Skip to main content
Springer Nature Link
Log in

An ordinal analysis for theories of self-referential truth

  • Published:
Archive for Mathematical Logic Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+
from $39.99 /Month
  • Starting from 10 chapters or articles per month
  • Access and download chapters and articles from more than 300k books and 2,500 journals
  • Cancel anytime
View plans

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Explore related subjects

Discover the latest articles and news from researchers in related subjects, suggested using machine learning.

References

  1. Afshari B., Rathjen M.: Reverse mathematics and well-ordering principles: a pilot study. Ann. Pure Appl. Log. 160, 231–237 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  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. Buchholz W., Feferman S., Pohlers W., Sieg W.: Iterated Inductive Definitions and Subsystems of Analysis: Recent Proof-Theoretical Studies Theories. Springer, Berlin (1981)

    MATH  Google Scholar 

  4. Cantini, A.: A note on a predicatively reducible theory of elementary iterated induction. Bollettino U.M.I., pp. 413–430 (1985)

  5. Cantini A.: On the relation between choice and comprehension principles in second order arithmetic. J. Symb. Log. 51, 360–373 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  6. Cantini A.: A theory of formal truth arithmetically equivalent to ID 1. J. Symb. Log. 55, 244–259 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  7. Feferman S.: Systems of predicative analysis. J. Symb. Log. 29, 1–30 (1964)

    Article  MATH  MathSciNet  Google Scholar 

  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. Feferman S.: Toward useful type-free theories, I. J. Symb. Log. 49, 75–111 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  10. Feferman S.: Reflecting on incompleteness. J. Symb. Log. 56, 1–49 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  11. Friedman H., Sheard M.: An axiomatic approach to self-referential truth. Ann. Pure Appl. Log. 33, 1–21 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  12. Friedman H., Sheard M.: Elementary descent recursion and proof theory. Ann. Pure Appl. Log. 71, 1–45 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  13. Halbach V.: Truth and reduction. Erkenntnis 53, 97–126 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  14. Halbach, V.: A system of complete and consistent truth, Notre Dame J. Formal Log. 35 (1994).

  15. Hilbert D., Bernays P.: Grundagen der Mathematik I. Springer, Berlin (1939)

    Google Scholar 

  16. Jäger G., Strahm T.: Some theories with positive induction of ordinal strength φω0. J. Symb. Log. 61, 818–842 (1996)

    Article  MATH  Google Scholar 

  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)

    MATH  MathSciNet  Google Scholar 

  18. Leigh, G.: Reflecting on Truth, PhD thesis, University of Leeds, (forthcoming)

  19. McGee V.: How truth-like can a predicate be? A negative result. J. Philos. Log. 14, 399–410 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  20. Probst D.: The proof-theoretic analysis of transfinitely iterated quasi least fixed points. J. Symb. Log. 71, 721–726 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  21. Rathjen M.: The role of parameters in bar rule and bar induction. J. Symb. Log. 56, 715–730 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  22. Schütte K.: Proof Theory. Springer, Berlin (1977)

    MATH  Google Scholar 

  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. Sheard M.: A guide to truth predicates in the modern era. J. Symb. Log. 59, 1032–1053 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  25. Sheard M.: Weak and strong theories of truth. Stud. Logica 68, 89–101 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  26. Simpson S.: Subsystems of Second Order Arithmetic. Springer, Berlin (1998)

    Google Scholar 

  27. Smorynski C.: The incompleteness theorems. In: Barwise, J. (eds) Handbook of Mathematical Logic, pp. 821–865. North-Holland, Amsterdam (1977)

    Chapter  Google Scholar 

  28. Takeuti G.: Proof Theory, 2nd edn. North-Holland, Amsterdam (1987)

    MATH  Google Scholar 

  29. Tarski A.: Der Wahrheitsbegriff in den formalisierten Sprachen. Stud. Philos. 1, 261–404 (1936)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Pure Mathematics, University of Leeds, Leeds, LS2 9JT, UK

    Graham Emil Leigh & Michael Rathjen

Authors
  1. Graham Emil Leigh
    View author publications

    Search author on:PubMed Google Scholar

  2. Michael Rathjen
    View author publications

    Search author on:PubMed Google Scholar

Corresponding author

Correspondence to Graham Emil Leigh.

Additional information

Research of the first author was partially supported by a University of Leeds Research Scholarship and the Marie Curie Early Stage Training Network MATHLOGAPS (MEST-CT-2004-504029). Research of the second author was supported by a Royal Society International Joint Projects award 2006/R3. The second author would like to thank the Swedish Collegium for Advanced Study in Uppsala for providing an excellent research environment for the completion of this paper.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Leigh, G.E., Rathjen, M. An ordinal analysis for theories of self-referential truth. Arch. Math. Logic 49, 213–247 (2010). https://doi.org/10.1007/s00153-009-0170-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00153-009-0170-2

Keywords

Mathematics Subject Classification (2000)