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
Citations
More filters
Posted Content
Twist-Valued Models for Three-valued Paraconsistent Set Theory
TL;DR: It is argued that the implication operator of LPT0 is more suitable for a paraconsistent set theory than the implication of PS3, since it allows for genuinely inconsistent sets w such that [(w = w)] = 1/2 .
Journal ArticleDOI
Set Theoretical Forcing in Quantum Mechanics and AdS/CFT Correspondence
TL;DR: In this article, a connection of Set Theoretical Forcing with Quantum Mechanical lattice of projections over some separable Hilbert space is made, where the basic ingredient of the construction is the rule of indistinguishability of Standard and some Nonstandard models of Peano Arithmetic.
Journal ArticleDOI
A Bridge Between Q-Worlds
TL;DR: In this paper, the authors provide a unifying framework that allows us to better understand the relationship between different Q-worlds, and define a general method for transferring concepts and results between TQT and QST, thereby significantly increasing the expressive power of both approaches.
Book ChapterDOI
The History of Quantum Logic
TL;DR: This chapter discusses the Birkhoff-von Neumann idea of quantum logic, which is related to deep mathematical discoveries in the mid thirties, to the history of quantum mechanics in the twenties, and to conceptual difficulties in connection with the frequency-interpretation of quantum probability.
Journal ArticleDOI
Orthomodular-Valued Models for Quantum Set Theory
TL;DR: This paper unifies 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, to a general form holding in every orthodular-valued model.
References
More filters
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.