EMPIRICAL SET-THEORY

We share with Foulis and Randall the evangel that it is not orthomodular posets or the like, but manuals of operations that are of primary importance in the foundations of the empirical sciences. In sharp 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 Davis and of Takeuti, as a building block of our empirical set theory. Just as a smooth manifold is covered by open subsets of a Euclidean space interconnected by smooth mappings, our empirical set theory is covered by the Scott-Solovay universes V(B) over complete Boolean algebras B interconnected by geometric morphisms. Using the nomenclature of topos theory, our empirical set theory is a subcategory of the category BIop of Boolean localic toposes and geometric morphisms. It is shown that in this set theory observables can be identified with real numbers. This is the first step of formal development of Davis' ambitious program.