Proof-Theoretic Relations between Higman's and Kruskal's theorem, and Independence Results for Tree-like Structures

[2511.11297] Proof-Theoretic Relations between Higman's and Kruskal's theorem, and Independence Results for Tree-like Structures Skip to main content Search Submit Donate Log in Mathematics > Logic arXiv:2511.11297 (math) [Submitted on 14 Nov 2025 ( v1 ), last revised 28 Aug 2026 (this version, v3)] Title: Proof-Theoretic Relations between Higman's and Kruskal's theorem, and Independence Results for Tree-like Structures Authors: Gabriele Buriola , Andreas Weiermann View PDF HTML (experimental) Abstract: Higman's lemma and Kruskal's theorem are two of the most celebrated results in the theory of well quasi-orders. In his seminal paper G. Higman obtained what is known as Higman's lemma as a corollary of a more general theorem, dubbed here Higman's theorem. While the lemma deals with finite sequences over a well quasi-order, the theorem is about abstract operations of arbitrary high arity. J.B. Kruskal was fully aware of this broader framework: in his seminal paper, he not only applied Higman's lemma at crucial points of his proof but also followed Higman's proof schema. At the conclusion of the paper, Kruskal noted that Higman's theorem is a special case, restricted to trees of finite degree, of his own tree theorem. Although he provided no formal reduction, he included a glossary translating concepts between the tree and algebraic settings. The equivalence between these versions was later clarified by D. Schmidt and M. Pouzet. In this work, we revisit that equivalence to illuminate the proof-theoretical relationships between the two theorems within the base system $RCA_0$, of reverse mathematics. Moreover, some independence results over first- and second-orders are treated. In particular, tree-like structures, involving either Ackermannian terms or exponential expressions, are studied unveiling well-foundedness properties that are independent from Peano arithmetic and relevant fragments of second-order arithmetic. Subjects: Logic (math.LO) MSC classes: 03B30, 03F35, 05C05, 06A07 Cite as: arXiv:2511.11297 [math.LO] (or arXiv:2511.11297v3 [math.LO] for this version) https://doi.org/10.48550/arXiv.2511.11297 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Gabriele Buriola [ view email ] [v1] Fri, 14 Nov 2025 13:32:30 UTC (31 KB) [v2] Thu, 5 Feb 2026 17:09:53 UTC (32 KB) [v3] Fri, 28 Aug 2026 12:08:33 UTC (76 KB) Full-text links: Access Paper: View PDF HTML (experimental) TeX Source view license Current browse context: math.LO < prev | next > new | recent | 2025-11 Change to browse by: math References & Citations SciX Google Scholar Semantic Scholar export BibTeX citation Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab)

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