Scott Spectral Gaps are Bounded for Linear Orderings
Fuente:
arXiv
Guardado en:
| Autores principales: | , |
|---|---|
| Formato: | Preprint |
| Publicado: |
2024
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866918005911322624 |
|---|---|
| author | Gonzalez, David Harrison-Trainor, Matthew |
| author_facet | Gonzalez, David Harrison-Trainor, Matthew |
| contents | We demonstrate that any $Π_α$ sentence of the infinitary logic $L_{ω_1 ω}$ extending the theory of linear orderings has a model with a $Π_{α+4}$ Scott sentence and hence of Scott rank at most $α+3$. In other words, the gap between the complexity of the theory and the complexity of the simplest model is always bounded by $4$. This contrasts the situation with general structures where for any $α$ there is a $Π_2$ sentence all of whose models have Scott rank $α$. We also give new lower bounds, though there remains a small gap between our lower and upper bounds: For most (but not all) $α$, we construct a $Π_α$ sentence extending the theory of linear orderings such that no models have a $Σ_{α+2}$ Scott sentence and hence no models have Scott rank less than or equal to $α$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_12084 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Scott Spectral Gaps are Bounded for Linear Orderings Gonzalez, David Harrison-Trainor, Matthew Logic We demonstrate that any $Π_α$ sentence of the infinitary logic $L_{ω_1 ω}$ extending the theory of linear orderings has a model with a $Π_{α+4}$ Scott sentence and hence of Scott rank at most $α+3$. In other words, the gap between the complexity of the theory and the complexity of the simplest model is always bounded by $4$. This contrasts the situation with general structures where for any $α$ there is a $Π_2$ sentence all of whose models have Scott rank $α$. We also give new lower bounds, though there remains a small gap between our lower and upper bounds: For most (but not all) $α$, we construct a $Π_α$ sentence extending the theory of linear orderings such that no models have a $Σ_{α+2}$ Scott sentence and hence no models have Scott rank less than or equal to $α$. |
| title | Scott Spectral Gaps are Bounded for Linear Orderings |
| topic | Logic |
| url | https://arxiv.org/abs/2411.12084 |