On the computability of optimal Scott sentences
Rachael Alvir, Barbara Csima, Matthew Harrison-Trainor
TL;DR
The paper investigates the computability of optimal Scott sentences for countable structures in $\mathcal{L}_{\omega_1\omega}$. It proves the existence of a computable structure with a $\Pi_2$ Scott sentence but no computable $\Pi_3$ Scott sentence, and deduces that there can be no computable $\Sigma_4$ Scott sentence in such cases; it also shows that there is no effective, uniform characterization of computable structures having computable $\Pi_n$ Scott sentences, via $\Pi^1_1$-$m$-completeness of the corresponding index sets. A simplifying Remark with $\mathcal{A}\cdot\omega$ connects $\Pi_\alpha$-Scott sentences to their counterparts under the Hopf-like construction, while a tailored construction diagonalizes against all computable $\Pi_3$ (and higher) sentences to block computable higher-level Scott sentences. The results extend to general $\Pi_n$ levels through Marker extensions and jump inversions, yielding a suite of corollaries about Scott families, pseudo-Scott sentences, and effective descriptive set-theoretic properties in Mod$(\mathcal{L})$. Overall, the work clarifies the gap between logical definability of structures and their computable realizations, revealing inherent limits to effective characterizations of optimal Scott sentence complexity.
Abstract
Given a countable mathematical structure, its Scott sentence is a sentence of the infinitary logic $\mathcal{L}_{ω_1 ω}$ that characterizes it among all countable structures. We can measure the complexity of a structure by the least complexity of a Scott sentence for that structure. It is known that there can be a difference between the least complexity of a Scott sentence and the least complexity of a computable Scott sentence; for example, Alvir, Knight, and McCoy showed that there is a computable structure with a $Π_2$ Scott sentence but no computable $Π_2$ Scott sentence. It is well known that a structure with a $Π_2$ Scott sentence must have a computable $Π_4$ Scott sentence. We show that this is best possible: there is a computable structure with a $Π_2$ Scott sentence but no computable $Σ_4$ Scott sentence. We also show that there is no reasonable characterization of the computable structures with a computable $Π_n$ Scott sentence by showing that the index set of such structures is $Π^1_1$-$m$-complete.