Empirical set theory
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.
complete Boolean algebra of observable events
scientific article; zbMATH DE number 445103
scientific article