Skip to main content Accessibility help

Classical and constructive hierarchies in extended intuitionistic analysis

  • Joan Rand Moschovakis (a1)


This paper introduces an extension of Kleene's axiomatization of Brouwer's intuitionistic analysis, in which the classical arithmetical and analytical hierarchies are faithfully represented as hierarchies of the domains of continuity. A domain of continuity is a relation R(α) on Baire space with the property that every constructive partial functional defined on {α: R(α)} is continuous there. The domains of continuity for coincide with the stable relations (those equivalent in to their double negations), while every relation R(α) is equivalent in to ∃β A(α, β) for some stable A(α, β) (which belongs to the classical analytical hierarchy).

The logic of is intuitionistic. The axioms of include countable comprehension, bar induction, Troelstra's generalized continuous choice, primitive recursive Markov's Principle and a classical axiom of dependent choices proposed by Krauss. Constructive dependent choices, and constructive and classical countable choice, are theorems, is maximal with respect to classical Kleene function realizability, which establishes its consistency. The usual disjunction and (recursive) existence properties ensure that preserves the constructive sense of “or” and “there exists.”


Corresponding author

721 24th Street, Santa Monica, CA 90402, USA


Hide All
[1] Beeson, M. J., Problematic principles in constructive mathematics, Logic colloquium '80 (Lascar, D. van Dalen, D. and Smiley, T. J., editors), North-Holland, Amsterdam, 1982.
[2] Beeson, M. J., Foundations of Constructive Mathematics, Springer, Berlin, 1985.
[3] Bishop, E., Foundations of Constructive Analysis, McGraw-Hill, New York, 1967.
[4] Diaconescu, R., Axiom of choice and complementation, Proceedings of the American Mathematical Society, vol. 51 (1975), pp. 176178.
[5] Gödel, K., On intuitionistic arithmetic and number theory, Kurt Gödel: Collected Works, Volume I (Feferman, S. et al., editors), Oxford University Press, New York, 1993, translated by Bauer-Mengelberg, S. and van Heijenoort, J., pp. 287295.
[6] Kleene, S. C., Introduction to Metamathematics, North-Holland, Amsterdam, 1952.
[7] Kleene, S. C., Realizability and Shanin's algorithm for the constructive deciphering of mathematical sentences, Logique et Analyse, Nouvelle Série, vol. 3 (1960), pp. 154165.
[8] Kleene, S. C., Classical extensions of intuitionistic mathematics, Logic, methodology and philosophy of science, proceedings of the 1964 international congress (Amsterdam) (Bar-Hillel, Y., editor), North-Holland, 1965, pp. 3144.
[9] Kleene, S. C., Formalized recursive functionals and formalized realizability, Memoirs, no. 89, American Mathematical Society, 1969.
[10] Kleene, S. C. and Vesley, R. E., The Foundations of Intuitionistic Mathematics, Especially in Relation to Recursive Functions, North-Holland, Amsterdam, 1965.
[11] Krauss, P. H., A constructive interpretation of classical mathematics, Mathematische Schriften Kassel, preprint No. 5/92, 1992.
[12] Moschovakis, J. R., Disjunction, existence and λ-elimination, Ph.D. thesis , University of Wisconsin, 1965.
[13] Moschovakis, J. R., Can there be no non-recursive functions, this Journal, vol. 36 (1971), pp. 309315.
[14] Moschovakis, J. R., Analyzing realizability by Troelstra's methods, Annals of Pure and Applied Logic, vol. 114 (2002), pp. 203225.
[15] Moschovakis, Y. N., Descriptive Set Theory, North-Holland, Amsterdam, 1980.
[16] Troelstra, A. S. (editor), Metamathematical Investigation of Intuitionistic Arithmetic and Analysis, Lecture Notes in Mathematics, no. 344, Springer Verlag, Berlin, Heidelberg, New York, 1973, with contributions by Troelstra, A. S., Smorynski, C. A., Zucker, J. I. and Howard, W. A..
[17] Troelstra, A. S., Intuitionistic extensions of the reals II, Logic Colloquium '80 (Amsterdam) (Lascar, D. van Dalen, D. and Smiley, T. J., editors), North-Holland, 1982, pp. 279310.
[18] Troelstra, A. S. and van Dalen, D., Constructivism in Mathematics: An Introduction, vol. 1 and 2, North-Holland, Amsterdam, 1988.
[19] Veldman, W., Investigations in intuitionistic hierarchy theory, Ph.D. thesis , Catholic University of Nijmegen, 1981.
[20] Veldman, W., A survey of intuitionistic descriptive set theory, Mathematical Logic, Proceedings of the Heyting Conference 1988 (New York and London) (Petkov, P. P., editor), Plenum Press, 1990, pp. 155174.
[21] Veldman, W., The Borel hierarchy and the projective hierarchy in intuitionistic mathematics, University of Nijmegen Department of Mathematics Report No. 0103, 03 2001.
[22] Veldman, W., On some sets that are not positively Borel, University of Nijmegen Department of Mathematics Report No. 0201, 01 2002.
[23] Veldman, W., On the persistent difficulty of disjunction, Proceedings of the XIth International Congress on Logic, Methodology and Philosophy of Science, held in Cracow, Poland, 1999, forthcoming.

Related content

Powered by UNSILO

Classical and constructive hierarchies in extended intuitionistic analysis

  • Joan Rand Moschovakis (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.