Strange Structures from Computable Model Theory

Abstract

Let $L$ be a countable language, let $\mathcal{I}$ be an isomorphism-type of countable $L$-structures, and let $a\in 2^{\omega }$. We say that $\mathcal{I}$ is $a$-strange if it contains a computable-from-$a$ structure and its Scott rank is exactly ${\omega }_1^a$. For all $a$, $a$-strange structures exist. Theorem (AD): If $\mathcal{C}$ is a collection of ${\aleph }_1$ isomorphism-types of countable structures, then for a Turing cone of $a$’s, no member of $\mathcal{C}$ is $a$-strange.