Improving a Bounding Result That Constructs Models of High Scott Rank

Abstract

Let $T$ be a theory in a countable fragment of ${\mathcal{L}}_{{\omega }_1,\omega }$ 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 $T_{\gamma }\subseteq T_{\delta }$ with $T_{\gamma }$- and $T_{\delta }$-theories on level $\gamma$ and $\delta$, respectively. Then if $⌜p⌝\left(T_{\delta }\right)$ is a formula that defines a type for $T_{\delta }$, our predecessor function provides a formula for defining its subtype in $T_{\gamma }$.

By constructing this predecessor function, we weaken an assumption for Sacks’s result.