Effective Construction of Models

Publisher Summary This chapter discusses the effective construction of models. The models considered in the chapter are all countable, with universe ω, and the languages are all recursive. A type is a set of formulas all having precisely the same set of free variables. The chapter presents Solovay's results showing that the degrees of non-standard models of true arithmetic are precisely the degrees of enumerations of Scott sets containing the arithmetic sets. Solovay's proof makes use of the fact that if R is an enumeration of a Scott set containing the arithmetic sets, then the theory of true arithmetic is recursive in R.