Book ChapterDOI
Quantum Set Theory
Gaisi Takeuti
- pp 303-322
TLDR
This paper studies set theory based on quantum logic, which is the lattice of all closed linear subspaces of a Hilbert space and shows the fact that there are many complete Boolean algebras inside quantum logic.Abstract:
In this paper, we study set theory based on quantum logic. By quantum logic, we mean the lattice of all closed linear subspaces of a Hilbert space. Since quantum logic is an intrinsic logic, i.e. the logic of the quantum world, (cf. 1) it is an important problem to develop mathematics based on quantum logic, more specifically set theory based on quantum logic. It is also a challenging problem for logicians since quantum logic is drastically different from the classical logic or the intuitionistic logic and consequently mathematics based on quantum logic is extremely difficult. On the other hand, mathematics based on quantum logic has a very rich mathematical content. This is clearly shown by the fact that there are many complete Boolean algebras inside quantum logic. For each complete Boolean algebra B, mathematics based on B has been shown by our work on Boolean valued analysis 4, 5, 6 to have rich mathematical meaning. Since mathematics based on B can be considered as a sub-theory of mathematics based on quantum logic, there is no doubt about the fact that mathematics based on quantum logic is very rich. The situation seems to be the following. Mathematics based on quantum logic is too gigantic to see through clearly.read more
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Discrete quantum structures
TL;DR: Weaver's quantum predicate logic as discussed by the authors is a generalization of the quantum propositional logic of Birkhoff and von Neumann, which provides canonical quantum generalizations for a large class of properties.
Journal ArticleDOI
The representation of Takeuti's\(\begin{array}{*{20}c} \parallel \\ \_ \\ \end{array} \)-operator
TL;DR: In this paper, the authors investigated the connection between structural properties of orthomodular lattices and the existence of two-valued homomorphisms on the lattice and showed that the degree in which the elements of a lattice behave classically corresponds to the degree to which the homomorphism behaves classically.
Journal ArticleDOI
Quantum set theory extending the standard probabilistic interpretation of quantum theory (extended abstract)
TL;DR: In this article, the quantum set theory developed by Takeuti and the present author is used to systematically extend the probabilistic interpretation of quantum theory to define the probability of equality relation for a pair of arbitrary observables.
Journal ArticleDOI
From Boolean Valued Analysis to Quantum Set Theory: Mathematical Worldview of Gaisi Takeuti
TL;DR: This paper analyzes Gaisi Takeuti’s mathematical world view underlying his program from two perspectives: set theoretical foundations of modern mathematics and extending the notion of sets to multi-valued logic.
Journal ArticleDOI
The Paradox of Classical Reasoning
TL;DR: In this paper , it was shown that classical and quantum formalisms cannot be in a hierarchical relation, that is, include one another, and this fact puts in doubt the quantum-classical correspondence and undermines the reductionist approach to the physical world.
References
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Journal ArticleDOI
The Logic of Quantum Mechanics
TL;DR: In this article, it was shown that even a complete mathematical description of a physical system S does not in general enable one to predict with certainty the result of an experiment on S, and in particular one can never predict both the position and the momentum of S, (Heisenberg's Uncertainty Principle) and most pairs of observations are incompatible, and cannot be made on S simultaneously.
Book ChapterDOI
The Logic of Quantum Mechanics
TL;DR: In this article, it was shown that even a complete mathematical description of a physical system S does not in general enable one to predict with certainty the result of an experiment on S, and that in particular one can never predict both the position and the momentum of S, (Heisenberg's Uncertainty Principle).
Journal ArticleDOI
A relativity principle in quantum mechanics
TL;DR: In this article, it is suggested that the real numbers of such a model can be taken to be self-adjoint operators which can be resolved in terms of projections belonging to the Boolean algebra and that quantum theory involves a relativity principle with Takeuti's Boolean algebras serving as reference frames.