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  1.  67
    Transfer Principle in Quantum Set Theory.Masanao Ozawa - 2007 - Journal of Symbolic Logic 72 (2):625 - 648.
    In 1981, Takeuti introduced quantum set theory as the quantum counterpart of Boolean valued models of set theory by constructing a model of set theory based on quantum logic represented by the lattice of closed subspaces in a Hilbert space and showed that appropriate quantum counterparts of ZFC axioms hold in the model. Here, Takeuti's formulation is extended to construct a model of set theory based on the logic represented by the lattice of projections in an arbitrary von Neumann algebra. (...)
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  2.  58
    Quantum Reality and Measurement: A Quantum Logical Approach.Masanao Ozawa - 2011 - Foundations of Physics 41 (3):592-607.
    The recently established universal uncertainty principle revealed that two nowhere commuting observables can be measured simultaneously in some state, whereas they have no joint probability distribution in any state. Thus, one measuring apparatus can simultaneously measure two observables that have no simultaneous reality. In order to reconcile this discrepancy, an approach based on quantum logic is proposed to establish the relation between quantum reality and measurement. We provide a language speaking of values of observables independent of measurement based on quantum (...)
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  3.  86
    Reconstructing Bohr’s Reply to EPR in Algebraic Quantum Theory.Masanao Ozawa & Yuichiro Kitajima - 2012 - Foundations of Physics 42 (4):475-487.
    Halvorson and Clifton have given a mathematical reconstruction of Bohr’s reply to Einstein, Podolsky and Rosen (EPR), and argued that this reply is dictated by the two requirements of classicality and objectivity for the description of experimental data, by proving consistency between their objectivity requirement and a contextualized version of the EPR reality criterion which had been introduced by Howard in his earlier analysis of Bohr’s reply. In the present paper, we generalize the above consistency theorem, with a rather elementary (...)
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  4.  9
    Forcing in Nonstandard Analysis.Masanao Ozawa - 1994 - Annals of Pure and Applied Logic 68 (3):263-297.
    A nonstandard universe is constructed from a superstructure in a Boolean-valued model of set theory. This provides a new framework of nonstandard analysis with which methods of forcing are incorporated naturally. Various new principles in this framework are provided together with the following applications: An example of an 1-saturated Boolean ultrapower of the real number field which is not Scott complete is constructed. Infinitesimal analysis based on the generic extension of the hyperreal numbers is provided, and the hull completeness theorem (...)
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  5.  51
    Scott Incomplete Boolean Ultrapowers of the Real Line.Masanao Ozawa - 1995 - Journal of Symbolic Logic 60 (1):160-171.
    An ordered field is said to be Scott complete iff it is complete with respect to its uniform structure. Zakon has asked whether nonstandard real lines are Scott complete. We prove in ZFC that for any complete Boolean algebra B which is not (ω, 2)-distributive there is an ultrafilter U of B such that the Boolean ultrapower of the real line modulo U is not Scott complete. We also show how forcing in set theory gives rise to examples of Boolean (...)
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  6.  10
    A Bridge Between Q-Worlds.Benjamin Eva, Masanao Ozawa & Andreas Doering - forthcoming - Review of Symbolic Logic:1-40.
    Quantum set theory and topos quantum theory 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 (...)
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  7.  14
    Statistical Inference and Quantum Measurement.Masanao Ozawa - 1989 - Annals of the Japan Association for Philosophy of Science 7 (4):185-194.
  8.  9
    Quantum State Reduction and the Repeatability Hypothesis.Masanao Ozawa - 2003 - Annals of the Japan Association for Philosophy of Science 11 (2):107-121.
  9.  3
    Set Theory and Lesniewski's Ontology.Masanao Ozawa & Toshiharu Waragai - 1985 - Annals of the Japan Association for Philosophy of Science 6 (5):261-272.