{
  "abstract": "We introduce a novel extension of reflection calculus, $\\mathsf{RC_{BI}}$, to analyze a family of extensions of $\\mathsf{ACA}_0$ generated by uniform $ω$-model and syntactic reflection principles, together with several other related operators. The union of these theories coincides with the theory of Bar Induction ($\\mathsf{BI}$). We prove that $\\mathsf{RC_{BI}}$ is arithmetically sound, and that it is sound and complete with respect to a natural class of Kripke models. Leveraging these modal semantics, we demonstrate that the class of theories denoted by $\\nabla$-free closed formulas is a pre-wellordering with respect to $Π^1_1$-soundness and consistency strength. This allows us to establish an ordinal analysis for $\\mathsf{BI}$ and its fragments, analogous to Beklemishev's analysis of $\\mathsf{PA}$. The structural properties of $\\mathsf{RC_{BI}}$ naturally give rise to a novel ordinal notation system for the Bachmann-Howard ordinal which differs substantially from existing representations.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.20115",
  "author_total": 2,
  "authors": [
    {
      "name": "Fedor Pakhomov"
    },
    {
      "name": "Vlad Lazar"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/82w57m7gmb42v.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.20115",
  "fulltext_url": null,
  "id": "82w57m7gmb42v",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.20115",
  "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"
    ],
    [
      "openalex",
      "2608|Advanced Topology and Set Theory"
    ]
  ],
  "openalex": "W7221851284",
  "pdf_url": "https://arxiv.org/pdf/2610.20115",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-14",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/82w57m7gmb42v.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:35Z",
      "native_id": "2610.20115",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.20115"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.LO",
      "subfield": "logic-foundations"
    }
  ],
  "title": "A Reflection Calculus for Bar Induction and the Linearity of Consistency Strength Ordering",
  "title_text": "A Reflection Calculus for Bar Induction and the Linearity of Consistency Strength Ordering",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:35Z",
  "url": "https://research.draconitic.com/paper/82w57m7gmb42v",
  "venue": "arXiv",
  "versions": []
}