Notre Dame Journal of Formal Logic

Classical Consequences of Continuous Choice Principles from Intuitionistic Analysis

François G. Dorais


The sequential form of a statement


is the statement


There are many classically true statements of the form (†) whose proofs lack uniformity, and therefore the corresponding sequential form is not provable in weak classical systems. The main culprit for this lack of uniformity is of course the law of excluded middle. Continuing along the lines of Hirst and Mummert, we show that if a statement of the form (†) satisfying certain syntactic requirements is provable in some weak intuitionistic system, then the proof is necessarily sufficiently uniform that the corresponding sequential form is provable in a corresponding weak classical system. Our results depend on Kleene’s realizability with functions and the Lifschitz variant thereof.

Article information

Notre Dame J. Formal Logic, Volume 55, Number 1 (2014), 25-39.

First available in Project Euclid: 20 January 2014

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 03B20: Subsystems of classical logic (including intuitionistic logic)
Secondary: 03B30: Foundations of classical theories (including reverse mathematics) [See also 03F35] 03F35: Second- and higher-order arithmetic and fragments [See also 03B30] 03F55: Intuitionistic mathematics

intuitionistic analysis second-order arithmetic reverse mathematics realizability choice principles


Dorais, François G. Classical Consequences of Continuous Choice Principles from Intuitionistic Analysis. Notre Dame J. Formal Logic 55 (2014), no. 1, 25--39. doi:10.1215/00294527-2377860.

Export citation


  • [1] Brouwer, L. E. J., “Über Definitionsbereiche von-Funktionen,” Mathematische Annalen, vol. 97 (1927), pp. 60–75. English translation in J. van Heijenoort, From Frege to Gödel: A Source Book in Mathematical Logic, 1879–1931, Harvard University Press, Cambridge, Mass., 1967, pp. 446–63.
  • [2] Dorais, F. G., J. L. Hirst, and P. Shafer, “Reverse mathematics, trichotomy and dichotomy,” Journal of Logic and Analysis, vol. 4 (2012), no. 13.
  • [3] Hirst J. L., “Representations of reals in reverse mathematics,” Bulletin of the Polish Academy of Sciences, Mathematics, vol. 55 (2007), pp. 303–16.
  • [4] Hirst, J. L., and C. Mummert, “Reverse mathematics and uniformity in proofs without excluded middle,” Notre Dame Journal of Formal Logic, vol. 52 (2011), pp. 149–62.
  • [5] Kleene, S. C., and R. E. Vesley, The Foundations of Intuitionistic Mathematics, Especially in Relation to Recursive Functions, North-Holland, Amsterdam, 1965.
  • [6] Kohlenbach, U., “Higher order reverse mathematics,” pp. 281–95 in Reverse Mathematics 2001, vol. 21 of Lecture Notes in Logic, Association for Symbolic Logic, La Jolla, Calif., 2005.
  • [7] Kohlenbach, U., Applied Proof Theory: Proof Interpretations and Their Use in Mathematics, Springer Monographs in Mathematics, Springer, Berlin, 2008.
  • [8] Simpson, S. G., Subsystems of Second Order Arithmetic, 2nd edition, Perspectives in Logic, Cambridge University Press, Cambridge; Association for Symbolic Logic, Poughkeepsie, N.Y., 2009.
  • [9] Troelstra, A. S., ed., Metamathematical Investigation of Intuitionistic Arithmetic and Analysis, vol. 344 of Lecture Notes in Mathematics, Springer, Berlin, 1973.
  • [10] van Oosten, J., “Lifschitz’ realizability,” Journal of Symbolic Logic, vol. 55 (1990), pp. 805–21.