Iterated Inductive Definitions Revisited

Chapter
Part of the Outstanding Contributions to Logic book series (OCTR, volume 13)

Abstract

In this paper we revisit our contribution to Lecture Note 897 and present a computation of the prooftheoretic ordinals of formal theories for iterated inductive definitions together with a characterization of their provably recursive functions. The techniques used here differ essentially from the original ones. Sections 15 contain a general survey of the connections between recursion theoretic and proof theoretic ordinals and roughly recap some basic facts on iterated inductive definitions. Beginning with Sect. 6 the paper becomes more technical. After introducing the semi–formal system for iterated inductive definition and the necessary ordinals we compute the ordinal spectrum of the formal theory for iterated inductive definitions and characterize their provably total functions in terms of a subrecursive hierarchy. The last section briefly discusses the foundational aspects of the paper.

Keywords

Inductive definitions Cut–elimination Ordinal analysis Provably recursive functions. 

2010 Mathematics Subjects Classification

03F03 03F05 03F15 03D20 03D60 03D70 

References

  1. 1.
    Avigad, J., Towsner, H.: Functional interpretation and inductive definitions. J. Symb. Logic 74(4), 1100–1120 (2009)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Beckmann, A., Pohlers, W.: Application of cut-free infinitary derivations to generalized recursion theory. Ann. Pure Appl. Logic 94, 1–19 (1998)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Blankertz, B., Weiermann, A.: How to characterize provably total functions by the Buchholz operator method. Lecture Notes in Logic, vol. 6. Springer, Heidelberg (1996)Google Scholar
  4. 4.
    Blankertz, B.: Beweistheoretischen Techniken zur Bestimmung von \(\Pi ^0_2\)–Skolem Funktionen, Dissertation, Westfälische Wilhelms-Universität, Münster (1997)Google Scholar
  5. 5.
    Buchholz, W., Feferman, S., Pohlers, W., Sieg, W. (eds.): Iterated Inductive Definitions and Subsystems of Analysis: Recent Proof-theoretical Studies, Lecture Notes in Mathematics, vol. 897. Springer, Heidelberg (1981)Google Scholar
  6. 6.
    Buchholz, W.: Another reduction of classical \({I\!D}_\nu \) to constructive \({I\!D}_\nu \). In: Schindler, R. (ed.) Ways of Proof Theory, Ontos Mathematical Logic, vol.  2, Ontos Verlag, Frankfurt, Paris, Lancaster, New Brunswick, pp. 183–197 (2010)Google Scholar
  7. 7.
    Buchholz, W.: The \({\Omega }_{\mu +1}\)-rule, iterated inductive definitions and subsystems of analysis: recent proof-theoretical studies. In: Buchholz, W. et al. (eds.), Lecture Notes in Mathematics, vol. 897, pp. 188–233. Springer, Heidelberg (1981)Google Scholar
  8. 8.
    Buchholz, W.: A simplified version of local predicativity. In: Aczel, P. (ed.) Proof Theory, pp. 115–147. Cambridge University Press, Cambridge (1992)Google Scholar
  9. 9.
    Buchholz, W., Pohlers, W.: Provable well orderings of formal theories for transfinitely iterated inductive definitions. J. Symb. Logic 43, 118–125 (1978)CrossRefMATHGoogle Scholar
  10. 10.
    Buchholz, W., Cichon, E.A., Weiermann, A.: A uniform approach to fundamental sequences and hierarchies. Math. Logic Quart. 40, 273–286 (1994)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Feferman, S., Sieg, W.: Inductive definitions and subsystems of analysis. In: Buchholz, W. et al., (ed.), Iterated Inductive Definitions and Subsystems of Analysis: Recent Proof-theoretical Studies, Lecture Notes in Mathematics, vol.  897, pp. 16–77. Springer, Heidelberg (1981)Google Scholar
  12. 12.
    Feferman, S.: Formal theories for transfinite iteration of generalized inductive definitions and some subsystems of analysis. In: Kino, A. et al., (ed.) Intuitionism and Proof Theory, Studies in Logic and the Foundations of Mathematics, North-Holland Publishing Company, Amsterdam, pp. 303–326 (1970)Google Scholar
  13. 13.
    Feferman, S.: Preface: how we got from there to here. In: Buchholz W. et al., (eds.) Iterated Inductive Definitions and Subsystems of Analysis: Recent Proof-theoretical Studies, Lecture Notes in Mathematics, vol.  897, pp. 1–15. Springer, Heidelberg (1981)Google Scholar
  14. 14.
    Feferman, S.: The proof theory of classical and constructive inductive definitions. A forty year saga, 1968–2008. In: Schindler, R. (ed.) Ways of Proof Theory, Ontos Mathematical Logic, vol. 2, Ontos Verlag, Frankfurt, Paris, Lancaster, New Brunswick, pp. 79–95 (2010)Google Scholar
  15. 15.
    Gentzen, G.: Beweisbarkeit und Unbeweisbarkeit von Anfangsfällen der transfiniten Induktion in der reinen Zahlentheorie. Math. Ann. 119, 140–161 (1943)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Hilbert, D., Die Grundlagen der Mathematik. Vortrag gehalten auf Einladung des Mathematischen Seminars im Juli 1927 in Hamburg. In: Hamburger Mathematische Einzelschriften, vol. 5, pp. 1–21. Heft (1928)Google Scholar
  17. 17.
    Howard, W.A.: A system of abstract constructive ordinals. J. Symb. Logic 37, 355–374 (1972)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Jäger, G., Pohlers, W.: Eine beweistheoretische Untersuchung von \(({\Delta }^1_2\) -CA\()+(\)BI\()\) und verwandter Systeme. Bayerische Akademie der Wissenschaften, Sitzungsberichte 1982, pp. 1–28 (1983)Google Scholar
  19. 19.
    Jäger, G.: Theories for Admissible Sets. A Unifying Approach to Proof Theory, Studies in Proof Theory, Lecture Notes, vol.  2. Bibliopolis, Naples (1986)Google Scholar
  20. 20.
    Jäger, G., Strahm, T.: Upper bounds for metapredicative Mahlo in explicit mathematics and admissible set theory. J. Symb. Logic 66(2), 935–958 (2001)Google Scholar
  21. 21.
    Kreisel, G.: Generalized inductive definitions, Reports of seminars on the foundation of Analysis. (Stanford Report), mimeographed, vol. section III (1963)Google Scholar
  22. 22.
    Martin-Löf, P.: Hauptsatz for the intuitionistic theory of iterated inductive definitions. In: Fenstad, J.E. (ed.) Proceedings of the 2nd Scandinavian Logic Symposium, Studies in Logic and the Foundations of Mathematics, pp. 179–216, vol.  63. North-Holland Publishing Company, Amsterdam (1971)Google Scholar
  23. 23.
    Möllerfeld, M.: Systems of inductive definitions, Ph.D. thesis, Münster (2003)Google Scholar
  24. 24.
    Moschovakis, Y.N.: Axioms for computation theories – first draft. In: Gandy, R.O., Yates, C.M.E. (eds.), Logic colloquium ’69 Studies in Logic and the Foundations of Mathematics, vol. 61, pp. 199–255. North-Holland Publishing Company, Amsterdam (1971)Google Scholar
  25. 25.
    Moschovakis, Y.N.: Elementary Induction on Abstract Structures, Studies in Logic and the Foundations of Mathematics, vol. 77. North-Holland Publishing Company, Amsterdam (1974)Google Scholar
  26. 26.
    Pohlers, W., Stegert, J.C.: Provably recursive functions of reflections. In: Berger, U. et al. (eds.), Logic, Construction, Computation, Ontos Mathematical Logic, vol. 3, pp. 381–474. Ontos Verlag, Frankfurt (2012)Google Scholar
  27. 27.
    Pohlers, W.: An upper bound for the provability of transfinite induction, \(\models \). In: Diller, J., Müller, G.H. (eds.) ISILC Proof Theory Symposium, Lecture Notes in Mathematics, vol. 500, pp. 271–289. Springer, Heidelberg (1975)Google Scholar
  28. 28.
    Pohlers, W.: Beweistheorie der iterierten induktiven Definitionen. Ludwig Maximilians-Universität, München, München, Habilschrift (1977)Google Scholar
  29. 29.
    Pohlers, W.: From subsystems of analysis to subsystems of set theory. In: Kahle, R. et al., (eds.), Advances in Proof Theory, Progress in Computer Science and Applied Logic 28, pp. 319–338. Birkhäuser Verlag (2016)Google Scholar
  30. 30.
    Pohlers, W.: Proof-theoretical analysis of ID\(_{\nu }\) by the method of local predicativity. In: Buchholz, W. et al., (eds.) Iterated Inductive Definitions and Subsystems of Analysis: Recent Proof-theoretical Studies, Lecture Notes in Mathematics, vol. 897, pp. 261–357. Springer, Heidelberg (1981)Google Scholar
  31. 31.
    Pohlers, W.: Semi–formal calculi and their applications. In: Kahle, R., Rathjen, M. (eds.), Gentzen’s Centenary: The Quest for Consistency, pp. 195–232. Springer, Berlin (2015)Google Scholar
  32. 32.
    Pohlers, W.: Ordinals connected with formal theories for transfinitely iterated inductive definitions. J. Symb. Logic 43, 161–182 (1978)MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Pohlers, W.: Proof Theory, The First Step into Impredicativity. Universitext, Springer, Berlin (2009)Google Scholar
  34. 34.
    Pohlers, W.: Hilbert’s programme and ordinal analysis. In  Probst, D., Schuster, P. (eds.), Concepts of Proof in Mathematics, Philosophy and Computer Science, Ontos Mathematical Logic, DeGruyter, pp. 291–322 (2016)Google Scholar
  35. 35.
    Rathjen, M.: Proof theory of reflection. Ann. Pure Appl. Logic 68, 181–224 (1994)MathSciNetCrossRefMATHGoogle Scholar
  36. 36.
    Rathjen, M.: An ordinal analyis of parameter free \({\Pi }_2^1\)-comprehension. Arch. Math. Logic 48(3), 263–362 (2005)Google Scholar
  37. 37.
    Rathjen, M.: An ordinal analyis of stability. Arch. Math. Logic 48(2), 1–62 (2005)Google Scholar
  38. 38.
    Schindler, R. (ed.): Ways of Proof Theory, Ontos Mathematical Logic, vol. 2. Ontos Verlag, Frankfurt (2010)Google Scholar
  39. 39.
    Schütte, K.: Proof theory, Grundlehren der mathematischen Wissenschaften, vol. 225. Springer, Heidelberg (1977)Google Scholar
  40. 40.
    Sieg, W.: Inductive definitions, constructive ordinals and normal derivations. In: Buchholz, W. et al. (eds.), Iterated inductive Definitions and Subsystems of Analysis: Recent Proof-theoretical Studies, Lecture Notes in Mathematics, vol.  897, pp. 143–187. Springer, Heidelberg (1981)Google Scholar
  41. 41.
    Spector, C.: Provably recursive functionals of analysis: a consistency proof of analysis by an extension of principles formulated in current intuitionistic mathematics. In: Dekker, J.C.E. (ed.), Recursive Function Theory, Proceedings of Symposia in Pure Mathematics, vol.  5, pp. 1–27. American Mathematical Society, Providence (1962)Google Scholar
  42. 42.
    Stegert, J.-C.: Ordinal proof theory of Kripke–Platek set theory augmented by strong reflection principles, Ph.D. thesis, Westfälische Wilhelms-Universität, Münster (2011)Google Scholar
  43. 43.
    Zucker, J.I.: Iterated inductive definitions, trees, and ordinals. In: Troelstra, A.S. (ed.), Metamathematical Investigation of Intuitionistic Arithmetic and Analysis, Lecture Notes in Mathematics, vol. 344, pp. 392–453, Springer, Heidelberg (1973)Google Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2017

Authors and Affiliations

  1. 1.Institute for Mathematical Logic and Foundational Research Universität MünsterMünsterGermany

Personalised recommendations