{
  "abstract": "Many explicit constructions in algebra are not first-order definable in classical finitary logic, but are, however, definable in infinitary logic. The aim of this paper is to show how infinitary logic can be used to lift model-theoretic results in arithmetic to a quantifier-sensitive form. We apply this idea to a deep result of Jahnke and Kartas on the model theory of perfectoid fields. Our main result is that a first-order sentence $ϕ$ holds in the tilt $k^{\\flat}$ of a perfectoid field $k$ if and only if a certain sentence $ϕ^{\\natural}$ holds in a non-principal ultrapower of $k$, where $ϕ^{\\natural}$ does not depend on $k$ and is only slightly more complicated than $ϕ$ itself. In particular, tilting preserves the Lévy hierarchy in a precise sense. To this end, we prove that the Standard Decomposition of a valued field at a constant $t$ is infinitarily interpretable by formulas of rank at most 1.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.41064",
  "author_total": 1,
  "authors": [
    {
      "name": "Leo Gitin"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1rc81m3ktkcxr.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.41064",
  "fulltext_url": null,
  "id": "1rc81m3ktkcxr",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.41064",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.LO"
    ],
    [
      "arxiv",
      "math.NT"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2610.41064",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1rc81m3ktkcxr.ris",
  "sources": [
    {
      "first_seen": "2026-11-02T12:21:20Z",
      "native_id": "2610.41064",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.41064"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.LO",
      "subfield": "logic-foundations"
    }
  ],
  "title": "The complexity of tilting",
  "title_text": "The complexity of tilting",
  "type": "preprint",
  "updated_at": "2026-11-04T15:20:52Z",
  "url": "https://research.draconitic.com/paper/1rc81m3ktkcxr",
  "venue": "arXiv",
  "versions": []
}