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A Reflection Calculus for Bar Induction and the Linearity of Consistency Strength Ordering

  • Fedor Pakhomov
  • Vlad Lazar
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Abstract

We introduce a novel extension of reflection calculus, RCBI, to analyze a family of extensions of ACA0 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 (BI). We prove that RCBI 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 ∇-free closed formulas is a pre-wellordering with respect to Π11-soundness and consistency strength. This allows us to establish an ordinal analysis for BI and its fragments, analogous to Beklemishev's analysis of PA. The structural properties of RCBI naturally give rise to a novel ordinal notation system for the Bachmann-Howard ordinal which differs substantially from existing representations.

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Cite

BibTeXRIS
BibTeX
@misc{pakhomov2026reflection,
  title = {{A Reflection Calculus for Bar Induction and the Linearity of Consistency Strength Ordering}},
  author = {Fedor Pakhomov and Vlad Lazar},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.20115},
  eprint = {2610.20115},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.20115},
}
RIS
TY  - UNPB
TI  - A Reflection Calculus for Bar Induction and the Linearity of Consistency Strength Ordering
AU  - Fedor Pakhomov
AU  - Vlad Lazar
PY  - 2026
DA  - 2026/10/14
PB  - arXiv
DO  - 10.48550/arxiv.2610.20115
UR  - https://arxiv.org/abs/2610.20115
AB  - We introduce a novel extension of reflection calculus, RC_BI, to analyze a family of extensions of 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 (BI). We prove that 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 ∇-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 BI and its fragments, analogous to Beklemishev's analysis of PA. The structural properties of RC_BI naturally give rise to a novel ordinal notation system for the Bachmann-Howard ordinal which differs substantially from existing representations.
ER  -