Advertisement

Archive for Mathematical Logic

, Volume 56, Issue 5–6, pp 453–474 | Cite as

Models of weak theories of truth

  • Mateusz ŁełykEmail author
  • Bartosz Wcisło
Open Access
Article

Abstract

In the following paper we propose a model-theoretical way of comparing the “strength” of various truth theories which are conservative over \( PA \). Let \({\mathfrak {Th}}\) denote the class of models of \( PA \) which admit an expansion to a model of theory \({ Th}\). We show (combining some well known results and original ideas) that
$$\begin{aligned} {{\mathfrak {PA}}}\supset {\mathfrak {TB}}\supset {{\mathfrak {RS}}}\supset {\mathfrak {UTB}}\supseteq \mathfrak {CT^-}, \end{aligned}$$
where \({\mathfrak {PA}}\) denotes simply the class of all models of \( PA \) and \({\mathfrak {RS}}\) denotes the class of recursively saturated models of \( PA \). Our main original result is that every model of \( PA \) which admits an expansion to a model of \( CT ^-\), admits also an expansion to a model of \( UTB \). Moreover, as a corollary to one of the results (brought to us by Carlo Nicolai) we conclude that \( UTB \) is not relatively interpretable in \( TB \), thus answering the question from Fujimoto (Bull Symb Log 16:305–344, 2010).

Keywords

Axiomatic truth theories Satisfaction classes Models of Peano arithmetic Recursive saturation 

Mathematics Subject Classification

03C62 03H15 

Notes

Acknowledgements

Funding was provided by National Science Centre in Cracow (NCN) (Grant No. DEC-2011/01/B/HS1/03910).

References

  1. 1.
    Enayat, A., Visser, A.: New constructions of satisfaction classes. Logic Group Preprint Series 303Google Scholar
  2. 2.
    Field, H.: Deflationist views of meaning and content. Mind New Ser. 103, 249–285 (1994)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Fujimoto, K.: Relative truth definability of axiomatic truth theories. Bull. Symb. Log. 16, 305–344 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Halbach, V.: Axiomatic Theories of Truth. Cambridge University, Cambridge (2011)CrossRefzbMATHGoogle Scholar
  5. 5.
    Horsten, L.: Levity. Mind 118, 555–581 (2009)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Kaufmann, M.: A rather classless model. Proc. Am. Math. Soc. 62, 330–333 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Kaye, R.: Models of Peano Arithmetic. Oxford University Press, New York (1991)zbMATHGoogle Scholar
  8. 8.
    Ketland, J.: Deflationism and Tarski’s paradise. Mind 108, 69–94 (1999)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Kossak, R., Schmerl, J.H.: The Structure of Models of Peano Arithmetic. Clarendon, Oxford (2006)CrossRefzbMATHGoogle Scholar
  10. 10.
    Leigh, G.: Conservativity for theories of compositional truth via cut elimination. J. Symb. Log. 80, 845–865 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Schmerl, J.: Recursively saturated, rather classless models of Peano arithmetic. Lect. Notes Math. 859, 268–282 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Shapiro, S.: Proof and truth: through thick and thin. J. Philos. 95, 493–521 (1998)MathSciNetGoogle Scholar
  13. 13.
    Shelah, S.: Models with second order properties II. Trees with no undefined branches. Ann. Math. Log. 14, 73–87 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Smith, S.T.: Nonstandard definability. Ann. Pure Appl. Log. 42, 21–43 (1989)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© The Author(s) 2017

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.University of WarsawWarsawPoland

Personalised recommendations