{
  "abstract": "We bound the eventual order types of Skolem functions below finite exponential towers. Writing $E_0(u)=u$, $E_{n+1}(u)=2^{E_n(u)}$, and $ω_0=1$, $ω_{k+1}=ω^{ω_k}$, the argument gives \\[ |\\Sk_{<E_n(x^m)}|<ω_{r_n},\\qquad r_n=2+\\frac{n(n+3)}2\\quad(n\\ge1,\\ m\\ge2\\text{ fixed}). \\] For triple towers we obtain the sharper bound $|\\Sk_{<E_3(x^m)}|<ω_{10}$ for every fixed $m\\ge1$. The proof combines comparisons of asymptotic expansions with finite recursive decompositions and ordinal estimates for ordered sums and products. The analytic part is developed in the classical field of logarithmic-exponential series. We prove the required discreteness, support and truncation statements in this setting. The external inputs are the series construction of van den Dries--Macintyre--Marker and the order and ordinal estimates recalled from Berarducci--Mamino. These bounds imply $|\\Sk|=\\varepsilon_0$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.12740",
  "author_total": 1,
  "authors": [
    {
      "name": "Andreas Weiermann"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bme9rrz1r8f1n.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.12740",
  "fulltext_url": null,
  "id": "bme9rrz1r8f1n",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.12740",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.LO"
    ],
    [
      "openalex",
      "2608|Advanced Topology and Set Theory"
    ]
  ],
  "openalex": "W7212535575",
  "pdf_url": "https://arxiv.org/pdf/2609.12740",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-11",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bme9rrz1r8f1n.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:11Z",
      "native_id": "2609.12740",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.12740"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.LO",
      "subfield": "logic-foundations"
    }
  ],
  "title": "Finite-tower bounds for Skolem functions",
  "title_text": "Finite-tower bounds for Skolem functions",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:11Z",
  "url": "https://research.draconitic.com/paper/bme9rrz1r8f1n",
  "venue": "arXiv",
  "versions": []
}