EMPIRICAL SETS

The pillar concept of Foulis and Randall's school is surely that of a manual of operations. They chose to regard an operation as the set of possible outcomes, thereby taking a manual of operations to be a family of partially overlapping operations. Our previous work is a development of their ideas in two points. First, each operation is represented not by the set of possible outcomes, but by the complete Boolean algebra of observable events. Second, since each complete Boolean algebra B possesses the Scott-Solovay model V-(B) of classical set theory as its higher-order companion, the Scott-Solovay universes of all the operations in the manual lump together into a family of Boolean set theories interconnected by geometric morphisms, which we suggestively designated an empirical set theory. The principal concern of this paper is to show how to get a cross-operational set concept by choosing an internal set within V-(B) for each operation B in the manual and bundling them up. The resulting structure is denominated an empirical set. We show that the category of empirical sets is complete, is cocomplete, has a subobject classifier for well-rounded subobjects, and has exponentials only for degraded objects.