Effective Construction of Models
Studies in logic and the foundations of mathematics · 1986
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.