Archive for Mathematical Logic

, Volume 49, Issue 2, pp 213–247 | Cite as

An ordinal analysis for theories of self-referential truth

Article

Abstract

The first attempt at a systematic approach to axiomatic theories of truth was undertaken by Friedman and Sheard (Ann Pure Appl Log 33:1–21, 1987). There twelve principles consisting of axioms, axiom schemata and rules of inference, each embodying a reasonable property of truth were isolated for study. Working with a base theory of truth conservative over PA, Friedman and Sheard raised the following questions. Which subsets of the Optional Axioms are consistent over the base theory? What are the proof-theoretic strengths of the consistent theories? The first question was answered completely by Friedman and Sheard; all subsets of the Optional Axioms were classified as either consistent or inconsistent giving rise to nine maximal consistent theories of truth.They also determined the proof-theoretic strength of two subsets of the Optional Axioms. The aim of this paper is to continue the work begun by Friedman and Sheard. We will establish the proof-theoretic strength of all the remaining seven theories and relate their arithmetic part to well-known theories ranging from PA to the theory of \({\Sigma^1_1}\) dependent choice.

Keywords

Theories of truth Relative consistency and interpretations Ordinal analysis Cut elimination 

Mathematics Subject Classification (2000)

03F03 03F25 03F05 03B42 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Afshari B., Rathjen M.: Reverse mathematics and well-ordering principles: a pilot study. Ann. Pure Appl. Log. 160, 231–237 (2009)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Afshari, B., Rathjen, M.: A note on the theory of positive induction, \({{\rm ID}_1^*}\), Arch. Math. Logic (2009). doi: 10.1007/s00153-009-0168-9
  3. 3.
    Buchholz W., Feferman S., Pohlers W., Sieg W.: Iterated Inductive Definitions and Subsystems of Analysis: Recent Proof-Theoretical Studies Theories. Springer, Berlin (1981)MATHGoogle Scholar
  4. 4.
    Cantini, A.: A note on a predicatively reducible theory of elementary iterated induction. Bollettino U.M.I., pp. 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.
    Cantini A.: A theory of formal truth arithmetically equivalent to ID 1. J. Symb. Log. 55, 244–259 (1990)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Feferman S.: Systems of predicative analysis. J. Symb. Log. 29, 1–30 (1964)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Feferman S.: Iterated Inductive Fixed-Point Theories: Application to Hancock’s Conjecture. Patras Logic Symposion, pp. 171–196. North-Holland, Amsterdam (1982)Google Scholar
  9. 9.
    Feferman S.: Toward useful type-free theories, I. J. Symb. Log. 49, 75–111 (1984)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Feferman S.: Reflecting on incompleteness. J. Symb. Log. 56, 1–49 (1991)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Friedman H., Sheard M.: An axiomatic approach to self-referential truth. Ann. Pure Appl. Log. 33, 1–21 (1987)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Friedman H., Sheard M.: Elementary descent recursion and proof theory. Ann. Pure Appl. Log. 71, 1–45 (1995)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Halbach V.: Truth and reduction. Erkenntnis 53, 97–126 (2000)MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Halbach, V.: A system of complete and consistent truth, Notre Dame J. Formal Log. 35 (1994).Google Scholar
  15. 15.
    Hilbert D., Bernays P.: Grundagen der Mathematik I. Springer, Berlin (1939)Google Scholar
  16. 16.
    Jäger G., Strahm T.: Some theories with positive induction of ordinal strength φω0. J. Symb. Log. 61, 818–842 (1996)MATHCrossRefGoogle Scholar
  17. 17.
    Kreisel G.: The axiom of choice and the class of hyperarithmetic functions. Koninklijke Nederlands Akadenie van Wetenschappen, proceedings, ser. A. 65, 307–319 (1962)MATHMathSciNetGoogle Scholar
  18. 18.
    Leigh, G.: Reflecting on Truth, PhD thesis, University of Leeds, (forthcoming)Google Scholar
  19. 19.
    McGee V.: How truth-like can a predicate be? A negative result. J. Philos. Log. 14, 399–410 (1985)MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Probst D.: The proof-theoretic analysis of transfinitely iterated quasi least fixed points. J. Symb. Log. 71, 721–726 (2006)MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Rathjen M.: The role of parameters in bar rule and bar induction. J. Symb. Log. 56, 715–730 (1991)MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Schütte K.: Proof Theory. Springer, Berlin (1977)MATHGoogle Scholar
  23. 23.
    Schwichtenberg H.: Proof theory: some applications of cut-elimination. In: Barwise, J. (eds) Handbook of Mathematical Logic, North-Holland, Amsterdam (1977)Google Scholar
  24. 24.
    Sheard M.: A guide to truth predicates in the modern era. J. Symb. Log. 59, 1032–1053 (1994)MATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Sheard M.: Weak and strong theories of truth. Stud. Logica 68, 89–101 (2001)MATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Simpson S.: Subsystems of Second Order Arithmetic. Springer, Berlin (1998)Google Scholar
  27. 27.
    Smorynski C.: The incompleteness theorems. In: Barwise, J. (eds) Handbook of Mathematical Logic, pp. 821–865. North-Holland, Amsterdam (1977)CrossRefGoogle Scholar
  28. 28.
    Takeuti G.: Proof Theory, 2nd edn. North-Holland, Amsterdam (1987)MATHGoogle Scholar
  29. 29.
    Tarski A.: Der Wahrheitsbegriff in den formalisierten Sprachen. Stud. Philos. 1, 261–404 (1936)Google Scholar

Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  1. 1.Department of Pure MathematicsUniversity of LeedsLeedsUK

Personalised recommendations