### The generalised type-theoretic interpretation of constructive set theory

Peter Aczel and Nicola Gambino
Source: J. Symbolic Logic Volume 71, Issue 1 (2006), 67-103.

#### 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.

First Page:
Primary Subjects: 03F25, 03F50
We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber.

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

### 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.
Mathematical Reviews (MathSciNet): MR519801
Zentralblatt MATH: 0481.03035
-------- 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.
Mathematical Reviews (MathSciNet): MR717236
Zentralblatt MATH: 0529.03035
-------- 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.
Mathematical Reviews (MathSciNet): MR874778
Zentralblatt MATH: 0624.03044
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.
Mathematical Reviews (MathSciNet): MR2044527
Zentralblatt MATH: 1054.03036
Digital Object Identifier: doi:10.1007/3-540-45842-5_1
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.
Mathematical Reviews (MathSciNet): MR2081047
Digital Object Identifier: doi:10.1093/logcom/14.4.447
Zentralblatt MATH: 1050.03016
S. Awodey and M. Warren Predicative algebraic set theory, Theory and applications of categories, vol. 15 (2005), no. 1, pp. 1--39.
Mathematical Reviews (MathSciNet): MR2210574
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.
Mathematical Reviews (MathSciNet): MR2013394
Digital Object Identifier: doi:10.1016/S0168-0072(03)00052-6
Zentralblatt MATH: 1070.03041
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.
Mathematical Reviews (MathSciNet): MR347565
Digital Object Identifier: doi:10.2307/2272068
Zentralblatt MATH: 0278.02045
-------- Set theoretic foundations of constructive analysis, Annals of Mathematics, vol. 105 (1977), pp. 1--28.
Mathematical Reviews (MathSciNet): MR434784
Digital Object Identifier: doi:10.2307/1971023
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.
Mathematical Reviews (MathSciNet): MR2179072
-------- Heyting-valued interpretations for Constructive Set Theory, Annals of Pure and Applied Logic, vol. 137 (2006), no. 1--3, pp. 164--188.
Mathematical Reviews (MathSciNet): MR2182102
Digital Object Identifier: doi:10.1016/j.apal.2005.05.021
Zentralblatt MATH: 1077.03038
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.
Mathematical Reviews (MathSciNet): MR555552
Zentralblatt MATH: 0419.03033
-------- Forcing in intuitionistic systems without power-set, Journal of Symbolic Logic, vol. 48 (1983), no. 3, pp. 670--682.
Mathematical Reviews (MathSciNet): MR716628
Digital Object Identifier: doi:10.2307/2273459
Zentralblatt MATH: 0595.03056
E. R. Griffor and M. Rathjen The strength of some Martin-Löf type theories, Archive for Mathematical Logic, vol. 33 (1994), pp. 347--385.
Mathematical Reviews (MathSciNet): MR1308848
Digital Object Identifier: doi:10.1007/BF01278464
B. Jacobs Categorical logic and type theory, North-Holland,1999.
Mathematical Reviews (MathSciNet): MR1674451
Zentralblatt MATH: 0911.03001
P. T. Johnstone Stone spaces, Cambridge University Press,1982.
Mathematical Reviews (MathSciNet): MR698074
Zentralblatt MATH: 0499.54001
R. S. Lubarsky Independence results around constructive ZF, Annals of Pure and Applied Logic, vol. 132 (2005), no. 2-3, pp. 209--225.
Mathematical Reviews (MathSciNet): MR2110824
Digital Object Identifier: doi:10.1016/j.apal.2004.08.002
Zentralblatt MATH: 1068.03045
S. MacLane and I. Moerdijk Sheaves in Geometry and Logic, Springer,1992.
Mathematical Reviews (MathSciNet): MR1300636
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.
Mathematical Reviews (MathSciNet): MR2188638
Zentralblatt MATH: 02247253
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.
Mathematical Reviews (MathSciNet): MR1678360
Zentralblatt MATH: 0896.03051
-------- 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.
Mathematical Reviews (MathSciNet): MR769301
I. Moerdijk and E. Palmgren Wellfounded trees in categories, Journal of Pure and Applied Logic, vol. 104 (2000), pp. 189--218.
Mathematical Reviews (MathSciNet): MR1778939
Digital Object Identifier: doi:10.1016/S0168-0072(00)00012-9
Zentralblatt MATH: 1010.03056
-------- 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.
Mathematical Reviews (MathSciNet): MR1879412
Digital Object Identifier: doi:10.1016/S0168-0072(01)00079-3
Zentralblatt MATH: 0999.03061
J.R. Myhill Constructive Set Theory, Journal of Symbolic Logic, vol. 40 (1975), no. 3, pp. 347--382.
Mathematical Reviews (MathSciNet): MR381941
Digital Object Identifier: doi:10.2307/2272159
Zentralblatt MATH: 0314.02045
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.
Mathematical Reviews (MathSciNet): MR1859469
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.
Mathematical Reviews (MathSciNet): MR2194246
Digital Object Identifier: doi:10.2178/jsl/1129642124
Zentralblatt MATH: 1100.03046
Project Euclid: euclid.jsl/1129642124
-------- Replacement versus Collection in Constructive Zermelo-Fraenkel Set Theory, Annals of Pure and Applied Logic, vol. 136 (2005), no. 1--2, pp. 156--174.
Mathematical Reviews (MathSciNet): MR2162852
Digital Object Identifier: doi:10.1016/j.apal.2005.05.010
Zentralblatt MATH: 1073.03030
-------- 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.
Mathematical Reviews (MathSciNet): MR2207359
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.
Mathematical Reviews (MathSciNet): MR1998402
Digital Object Identifier: doi:10.1002/malq.200310054
Zentralblatt MATH: 1042.03040
G. Sambin Intuitionistic formal spaces--A first communication, Mathematical Logic and its Applications (D. Skordev, editor), Plenum,1987, pp. 87--204.
Mathematical Reviews (MathSciNet): MR945195
Zentralblatt MATH: 0703.03040
-------- Some points in formal topology, Theoretical Computer Science, vol. 305 (2003), no. 1--3, pp. 347--408.
Mathematical Reviews (MathSciNet): MR2013578
Digital Object Identifier: doi:10.1016/S0304-3975(02)00704-1
Zentralblatt MATH: 1044.54001
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.
Mathematical Reviews (MathSciNet): MR1686868