Notre Dame Journal of Formal Logic

On Interpretability in the Theory of Concatenation

Vítězslav Švejdar


We prove that a variant of Robinson arithmetic Q with nontotal operations is interpretable in the theory of concatenation TC introduced by A. Grzegorczyk. Since Q is known to be interpretable in that nontotal variant, our result gives a positive answer to the problem whether Q is interpretable in TC . An immediate consequence is essential undecidability of TC .

Article information

Notre Dame J. Formal Logic, Volume 50, Number 1 (2009), 87-95.

First available in Project Euclid: 19 January 2009

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Zentralblatt MATH identifier

Primary: 03B25: Decidability of theories and sets of sentences [See also 11U05, 12L05, 20F10] 03F25: Relative consistency and interpretations

concatenation interpretability Robinson arithmetic essential undecidability


Švejdar, Vítězslav. On Interpretability in the Theory of Concatenation. Notre Dame J. Formal Logic 50 (2009), no. 1, 87--95. doi:10.1215/00294527-2008-029.

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  • [][1] Čačić, V., P. Pudlák, G. Restall, A. Urquhart, and A. Visser, ``Decorated linear order types and the theory of concatenation,'' 2007.
  • [2] Ganea, M., "Arithmetic on semigroups", The Journal of Symbolic Logic, vol. 74 (2009), pp. 265--78.
  • [3] Grzegorczyk, A., "Undecidability without arithmetization", Studia Logica, vol. 79 (2005), pp. 163--230.
  • [4] Grzegorczyk, A., and K. Zdanowski, "Undecidability and concatenation", pp. 72--91 in Andrzej Mostowski and Moundational Studies, edited by A. Ehrenfeucht, V. W. Marek, and M. Srebrny, IOS, Amsterdam, 2008.
  • [5] Hájek, P., "Mathematical fuzzy logic and natural numbers", Fundamenta Informaticae, vol. 81 (2007), pp. 155--63.
  • [6] Quine, W. V., "Concatenation as a basis for arithmetic", The Journal of Symbolic Logic, vol. 11 (1946), pp. 105--14.
  • [7] Švejdar, V., "An interpretation of Robinson arithmetic in its Grzegorczyk's weaker variant", Fundamenta Informaticae, vol. 81 (2007), pp. 347--54.
  • [8] Tarski, A., Undecidable Theories, Studies in Logic and the Foundations of Mathematics. North-Holland Publishing Company, Amsterdam, 1953. In collaboration with A. Mostowski and R. M. Robinson.
  • [9] Visser, A., "An overview of interpretability logic", pp. 307--59 in Advances in Modal Logic I (AiML'96, Berlin), edited by M. Kracht, M. de Rijke, H. Wansing, and M. Zakharyaschev, vol. 87 of CSLI Lecture Notes, CSLI Publications, Stanford, 1998.
  • [10] Visser, A., "Growing commas: A study of sequentiality and concatenation", Notre Dame Journal of Formal Logic, vol. 50 (2009), pp. 61--85.