Improving a Bounding Result That Constructs Models of High Scott Rank

Abstract

Let T be a theory in a countable fragment of Lω1,ω whose extensions in countable fragments have only countably many types. Sacks proves a bounding theorem that generates models of high Scott rank. For this theorem, a tree hierarchy is developed for T that enumerates these extensions. In this paper, we effectively construct a predecessor function for formulas defining types in this tree hierarchy as follows. Let Tγ ⊂ T with T - and Tγ -theories on level and γ, respectively. Then if p (Tγ) is a formula that defines a type for Tγ , our predecessor function provides a formula for defining its subtype in T . By constructing this predecessor function, we weaken an assumption for Sackss result.