Scott Ranks of Classifications of the Admissibility Equivalence Relation

TL;DR

This paper investigates the Scott ranks of classifications of the admissibility equivalence relation, establishing a connection between the Scott rank of certain structures and the admissibility hierarchy in descriptive set theory.

Contribution

It introduces a method to determine the Scott rank of structures classified by admissibility equivalence, linking it to the admissibility hierarchy and providing new insights into their complexity.

Findings

Existence of structures with Scott rank exactly +1 for given admissibility levels.

Establishment of a correspondence between  and Scott ranks of certain classifications.

Advancement in understanding the complexity of classification problems in logic.

Abstract

Let $\mathscr{L}$ be a recursive language. Let $S(\mathscr{L})$ be the set of $\mathscr{L}$-structures with domain $\omega$. Let $\Phi : {}^\omega 2 \rightarrow S(\mathscr{L})$ 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 $\Phi(x) \approx_{\mathscr{L}} \Phi(y)$. Then there is some $x \in {}^\omega 2$ so that $\mathrm{SR}(\Phi(x)) = \omega_1^x + 1$.