Abstract
In Das Kontinuum, Weyl showed how a large body of classical mathematics could be developed on a purely predicative foundation. We present a logic-enriched type theory that corresponds to Weyl’s foundational system. A large part of the mathematics in Weyl’s book — including Weyl’s definition of the cardinality of a set and several results from real analysis — has been formalised, using the proof assistant Plastic that implements a logical framework. This case study shows how type theory can be used to represent a non-constructive foundation for mathematics.
This work is partially supported by the UK EPSRC research grants GR/R84092 and GR/R72259 and EU TYPES grant 510996.
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.: A Modular Hierarchy of Logical Frameworks. PhD thesis, University of Manchester (2004)
Aczel, P., Gambino, N.: Collection principles in dependent type theory. In: Callaghan, P., Luo, Z., McKinna, J., Pollack, R. (eds.) TYPES 2000. LNCS, vol. 2277, pp. 1–23. Springer, Heidelberg (2002)
Bertot, Y., Castéran, P.: Interactive Theorem Proving and Program Development: Coq’Art: The Calculus of Inductive Constructions. In: Texts in Theoretical Computer Science, Springer, Heidelberg (2004)
Coquand, Th., Huet, G.: The calculus of constructions. Information and Computation 76(2/3) (1988)
Callaghan, P.C., Luo, Z.: An implementation of typed LF with coercive subtyping and universes. J. of Automated Reasoning 27(1), 3–27 (2001)
Feferman, S.: Weyl vindicated. In: In the Light of Logic, pp. 249–283. Oxford University Press, Oxford (1998)
Feferman, S.: The significance of Hermann Weyl’s Das Kontinuum. In: Hendricks, V. et al. (eds.) Proof Theory – Historical and Philosophical Significance of Synthese Library, vol. 292 (2000)
Gambino, N., Aczel, P.: The generalised type-theoretic interpretation of constructive set theory. J. of Symbolic Logic 71(1), 67–103 (2006)
Gonthier, G.: A computer checked proof of the four colour theorem (2005)
Harper, R., Honsell, F., Plotkin, G.: A framework for defining logics. Journal of the Association for Computing Machinery 40(1), 143–184 (1993)
Luo, Z.: Computation and Reasoning: A Type Theory for Computer Science. Oxford University Press, Oxford (1994)
Luo, Z.: A type-theoretic framework for formal reasoning with different logical foundations. In: Okada, M., Satoh, I. (eds.) Proc of the 11th Annual Asian Computing Science Conference. Tokyo (2006)
Martin-Löf, P.: Intuitionistic Type Theory. Bibliopolis (1984)
Nordström, B., Petersson, K., Smith, J.: Programming in Martin-Löf’s Type Theory: An Introduction. Oxford University Press, Oxford (1990)
Poincaré, H.: Les mathématiques et la logique. Revue de Métaphysique et de Morale 13, 815–835; 14, 17–34; 14, 294–317 (1905-1906)
Pfenning, F., Schürmann, C.: System description: Twelf — a meta-logical framework for deductive systems. In: Ganzinger, H. (ed.) Automated Deduction - CADE-16. LNCS (LNAI), vol. 1632, pp. 202–206. Springer, Heidelberg (1999)
Simpson, S.: Subsystems of Second-Order Arithmetic. Springer, Heidelberg (1999)
Weyl, H.: Das Kontinuum. Translated as [Wey87] (1918)
Weyl, H.: The Continuum: A Critical Examination of the Foundation of Analysis. Translated by Pollard, S., Bole, T. Thomas Jefferson University Press, Kirksville, Missouri (1987)
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2007 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Adams, R., Luo, Z. (2007). Weyl’s Predicative Classical Mathematics as a Logic-Enriched Type Theory. In: Altenkirch, T., McBride, C. (eds) Types for Proofs and Programs. TYPES 2006. Lecture Notes in Computer Science, vol 4502. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-74464-1_1
Download citation
DOI: https://doi.org/10.1007/978-3-540-74464-1_1
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-74463-4
Online ISBN: 978-3-540-74464-1
eBook Packages: Computer ScienceComputer Science (R0)