Formula balancing and continuously valuated models

Abstract

Uniform spaces can be Cauchy-completed; and if the base space was a first-order structure, this structure can be naturally extended to the completion. While common in algebra, this construction has been more recently used to produce new models of special set theories. We investigate here a natural way to twist the semantics of any structure according to a uniformity on its universe. We use it to relate the (classical first-order) theories of structures and dense substructures and apply it to the case of Cauchy-completions.