Index sets for classes of high rank structures

Abstract

This paper calculates, in a precise way, the complexity of the index sets for three classes of computable structures: the class $K_{\omega^{CK}_1}$ of structures of Scott rank $\omega^{CK}_1$, the class $K_{\omega^{CK}_1+1}$ of structures of Scott rank $\omega^{CK}_1+1}$, and the class $K$ of all structures of non-computable Scott rank. We show that $I(K)$ is $m$-complete $\Sigma^1_1$, $I(K_{\omega^{CK}_1})$ is $m$-complete $\Pi^0_2$ relative to Kleene’s $\mathcal{O}$, and $I(K_{\omega^{CK}_1+1})$ is $m$-complete $\Sigma^0_2 relative to $\mathcal{O}$.