Skip to main content Accessibility help

Interpreting classical theories in constructive ones

  • Jeremy Avigad (a1)


A number of classical theories are interpreted in analogous theories that are based on intuitionistic logic. The classical theories considered include subsystems of first- and second-order arithmetic, bounded arithmetic, and admissible set theory.



Hide All
[1]Aczel, Peter, The strength of Martin-Löf's intuitionistic type theory with one universe, Proceedings of the symposiums of mathematical logic, Oulu 1974 and Helsinki 1975 (Miettinen, Seppo and Väänänen, Jouko, editors), Department of Philosophy, University of Helsinki, 1977.
[2]Aczel, Peter, The type theoretic interpretation of constructive set theory, Logic colloquium '77 (Macintyre, A., Pacholski, L., and Paris, J., editors), North-Holland, Amsterdam, 1978, pp. 5566.
[3]Aczel, Peter, The type theoretic interpretation of constructive set theory: choice principles, L. E. J. Brouwer centenary symposium (Troelstra, A. S. and van Dalen, D., editors), North-Holland, Amsterdam, 1982, pp. 140.
[4]Avigad, Jeremy and Sommer, Richard, The model-theoretic ordinal analysis of predicative theories, this Journal, vol. 64 (1999), pp. 327349.
[5]Barwise, Jon, Admissible sets and structures, Springer, Berlin, 1975.
[6]Beeson, Michael J., Foundations of constructive mathematics, Springer, Berlin, 1985.
[7]Buchholz, Wilfried, The Ωμ+1-rule, [8], 1981, pp. 188233.
[8]Buchholz, Wilfried, Feferman, Solomon, Pohlers, Wolfram, and Sieg, Wilfried, Iterated inductive definitions and subsystems of analysis: Recent proof-theoretical studies, Lecture Notes in Mathematics, no, 897, Springer, Berlin, 1981.
[9]Burr, Wolfgang, Fragments of Heyting-arithmetic, preprint, 1998.
[10]Cook, Stephen A. and Urquhart, Alasdair, Functional interpretations of feasibly constructive arithmetic, Annals of Pure and Applied Logic, vol. 63 (1993), pp. 103200.
[11]Coquand, Thierry and Hofmann, Martin, A new method for establishing conservativity of classical systems over their intuitionistic version, Mathematical Structures in Computer Science, vol. 9 (1999), pp. 323333.
[12]Friedman, Harvey M., Subsystems of set theory and analysis, Ph.D. thesis, Massachusetts Institute of Technology, 1967.
[13]Friedman, Harvey M., The consistency of classical set theory relative to a set theory with intuitionistic logic, this Journal, vol. 38 (1973), no. 2, pp. 315319.
[14]Friedman, Harvey M., Classically and intuitionistically provable functions, Higher set theory (Miiller, H. and Scott, D., editors). Lecture Notes in Mathematics, no. 669, Springer, Berlin, 1978, pp. 2127.
[15]Griffor, E. and Rathjen, M., The strength of some Martin-Löf type theories, Archive for Mathematical Logic, vol. 33 (1994), pp. 347385.
[16]Jäger, Gerhard, Theories for admissible sets: A unifying approach to proof theory, Bibliopolis, Napoli, 1986.
[17]Martin-Löf, Per, An intuitionistic theory of types, 1972, manuscript, reprinted in Twenty-five years of constructive type theory, Oxford University Press, 1998, pp. 127172.
[18]Palmgren, Erik, Type-theoretic interpretation of iterated, strictly positive inductive definitions, Archive for Mathematical Logic, vol. 32 (1992), pp. 7599.
[19]Parikh, Rohit, Existence and feasibility in arithmetic, this Journal, vol. 36 (1971), pp. 494508.
[20]Rathjen, Michael, The proof-theoretic characterization of the primitive recursive set functions, this Journal, vol. 57 (1992), pp. 954969.
[21]Simpson, Stephen G., Subsystems of second-order arithmetic, Springer, Berlin, 1998.
[22]Troelstra, A. S., Metamathematical investigation ofintuitionistic arithmetic and analysis, Lecture Notes in Mathematics, no. 344, Springer, Berlin, 1973.
[23]van Dalen, Dirk, Logic and structure, third ed., Springer, Berlin, 1997.

Related content

Powered by UNSILO

Interpreting classical theories in constructive ones

  • Jeremy Avigad (a1)


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed.