Abstract
Working in the fragment of Martin-Löfs extensional type theory [12] which has products (but not sums) of dependent types, we consider two additional assumptions: firstly, that there are (strong) equality types; and secondly, that there is a type which is universal in the sense that terms of that type name all types, up to isomorphism. For such a type theory, we give a version of Russell's paradox showing that each type possesses a closed term and (hence) that all terms of each type are provably equal. We consider the kind of category theoretic structure which corresponds to this kind of type theory and obtain a categorical version of the paradox. A special case of this result is the degeneracy of a locally cartesian closed category with a morphism which is generic in the sense that every other morphism in the category can be obtained from it via pullback.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
H. J. Boom, The Russell Paradox in an impredicative intuitionistic theory of types, Concordia Computer Science Technical Report CSD-87-03, Montréal, July 1987.
R. Burstall and B. Lampson, A kernel language for abstract data types and modules, In: G. Kahn et al (eds.), Semantics of Data Types, Lecture Notes in Computer Science No. 173, Springer-Verlag, Berlin-Heidelberg, 1984, pp. 1–50.
L. Cardelli, A polymorphic λ-calculus with Type: Type, SRC Report, DEC, Palo Alto, 1986.
Th. Coquand, An analysis of Girard's paradox, In: Proc. 1st Annual Symp. Logic in Computer Science, IEEE Computer Society Press, Washington, 1986, pp. 227–236.
J. Cartmell, Generalized algebraic theories and contextual categories, Annals of Pure and Applied Logic 32 (1986), pp. 209–243.
J.-Y. Girard, Interprétation fonctionelle et élimination des coupures dans l'arithmétique d'ordre supérieur, Thèse de Doctorat d'Etat, Université de Paris, 1972.
D. J. Howe, The computational behaviour of Girard's paradox, In: Proc. 2nd Annual Symp. Logic in Computer Science, IEEE Computer Society Press, Washington, 1987, pp. 205–214.
H. Huwig and A. Poigné, A note on inconsistencies caused by fixpoints in a cartesian closed category, Theoretical Computer Science, to appear.
J. M. E. Hyland and A. M. Pitts, The theory of constructions: categorical semantics and topos-theoretic models, In: J. W. Gray and A. Ščedrov (eds.), Categories in Computer Science and Logic, Contemporary Math., Amer. Math. Soc., Providence RI, to appear.
G. Longo, On Church's formal theory of functions and functionals. The lambda calculus: connections to higher type recursion theory, proof theory, category theory, Annals of Pure and Applied Logic, to appear.
J. Lambek and P. J. Scott, Introduction to Higher Order Categorical Logic, Cambridge Studies in Advanced Mathematics No. 7, Cambridge University Press, 1986.
P. Martin-Löf, Constructive mathematics and computer programming, In: L. J. Cohen et al. (eds.), Logic, Methodology and Philosophy of Science VI, North-Holland, Amsterdam, 1982, pp. 153–175.
A. R. Meyer and M. B. Reinhold, “Type” is not a type, In: Conference record of 13th Annual ACM Symposium on Principles of Programming Languages, ACM, SIGACT, SIGPLAN, 1986, pp. 287–295.
R. A. G. Seely, Locally cartesian closed categories and type theory, Math. Proc. Camb. Philos. Soc. 95 (1984), pp. 33–48.
P. Taylor, Recursive domains, indexed category theory and polymorphism, Ph. D. Thesis, University of Cambridge, 1986.
A. S. Troelstra, On the syntax of Martin-Löf's theories, Theoretical Computer Science 51 (1987), pp. 1–26.
Author information
Authors and Affiliations
Additional information
The financial support of the Royal Society in London is gratefully acknowledged.
Rights and permissions
About this article
Cite this article
Pitts, A.M., Taylor, P. A note on russell's paradox in locally cartesian closed categories. Stud Logica 48, 377–387 (1989). https://doi.org/10.1007/BF00370830
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF00370830