Freund, A, Rathjen, M orcid.org/0000-0003-1699-4778 and Weiermann, A (2022) Minimal bad sequences are necessary for a uniform Kruskal theorem. Advances in Mathematics, 400. 108265. ISSN 0001-8708
Abstract
Kruskal's theorem on trees is a classical result of combinatorics with important applications in computer science. The minimal bad sequence argument of Nash-Williams yields a proof that is very elegant but not as elementary as possible, by previous results from mathematical logic. In the present paper, we formulate a uniform Kruskal theorem, which extends the original theorem from trees to general recursive data types. Our main result shows that the minimal bad sequence argument does yield the most elementary proof of this uniform Kruskal theorem. So in the uniform case, the elegant proof coincides with the most elementary one. In more precise and technical terms, we work in the framework of reverse mathematics, where we show that the uniform Kruskal theorem is equivalent to -comprehension, over extended by the chain-antichain principle. As a by-product of our investigation, we obtain uniform proofs of several Kruskal-type independence results.
Metadata
Item Type: | Article |
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Authors/Creators: |
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Copyright, Publisher and Additional Information: | © 2022 Elsevier Inc. All rights reserved. This is an author produced version of an article published in Advances in Mathematics. Uploaded in accordance with the publisher's self-archiving policy. |
Keywords: | primary; 03B30; secondary; 05C05; 06A07; 68Q42; 03F35 |
Dates: |
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Institution: | The University of Leeds |
Academic Units: | The University of Leeds > Faculty of Engineering & Physical Sciences (Leeds) > School of Mathematics (Leeds) > Pure Mathematics (Leeds) |
Funding Information: | Funder Grant number John Templeton Foundation (US) 60842 |
Depositing User: | Symplectic Publications |
Date Deposited: | 18 Mar 2022 12:09 |
Last Modified: | 25 Feb 2023 01:13 |
Status: | Published |
Publisher: | Elsevier |
Identification Number: | 10.1016/j.aim.2022.108265 |
Open Archives Initiative ID (OAI ID): | oai:eprints.whiterose.ac.uk:184735 |