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A Buchholz derivation system for the ordinal analysis of KP + Π3-reflection

  • Markus Michelbrink (a1)
Abstract
Abstract

In this paper we introduce a notation system for the infinitary derivations occurring in the ordinal analysis of KP + Π3-Reflection due to Michael Rathjen. This allows a finitary ordinal analysis of KP + Π3-Reflection. The method used is an extension of techniques developed by Wilfried Buchholz, namely operator controlled notation systems for RS-derivations. Similarly to Buchholz we obtain a characterisation of the provably recursive functions of KP + Π3-Reflection as <-recursive functions where < is the ordering on Rathjen's ordinal notation system . Further we show a conservation result for -sentences.

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[1]Toshiyasu Arai , Proof theory for theories of ordinals I: recursively Mahlo ordinals, Annals of Pure and Applied Logic, vol. 122 (2003), no. 1–3, pp. 185.

[2]Toshiyasu Arai , Proof theory for theories of ordinals II: Π-3-reflection, Annals of Pure and Applied Logic, vol. 129 (2004), no. 1–3, pp. 3992.

[3]Jon Barwise , Admissible set and structures, Springer Verlag, Berlin, Heidelberg, New York, 1975.

[5]Wilfried Buchholz , Notation systems for infinitary derivations, Archive for Mathematical Logic, vol. 30 (1991).

[6]Wilfried Buchholz , Explaining Gentzen's consistency proof within infinitary proof theory, Computational Logic and Proof Theory, 5th Kurt Gödel Colloqium, KGC'97 (G. Gottlob , A. Leitsch , and D. Mundici , editors), Lecture Notes in Computer Science, vol. 1298, Springer Verlag, 1997.

[7]Wilfried Buchholz , Explaining the Gentzen-Takeuti reduction steps: A second-order system, Archive for Mathematical Logic, vol. 40 (2001), no. 4, pp. 255272.

[8]Wilfried Buchholz , Finitary treatment of operator controlled derivations, Mathematical Logic Quarterly, vol. 47 (2001), no. 3, pp. 363396.

[11]Petr Hajek and Pavel Pudlak , Metamathematics of first-order arithmetic, Springer Verlag, Berlin, 1993.

[12]David Hilbert and Paul Bernays , Grundlagen der Mathematik, Zweite Auflage, Springer Verlag, Berlin, Heidelberg, New York, 1968.

[14]Thomas Jech , Set theory, Springer Verlag, Berlin, Heidelberg, New York, 1997.

[16]Azriel Levy , The size of the indescribable cardinals, Proceedings of Symposia in Pure Mathematics, vol. XII, Part 1, 1971.

[18]Wolfram Pohlers , Proof theory: An introduction, Lecture Notes in Mathematics, vol. 1407, Springer Verlag, Berlin, 1989.

[22]Michael Rathjen , Proof theory of reflection, Annals of Pure and Applied Logic, vol. 68 (1994), pp. 181224.

[23]Michael Rathjen , Recent advances in ordinal analysis:– CA and related systems, The Bulletin of Symbolic Logic, vol. 1 (1995).

[24]Michael Rathjen , An ordinal analysis of stability, Archive for Mathematical Logic, vol. 44 (2005), no. 1, pp. 162.

[26]Kurt Schütte , Proof theory, Springer Verlag, Berlin, Heidelberg, New York, 1977.

[29]A. S. Troelstra , Metamathematical investigation of intuitionistic arithmetic and analysis, Springer Verlag, Berlin, 1973.

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The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
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