A Many-Valued Generalization of the Ultrapower Construction

Abstract

D. Scott in [1969] remarked: ‘The idea of constructing Boolean-valued models could have been (but was not) discovered as a generalization of the ultraproduct method used now so often to obtain nonstandard models for ordinary analysis. Roughly, we can say that ultraproducts use the standard Boolean algebras (the power-set Boolean algebras) to obtain models elementarily equivalent to the standard model, whereas the Boolean method allows the nonstandard complete algebras (such us the Lebesgue algebra of measurable sets modulo sets of measure zero or the Baire algebra of Borel sets modulo sets of the first category.) Thus the Boolean method leads to nonstandard nonstandard models that are not only not isomorphic to the standard model but are not even equivalent. Nevertheless, they do satisfy all the usual axioms and deserve to be called models of analysis.’