Towards a Paraconsistent Quantum Set Theory
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TLDR
In this paper, a connection between quantum set theory and topos quantum theory was established by Ozawa, Takeuti and Titani, who studied algebraic valued set-theoretic structures whose truth values correspond to clopen subobjects of the spectral presheaf of an orthomodular lattice of projections onto a given Hilbert space.Abstract:
In this paper, we will attempt to establish a connection between quantum set theory, as developed by Ozawa, Takeuti and Titani, and topos quantum theory, as developed by Isham, Butterfield and Doring, amongst others. Towards this end, we will study algebraic valued set-theoretic structures whose truth values correspond to the clopen subobjects of the spectral presheaf of an orthomodular lattice of projections onto a given Hilbert space. In particular, we will attempt to recreate, in these new structures, Takeuti's original isomorphism between the set of all Dedekind real numbers in a suitably constructed model of set theory and the set of all self adjoint operators on a chosen Hilbert space.read more
Citations
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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
A Generalisation of Stone Duality to Orthomodular Lattices
Sarah Cannon,Andreas Döring +1 more
TL;DR: In this article, it was shown that the assignment of a complete orthomodular lattice L to a spectral presheaf is contravariant functorial, and that the clopen subobjects form a complete bi-Heyting algebra.
Journal ArticleDOI
A Bridge between Q-Worlds
TL;DR: A unifying framework is provided that allows to better understand the relationship between different Q-worlds, and a general method for transferring concepts and results between TQT and QST is defined, thereby significantly increasing the expressive power of both approaches.
Journal ArticleDOI
Quantum set theory: Transfer Principle and De Morgan's Laws
Masanao Ozawa,Masanao Ozawa +1 more
TL;DR: In this article, a new truth value assignment for bounded quantifiers that satisfies De Morgan's Laws was proposed, and it was shown that this assignment has a lower bound determined by the commutator, a projection-valued degree of commutativity, of constants in the formula.
References
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Journal ArticleDOI
Transfinite numbers in paraconsistent set theory
TL;DR: This paper begins an axiomatic development of naive set theory—the consequences of a full comprehension principle—in a paraconsistent logic, and indicates how later developments of cardinal numbers will lead to Cantor’s theorem, the existence of large cardinals, and a counterexample to the continuum hypothesis.
Journal ArticleDOI
Transfer principle in quantum set theory
TL;DR: In this article, a quantum set theory based on quantum logic represented by the lattice of closed subspaces in a Hilbert space is introduced and a transfer principle is established that enables us to transfer theorems of ZFC to their quantum counterparts holding in the model.
Journal ArticleDOI
A lattice-valued set theory
TL;DR: In this paper, a lattice-valued set theory is formulated by introducing the logical implication $\to$ which represents the order relation on the lattice.
Book ChapterDOI
Topos-based logic for quantum systems and bi-Heyting algebras
TL;DR: In this article, the authors associate a complete bi-Heyting algebra to each quantum system, described by a von Neumann algebra of physical quantities, with contextualised propositions about the values of the physical quantities of the quantum system.
Journal ArticleDOI
Self-adjoint operators as functions i: lattices, galois connections, and the spectral order
Andreas Döring,Barry Dewitt +1 more
TL;DR: In this article, the authors developed a new perspective on quantum observables and showed that observables of a quantum system, described by self-adjoint operators in a von Neumann algebra or affiliated with it in the unbounded case, form a conditionally complete lattice when equipped with spectral order.