Notre Dame Journal of Formal Logic

Nonstandard Functional Interpretations and Categorical Models

Amar Hadzihasanovic and Benno van den Berg

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Recently, the second author, Briseid, and Safarik introduced nonstandard Dialectica, a functional interpretation capable of eliminating instances of familiar principles of nonstandard arithmetic—including overspill, underspill, and generalizations to higher types—from proofs. We show that the properties of this interpretation are mirrored by first-order logic in a constructive sheaf model of nonstandard arithmetic due to Moerdijk, later developed by Palmgren, and draw some new connections between nonstandard principles and principles that are rejected by strict constructivism. Furthermore, we introduce a variant of the Diller–Nahm interpretation with two different kinds of quantifiers, similar to Hernest’s light Dialectica interpretation, and show that one can obtain nonstandard Dialectica by weakening the computational content of the existential quantifiers—a process called herbrandization. We also define a constructive sheaf model mirroring this new functional interpretation, and show that the process of herbrandization has a clear meaning in terms of these sheaf models.

Article information

Notre Dame J. Formal Logic, Volume 58, Number 3 (2017), 343-380.

Received: 5 February 2014
Accepted: 28 October 2014
First available in Project Euclid: 19 April 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 03H15: Nonstandard models of arithmetic [See also 11U10, 12L15, 13L05]
Secondary: 03F25: Relative consistency and interpretations 03G30: Categorical logic, topoi [See also 18B25, 18C05, 18C10]

nonstandard arithmetic functional interpretations categorical logic Dialectica interpretation proof mining constructive mathematics


Hadzihasanovic, Amar; van den Berg, Benno. Nonstandard Functional Interpretations and Categorical Models. Notre Dame J. Formal Logic 58 (2017), no. 3, 343--380. doi:10.1215/00294527-3870348.

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  • [1] Avigad, J., and J. Helzner, “Transfer principles in nonstandard intuitionistic arithmetic,” Archive for Mathematical Logic, vol. 41 (2002), pp. 581–602.
  • [2] Berger, U., “Uniform Heyting arithmetic,” Annals of Pure and Applied Logic, vol. 133 (2005), pp. 125–48.
  • [3] Blass, A., “Two closed categories of filters,” Fundamenta Mathematicae, vol. 94 (1977), pp. 129–43.
  • [4] Butz, C., “Saturated models of intuitionistic theories,” Annals of Pure and Applied Logic, vol. 129 (2004), pp. 245–75.
  • [5] Diller, J., and W. Nahm, “Eine Variante zur Dialectica-Interpretation der Heyting-Arithmetik endlicher Typen,” Archive for Mathematical Logic, vol. 16 (1974), pp. 49–66.
  • [6] Gödel, K., “Über eine bisher noch nicht benützte Erweiterung des finiten Standpunktes,” Dialectica, vol. 12 (1958), pp. 280–87.
  • [7] Hernest, M.-D., “Light functional interpretation: An optimization of Gödel’s technique towards the extraction of (more) efficient programs from (classical) proofs,” pp. 477–92 in Computer Science Logic, edited by L. Ong, vol. 3634 of Lecture Notes in Computer Science, Springer, Berlin, 2005.
  • [8] Ishihara, H., “Reverse mathematics in Bishop’s constructive mathematics,” Philosophia Scientiæ (Paris), vol. 6 (2006), pp. 43–59.
  • [9] Johnstone, P. T., Sketches of an Elephant: A Topos Theory Compendium, Vol. 1, vol. 43 of Oxford Logic Guides, Oxford University Press, Oxford, 2002.
  • [10] Kohlenbach, U., Applied Proof Theory: Proof Interpretations and Their Use in Mathematics, Springer, Berlin, 2008.
  • [11] Kohlenbach, U., and P. Oliva, “Proof mining: A systematic way of analyzing proofs in mathematics,” Proceedings of the Steklov Institute of Mathematics, vol. 242 (2003), pp. 136–64.
  • [12] Kuroda, S., “Intuitionistische Untersuchungen der formalistischen Logik,” Nagoya Mathematical Journal, vol. 2 (1951), pp. 35–47.
  • [13] Lifschitz, V., “Calculable natural numbers,” pp. 173–90 in Intensional Mathematics, vol. 113 of Studies in Logic and the Foundations of Mathematics, North-Holland, Amsterdam, 1985.
  • [14] Mac Lane, S., and I. Moerdijk, Sheaves in Geometry and Logic: A First Introduction to Topos Theory, corrected reprint of the 1992 edition, Universitext, Springer, New York, 1994.
  • [15] McCarty, C., “Variations on a thesis: Intuitionism and computability,” Notre Dame Journal of Formal Logic, vol. 28 (1987), pp. 536–80.
  • [16] Moerdijk, I., “A model for intuitionistic non-standard arithmetic,” Annals of Pure and Applied Logic, vol. 73 (1995), pp. 37–51.
  • [17] Nelson, E., “Internal set theory: A new approach to nonstandard analysis,” Bulletin of the American Mathematical Society, vol. 83 (1977), pp. 1165–98.
  • [18] Oliva, P., “Unifying functional interpretations,” Notre Dame Journal of Formal Logic, vol. 47 (2006), pp. 263–90.
  • [19] Oliva, P., “Functional interpretations of linear and intuitionistic logic,” Information and Computation, vol. 208 (2010), pp. 565–77.
  • [20] Palmgren, E., “A constructive approach to nonstandard analysis,” Annals of Pure and Applied Logic, vol. 73 (1995), pp. 297–325.
  • [21] Palmgren, E., “A sheaf-theoretic foundation for nonstandard analysis,” Annals of Pure and Applied Logic, vol. 85 (1997), pp. 69–86.
  • [22] Palmgren, E., “Developments in constructive nonstandard analysis,” Bulletin of Symbolic Logic, vol. 4 (1998), pp. 233–72.
  • [23] Palmgren, E., “Constructive nonstandard representations of generalized functions,” Indagationes Mathematicae, vol. 11 (2000), pp. 129–38.
  • [24] Palmgren, E., “Unifying constructive and nonstandard analysis,” pp. 167–83 in Reuniting the Antipodes—Constructive and Nonstandard Views of the Continuum (Venice, 1999), vol. 306 of Synthese Library, Kluwer, Dordrecht, 2001.
  • [25] Robinson, A., Non-Standard Analysis, reprint of the second (1974) edition, Princeton Landmarks in Mathematics, Princeton University Press, Princeton, 1996.
  • [26] Schmieden, C., and D. Laugwitz, “Eine Erweiterung der Infinitesimalrechnung,” Mathematische Zeitschrift, vol. 69 (1958), pp. 1–39.
  • [27] Troelstra, A. S., “Models and computability,” pp. 97–174 in Metamathematical Investigation of Intuitionistic Arithmetic and Analysis, vol. 344 of Lecture Notes in Mathematics, Springer, Berlin, 1973.
  • [28] Troelstra, A. S., “Realizability,” pp. 407–73 in Handbook of Proof Theory, edited by S. R. Buss, vol. 137 of Studies in Logic and the Foundations of Mathematics, North-Holland, Amsterdam, 1998.
  • [29] van den Berg, B., “The Herbrand topos,” Mathematical Proceedings of the Cambridge Philosophical Society, vol. 155 (2013), pp. 361–74.
  • [30] van den Berg, B., E. Briseid, and P. Safarik, “A functional interpretation for nonstandard arithmetic,” Annals of Pure and Applied Logic, vol. 163 (2012), pp. 1962–94.
  • [31] van Oosten, J., “Basic category theory,” preprint,