Hostname: page-component-745bb68f8f-v2bm5 Total loading time: 0 Render date: 2025-01-18T13:37:34.944Z Has data issue: false hasContentIssue false
  • English
  • Français

Continuity and nondiscontinuity in constructive mathematics

Published online by Cambridge University Press:  12 March 2014

Hajime Ishihara
Hajime Ishihara*
Affiliation:
Faculty of Integrated Arts and Sciences, Hiroshima University, Hiroshima 730, Japan
Get access Rights & Permissions [Opens in a new window]

Abstract

The purpose of this paper is an axiomatic study of the interrelations between certain continuity properties. We show that every mapping is sequentially continuous if and only if it is sequentially nondiscontinuous and strongly extensional, and that “every mapping is strongly extensional”, “every sequentially nondiscontinuous mapping is sequentially continuous”, and a weak version of Markov's principle are equivalent. Also, assuming a consequence of Church's thesis, we prove a version of the Kreisel-Lacombe-Shoenfield-Tseĭtin theorem.

Keywords

Sequentially continuoussequentially nondiscontinuousweak version of Markov's principleKreisel-Lacombe-Shoenfield-Tseĭtin theorem
Type
Research Article
Information
The Journal of Symbolic Logic , Volume 56 , Issue 4 , December 1991 , pp. 1349 - 1354
Copyright
Copyright © Association for Symbolic Logic 1991

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Beeson, M., The nonderivability in intuitionistic formal systems of theorems on the continuity of effective operations, this Journal, vol. 40 (1975), pp. 321–346.Google Scholar
[2]Bishop, E., Foundations of constructive analysis, McGraw-Hill, New York, 1967.Google Scholar
[3]Bishop, E. and Bridges, D., Constructive analysis, Springer-Verlag, Berlin, 1985.CrossRefGoogle Scholar
[4]Bridges, D. and Ishihara, H., Linear mappings are fairly well-behaved, Archiv der Mathematik, vol. 54(1990), pp. 558–562.CrossRefGoogle Scholar
[5]Bridges, D. and Richman, F., Varieties of constructive mathematics, London Mathematical Society Lecture Note Series, vol. 97, Cambridge University Press, Cambridge, 1987.CrossRefGoogle Scholar
[6]Ishihara, H., A constructive closed graph theorem (preprint).Google Scholar
[7]Kreisel, G., Lacombe, D. and Shoenfield, J., Fonctionnelles récursivement définissables et fonctionnelles récursives, Comptes Rendus Hebdomadaires des Séances de l'Académie des Sciences, vol. 245 (1957), pp. 399–402.Google Scholar
[8]Kushner, B. A., Lectures on constructive mathematical analysis, “Nauka”, Moscow, 1973; English translation, American Mathematical Society, Providence, Rhode Island, 1984.Google Scholar
[9]Mandelkern, M., Constructive continuity, Memoirs of the American Mathematical Society, no. 277, American Mathematical Society, Providence, Rhode Island, 1983.CrossRefGoogle Scholar
[10]Mandelkern, M., Constructive complete finite sets, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 34 (1988), pp. 97–103.CrossRefGoogle Scholar
[11]Orevkov, V. P., Equivalence of two definitions of continuity, Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR (LOMI), vol. 20 (1971), pp. 145–159, 286; English translation, Journal of Soviet Mathematics, vol. 1 (1973), pp. 92–99.Google Scholar
[12]Richman, F., Church's thesis without tears, this Journal, vol. 48 (1983), pp. 797–803.Google Scholar
[13]Troelstra, A. S. and Van Dalen, D., Constructivism in mathematics, Vol. 1, North-Holland, Amsterdam, 1988.Google Scholar
[14]Troelstra, A. S. and Van Dalen, D., Constructivism in mathematics, Vol. 2, North-Holland, Amsterdam, 1988.Google Scholar
[15]Tseĭtin, G. S., Algorithmic operators in constructive complete separable metric spaces, Doklady Akademii Nauk SSSR, vol. 128 (1959), pp. 49–52, (Russian).Google Scholar