Proof theory for theories of ordinals. I: Recursively Mahlo ordinals
type: Item
instance of: scholarly article
title: Proof theory for theories of ordinals. I: Recursively Mahlo ordinals
zbMATH Open document ID: 1032.03049
DOI: 10.1016/S0168-0072(03)00020-4
author: Toshiyasu Arai
published in: Annals of Pure and Applied Logic
publication date: 2003-08-13T00:00:00Z
review text: In this paper, the author presents a proof-theoretic analysis of a theory of ``recursively Mahlo ordinals in the form of \(\Pi_2\)-reflecting on \(\Pi_2\)-reflecting ordinals''. As notation system, a subsystem of a system of ordinal diagrams, previously introduced by the same author [Arch. Math. Logic 39, 353-391 (2000; Zbl 0977.03030)], is used. There is no direct set-theoretic interpretation of the notation system, but it was extracted out of the cut-elimination proofs with some additional conditions to ensure foundation. In contrast to the proof-theoretic analysis of KPM by \textit{M. Rathjen} [Arch. Math. Logic 30, 377-403 (1991; Zbl 0727.03036)], which is carried out in the ``Schütte style'', the analysis here follows the ``Gentzen-Takeuti style'' [for the relation of the two approaches, cf., e.g., \textit{W. Buchholz}, Arch. Math. Logic 40, 255-272 (2001; Zbl 1007.03051)]. For an extension of the approach to \(\Pi_3\) reflection, one can consult recent work of \textit{M. Michelbrink} [Zur endlichen Behandlung der Beweistheorie schwacher Fragmente der Mengenlehre: KP + \(\Pi_3\)-Reflexion, PhD thesis, Universität Hannover, Fachbereich Mathematik und Informatik (2001), \url{http://www.cs.swan.ac.uk/~csmichel/Articles/Dissertation.pdf}].
reviewed by: Reinhard Kahle
Mathematics Subject Classification ID: 03F35
Mathematics Subject Classification ID: 03F15
zbMATH DE Number: 1963862
zbMATH Keywords: proof-theoretic ordinals
zbMATH Keywords: ordinal diagrams
zbMATH Keywords: reflecting ordinals
zbMATH Keywords: Mahlo ordinals
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1963862