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Book ChapterDOI

Quantum Set Theory

01 Jan 1981-pp 303-322
TL;DR: 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.
Citations
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Book ChapterDOI
TL;DR: In this paper, it was shown that constructing a theory of physics is equivalent to finding a representation in a topos of a certain formal language that is attached to the system. But the problem of quantum topos is different from that of quantum quantum physics.
Abstract: The goal of this paper is to summarise the first steps in developing a fundamentally new way of constructing theories of physics. The motivation comes from a desire to address certain deep issues that arise when contemplating quantum theories of space and time. In doing so we provide a new answer to Heidegger's timeless question ``What is a thing?''. Our basic contention is that constructing a theory of physics is equivalent to finding a representation in a topos of a certain formal language that is attached to the system. Classical physics uses the topos of sets. Other theories involve a different topos. For the types of theory discussed in this paper, a key goal is to represent any physical quantity $A$ with an arrow $\breve{A}_\phi:\Si_\phi\map\R_\phi$ where $\Si_\phi$ and $\R_\phi$ are two special objects (the `state-object' and `quantity-value object') in the appropriate topos, $\tau_\phi$. We discuss two different types of language that can be attached to a system, $S$. The first, $\PL{S}$, is a propositional language; the second, $Ł{S}$, is a higher-order, typed language. Both languages provide deductive systems with an intuitionistic logic. With the aid of $\PL{S}$ we expand and develop some of the earlier work (By CJI and collaborators.) on topos theory and quantum physics. A key step is a process we term `daseinisation' by which a projection operator is mapped to a sub-object of the spectral presheaf $\Sig$--the topos quantum analogue of a classical state space. The topos concerned is $\SetH{}$: the category of contravariant set-valued functors on the category (partially ordered set) $\V{}$ of commutative sub-algebras of the algebra of bounded operators on the quantum Hilbert space $\Hi$.

152 citations


Cites background from "Quantum Set Theory"

  • ...For let P and Q be propositions, represented by the subsets SP and (24)If the distributive law is dropped we could move towards the quantum-set ideas of [71]; or, perhaps, the ideas of non-commutative geometry instigated by Alain Connes [14]....

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  • ...For example, Baez’s advocation of n-categories [5, 6]; ‘categorial quantum theory’ [2, 73]; Takeuti’s theory(19) of ‘quantum sets’ [71]; and Crane’s work on categorial models of space-time [17]....

    [...]

Book ChapterDOI
TL;DR: In this article, the authors summarise the first steps in developing a fundamentally new way of constructing theories of physics and provide a new answer to Heidegger's timeless question "What is a thing?"
Abstract: The goal of this article is to summarise the first steps in developing a fundamentally new way of constructing theories of physics. The motivation comes from a desire to address certain deep issues that arise when contemplating quantum theories of space and time. In doing so we provide a new answer to Heidegger’s timeless question “What is a thing?”.

129 citations

Journal ArticleDOI
TL;DR: The author hopes that this paper will be a useful map for AI researchers who are going to explore further and deeper connections between AI and quantum computation as well as quantum theory although some parts of the map are very rough and other parts are empty, and waiting for the readers to fill in.

81 citations


Cites methods from "Quantum Set Theory"

  • ...This allows us to use rich mathematical methods developed in Birkhoff-von Neumann quantum logic [17] and Takeuti’s quantum set theory [96]....

    [...]

Journal ArticleDOI
TL;DR: The notion of orthomodular lattice-valued (quantum) automaton is introduced and the Kleene theorem about equivalence of regular expressions and finite automata is generalized into quantum logic.

52 citations


Cites background or methods from "Quantum Set Theory"

  • ...Furthermore, Titani and Kozawa [74] provided a representation of unitary operators by complex numbers in V P (H)....

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  • ...Thus, a potential way of establishing link between these quantum automata and our orthomodular lattice-valued automata is to use Takeuti, Titani and Kozawa’s representation of observables and unitary operators by real and complex numbers in the universe V P (H) of quantum set theory, where P (H) is the orthomodular lattice of closed linear subspaces of a Hilbert space H [71, 74]....

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  • ...Kozawa [74] with a strong implication corresponding to the order in the lattice of truth values....

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Journal ArticleDOI
TL;DR: In this paper, the authors present a bibliography containing 1851 references on axiomatic structures underlying quantum mechanics, with stress on varieties of algebraico-logical, probabilistic, and operational structures for which the term quantum logics is adopted.
Abstract: The bibliography contains 1851 references on axiomatic structures underlying quantum mechanics, with stress on varieties of algebraico-logical, probabilistic, and operational structures for which the term quantum logics is adopted. An index of about 250 keywords picked out from the titles is included and statistics about papers, journals, and authors are presented. Monographs and proceedings on the subject are noted.

49 citations

References
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Journal ArticleDOI
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.
Abstract: One of the aspects of quantum theory which has attracted the most general attention, is the novelty of the logical notions which it presupposes It asserts 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 with certainty both the position and the momentum of S, (Heisenberg’s Uncertainty Principle) It further asserts that most pairs of observations are incompatible, and cannot be made on S, simultaneously (Principle of Non-commutativity of Observations)

2,315 citations

Book ChapterDOI
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).
Abstract: One of the aspects of quantum theory which has attracted the most general attention, is the novelty of the logical notions which it presupposes. It asserts 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 with certainty both the position and the momentum of S, (Heisenberg’s Uncertainty Principle). It further asserts that most pairs of observations are incompatible, and cannot be made on S, simultaneously (Principle of Non-commutativity of Observations).

1,289 citations

Book
21 Aug 1978

156 citations

Journal ArticleDOI
Martin Davis1
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.
Abstract: Takeuti has studied models of axiomatic set theory in which the “truth values” are elements of a complete Boolean algebra of projections on closed subspaces of a Hilbert space, and has found 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. It is suggested that this is the mathematical source of the replacement of real quantities by operators in quantizing a classical description, and that quantum theory involves a relativity principle with Takeuti's Boolean algebras serving as reference “frames.”

92 citations