Further First-Order Features

Abstract

Over the years many fundamental and important results have been established for classical first-order logic. This chapter is devoted to what are probably the most basic of these, ranging from Herbrand’s theorem to Beth’s definability theorem. The Model Existence Theorem plays a role in establishing many of the results of this chapter, thus proving its versatility. It has one big drawback, though—proofs that use it are non-constructive in nature. For some of the things we are interested in, such as Herbrand’s theorem, it is important to have a constructive proof as well, and this leads us to the beginnings of proof theory, supplementing the semantic methods that we initially rely on.