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 $\alpha \in C$.

arXiv comment3 pages