Abstract
This paper proves the non-derivability of induction in second order dependent type theory (λP2). This is done by providing a model construction for λP2, based on a saturated sets like interpretation of types as sets of terms of a weakly extensional combinatory algebra. We give counter-models in which the induction principle over natural numbers is not valid. The proof does not depend on the specific encoding for natural numbers that has been chosen (like e.g. polymorphic Church numerals), so in fact we prove that there can not be an encoding of natural numbers in λP2 such that the induction principle is satisfied. The method extends immediately to other data types, like booleans, lists, trees, etc.
In the process of the proof we establish some general properties of the models, which we think are of independent interest. Moreover, we show that the Axiom of Choice is not derivable in λP2.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
H.P. Barendregt, Combinatory Logic and the Axiom of Choice, in Indagationes Mathematicae, vol. 35,nr. 3, pp. 203–221.
H.P. Barendregt, Typed lambda calculi. In Handbook of Logic in Computer Science, eds. Abramski et al., Oxford Univ. Press.
S. Berardi, Encoding of data types in Pure Construction Calculus: a semantic justification, in Logical Environments, eds. G. Huet and G. Plotkin, Cambridge University Press, pp 30–60.
Th. Coquand, Metamathematical investigations of a calculus of constructions. In Logic and Computer Science, ed. P.G. Odifreddi, APIC series, vol. 31, Academic Press, pp 91–122.
Th. Coquand and G. Huet, The calculus of constructions, Information and Computation, 76, pp 95–120.
J.H. Geuvers, Logics and Type systems, Ph.D. Thesis, University of Nijmegen, Netherlands.
J.H. Geuvers, Extending models of second order logic to models of second order dependent type theory, Computer Science Logic, Utrecht, eds. D. van Dalen and M. Bezem, LNCS 1258, 1997, pp 167–181.
J.M.E. Hyland and C.-H. L. Ong, Modified realizability toposes and strong normalization proofs. In Typed Lambda Calculi and Applications, Proceedings, eds. M. Bezem and J.F. Groote, LNCS 664, pp. 179–194, Springer-Verlag, 1993.
J.-Y. Girard, Y. Lafont and P. Taylor, Proofs and types, Camb. Tracts in Theoretical Computer Science 7, Cambridge University Press.
G. Longo and E. Moggi, Constructive Natural Deduction and its “Modest” Interpretation. Report CMU-CS-88-131.
M. Stefanova and J.H. Geuvers, A Simple Model Construction for the Calculus of Constructions, in Types for Proofs and Programs, Int. Workshop, Torino, eds. S. Berardi and M. Coppo, LNCS 1158, 1996, pp. 249–264.
T. Streicher, Independence of the induction principle and the axiom of choice in the pure calculus of constructions, TCS 103(2), pp 395–409.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2001 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Geuvers, H. (2001). Induction Is Not Derivable in Second Order Dependent Type Theory. In: Abramsky, S. (eds) Typed Lambda Calculi and Applications. TLCA 2001. Lecture Notes in Computer Science, vol 2044. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45413-6_16
Download citation
DOI: https://doi.org/10.1007/3-540-45413-6_16
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-41960-0
Online ISBN: 978-3-540-45413-7
eBook Packages: Springer Book Archive