Scott sentence complexities of linear orderings
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866914319740960768 |
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| author | Gonzalez, David Rossegger, Dino |
| author_facet | Gonzalez, David Rossegger, Dino |
| contents | 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 $Π^{\mathrm{in}}_α$ complexities. We then take a more direct approach and exhibit linear orderings of all Scott complexities except $Σ^{\mathrm{in}}_{3}$ and $Σ^{\mathrm{in}}_{λ+1}$ for $λ$ a limit ordinal. We show that the former can not 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2305_07126 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Scott sentence complexities of linear orderings Gonzalez, David Rossegger, Dino Logic 03C57, 03D45, 03C70, 03E15 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 $Π^{\mathrm{in}}_α$ complexities. We then take a more direct approach and exhibit linear orderings of all Scott complexities except $Σ^{\mathrm{in}}_{3}$ and $Σ^{\mathrm{in}}_{λ+1}$ for $λ$ a limit ordinal. We show that the former can not 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. |
| title | Scott sentence complexities of linear orderings |
| topic | Logic 03C57, 03D45, 03C70, 03E15 |
| url | https://arxiv.org/abs/2305.07126 |