Construction of a valued field whose valuation ring is definable but neither $\exists\forall\exists$ nor $ \forall\exists\forall$-definable in the language of rings
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866915437305921536 |
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| author | Khani, Mohsen Shirani, Shaghayegh Yadegari, Zahra Zarei, Afshin |
| author_facet | Khani, Mohsen Shirani, Shaghayegh Yadegari, Zahra Zarei, Afshin |
| contents | We give an example of a valued field $(K,A)$ such that the valuation ring $A$ is definable by an $L_{\text{ring}}$-formula without parameters, but there is no $\exists\forall\exists$ or $\forall\exists\forall$-formula in $L_{\text{ring}}$ to define it. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_06876 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Construction of a valued field whose valuation ring is definable but neither $\exists\forall\exists$ nor $ \forall\exists\forall$-definable in the language of rings Khani, Mohsen Shirani, Shaghayegh Yadegari, Zahra Zarei, Afshin Logic 03C60, 03C40, 12L12, 12J10 We give an example of a valued field $(K,A)$ such that the valuation ring $A$ is definable by an $L_{\text{ring}}$-formula without parameters, but there is no $\exists\forall\exists$ or $\forall\exists\forall$-formula in $L_{\text{ring}}$ to define it. |
| title | Construction of a valued field whose valuation ring is definable but neither $\exists\forall\exists$ nor $ \forall\exists\forall$-definable in the language of rings |
| topic | Logic 03C60, 03C40, 12L12, 12J10 |
| url | https://arxiv.org/abs/2508.06876 |