Abstract
This paper consists of two parts. In the first part we argue that an appropriate “naive type theory” should replace naive set theory (as understood in Halmos’ book) in everyday mathematical practice, especially in teaching mathematics to Computer Science students. In the second part we make the first step towards developing such a theory: we discuss a certain pure type system with powerset types. While the system only covers very initial aspects of the intended theory, we believe it can be used as an initial formalism to be further developed. The consistency of this basic system is established by proving strong normalization.
Partly supported by the Polish Government Grant 3 T11C 002 27, and by the EU Coordination Action 510996 “Types for Proofs and Programs”.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Adams, R., Luo, Z.: Weyl’s predicative classical mathematics as a logic-enriched type theory. In: Altenkirch, T., McBride, C. (eds.) TYPES 2006. LNCS, vol. 4502, Springer, Heidelberg (2007)
Andrews, P.B.: An Introduction to Mathematical Logic and Type Theory: To Truth Through Proof, 2nd edn. Applied Logic Series, vol. 27. Kluwer Academic Publishers, Dordrecht (2002)
Barendregt, H.P.: Lambda calculi with types. In: Abramsky, S., Gabbay, D.M., Maibaum, T.S.E. (eds.) Handbook of Logic in Computer Science, vol. II, pp. 117–309. Oxford University Press, Oxford (1992)
Barthe, G.: Extensions of pure type systems. In: Dezani-Ciancaglini and Plotkin [10], pp. 16–31
Bertot, Y., Castéran, P.: Interactive Theorem Proving and Program Development. Coq’Art: The Calculus of Inductive Constructions. Texts in Theoretical Computer Science. An EATCS Series, Springer, Heidelberg (2004)
de Bruijn, N.G.: A survey of the project automath. In: Seldin, J.P., Hindley, J.R. (eds.) To H.B. Curry: Essays on Combinatory Logic, Lambda Calculus and Formalism, pp. 579–606. Academic Press, London (1980)
Chrząszcz, J., Sakowicz, J.: Papuq: A Coq assistant (manuscript, 2007)
Church, A.: A formulation of the simple theory of types. Journal of Symbolic Logic 5(2), 56–68 (1940)
Constable, R.L.: Naive computational type theory. In: Schwichtenberg, H., Steinbruggen, R. (eds.) Proof and System-Reliability, pp. 213–259. Kluwer Academic Press, Dordrecht (2002)
Dezani-Ciancaglini, M., Plotkin, G. (eds.): TLCA 1995. LNCS, vol. 902. Springer, Heidelberg (1995)
Farmer, W.M.: A partial functions version of Church’s simple theory of types. Journal of Symbolic Logic 55(3), 1269–1291 (1990)
Farmer, W.M.: A simple type theory with partial functions and subtypes. Annals of Pure and Applied Logic 64, 211–240 (1993)
Farmer, W.M.: A basic extended simple type theory. Technical Report 14, McMaster University (2003)
Farmer, W.M.: The seven virtues of simple type theory. Technical Report 18, McMaster University (2003)
Farmer, W.M.: Formalizing undefinedness arising in calculus. In: Basin, D., Rusinowitch, M. (eds.) IJCAR 2004. LNCS (LNAI), vol. 3097, pp. 475–489. Springer, Heidelberg (2004)
Geuvers, H.: The Church-Rosser property for beta-eta-reduction in typed lambda calculi. In: Logic in Computer Science, pp. 453–460 (1992)
Geuvers, H.: Private communication (2006)
Halmos, P.R.: Naive Set Theory. Van Nostrand, 1960. Reprinted by Springer, Heidelberg (1998)
Hurkens, A.J.C.: A simplification of Girard’s paradox. In: Dezani-Ciancaglini and Plotkin [10], pp. 266–278
Jensen, R.B.: On the consistency of a slight(?) modification of Quine’s NF. Synthese 19, 250–263 (1969)
Kamareddine, F., Laan, T., Nederpelt, R.: Types in logic and mathematics before 1940. Bulletin of Symbolic Logic 8(2), 185–245 (2002)
Luo, Z.: A type-theoretic framework for formal reasoning with different logical foundations. In: Okada, M., Satoh, J. (eds.) ASIAN 2006. LNCS, vol. 4435, pp. 214–222. Springer, Heidelberg (2006)
Quine, W.V.: New foundations for mathematical logic. American Mathematical Monthly 44, 70–80 (1937)
Sørensen, M.H., Urzyczyn, P.: Lectures on the Curry-Howard Isomorphism. Elsevier, Amsterdam (2006)
Weyl, H.: The Continuum. Dover, Mineola, NY (1994)
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2008 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Kozubek, A., Urzyczyn, P. (2008). In the Search of a Naive Type Theory. In: Miculan, M., Scagnetto, I., Honsell, F. (eds) Types for Proofs and Programs. TYPES 2007. Lecture Notes in Computer Science, vol 4941. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-68103-8_8
Download citation
DOI: https://doi.org/10.1007/978-3-540-68103-8_8
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-68084-0
Online ISBN: 978-3-540-68103-8
eBook Packages: Computer ScienceComputer Science (R0)