Hostname: page-component-7c8c6479df-p566r Total loading time: 0 Render date: 2024-03-26T20:59:00.626Z Has data issue: false hasContentIssue false

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

Published online by Cambridge University Press:  12 March 2014

Markus Michelbrink*
Affiliation:
Department of Computer Science, University of Wales Swansea, Singleton Park, Swansea, SA2 8PP, United Kingdom, E-mail: m.michelbrink@swansea.ac.uk

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.

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 2006

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Arai, Toshiyasu, Proof theory for theories of ordinals I: recursively Mahlo ordinals, Annals of Pure and Applied Logic, vol. 122 (2003), no. 1–3, pp. 185.CrossRefGoogle Scholar
[2]Arai, Toshiyasu, Proof theory for theories of ordinals II: Π-3-reflection, Annals of Pure and Applied Logic, vol. 129 (2004), no. 1–3, pp. 3992.CrossRefGoogle Scholar
[3]Barwise, Jon, Admissible set and structures, Springer Verlag, Berlin, Heidelberg, New York, 1975.CrossRefGoogle Scholar
[4]Buchholz, Wilfried, A simplified version of local predicativity, Proof theory (Aczel, Peter, Simmons, Harold, and Wainer, Stan, editors), Cambridge University Press, Leeds, 1990.Google Scholar
[5]Buchholz, Wilfried, Notation systems for infinitary derivations, Archive for Mathematical Logic, vol. 30 (1991).CrossRefGoogle Scholar
[6]Buchholz, Wilfried, Explaining Gentzen's consistency proof within infinitary proof theory, Computational Logic and Proof Theory, 5th Kurt Gödel Colloqium, KGC'97 (Gottlob, G., Leitsch, A., and Mundici, D., editors), Lecture Notes in Computer Science, vol. 1298, Springer Verlag, 1997.Google Scholar
[7]Buchholz, Wilfried, Explaining the Gentzen-Takeuti reduction steps: A second-order system, Archive for Mathematical Logic, vol. 40 (2001), no. 4, pp. 255272.CrossRefGoogle Scholar
[8]Buchholz, Wilfried, Finitary treatment of operator controlled derivations, Mathematical Logic Quarterly, vol. 47 (2001), no. 3, pp. 363396.3.0.CO;2-P>CrossRefGoogle Scholar
[9]Buchholz, Wilfried and Schütte, Kurt, Proof theory of impredicative subsystems of analysis, Bibliopolis, Napoli, 1988.Google Scholar
[10]Drake, Frank, Set theory: An introduction to large cardinals, North-Holland Publishing Company, Amsterdam, London, 1974.Google Scholar
[11]Hajek, Petr and Pudlak, Pavel, Metamathematics of first-order arithmetic, Springer Verlag, Berlin, 1993.CrossRefGoogle Scholar
[12]Hilbert, David and Bernays, Paul, Grundlagen der Mathematik, Zweite Auflage, Springer Verlag, Berlin, Heidelberg, New York, 1968.CrossRefGoogle Scholar
[13]Jäger, Gerhard and Pohlers, Wolfram, Eine beweistheoretische Untersuchung von – CA + BI und verwandter Systeme, Technical report, 1982.Google Scholar
[14]Jech, Thomas, Set theory, Springer Verlag, Berlin, Heidelberg, New York, 1997.CrossRefGoogle Scholar
[15]Rogers, Hartley Jr., Theory of recursive functions and effective computability, McGraw-Hill Book Company, New York, St. Louis, San Francisco, Toronto, London, Sydney, 1967.Google Scholar
[16]Levy, Azriel, The size of the indescribable cardinals, Proceedings of Symposia in Pure Mathematics, vol. XII, Part 1, 1971.CrossRefGoogle Scholar
[17]Michelbrink, Markus, Zur endlichen Behandlung der Beweistheorie schwacher Fragmente der Mengenlehre: KP + Π3-Reflexion, Ph.D. thesis, University of Hannover, 2001.Google Scholar
[18]Pohlers, Wolfram, Proof theory: An introduction, Lecture Notes in Mathematics, vol. 1407, Springer Verlag, Berlin, 1989.CrossRefGoogle Scholar
[19]Rathjen, Michael, An ordinal analysis of parameter free -comprehension: Part I.Google Scholar
[20]Rathjen, Michael, Eine Ordinalzahlanalyse der Π3-Reflexion, Ph.D. thesis, Westfälische Wilhelms Universität Münster, Münster, 1992.Google Scholar
[21]Rathjen, Michael, Admissible proof theory and beyond, Logic, methodology and philosophy of science IX (Prawitz, D., Skyrms, B., and Westerstahl, , editors), Elsevier Science B.V., 1994.Google Scholar
[22]Rathjen, Michael, Proof theory of reflection, Annals of Pure and Applied Logic, vol. 68 (1994), pp. 181224.CrossRefGoogle Scholar
[23]Rathjen, Michael, Recent advances in ordinal analysis:– CA and related systems, The Bulletin of Symbolic Logic, vol. 1 (1995).CrossRefGoogle Scholar
[24]Rathjen, Michael, An ordinal analysis of stability, Archive for Mathematical Logic, vol. 44 (2005), no. 1, pp. 162.CrossRefGoogle Scholar
[25]Schütte, Kurt, Beweistheoretische Erfassung der unendlichen Induktion in der Zahlentheorie, Mathematische Annalen, vol. 122 (1951).Google Scholar
[26]Schütte, Kurt, Proof theory, Springer Verlag, Berlin, Heidelberg, New York, 1977.CrossRefGoogle Scholar
[27]Skolem, T., Begründung der elementaren Arithmetik durch die rekurrierende Denkweise ohne Anwendung scheinbarer Veränderlicher mit unendlichen Ausdehnungsbereich, From Frege to Gödel. A source book in mathematical logic 1879–1931 (Heienoort, J. van, editor), Harvard University Press, 1967.Google Scholar
[28]Takeuti, Gaisi, Proof theory, 2 ed., Elsevier Science Publishers B.V., Amsterdam, New York, Oxford, Tokyo, 1987.Google Scholar
[29]Troelstra, A. S., Metamathematical investigation of intuitionistic arithmetic and analysis, Springer Verlag, Berlin, 1973.CrossRefGoogle Scholar
[30]Troelstra, A. S. and Schwichtenberg, H., Basic proof theory, Cambridge University Press, 1996.Google Scholar
[31]Troelstra, A. S. and Van Dalen, D., Constructivism in mathematics. An introduction, vol. I, Elsevier Science Publishers B.V., Amsterdam, New York, Oxford, Tokyo, 1988.Google Scholar
[32]Tupailo, Sergei, Finitary reductions for local predicativity, 1: recursively regular ordinals, Logic Colloquium 98 (Pudlak, P.Buss, S., Hajek, P., editor), 2000, pp. 465499.Google Scholar