Model Theory for Dissecting Recursion Theory

Abstract

This chapter provides an introduction to model theory for dissecting recursion theory. The chapter discusses several completeness theorems. Predicate calculus with equality, ω -logic, infinitary &'s and V's, the infinity quantifier, and the uncountability quantifier, are described in the chapter. The chapter describes compactness, omitting types, and omitting compactifiable types. It also describes application to uncountability quantifier and barwise compactness. The chapter focuses on how a consistent set φ of sentences has a model. The key to this demonstration is the formation of a set φ ∞ of sentences with the several properties discussed in the chapter. The chapter discusses a fact that when one adds two new quantifiers UCx and CCx to a countables language with the intended meanings of “for uncountably many x ” and “for cocountably many x .” The additional rules needed for these quantifiers are finitary.