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| In 1981, Takeuti introduced quantum set theory as a 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. In 2007, we extended the Takeuti formulation to construct a model of set theory based on the logic represented by the lattice of projections in an arbitrary von Neumann algebra and established a transfer principle that modifies every theorem of ZFC to a true statement for the model. Here, we discuss the following problem: In what model every theorem of ZFC holds with probability one in a given state. We call such a model as a beable universe. We determine all beable universes maximal under the condition that it contains a given observable and it is implicitly defined by the given state and observables. We also discuss the relation between those notions and a modal interpretation of quantum mechanics developed by Bub, Clifton, and Halvorson. |
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