Stone duality for first order logic

Abstract

Publisher Summary This chapter presents a general theory on the relationship of syntax and semantics of first-order logic. The theory has a close-formal relationship to the Stone-duality theory for Boolean algebras. It subsumes the Godel completeness theorem that occupies a place in it and is analogous to that of the Stone representation inside Stone duality. The theory makes an essential use of ultraproducts. The theory is formulated in the language of category theory. Categories appear in three ways: (1) (first order) theories themselves are made into categories (pretoposes), (2) the collection of models of a fixed theory is made into a category and endowed with an additional structure derived from ultraproducts, resulting in the “ultracategory” of models, and (3) pretoposes on the one hand, and ultracategories on the other, are organized into categories (actually: two categories), and the main result in its final form is stated, in terms of a comparison between these two categories.