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DISJUNCTIONS WITH STOPPING CONDITIONS

Part of: Proof theory and constructive mathematics Philosophical aspects of logic and foundations Nonstandard models

Published online by Cambridge University Press:  05 January 2021

ROMAN KOSSAK  and
BARTOSZ WCISŁO
ROMAN KOSSAK
Affiliation:
THE GRADUATE CENTER CITY UNIVERSITY OF NEW YORK 365 FIFTH AVENUE, NEW YORK, NY10016, USAE-mail: RKossak@gc.cuny.edu
BARTOSZ WCISŁO
Affiliation:
INSTITUTE OF MATHEMATICS POLISH ACADEMY OF SCIENCES UL. ŚNIADECKICH 800-656WARSAW, POLANDE-mail: bar.wcislo@gmail.com
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Abstract

We introduce a tool for analysing models of $\text {CT}^-$ , the compositional truth theory over Peano Arithmetic. We present a new proof of Lachlan’s theorem that the arithmetical part of models of $\text {CT}^-$ are recursively saturated. We also use this tool to provide a new proof of theorem from [8] that all models of $\text {CT}^-$ carry a partial inductive truth predicate. Finally, we construct a partial truth predicate defined for a set of formulae whose syntactic depth forms a nonstandard cut which cannot be extended to a full truth predicate satisfying $\text {CT}^-$ .

Keywords

truth theoriestruth predicatessatisfaction classesPeano Arithmeticmodels of arithmeticrecursive saturation

MSC classification

Primary: 03H15: Nonstandard models of arithmetic
Secondary: 03F30: First-order arithmetic and fragments 03A99: None of the above, but in this section
Type
Articles
Information
Bulletin of Symbolic Logic , Volume 27 , Issue 3 , September 2021 , pp. 231 - 253
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Copyright
© 2021, Association for Symbolic Logic

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References

Cieśliński, C., ŁeŁyk, M., and WcisŁo, B., Models of ${{{PT}}\hspace{1.5pt}}^{-}$ with internal induction for total formulae . The Review of Symbolic Logic , vol. 10 (2017), no. 1, pp. 187–202.CrossRefGoogle Scholar
Enayat, A. and Pakhomov, F., Truth, disjunction, and induction. Archive for Mathematical Logic , vol. 58 (2019), pp. 753–766.Google Scholar
Enayat, A. and Visser, A., New constructions of satisfaction classes , Unifying the Philosophy of Truth (T. Achourioti, H. Galinon, J. Martínez Fernández, and K. Fujimoto, editors), Springer, Dordrecht, 2015, pp. 321–335.CrossRefGoogle Scholar
Kaye, R., Models of Peano Arithmetic , Clarendon Press, Oxford, 1991.Google Scholar
Kotlarski, H., Bounded induction and satisfaction classes . Zeitschrift für mathematische Logik und Grundlagen der Mathematik , vol. 32 (1986), pp. 531–544.CrossRefGoogle Scholar
Lachlan, A. H., Full satisfaction classes and recursive saturation . Canadian Mathmematical Bulletin , vol. 24 (1981), pp. 295–297.CrossRefGoogle Scholar
ŁeŁyk, M., Axiomatic Theories of Truth, Bounded Induction, and Reflection Principles . Ph.D. disseration, University of Warsaw, 2017.Google Scholar
ŁeŁyk, M. and WcisŁo, B., Models of weak theories of truth . Archive for Mathematical Logic , vol. 56 (2017), no. 5, pp. 453–474.CrossRefGoogle Scholar
Smith, S., Nonstandard syntax and semantics and full satisfaction classes for models of arithmetic , Ph.D. thesis, Yale University, 1984.Google Scholar
Smith, S. T., Nonstandard definability . Annals of Pure and Applied Logic , vol. 42 (1989), no. 1, pp. 21–43.CrossRefGoogle Scholar
WcisŁo, B., Understanding the strength of the compositional truth , Ph.D. thesis, University of Warsaw, 2018.Google Scholar