On Stability and Existence of Models in Abstract Elementary Classes
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866917768927903744 |
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| author | Mazari-Armida, Marcos Vasey, Sebastien Yang, Wentao |
| author_facet | Mazari-Armida, Marcos Vasey, Sebastien Yang, Wentao |
| contents | For an abstract elementary class $\mathbf{K}$ and a cardinal $λ\geq LS(\mathbf{K})$, we prove under mild cardinal arithmetic assumptions, categoricity in two succesive cardinals, almost stability for $λ^+$-minimal types and continuity of splitting in $λ$, that stability in $λ$ is equivalent to the existence of a model in $λ^{++}$. The forward direction holds without any cardinal or categoricity assumptions, this result improves both [Vas18b, 12.1] and [MaYa24, 3.14].
Moreover, we prove a categoricity theorem for abstract elementary classes with weak amalgamation and tameness under mild structural assumptions in $λ$. A key feature of this result is that we do not assume amalgamation or arbitrarily large models. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2406_15263 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On Stability and Existence of Models in Abstract Elementary Classes Mazari-Armida, Marcos Vasey, Sebastien Yang, Wentao Logic Primary: 03C48. Secondary: 03C45, 03C52, 03C55 For an abstract elementary class $\mathbf{K}$ and a cardinal $λ\geq LS(\mathbf{K})$, we prove under mild cardinal arithmetic assumptions, categoricity in two succesive cardinals, almost stability for $λ^+$-minimal types and continuity of splitting in $λ$, that stability in $λ$ is equivalent to the existence of a model in $λ^{++}$. The forward direction holds without any cardinal or categoricity assumptions, this result improves both [Vas18b, 12.1] and [MaYa24, 3.14]. Moreover, we prove a categoricity theorem for abstract elementary classes with weak amalgamation and tameness under mild structural assumptions in $λ$. A key feature of this result is that we do not assume amalgamation or arbitrarily large models. |
| title | On Stability and Existence of Models in Abstract Elementary Classes |
| topic | Logic Primary: 03C48. Secondary: 03C45, 03C52, 03C55 |
| url | https://arxiv.org/abs/2406.15263 |