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| Main Authors: | , , |
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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2406.01836 |
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| _version_ | 1866909215897944064 |
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| author | Gonzalez, David Harrison-Trainor, Matthew Ho, Meng-Che "Turbo" |
| author_facet | Gonzalez, David Harrison-Trainor, Matthew Ho, Meng-Che "Turbo" |
| contents | For any limit ordinal $λ$, we construct a linear order $L_λ$ whose Scott complexity is $Σ_{λ+1}$. This completes the classification of the possible Scott sentence complexities of linear orderings. Previously, there was only one known construction of any structure (of any signature) with Scott complexity $Σ_{λ+1}$, and our construction gives new examples, e.g., rigid structures, of this complexity.
Moreover, we can construct the linear orders $L_λ$ so that not only does $L_λ$ have Scott complexity $Σ_{λ+1}$, but there are continuum-many structures $M \equiv_λL_λ$ and all such structures also have Scott complexity $Σ_{λ+1}$. In contrast, we demonstrate that there is no structure (of any signature) with Scott complexity $Π_{λ+1}$ that is only $λ$-equivalent to structures with Scott complexity $Π_{λ+1}$.
Our construction is based on functions $f \colon \mathbb{Z}\to \mathbb{N}\cup \{\infty\}$ which are almost periodic but not periodic, such as those arising from shifts of the $p$-adic valuations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_01836 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Scott analysis, linear orders and almost periodic functions Gonzalez, David Harrison-Trainor, Matthew Ho, Meng-Che "Turbo" Logic For any limit ordinal $λ$, we construct a linear order $L_λ$ whose Scott complexity is $Σ_{λ+1}$. This completes the classification of the possible Scott sentence complexities of linear orderings. Previously, there was only one known construction of any structure (of any signature) with Scott complexity $Σ_{λ+1}$, and our construction gives new examples, e.g., rigid structures, of this complexity. Moreover, we can construct the linear orders $L_λ$ so that not only does $L_λ$ have Scott complexity $Σ_{λ+1}$, but there are continuum-many structures $M \equiv_λL_λ$ and all such structures also have Scott complexity $Σ_{λ+1}$. In contrast, we demonstrate that there is no structure (of any signature) with Scott complexity $Π_{λ+1}$ that is only $λ$-equivalent to structures with Scott complexity $Π_{λ+1}$. Our construction is based on functions $f \colon \mathbb{Z}\to \mathbb{N}\cup \{\infty\}$ which are almost periodic but not periodic, such as those arising from shifts of the $p$-adic valuations. |
| title | Scott analysis, linear orders and almost periodic functions |
| topic | Logic |
| url | https://arxiv.org/abs/2406.01836 |