Let be a theory in a countable fragment of 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 that enumerates these extensions.
In this paper, we effectively construct a predecessor function for formulas defining types in this tree hierarchy as follows. Let with - and -theories on level and , respectively. Then if is a formula that defines a type for , our predecessor function provides a formula for defining its subtype in .
By constructing this predecessor function, we weaken an assumption for Sacks’s result.
"Improving a Bounding Result That Constructs Models of High Scott Rank." Notre Dame J. Formal Logic 57 (1) 59 - 71, 2016. https://doi.org/10.1215/00294527-3328289