Journal of Symbolic Logic

The generalised type-theoretic interpretation of constructive set theory

Peter Aczel and Nicola Gambino

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Abstract

We present a generalisation of the type-theoretic interpretation of constructive set theory into Martin-Löf type theory. The generalisation involves replacing Martin-Löf type theory with a new type theory in which logic is treated as primitive instead of being formulated via the propositions-as-types representation. The original interpretation treated logic in Martin-Löf type theory via the propositions-as-types interpretation. The generalisation involves replacing Martin-Löf type theory with a new type theory in which logic is treated as primitive. The primitive treatment of logic in type theories allows us to study reinterpretations of logic, such as the double-negation translation.

Article information

Source
J. Symbolic Logic Volume 71, Issue 1 (2006), 67-103.

Dates
First available in Project Euclid: 22 February 2006

Permanent link to this document
http://projecteuclid.org/euclid.jsl/1140641163

Digital Object Identifier
doi:10.2178/jsl/1140641163

Zentralblatt MATH identifier
05038888

Mathematical Reviews number (MathSciNet)
MR2210056

Subjects
Primary: 03F25: Relative consistency and interpretations 03F50: Metamathematics of constructive systems

Keywords
Constructive Set Theory Dependent Type Theory

Citation

Gambino, Nicola; Aczel, Peter. The generalised type-theoretic interpretation of constructive set theory. Journal of Symbolic Logic 71 (2006), no. 1, 67--103. doi:10.2178/jsl/1140641163. http://projecteuclid.org/euclid.jsl/1140641163.


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References

  • P. Aczel The type theoretic interpretation of constructive set theory, Logic Colloquium '77 (A. MacIntyre, L. Pacholski, and J. Paris, editors), North-Holland,1978, pp. 55--66.
  • -------- The type theoretic interpretation of constructive set theory: choice principles, The L. E. J. Brouwer Centenary Symposium (A. S. Troelstra and D. van Dalen, editors), North-Holland,1982, pp. 1--40.
  • -------- The type theoretic interpretation of constructive set theory: inductive definitions, Logic, methodology and philosophy of science VII (R. Barcan Marcus, G.J.W. Dorn, and P. Weinegartner, editors), North-Holland,1986, pp. 17--49.
  • P. Aczel and N. Gambino Collection principles in Dependent Type Theory, Types for proofs and programs (P. Callaghan, Z. Luo, J. McKinna, and R. Pollack, editors), Lecture Notes in Computer Science, vol. 2277, Springer,2002, pp. 1--23.
  • P. Aczel and M. Rathjen Notes on Constructive Set Theory, Technical Report 40, Mittag-Leffler Institute, The Swedish Royal Academy of Sciences,2001, available from the first author's web page at http://www.cs.man.ac.uk/~petera/papers.html.
  • S. Awodey and A. Bauer Propositions as [types], Journal of Logic and Computation, vol. 14 (2004), no. 4, pp. 447--471.
  • S. Awodey and M. Warren Predicative algebraic set theory, Theory and applications of categories, vol. 15 (2005), no. 1, pp. 1--39.
  • T. Coquand, G. Sambin, J. M. Smith, and S. Valentini Inductively generated formal topologies, Annals of Pure and Applied Logic, vol. 124 (2003), no. 1-3, pp. 71--106.
  • H. M. Friedman The consistency of classical set theory relative to a set theory with intuitionistic logic, Journal of Symbolic Logic, vol. 38 (1973), pp. 315--319.
  • -------- Set theoretic foundations of constructive analysis, Annals of Mathematics, vol. 105 (1977), pp. 1--28.
  • N. Gambino Types and sets: a study on the jump to full impredicativity, Laurea dissertation, Dipartimento di Matematica Pura e Applicata, Università degli Studi di Padova, 1999.
  • -------- Sheaf interpretations for generalised predicative intuitionistic systems, Ph.D. thesis, University of Manchester,2002, available from the author's web page.
  • -------- Presheaf models for constructive set theory, From Sets and Types to Topology and Analysis (L. Crosilla and P. Schuster, editors), Oxford University Press,2005, pp. 62--77.
  • -------- Heyting-valued interpretations for Constructive Set Theory, Annals of Pure and Applied Logic, vol. 137 (2006), no. 1--3, pp. 164--188.
  • R. J. Grayson Heyting-valued models for Intuitionistic Set Theory, Applications of sheaves (M. P. Fourman, C. J. Mulvey, and D. S. Scott, editors), Lecture Notes in Mathematics, vol. 753, Springer,1979, pp. 402--414.
  • -------- Forcing in intuitionistic systems without power-set, Journal of Symbolic Logic, vol. 48 (1983), no. 3, pp. 670--682.
  • E. R. Griffor and M. Rathjen The strength of some Martin-Löf type theories, Archive for Mathematical Logic, vol. 33 (1994), pp. 347--385.
  • B. Jacobs Categorical logic and type theory, North-Holland,1999.
  • P. T. Johnstone Stone spaces, Cambridge University Press,1982.
  • R. S. Lubarsky Independence results around constructive ZF, Annals of Pure and Applied Logic, vol. 132 (2005), no. 2-3, pp. 209--225.
  • S. MacLane and I. Moerdijk Sheaves in Geometry and Logic, Springer,1992.
  • M. E. Maietti The type theory of categorical universes, Ph.D. thesis, Dipartimento di Matematica Pura e Applicata, Università degli Studi di Padova,1998, available from the author's web page.
  • -------- Modular correspondence between dependent type theories and categories including pretopoi and topoi, Mathematical Structures in Computer Science, to appear.
  • M. E. Maietti and G. Sambin Towards a minimalistic foundation for constructive mathematics, From sets and types to topology and analysis (L. Crosilla and P. Schuster, editors), Oxford University Press,2005, pp. 91--114.
  • M. Makkai First-order logic with dependent sorts, with applications to category theory, available from the author's web page,1995.
  • -------- Towards a categorical foundation of mathematics, Logic Colloquium '95 (J. A. Makowsky and E. V. Ravve, editors), Lecture Notes in Logic, vol. 11, Association for Symbolic Logic,1998, pp. 153--190.
  • -------- On comparing definitions of weak $n$-category, available from the author's web page,2001.
  • P. Martin-Löf Intuitionistic type theory --- Notes by G. Sambin of a series of lectures given in Padua, June 1980, Bibliopolis,1984.
  • I. Moerdijk and E. Palmgren Wellfounded trees in categories, Journal of Pure and Applied Logic, vol. 104 (2000), pp. 189--218.
  • -------- Type theories, toposes and Constructive Set Theory: predicative aspects of AST, Annals of Pure and Applied Logic, vol. 114 (2002), no. 1-3, pp. 155--201.
  • J.R. Myhill Constructive Set Theory, Journal of Symbolic Logic, vol. 40 (1975), no. 3, pp. 347--382.
  • B. Nordström, K. Petersson, and J. M. Smith Martin-Löf Type Theory, Handbook of Logic in Computer Science (S. Abramski, D. M. Gabbay, and T. S. E. Maibaum, editors), vol. 5, Oxford University Press,2000.
  • M. Rathjen The disjunction and related properties for Constructive Zermelo-Frankel Set Theory, Journal of Symbolic Logic, vol. 70 (2005), no. 4, pp. 1233--1254.
  • -------- Replacement versus Collection in Constructive Zermelo-Fraenkel Set Theory, Annals of Pure and Applied Logic, vol. 136 (2005), no. 1--2, pp. 156--174.
  • -------- Realizability for Constructive Zermelo-Fraenkel Set Theory, Logic colloquium '03 (J. Väänänen and V. Stoltenberg-Hansen, editors), Lecture Notes in Logic, vol. 24, Association for Symbolic Logic and AK Peters,2006, pp. 282--314.
  • M. Rathjen and R. S. Lubarsky On the regular extension axiom and its variants, Mathematical Logic Quarterly, vol. 49 (2003), no. 5, pp. 511--518.
  • G. Sambin Intuitionistic formal spaces--A first communication, Mathematical Logic and its Applications (D. Skordev, editor), Plenum,1987, pp. 87--204.
  • -------- Some points in formal topology, Theoretical Computer Science, vol. 305 (2003), no. 1--3, pp. 347--408.
  • G. Sambin and S. Valentini Building up a toolbox for Martin-Löf's type theory: subset theory, Twenty-five years of Constructive Type Theory (G. Sambin and J. M. Smith, editors), Oxford University Press,1998, pp. 221--244.