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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2605.28299 |
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| _version_ | 1866917540346724352 |
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| author | Ramsey, Nicholas |
| author_facet | Ramsey, Nicholas |
| contents | We analyze a construction of Cherlin, van den Dries, and Macintyre to code graphs in PAC fields. We show that, in many cases, model-theoretic properties of the graph are preserved in the passage from the graph to the field. As a corollary, we show that the SOP$_{n}$ hierarchy is strict in the class of fields. The main ingredient is a detailed treatment of the model theory of inverse systems of certain profinite groups that code graphs and can be realized as the absolute Galois groups of PAC fields. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_28299 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Classification Theory and the Construction of PAC Fields Ramsey, Nicholas Logic We analyze a construction of Cherlin, van den Dries, and Macintyre to code graphs in PAC fields. We show that, in many cases, model-theoretic properties of the graph are preserved in the passage from the graph to the field. As a corollary, we show that the SOP$_{n}$ hierarchy is strict in the class of fields. The main ingredient is a detailed treatment of the model theory of inverse systems of certain profinite groups that code graphs and can be realized as the absolute Galois groups of PAC fields. |
| title | Classification Theory and the Construction of PAC Fields |
| topic | Logic |
| url | https://arxiv.org/abs/2605.28299 |