Archive for Mathematical Logic

, Volume 42, Issue 6, pp 515–552 | Cite as

Proof-theoretic analysis by iterated reflection

Article

Abstract.

Progressions of iterated reflection principles can be used as a tool for the ordinal analysis of formal systems. We discuss various notions of proof-theoretic ordinals and compare the information obtained by means of the reflection principles with the results obtained by the more usual proof-theoretic techniques. In some cases we obtain sharper results, e.g., we define proof-theoretic ordinals relevant to logical complexity Π10 and, similarly, for any class Π n 0 . We provide a more general version of the fine structure relationships for iterated reflection principles (due to U. Schmerl [25]). This allows us, in a uniform manner, to analyze main fragments of arithmetic axiomatized by restricted forms of induction, including IΣ n , IΣ n , IΠ n and their combinations. We also obtain new conservation results relating the hierarchies of uniform and local reflection principles. In particular, we show that (for a sufficiently broad class of theories T) the uniform Σ1-reflection principle for T is Σ2-conservative over the corresponding local reflection principle. This bears some corollaries on the hierarchies of restricted induction schemata in arithmetic and provides a key tool for our generalization of Schmerl's theorem.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    L.D. Beklemishev L.D.: Remarks on Magari algebras of PA and IΔ0+ EXP. In P. Agliano and A. Ursini, editors, Logic and Algebra, pages 317–325. Marcel Dekker, New York, 1996.Google Scholar
  2. 2.
    Beklemishev L.D.: Induction rules, reflection principles, and provably recursive functions. Annals of Pure and Applied Logic, 85, 193–242 (1997)Google Scholar
  3. 3.
    Beklemishev L.D.: Notes on local reflection principles. Theoria, 63, 139–146 (1997)Google Scholar
  4. 4.
    Beklemishev L.D.: A proof-theoretic analysis of collection. Archive for Mathematical Logic, 37, 275–296 (1998)Google Scholar
  5. 5.
    Beklemishev L.D.: Parameter free induction and provably total computable functions. Theoretical Computer Science, 224, 13–33 (1999)Google Scholar
  6. 6.
    Beklemishev L.D.: Provability algebras and proof-theoretic ordinals, I. Logic Group Preprint Series 208, University of Utrecht, 2001. http://www.phil.uu.nl/ preprints.html. To appear in Annals of Pure and Applied Logic Google Scholar
  7. 7.
    Feferman S.: Arithmetization of metamathematics in a general setting. Fundamenta Mathematicae, 49, 35–92 (1960)Google Scholar
  8. 8.
    Feferman S.: Transfinite recursive progressions of axiomatic theories. Journal of Symbolic Logic, 27, 259–316 (1962)Google Scholar
  9. 9.
    Gaifman H., Dimitracopoulos C.: Fragments of Peano's arithmetic and the MDRP theorem. In Logic and algorithmic (Zurich, 1980), (Monograph. Enseign. Math., 30), pages 187–206. Genève, University of Genève, 1982Google Scholar
  10. 10.
    Goryachev S.: On interpretability of some extensions of arithmetic. Mat. Zametki, 40, 561–572 (1986) In Russian. English translation in Math. Notes, 40Google Scholar
  11. 11.
    Hájek P., Pudlák P.: Metamathematics of First Order Arithmetic. Springer-Verlag, Berlin, Heidelberg, New York, 1993Google Scholar
  12. 12.
    Kaye, R., Paris, J., Dimitracopoulos, C.: On parameter free induction schemas. Journal of Symbolic Logic 53, 1082–1097 (1988)MathSciNetMATHGoogle Scholar
  13. 13.
    Kreisel, G.: Wie die Beweistheorie zu ihren Ordinalzahlen kam und kommt. Jahresbericht der Deutschen Mathematiker-Vereinigung, 78, 177–223 (1977)Google Scholar
  14. 14.
    Kreisel, G., Lévy, A.L.: Reflection principles and their use for establishing the complexity of axiomatic systems. Zeitschrift f. math. Logik und Grundlagen d. Math., 14, 97–142 (1968)Google Scholar
  15. 15.
    Leivant, D.: The optimality of induction as an axiomatization of arithmetic. Journal of Symbolic Logic 48, 182–184 (1983)MathSciNetMATHGoogle Scholar
  16. 16.
    Lindström, P.: The modal logic of Parikh provability. Tech. Rep. Filosofiska Meddelanden, Gröna Serien~5, Univ. Göteborg, 1994Google Scholar
  17. 17.
    Möllerfeld, M.: Zur Rekursion längs fundierten Relationen und Hauptfolgen. Diplomarbeit, Institut für Mathematische Logik, Westf. Wilhelms-Universität, Münster, 1996Google Scholar
  18. 18.
    Ono, H.: Reflection principles in fragments of Peano Arithmetic. Zeitschrift f. math. Logik und Grundlagen d. Math., 33, 317–333 (1987)Google Scholar
  19. 19.
    Parikh, R.: Existence and feasibility in arithmetic. Journal of Symbolic Logic 36, 494–508 (1971)MATHGoogle Scholar
  20. 20.
    Parsons, C.: On a number-theoretic choice schema and its relation to induction. In A. Kino, J. Myhill, and R.E. Vessley, editors, Intuitionism and Proof Theory, pages 459–473. North Holland, Amsterdam, 1970Google Scholar
  21. 21.
    Parsons, C.: On n-quantifier induction. Journal of Symbolic Logic, 37, 466–482 (1972)Google Scholar
  22. 22.
    Pohlers, W.: A short course in ordinal analysis. In A. Axcel and S. Wainer, editors, Proof Theory, Complexity, Logic, pages 867–896. Oxford University Press, Oxford, 1993Google Scholar
  23. 23.
    Pohlers, W.: Subsystems of set theory and second order number theory. In S.R. Buss, editor, Handbook of Proof Theory, pages 210–335. Elsevier, North-Holland, Amsterdam, 1998Google Scholar
  24. 24.
    Rose, H.E.: Subrecursion: Functions and Hierarchies. Clarendon Press, Oxford, 1984Google Scholar
  25. 25.
    Schmerl, U.R.: A fine structure generated by reflection formulas over Primitive Recursive Arithmetic. In M. Boffa, D. van Dalen, and K. McAloon, editors, Logic Colloquium'78, pages 335–350. North Holland, Amsterdam, 1979Google Scholar
  26. 26.
    Smoryński, C.: The incompleteness theorems. In J. Barwise, editor, Handbook of Mathematical Logic, pages 821–865. North Holland, Amsterdam, 1977Google Scholar
  27. 27.
    Sommer, R.: Transfinite induction within Peano arithmetic. Annals of Pure and Applied Logic 76, 231–289 (1995)CrossRefMathSciNetMATHGoogle Scholar
  28. 28.
    Turing, A.M.: System of logics based on ordinals. Proc. London Math. Soc. ser. 2, 45, 161–228 (1939)Google Scholar
  29. 29.
    Visser, A.: Interpretability logic. In P.P. Petkov, editor, Mathematical Logic, pages 175–208. Plenum Press, New York, 1990Google Scholar
  30. 30.
    Wilkie, A., Paris, J.: On the scheme of induction for bounded arithmetic formulas. Annals of Pure and Applied Logic 35, 261–302 (1987)MathSciNetMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  1. 1.Steklov Mathematical Institute Gubkina 8MoscowRussia

Personalised recommendations