Scott Ranks of Classifications of the Admissibility Equivalence Relation

Abstract

Let $\mathcal{L}$ be a recursive language. Let $S(\mathcal{L})$ be the set of $\mathcal{L}$-structures with domain $\omega$. Let $\Phi :{}^{\omega }2\to S(\mathcal{L})$ be a ${\Delta }_1^1$ function with the property that for all $x,y\in {}^{\omega }2$, ${\omega }_1^x={\omega }_1^y$ if and only if $\Phi (x){\approx }_{\mathcal{L}}\Phi (y)$. Then there is some $x\in {}^{\omega }2$ so that $\mathrm{SR}(\Phi (x))={\omega }_1^x+1$.