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.
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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)
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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
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DOI: https://doi.org/10.1007/978-3-540-74464-1_1
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