Work

Scott Sentence Complexities of Linear Orderings

Year: 2024

Type: article

Abstract: We study possible Scott sentence complexities of linear orderings using two approaches. First, we investigate the effect of the Friedman–Stanley embedding on Scott sentence complexity and show that it only preserves $\Pi ^{\mathrm {in}}_{\alpha }$ complexities. We then take a more direct approach and exhibit linear orderings of all Scott sentence complexities except $\Sigma ^{\mathrm {in}}_{3}$ and $\Sigma ^{\mathrm {in}}_{\lambda +1}$ for $\lambda $ a limit ordinal. We show that the former cannot be the Scott sentence complexity of a linear ordering. In the process we develop new techniques which appear to be helpful to calculate the Scott sentence complexities of structures.

Source: Journal of Symbolic Logic

DOI: 10.1017/jsl.2024.59

Authors: DAVID GONZALEZ, DINO ROSSEGGER

Institutions: University of California System, University of California, Berkeley, TU Wien

Language: English

Topic: Advanced Topology and Set Theory

Subfield: Geometry and Topology

Field: Mathematics

Domain: Physical Sciences

Open Access status: green

Funder: European Commission

Award: Algorithmic complexity of structures and their equivalence relations