The number of models of a fixed Scott rank, for a counterexample to the analytic Vaught conjecture

Abstract

We show that if $\gamma \in \omega \cup \{\aleph_{0}\}$ and $\mathcal{A}$ is a counterexample to the analytic Vaught conjecture having exactly $\gamma$ many models of Scott rank $\omega_{1}$, then there exists a club $C \subseteq \omega_{1}$ such that $\mathcal{A}$ has exactly $\gamma$ many models of Scott rank $\alpha$, for each $\alpha \in C$.