A dichotomy for $T$-convex fields with a monomial group
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866913623131029504 |
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| author | Kaplan, Elliot Kesting, Christoph |
| author_facet | Kaplan, Elliot Kesting, Christoph |
| contents | We prove a dichotomy for o-minimal fields $\mathcal{R}$, expanded by a $T$-convex valuation ring (where $T$ is the theory of $\mathcal{R}$) and a compatible monomial group. We show that if $T$ is power bounded, then this expansion of $\mathcal{R}$ is model complete (assuming that $T$ is), it has a distal theory, and the definable sets are geometrically tame. On the other hand, if $\mathcal{R}$ defines an exponential function, then the natural numbers are externally definable in our expansion, precluding any sort of model theoretic tameness. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2305_07749 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A dichotomy for $T$-convex fields with a monomial group Kaplan, Elliot Kesting, Christoph Logic Primary 03C64. Secondary 03C10, 12J10 We prove a dichotomy for o-minimal fields $\mathcal{R}$, expanded by a $T$-convex valuation ring (where $T$ is the theory of $\mathcal{R}$) and a compatible monomial group. We show that if $T$ is power bounded, then this expansion of $\mathcal{R}$ is model complete (assuming that $T$ is), it has a distal theory, and the definable sets are geometrically tame. On the other hand, if $\mathcal{R}$ defines an exponential function, then the natural numbers are externally definable in our expansion, precluding any sort of model theoretic tameness. |
| title | A dichotomy for $T$-convex fields with a monomial group |
| topic | Logic Primary 03C64. Secondary 03C10, 12J10 |
| url | https://arxiv.org/abs/2305.07749 |