Infinitary Logics and Abstract Elementary Classes
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
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2020
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| _version_ | 1866918219431804928 |
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| author | Shelah, Saharon Villaveces, Andrés |
| author_facet | Shelah, Saharon Villaveces, Andrés |
| contents | We prove that every abstract elementary class (a.e.c.) with LST number $κ$ and vocabulary $τ$ of cardinality $\leq κ$ can be axiomatized in the logic ${\mathbb L}_{\beth_2(κ)^{+++},κ^+}(τ)$. In this logic an a.e.c. is therefore an EC class rather than merely a PC class. This constitutes a major improvement on the level of definability previously given by the Presentation Theorem. As part of our proof, we define the \emph{canonical tree} $\mathcal S={\mathcal S}_{\mathcal K}$ of an a.e.c. $\mathcal K$. This turns out to be an interesting combinatorial object of the class, beyond the aim of our theorem. Furthermore, we study a connection between the sentences defining an a.e.c. and the relatively new infinitary logic $L^1_λ$.} |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2010_02145 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Infinitary Logics and Abstract Elementary Classes Shelah, Saharon Villaveces, Andrés Logic 03C48, 03C75, 03C40 We prove that every abstract elementary class (a.e.c.) with LST number $κ$ and vocabulary $τ$ of cardinality $\leq κ$ can be axiomatized in the logic ${\mathbb L}_{\beth_2(κ)^{+++},κ^+}(τ)$. In this logic an a.e.c. is therefore an EC class rather than merely a PC class. This constitutes a major improvement on the level of definability previously given by the Presentation Theorem. As part of our proof, we define the \emph{canonical tree} $\mathcal S={\mathcal S}_{\mathcal K}$ of an a.e.c. $\mathcal K$. This turns out to be an interesting combinatorial object of the class, beyond the aim of our theorem. Furthermore, we study a connection between the sentences defining an a.e.c. and the relatively new infinitary logic $L^1_λ$.} |
| title | Infinitary Logics and Abstract Elementary Classes |
| topic | Logic 03C48, 03C75, 03C40 |
| url | https://arxiv.org/abs/2010.02145 |