A Topological-algebraic Approach to the Compactness Theorem of Classical Logic

Abstract

There are some methods of proof of the compactness theorem for classical logic which bypass the completeness theorem. Among them are the purely topological one, the purely algebraic one, and the hybrid one. These methods make essential use of either Tychonoff's Theorem, the concept of ultraproducts or the concept of Cantor sets as topological spaces. Instead of these conceptual tools, the paper provides the theorem with a method of proof that appeals to the concept of Stone spaces of Boolean algebras. In connection with a classical logical system (a propositional calculus or a predicate calculus), the method consists of five components. Firstly, the problem of the compactness of the logical system is reduced to that of the compactness of some topological space. Secondly, what is called the Lindenbaum algebra of the system is set up, which is in fact a Boolean algebra. Thirdly, it has to be shown that the Stone space of the Boolean algebra is compact. Fourthly, the set of sentences whose equivalent classes are members of the Stone space is shown to be satisfiable or simultaneously true. Finally, a homeomorphism is constructed between the topological space and the compact Stone space. Additionally, the method admits of a natural generalisation to the proof of the compactness theorem for modal logic.