Boolean Sentence Algebras: Isomorphism Constructions

Abstract

Associated with each first-order theory is a Boolean algebra of sentences and a Boolean space of models. Homomorphisms between the sentence algebras correspond to continuous maps between the model spaces. To what do recursive homomorphisms correspond? We introduce axiomatizable maps as the appropriate dual. For these maps we prove a Cantor-Bernstein theorem. Duality and the Cantor-Bernstein theorem are used to show that the Boolean sentence algebras of any two undecidable languages or of any two functional languages are recursively isomorphic where a language is undecidable iff it has at least one operation or relation symbol of two or more places or at least two unary operation symbols, and a language is functional iff it has exactly one unary operation symbol and all other symbols are for unary relations, constants, or propositions.