Archive for Mathematical Logic

, Volume 49, Issue 2, pp 275–281 | Cite as

A note on the theory of positive induction, \({{\rm ID}^*_1}\)

Article

Abstract

The article shows a simple way of calibrating the strength of the theory of positive induction, \({{\rm ID}^{*}_{1}}\) . Crucially the proof exploits the equivalence of \({\Sigma^{1}_{1}}\) dependent choice and ω-model reflection for \({\Pi^{1}_{2}}\) formulae over ACA 0. Unbeknown to the authors, D. Probst had already determined the proof-theoretic strength of \({{\rm ID}^{*}_{1}}\) in Probst, J Symb Log, 71, 721–746, 2006.

Mathematics Subject Classification (2000)

03B30 03F25 03F35 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Aczel, P.: The Strength of Martin-Löf’s Type Theory with One Universe. Technical report, Department of Philosophy, University of Helsinki (1977)Google Scholar
  2. 2.
    Afshari B.: Relative Computability and the Proof-Theoretic Strengths of Some Theories. PhD thesis. University of Leeds, UK (2008)Google Scholar
  3. 3.
    Buchholz W., Feferman S., Pohlers W., Sieg W.: Iterated Inductive Definitions and Subsystems of Analysis: Recent Proof-Theoretical Studies Theories. Springer-Verlag, Berlin, Heidelberg (1981)MATHGoogle Scholar
  4. 4.
    Cantini, A.: A Note on a Predicatively Reducible Theory of Elementary Iterated Induction. Bollettino U.M.I 413–430 (1985)Google Scholar
  5. 5.
    Cantini A.: On the relation between choice and comprehension principles in second order arithmetic. J. Symb. Log. 51, 360–373 (1986)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Feferman S.: Iterated Inductive Fixed-Point Theories: Application to Hancock’s Conjecture, Patras Logic Symposion, pp. 171–196. North-Holland, Amsterdam (1982)Google Scholar
  7. 7.
    Friedman, H.: Subtheories of Set Theory and Analysis, Dissertation, MIT (1967)Google Scholar
  8. 8.
    Friedman, H.: Theories of Inductive Definitions, Unpublished notes (1969)Google Scholar
  9. 9.
    Friedman H., Sheard M.: An axiomatic approach to self-referential truth. Ann. Pure Appl. Log. 33, 1–21 (1987)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Jäger G., Strahm T.: Some theories with positive induction of ordinal strength \({\varphi\omega 0}\) . J. Symb. Log. 61, 818–842 (1996)MATHCrossRefGoogle Scholar
  11. 11.
    Kreisel, G.: Generalized Inductive Definitions. Technical report, Stanford University (1963)Google Scholar
  12. 12.
    Leigh, G., Rathjen, M.: An ordinal analysis for theories of self-referential truth. To appear in Arch. Math. Logic. doi: 10.1007/s00153-009-0170-2
  13. 13.
    Probst D.: The proof-theoretic analysis of transfinitely iterated quasi least fixed points. J. Symb. Log. 71, 721–746 (2006)MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Simpson S.G.: \({\Sigma^{1}_{1}}\) and \({\Pi^{1}_{1}}\) transfinite induction. In: Dalen, D., Lascar, D., Smiley, TJ (eds) Logic Colloquium ’80, pp. 239–253. North-Holland, Amsterdam (1980)Google Scholar
  15. 15.
    Simpson S.G.: Subsystems of Second Order Arithmetic. Springer-Verlag, Berlin, Heidelberg (1999)MATHGoogle Scholar

Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  1. 1.School of MathematicsUniversity of LeedsLeedsUK

Personalised recommendations