the generalized transfer principle. Moreover, we determine all the polynomially definable operations for which the generalized transfer principle holds. This result allows us to abandon the Sasaki arrow originally assumed for Takeuti’s model and leads to a much more flexible approach to quantum set theory.
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DOI 10.1017/s1755020317000120
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References found in this work BETA

Set Theory and the Continuum Hypothesis.Kenneth Kunen - 1966 - Journal of Symbolic Logic 35 (4):591-592.
An Axiom System for the Modular Logic.Jerzy Kotas - 1967 - Studia Logica 21 (1):17 - 38.
The Principles of Quantum Mechanics.P. A. M. Dirac - 1936 - Revue de Métaphysique et de Morale 43 (2):5-5.
Transfer Principle in Quantum Set Theory.Masanao Ozawa - 2007 - Journal of Symbolic Logic 72 (2):625 - 648.
Implication Connectives in Orthomodular Lattices.L. Herman, E. L. Marsden & R. Piziak - 1975 - Notre Dame Journal of Formal Logic 16 (3):305-328.

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Citations of this work BETA

Generalized Algebra-Valued Models of Set Theory.Benedikt Löwe & Sourav Tarafder - 2015 - Review of Symbolic Logic 8 (1):192-205.
A Bridge Between Q-Worlds.Benjamin Eva, Masanao Ozawa & Andreas Doering - forthcoming - Review of Symbolic Logic:1-40.
Quantum Set Theory: Transfer Principle and De Morgan's Laws.Masanao Ozawa - 2021 - Annals of Pure and Applied Logic 172 (4):102938.

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