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  1. Scott Sentence Complexities of Linear Orderings.David Gonzalez & Dino Rossegger - forthcoming - Journal of Symbolic Logic:1-30.
    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 (...)
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  • New jump operators on equivalence relations.John D. Clemens & Samuel Coskey - 2022 - Journal of Mathematical Logic 22 (3).
    We introduce a new family of jump operators on Borel equivalence relations; specifically, for each countable group [Formula: see text] we introduce the [Formula: see text]-jump. We study the elementary properties of the [Formula: see text]-jumps and compare them with other previously studied jump operators. One of our main results is to establish that for many groups [Formula: see text], the [Formula: see text]-jump is proper in the sense that for any Borel equivalence relation [Formula: see text] the [Formula: see (...)
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  • On bi-embeddable categoricity of algebraic structures.Nikolay Bazhenov, Dino Rossegger & Maxim Zubkov - 2022 - Annals of Pure and Applied Logic 173 (3):103060.
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  • The Structural Complexity of Models of Arithmetic.Antonio Montalbán & Dino Rossegger - 2024 - Journal of Symbolic Logic 89 (4):1703-1719.
    We calculate the possible Scott ranks of countable models of Peano arithmetic. We show that no non-standard model can have Scott rank less than $\omega $ and that non-standard models of true arithmetic must have Scott rank greater than $\omega $. Other than that there are no restrictions. By giving a reduction via $\Delta ^{\mathrm {in}}_{1}$ bi-interpretability from the class of linear orderings to the canonical structural $\omega $ -jump of models of an arbitrary completion T of $\mathrm {PA}$ we (...)
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  • On categoricity of scattered linear orders of constructive ranks.Andrey Frolov & Maxim Zubkov - 2025 - Archive for Mathematical Logic 64 (1):279-297.
    In this article we investigate the complexity of isomorphisms between scattered linear orders of constructive ranks. We give the general upper bound and prove that this bound is sharp. Also, we construct examples showing that the categoricity level of a given scattered linear order can be an arbitrary ordinal from 3 to the upper bound, except for the case when the ordinal is the successor of a limit ordinal. The existence question of the scattered linear orders whose categoricity level equals (...)
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