Empirical set theory
type: Item
instance of: scholarly article
title: Empirical set theory
zbMATH Open document ID: 0779.03020
DOI: 10.1007/BF00675196
author: Hirokazu Nishimura
published in: International Journal of Theoretical Physics
publication date: 1993-11-30T00:00:00Z
review text: We share with \textit{D. J. Foulis} and \textit{C. H. Randall} [J. Math. Phys. 13, 1667-1675 (1972; Zbl 0287.60002) and ibid. 14, 1472-1480 (1973; Zbl 0287.60003)] the conviction that it is not orthomodular posets or the like, but manuals of operations that are of primary significance in the foundations of empirical sciences. In contrast to them we regard an operation not as a set of possible outcomes but as a complete Boolean algebra of observable events, which we adopt, following the lines of \textit{M. Davis} [Int. J. Theor. Phys. 16, 867-874 (1978; Zbl 0392.03040)] and \textit{G. Takeuti} [Proc. Int. Symp. Found. Quantum Mechanics (S. Kamefuchi et al. (ed.)) Tokyo 1983, 256-260 (1984)], as a building block of our empirical set theory. Differing from \textit{L. N. Stout} [Manuscr. Math. 28, 379-403 (1979; Zbl 0409.03039)], we prefer geometric morphisms to logical morphisms as interconnecting machinery. Thus, just as a smooth manifold is a family of open subsets of a Euclidean space interconnected by smooth mappings, our empirical set theory is, roughly speaking, a family of Scott-Solovay universes \(V^{(B)}\) over complete Boolean algebras \(B\) interconnected by geometric morphisms. Using the nomenclature of topos theory, it is a subcategory of Boolean localic toposes and geometric morphisms, where two parallel geometric morphisms are identified provided that they are naturally equivalent. Thereof observables in the sense of quantum mechanics are identified with real numbers of our set theory.
reviewed by: Hirokazu Nishimura
Mathematics Subject Classification ID: 03E70
Mathematics Subject Classification ID: 18B25
Mathematics Subject Classification ID: 81P10
zbMATH DE Number: 445103
zbMATH Keywords: complete Boolean algebra of observable events
zbMATH Keywords: geometric morphisms
zbMATH Keywords: Scott-Solovay universes
zbMATH Keywords: Boolean localic toposes
zbMATH Keywords: observables
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 445103